Laser Age in Optics [L._v_tarasov]

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L. V. Tarasov

Laser Age in OpticsTranslatedfrom the Russianby V. Kisin, Cando Sc. (phys.)

Mir PublishersMoscow

First published 1981Revised from the 1977 Russian edition

© 1-11JlaTC.'lbCT130 «npOCBeIllCIIMe»), 1977

© English translation, Mir Publishers. 1981

Contents

Preface

From incoherent to coherent optics

1. Waves and their interference / 92. Is interference an inevitable result of wave superposi-

tion? /20

3. How can we generate coherent light waves? /36

4. The laser: working principles / 46

5. Lasers as sources of coherent optical radiation /59

Optical holography

6. Formation of optical images /69

7. Holography: elementary examples / 75

8. Holographic laboratory / 87

9. Advantages and possibilities of holography / 96

10. Holographic interferometry /107

II. Computer technology and holography / 118

Contents

Nonlinear optics

12. A few words about optical characteristics of themedium {128

13. Can the optical properties of a medium depend uponthe intensity of the radiation? {139

14. Intensity-dependent transparency of the medi-um {149

IS. Self-focusing of light {I 57

16. Optical transitions {161

17. Transformation of one light wave into theother/I 70

18. The principle of operation of the parametric lightoscillator {181

19. Nonlinear optics and progress in laser technology {189

Historical background

6 Preface

There can be no doubt that the laser representsone of the most remarkable scientific and technicalmilestones of the 20th century. The dramatic growth oflaser technology began in 1960, wilen the first successful laser was reported. Lasers are now used in themost diverse fields: biology and medicine, cyberneticsand computer circuitry, communications and radarsystems, industrial processes, and measurements ofvarious types.

A laser is a very special light source, greatlydifferent from incandescent lamps, l1uorescent lights,and so forth. In contrast to other sources of light, thelaser's radiation is characterized by a high degree ofordering of the light field. This idea is expressed bysaying that it has a high degree of coherence. A lasercan be treated as a sort of an "optical radiotransmitter"; in comparison, all other light sourcesgenerate only "optical noise".

Until the advent of the laser, the radiofrequencyrange and the optical range differed greatly withrespect to coherence: radiophysics widely used coherentwaves while optics had only incoherent light at itsdisposal. A textbook was the only place where light

Preface 8

could "exist" as sine waves. Such light waves becamereal only when the laser was invented.

Laser optics is the optics of coherent radiation.Although this new branch appeared only twenty yearsago, it has already given rise to a number of surprises:novel and very unusual optical phenomena were foundand then put to use in extremely interesting applications.

The two newest directions of coherent optics areespecially significant: optical holography and nonlinearoptics. These two make up the subject of this book.

From Incoherentto Coherent Optics

1. Waves and Their Interference

It should be helpful for our purposes to precedethe discussion of optics by considering some verygeneral properties of waves and their interference. Thephysical nature of waves can be ignored for themoment.

Mpnochromatic plane wave. A monochromatic planewave propagating without attenuation is the simplesttype of wave. It can be described by the formula

The direction of x is assumed posItIve in the directionof propagation of the wave. In the case of elasticwaves, p (x, t) is the displacement of particles of themedium at the point x from the equilibrium position,at each moment t; v is the wave frequency; A is theamplitude; and v is the velocity of wave propagation.

Equation (1.1) can be derived in a very simplemanner. Let the source of harmonic oscillations belocated at the origin of the reference frame (at pointx = 0), where the displacement at moment t is Po(t) =

= A cos (2rrvt). Consider now a point at a distance

(1.1)

By introducing the wavelength').. = vlv. onerewrites Eq. (1.1) in the form

p(x, t)=ACOS[21t(vt- ~)J (1.2)

Equations (1.1) and (1.2) describe a monochromaticplane undamped wave.

This wave is said to be monochromatic because itinvolves a single frequency v. Furthermore, it is planesince the displacement p is a function of a singlespatial coordinate (namely, x); the wave front (i. e. thelocus of points all of which have the same phase atany moment) is a plane perpendicular to the axis x.And finally, the wave in question is undamped since itsamplitude A is constant at all points along thepropagation direction. The energy of oscillation IS

known to be proportional to the square of theamplitude. Consequently, the constancy of theamplitude along the propagation direction signifies thatthe wave energy is transferred from one point to thenext without losses, that is, no damping takes place.

Further in the text we use a quantity calledintensity, denoted by 1 and defined as the amount ofenergy transported by a wave per unit time acrossa unit area oriented normally to the propagationdirection at the point of observation. In complete

Fig. 1

11

Spherical wave

«--1- .....". ..... \/ \ \

I I / :(\\\

~F~~(Plane wave

1. Waves and Their Interference

analogy to the energy of oscillations the intensity ofa wave is proportional to its amplitude squared:

1 ~ A 2 (1.3)

Monochromatic spherical wave. A point sourceplaced in an isotropic medium (a medium whoseproperties are independent of direction) generates a wavewith a spherical front, referred to as a spherical wave.The difference between plane and spherical waves isshown in Fig. 1 (arrows indicate the directions ofpropagation, and dashed lines trace wavefrontcross-sections). Note that only the plane wave can besaid to have a definite direction of propagation.

Under the assumption that a spherical wave ismonochromatic and its energy is not absorbed by themedium, the corresponding wave equation is

p (r, t) = ~ cos [21t (vt - ~)] (1.4)

Here r is the distance between the source and theobservation point, also termed the radius of the

10From Incoherent to Coherent Optics

x from the origin. A wave takes the time xlv to coverthis distance. Consequently, the displacement p (x, t) atpoint x and at time t must be identical to that atpoint 0 and at time t - (xlv). Hence, p (x, t) =

From Incoherent to Coherent .J}ptics 12 1. Waves and Their Interference 13

spherical wave front; a is the oscillation amplitudeclose to the source; and air is the wave amplitude ata distance r from the source. As r increases, the areaof the wave front (i. e. of the sphere) risesproportionally to r 2

, and the wave intensityconsequently diminishes as 1/r2

, since the total energytransported by the wave per unit time across the wholesphere does not depend, in a nonabsorbing medium, onthe sphere radius. Hence, the wave amplitude must beproportional to l/r. Note that wave attenuation in theabove case is not caused by absorption in the mediumbut only by wavefront divergence. A plane wave haszero divergence, and for a spherical wave the divergenceis of maximum value.

v . The ratio ~v/vo characterizes the degree ofn~nrnonochrornaticity of the real wave in question.A wave is termed quasimonochromatic, that is, nearlymonochromatic, if ~v/vo « 1.

In what follows we consider a real wave, in thesense outlined above, as a sum of monochromaticplane waves. An individual monochromatic plane w.aveis characterized by a definite frequency and a defimtedirection of propagation (no divergence). A real wavemay be only partially monochromatic and to a certaindegree divergent. We have already mentioned thedegree of nonmonochromaticity as a characteristic of a realwave. Likewise, the degree of divergence of a wave canbe introduced. These characteristics are discussed laterIII the text for light waves.

Interference of waves. Interference is known to takeplace when two (or more) waves are superposed. Inorder to find out the essential features of thisphenomenon, consider the simplest case of twomonochromatic plane waves with equal frequencies. Let usanalyze two situations.

The first example is the superposition of a wavedescribed by Eq. (1.2) and a similar one propagating inthe opposite direction. The second wave is described bythe same equation (1.2) but with the minus sign infront of x/A replaced by a plus. Mathematically, thissuperposition takes the form

p (x, t) = A cos [2n (vt - ~)J+ A cos [2n (vt + ~)J

Real waves. Rigorously speaking, oscillationsgenerated by real sources of waves are never harmonic,so that real waves are never monochromatic. A realwave front may have a very complicated shape, farfrom planar or spherical. In addition, the shape maychange in the course of wave propagation. Finally, itshould be kept in mind that to a certain extent all realmediums are absorbing, so that the attenuation ofa real wave is related not only to its divergence butalso to its absorption in the medium.

It is essential, however, that any real wave can berepresented mathematically as a sum of a set ofdistinct plane or spherical waves.

Any nonharmonic oscillation is representable asa sum of harmonic vibrations with different frequenciesand amplitudes. Let us assume that a real osciliatiOlt isgiven by a sum of harmonic oscillations with frequencies in the interval ~v around a mean frequency

2nxp (x, t) = 2A cos (2nvt) cos -')..,- (1.5)

x = ~(~ + n)2 2-

This is an equation of the so-called standing wave. Itshould be emphasized that some points on a standingwave are at rest. These are the points for whichcos (21tx/A.) = 0; they are called the nodes of a standingwave, and their coordinates are

15

x

/-,I \

I \ StandingI ,...-, \ wave

~/'~ x

Fig. 2

\' 4 /1\ ' ..... _/ I\ /\\ 5 /, ../

Thin straight lines in Fig. 3a trace the system ofwave fronts of the interfering waves considered independently. Heavy solid lines trace the fronts on which thephase of the wave is equal to 21tn (n are integers) sothat the cosine in Eq. (1.2) is unity, while dashed linesare fronts with phase 1t (2n + 1) so that the cosine isminus unity. The two indicated wave fronts intersecteither with identical phase (points Db D 2 , D 3 , E b E 2 ,

E3

oscillating with amplitude 2A), or exactly out-of-phase (points Ab A 2 , A 3 , A 4 , Bb B 2, Cb C2 in whichamplitude is zero). As the waves propagate, points Db

D 2, D3 move along the line DD, while points E b E2 ,

E 3 move along EE. Correspondingly, other points tracethe lines AA, BB, cc.

1. Waves and Their Interference14From Incoherent to Coherent Optics

where n are integers.At the same time there are points (located halfway

between each pair of neighbouring nodes) whoseamplitude of oscillation is twice that of each of thewaves; these are called the antinodes of the wave.

There is definitely much to be surprised about inthis result: we have found that the interference(summation) of two travelling waves propagating inopposite directions results in the energy of these wavesnot being transferred at all to some fixed points ofthe medium, while to other points it is transferred inapparently excessive amounts (indeed, an amplitude of2A at the antinodes implies that the energy ofoscillation at these points is 4A 2, and therefore is twicethe total energy of oscillation of points close to thesources of both waves).

A standing wave is the simplest example of waveinterference. The difference between a standing and anordinary (travelling) wave is illustrated in Fig. 2, wherethe displacement p is plotted as a function ofcoordinate x for five different moments of time.

Consider now the case of superposition of twomonochromatic plane waves propagating at an angle rx.The waves' frequencies and amplitudes are identical(Fig. 3).

o

17

I

BI

B~ --- ----

D D

A I A'-- ---- ----

E E

CI

C--- ----

o

Let us cut the interference region by a planeperpendicular to the plane of the drawing. The trace ofthis plane is shown in Fig. 3a as line 00; the plane"faces" the reader in Fig. 3b (dashed lines indicate theregions, with zero intensity of the resultant wave, andsolid lines show regions of maximum intensity). Thedistance between neighbouring lines of zero intensity is

1. Waves and Their Interference16

Fig. 3

From Incoherent to Coherent Optics

10

. As a result of interference, therefore, stationaryregIOns are formed where points are at rest andregions are formed where points oscillate a; twice theamplitude..These regions are traced in Fig. 3a by dashed ~nd s?hd heavy lines, respectively (in reality, thesestraight lmes are sections of planes of constantamplitude).

The general arrangement necessary to obtain aninterference pattern is shown in Fig. 3d. Two wave

denoted by D; obviously, the distance between themaximum intensity lines is the same.

In order to determine D, consider Fig. 3c, which isa fragment of Fig. 3a. It is easily seen that D2A4 = A,angle B 2D 2 A 4 is 90°. and angles D 2B 2A 3 and A 3B 2A 4

are each equal to rJ./2. Hence, B2A 4 = D2A4/sin rJ. == A/sin rJ.. Moreover,

19Waves and Their Interference

trains (their boundaries are shown arbitrarily) interferewithin a certain volume shown in the figure by rhombAOAO. Solid horizontal lines trace regions of maximumintensii,y and dashed lines trace zero-intensity regions.The figure demonstrates that interference results ina spatial redistribution of energy. Outside of rhombAOAO the energy of oscillations is uniformly distributed within each beam of waves, while the distributionwithin the interference region is essentially nonuniform,with energy concentrating in the vicinity of solidhorizontal lines in the figure.

2"

Interference: Summary. The examples discussedabove demonstrate that, first of all, superposition ofwaves results in a spatial redistribution of theoscillation energy; in other words, the intensity ofwaves is redistributed in space. Regions with zerointensity and those with intensity above the intensity ofoverlapping waves are produced. Second, this spatialredistribution of energy is found to be time-independ~nt. This means that a stable pattern of fixedlllterference fringes is formed.

The phenomenon of wave interference reduces therefore to a spatial redistribution of intensities of waves,~hich results in formation of a fixed pattern ofIllterference fringes.

It must be noted in conclusion that interference oftwo plane waves constitutes one of the elementarycases of interference. It is also' possible to analyze the?ases o.f interfering plane and spherical waves, twoIllterfenng spherical waves, or waves with still morecomplex wavefronts. Obviously the more complicatedthe wavefronts, the more com~lex the interference fringepatterns are.

18

(1.6)

(1.7)

. rJ.2 Slll

2

rJ.Acos 2

sin rJ.


Finally we obtain

Note that for rJ. = 1C we have D = 1../2. This is thedistance between the neighbouring nodes of the standing wave.

An interference pattern is often observed ina plane perpendicular to the direction of propagationof one of the plane waves. Such a plane is, forexample, plane 0 10 1 in Fig. 3a. Obviously, the distanced between the lines of zero intensity in this plane isthe segment B2A4 (see Fig. 3c). Therefore,

Ad = B 2 A 4 = -.

Sill rJ.

the X-ray range, followed at shorter wavelengths by therange of y-radiation. The optical range is representedby radiation emitted by molecules and atoms, and theX-ray r\nge by that of atoms and nuclei.

The optical range is subdivided in its turn intothree parts: visible rad1ation (Ie = 0.75-0.4 /lm), infraredradiation (Ie > 0.75 /lm), and ultraviolet radiation (Ie << 0.4 /lm). The infrared (IR) part of the spectrum ismostly the radiation of molecules, while the visible andultraviolet (UV) parts are represented by emission from'atoms.

One important feature must be emphasized in thiscontext: the main distinction between the electromagnetic waves of the optical and short-waveranges and those of the radio and microwave ranges liesin the process of wave generation. In the case of radioand microwave frequencies the process of generation isbased on regularly repeated motions of electrons inoscillatory systems, while generation in the opticalrange is produced by quantum transitions in individu'aJmolecules and atoms.

A brief reminder concerning electromagnetic wavesis in order here.

Two vectors oscillate in an electromagnetic wave:-> ->

the electric and magnetic vectors E and B, respectively,The first of these is called the vector of electric fieldstrength, and the second the magnetic induction vector.The vectors are mutually perpendicular, and normal tothe direction of wave propagation (electromagneticwaves are transverse). The wave propagation velocity ina medium is


2. Is Interferencean Inevitable Resultof Wave Superposition?

Superposition of two monochromatic plane wavesalways produces an interference pattern. Is a similarstatement correct for any two arbitrary real waves? Isinterference an inevitable corollary of superposition ofreal waves?

The answer is no. There are numerous physicalphenomena in which superposition of waves producesno interference. This is the field of optics, i. e. of lightwaves. Our day-to-day experience indicates that asa rule no interference is observed when light wavessuperpose. The resultant intensity of several superposedlight waves is simply the sum of intensities of thecomponent waves. (Before the advent of the laser, veryspecial conditions were required to observe interferencepatterns, such as thin film colour fringes or Newton'srings.) The reason lies in specific features of the rangeof wavelengths that are used in optics

The spectrum of electromagnetic waves. We remindthe reader that the spectrum of electromagnetic waves(also referred to as the radiation spectrum) is separatedinto two main parts: waves emitted by electricoscillators (Ie > 102 /lm) and those emitted bymolecules, atoms, and nuclei (Ie < 102 /lm). The firstpart includes, among other types of radiation, radiowaves and microwave radiation, which lies directly nextto the boundary wavelength Ie = 102 /lm. The secondpart of the spectrum begins with the optical range,which covers wavelengths from 102 to 10 - 2 /lID. Therange of wavelengths below 10 - 2 J-lm corresponds to

202. Is Interference an Inevitable Result?

cv =

n

21

(2.1)

From Incoherent to Coherent Optics 22 2. Is Interlerence an Inevitable Result? 23

where n is the absolute refractive index of the medium,and c is the velocity of light in a vacuum (c == 3· 108 m/s). --+ --+

The vectors E and B are equivalent components ofan electromagnetic wave, but the photochemical,photoelectric, and physiological effects of light are

--+mostly produced by the electric vector E. For this

--+reason we restrict the following discussion to E.

Figure 4 gives a clear representation of a plane--+

electromagnetic wave. It shows that the vectors E andB oscillate in different planes and are in phase. One of

these planes is chosen as the plane of polarization ofthe wave. The electric vector being more important for

--+applications, we choose to consider the E-plane as thepolarization plane.

How light is generated. Processes leading toemission of light are numerous and varied. Forinstance, light is emitted by a decelerated chargedparticle, such as an electron, as a result of interactionwith atomic and interatomic fields (so-calledbremsstrahlung), by an electron moving in a medium ata velocity above the velocity of propagation ofelectromagnetic waves in this medium(Vavilov-Cherenkov radiation), and in electron-positroncollisions (annihilation emission). The most typicalprocess of light generation, however, is that oftransition from excited states to non-excited (or rather,less excited) states in atoms (or molecules) of theemitting substance. This mechanism is operative whenlight is emitted by a match flame, an incandescentlamp, a fluorescent light, and finally a laser. For thisreason, our attention will be concentrated exclusivelyon this mechanism of light generation.

Whatever the mechanism, light is always emitted inthe form of very specific particles, so-called photons.This becomes obvious when the process of lightemission is treated as a result of atomic (or molecular)transitions from excited to non-excited (ground) states:indeed, a transition in an individual atom or moleculegenerates an individual light particle, that is, a photon.

The possible values of energy of an atom (ormolecule) are discrete, which fact allows us to refer toa system of atomic (or molecular) energy levels. Let usassume that an atom undergoes a transition froma state with energy E2 to a state with lower energy E 1

(from the E2 energy level to the E 1 level). Thistransition generates a photon with energy E = E2 - E1•

It is readily apparent that a reverse transition mustresult in the elimination (absorption) of a photon withenergy E2 - E1 . The set of energies of photons thatcan be generated (or absorbed) by an atom is easily

Fig. 4

E

B

From Incuherent to Coherent Optics 24 2. Is Interference an Inevitable Result? 25

E = n

predicted once the system of energy levels of the atomin question is known.

Consider as an example the hydrogen atom,making use of the model that, as the reader is wellaware, was suggested by the great Danish physicistNiels Bohr. According to this model, the energy levelsin a hydrogen atom are given by the formula

41t2me4

h2n2

where m is the electron mass, e IS ItS charge, n· is aninteger, and h is a universal physical constant termedPlanck's constant (h = 6.6 ·10 - 34 J. s). Energy levels inthe hydrogen atom are shown in Fig. 5. Usually theyare referred to as the electron energy levels in thehydrogen atom. Level E 1 corresponds to the ground

Fig. 5

E

o

(2.2)

,I

state of the atom, in which an electron circles the nucleus in an orbit of minimum radius. Levels E2 • E 3 • andso on, denote excited states of the atom. The zeropoint of the energy axis (the origin of the referenceframe) corresponds to the ionized state in which theelectron is outside of the atom. The energy ofionization of a hydrogen atom is given by

4 24E'= 1tme =13.55eV

h2

(1 eV = 1.6.10- 19 J). Equation (2.2) indicates that theenergies of photons that can be emitted by a hydrogenatom are found from the expression

_ 41t2me

4 (_1 1_) (2.3)€ - h2 n2 e

where nand k are integers and n < k. Assuming n = 1and varying k, we obtain the possible photon energiesproduced by transitions to the ground state, that is, tothe level E1• Assuming n = 2, we obtain the spectrumof photons emitted in transitions to the level E2•

Assuming n = 3, we arrive at the spectrum fortransitions to the level E3 , and so on.

Photons. Microparticles, and photons among them,are very peculiar physical objects; their behaviour isdescribed not by classical mechanics but by so-calledquantum mechanics. Many of the customary, age-oldconcepts formed as a result of observation andinvestigation of ordinary (classical) objects have to bediscarded when microparticles are analyzed. Forexample, many questions lose their physical meaning,

where c stands for the velocity of light in the medium.And finally, the photon's polarization must be

determined. This characteristic is quite similar to thecorresponding one of a light wave. For instance, onecan speak of a photon polarized in a specific plane.Two independent states of polarization of a light waveare known (we mean the states of wave polarization intwo mutually perpendicular planes). Correspondingly,we refer to two states of polarization of a photon andcharacterize them by a parameter G that assumes thevalue 1 for one state and 2 for the other one.

How can one picture a photon's polarization? Ina word, what is it? We are used to constructinga clear, demonstrative image, a model ofa phenomenon. However, it is very often impossible (inprinciple!) to work out descriptive images in the worldof microparticles. In particular, it is impossible todescribe the difference in the "appearance" of G = 1and G = 2 photons. What, then, is the meaning of

(2.5)

27

E = hv

It can be said that a monochromatic plane wave IS anensemble of photons in the same state. Different

Several photons are said to be in the same state ifthey have identical sets of the four characteristics Px,Py, p" G. This set can be regarded, therefore, asa characterization of a photon state. A photon statechanges if at least one of the four characteristicschanges.

It is important that the characteristics of a photonstate correspond to those of a monochromatic planewave. Thus, a photon's momentum is directed alongthe wave propagation direction, and its polarization ist~at of the wave. As for the energy of the photon, it isgiven by the wave's frequency:


polarization with respect to photons? The onlylegitimate answer is the following: if a photon is"extracted" from a light wave with polarization G = 1,its polarization will likewise be G = 1; if, however, it isextracted from a light wave with polarization G = 2, itspolarization is G = 2. Nothing more specific (moredescriptive) can be read into the concept of photonpolarization.

Consequently, for a photon to be described, fourquantities must be determined: three projections ofmomentum (Px, P)" pz) and G, which defines the photon'spolarization. As can be seen from Eq. (2.4), thisimmediately gives the photon's energy:

26

(2.4)E

P =c


such as: What does a photon look like? What are itscomponents? What is its size?

We avoid the above "head-on" questions bychoosing a bypass: what should be known abouta photon in order for it to be "fixed" (described)?

First, the photon's energy E, and second, itsdirection of propagation must be determined. This isequivaltnt to determining the photon's momentum P.The momentum of a photon coincides with itsdirection of propagation, and the magnitude of themomentum is related to energy by the formula

From Incoherent to Coherent Optics28 2. Is Interference an Inevitable Result? 29

photon states correspond to different monochromaticplane waves.

Substitution of Eq. (2.5) into (2.4) yields

hv hp = ~ = T (2.6)

Equ~tions (2.5) and (2.6) point to the particle-wavedualIty of a photon's property: they' relate thecorpuscular (1':, p) and wave (v, A) characteristics ofa photon.

Fermions and bosons. The number of microparticlesknown nowadays is quite impressive. It includes abouttwo dozen elementary particles (photon, neutrino,electr~n, proton, neutron, etc.), about the same numberof an.tlpar~icles, as well as various atoms and atomicnucleI. It IS noteworthy that all microparticles in natureare c1~arly classified into two groups by the characterof t~e~r ,behaviour in .an ensemble of similar particles (aphY~I~lst s way of saymg the same is: "by theirstatIstIcal properties"). Figuratively speaking, particles ofthe first group are extremely individualistic: oncea state is occupied by a particle, no other particle ofthe same type can occupy the same state. In otherwords, these particles are governed by theone-state-one-particle rule. Microparticles of the secondgroup behave quite differently: in an ensemble the'number of particles per state is not only unlimited, butthe gre~ter the ~umber of particles already occupyinga stat~ IS, .the hIgher the probability of findinga partIcle m this state.

Particles of th.e first group are called fermions (inhonour of the ItalIan physicist Enrico Fermi), and

those of the second group~bosons (in honour of theIndian physicist Bose). A particle is either a fermion ora boson. This fundamental fact finds certainexplanations in physics, but they are outside the scopeof this book. A feature of extreme importance here isthat photons are bosons. It will be demonstrated belowthat this fact (photons being governed by the Bosestatistics) plays the most important role in opticalphenomena.

Note that electrons are fermions, which explainsthe well-known fact that each energy level of an atomcannot contain more than a specific number ofelectrons (2, 8, 18, ...). The reason is: each atomic levelcorresponds to a specific number of electron states(2, 8, 18, ... ). If electrons were not kept from occupyingthe same state in numbers above unity, all electronswould certainly be found on the lowest energy level; asa result, the existing variety of chemical elements woulddisappear. And this would mean the end of our world!

Photons and light waves. How can we reconcile thedescription of optical radiation in terms of photonswith the existence of light waves? How can we "pass"from photons to light waves?

We have mentioned above that relationships (2.5)and (2.6) reflect the particle-wave duality of photons. Inother words, a photon, as any other microparticle,possesses both corpuscular and wave properties. Thewave properties of photons, however, are not sufficientto explain the existence of light waves. There is yetanother reason, and a very important one: the behaviour of photons in an ensemble, in other words, theBose statistics of photons.

C?wing .to. these statistics, photons can populate anystate Ill. unhmIt~d numbers. Moreover, the higher thepopulatIon densIty of a state is, the higher theprobability of occupying the state.

Let us mentally select a photon pcr-state, that is,a st~te .characteri.zed by a certain momentum p andpolarIzatIon cr. ThIs state corresponds to a definitemonochromatic plane light wave (referred to, hereafter,as the pcr-wave) with frequency v = E/h = pc/h. Let N

podenote the number of photons per unit volume ina pcr-state.

If

that is, if the chosen photon state contains a very largenumber of photons, then the discontinuous nature("~ranul~ri~y") of the radiation can be neglected andthIS radIatIOn can be treated as a "continuous medium"(as. a light ,,:av.e). If, however, condition (2.7) is notsatIsfied, radIatIOn must be considered as discrete, andthe ensemble of photons cannot be treated as a lightwave.

Recapitulating, a pcr light wave exists if condition(2.7) ,?olds; and if (2-:7) is violate?, the term "pcr lightwave becoI?es meamngle~s. A hIgh population densityof p~~tons III a pcr-state IS the necessary and sufficientCOndItIOn of the existence of a pcr light wave.

It . ca~ be. readily seen that classical photon wavescan eXIst (I. e. hght waves can exist) as a directcorollary of the Bose statistics of photons. Were thisnot the case, that is, were photons fermions then notmore than a single photon could occupy an~ one


Npo »

30

(2.7)


photon state. Condition (2.7) would then be violated. Itis worth mentioning that this is the reason whyclassical electron waves cannot be formed.

"Disordered" light waves. Wave trains. Whengeneration of optical radiation is considered, differentphoton states have to be taken into account. Indeed,individual atoms (or molecules) of the emitting mediumgenerate photons in a substantially independentmanner; hence, the photons that appear differ inenergy, momentum orientation, and polarization.Radiation "made up" of such different photons ("disordered radiation") cannot be treated asa monochromatic plane wave. A spread of photonsover available states is characteristic of such "disordered" light waves.

A "disordered" light wave is often modelled bya set of so-called wave trains. Let us assume that thephotons of which the radiation consists can beseparated (mentally, of course) into groups eachcomprising a sufficiently large number of photons inidentical states. The groups differ in the characteristicsof photon states and in the degree of population ofeach state. Each such group is a wave train. In thesimplest case, a wave train is illustrated by a "segment"of a monochromatic plane wave with characteristicscorresponding to those of the photon state in questi?n;the length of the train is determined by the populatIOnof the state (by the number of photons in the wavetrain): the greater this number, the greater is itssimilarity to a corresponding monochromatic planewave. A wave train is schematically drawn in Fig. 6 (tdenotes the train duration, and tc-its spatial length).

31

From Incoherent to Coherent Optics 32 2. Is Interference an Inevitable Result? 33

Coherence of a light wave as a measure of itsability to interfere. The more "disordered" a light wave,the less capable it is of forming an interference pattern.In actual conditions, the ability of a wave to forminterference fringes can be evaluated by measuring thecontrast of the interference pattern, that is, the ratio

An extremely important parameter, the degree ofcoherence, is introduced to characterize this ability oflight to form interference patterns. The higher thisdegree, the greater its ability is. Correspondingly,a decrease in the degree of coherence corresponds toincreased "disordering" of light.

Monochromatic plane waves are ideally coherent.Their ability to interfere is maximum, as is their"ordering". We may assume that the opposite case canalso be realized, that is the case of ideally incoherent(absolutely "disordered") waves completely unable toform interference patterns. Obviously, actual situationscorrespond to various intermediate cases. Rigorously

(2.8)

Fig. 6

II - 12

II + 12

where II is the intensity at the centre of a brightinterference fringe, and 12 is that at the centre ofa dark fringe.

Contrast is at a maximum (11 = 1) when 12 = 0,and at a minimum (11 = 0) when II = 12 , In this caseinterference simply does not exist.

The higher the pattern contrast 11 (the closer it isto unity), the higher the wave's ability to forminterference patterns is. We see, therefore, that numerous intermediate cases corresponding to differentdegrees of ability to interfere can be found between thelimiting cases of ideal and destructive interference.These cases are illustrated in Fig. 7 (the ability tointerfere diminishes from a to c).

'l'C

(a) (b) (c)

The degree of coherence of light waves and thecharacter of photon populations. We want to clarify nowwhat determines the degree of coherence of a lightwave, that is, its ability to form interferencefringes.

The decisive factor is the character of photonpopulation of photon states. If coherence is ideal (thecase of a monochromatic plane wave), all the photons

speaking, both ideally coherent and ideally incoherentwaves are abstractions, and only partially coherentwaves are a physical reality.

The degree of coherence of electromagnetic wavesin the radio and microwave ranges is quite highbecause the process of generation of these waves isbased on regularly repeated motions of electrons.Substantially lower coherence in the optical range iscaused by the specific process of generation in thisrange (radiation of atoms and molecules). The opticsexisting before the advent of the laser is often calledthe incoherent optics; the term is definitely a veryconventional one, since a small degree of coherenceshould have been mentioned. A rigorous approach wouldrequire that a higher degree of coherence be indicatedfor fluorescent lights than for incandescent lamps.

Lasers led to a revolution in optics. It was foundthat radiation by atoms and molecules can also besufficiently coherent. The laser age in optics is the ageof coherent optics.

At present, progress in coherent optics makes itpossible to extend the methods of traditional radiophysics to the range of optics. The new branch ofscience that resulted from this extension is calledradiooptics.

,*

352. Is Interlerence an Inevitable Result?

are in the same state, that is, have the same energy,the same direction of momentum, and the samepolarization. On the contrary, ideal incoherence meanssufficiently uniform distribution of photons overdifferent states. The higher the selectiVity (non-uniformity) in the photon population of the availablestates, or in other words, the greater the populationdensity of some states at the expense of other states(which remain unoccupied or nearly unoccupied), thehigher the radiation's coherence. In practical terms thismeans that the higher the degree of coherence ofoptical radiation is, the lower its nonmonochromaticity,the smaller its divergence, and the higher the degree ofpolarization.

The degree of nonmonochromaticity of a wave (seeSec. 1) is given by the ratio ~ = I'1v/vo, where Vo is themean frequency and I'1v is the frequency rangecharacterizing the "spread" in photon energies (if this"spread" is denoted by 1'16, then I'1v = 1'16/h). Thedegree of divergence is characterized by the angle ofthe cone in which the wave propagates; this angle isreferred to as the divergence angle. The smaller theangle of divergence, the closer the wavefront's shape isto planar. In practical cases the degree of polarization?f light waves is found by means of polarizers, formstance by means of a special crystal that transmitsonly waves with a definite plane of oscillation of theelectric field strength (fixed by the orientation of thecrystal). The intensity of light transmitted by thepolarizer is measured for different orientations of thecrystal with respect to the beam propagation direction,and the minimum (I2) and maximum (II) intensities arerecorded. The degree of polarization is given then by


1~ - - (2.10)

'tvo

the ratio

x = 11 - 12 (2.9)11 + 12

Finally it should be noted that there existsa simple relationship between the degree ofnonmonochromaticity of a light wave, ~, and the timeduration, 't, of wave trains:

From Incoherent to Coherent Optics 363. How can We Generate Coherent Light Waves?

same train and thus obtain interference. The principalarrangement of an experiment making such interferencepossible is given in Fig. 8 (A is the source of light;B is a semitransparent mirror; C is a completelyreflecting mirror; and D is a screen to observe theinterference pattern). Mirror B splits the wave train,and mirror C realizes the superposition of the splitparts of the train at point D. The following conditionmust be satisfied in order that parts of the same wavetrain superpose at point D:

37

Fig. 8

where T is the wave train duration (coherence time), Lis the difference between paths travelled by the parts ofthe wave train from the splitting point to the finalpoint (in the case in question this is the differencebetween IBCI + ICDI and IBDI). Interference is observedif condition (3.1) is satisfied.

Various methods are used in pre-laser optics torealize the principle illustrated in Fig. 8: Fresnel'sbiprism, Michelson's interferometer, observation of

(3.1 )L«TC

3. How Can We GenerateCoherent Light Waves?Coherent effects in pre-laser optics. It was

mentioned above that any light wave is characterizedby a certain degree of coherence. This is equally truefor waves emitted by ordinary (non-laser) light sources,however small the degree of coherence of a light wave,in principle it can always be used to generate aninterference pattern.

The truth of this can be demonstrated by usingthe concept of wave trains. Obviously, different wavetrains do not give rise to interference when superposed;it is nevertheless possible to superpose parts of the

The quantity 't is generally referred to, in the specialliterature, as the coherence time. This is a veryimportant characteristic of the degree of coherence ofa light wave. Obviously, the longer the wave trains, thehigher the degree of coherence; this is in perfectagreement with the dependence of coherence on thedegree of selective population of photon states.

From Incoherent to Coherent Optics 38 3. How can We Generate Coherent Li/?ht Waves? 39

Newton's rings and thin film colour fringes, and so on.Interference fringes are observable in all these casesbecause L is sufficiently small (note that the train lengthin ordinary light sources is normally not more thanone centimetre or even a fraction of a centimetre).

Photon population of states must be a controlledfactor. The pertinent questions are: how can wegenerate a sufficiently coherent wave, and how can weincrease the degree of coherence of a wave?

In principle, the answer is clear. It is necessary forthe photons to occupy only selected states. This in turnrequires that the photon population be a controlledcharacteristic.

Such control is indeed possible owing to the"boson nature" of photons, that is, to their tendency topopulate predominantly just those states that alreadyha.ve a sufficiently high population density. Obviously,thiS property of photons can in principle be used toaccumulate photons in specific states.

Let us discuss the methods by which the problemcan be solved in actual conditions.

In ordinary conditions (in the initial state) allactive centres have energy E b that is, are at level 1.Assume now that somehow the active centres areexcited so that a large enough number is raised tolevel 3.

A microparticle cannot remain in an excited statefor an indefinitely long time. Sooner or later, itinevitably undergoes a spontaneous transition to one ofthe less excited states; in the case in question, thismeans a transition to level 2 and then to level 1 (ordirectly from level 3 to level 1). The released energy ofexcitation is then either transferred to other particles ofthe medium or emitted as a photon. It is impossible,in principle, to indicate the moment of time when anactive centre spontaneously reduces its energy, but theprobability of such a transition per unit time can begiven. Note that this probability is independent of boththe choice of the active centre and the time it hasalready spent in the excited state.

Let us denote the probability of spontaneoustransition from level 3 to level 2 (the 3 --+ 2 transition)by 1/132, and that of the transition 3 --+ 1 by 1/131 , By

Generation o{ inverted population o{ the levels inactive centres. First of all, we want a materialcontaining particles (atoms, ions. or molecules) witha special system of energy levels (also called terms).Such a material is called an active medium, and theparticles are referred to as active centres. Figure9 Illustrates a characteristic system of levels of anactive centre. It contains three levels (all other levels ofthe active centre play an insignificant role as far as theprocesses in question are concerned).

Fig. 9

3

2>.~

'"<==~

Lasing levels

is satisfied. This means that active centres must leavelevel 3 fairly quickly and pass predominantly to level 2,where they must be "trapped" for some time (as a rule,I3Z ~ IO- Ms, and IZI ~ 1O- 4 _IO- z s). If (3.2) holds,active centres on level 2 can be accumulated innumbers exceeding those on level 1. It is said in thiscase that an inverted population of levels 1 and 2 isreached.

Stimulated (forced) emission of photons. Let usassume that a photon passes in the vicinity of activecentres with inverse population of levels 1 and 2, andthat the photon's energy E is equal to the difference inenergies of these levels, E = £z - £1' What result canbe expected?

First, the photon may be absorbed by one ofthose active centres that are still on level 1, so that thenumber of active centres on level 2 increases by unity.Second, the photon in question may "stimulate" one ofthe active centres on level 2 to drop to level 1; thisgenerates one additional photon.

The physical nature of the second process is clear.It results in an increase in the number of photons ina certain photon state, whereby the photon emitted byan active centre is added to the initial photon. This isa direct consequence of the bosonic nature of photons,

3. How can We Generate Coherent LiRht Waves? 41

that is, their tendency to accumulate in the samephoton state. This fundamental property "makes" thephotons "trigger" such transitions in the material,which produce new (secondary) photons. It must beemphasized that the secondary photon is in exactly thesame state as the stimulating photon.

This process of photon emission differs in principlefrom spontaneous emission because here the transitionis controlled by an external photon that "triggers" theemission. Consequently, one distinguishes between thespontaneous and stimulated (forced) emission.

The author once encountered a simile comparingstimulated emission with a process in which a pearfalling off a tree shakes the branches and therebyforces down other pears in its wake. This comparisonis justified, in a certain sense, since it demonstrates thedifference between spontaneous and stimulated emission.The fall of the first pear happens as if "by itself', andwe get a "spontaneous pear". The "stimulated pears",on the other hand, fall not by themselves but owing toa controlling factor: they are shaken off by other pears.In analogy to the case of photons, a "spontaneouspear" may serve as a trigger for the generation of"stimulated pears". This analogy should not, however,be taken too literally. For instance, it must be kept inmind that before being "triggered", a pear is an entitypresent on the branch, while a Stimulated photon isgenerated in the very process of emission (it did notexist earlier). The initial photon does not actually"force another to follow suit", but triggers anappropriate transition in an active centre, whichgenerates a stimulated photon.

The following interpretation can be given to theprocess of stimulated emission in the "language" of

I•

},

40

(3.2)

convention, constants I3 Z and I31 are called thelifetimes of the active centre on level 3 with respect totransitions 3 -+ 2 and 3 -+ 1. We also introduce theprobability of transition 2 -+ 1 and denote it by IfIzl'

The system of energy levels of an active centremust be such that the condition


From Incoherent to Coherent Optics 42 3. How can We Generate Coherent Light Waves? 43

trains. An initial train interacts with excited activecentres and "triggers" transitions generating secondarytrains. These secondary trains are generated essentiallyin phase with the primary one, which can be said toform a new, longer train. This lengthening of a trainsignifies an increase in the degree of coherence of thewave, requiring at the same time that the primary andsecondary trains be in phase.

It must be emphasized that the generation ofa stimulated photon in exactly the same state as thatof an initial photon and the emission of a stimulatedtrain in phase with an initial train are two methods ofdescribing the same phenomenon.

Competition between the processes of absorption andstimulated emission. Thus, we have established that twodifferent and competitive processes are possible whenphotons with energy g = E 2 - E 1 interact with activecentres (see Fig. 9): absorption of photons, or stimulatedemission of additional photons. The greater the numberof active centres in the appropriate initial state thehigher the probability of each of these two processes IS.

If the number of active centres on level 1 is greaterthan that on level 2, absorption becomes a moreprobable process. If, on the contrary, there are moreactive centres on level 2, stimulated emission becomesmore probable. •

Consequently, the inverted population of levels1 and 2 is a necessary condition for stimulatedemission to exceed absorption. Absorption normallyproceeds at a higher rate than stimulated emission(higher levels usually have lower population density), sothat a light wave travelling in a medium is dampedout. If, however, the population of active centres is

inverted, a light wave may be amplified in an activemedium. Obviously, this wave must be coherent andhave the frequency v = (E 2 - Ed/h.

But what can produce a coherent light wave, thewave that ultimately is our goal?

The idea of selectivity for photon states. Eachphoton emitted by inevitable spontaneouS transitions2 ---> 1 may be absorbed or may stimulate the emissionof new photons. The emission of new photonspredominates in a medium with an inverted populationof levels 1 and 2. In other words, spontaneouslyemitted photons trigger numerous stimulatedemissions.

Unfortunately, active centres emit "spontaneous"photons independently and therefore in arbitrary states.First of all, the photons differ in the direction of theirmomenta. It is clear that each stimulated photon is inthe state of the corresponding initial photon, so thata "spread" in states of spontaneous photons results ina "spread" of stimulated photons as well. Obviously,such stimulated radiation cannot have a high degree ofcoherence.

Let us assume, however, that we have succeeded increating favourable conditions for stimulated emissionfor some (rather few) photon states, and unfavourableconditions for all remaining states. (For the sake ofbrevity, we shall use the terms "favoured" and "nonfavoured" states.) Then a large number of photons aregenerated by photons that are spontaneously emitted in"favoured" states, while the photons spontaneouslygenerated in "non-favoured" states quite soon leave thefield without stimulating appreciable numbers ofphotons.


Evidently, the flux of photons in "favoured" statesforms highly coherent optical radiation emitted by theactive medium. (In terms of Fig. 9, the radiation isgenerated by the 2 ---> 1 transition, levels 2 and 1 beingreferred to as lasing levels.)

The lower the number of "favoured" photon states,the more pronounced they are and the more"suppressed" the remaining photon states are, the betterthe selectivity of emission is and the higher thecoherence of the emitted radiation. If it were possibleto single out just one photon state, the generated lightwave would be ideally coherent, that is, it would bea monochromatic plane wave with a fixed polarization.

How can we realize selectivity for photon states?Several approaches may be suggested for realizingselectivity. For instance, directional selectivity is achieved by preparing the active medium in the shape ofa long rod with a comparatively small cross-section.Spontaneous photons with momentum parallel to therod's axis are obviously favoured, since they caninteract with a higher number of active centres andtherefore stimulate an intensive avalanche of stimulatedphotons. On the other hand, those spontaneousphotons with momenta at an angle to the rod's axisare soon leaving the active medium. The path withinthe active medium travelled by favoured photons maybe increased still further by placing partially reflectingmirrors at the end faces of the rod. The mirrors returnpart of the radiation back to the active medium andtherefore enhance the effect of stimulated emission forthe photons that are forced to travel over longer pathsin the active medium. This in fact is the idea of an

443. How can We Generate Coherent LiRht Waves?

optical resonator, an essential element of a laser, whichwill be discussed later in the book.

The selectivity of photon energies is furnished byactive centres with an appropriate system of energylevels. In real situations, however, the system of levelsof an active centre is always richer in terms than theone shown in Fig. 9. It is possible to suppress theunnecessary levels and the transitions between them by,for instance, making the reflection coefficient of theabove-mentioned mirrors frequency-dependent. Suchmirrors create a selectivity of emission over photonstates.

Conclusions. In order to obtain highly coherentoptical radiation, it is necessary, first, to. excite activecentres and produce an inverted populatIOn of thelasing levels, and second, to create selectivity for photonstates. The inverted population of the lasing levels ofactive centres is required in order that stimulated..emission predominate over absorption. And selectlVlty isrequired for the effect of stimulated emission to. besignificant only for a few photon state~ (otherWise ?osufficiently coherent radiation can possibly be obtamed).

High coherence of laser emission is based, fi.rst, onthe nature of stimulated emission, which makes Itpossible to accumulate photons in selected photonstates and second, on the creation of selectivity ofemission in the medium, which permits the utilizationof such accumulation only for a few photon states.

45

Fig. 10

From Incoherent to Coherent Optics 47

(b) crystals and glasses doped by special ions(solid-state lasers);

(c) liquids (liquid lasers);(d) semiconductors (semiconductor la~ers).

Only gas and solid-state lasers are discussed in thisbook. .

Normally the active medium of a gas laser ISa mixture of several gases; atoms or molecules of oneof them are active centres while other gaseouscomponents serve to produce population inversi?n onthe lasing levels of the active centres. One possiblemixture, for instance, is helium and neon (neon atomsare active centres). This mixture is placed in a gasdischarge tube at low pressure: neon at a pressure ofabout 10 Pa and helium at about 100 Pa.

The active medium of a solid-state laser isnormally a rod with a circular cross-secti~n, dopedwith special ions that play the role of aC~IVe ~entres.

A classical example of a solid lasing medIUm IS a rubyrod 3 to 20 mm in diameter and 5 to 30 cm long.Ruby is crystalline alumina (AI20 3 ) dope? w~th chromium ions (from 0.05 to 0.5%). Note that It IS thiSimpurity (dopant) that gives ruby its typical colour(from pink to deep red).

Systems of pumping vary, and to a la:ge .exte.ntdepend on the type of active medium. EXCitatIOn In gaslasers is realized in the simplest manner by means ofelectric discharge in the active medium (t.ypi~ally. glowdischarge). In this case the energy of excltatIo~ .IStransferred to active centres as a result of collISIOnswith particles in the gas discharge plasma.

In the case of the helium-neon gas laser, thedischarge .is dc glow discharge. A simplifi.ed diagra:n ofenergy levels in neon and helium atoms IS shown In

4. The Laser: Working Principles46

o

possible additional elements

system of excitation of medium

o

4. The Laser:Working Principles

A functional schematic of a laser is shown inFig. 10. The active medium and theadditional elements are located inside the opticalresonator. The resonator singles out an optical axis 00of the laser; the emitted radiation propagates along00. Note that a laser can emit radiation both ina single direction and in two opposite directions alongthe optical axis.

A laser is triggered by initiating its pumpingsystem. This system provides excitation of activecentres, and an inverted population of lasing levelsbuilds up. The optical resonator (together with someadditional elements) provides selectivity over photonstates. As a result, a highly coherent radiation, calledthe laser radiation, appears along the axis 00.

Active mediums and methods of excitation. Thefollowing active mediums are used in lasers:

(a) gases and mixtures of gases (gas lasers);

From Incoherent to Coherent Optics 484. The Laser: Working Principles 49

Fig. 11

Fig. 11; the lasing levels are levels 2 and 3. Because ofcollisions with electrons in the gas-discharge plasma,neon and helium atoms are excited to level 1 (heliumatoms) and levels 2 and 3 (neon atoms). In theimpacts, electrons transfer to the atoms a fraction oftheir kinetic energy (electron excitation). There is,however, another possibility to excite neon atoms tolevel 3: resonance transfer of energy from excited helIum at~ms to ne~n atoms in the ground state, wherebythe helium atom IS de-excited while the neon atomundergoes a transition from the ground state to one ofthe excited levels. This energy transfer takes placebecause the corresponding energy levels in helium andneon (1 and 3) are almost at the same distance from~he ground leve.l, and also because the helium densityIII the mIxture IS sufficiently high (this prevents theunfavourable transfer of energy from neon to helium,since the probability of an atomic process isproportional, among other factors, to the number ofatoms in the initial state). Taking into accountcollisions both with electrons and with excited heliumatoms, we conclude that level 3 must be populated

He

3

2

Ne

with higher probability than level 2; hence, levels2 and 3 in neon atoms must be inverselypopulated.

Excitation in solid-state lasers is carried out byirradiating the lasing rod with light from '\ sufficientlypowerful light source, such as a pulse lamp (opticalpumping). In this case active centres are excited owingto the absorption of photons emitted by the pumplamp. Obviously, lasers with optical pumping can beconsidered as converters of optical radiation, converting,for instance, the incoherent radiation of a pulse lampinto highly coherent laser emission.

Optical resonator. An optical resonator is realizedas a system of mirrors. In a more general sense, itincludes not only a system of mirrors but alsoeverything inside this system, including the active medium. Figure 12 schematically shows examples of opticalresonators: (a)-a simple linear resonator; (b)-acoupled linear resonator; (c)-a ring resonator(00 - the optical axis of the laser, fixed in space bythe system of resonator mirrors).

Mirrors may be coated with dielectric or metallayers. At least one of the mirrors must be partiallytransparent with respect to the emitted radiation; inthe case of metal coating, the problem is solved bymaking a hole at the centre of the mirror. Thetechnology of manufacturing laser mirrors is very

- complicated; a coating is deposited onto a substrate bysuccessive thin layers. Resonator mirrors of gas lasersare usually mounted at both ends of the gas-dischargetube and are not linked to it in a rigid manner.Mirrors in solid-state lasers are typically formed onspecially prepared end faces of the active-medium rod.4-190

From Incoherent to Coherent Optics 50 4. The Laser: Working Principles 51

Returning to the optical resonator, we can nowformulate that it singles out a direction in space forwhich the losses are minimized and the condition ofgeneration is satisfied. Here we see the role of theoptical resonator revealed as that of an elementcreating selectivity for photon states.

Fig. 13

A prism inside a resonator. Additional elementswithin the optical resonator also serve to maintain theconditions of selectivity. For example, a prism insidethe resonator provides energy selectivity.

Let us assume that the active medium generates attwo frequencies, VI and Vl, simultaneously, and that theV 1 emission must be suppressed. Various methods couldbe used for this purpose. One of them is based onresorting to the phenomenon of dispersion of light, thatis, the dependence of the refractive index on the frequency (If light waves.

If a prism is placed inside the resonator, the lightwaves with different frequencies emitted from the activemedium are spatially separated by the prism (Fig. 13).If the right-hand mirror (in the figure) of the resonatoris perpendicular to the propagation direction of the VI

wave, then the Vl wave is incident on the mirror

mirrors

(c)

mirrors

Fig. 12

mirror

mirror

As mentioned above, an optical resonator definesthe direction in which radiation is emitted. Theprocesses of stimulated emission are able to compensatefor absorption only for this direction.

When radiation inside a real laser is considered, itis necessary to take into account not only stimulatedemission and photon absorption processes by activecentres but also many other processes causingradiati~n losses (such as scattering and absorption bynon-active centres). Laser oscillation is sustained only ifstimulated emission by active centres compensates notonly for the absorption by these centres but for allother losses as well.

The concept of "favoured" photon states introducedin the preceding Section can now be elaborated.Namely, the "favoured" states are those states Torwhich the losses are eliminated to a maximum degree.

oiE~~ ~~'=?+f-(a) .. (b)

actIve medIUm

(4.1)

(4.2)

534. The Laser: Working Principles

The refracted beam is polarized predominantly in theplane of incidence, while the reflected beam tends to bepolarized in the plane normal to it.

If, however, the angle of incidence y satisfies thecondition

Fig. 14

tan y = n

then the refracted and reflected beams are perfectlypolarized. The angle of incidence y satisfying thecondition (4.2) is called the Brewster angle (Fig. 14).Dots and arrows in the figure indicate the direction ofoscillation of the vector of electric field (arrows denoteoscillation within the plane of the drawing, anddots-normal to this plane). It can be easily foundthat for an angle of incidence equal to the Brewsterangle, the reflected and refracted beams are at rightangles. Indeed, Eqs. (4.1) and (4.2) yield that sin ~ =

52

sin y--=nsin ~

From fncoherent to Coherent Optics

where n is the refractive index of the medium.Experiments demonstrate that reflection and

refr~ction change not only the direction of propagationof light beams but their polarization as well. In thegeneral case, an unpolarized light beam is transformedinto partially polarized refracted and reflected beams.

. . What is the Brewster angle? When a light beam is~n~ldent on the surface of a medium at a certain angle,It IS transformed into two beams, one reflected and onerefracted. The first of them propagates outward at anangle y equal to the incidence angle, and the second isrefracted into the medium at an angle ~ satisfying therelationship

obliquely and is not returned into the active mediumafter reflection. Consequently, only photons with energyhV I are returned into the active medium, while thosewith energy hVl leave the system immediately afterreflection. The prism clearly realizes the selectivityeffect: photon states with energy hV I are "favoured"while those with energy hVl are "non-favoured".

Note that although Figure 13 shows two lightwaves spatially separated by the prism, in reality thereis no wave with frequency Vl' The prism damps thiswave so effectively that it is simply not emitted at all.If the prism is removed, both waves are generated; ifthe prism is there, the generation of one of the wavesis damped out. The correct interpretation of Fig. 13 is:if a wave with frequency Vl appears, it is immediatelysuppressed by the prism.

From Incoherent to Coherent Optics 54 4. The Laser: Working Principles 55

= cos y. Hence, angles ~ and yare complementary,that is, together make a right angle.

plane. Clearly, the wave polarized normal to the "planeof oscillation" is completely reflected by the windowplate while that polarized in this plane is transmittedby the plane-parallel window plate in the samedirection. This means that the first of these waves isimmediately shut out of the picture while the secondone is repeatedly reflected by the resonator mirrors andcorrespondingly passes repeatedly through the activemedium. This system "favours", therefore, the photonstate with polarization in the "plane of oscillation" andis "unfavourable" to the states polarized normally tothis plane. The emitted radiation is thus polarized inthe "plane of oscillation".

Note that this device kills two birds with one stone:first, we obtainplane-polari~~!Lll!~er-e~.J.Si~9n::-!nQ--·--·secoi!a,-we-etiminate-tu"Sses that would be caused"yreflection at the surfaces of "tile-tubewincfow~plates.Indeed, the light wave thiiCwouldhave-been reflectedby these surfaces by virtue of its polarization is simplynot emitted (Brewster windows make attenuation of thiswave too high.)

Basic modes of laser oscillation. Three basic modes aredistinguished: continuous oscillation, pulsed mode offree oscillation, and pulsed mode with controlled losses(Q-switching, or the so-called giant-pulse mode ofoscillation). The continuous mode is typical for gaslasers, and pulsed modes are mostly employed insolid-state lasers. It must be mentioned, however, thatif necessary, any type of laser can be made to operatein any of these modes of oscillation.

In the case of continuous oscillation, a laser emitsa continuous light beam with constant power. Thismode requires a continuous steady-state pumping of the

--;;'f1/f----,----Jr+1-+-+-0

Fig. 15

Why do we orient the end face of the gas discharge tubeat the Brewster angle? The planes of the end faces ofgas-laser discharge tubes are usually tilted at an angleso that the normal to the end face and the optical axisare at the Brewster angle corresponding to therefractive index of the material of which the end-facewindow is made. Figure 15 gives a schematic of oneend of a gas discharge tube (00 - optical axis; andy is the Brewster angle). A similar angled window isinstalled at the opposite end of the tube.

The end face of the discharge tube is oriented at theBrewster angle in order to realize the conditions ofpolarization selectivity of emission in the gas laser. Leta non-polarized light wave be incident on theplane-parallel window (end plate) of the tube along itsaxis. An unpolarized wave can be expressed as a sumof two plane-polarized waves one of which is polarized,for example, in the plane drawn through the normal tothe window and the tube axis (let us refer to it as the"plane of oscillation"), and the other normal to this

o

I!

Giant pulses. This special mode of laser oscillation willbe discussed in some detail (another term: Q-switching,i. e. modulation of losses). Light pulses emitted in thismode are indeed giant, since their peak power reaches109 W. For comparison we remind the reader that thebiggest hydroelectric power plant in the world(Krasnoyarsk Power Plant on the Yenisei river in theUSSR) has a power of 6.109 W. Such pulses, however,have very short duration, on the order of 10- 8 s, so

active medium. This pumping can be produced bysteady-state discharge in a gas or by a continuouslyoperating pumping lamp (in the case of optical pumping).

In pulsed free oscil1ation the emission takes the formof periodic (and sometimes irregularly repeated) lightpulses each 10 - 6_10 - 3 S long, emitted at a frequencyof 10Hz to 10 kHz. The pulsed character of emissionis a result of the pulsed operation of the pumpingsystem (a pulsed lamp, or pulsed discharge in thegas). This pumping creates population inversion onactive centres periodical1y and for short periodsonly.

The pulsed mode of oscillation enables us to achieveconsiderable concentration of light energy, and toobtain, within short time intervals, substantial poweroutput. If, for example, a laser emits 10 light pulsesper second, each pulse 10 - 4 S long, with energy perpulse 1 J, then the maximum power achieved (referredto as the peak power) is not less than 104 W, whilethe average power of emission is only 10 W.The concentration of light energy reaches a maximumin the giant-pulse mode of oscillation.

.) There are also so-called axial mode-locking oscillations. whichmake it possible to obtain especially short light pulses 10 - 11 to 10 - 12S

long. The peak power achieved in such ultrashort pulses reaches1012 W. Ultrashort light pulses are formed owing to a peculiarinterference effect that redistributes light energy within each giant pulseand concentrates this energy into several ultrashort (superpowerful)pulses.

574. The Laser: Working Principles

that the total energy of light emitted per pulse is onlyabout 1 to 10 J *.

Giant-pulse mode of oscillation can be realized bycontrol1ing losses inside the resonator. If these lossesare somehow sufficiently increased for a certain time,oscillation cannot develop. Consequently, the system ofpumping builds a considerable overpopulation of theupper lasing level in active centres. If, however, thelosses are then sharply brought down, the process ofstimulated emission by the "favoured" photon statesdevelops in an avalanche manner and gives rise toa short light pulse with very high peak power.

In order to control the losses in a resonator (tocontrol its Q-factor), a so-cal1ed optical switch isintroduced into it. In the simplest case, it is anelementary chopper that rotates and thereforeperiodical1y interrupts the light beam.

In particular, one of the mirrors of the opticalresonator can be made rotating. In this case losses aresmal1 (Q is high) only during the short interval whenthe rotating mirror is nearly perpendicular to theoptical axis of the laser; hence, this is the time whena giant pulse can be emitted.

It is essential for the chopper of the beam or fora rotating mirror to be synchronized in time withpump pulses, for instance, with the pulses of a pump

;,


From Incoherent to Coherent Optics 58 5. Lasers as Sources of Coherent Optical Radiation 59

5. Lasers as Sourcesof Coherent Optical Radiation

This Section is chiefly for reference. A readerinterested mostly in the physical side of the topic cansafely skip it on the first reading.

on time of the power of the emitted radiation; this isthe shape of the giant pulse. The pulse is emitteddirectly after the drop in losses occurs. The figureclearly demonstrates that the moment to is chosen insynchronism with the pumping pulse. The process canbe repeated if losses are again increased after the pulseis emitted.

It should be emphasized that emission of a giant pulseis meaningful only if the drop in losses is very sharp.From this standpoint, rotating optical switches are notvery suitable: as all other devices based on mechanicalmotion, these Q-switches have prohibitively high inertia.So-called electrooptical and passive (phototropic)switches, which are optical switches without moving(rotating) parts, are more effective. In electroopticalswitches an external electric field modulatestransparency (and consequently, losses) of the switch; inpassive switches (also called photon-activated switches)losses are modulated by the generated laser emission.Passive switches are especially promising; they arediscussed in detail in Chapter 3.

Finally, we note that rotating optical switches enableus to generate light pulses with power up to 10 7 W(pulse duration not less than 10 - 7 s). Shorter and,therefore, more powerful light pulses are obtained withelectrooptical and photon-activated switches.

t..

c

~1-----+:---------

Fig. 16

lamp: the steep decrease of losses must take placeimmediately after a substantial population inversion ofthe lasing levels is achieved.

Figure 16 shows three curves. Curve A plots thepumping intensity as a function of time, that is, givesthe shape of the pumping pulse. Curve B plotsa quantity characterizing losses, also as a function oftime. 'Fhe curve shows that losses dropped sharply attime moment to. Finally, curve C plots the dependence

11 ~~ ._---.-.!---=-------I to

----'\'~'---

toIIII

From Incoherent to Coherent Optics 605. Lasers as Sources of Coherent Optical Radiation 61

Several typical types of lasers. The number of lasersystems in use today, differing in active mediums andmethods of pumping, runs into the hundreds. Onlythree types will be mentioned here: the helium-neongas laser, and two solid-state lasers, the ruby crystallaser. (historically the first to appear) and the yttriumalummum garnet laser (at present the most widely usedsolid-state laser).. Helium-neon gas laser. Figure 17 gives a generalIdea of the construction of helium-neon lasers. Theschematic shows the gas-discharge tube and mirrors ofthe resonator. The active medium consists of a mixtureof helium and neon contained in the discharge capillary(internal diameter from 1 to 10 mm). The reader recallsthat neon atoms are active centres, and helium is anauxiliary ~as serving to produce an inverted populationof the lasmg levels of the active centres. The schematicclearly shows that end faces of the discharge tube arenot perpendicular to the tube axis; in fact, the normalto a window plate is at the Brewster angle to the tubeaxis.

Helium-neon lasers have emission power on theorder of 10 mW and an efficiency not higher than0.1 %. The lasers operate in the continuous wave mode(CW). The basic generated wavelength is 0.63 Ilm (redbeam). The lasers may also oscillate in the infraredrange at wavelengths of 1.15 Ilm, 3.39 Ilm, and someothers.

A solid-state laser with optical pumping is shownin Fig. 18. The pumping lamp and the laser rod arelocated inside the reflector parallel to each other andpass through the focuses of the elliptical cross-sectionof the reflector. This achieves a high concentration ofthe light flux of the pump lamp on the laser rod.

Concentration of active centres in solid-state lasersis by several orders of magnitude higher than in gaslasers (1017 _1020 em - 3). As a result, population of theupper lasing level is much higher and the emittedpower is sufficiently high for a relatively small length

Fig. 17 Fig. 18

specially machined end faces

elliptical pump reflectorpulsed pumping lamp

laser rodanodecathode

resonatormirror reso nato rmirrori ."~,;:r~~d~isiChiairg~e~c=ap:il:la~ry~-;=-=-=-=-:=-=-=-=-=-~·~-~~B:re~w~st:e~r

WIndow window

The wonderful laser beam. Imagine a helium-neonlaser operating in a darkened laboratory. The rich redcolour of the beam is a wonderful sight in thesemidarknes~ of the r?om. The beam looks very un~sual:. no dIvergence (WIdening) is noticeable, andmtensIty IS practically constant. One can placea number of reflecting mirrors in its way and make thebeam ~race a ~igzag path within the laboratory. Theresult IS magnIficent: darkness crossed in all directionsb~ bright-red filaments. If the beam diameter is magnIfied by means of a lens and then the beam is thrown~n a scree~, such as a sheet of paper, a very unusuallight spot IS .observed: it. "speckles", dark and brightspots appeanng and vanIshing.

The unusual behaviour of the laser beam isproduced exclusively by its high degree of coherence.The first corollary is a very low divergence of thebeam, and consequently, almost constant intensity aswe move away from the laser. The richness of the

beam's red colouring is due to the high degree ofmonochromaticity of the emission.

The speckle pattern on the light spot is also causedby the coherence of emission. Light and dark specks

apI't:ar because of the interference of coherenfbeams'--reflected to the ooserver's,oeyes from different points ofthe spot. Sli.ghtunconsclous motions of the observer'shead change the angle at which parts of the spot areseen and modify the conditions of interference, so thatbright spots are turned into dark ones, and vice versa.

Characteristics of coherence of gas and solid-statelaser emission. Different lasers emit radiation witha different degree of coherence. In order to characterizethe degree of coherence quantitatively, we consider,.first, the coherence time t, and second, the angle ofdivergence of the light beam.

The best coherence is found in gas laser emission.Here the coherence time reaches about 10 - 3 s. Thismeans that the wave train length (the product tc)comes up to 105 m (up to 100 km!). It can bementioned for comparison that in non-laser lightsources the wave train length is shorter by at leastseven orders of magnitude! The angle of divergence ofthe gas laser emission is only one minute.

Solid-state lasers emit radiation with a lowerdegree of coherence. Typical coherence time is 10- 6 s(wave train length about 100 m), and an angle of divergence on the order of 10 minutes.

The emission of solid-state lasers is less coherentin comparison with that of gas lasers becausespontaneous photons in these lasers are spread ina wider range of states, and because it is more difficultto achieve high selectivity conditions for photon states.


of the active medium. As a rule, solid-state lasersoperate in the pulsed mode of oscillation.

Historica.lly, the ~rst laser was the solid-state rubylaser (crystallme alumma doped with chromium ionsserving as active centres). The laser emits at thewavelength of 0.69 Ilm; average emitting power is onthe order .of 1 Wand efficiency is up to 1%.

~ solid-state laser widely used now is the yttriumalUI!unum ga.rnet laser. The active medium (lasant) isyttnum.-alummum garnet (Y3A15012) doped withaPl?roxlmately 3% neodymium ions, which serve asactive ce~tres. T~e laser oscillates at the wavelength of1.06 Ilm (m the mfrared range), with an emitting powerfrom 10 to 200 Wand efficiency up to 1%.

62

r(

I'I

I"

5. Lasers as Sources of Coherent Optical Radiation 63


One of the reasons for a higher spread of photonsover the states lies in the difficulties involved in growing a solid active medium with sufficiently uniformoptical properties and in creating sufficiently uniformoptical excitation of the lasing rod. Furthermore, the"spread" in photon energies is additionally increased bya comparative abundance of energy levels in the solidstate.

High selectivity of photon states is also achievedwith difficulties, partly because of the impossibility ofobtaining sufficiently long lasing rods. Obviously, thelonger the rod (the longer the optical resonator) thebetter the selectivity over orientation of photonmomentums is realized.

In the case of gas lasers all these difficulties are infact absent since a gaseous medium is very uniform(homogeneous) and the gas-discharge excitation can alsobe made very uniform. At the same time, low densityof the gas is not an obstacle to increasing theresonator's length.

The range of wavelengths "mastered" by lasertechnology. The available lasers emit light waves in thevisible, infrared, and near ultraviolet spectral ranges.The IR range is effectively "mastered" by using activemediums operating on transitions between energy levelsin molecules, while UV lasers are built on the basis oftransitions between energy levels in ions.

At the present moment the range of the wave-lengths is from 0.2 !-lm to approximately 100 !-lm. Highlycoherent high-intensity radiation can actually beobtained on any wavelength within the indicated range.Current research effort is aimed at further broadeningthe "laser range", especially at lowering the short-

II1

I

wavelength limit, down to the X-ray range of thespectrum. Tremendous difficulties will have to beovercome before this problem is successfully solved.

The fields in which lasers are used are multiplying.Two main approaches to the application of lasers inscience and industry can be indicated: first, coherentoptical radiation as a factor affecting ~aterials, andsecond, coherent optical radiation as a means for thetransmission and processing of data (so-calledinformation-oriented applications).

Effects on materials. The high coherence of laseremission makes it possible to realize a tremendousspatial concentration of light power, such as 1013 W ina space with linear dimensions of only 1 !-lm. Radiationof such intensity can cut metal, produce microwelding,drill microscopic holes through diamond crystals, andso on.

Today Garin's fantastic hyperboloid has "turnedinto" a real laser that is widely used in industrialprocesses for high-precision treatment of materials.

Many complex, fine operations in surgery todayare conducted not only with the traditional scalpel andlancet, but also with the laser beam. As an example,we mention the operations of welding of detachedretina on the fundus oculi, penetration of blood vesselsin the eye for treating glaucoma, destruction ofmalignant tumors, and so on.

Laser radiation may, without destroying a material,considerably change its properties and above all itsoptical properties (refractive index, dielectricsusceptibility, transparency, etc.).

Information-oriented applications. Information can betransmitted if a coherent electromagnetic wave

5-190


(so-called carrier wave) is appropriately modulated; thismeans that one has to "occupy" a frequency range ~v

whose width depends on the character of theinformation to be transmitted. For instance,transmission of speech or music requires that thecarrier wave be modulated by sound vibrations. Therange of frequencies audible to the human ear coversthe range from 16 to 2· 104 Hz, so that modulation ofthe carrier wave must occupy a frequency band ~v ~

~ 104 Hz (a narrower band, about 103 Hz, is sufficientfor transmission of speech only). Transmission ofimages (TV videosignals) needs a much wider frequencyband, about 107 Hz.

The required bandwidth is easy to estimate. Indeed, normal visual perception of a moving picturerequires that frames be projected at a frequency ofabout 50 Hz. If the electron beam scanning the cathoderay tube (CRT) screen covers it in 400 lines, thismeans that lines are "switched" with frequency of 50 xx 400 = 2.104 Hz. Finally, if each line is divided into

400 ""image points", the necessary frequency of electronbeam modulation becomes 400 x 2.104 = 8.106 Hz.

Obviously, the frequency of modulation must besubstantially lower than that of the carrier wave. Thehigher the latter, the wider is the frequency band thatcan be used for modulation. The development ofcoherent radiation sources in the optical frequencyrange makes it possible to increase the frequency bandfor modulation, ~v, approximately to 1012 Hz; thismeans that up to 109 telephone conversations or up to105 television programmes could be transmittedsimultaneously on a single laser beam. In other words,the "information capacity" of coherent radiation in theoptical range is tremendous.

\.

It should be mentioned that the high informationcapacity of the optical range can be fully utilized onlyif we solve the problem of modulation of opticalradiation at frequencies on the order of 1012 Hz (andeven higher). This is an extremely difficult problem.Modulation frequencies realizable when this book wasbeing prepared reached "only" 109 to 1010 Hz.

Laser technology is promising not only from thestandpoint of transmission of data, but also with respectto data reception and processing. Lasers are successfullyused today to measure distances and velocities withextremely high accuracy, to detect internal stresses anddefects in the framework of non-destructive structuralcontrol techniques, to detect weak underground shocks,to measure drift of continents, and so on. Lidars, thatis, optical locators, are used on an ever increasingscale. The methods developed in optical holography(Sections 6-11) are being applied more and more toprocess information by means of coherent opticalradiation (Sec. 11).

Problems in the near future. The fields to whichlasers make a significant contribution are growing inparallel with the progress in the domain of lasertechnology. In some fields lasers already have a highstanding, in other areas only first applications haveappeared, while in still other ones the possibilities oftheir application are only at the appraisal stage.

One of the most pressing and important problems,and one that is being solved now, is that of controllednuclear fusion. Very serious arguments point to thepossibility of successful solution of this problem ifsuperpowerful laser pulses are used. This approach isunder scrutiny in the USSR, USA, and othercountries.

5"


Another important problem is connected with theneed to increase the speed of operation and thememory capacity of computers. This is demanded bythe needs of scientific and industrial progress, and bythe necessity to manage industry and economy of thecountry as a whole.

It can be expected that we shall witnessa qualitative breakthrough in this direction very soon;it is anticipated that purely electronic techniques ofdata processing will be replaced by a combination ofelectronic and optical methods, that is, traditionalelectronics will be replaced by optoelectronics. Sucha transition is only possible after a massive penetrationof lasers into cybernetics and computers. The first stepsin this direction are already a reality.

The prospects promised by coherent optics arespectacular. In the chapters to follow the reader isinvited to make a better acquaintance with some of themost exciting and promising branches of this noveldomain of modern physics.

68Optical Holography

6. Formation of Optical ImagesLet us place an arbitrary object in front of

a screen and illuminate it. Light beams reflected byeach point of the object illuminate all points of thescreen (Fig. 19); the beams scattered by different pointsof the object are "tangled up". As a result, the screenis illuminated more or less uniformly. An image of theobject can only be obtained if the light beams aresomehow "disentangled" and the patterns of raysscattered by the object are "ordered".

Pinhole camera. The light rays are easily orderedby placing an opaque sheet with a pinhole in itbetween the screen and the object. This is the idea of

Fig. 19

Optical Holography 70 6. Formation of Optical Images 71

the pinhole camera, which consists of a dark box witha pinhole in one of the walls; the wall opposite to thepinhole serves as a screen.

Each point of the object sends only a narrow lightbeam through the hole, which produces the image ofthis point on the screen. Hence, an inverted image ofthe object is seen (Fig. 20).

For an image to be sharply defined, the pinholemust be very small, of a diameter from 0.5 to 0.1 mm.Obviously, the smaller the pinhole the lower theirradiance of the image is. This is the reason for thedark box, and this is why the object must be verybrightly illuminated.

Pinhole cameras utilize a very insignificant fractionof the light flux reflected by the object, and thereforewere never widely used as image projectors.

Lens systems. Photography. The most widespreadmethod of obtaining optical images is based on lenssystems.

Figure 21 shows how a lens placed between anobject and a screen collects, on a single point of thescreen, all the light rays that are scattered by this

Fig. 20

point of the object onto the whole surface of the lens.The image is formed by a much larger light flux thanin the case of the pinhole camera. One of the factorsdetermining this light flux is the lens diameter.

The photographic method is based on lenses. Theimage of the photographed object is recorded on photographic plates or films. Absorption of light in thelight-sensitive layer results in chemical reactions and information of a latent image, transformed to a visibleone by the process of developing. The image is thenfixed and may be kept for indefinitely long periods.

A screen with a light-sensitive layer that, after anappropriate treatment, retains the image of an objectfor long periods of time is referred to as a photodetector.

Holography. This is a fundamentally new methodof recording optical images; the progress in holographyduring the last decade was tremendous. The origin ofthe term will be clarified later.

Let us analyze Fig. 22. An object is placed in frontof a photodetector and illuminated with a coherentlight wave emitted by a laser. The light wave reflected

Fig. 21

Optical Holography 72 6. Formation of Optical Images 73

by the object (the object wave) falls on the screen. Inaddition, another wave (also coherent, emitted by thesame laser) is also directed at the screen. The auxiliarylight wave shown in the figure is a plane wave incidenton the screen at an angle \1..

The figure shows that the light rays scattered bythe object are still "tangled": each point of the objectsends rays to all points of the photodetector (screen). Itis essential that the experimenter makes no attempt to"disentangle" the light beams by additional devices(such as a pinhole screen or a lens) and does not tryto create "order" in the pattern of scattered rays. Nowonder that the photodetector with the imagedeveloped looks like an inadvertently exposed film;even the sharpest eye fails to detect the slightest traceof the object's image.

This seemingly "spoilt negative" possesses, however,one spectacular property: it has memorized the encoded(enciphered) image of the object. The decoding of thiscipher, that is, the visualization of the image, is straight-

Fig. 22

forward. The image is reconstructed by illuminatingthe "negative" with a coherent wave identical to thatused as the auxiliary one. In the case of question, thewave must be plane and be incident on the screen atan angle \1..

Once such a wave is directed at the "spoiltnegative", its surface miraculously becomes transparentand the observer finds himself looking at the object'simage as if through a "window" (its area is that of thephotodetector).

Recording and reconstruction of a hologram. This"negative" is called the hologram of the object. Itrecords the interference pattern formed by thesuperposition of two coherent light waves: onescattered by the object (the object, or signal wave) andthe other, by auxiliary (the reference wave). Holographyis in principle an interference-based technique; it islogical therefore that light waves with a high degree ofcoherence are required for its realization.

The interference pattern recorded by a hologramhas a very detailed fine structure. Two interferencefringes on a hologram may be spaced by only 0.001 mm,so that this fine structure is not resolvable by thenaked eye. Obviously, the spatial resolution of thephotodetector material must be sufficiently high forrecording such detailed patterns. Spatial resolution ismeasured by the maximum number of parallel lines perunit length (uS"ually one millimetre) that can be distinguished on a material [Fig. 23 illustrates two cases ofdifferent spatial resolution: in case (a) it is twice ashigh as in case (b)]. The recording of hologramsrequires that the spatial resolution of materials be notworse than 1000 lines per millimetre.

Optical Holography 74 7. Holography: Elementary Examples 75

(7.1)

Information about the object is recorded on ahologram as an interference pattern. When a hologram isilluminated with a coherent light wave identical to thereference wave (the readout wave), this wave is diffract-ed by the system of interference fringes that is fixed onthe hologram and acts as a sort of a diffraction grating. This diffraction reconstructs (makes observable) theobject's image recorded on the hologram.

Note that the condition of identity of the readoutand reference wavefronts does not include the identityof wavelengths. Recording and reconstruction of a hologram can be carried out at different wavelengths. Theeffect will be a change in the scale of the reconstructedimage.

We see that holography, in contrast to earliermethods of image formation, is a two-stage process. In

Fig. 23

the first stage a holographic "photo" is taken (the hologram is recorded); in the second stage the image ofthe object is recol1structed from the hologram (thereadout of the image). The formation (recording) of thehologram is based on the interference of light waves,while the reconstruction is based on the diffraction ofthese waves.

In conclusion, we want to underline an importantfeature of the holographic method of image formation.It consists in the fact that the "disentangling" of lightrays necessary for the reconstruction of the image (a) iscarried out only after the image has been recorded onthe photodetector, and (b) is in a certain senseautomatic, since this "disentangling" is realized by thereadout light wave.

7. Holography:Elementary ExamplesExample 1. The process of hologram recording is

illustrated in Fig. 24a. Both light waves in the figureare monochromatic plane waves with identical frequencies, the reference wave propagating normal to thehologram plane and the object wave at an angle a.to it.

Interference of two monochromatic plane wavespropagating at an angle a. was discussed in Section 1.By using equation (1.7), we conclude that a hologramrecords an interference pattern formed by a system ofequidistant parallel flinges spaced by

A.d=-.-

sm a.

Optical Holography 767. Holography.' Elementary Examples 77

. Phot~proce.ssing of the hologram (developing, fix-mg, washmg) yields a plate with alternating transparentand opaque parallel fringes. Such a plate can be regarded as a di(fi-action grating with period d calculated byEq. (7.1). This period is approximately 0.001 mm; theslits on such a grating can only be seen if magnified(Fig. 25).

This hologram corresponds to an object reflectingthe plane wave. A plane mirror of sufficient dimensionscould serve as such an object. It can be said, therefore,that in this example we are dealing with the hologramof a plane mirror.

The process of hologram reconstruction is clarifiedin Fig. 24b. Diffraction of a wave on a diffraction grating is described by the formula

where k = 0, 1, 2, ... ; d is the grating space; and <Pkdenotes the angles between the normal to the gratingplane and the directions at the so-called principaldiffraction maximums (directions in which diffractedwaves propagate). The value k = 0 corresponds to theundiffracted wave; k = 1 corresponds to two maindiffracted waves. We neglect the remaining diffractedwaves (k ~ 2) because of their low intensity. Accordingto Eq. (7.2), the following is true for main diffractedmaximums:

d sin <PI = Ie (7.3)

The angle of diffraction is equal to the angle ofinterference. Let us refer to angle <PI at the firstprincipal diffraction maximum as the angle of

d sin <Pk = kA

Fig. 24

referencelight wave

(7.2)

readoutlight wave

unditTracted partof readout wave

------t·IIE-~~----:..:.-=..:::;

..light wavesproduced bydiffraction ofreadout wave

(b)

Fig. 25

Optical Holography 78 7. Holography.' Elementary Examples 79

Obviously, in the general case the angle ofinterference varies over the area of the hologram. Theangle of diffraction varies correspondingly. The essentialpoint is that the equality of the two angles holds inany point of the hologram.

Note that, strictly speaking, instead of a single

diffraction, and to angle \'I. between the directions oftwo interfering waves as the angle of interference. Bycomparing Eqs. (7.1) and (7.3) we obtain that thediffraction angle is equal to that of interference:

<PI = \'I. (7.4)

point, one must consider at least a small area on thehologram. Whatever the complexity of the interferencepattern, each small segment of the pattern can be regarded as a diffraction grating with a definite gratingspace. Figure 26 shows a fraction of an interferencepattern recorded on a hologram. It illustrates that thegrating space close to point A is dA , while that in thevicinity of B is dB.

Example 2 (point source hologram). The process ofhologram recording is shown in Fig. 27a. The hologramreconstruction is shown in Fig. 27b. figure 27billustrates the rule stated above: the angles of

Fig. 26

Fig. 27

poin! sourceobject

(a)

hologram

II

cf/

virtualimage

(b)

main diffractedwaves

Optical Holography

diffraction and interference are equal at any point ofthe hologram. Two points, A and B, are selected in thefigure; hence, <PIA = CXA, and <PtB = cx.!.

Figure 27 demonstrates that two images arereconstructed from a hologram: one virtual and theother real. The observer finds the virtual image in thesame place with respect to the hologram as wasoccupied by the object in the process of hologramrecording. The real image is, in this case, symmetricalto the virtual one.

The system of interference fringes in a hologram ofa point object is more complicated than in the firstexample: the fringes are not straight and the spacingsbetween them vary as a function of the angle ofincidence of the object light beam. Figure 28 illustrateswhat the pattern is like.

A hologram of a point object is the Fresnel zoneplate. So-called Fresnel zone plates are a very familiar

Fig. 28

80

,

7. Holography: Elementary Examples

object in optics. In the simplest case a Fresnel p~ate isa system of alternating transparent and opaque nngswhose width diminishes, according to a specIal formula,as their radius increases (Fig. 29). The Fresnel zoneplate is an interference device that in principle is .similar to the diffraction grating. Let a monochromatIcplane light wave be incident on such a plate.Interference of different waves diffracted by the annularslits of the plate focuses the light at a· specific pointbehind the plate. Hence, a zone plate is in fact

Fig. 29

81

Optical Holography 82 7. Holography: Elementary Examples 83

a planar (two-dimensional) equivalent of a converginglens. This property of Fresnel plates is very wellknown, but wide use of these plates in optics wasimpossible because of difficulties encountered inmanufacturing them (the number of rings necessary tohave a good "lens" is quite high).

Holography pointed to a very attractive method ofsufficiently simple production of high-quality Fresnelzone plates. It was found that the Fresnel zone plate isa hologram of a point source (point object).

The hologram illustrated in Fig. 28 is not reallyidentical to the Fresnel zone plate, which can beobtained if the conditions of recording are modified:the point object must be placed as shown in Fig. 30..In this case the hologram will take the form shown mFig. 29.

Let us denote by L the distance from the objectto the hologram, and by r the distance from theobservation point to the centre of the hologram. IfL » r, then Eq. (7.1) yields a formula for the width ofinterference fringes:

d = -J::- ~ _A_= AL (7.5)sm a. tan a. r

Fig. 30

The width of the rings is thus inversely proportional totheir radius.

The Fresnel zone plate as a planar equivalent ofa lens. We have mentioned above that the Fresnel zoneplate is a planar equivalent of a lens. This is easilyproved if the zone plate is treated as a hologram ofa point object.

Let us illuminate the hologram of Fig. 29 bya readout wave. Two diffracted light waves are produced,one giving a virtual image of the point object andthe other forming a real image (Fig. 31). The waveforming the real image is shown in the figure by dashhatching. It is apparent that the diagram in Fig. 31corresponds to a situation in which a plane light wave,incident on a converging lens, is focussed into a point.A hologram of the type of Fresnel zone plate possesses,therefore, the focussing properties of a lens.

The above example allows us to draw animportant conclusion: in addition to serving asa "store" of the encoded image of the object, a holo-

Fig. 31

././

./,-<' ..... .....

..........

6'

, J

Optical Holography 847. Holography: Elementary Examples 85

gram may be used as a "transformer" of light waves.A hologram reconstructs not just an image of anobject, but a light wave with an appropriate wavefront.In other words, holograms may be used as planarequivalents of various optical elements (lenses amongthem) and their combinations.

Since a hologram is merely a pattern on a plane(even if a very complicated one in the general case), itcan in principle be artificially reproduced. Not only theFresnel zone-plate pattern but much more complicatedinterference patterns could be produced, giving usplanar equivalents of the most varied optical elements.

..1 hol0m:f!!!! reconstructs the objecl,"Y!Jvefront.Jl'ldnot the object's image. Let us go back to the examples

-snownTn""-Flgs:24-"--ii-na 27. Among the waves producedbehind the hologram in both cases of imagereconstruction there is always a light wave identical tothe object wave.

We have mentioned above that the hologram readout phase reconstructs not only the image but theobject wave as welL This fact deserves special attention.

Take a light wave propagating away from theobject (Fig. 27a). An observer sees the object if thiswave is recorded by his eye. Let us record the objecton a hologram, and then remove the object.Illumination of this hologram by a readout light wavegenerates several waves, among them the same objectwave. And, although the object was removed, the lightwave scattered by this object is reconstructed.Consequently, an observer receives the image of theobject as that illuminating the object itself.

It would be more correct, therefore, to say thata hologram stores not just an encoded image of the

object but the object light wave. It is said sometimesthat recording of a hologram "freezes" the object wave,while the reconstruction "unfreezes" it.

Transition from a point object to a three-dimensionalone. It is evidently difficult to discuss the differencebetween a point object and its point image. Threedimensional (actual) objects have to be considered.

It is fairly obvious that the fact itself of hologramreconstruction (reconstruction of the object light wave)does not depend upon whether an object is threedimensional or not. Therefore we conclude that if it isthree-dimensional, the observer looking at thereconstructed image does not see a two-dimensionalpicture (as in photography) but something threedimensional and realistic; in other words, he seessomething looking very much like the object did in theprocess of the hologram recording. If the observer tiltshis head, he notices other objects behind the first one,or new details that were not noticeable before. Thismeans that the observer receives a quite realistic threedimensional picture.

It is normal to say that a hologram reconstructsa three-dimensional image of the object. This is undoubtedly true. However, this statement is somewhatincomplete and "underestimates" the possibilities of holography. A hologram reconstructs the real object wave,which is more than an image, even a three-dimensionalimage. A real light wave is something one can operatewith (for instance, it can be made to interfere withanother wave) while an image can only be recorded.

Two arrangements of hologram reconstruction. Thetransition from a point object to a 3-D object can be

Optical Holography 86 8. Holographic Laboratory 87

realized, of course, by establishing a correspondencebetween the 3-D object and a set of, for example, threepoints of the image. This evidently demonstrates thatan object reconstructed by a hologram is threedimensional. We find, in addition, that only the virtualimage has all the features of the real object; the realimage looks as if it is "turned inside out", that is,points farther from the observer are imaged closer tohim. Such images are called pseudoscopic.

If the reconstruction geometry is changed so thatthe readout wave is directed opposite to the referencewave, the real image is an exact copy of the objectwhile the virtual image becomes pseudoscopic.

)

Two geometric arrangements are therefore possiblefor reconstruction of a hologram: (1) the readout waveand reference wave are identical (the arrangement wasused in all the examples discussed so far); (2) the readout wave is inverted with respect to the reference wave.In the first arrangement one operates with the virtualimage, and in the second - with the real one. Figure32 shows: a - geometry of hologram recording;b - normal (ordinary) reconstruction geometry;c-- inverted reconstruction geometry.

(b)

Fig. 32

8. Holographic Laboratory

Typical arrangement for hologram recording. Differentgeometric arrangements can be used for record-ing a hologram. One of these arrangements (withbilateral illumination of the object) is shown in Fig. 33.Laser is a source of all the light waves used in thisarrangement; semitransparent mirrors split the laserbeam twice thus forming two waves illuminating theobject and the reference wave.

In essence, holography is a lensless method ofimage formation, although lenses are used in thearnlngement shown. They play an auxiliary role (serveas light beam expanders).

---

(c)

Laser. The laser is an irreplaceable instrument ina holographic laboratory. Indeed, sufficiently highcoherence is the basic requirement for the light source.The required degree of coherence of the laser radiationis determined by the type of the object and the choiceof the geometric arrangement of hologram recording.

where 't is the coherence time, and L is the maximumpath length difference for two light beams propagatingin the chosen arrangement from the laser to the hologram. In order to estimate L, one must take intoaccount all possible beam trajectories in the chosenarrangement and to trace them, including all reflectionsand refractions on the path of the beam from the laserto the hologram. Condition (8.1) has the same meaningas condition (3.1): interference of two light beamsrequires in both cases that the path difference for the

Optical Holography 89

two beams reaching the screen be smaller than thewave train length.

Another requirement for the laser emission is thatits intensity be sufficiently high. Together with thesensitivity of the light-sensitive material used to recordthe hologram, this characteristic determines the requiredtime of exposure.

Holography often uses gas lasers (as a rule heliumneon lasers) operating in the continuous wave (CW)mode. Although their coherence is high, the emittingpower is comparatively low (see Sec. 5). Consequently,the time of exposure has to be large, so that movingobjects cannot be holographed.

Holography also employs pulsed solid-state lasers.Their emission is less coherent than that of gas lasers,but they have high peak power, which makes itpossible to cut down the exposure to the light pulselength (for instance, to 10 - 3 s). Pulsed illumination isused to holograph moving objects and to record thedevelopment of a process in time.

It should be mentioned in conclusion that therequirements for the coherence of laser emission can berelaxed by reducing the depth of field recorded on thehologram and by better compensation of the pathlength differences of the light beams involved.

8. Holographic Laboratory

Measurement bench. The laser, object, hologram,and all necessary optical elements used in the system(reflecting mirrors, splitter plates, lenses, prisms, etc.) arefixed in predetermined positions on a measurementbench, which normally consists of a massive steel plateof sufficient area (for example, 2 x 2 m).

If a hologram is recorded by a gas laser, themeasurement bench must satisfy very severe

IiI

88

(8.1)

hologram

semitransparentmirror

Fig. 33

completelyreflecting mirror

The following condition must hold:

L« 'tc

Optical HoloRraphy 90 8. Holographic Laboratory 91

requirements with respect to possible displacements ofthe elements during the exposure: the positions of theelements must be fixed to within 1/4 of a wavelength(in other words, with the accuracy of 0.1 /lm). Violationof this condition results in blurring of the interferencepattern on the hologram.

It might seem that once the elements are fixed,nothing can cause their displacement. This is wrong.The elements shift because of vibrations of the walls,floor, and building foundation. Any city is rich incauses for these vibrations: traffic, large industry, andso on. As a rule, such vibrations are not felt;nevertheless, they are a reality, and in spite of beingseemingly negligible, are quite capable of breaking thesevere requirement of immovability of elements on theholographic bench.

In order to eliminate vibrations, optical elementsare mounted on sufficiently heavy foundations withshock absorbers (for instance, the steel plate may beplaced on air-filled inner tubes of car wheels).

Holographic recording materials. It was mentionedabove that materials for hologram recording mustresolve not less than 1000 lines per millimetre, whichmeans a rather high spatial resolution.

The reader will recall that photoemulsion consistsof minute grains of silver bromide suspended ina transparent gelatin layer. Consequently, a developedimage consists of separate "spots" (is built of specific"bricks"). Features of the image smaller than these"spots" are therefore indistinguishable. It is thus clearthat increasing the resolution calls for photographicmaterials with finer grain structure. It should always bekept in mind that diminishing the grains invariably

entails impaired sensitivity (indeed, a photon absorbedby emulsion affects a grain as a whole, so that thelarger the grains are the smaller the number ofphotons required to form an image). The developmentof photographic materials for holography with highresolution and at the same time high sensitivity is notan easy technical problem.

Photographic films now in use in holography havea resolution of 1500 to 2000 mm - 1 and sensitivity onthe order of 10 - 2 J/cm 2

• There is also an organicphotographic material, so-called photoresist, enablingone to achieve a resolution of 3000 mm - 1 ata sensitivity of 10 - 2 J/cm 2

. Experimental photographicplates were reported with especially fine-grain structure,realizing a resolution of 5000 mm -1.

Can a hologram be erased? One shortcomingcommon to all photographic materials is that they arenot meant for reuse. A hologram recorded on a photographic plate cannot be erased and a new hologramcannot be recorded in its place. As a result, photographic materials are classified as irreversiblerecording mediums.

But are there any reversible recording mediumsthat allow erasing and repeated recording of holo-grams? Yes, such mediums exist. Let us consider someexamples.

Holograms on magnetic tape. Let a laser beam fallon some point of a magnetized ferromagnetic film andheat the illuminated spot to a temperature above theCurie point. The film in this spot undergoesa transition from a ferromagnetic to paramagnetic stateand loses its magnetization. Once the laser illuminationis removed, the spot in question cools down, returns to

Optical Holography92

In order to erase the image recorded on a film, itis sufficient to switch on an external magnetic field andmagnetize the whole film in a chosen direction, afterwhich the film is again ready to memorize a newhologram.

The time required to obtain a hologram ona photographic material is determined by the time ofthe photographic treatment (developing and fixation ofthe image). In the case of a magnetic film, however,this is the period of time required to heat the film andswitch its direction of magnetization. This time can bevery short, less than 10- 7 s.

Obviously, ferromagnetic films most suitable forhologram recording are those with a relatively lowCmie point. For example, a manganese-bismuth filmwith the Curie point of 180 uC is used (spatialresolution 1000 mm - 1, sensitivity to light 10 - 2 Jjcm 2 ).

Holograms on thermoplastic films. Thermoplastic isa name of specific transparent dielectrics that soften atcomparatively low temperatures (for instance, at 50 QC).A hologram can be recorded on the surface ofa thermoplastic as a relief pattern. Such a hologramremotely resembles a phonograph record.

In order to use thermoplastics for hologramrecording, a thermoplastic film is applied to a substrateconsisting of two layers, one of a semiconductor andthe other of a transparent conductor. The arrangementof the layers is shown in Fig. 35. A hologram isrecorded as follows.

First of all the surface of the thermoplastic film is.uniformly charged in darkness by means of glowdischarge; this creates a sort of a capacitor whose"plates" are the charged surface of the thermoplasticand the conducting layer (Fig. 36a).

fer.romagn~tism, ~nd is immediately magnetized byneIghbourIng regIons that were not irradiated. It is~ssen~ial that the direction of magnetization of theIllumll:lated ~rea is now opposite to that of the surroundIn~ re~lOns. The reason is clear from the followingmodel. sItuatIon: two rectangular bar magnets placed~longside on a slippery horizontal surface. This systemIS .unstable .while identical poles of the magnets are~dJacent (FIg.. 34a), and stable only if opposite poles areIn contact (FIg. 34b), because in the second con-figuration the magnetic field lines are cloSed.

We thus obtain that illumination of a ferro~agn~tic ~ilm results in flipping the magnetizationdIrectIon In places with sufficient irradiance. Thisprocess may be used to record a pattern, for instance,a pattern formed by interference fringes.

Fig. 34

(a)

(b)

I

8. Holographic Laboratory 93

Optical Holography94 8. Holographic Laboratory 95

Fig. 36

of thermoplastics. Spatial resolution in the latter can beas high as 1000 mm - 1.

Heating is sufficient to erase the recorded relief ona thermoplastic, after which a new hologram can berecorded.

Holograms on photochromic materials. Photochromicmaterials are glasses doped with special impurities ororganic polymers, capable of changing colouration ortransparency as a result of irradiation with light ina specific frequency range (usually in the UV or shortwavelength range of the visual part of the spectrum).This is related to the transfer of electrons between

+++++++++++++++++

~

(a)

(b)

Light

11111Light

111

The next stage is the illumination of the wholesand~ich with superposed reference and object waves.By vIrtue of photoconductivity, the semiconductor'sresistivity increases sharply where interference maximaare formed; consequently, the distance betweencapacitor "plate~" in these places "contracts" (Fig. 36b).Note. that the fIeld strength inside the capacitorremams unchanged (it depends only on the density of~he surface charge). At the same time, a decrease of themte~pl.a~e spacing at constant field strength results ina dImInIshed potential difference. Hence, the potentialof the illuminated areas of the surface decreases.

The. surface ~f t~e thermoplastic is then rechargedso that Its potentIal IS restored to the initial value overthe wh~le su.rface. This concentrates additional chargeon the Illummated areas (Fig. 36c).

Finally, the system is heated; the thermoplasticso~tens, and coulomb forces between charges shape there.hef on the surface of the thermoplastic in accordancewIth the charge distribution. Cooling of the film freezesthis relief (Fig. 36d).

.Obviously, .hologram .formation on thermoplasticsrequIres more. tI.me .than m the case of magnetic films.!he charact:r~stIc tIme of recording on a magnetic filmIS about 10 s, and reaches only 10 ~ 1 S in the case

Fig. 35

Jaye r-..z1l88lllll1i:8lll/Zll~layer of semiconductor I

r=.=.1::==±'='~

transparent glass substrate

+++ ++++++++++++++++++++++

~ (c)

(d)

"Holography" means "complete recording". A lightwave can be considered as a carrier of information thatis "recorded" in terms of wave parameters. It isconvenient to distinguish between the information"recorded" (contained) in the wave amplitude, and theinformation contained in its phase (its wavefront shape).Correspondingly, amplitude and phase information aredistinguished.

Let us discuss the retrieval of information froma light wave, and the recording of the informationcarried by this wave in a selected medium.

Let us emphasize first of all that any photodetector records the intensity of an incident lightwave, that is, its response is proportional to the squaredamplitude of the wave. This means that a photo-detector retrieves only amplitude information andphase information is thereby lost.

I[ we want to retrieve and record by a photodetector not only amplitude but phase information aswell, we have to show "cunning", namely we mustmake the light wave carrying information interfere withan auxiliary light wave. Our "ingeniousness" lies in thefact that the amplitude of the resultant light wave dependsalso on the relative phase of the original andauxiliary waves. Therefore, the photodetector recordingthe intensity of the resultant wave at the same timerecords not only the amplitude but also the phaseinformation of the analyzed wave.

The method of holography is based on just this"stratagem". By using the interference of waves, we canretrieve from the object wave and record on a photo-

Optikal Holography

impurity atoms which results from absorption ofradiation.

In order to restore the initial state in the materialit has to be either heated or irradiated with light of 'lower frequency. Normally, photochromic materials aredarkened by visual light and "bleached" by infraredlight.

Photochromic materials are attractive because oftheir high spatial resolution, up to 3000 mm - 1.

9. Advantagesand Possibilities of Holography

A photograph and a hologram. At first glance,a photograph is preferable to a hologram. Indeed,a photograph shows everything "clearly", while nothingcan be seen on a hologram. The latter has to beill~minated with laser light in order to yield an image.ThIS, of course, means certain amount of trouble. Itseems probable that those uncoded photographic~mages will re~ain preferable to encoded holographicImages for ordmary purposes and in daily life. Thesituatio.n is quite different in science and technology,where Inconveniences caused by the encoded characterof image.s .~n. holograms are more than outweighed bythe pOSSIbIlItIes opened to scientists and engineers bythe new technique. To be precise, it is this "encoded"character of the images that underlies the richpotentialities of holography.

A hologram enables us to reconstruct a real lightwa~e and therefore makes it possible to manipulateoptIcal fields (something absolutely impossible in photography). In addition, an image reconstructed by a holo-

i ,\.,',,~

9. Advantages and Possibilities of Holography

gram differs from a photograph in its threedimensionality, realistic and true-to-life nature.

97

7-190

Optical Holography 98 9. Advantages and Possibilities of Holography 99

detector practically all the information about theobject, including both amplitude and phase components.It is not accidental that the term "holography",translated from its Greek roots, means "completerecording".

Let us emphasize that if a hologram did notmemorize nearly all the information carried by theoriginal object wave, the reconstruction of the realobject wave from the hologram would be impossible inprinciple. The process of reconstructing the object wavefrom a hologram indeed resembles a sort of "unfreezing" of the light wave.

Reliability of holographic storage of information.A group of people wishing to keep a momento ofa meeting had first a photograph and then a hologramof the group taken. The two records of the event werestacked together and put away for long-term storage.A small fire broke out one day. Both the photographand the hologram were saved, but both lost about onefourth of their area. Some time later one of themembers of the group had to be identified bydocuments. When the group photograph was extracted,the face in question was missing; presumably, it hadbeen on the burnt portion. The hologram was resortedto. It was illuminated by a laser beam, and gave theimage of the whole group, including the person to beidentified. The hologram proved more reliable than thephotograph.

This scenario was invented in order to illustratethe following: destruction of a portion of a photographic image results in an irrepairable loss of informationcorresponding to a part (or parts) of the object; on thecontrary, destruction of a part of the corresponding

hologram affects the reconstructed image in anabsolutely different manner. Indeed, Fig. 27ademonstrates that the information about a point objectis recorded over the whole area of the hologram.Obviously, this is true of any point of a real object:each such point is recorded on the whole hologramand not in one of its points (the situation realized inphotography).

Consequently, destruction of a part of a hologramdoes not erase a specific portion of the image in thereconstruction. Practice shows that up to 9/10 of thearea of conventional hologram can be removed withoutappreciable loss; the only result is a diminishedresolution (sharpness) over the whole image. Thismeans loss of fine details in the image as a whole.

Actually, this is not surprising if one recalls thata hologram reconstructs not just the object's image butthe object wave. The area of the illuminated hologramdetermines the area of the reconstructed wavefront.Contraction of the illuminated area is equivalent tocontraction of the wavefront (as if we were regardinga remote scene through a gradually contracting window).A comparatively small loss in the wavefront areadoes not affect the quality of the reconstructed image;greater reduction of the area worsens the resolution.

A hologram is thus a very reliable method of datastorage. It is not impossible that in the future the mostvaluable information will be stored in holographicform.

Information is recorded on a hologram as aninterference pattern, that is, in an encoded form, and itcan be decoded only by a coherent light wave withexactly the same wavefront as in the reference wave.Consequently, the shape of the wavefront rep~esents the

7'

Optical Holography 100 9. Advantages and Possibilities of Holography 101

key without which the code cannot be broken and thehologram read out. Even the most ingenious decipherment specialist will fail if the wavefront shape IS

unknown (and this shape may happen to be veryunusual).

Information capacity of the hologram. Is it possibleto record several photographic images on the sameplate? In principle, yes; but who wants a photo withseveral superposed images? Superposition of severalprinted pages would be just as pointless. This reflectsthe restrictions inherent in the photographic method ofdata storage.

Holography eliminates these restrictions thus forc-ing us to correct our habitual concepts. The same hologram may contain a number of consecutively recordedscenes (interference patterns) that can be reconstructedindependently. Indeed, reconstruction of an imagerecorded on the hologram requires employing a readoutwave with a wavefront structure identical to that of thereference wave and incident on the hologram in exactlythe same manner.

Assume, for the sake of simplicity, that thereference wave is plane. Let us record on the hologramdifferent scenes each time changing the angle at whichthe reference wave is incident on the holographic plate(this can be done, for example, by rotating tht> hologram in the reference beam). Evidently, reconstructionof a specific scene only requires that the hologram beproperly oriented with respect to a plane readout wave.The admissible number of recorded images on the samehologram rises considerably if we take into account thepossibility of changing not only the hologramorientation with respect to the reference wave but also

the possibility of changing the wavefront shape of thiswave.

Estimates indicate that a single hologram with anarea of about 100 cm2 may contain (under thecondition of unhindered reconstruction) at least onevolume of the Greater Soviet Encyclopaedia (orEncyclopaedia Britannica, for that matter)! This pointsto the extremely high information capacity of holograms.

Taken together, this high information capacity ofholograms and high reliability of hologram storageenable us to forecast that in the future bookdepositories may be replaced by hologram depositories.Instead of bulky volumes, which in addition are easilyharmed, we might use miniature cassettes with holograms.

A moment stopped dead. A photo shows a diversuspended in the air several metres above the water III

an unbelievable pose. Another photograph givesa magnified view of space filled with moving dustparticles; the specks are frozen in the positions inwhich they were caught at the moment of exposure. Itis said that photography is capable of "stopping time".

Let us assume, however, that we want to find outthe position of the diver's left hand. Alas, it is notvisible in the photo. We want to scrutinize in greaterdetail the dust specks in the background, but cannotbecause they are blurred. Obviously, there is nothing tobe done about it.

Now suppose that instead of having been photographed, the diver and the scene with moving dustparticles were recorded on holograms. Thereconstruction in continuous laser light produces a real

,.

Optical Holography

light wave identical to that scattered by the object atthe recording stage. And the visual perception showsthe diver who seems actually hanging in the air. Thesame is true of the dust-particle scene. Now we canlook at the diver from different viewpoints, can bringeither the nearest or the farthest dust specks into thefocus of a microscope, look at them from differentobservation points, and so on.

lt is thus clear that in contrast to a photograph,a hologram, "stopping time", enables us to extractmuch more complete information about the object ata given moment of time. A hologram records andcontinuously reconstructs the structure of optical fieldsthat existed only during the exposure. Consequently,holography provides a unique opportunity of multipurpose processing of optical information when theactual optical fields in question are already replaced bysubsequent fields (post-experiment data processing).

Holography and data processing. Optical holographyis in fact a little more than fifteen years old.Nevertheless, specialists in the field of data processingdevote much of their attention to holography. Let usconsider some of the possible applications of holography in this field.

Pattern recognition. Pattern recognition is animportant problem in cybernetics. How can werecognize a specified letter in a text? How can weselect a flawed element in a set of seemingly identicalparts? How can we recognize an anticipated signal inan ensemble of different signals at an instrument'sinput? These are specific examples from the realm ofpattern recognition.

102 9. Advantages and Possibilities of Holography

Holography is one of the promising methods ofpractical solution of this problem. As an example, letus consider how to use holography in order torecognize a letter in the text.

Let us choose "T" as this letter. In order to solvethe problem, we need to prepare a hologram of thesize of one letter of the text; the recording must bedone with the wave scattered by "T" serving asreference wave, and a wave from a bright source as theobject wave. A special device shifts the hologram alongthe lines of the text. Each time it faces a "T", a brightflash is observed, since the wave scattered by this "T"reconstructs the image of the bright source.

Associative retrieval of iriforrnation. Associativesearch is one of the principles inherent to humanmemory: we start by remembering a "detail", that is,some characteristic "feature" (associative identifier), andthis makes the whole picture "surface" in the memory.In other words, an associative search is thereconstruction of a whole from an individual feature.

Holography proves to be very suitable for thetechnical realization of the associative data search.

Assume now that a hologram was recorded without a reference wave, and only the object waveparticipated. Is it possible to reconstruct the image ofthe object?

Do not hurry with a negative answer. Indeed, wecan assume (and this is perfectly true) that theinterference pattern recorded in this case is formed asa result of superposition of light waves scattered bydifferent parts of the object. Let us consider the wavescattered by one element of the object (we will refer toit as the "characteristic feature") as the reference wave,and the ensemble of waves scattered by the remaining

103

Optical Holography

parts of the object as the object wave. For instance,wave in Fig. 37, reflected by feature of the object, canbe taken for the reference wave. Clearly, reconstructionof the object's image from the hologram is possible ifit is illuminated by the wave scattered by the"characteristic feature". This means that it is sufficientto "present" only a fraction of the object (its identifier)in order to reconstruct the whole image. In otherwords, the whole is reconstructed from its element, oran individual identifier. This search is clearlyassociative: if the hologram contains a number ofimages, only the one that contains the associativeidentifier is reconstructed.

We ascertain, therefore, that, first, it is possible towork with holograms recorded without a specialreference wave, and second, that holography beautifullysuits associative data search, the development of theassociative memory (associative storage).

Encoding and decoding of information. Let us recallthat a hologram carries information in an encoded

104

)

I

9. Advantages and Possibilities vI' Holography

(ciphered) form. Hologram recording is the encoding,and reconstruction is the decoding of this information.It is then logical to apply holography specially toencoding and decoding of data.

In order to encipher the information carried bya coherent light wave, it is sufficient to send this wavethrough a special plate changing the amplitude or frontshape of the wave. Such plates are termed code masks.

Two encoding techniques are known. In the firstmethod the code mask is placed in the path of thereference wave, and in the second method-in the pathof the object wave. Decoding in the first method isrealized with the arrangement shown in Fig. 32b (thesame mask used for encoding is placed in the readoutbeam, and in the same position). Decoding by thesecond method is shown in Fig. 32c. We recall thatthis arrangement reconstructs an undistorted real image.The essence of the second method is clarified inFig. 38. Coding is shown in Fig. 38a, and decoding inFig. 38b.

Fig. 38

105

Fig. 37

real image

(b)

Optical HoloRraphy 106 10. HoloRraphic Interferometry 107

A general notion of volume holography. Until nowwe assumed the photodetector to be two-dimensional.As long as the thickness of the light-sensitive layer iscomparable with the spacings between interferencefringes, this is true. If, however, the layer thickness ismuch greater, the photodetector reveals specific featuresof three-dimensional mediums. As a result, holographyis classified into ordinary, or two-dimensional holo-graphy, and three-dimensional, or volume holography. Theidea of volume holography was suggested by the Sovietscientist Denisyuk.

The pattern fixed by a light-sensitive film in theregion of superposition of the object and referencebeams is one of interference fringes, while a lightsensitive volume fixes a system of interference surfaces.In the first case the result is a plane hologram, and inthe second a volume hologram.

A system of interference surfaces of a volume hologram can reflect light waves, and this is indeedobserved at the hologram reconstruction stage. Thewaves reflected by these surfaces interfere constructivelyonly if their phases are in step. This means thata volume hologram manifests selectivity with respect tothe readout wavelength: obviously, the synchronizationof phases is realized only for the wavelength used torecord the hologram. It becomes possible, therefore, tocarry out hologram reconstruction in white light (sunlinght or light of an ordinary incandescent lamp), withthe hologram "selecting" out of the continuousspectrum the very wavelength that can reconstruct therecorded image.

If several fixed wavelengths (emitted by severallasers) are used to record the hologram, reconstructionof this hologram in white light singles out the same

wavelengths. This is a method of obtaining colourimages.

Holography around us. It has been mentionedabove that it is hardly likely that holography willreplace photographic methods in everyday situations; itsmaximum effect is expected in the scientific andtechnical domains. This does not mean, nevertheless,that people whose occupations are far from eitherscience or technology will not feel the impact of holographic "miracles".

Even today holographic images with nearly a 3600

view are used for impressive advertizing. It can bepredicted that such images will be widely applied inthe theatre and circus. It was experimentallydemonstrated that holographic cinema is a feasibleproposition. Time may come in the near future whena spectator in a movie theatre watches "live" threedimensional holographic images: absolutely "realistic"people (as in today's theatre) amidst absolutely "real"scenery (more real than that in the theatre). It is notunlikely that our generation will see the appearance ofholographic television.

In other words, holography may be expected toCome into our daily life, our chores, and our leisure,and in the not very distant future at that.

10. Holographic Interferometry

The principle of holographic interferometry. As welearned earlier, holography is capable of "stoppingtime" at a chosen moment: the light wave that wasreflected by the object at the moment of recording canbe fixed and then reconstructed. We can go even fur-

Optical Holography 108 10. Holographic Inter(erometrr 109

ther than that: not just "stop", but "juxtapose" twodistinct moments of time. This requires recording theobject at two time moments on the same hologram.When this hologram is reconstructed, two light wavesare produced simultaneously: one scattered by theobject at moment one, and one scattered at momenttwo. Both reconstructed waves are absolutely real, andtherefore can interfere. Observation of the interferenceof these two light waves is the essence of holographicinterferometry.

Note that holographic interferometry actually dealswith several interference patterns. When an image isrecorded twice and then reconstructed, threeinterference patterns are produced. The first pattern isformed by the hologram recording at moment 1; thesecond pattern comes from the repeated recording onthe same hologram (the two patterns superpose on thefilm and are then fixed); finally, the third pattern isa result of interference of waves reconstructed from thefirst and second interference patterns. In contrast to thefirst two, this last pattern is usually clearly visible tothe eye. It is observed as a system of interferencefringes over the object's image reconstructed from thehologram. This reconstructed image is called the holographic interferogram. An example is schematized inFig. 39 (a system of interference fringes covers an Imageof a metal plate).

The following procedure should be followed toproduce such holograms. The object (plate in this case)is fixed on the measurement bench and recorded ona hologram. Note that the moment for this recordingmay be arbitrary since the plate is immovable anyway.Then the plate is strained by applying some mechanicalstress (interesting results are obtained if the plate is

simply pressed to the bench surface in some of itspoints). After this the strained plate is recorded on thesame hologram. Reconstruction produces two objectwaves, one representing the unstrained plate andanother the loaded plate. By virtue of interference ofthese waves the resultant image contains interferencefringes (something like the pattern in Fig. 39).Obviously, the extent and character of deformation ofthe plate over its surface can be deduced from thewidth and distribution of the interference fringes.

Different types of holographic interferometry.A process fundamental for the method of holographicinterferometry is the interference of two light waves ofwhich at least one is reconstructed from a hologram(sometimes the two waves are said to be compared).

Fig, 39

Optical Holography 110 10. Holographic Interferometry 111

applies to the hologram, so that it must be eitherprocessed on the spot or its material must be suchthat special processing is unnecessary.

The interferogram obtained with this techniqueenables us to estimate the changes on the surface ofthe object that have occurred by the moment of

Two techniques are possible here. In the first one, bothwaves are reconstructed from a hologram (this versionwas discussed above). In the second version, only oneof the waves is reconstructed from a hologram whilethe other is scattered directly by the object. Let usanalyze these techniques in more detail.

First version. Both interfering (compared) waves arereconstructed from a hologram on which the objectwas recorded twice (double-exposure interferometry). Thepattern of fringes on the object's image represents thechanges that occurred on the surface during theinterval between two exposures. For this reason thetechnique is also referred to as time-lapse (orlapsed-time) interferometry.

It is extremely important that the position of theobject during the second exposure be absolutelyidentical to that during the first exposure. This enablesthe two images recorded from the hologram to beprecisely "inserted" one into the other.

The double-exposure technique is illustrated inFig. 40. Figure 40a shows the hologram recording stage.The object's deformation is grossly exaggerated for thesake of clarity. The process of reconstruction is shownin Fig.40b.

Second version. One of the interfering (compared)light waves is reconstructed from a hologram that wasexposed only once. The second wave is scattered byth~ object itself, which is present when the hologram isbemg reconstructed. In order to achieve interferencethe object must be precisely "inserted" into the virt~alimage reconstructed by the hologram. This requires~hat the object remain fixed on the bench during themterval between the hologram recording and theobservation of the interferogram. The same requirement

Fig. 40'

object during1st and 2nd hologram

e;~rreh~--1\ 1\1\11 -\1 1/

X~J ,~v--.

(b)

Fig. 41


observation. Furthermore, these changes can be followedas functions of time: the interference fringes may bemodified in the process of observation. This techniqueis called therefore real-time interferometry (or livefringeinterferometry).

The arrangement for real-time interferometry isillustrated in Fig. 41. The process of recording is shownin Fig. 41a; the reconstruction is illustrated in Fig. 41b,that is, observation of "live" fringes.

A comparison of Figs. 40 and 41 helps in clarifyingthe difference between the double-exposure andreal-time methods of holographic interferometry.

Practical applications of holographic interferometry.The progress of technology demands ever increasingprecision in manufacturing various products. Thisprecision has to be verified. When this problem istackled, any dismantling (in other words, destruction) ofa unit is often out of the question, and even handlingof the unit by "feelers", gauges, and so on is absolutelyforbidden. Consequently, the problem has to be solvedby non-destructive, contactless means of inspection.Holographic interferometry offers one of such means.

Assume that an object is placed in workingconditions and undergoes various mechanical loads. Weneed to know what the distribution of internal stress inthe object is, what regions are stress-concentrators andare therefore fraught with danger of failure. Holographic interferometry enables us to determine the degreeand nature of deformation of the observed surface, thatis, to extract the data necessary for calculating theinternal stress distribution.

Obviously, internal stress induced in the object bychanges in temperature can be analyzed in the same

manner. For instance, this is important for inspectionof welded seams of metals with different coefficients ofthermal expansion.

An object may have hidden internal defects (cracks,cavities, unwelded internal joints, etc.). In the case ofmetal objects, X-ray transmission methods are helpless.In some situations acoustic techniques fail also. Holographic interferometry may be applied to stressedobjects, so that the interferogram shows the distributionof deformations over the surface, from which it ispossible to establish the presence of defects, todetermine their types, and even to locate them.

As an example, let us consider an object consistingof two welded metal plates; unwelded areas arepossible on the internal surface of the seam. The objectis statically strained. An interferogram (Fig. 42) clearlyshows the regions where the regularity in the

Fig. 42

Optical HoloRraphy 114 10. Holographic Interferometry 115

Fig. 43

Fig. 44

Consider an example of holographic interferometryof a transparent object by means of the doubleexposure technique. Let this object be a gas-filledchamber that is traversed by a bullet. The firstexposure is taken before the shot, and the second during the passage of the bullet through the chamber (thelaser pulse must be automatically triggered by thebullet's penetration into the chamber). Two light wavesare reconstructed by the hologram; in one of themphase changes are determined by the gas in the unperturbed chamber, and in the other by the shock wavesgenerated by the bullet. Note that shock waves changethe gas density, and therefore the optical pathlength.Interference of light waves produces an interferogramwhose structure characterizes density changes in the gasbetween exposures. The interferogram obtained ispresented in Fig. 44.

reflecting mirror

transparent object

splitter mirror

Holographic interferometry of transparent objects.The arrangement for recording holograms oftransparent objects is shown in Fig. 43. The diagramdoes not show micro-objectives and individual lensesused to expand the light beams. When a light wavepasses through a transparent object, it is modulated byspecific phase information; the object modifies thepathlength of optical waves (the pathlength dependson the refractive index, and therefore on the density ofthe material) and, as a result, the phase of the waves.For this reason transparent objects are sometimesreferred to as phase objects, and holograms of suchobjects - as phase holograms.

interference pattern is interrupted, which indicates thepresence of internal defects in these places.

Holographic interferometry is also used to monitorthe dimensions and shape of objects, and the quality ofmachining of their surface. For this purpose one of thecompared waves is scattered by the object to beanalyzed and the other is reconstructed by a hologramon which a reference object was previously recorded.

laser


To summarize, holographic interferometry makes itpossible to obtain instantaneous and very informativeimages of bulk distribution of density in gases, that is,to study in detail a number of gas dynamicsphenomena.

Holographic interferometry of vibrating objects. Holographic interferometry is also widely used to analyzevibration of the surface of a number of objects.

In principle, an interferogram of a vibrating surfaceis very easy to obtain. It is sufficient to expose thefilm for a time much longer than the vibration period.The resultant hologram mostly reconstructs the twolight waves that correspond to the two extremepositions of the oscillating portions of the surface.Interference of these two waves produces an interferogram typical of the given oscillating surface (so-calledtime-averaged interferometry).

In order to clarify these statements, let us considerFig. 45, in which a sine wave characterizing theoscillatory motion is approximated by a step function.If this step curve really described the process of

Fig. 45

--/ "I \

\ / t\ /

'--t, t2

vibration, the hologram would indeed reconstruct onlytwo object waves, one corresponding to the exposureduring the interval t 1 and the other to the exposureduring t l . In fact, this would be identical to thedouble-exposure method. Obviously, this must ingeneral remain correct when the step function isreplaced by a sine wave.

An interferogram of the vibrating surface ofa circular membrane is shown as an example inFig. 46.

It should be noted in conclusion that the examplesgiven above point conclusively to the practical significance of holographic interferometry. This technique issuccessfully applied to nondestructive control andinspection of both small and large objects (fromminiature electronic devices and their assemblies to build-

Fig. 46

Optical Holor;raphy 118 11. Computer Technology and Holography 119

ings), as well as to detailed investigations of processesin gaseous mediums. In many cases the results obtainedby means of holographic interferometry are unique.

11. Computer Technologyand HolographyThe links between computer science and holography

are now well established and are developing fruit-fully. This will be illustrated by a number of examples.

Computer-generated holograms. It has been saidalready that a hologram is, as a rule, a "pattern" ona plane. This is indeed an interference pattern formedin the hologram plane as a result of superposition ofthe object and reference waves. Although this pattern iscomplicated and involves minute details, in principle itcan be reproduced by artificial means. Such syntheticholograms are now generated by computers.

The process of synthesis of a computer-generatedhologram consists of several stages. First, theinformation on the shape of the surface of the object,whose hologram' must be synthesized, is fed intoa computer. The computer then calculates theamplitude and phase (i. e. the wavefront) of the objectwave, that is, the wave that would be scattered by theobject. To be precise, the computer calculates theamplitude and phase of the object wave in the plane ofthe future hologram. As the next step, the computersuperposes the "object" wave and the "reference" wave,that is, computes the intensity distribution in the planeof the hologram. Finally, the computer transmits thispattern to a display device (such as a cathode-ray

tube), which graphically reproduces the obtainedintensity distribution. The obtained "pattern" is thenphotographed. The developed negative represents asynthetic hologram.

At least two aspects make computer-generated holograms extremely interesting.

First, such holograms enable us to obtain visual3-D "reconstructions" of imagined objects. For instance,one can reconstruct in three dimensions a "model" ofan object still at the design stage. Such a model mayeliminate the necessity of making an actual (material)model. One may visually "reconstruct" an object thatnever existed in real conditions.

Second, computer-generated holograms can be usedto reconstruct light waves with a specified wavefront.This means that a specially computed and manufacturedhologram may function as an optical element thattransforms in a predetermined manner the incident(readout) light wave. The simplest example is theFresnel zone-plate hologram, which is known toconstitute a plan;j equivalent of the lens. It should bevery attractive to substitute a set of optical elements(lenses, diaphragms, diffraction gratings, and so on)arranged in a specific manner by a single hologram

Calculation and manufacture of synthesized holograms form a new branch in optical holography,referred to as digital holography. In the general sense,digital holography comprises an analysis and synthesisof coherent optical fields by means of computers. Noother field of holographic applications can use to suchan extent the potentialities of controlling light waves.

Holography helps to manufacture microcircuits. Thefirst digital computers appeared at the very beginning

Optical Holography 120 11. Computer Technology and Holography 121

of the 1950's. These were so-called first-generationcomputers. Their logical elements were electron tubes.The speed of operation of the first generation ofcomputers was relatively low, from 102 to 104

operations per second. The main memory capacity ofthese computers was on the order of 104 bits *).

Computers of the first generation were used forcomputations in problems of a purely scientific orcommercial nature.

At the end of the 50's the second generation ofcomputers was born. This generation was based onsemiconductor triodes (transistors) used as logicelements. The rate of operation climbed to about 105

operations per second, and memory capacity increasedto about 105 bits. Computers started processing largedata arrays.

The third generation of cOmputers, developed inthe middle of the 60's, brought about a new era in thecomputer world: the era of data processing. It becamepossible to use computers to represent data in theinformative format, to retrieve the required informationfrom large data arrays, and to process this information.New computers can be given the task of controllingvarious processes. Third-generation computers havea very high speed of operation, up to 107 operationsper second, and memory capacity up to 106 bits. Thelogic elements of these computers are based onsemiconductor integral circuits (so-called microcircuits).

Microcircuits are also widely used in latergenerations of computers, and not only as logic

.) I bit is an elementary unit of information (the informationsufficient to select one of two equiprobable events). Note forcomparison that one volume of the Greater Soviet Encyclopediacontains about 107 bits of information.

'.1

,

elements, but also as memory devices. Microelectronicshas become a decisive component of modern computertechnology.

This brings to the fore the technological aspect ofmanufacturing microcircuits. A film microcircuit isa semiconductor film doped on predetermined areas byappropriate impurity atoms. These areas form anintricate and very detailed "pattern" on the surface,with some of the details being only 10 11m in size.A microcircuit is produced by using a special mask,that is, a very thin metal plate with holes reproducingthe mentioned "pattern" (impurities are introduced intothe semiconductor film through the holes in the mask).One of the techniques of preparation of such masks isbased on photolithography.

The diagram of a future microcircuit is first magnified to a scale of 10: 1 or more. This magnified diagram is photographed with high resolution, and thenprinted at a scale of 1 : 1, which gives the so-calledintegrated circuit photomask. The mask itself is a copyof the photomask. For the copying, a thin metal plateis coated with a special light-sensitive lacquer(photoresist). If the photomask is placed over thephotoresist and illuminated, the open areas of thephotoresist layer are exposed in accordance with theconfiguration of the pattern on the photomask. This isthe so-called contact method; another technique oftransferring the pattern from the photomask to thephotoresist layer, the so-called projection method,consists in the optical projection of the photomaskdiagram onto the photoresist.

Unexposed areas on the photoresist are then washed out,and the photoresist-coated metal film undergoeschemical etching. The etchant removes only those areas

Optical Holo[{raphy 122 11. Computer Technology and Holography 123

Fig. 47

This aspect makes the method of holography verypromising. Together with the advantages of theprojection method (in fact, holography is a variety ofthis technique), holography produces high resolution (upto 1 /lm) over a large area (up to 100 cm2

).

A holographic setup for printing the microcircuitdiagram on the photoresist layer consists of twosections. First of all, the hologram of the photomask isrecorded (see Fig. 47a). Then the hologram, after properprocessing operations, is reconstructed in anarrangement that reconstructs an undistorted real image.To achieve this, the readout wave is directed on thehologram in the direction opposite to that of thereference wave (see Fig. 47b).

In addition to high resolution over a large area,the holographic technique of microcircuit exposure hasa number of advantages directly related to theinterference nature of the method. For instance, wecould mention low sensitivity of photomask hologramsto damage, the possibility of having a set ofphotomasks on a single hologram, the possibility ofrecording on curvilinear surfaces (by forming a 3-Dimage), and so on.

The holographic memory matrix: the key element ofa holographic data storage device. At present thecomputer systems are more and more pervaded byoptical methods of data processing. A trend isdescernible of gradual "transformation" of digitalcomputers into computer devices of a quite novel class,namely into optoelectronic systems. Data processingand data storage will be realized in these systems bymeans of both electric and coherent optical signals.

The first steps toward optoelectronic computers are

read~~mlightwave

hologram

(b)

layer of photoresist

layer ofphotoresist

hologramq:==f:==f:=:J

~':!':'';:iC::====::::::~1

readoutlightwave

on the film that are not protected by the photoresistlayer, so that the result is the mask necessary forproduction of microcircuits.

Progress in computer technology results inconstantly increasing requirements for microcircuits:higher area and finer details of the "pattern". Atpresent large integral circuits require microcircuits witha pattern over an area with diameter of about 5 cmand with resolution of the order of 1 /lm. Highresolution over a large area is provided by the contactmethod of photomask application. Unfortunately, thismethod has certain shortcomings. The necessity of closecontact of the photomask and photoresist layers resultsin wear and tear of the former. In addition, the method is rather difficult to automate.

(a)


Fig. 49

holographicmemorymatrix

transparency

converginglenses

scanner

laser

a

stores an interference pattern that is formed as a resultof interference of the reference wave and a wavemodulated by some signal. In the case of associativememory, the interference pattern results fromsuperposition of the light wave modulated by the datastored and that modulated by the identification signal(associative signal).

Structure of the holographic data storage. Figure 49shows one possible arrangement for informationrecording in the holographic memory.

One possible system of a transparency wh~se

transmission is controlled by the external electnc fieldis a matrix of liquid-crystal cells. Liquid crystals arespecific organic dielectrics that possess, in a certaintemperature range, properties intermediate betwee?those of crystals and ordinary liquids. If voltage .IS

applied to a cell of the transparency, a r:cuhar h~dr?

dynamic effect is produced in this cell (Ill the hqUId

Fig. 48

already taken. An example is found in computers thatuse an optical memory comprising holographicelements: computers with holographic data storage.Such systems are expected to have a great futurebecause of the high capacity, reliability of storage, andhigh speed of data retrieval from the memory. Furthermore, holographic data storage seems to be themost efficient way of realizing associative memory III

computers.Holographic data storage is based on employing

a holographic memory matrix. This matrix is composedof an array of small holograms 2 to 5 mm in diameter(see Fig. 48). Each of the holograms may carry from104 to 106 bits of information. Each mini-hologram


Fig. 50

crystal): interaction of the external field and the electriccharges formed in the bulk of the liquid crystalproduces turbulent macroscopic motions. This turbUle~ce results in light scattering and, hence, in cello~aclty. Thus, a layer of liquid crystal only 10 11mthIck attenuates the light intensity by a factor of morethan 10 at a voltage of 10 V. The cross-section ofa liquid-crystal transparency is illustrated in Fig. 50.

Let us return to the diagram of the holographicdata storage device shown in Fig. 49. Assume that thescanner directs the laser beam to hologram "a" inmatrix (see the figure). The readout of this hologramreconstructs a divergent light beam (indicated in th"efigure by curvilinear hatching). At the same time, the~lOlogram transmits the undisturbed laser beam, whichIS sent by the left-hand lens through the centre of theright-hand lens and onto the hologram "a" of thememory m.atrix (this ?eam is traced in the figure bya double lme). The dIvergent light wave is spatially

modulated by transparency and is also incident on hologram "a" of matrix. The interference of this wave withthe light beam drawn by the double line records theinformation channeled to transparency into hologram"a" of matrix.

Let us change the information sent to transparency(by changing the set of electric signals applied to thetransparency cells) and at the same time send the laserbeam by means of scanner to a different hologram inmatrix (see dashed lines in Fig. 49). Clearly, this newinformation is recorded on a different hologram inmatrix. If scanner is synchronized with the devicesending electric signals to transparency, the wholememory matrix can be gradually filled with thenecessary information.

The above scheme was given as an example. Itcontains, however, all the basic elements involved inany holographic data storage device: a laser, a scanner,a transparency (we can term it the data input matrix),and a holographic memory matrix.

The examples discussed in this section sufficientlydemonstrate the diverse relationships between thecomputer science and optical holography. We see that,on one hand, progress in computers helps thedevelopment of holography (the example of computergenerated holograms) and, on the other hand, holo-graphy participates in bringing new advances in computersystems. The contribution of holography to computercircuitry is determined both by its participation inimproving the technology of computer manufacturing(the example of microcircuit technology), and by theimportant role that holographic data storage isexpected to play in the future of optoelectroniccomputer systems.

electrodes

asherglass-metal w

-.......

--- -------

- I----

I--- I- -

glass plate

liquid-cIYstaIcells

conductinglayer

Nonlinear Optics 12. About Optical Characteristics of the Medium 129

dipole moment. is the vector

Fig. 51

(12.2)

(12.1)

(12.3)

The factor x characterizes the medium and is termedits polarizability, or dielectric susceptibility.

---> --->p = qd

In the case of zero external electric field, dipolemoments point in random directions (Fig. 52a).

--->An external electric field E is a factor that tends

to orient the dipole moments in the direction of thefield. This trend is counteracted, however, by thethermal motion of molecules. As a result, the externalfield achieves only a partial orientation of dipolemoments, as shown schematically in Fig. 52b.

Let us denote the sum of all dipole moment--->

vectors per unit volume of the dielectric by P.

where the subscript i enumerates individual dipolemoments. Obviously, this vector is zero in the caseshown in Fig. 52a, and distinct from zero in thesituation of Fig. 52b (its direction evidently coincides

---> --->with that of E). The magnitude of P must be thegreater the stronger the orienting effect of the external

--->field is. This means that P gives a quantitative measureof polarization of dielectrics. It is called the polarizationvectdr.

The orienting effect of the external field onmolecular dipoles depends both on the properties ofa medium and on field strength. Let us write therefore

9-190

-q

12. A Few WordsAbout Optical Characteristicsof the Medium

Polarization of dielectrics. A dielectric can bepolarized if it is placed, for example, between chargedplates of a capacitor. Let us recall what processes takeplace inside the dielectric.

Assume that the dielectric is composed of polarmolecules representing electric dipoles. Each such dipoleis characterized by a physical vector quantity known asthe electric dipole moment. Figure 51 schematicallyshows a dipole molecule as a system of two pointcharges + q and - q at a distance d from each other.

--->The vector d is directed from the negative charge tothe positive counterpart. By definition, the electric

+q

------....... ""- --- .......

\Bf-------~j........ -... --- ......-----

Light polarizes dielectric mediums. We assumedabove that the dielectric was polarized in the electricfield of a capacitor. This capacitor is in fact quiteunnecessary. Dielectrics are polarized just as well bythe electric field of a light wave propagating in them.

131

(12.4)

(12.5)

n = V;E = 1 + 41tx

12. About Optical Characteristics of the Medium

Note that interrelationship between the refractiveindex of the medium, its dielectric permittivity, anddielectric susceptibility gives an additional argument in

-+In this case vector E in Eq. (12.3) is the vector ofelectric field strength of the light wave.

In what follows, it is this polarization that will berelevant in this context. The dielectric susceptibilityx will be regarded therefore as one of the opticalcharacteristics of the medium.

Two points must be emphasized when polarizationof the medium by light waves is Siscussed. First, thedirection of oscillation of vector E must be adequatelyregulated, that is, the light itself must be polarized.Second, the light wave electric field, in contrast to thatof a capacitor, varies with time, and at very high frequency at that. It becomes necessary, therefore, to takeinto account a sort of electric "inertia" of the medium,

-+ -+when response P of the medium follows E with some

-+time lag. In rigorous terms, polarization P at a given

-+moment is determined by field E at the precedingmoments of time.

Optical characteristics of the medium. Two moreoptical characteristics of the medium will be consideredin addition to the dielectric susceptibility x: dielectricpermittivity E and absolute refractive index n. The threeoptical characteristics are coupled by the followingrelationships:

130

+(b)

Fig. 52

Nonlinear Optics

Note that Eq. (12.3) is an example of a materialequation, that is of a relationship between an externalfactor and the "response" of the medium to this factor.

-+In the caso under discussion, field strength E is the

-+external factor and polarization vector P of the me-dium is its "response" to this factor.

Equation (12.3) is only one example of materialequations. The other example, well known to thereader, is Ohm's law, in which the electric field strength(or the potential difference which determines thefield strength) is an external factor and current density(or current) is the "response".

(al

Nonlinear Optics

favour of the electromagnetic nature of opticalphenomena.

What factors determine the refractive index? Onone hand, the refractive index is a function ofproperties of the refractive medium (its chemicalcomposition, phase composition, and temperature). Onthe other hand, it is determined by the characteristicsof the radiation:

frequency of the light wave (dispersion of light),and polarization of light.

Moreover, the refractive index in crystals may dependon the direction of propagation of the light wave.

Let us analyze in more detail the refractive indexof crystals as a function of polarization and directionof propagation of light waves. This will take us intothe field of crystal optics.

132 12. About Optical Characteristics of the Medium

identical for directions OA and OA1, and will differ fordirections OA and OB.

The discussion to follow will concern only uniaxialcrystals, owing to the predominant position they takein nonlinear optics.

Let us analyze propagation of light in a uniaxialcrystal. The difference with an isotropic mediumconsists in the fact that two monochromatic waves, one"ordinary" and the other "extraordinary", bothplane-polarized and having the same frequency,propagate simultaneously in any direction in a uniaxialcrystal. It is of principal significance that each of thesetwo waves has its own propagation velocity and, asa result, its own refractive index.

What is the difference between the ordinary andextraordinary waves? First, the plane of polarization

Fig. 53

133

Crystal optics. Crystal optics studies opticalproperties of crystals. Crystals are known to beanisotropic, which means that their physical properties,and optical properties among them, are functions ofdirection inside the crystal. The dependencies are foundto be essentially different in different types of crystals.Let us choose as an example a fairly widespread typeof crystals, which is referred to in crystal optics asuniaxial.

Each uniaxial crystal has an inherent directiontermed its principal axis. Optical properties of suchcrystals remain unaltered if the direction in which theyare measured is rotated around the principal axis(Fig. 53). Any other change in the direction will changethe properties. For instance, the refractive index will be

--.-/"

/{

\

"......

'-- B

Nonlinear Optics 134 12. About Optical Characteristics of the Medium 135

(the plane in which oscillates the electric field vector)of the extraordinary wave lies in a plane drawnthrough the principal axis of the crystal and the directionof wave propagation (sOrcal1ed cardinal section plane),while the plane of polarization of the ordinary wave isperpendicular to this plane. The situation is clarified inFig. 54 which shows separately the cases of ordinary (b)and extraordinary (a) waves. Arrows in the figureindicate the direction of oscillation of the electricvector of the light wave; 00 is the principal axis; OAis the chosen direction of wave propagation; thecardinal section plane is hatched.

Second, the refractive index of the ordinary wave(denoted by no) is independent of direction, while thatof the extraordinary wave (denoted by ne) is a function

(12.6)

where r:J. is an angle between the principala xis in thecrystal and the wave propagation direction; nl and n2are optical parameters of a given uniaxial crystal cal1edits principal indexes of refraction.

Note that the above comment on the differencebetween the indexes of refraction for the directions OAand OB (Fig. 53) was meant at the extraordinary wave:it is obvious that directions OA and OB differ in thevalue of r:J..

of direction:

Refractive index as a function of ex. Expression(12.6) gives the refractive index of the extraordinarywave, ne, for any value of angle r:J.. Let us draw a line00 on a sheet of paper, and assume it to be theprincipal axis of a crystal. Now draw a segment OA atan angle r:J. to 00; the length of OA (in arbitrary units)is equal to the refractive index ne for the same angle r:J..

Now vary r:J. from 0 to 360° and correspondingly rotateOA; the length of OA must be given by Eq. (12.6). Theend point of OA (i.e. point A) will ultimately trace anel1ipse (Fig. 55a). It is readily found that ne at r:J. = 0(and at r:J. = 180°) is equal to one of the principalrefractive indexes (n e = nt!, and at ex = 90° (and ex == 270°) it is equal to other principal refractive index

(n e = n2)·The obtained el1ipse is a section of the

refractive index surface, sometimes cal1ed the indicatrix,of the extraordinary wave. The indicatrix in Fig. 55a isdrawn in the cardinal section plane.

A

o

A

(a)

Fig. 54

o

Nonlinear Optics 136 12. About Optical Characteristics of the Medium 137

The transition from the cardinal section plane, tothe three-dimensional space is very simple since crystalrotation around the principal axis changes nothing inthe physics of the phenomenon. Consequently, therefractive index surface of the extraordinary wave is theellipsoid of revolution (Fig. 55b). If this ellipsoid isintersected by a plane passing through the principal axisof the crystal, we obtain the picture of Fig. 55a.

This analysis dealt with the refractive index surfaceof the extraordinary wave. As for the ordinary wave,its indicatrix is obviously spherical (circular incross-section). It is important that the ordinary andextraordinary waves have identical refractive indexesin the direction of the principal axis; consequently, theradius of the refractive index surface of the ordinarywave is nl' If we take account of both theextraordinary and ordinary waves, Figure 55a will betransformed to Figure 56 which shows, among other

Fig. 55

tI

things, that the difference between refractive indexes ofthe ordinary and extraordinary waves reaches maximumat \J. = 90° (and \J. = 270°).

The refractive index surface is an important opticalcharacteristic of a crystal. It provides a simple methodof finding the refractive index in any direction of wavepropagation. For this, it is sufficient to draw a straightline from the indicatrix centre in the selected direction.The distance from the centre point to the intersectionwith the refractive index surface of the extraordinarywave yields the required value of ne, and the distanceto the surface corresponding to the ordinary wave isthat of no.

The reader remembers that the extraordinary andordinary waves are polarized in different planes (in mutually perpendicular planes). This means that Figure 56also shows the refractive index as a function ofpolarization.

Fig. 56

o to

EXlraordinarywave

Nonlinear Optics 13. Dependence upon the Intensity of the Radiation 139

(a)

Light dispersion affects the dimensions (but not theshape) of the refractive index surface; in other words,ne and no are functions of frequency. Sections drawnby dashed lines in Figure 57 correspond to higher frequency than those drawn by solid lines.

Figures 55 - 57 are drawn on the assumption thatthe principal refractive indexes of the crystal satisfy theinequality nl > n2. It is customary to refer to uniaxialcrystals of this type as negative uniaxial crystals. If,however, n l < n2' a crysjil is called the positiveuniaxial crystal. Reference index surfaces of a negative(a) and positive (b) uniaxial crystals are compared inFigure 58.

13. Can the Optical Propertiesof a Medium Depend uponthe Intensity of the Radiation?The electric field of light waves is "gaining strength".

Nothing was said until now about opticalcharacteristics of the mediums as functions of the lightwave intensity. Before the advent of the laser,incoherent optics correctly assumed that opticalparameters of mediums are independent of the intensityof the light propagating in these mediums. Theessential fact is that the electric field strength in fieldsemitted by non-laser light sources is always muchsmaller thaI} field strengths of interatomic and atomicelectric fields. Non-laser light sources generate fieldswith electric field strengths not exceeding 103 V/cm,while atomic fields are characterized by field strengthsof the order of 107 to 1010 V/cm. It is only naturalthat with this ratio of field strengths, the light wave is

o

(b)

\\\

\ '\

" "...... ,.....""'-

Fig. 57

o

o

Fig. 58

Nonlinear Optics 140 13. Dependence upon the Intensity of the Radiation 141

not intensive enough to affect atomic fields and withthem the optical parameters.

Lasers have drastically changed the situation.Extremely high spatial concentration of light powerbecame feasible owing to the high degree of coherenceof the laser radiation. This is achieved, in practicalterms, by very little divergence of the emitted beamand by the possibility to generate light pulses with veryhigh peak power (see Sec. 4). Lasers made it possibleto generate optical fields with field strength from 105

to 109 V/cm, which is already commensurate to that ofatomic electric fields.

It has become necessary, therefore, to take accountof the dependence of optical characteristics of the medium on the intensity of the light wave propagating init. This necessity forms something of a "watershed"between the old (pre-laser) and new (laser) optics. It iscustomary to refer to the former as the linear, and tothe latter as the nonlinear optics. It must be emphasiz-ed that dependence of optical parameters of the medium on the intensity of the light wave constitutes themost characteristic feature of nonlinear optics, onewhich distinguishes it from the linear optics.

What is the origin of the term "nonlinear optics". Ifthe laser radiation is sufficiently intensive (so thatoptical characteristics are functions of light intensity),susceptibility x stops being constant and becomesa function of field strength E in the light wave. Theoryshows that in the first approximation this function canbe expressed as a sum

(13.1 )

where Xo, XI> X2, ... are parameters of a mediumcharacterizing its polarizability.

Note that all optical characteristics of mediums(not only susceptibility but dielectric I?ermittiv~ty andrefractive index as well) become functions of fieldstrength in sufficiently intensiv~ light field~.

Substitution of Eq. (13.1) mto (12.3) yields thefollowing expression for polarization of the medium(vector notation is dropped for the sake ofsimplification) :

P = xoE + x1E2 + x2E3 + ... (13.2)

An important fact is that Eq. (13.2) is nonlinearwith respect to field strength in the light wave. Hencethe term nonlinear optics.

If field strength in the light field is sufficiently low,only the first term can be retained in Eq. (13.2):

E (13.3)P = X o

This situation precisely corresponds to the pre-laseroptics: polarization of the medium is des.cribed b~a linear formula (13.3). Hence the term lmear optICS.

The relation between wave field strength andpolarization of the medium is therefore linear if th.e ,light wave field strength is relatively low; the mediUm spolarizability is represented then by the parame~er Xo

called the linear susceptibility. If, however, the hghtfield strength in the laser beam is. sufficientl~. high, therelation in question becomes nonlmear; additiOnalparameters (XI> X2, ... ) referred to as non~inear

susceptibilities are then required to descnbepolarizability of the medium.


Our question now is: what will be the timedependence of the "response" of the medium?

In the case of a linear medium the "response" (i. e.polarization) will strictly follow the temporal changes inthe external signal. Indeed, substitution of Eq. (13.7)into the linear relation (13.3) yields

We conclude, therefore, that the nonlinearexpression for polarization of optically isotropicmediums will have, in addition to a linear term, onlya third-order "nonlinear term". In this case the mediumis said to have third-order nonlinearity. In the case ofanisotropic crystals, Eq. (13.6) is valid, and the mediumis said to have second-order nonlinearity.

(13.7)

(13.6)

(13.8)P(t) = xoEocos (27tvt)

What is the response of a nonlinear medium toexternal factors? We mentioned above that polarizationP of a medium is a response of the medium to anexternal factor, namely to the field strength of the lightwave. It is essential that the external factor underconsideration is a function of time; fora monochromatic light wave, this function is

E = Eo cos (27tvt)

or, in vector notation,

Second-order and third-order nonlinear mediums. Ifpolariz~tion of a m~dium is described by a nonlinearex~resslOn, the medIUm is called a "nonlinear medium".Stnctly speaking, any medium becomes "nonlinear" ifa consid~rably intensive optical radiation is passedthrough It. The nonlinearity is found in the dependenceof the medium's properties on light intensity.

Note that in optically isotropic mediums (such arenot only gases and liquids but some crystals as wellnamely. those with cubic symmetry of the lattice) the'expressl?n for polar

2ization does not comprise the

qua~ratIc term xlE (owing to symmetry); thenonlInear relation (13.2) then becomes

P = xoE + x2E 3(13.4)

--+ --+ --+

P = xoE + X2E2 E (13.5)

Both the quadratic and cubic "nonlinear terms"ca~ be p~esent only in the equation for opticallyamsotroplc crystals, for instance for uniaxial crystals.We hav.e to ~ake into account, however, that as a rulethe cubIC (thIrd-order) term is substantially smaller thanthe second-order one; hence, Eq. (13.2) is thensimplified to *)

.•J Note that Eq. (13.6) mathe~atically....is not rigorous. Owing toanIsotropy of the mediu m, vectors P and E are not parallel. Hence,we must consider no~ one but three simultaneous equations: eachPIoJectlOn of vector P is expressed via three prOjections of vectorE. We. consider that such a complication is out of place in a book!Ike thiS, and so .sacrifice some rigorousness and use mathematicallyslmphfied approxImate expression of the type (13.2) and (13.6). ThisslmphficatlOn IS called scalar.

A nonlinear medium behaves in quite a differentmanner. The point to be emphasized here is that theresponse of a nonlinear medium is a different (sic!)function of time than that describing the appliedexternal field. This is readily confirmed by substituting

Nonlinear Optics

Eq. (13.7) into a nonlinear relation (13.2). Thesubstitution yields

P(t} = xoEocos(21tvt} + xIE5COS2(21tvt} +

+ X2E6 cos 3(21tvt} + ...If we use the formulas cos2 CL = (1 + cos 2CL)/2

cos3 CL = (cos 3CL + 3 cos CL)/4, Eq (13.9) can betransformed to the form

P(t) = Po + PI cos (21tvt) + P2 cos (41tvt) +

+ P3 cos(61tvt) + ...where

144

(13.9)

and

(13.10)

(13.11)

J3. Dependence upon the Intensity of the Radiation

the second harmonic of polarization; the fourth termvaries at frequency 3v and is called the third harmonicof polarization.

The difference in the responses of the linear andnonlinear mediums to a monochromatic signal isclearly seen if one compares Figure 59 (the case ofa linear medium) with Figure 60 (the case ofa second-order-nonlinear medium). Figure 60 alsodemonstrates the above-mentioned components ofpolarization.

The third harmonic of polarization is absent if themedium is nonlinear to the second order. If themedium is third-order nonlinear, polarization containsno constant term and no second harmonic. In

Fig. 59

145

The response of the selected nonlinear medium tothe external perturbation represented bya monochromatic light wave of frequency v comprisesfour terms. The first one is independent of time; thesecond one follows the external perturbation (it iscalled the first or fundamental harmonic of polarization);the third one oscillates at frequency 2v and is called

(13.12)

(13.13)

(13.14)

p p


Nonlinear optical phenomena come to the fore. Weshould not forget, speaking about the time-dependentexternal factor (perturbation), that this factor is thelight wave propagating in the medium at some velocity

a general case, the response of a nonlinear medium isdetermined by the form of function (13.1). Obviously,the response of the medium would be still morecomplicated if this function included terms with higherpower of field strength E.

~P~ Second harmonic

o _~ /'\ /'\ /'\V~

ot Po •

Nonlinear optical phenomena as reponses of themedium to light waves. As follows from some of theabove remarks, all phenomena in nonlinear optics arecaused, in the long run, by changes in opticalproperties of the medium induced by sufficientlypowerful optical radiation. This change in the

v. The polarization wave propagates in the medium atthe same velocity. In linear mediums this wave has thesame frequency v as the light wave. In nonlinearmediums described by Eq. (13.2) several monochromaticpolarization waves are produced having frequencies v,2v, 3v.

A polarization wave can be regarded as a sort ofan "emitting antenna" moving at the velocity v throughthe medium. This "antenna" may emit a new lightwave. Let us refer to this new light wave as re-emitted.The frequency of the re-emitted light wave must beequal to the polarization wave frequency; hence,nonlinear mediums may re-emit light waves not only atfrequency v but also at other frequencies, such as 2vand 3v.

Nonlinear polarization of the medium can thuslead to a specific nonlinear-optics phenomenon:transmission of a light wave with frequency v isaccompanied by emission of light waves at frequencies2v and 3v. This phenomenon is referred to asgeneration of optical harmonics.

Generation of optical harmonics is only one ofphenomena of nonlinear optics due to nonlinearpolarization of the medium. Later we shall discussother phenomena in the domain of nonlinear optics,also caused by specifics of the polarization response ofa nonlinear medium to light waves.

ResponsePP

Fig. 60

10*

Nonlinear Optics 148 14. Intensity-dependent Transparency of the Medium 149

The photoelectric effect IS observed, therefore, if thecondition

induced transparency and induced opacity. We choosethese two to begin a more detailed discussion of effectsin nonlinear optics.

is satisfied. The frequency Vo = A/h is the lowestfrequency for which photoelectric effect is observable inthe material. It is called the threshold frequency.

Let us formulate the following question: could thephotoelectric effect be observed beyond the thresholdfrequency, that is for v < vo?

14. Intensity-dependent Transparencyof the MediumCould the photoelectric effect be observed beyond the

low-energy threshold? The physical essence of thephotoelectric effect is quite clear. In a few words, thephenomenon consists in ejection of electrons from thematter by the action of light. Assume that a photonwith energy hv is absorbed by an electron inside thematter. If the photon energy exceeds the workA required to remove an electron from this material(so-called work function), then the electron in questioncan leave the material. Energy of the ejected electron is

(14.2)

(14.1 )

hv ~ A

mv2

-- = hv - A2

properties of the medium can be treated asa "response" to the light wave.

"Responses" can be of different origin. Two typesare usually distinguished: the polarization responsealready mentioned, and the so-called "level populationresponse".

We have already indicated that the polarizationresponse is caused by nonlinear polarization of themedium induced by the incident light wave. Its fieldreorients electric dipole moments and also createsinduced dipole moments. The polarization response iscomparatively fast: its "inertia" is characterized bya fairly short time interval, down to 10 - 13 s.

The "level population response" is absolutelydifferent in nature. It is related to changes in thepopulation of energy levels again induced by the lightwave propagating through the medium. Since this typeof response involves transition of a large number ofparticles from one level to another, the response of themedium is found to be comparatively slow: its "inertia"is characterized by response time above 10 - 8 s.Consequently, the "level population response" cannot,in contrast to the polarization response, follow the lightwave field without a considerable time lag.

Thus, fast and slow "responses" of the medium toan external perturbation (light wave) are distinguished.The former is of polarization nature, and the latter isassociated with changes in population of energylevels.

Each of the two types of response is responsiblefor a specific group of nonlinear-optical effects. Thoserelated to polarization response were alreadymentioned. Among the effects based on the "levelpopulation response" are such nonlinear effects as

l{onlinear Optics 150 14. Intensity-dependent Transparency of the Medium 151

We find that the threshold frequency Vo = A/h doesnot constitute the lower bound any more. In otherwords, Vo cannot be considered, as before, as thelowest frequency at which the effect is observable.

Second, the statement that the occurrence ofphotoelectric emission is totally determined by thefrequency of the radiation and has nothing to do withits intensity, had to be discarded. It is apparent that

The theory of the photoelectric effect alwaysassumed that an electron can absorb one (only one!)photon. This assumption was in perfect agreement withthe experiment in the pre-laser optics. The advent ofthe laser changed the situation drastically. Light beams(and light pulses) now available to the experimenterreached extremely high intensity. The density ofphotons in such beams (pulses) was so high that anelectron could absorb simultaneously two (and evenmore!) photons. The well-known laws of thephotoelectric effect had therefore to be reconsidered.

First, the concept of the threshold frequency lostits significance. If, for instance, an electron absorbs notone but N photons at a time, each having energy ofhv, then Eq. (14.1) has to be replaced by the relation

mv 2

-- = Nhv - A2

This means that the photoelectric effect can beobserved if

Ahv~

N

(14.3)

(14.4)

the higher the radiation intensity is, the greater thephoton density; consequently, the greater the numberof photons that can be absorbed by one electron. Inview of this, it should be more logical to speak notabout the disappearance of the threshold frequencybut about its dependence on the intensity of thelight wave. The threshold frequency is found as

AVo (N) = Nh (14.5)

The higher the light intensity, the greater N is andhence, the lower the threshold frequency.

The coming of the laser thus opened a new pagein the history of the photoelectric effect. It becamepossible to observe the so-called multiquantumphotoelectric effect. This effect totally belongsto the realm of the nonlinear optics: it isessentially intensity-dependent.

Induced opacity of the medium. Both thesingle-quantum and multiquantum photoelectriceffects discussed above represent the so-calledexternal effect: absorption of light results inthe emission of the electrons out of the material. Anotherpossibility is the internal photoelectriceffect, when the absorption of light transferselectrons from lower energy levels to upper ones. Botheffects may be single- or multiquantum. Themultiquantum internal photoelectric effect is at thebasis of the induced opacity, one of the effects in nonlinearoptics.

Let us consider the physical meaning of thisphenomenon. We assume that light absorption occurs

NonUnear Optics 152 14. Intensity-dependent Transparency of the Medium 153

because photons are adsorbed by certain particles of themedium (by absorption centres). Figure 61 shows energylevels of an absorption centre, where for the sake ofsimplicity only two levels, Eland E 2' are taken intoconsideration. Before irradiation, that is in theinitial state, all absorption centres are at theground level E2 . Assume now that the medium isirradiated with light whose frequency is chosen to besuch that the photon energy be exactly one half of thedifference between the energies of the two levels: v == (E 1 - E 2 )/2h. A photon with energy hv =

= (E 1 - E2 )/2 cannot be absorbed by a centre sincethis energy is insufficient for transition from level E2 tolevel E1 ; hence, the medium is transparent to thisradiation.

Now let us raise the intensity without changingfrequency. Sufficiently high intensity provided only bythe laser makes it possible to realize absorption of twophotons simultaneously by a single absorption centre.This means that the centre gains energy 2hv = E 1 - E2

and jumps from level E2 to level E1• The medium istherefore capable of absorbing the optical radiation.

The simple example given above is a clear

Fig. 61

demonstration of the nonlinear phenomenon underconsideration. Photons are not absorbed, and themedium remains transparent, when intensity is keptlow. If intensity becomes sufficiently high, the photondensity in the incident radiation increases to an extentwhere groups of photons begin to interact withabsorption centres (groups in the above example arepairs of photons). Photon absorption is now possibleand the medium ceases to be transparent.

Induced transparency of the medium. Not onlyopacity but transparency as well can be induced bylight if its intensity is sufficiently high. In other words,an increase in light intensity which in some casesreduces transparency of the medium, in other casesmay produce an opposite effect, that is clarification ofthe medium.

In essence, the physical meaning of this effect canbe illustrated by using Fig. 61. But in this case weassume that the radiation frequency is v = (E 1 - E2 )/h.Photons with energy hv = E1 - E2 will of course heabsorbed by the absorption centres. If the photondensity in the radiation is sufficiently high, practicallyall absorption centres may be raised from level E2 tolevel E1 . If this situation is realized, the medium isunable to absorb light at frequency v = (E 1 - E2)/h:indeed, there is "nowhere" for the photons with energyhv to be absorbed. Absorption of light at frequencyv reaches saturation.

The effect of induced transparency is quiteimpressive: a powerful light pulse incident on anopaque medium makes it transparent almostinstantaneously, and is transmitted. Some time after theend of the pulse (this time is different in different

Nonlinear Optics 154 14. Intensity-dependent Transparency of the Medium 155

mediums, and may vary in a rather wide range, from10 - 8 to 10 - 2 s) the absorption centres returnspontaneously to the ground level £2 and the mediumregains its opacity, at least until the arrival of the nextpowerful light pulse.

5 is equal to hv, that is to the energy of photonsemitted by the active centres in transitions betweenlasing levels.

Fig. 63

Phototropic gate: principle of functioning. Theinduced transparency effect is used in phototropic gates(shutters) already mentioned in connection withrealization of the giant-pulse mode in lasers (see Sec. 4).A schematic of a laser with the phototropic gate isshown in Fig. 62 where 00 is an optical axis of thelaser.

The process which results in emission of a giantpulse will be analyzed stage-by-stage, for reasons ofconvenience. The stages are illustrated in Fig. 63. Twosystems of energy levels are shown for each stage:a system of three levels (levels 1, 2, 3) of the activecentre in the lasing medium, and a system of twolevels (levels 4 and 5) of the absorption centre in theinduced-transparency medium, that is in the gate. It isessential that the difference between energy levels 4 and

3

2(a)

1000000

3

(b)

100

h"

5

4

5

4

Fig. 62

4

3

1000000

2

(c)opticalresonatormirror

nonlinearmediumactive medium

opticalresonatormirror

pumping system

o-~fH\-r-++-+- ---- --1---+

The situation in Fig. 63a is the initial one (prior tothe arrival of the pumping pulse): all active centres areat levell, all absorption centres at level 4.

The situation of Fig. 63b occurs immediately afterthe pumping pulse: population of level 2 of the activecentres is sufficiently high (population of the lasinglevels is inverted). Note that the gate is opaque both incase (a) and in case (b); hence, attenuation for photonstates with energy hv is very high.

Situation (b) is unstable. Some of the active centresspontaneously return from level 2 to levell, emittingphotons with energy hv. When these photons passthrough the gate, they may be absorbed; as a result,absorption centres will start populating level 5, therebyclarifying the gate medium. This induced transparencyof the gate lowers the attenuation for photon stateswith energy hv, and at a certain moment the lasingcondition will be satisfied. From this moment on, theprocess develops with accelerating pace. An avalancheof stimulated photons with energy hv rapidly makes thegate completely transparent and, during its short life,generates a very narrow but tremendously powerfullight pulse - the giant pulse.

Figure 63c illustrates the state of the systemimmediately after the emission of a giant pulse: alllasing centres are back at levell, and all absorptioncentres are at level 5 which corresponds to themaximum transparency of the gate.

Active centres will gradually return to level 4. Asa result, the gate will restore its opacity and photonstates with energy hv will be again effectively damped.In other words, both the lasing medium and the gatewill be back at the initial configuration (a), ready torespond to another pumping pulse.

Susceptibility I< of a nonlinear medium is a function ofthe light wave field strength; equation (15.1) .the? statesthat refractive index must also depend on thiS fieldstrength. A nonlinear effect following from this is theself-focusing of intensive light beams.

Self-focusing does not alter the frequency of thelight wave. This means that in analyzing thepolarization response, we need to take account only ofthe term (13.12) which describes the fundamental

(15.1)

Nonlinear Optics 156 ~. Self~focusing of Light

Compared to other types of optical gates(mechanical or electrooptical), phototropic gates ha~ean important advantage, namely, they are automatIc.The gate is opened by the pumI?ing p~lse an? returnsto the "initial position" automatIcally, Immediately afterthe end of the giant pulse emission. The experimenterneed not monitor, for instance, "on" and "off' states ofelectric or magnetic fields, or worry aboutsynchronization of some rotating elements withpumping pulses, and so on. All this is made .unnecessary. As to design considerations, a phototropicgate is simplicity itself, with all complexities ."transferred" to the physics of the processes takmgplace in the nonlinear medium with inducedtransparency.

15. Self-focusing of LightRefractive index as a function of intensity of light.

Equations (12.4) and (12.5) cited above enable us towrite

n = til + 411:1<

157

I Nonlinear Optics 151r 15. Self-focusing of Light 159

harmonic, that is the one with frequency equal to thatof the light wave. By using Eqs. (13.10), (13.12), and(13.7), we can recast this harmonic in the form

(15.2)

This shows that instead of x, the expression for therefractive index (15.1) must include the sum Xo ++ iX2E~. Therefore,

or

(15.3)

where c/ = 1 + 41txo is the dielectric permittivity of thecorresponding linear medium, and Cnl = 31tX2E~ isa nonlinear increment in the expression for dielectricpermittivity. Expression (15.3) can be simplified if wetake into acc,mnt that Cnl« t-,. By using an

approximation formula V1 + P ~ 1 + ! p, valid forp « 1, we obtain from Eq. (15.3): 2

Selflocusing. The refractive index of a nonlinearmedium is therefore proportional to light intensity. Let.us demonstrate that this results in self-focusing of lightwaves.

The intensity of any real light beam is notconstant over its cross-section. Normally it peaks in thecentral area of the beam (in the vicinity of the opticalaxis of the beam) and falls off gradually away from thebeam axis. This means, according to Eq. (15.4), that therefractive index of the medium (and so its opticalthickness) must decrease away from the optical axis ofthe beam. Recalling Eq. (2.1), we conclude that the lightpropagation velocity must increase with the distanceaway from the beam axis.

Let us "picture" the light beam as a set of lightrays, as is customary in the geometrical optics. Theconclusion drawn above means that the rays fartherfrom the beam axis will have greater velocity.Consequently, a plane wavefront incident on the. materialbecomes concave in the process of wave propagatlOn(Fig. 64). In other words, the beam transforms itself asif by a converging lens! It "contracts" toward theoptical axis, in other words, it self-focuses.

(15.4)Fig. 64

Here n, = ~ is the refractive_ index of the linearmedium, and n/TJE5 = cn//(2!h, ) is the nonlinearincrement in the expression for the refractive index.This increment is essentially proportional to thesquared amplitude of the light wave, that is to itsintensity.

Nonlinear Optics 160 16. Optical Transitions 161

Self-focusing in practice. Figure 65 shows the wayin which a light beam self-focuses in an appropriatemedium. A light beam with diameter D, incident ona nonlinear medium, self-focuses over a distance L oand then propagates as a narrow light fibre. Thedistance L o can be estimated by an approximationformula

Lo ~ D

VllE6

The distance over which self-focusing develops is theshorter the higher the intensity and the greater thenonlinear susceptibility X2 are.

Self-focusing is one of the threshold phenomena III

nonlinear optics: it occurs when intensity reaches

Fig. 65

a certain limiting (threshold) value. The theory givesthe following estimate for the threshold intensity:

A2

I thr = 2 2n/llD

where A is the light wavelength. The formula showsthat the threshold intensity diminishes as the frequencyof radiation rises. Furthermore, it is the lower thegreater the nonlinear susceptibility X 2.

Self-focusing was mostly investigated in liquids:carbon disulphide, nitrobenzene, benzene, acetone, andsome others. The observed light fibres were 30 to50 Jlm in diameter, with the self-focusing distance Lobeing about 10 cm for the initial light-beam diameterof 0.5 mm.

Investigations show that light self-focusing is a verycomplicated phenomenon. It has been established, forexample, that the observed light fibre has still finerstructure - it separates into a number of still thinner"filaments" with diameters down to 5 Jlm.

16. Optical TransitionsNonlinear optics proved successful in a solution of

a problem which was very important both scientificallyand practically: transformation of one light wave intoanother. For instance, how to transform a coherentlight wave with frequency v into a coherent light wavewith frequency, say, 2v?

Let us approach this problem first in theframework of elementary processes; in other words, III

11-190

Single-photon and multi-photon transitions. Opticaltransitions are classified into single-photon andmultiphoton ones. Only one photon is emitted orabsorbed in a single-photon transition, while two ormore participate simultaneously in multiphotontransitions. The number of photons participating ina transition determines its multiplicity: there aretwo-photon transitions (multiplicity 2), three-photontransitions (multiplicity 3), and so on. Let us considerthe general case of a transition of multiplicity N. Thismeans that N photons participate in this process. Itmay happen that m photons are emitted and N - mabsorbed. Obviously, all possible types of multiplicity Nmultiphoton transitions will be exhausted if m is variedfrom zero to N.

It should be emphasized that a multiphotontransition cannot in principle be divided intoa temporal sequence of events; it must be treated assomething indivisible in time.

As an example, It~t us take a two-photon transitionin which two photons are absorbed. It would be wrongto assume that first one and then another photon isabsorbed. It is essential that both photons are absorbedsimultaneously. Otherwise we would have' to considernot one two-photon transition but two single-photontransitions.

We conclude that any multiphoton transition IS

a qualitatively different process in comparison toa sequence (a set) of single-photon transitions.

Nonlinear Optics

terms of elementary interactions between individualphotons and individual quantum systems.

There is no direct photon-to-photon interaction. Westart with mentioning a fact of principal importance:there is no direct interaction between photons. True,we mentioned earlier the tendency of photons topopulate first of all the states which already havesl,lfficiently dense population. In a sense, this can beregarded as a sort of mutual "attraction" betweenphotons. It should be kept in mind, however, that thetendency has nothing to do with the concept of "directinteraction" which causes scattering of particles on oneanother, absorption of some particles by other ones,and mutual transformations of particles, includingdecays. Photons are not scattered by photons, do notabsorb one another, and do not decay. Neitherelectromagnetic nor other known forces mediateinteractions between photons.

So there is no direct photon-to-photon interaction,and each time some photons are "transformed" intodifferent ones, one must speak about photonsinteracting through a "go-between". The role of theintermediary is played by the matter, or to be exact,by its particles, first of all, by electrons. In whatfollows we consider this intermediary as a microscopicobject characterized by a system of energy levels.

A photon and a microscopic object are in directinteraction. This means that the microscopic object canabsorb or emit photons (or absorb and emit themsimultaneously). In the process, it goes throughquantum transitions between specific energy levels.Photons being essential participants of these transitions,the transitions are said to be optical.

162 16. Optical Transitions

All processes of "transformation" of photons intoother photons (all processes of transformation of lightinto light) are therefore reduced to optical transitionsin microscopic objects. For this reason, they deservea more detailed discussion.

163

Nonlinear Optics

What is a "virtual level"? Fig. 66£1 shows twosingle-photon transitions: first one photon with energy

/ hv is absorbed and the microscopic system is raisedfrom level 1 to level 2, and then the second photon isabsorbed and the system is raised from level 2 to level3. But how do we symbolize a two-photon transitionin which two photons of energy hv are absorbed? Byconvention, it is shown as illustrated in Fig. 66b, wherethe dashed line denotes the so-called virtual level.

What is a "virtual level"? Recall, first of all, thata two-photon transition cannot be separated in timeinto two stages. Hence, the microscopic object cannotbe observed on the virtual level (otherwise we couldconsider two stages - one prior to and another afterthe observation moment). This constitutes the maindifference between a virtual and an ordinary energylevel.

Could it be concluded then that a virtual level is"non-existent" or "not real"? Indeed, a microscopicsystem on any really existing level can always beobserved, at least in principle.

Fig. 66

3


We are not going into a discussion of reality (orunreality) of virtual levels. The most important pointfor us is that both single- and multiphoton transitionsare a reality. Moreover, a system of familiar (real)energy levels is sufficient to describe the behaviour ofsingle-photon transitions, while in the case ofmultiphoton transitions this system is definitelyinsufficient and it is necessary to resort to a specificconcept - that of a virtual level. The example inFig. 66 is a clear illustration of the specific nature ofthis concept.

It should be noted in conclusion that one exampleof the two-photon absorption has already beendiscussed in Sec. 14 (the effect of induced opacity).

H ow is the microscopic object acting as anintermediary in light-to-light transformation processes?Let us consider some of the processes in whichphotons are "transformed" into photons in a differentstate. We shall begin with a process shown in Fig. 67.The microscopic system absorbs a photon with energyhv] and changes from level 1 to level 3. The system

Fig. 67

165

2

(a)

h"

h"

(b)

3

h"2 'VV'.f+2

Nonlinear Optics

then emits a photon with energy hvz and drops fromlevel 3 to level 2. The initial (primary) photon withenergy hVl is therefore "transformed" into the final(secondary) photon with energy hvz. The microscopicsystem has acted in this "transformation" as an"intermediary": indeed, its state has also been changed,from level 1 to level 2.

This role of an intermediary between photons (butnothing more than an intermediary) stands out stillmore cl~arly in the process shown in Fig. 68a. Themicroscopic system absorbs a photon with energy hvand changes from level 1 to level 2. Then it emitsa photon with the same energy and returns to level 1.The state of the microscopic system finally remainsunaltered, while the primary photon has been "turned"into the secondary one. The secondary photon has the

Fig. 68

-~---,.--2

(b)


same energy but of course may differ in both themomentum direction and polarization.

Let us turn now to the process shown in Fig. 68b(dashed line denotes a virtual level). In contrast to thetwo preceding single-photon processes, here we dealwith one two-photon event rather than twosingle-photon events. In principle, the case of Fig. 68aallows to observe the microscopic system on level2 (during the interval between the absorption of theprimary photon and the emission of the secondaryone). The situation of Fig. 68b is absolutely different:the microscopic system cannot be found on the virtuallevel in principle, since there is no "interval" betweenthe absorption of the primary and the emission of thesecondary photons. One cannot even state that theprimary photon absorption occurs prior to the emissionof the secondary one. The process of absorption andemission proceeds as something indivisible in time, sothat it would be meaningless to try and observe eventemporary changes in the state of the microscopicsystem.

The microscopic system of the above-discussedtwo-photon process may be said to behave as a very"tactful" intermediary staying very much "behind thescenes".

A process describing generation of the secondharmonic (SHG). The multiphoton processes in whichthe initial and the final states of the microscopicsystem are identical, are of special interest in th.:nonlinear optics. The two~photon process was discussedabove. Let us consider two three-photon process"s.

The first of them is shown in Fig. 69 (dasned linesindicate virtual levels). The microscopic system

167

Nonlinear Optics 168 16. Optical Transitions 169

A process describing parametric generation of light.Figure 70 represents a three-photon process in which

(here P: and Ii are the momenta of the absorbedphotons, and p is the momentum of the emitted one).

In nonlinear optics the above process is referred toas the second harmonic generation (SHG). It describesthe "transformation" of light with frequency v into lightwith double frequency, 2v. The SHG-process is treatedin more detail in Sec. 17.

(16.2')

•On one possible source of doubt. A reader may be

uncertain about whether the processes in Figs. 69 and

This process is often referred to as the parametricgeneration of light. It describes the "transformation" ofthe light wave with frequency v into two new lightwaves with frequencies VI and V2' In principle, any ofthese frequencies (for instance, vtl can be varied ina continuous manner from zero to v. The phenomenonof parametric generation of light will be analyzed inmore detail in Sections 17 and 18.

one photon with energy hv is absorbed and twophotons, one with energy hVI and the other with hV2,are emitted; the state of the microscopic system isunaltered. In a sense, this process can be treated asa "decay" of one (primary) photon into two new(secondary) ones. The photons participating in theprocess are again subject to the energy-momentumconservation:

hv = hV I + hV2 (16.2)(16.1 )

(16.1')

hv + hv = 2hv~ ~ ~

PI+P2=P

participates in a three-photon transition: two photons,-with energy hv each, are absorbed and one photonwith energy 2hv is emitted; the state of the microscopicsystem is unaltered. The microscopic system as an"intermediary" remaining "in the shadows", we canregard the process as a "direct transformation" of twophotons into one (two colliding photons merge intoa single one). The energy-momentum conservation lawis satisfied:

Fig. 70Fig. 69

------r--J\N"-- h"

-- --

h"

:!h" J\!\.r-- h"

---J-----h"l '\fVV-

-- -- ---

h" 2 'VV'.J---

Nonlinear Optics

70 indeed require to be mediated by an "intermediary".Can it be that the photons in these processes interactdirectly?

Indeed, we are tempted to assume that in someprocesses photons interact without intermediaries (asnumerous other particles do). In this case we could getrid of the concept of a virtual level. It would bepossible then to consider that a photon with energy hvin Fig. 70 decays into two photons hVj and hV2

without involving the microscopic system which remainsat the same energy level and does not take part invirtual transitions.

These arguments are, however, unacceptable.Experiments show that the processes shown in Figs. 69and 70 (and other processes in question) simply cannotoccur outside the medium! However deep the "shadow"in which the microscopic system is lurking, its role ofan intermediary is always decisive, in the sense that itdetermines the very possibility of realization ofa multiphoton process.

17. Transformation of One Light Waveinto the Other

Incoherent and coherent light-to-light transformationprocesses. The preceding subsection was devoted tovarious processes of light-to-light transformationillustrated by elementary interactions between photonsand the microscopic system. Sometimes transitions withabsorption of photons and those with emission ofphotons are sharply separated in time: they areaccompanied by changes in the state of the microscopicsystem (even if the initial and final states of this system

170 17. Transformation of One Light Wave into the Other

are identical). In other processes these transitions arenot separated in time, and no changes could bedetected in the state of the microscopic system; theenergy-momentum conservation in these processes holdsas though photons interact directly.

The processes of the first type are referred to asthe incoherent processes of light-to-light transformation,while the processes of the second type are said to becoherent processes. Let us analyze some specifics ofboth types of processes.

Incoherent processes. In the case of incoherentprocesses the primary light wave (pumping wave) isabsorbed and thereby causes changes in the populationof levels in the material. New quantum transitions inthe medium then result in emission of the secondarylight wave. Obviously, no interaction between thepumping wave and the secondary light wave is possiblein this case. Indeed, first the pumping wave raises themedium to the excited state and only after some timethe medium returns to the ground state and emits thesecondary light wave.

The process of generation of the laser emissionunder conditions of optical pumping is one example ofthe incoherent light-to-light transformation. Here theradiation of the flash lamp is the pumping wave, andthe coherent radiation generated in the lasant is thesecondary light wave. Another example is thephotoluminescence which is used in lamps customarilyreferred to as fluorescent lights.

Coherent processes. In contrast to incoherentprocesses, the acts of interaction of the medium withthe pumping wave and with the secondary wave cannotbe separated in time a.nd have to be regarded asa unified process (we remind that this constitutes the

171

Nonlinear Optics 172 17. Transformation of One Light Wave into the Other 173

The condition of wave synchronism in the case ofSHG. Assume that the directions of propagation of thepumping and the secondary wave in the case of SHGare identical, so that momentums of all photons inEq. (16.1') propagate in the same direction. This enablesus to replace the vector equation (16.1') by a scalarone:

where PI and P are momentums of the primary andsecondary photons, respectively.

We remind that the photon momentum in vacuumis expressed in terms of the radiation frequency bymeans of Eq. (2.6). In the case of a medium thisformula has to be slightly corrected by introducing theindex of refraction (frequency-dependent):

The requirement of matching the pumping waveparameters with those of the secondary wave isformulated as the condition of wave synchronism. In"photon terms" this condition is identical to the law ofmomentum conservation for photons participating inthe process. The wave synchronism condition isimportant in coherent processes: it is a necessarycondition of the efficient transfer of light energy fromthe pumping wave to the secondary one.

Let us elucidate the meaning of the wavesynchronism condition by analyzing an example of thesecond harmonic generation.

(17.2)

(17.1 )

hvP = -n(v)

c

main distinction of transitions involving virtual levels).This characteristic of coherent processes is essential intwo respects. First, it is impossible to detect any

-- changes in the state of the medium interacting withlight waves. Second, we can, in a sense, operate withdirect interaction between the pumping wave and thesecondary wave. No doubt, waves interact only via thematter and this interaction is determined by thematter's parameters. Nevertheless, the participation ofthe medium, though being essential in principle,remains virtual in character so that the light wavesinteract as if directly.

Interaction of the waves requires that the pumpingand secondary waves be matched in frequency, directionof propagation, and polarization. This means that eachof the interacting waves must obviously becharacterized by a definite frequency, propagationdirection, and polarization. Hence, coherent processesmust involve highly coherent light waves. It can besaid that all coherent processes are the processes oftransformation of coherent light to coherent light.

The importance of light coherence to coherentprocesses can be additionally clarified on the basis ofphoton concepts. For a coherent process to occur, it isnecessary that the momentum-energy conservation lawbe satisfied for the photons; consequently, the primaryand secondary photons must be in definite states, thatis in states with definite energy and momentum.Obviously, the greater the number of photons in therequired state and the smaller the spread of photonsover all other possible states, the more effective thecoherent process in question is. And reduction of thespread of photons over allowed states is identical toincreasing the coherence of the radiation (see Sec. 2).

Nonlinear Optics 174 17. Transformation of One Light Wave into the Other 175

Making use of Eqs. (17.2) and (16.1), recast (17.1) to theform

Take a uniaxial crystal.. .. Refractive index surfaceof a negative uniaxial crystal was shown in Sec. 12 (seeFig. 57). These surfaces are repeated in Fig. 71, withsolid curves tracing surfaces for frequency v and dashedcurves-for frequency 2v. The refractive index surfacesof the ordinary wave with frequency v and that of theextraordinary wave with frequency 2v intersect inpoints A and A\.

This means that if we select, for example, thedirection AA (going at an angle a. to the principal aXIs

(17.4)no (v) = ne (2v)

of the crystal), then the following condition is satisfiedfor light waves propagating in the direction AA:

Fig. 71

This is the synchronism condition for the SHG processin which the pumping wave is the ordinary wave, andthe second harmonic is the extraordinary wave. Thedirection AA is the direction of synchronism for theprocess in question.

What has to be done, therefore, in order to realizethe process of generation of the second harmonic?

This requires first of all a uniaxial crystal withsufficiently high nonlinear susceptibility 1'\. (This can bethe negative uniaxial crystal of potassiumdihydrophosphate, KH zP04 .) The crystal must be cutin the shape, for example, of a rectangularparallelepiped with the axis along the direction of

(17.3)II (v) = 11 (2v)

hv h2v2-n(v) = -n(2v)

c c

which yields, after factoring out identical multipliers,

This is the wave synchronism condition for the SHGprocess. According to this condition, efficient transfer oflight energy from the pumping wave to the secondharmonic requires that the two waves have identicalrefractive indices.

Obviously, Eq. (17.3) does not hold in the generalcase (because of the dispersion effect). So the practicallyimportant problem becomes: how to satisfy thiscondition. A satisfactory answer to this question wasnot immediately found. It proved to be veryinteresting: it suggested using the dependence of therefractive index on the direction in the crystal.

177Nonlinear Optics

synchronism for the given frequency v of the pumpingradiation.

Pumping must be realized with a laser. Thepumping wave must be plane-polarized, with the planeof polarization perpendicular to the plane of cardinalsection of the nonlinear crystal (the plane drawnthrough the principal axis of the crystal and theparallelepiped axis). This polarization of the pumpingwave is required in order to act as the ordinary wave(the plane of polarization of the ordinary wave isperpendicular to the cardinal section plane).

If these conditions are satisfied, propagation of thepumping wave with frequency v in a nonlinear crystalproduces an additional light wave: the second opticalharmonic. The direction of propagation of this wavecoincides with that of the pumping wave (although theopposite direction is also possible), its frequency istwice that of the primary radiation, and the plane ofpolarization coincides with the cardinal section plane asin the extraordinary wave. If a nonlinear crystal isseveral centimetres long, more than 10% of thepumping energy can be transformed into the secondharmonic radiation.

Classical interpretation of the second harmonicgeneration. Until now the SHG effect was described inphoton terms, that is by referring to the three-photonprocess shown in Fig. 69. It is not difficult, however, tosupply a purely classical explanation as well.

Let a coherent pumping wave (13.7) with frequencyv be incident on a quadratically nonlinear medium. Ifthe medium were linear, its polarization would changein time exactly as the pumping wave field strengthdoes i. e. with frequency v [see Eq. (13.1\)]. But

17617. Transformation 0( One Light Wave into the Other

polarization of nonlinear mediums contains the secondharmonic as well: the term ix\E5cos(41tvt) inEq. (13.10). Naturally, polarization oscillatin~ atfrequency 2v may result in re-emission of lIght at thisdouble frequency, that is in the emission of thesecondary light wave with frequency 2v.

It has already been mentioned in Sec. 13 that thewave of polarization (and with it the second harmonicof polarization) propagates in the m~diu~ with the .velocity of the pumping wave, that IS With the velOCityc/n(v). Energy transfer from the polarizatio~ wa.ve tothe re-emitted light wave will only be effiCient If wavepropagation velocities coincide. The velocity of thewave with frequency 2v being c/n(2v), the lightre-emission condition at this frequency becomes

n(v) = n(2vJ

which is a familiar condition of wave synchronism.This is a classical interpretation of the SHG effect

in nonlinear optics. Note that this interpretation bringsto the fore the role of the medium as an intermediaryin the interaction between the primary and secondarylight waves. Indeed, the interaction is "~ran.sferred"

along a "chain": pumping wave-polanzatIonwave - secondary light wave.

The process of the third harmonic gene:a~io~ canalso be easily outlined. In "photon terms" thiS ISa specific four-photon process in which three photo~s

with energies hv are annihilated and one pho~on Withenergy 3hv is produced. In term~ ~f the clas~lcal :-vaveconcept this is a result of re-emlSSlOn f?ll?wlllg dIrectlyfrom the existence of the third harmOnIC III thenonlinear polarization of the medium [see the term±X2Egcos(61tvt) in Eq. (13.10)].

12-190

(17.5)

Nonlinear Optics

The processes of generation of still higher ordersof optical harmonics - fourth, fifth, and so on, are alsopossible. It seems that these processes do not followfrom the formula (13.10). However, Eq. (13.10) wasbased on (13.1) which contains the terms of order nothigher than E2

; higher powers of E were neglected. Inprinciple, we could include into (13.1) severalhigher-order terms which would make it possible toanalyze higher-order harmonics in the nonlinearpolarization of the medium.

Nonlinear polarization of the medium allows mixingof frequencies. Let the polarization of a nonlinearmedium be represented by Eq. (13.6). We assume thattwo coherent light waves with unequal frequencies areincident on the medium: E 1 cos (2rcVlt) andE2 cos (2rcV2t). If the sum of these waves,

is substituted into Eq. (13.6), the final expression for thepolarization of the medium will contain a term

178 17. Transformation of One Light Wave into the Other

The fact that the expression for nonlinearpolarization of the medium contains the term (17.7)means that light can be re-emitted at frequencies VI ++ V2 and VI - V2' Hence, nonlinear mediums make itpossible to realize summation and subtraction of lightwave frequencies. We find that in the case in questioninteraction of waves with frequencies VI and V2 maygenerate secondary light waves with frequencies VI'+ V2

and VI - V2'

Equation (13.6) is the simplest expression for thepolarization of a nonlinear medium (nonlinearpolarization is described by a term quadratic in fieldstrength). In a more general case, polarizationexpression may include also the terms with E3

, E4, and

so on. With these terms, substitution into Eq. (13.5)results in polarization containing terms with frequenciesVnlll = nV 1 ± mV2, where nand m are integers. This.means that other types of frequency mixing are possiblein addition to summation and subtraction.

We note in conclusion that the difference VI - V2

may fall into the range of acoustic frequencies. In thissense one may speak of, for example, an opticalmethod of generation of ultrasonic waves.

179

Making use of the relation 2 cos rt cos ~ = cos (rt ++ ~) + cos (rt - ~), we transform Eq. (17.6) to thefollowing:

Pl.2 = ><I E 1E2cos [2rc (VI + v2) t] +

+ ><IE1E2 cos [2rc(v 1 - v2)t] (17.7)

A classical explanation of the parametric generationof light. The effect of parametric generation of lightwas analyzed in Sec. 16 in terms of photon concepts.A classical explanation of this phenomenon is alsopossible. It is based on the discussed above "mixing"of light waves in a medium with nonlinear polarization.

Assume that the medium is illuminatedsimultaneously by a high-intensity coherent light wave(pumping wave) Ep cos (2rcvt) and two low-intensity lightwaves as the first and second initial signals,

12'

Elements required to realize parametric generation oflight. A general set of requirements was in fact given inthe preceding Section. .

The first element is a uniaxial crystal Withrelatively high nonlinear susceptibility, cut according tothe requirements of the wave synchro~ism. This cr~stalis placed inside the optical resona~or m ord~r to directthe synchronism direction, for a gIVen combm~t1on o.ffrequencies VI and V2 = V - Vb along the optlcal.axlsof the resonator. This will favour photon states Withenergies hv and h(v - VI) = hV2' Now the resonator IS

illuminated Iby a coherent light wave with ~reque?cyV (pumping wave emitted by a la.ser). Th.e m~enslty ofthe pumping wave must be suffiCiently high m order tobring out the nonlinear properties of the crystal and toexceed the level of losses for favoured photon states.Furthermore, in accordance with the requirements ofthe wave synchronism, the pumping wave must beeither ordinary or extraordinary, that is it must bepolarized in a specific manner. And finally, there mustbe a device which makes it possible to "control thedirection of synchronism" thereby realizing smoothtunability of the frequency of secondary light waves.

Parametric light oscillator. A schema~ic of theparametric light oscillator is shown !n Fig. 72a(00 - optical axis of the system defmed by theresonator mirrors).

Figure 72b schematically shows t~e pumping wa~e

and secondary light waves generated m the parametnclight oscillator which is shown by a rectangle. The

Nonlinear Optics

condition VI + V2 = v. Let us refer to these weakwaves as the first and second initial signals,respectively. "Mixing" of the first initial signal with thepumping wave may result in a sufficiently intensivesecondary wave at frequency v - VI = V2' Likewise,"mixing" of the second initial signal with the pumpingwave may produce a secondary wave at frequency v - V2 = VI' We can therefore excite parametrically two

secondary light waves at frequencies VI and V2, at theexpense of part of the energy of the pumping wave.

Naturally, parametric generation of light requires,as any other coherent process, that the correspondingcondition of wave synchronism be satisfied. Materialsto be used are, as in the case of SHG, uniaxial crystalswith relatively high nonlinear susceptibility; moreover,the direction of synchronism is again found for thechosen combination of frequencies v and VI'

Initial signals required for "triggering" the processof parametric generation are always available in anyreal crystal in the form of an inevitable "background"which can be explained, among other factors, by thepresence of spontaneous photons. These very weaksignals are "distributed" over a spectrum of photonstates. The experimenter can, by varying thesynchronism direction, bring "into play" initial signalsof different frequencies and thus tune the frequencies VI

and V2 of the secondary light waves generated in theprocess, in a continuous manner.

The term parametric generation of light originatesfrom the similarity between this process and themethod of parametric excitation of oscillations widelyused in electronics. Parametric phenomena inelectronics occur in circuits involving nonlinearcapacitors, while in optics we have to use nonlinearcrystals.

180

.Id

18. Principle of Operation

18. The Principle of Operationof the Parametric Light Oscillator

181

Methods of tuning parametric light oscillators. Thereare several methods of controlling frequency ofsecondary waves generated in parametric lightoscillators.

Angle tuning. Let a nonlinear crystal be orientedinside the resonator in such way that its principal axisAA is at an angle !XI to the optical axis 00 (Fig. 73).In this case frequencies of the secondary waves, VI andV - VI' will be such that the synchronism direction willmake the angle !XI with the principal axis of the crystal.Let us slightly tilt the crystal, until the angle betweenthe principal axis of the crystal and the oscillator axis

Nonlinear Optics

pumping wave with frequency v propagates along theoptical axis 00. Two secondary waves with frequenciesVI and Vl = v - VI propagate along this axis in bothdirections. All interacting light waves in theconfiguration under discussion are directed along theoptical axis 00; in other words, momentums of allphotons involved in the process have identicaldirection.

The wave synchronism condition is satisfied, ina given crystal, for certain directions specific for eachcombination of frequencies VI and Vl = V - VI; thesedirections (directions of synchronism) are at a certainangle to the principal axis of the crystal. Withoutgoing into the wave synchronism condition for the

Fig. 72

mirror mirrorof optical of opticalresonator resonator

o-·~·~t-o(a)

182

~~....:~,

18. Principle oj" Operation

parametric generation of light (wave synchronism wasanalyzed in detail for the SHG case), let us note thatthis condition is satisfied only by special combinationsof ordinary and extraordinary waves. Differentcombinations are possible. In one of them anextraordinary pumping wave is used; one of thesecondary waves is extraordinary and the other isordinary. In another combination, an ordinary pumpingwave is used, both secondary waves beingextraordinary. We remind that in uniaxial crystals theordinary and extraordinary waves differ, first, in theshape of the refractive index surface and, second, inpolarization: the extraordinary wave is polarized in thecardinal section plane while the ordinary wave haspolarization normal to this plane (see Sec. 12).

In order to obtain secondary light waves atfrequencies VI and Vl = V - v" it is necessary toorient the nonlinear crystal inside the resonator in suchmanner that the optical axis 00 coincides with thesynchronism direction for the chosen combination offrequencies V and VI.

183

Nonlinear Optics 184 18. Principle of Operation 185

is 1'12 (dashed line in Fig. 73). The frequencies ofsecondary waves that are generated now are such thatthe synchronism direction is at the angle 1'12 to theprincipal axis of the crystal. It is thus possible tothe frequency of secondary light waves (in a certainrange, of course) by varying the angle 1'1, that is byrotating the nonlinear crystal with respect to the axis00.

Temperature tuning. We have mentioned in Sec. 12that the refractive index of the medium is alsoa function of temperature. This means that in uniaxialcryst~ls temperature must affect to some extent theshape of optical refractive index surfaces of theordinary and extraordinary waves. This must lead, inturn, to changes in the direction of wave synchronismfor any fixed combination of frequencies.

Consequently, the secondary wave frequency maybe tuned by varying crystal temperature, withoutrotating the nonlinear crystal inside the resonator. Asthe crystal is heated or cooled, new synchronismdirections, corresponding to new combinations of

A

Fig. 73

0-II"'-.::::---------iJ

-----_ I--..;

o

vary

secondary wave frequencies, will be coincident with thedirection of the optical axis 00.

It should be noted in conclusion that both thesemethods of tuning are based, in the long run, on thedependence of the synchronism direction on thefrequencies of interacting waves (dispersion of light isrevealed in the dependence of the refractive index onfrequency, see Sec. 12). The frequency of generated lightwaves is effectively controlled by directing the chosendirection of wave synchronism along the optical axis.

Is it possible to generate only one secondary lightwave? A characteristic feature of the parametric lightgeneration is the excitation of two secondary waves (attwo distinct frequencies). Assume, however, that light ofonly one frequency must be emitted (say, vd. How tosuppress the unwanted component of frequency v - VI?

Obviously, the simplest approach would be to passthe exit beam through a filter which realizes resonantabsorption at frequency V - VI but has almost noabsorption at VI' Unfortunately, this would mean thatpart of the pumping wave energy consumed to excitethe wave at frequency V - Vb is completely lost; thisenergy will simply heat the filter.

A wiser approach consists in introducing into theresonator an additional element which substantiallyincreases the level of losses for waves with frequencyV - VI but does not appreciably affect that of thewaves with frequencies V and VI' This is realized withminimum difficulties if v- and vI-waves areextraordinary and V - VI is an ordinary wave. In thiscase we can use the fact that polarization planes of theordinary and extraordinary waves are mutuallyperpendicular.

Fig. 74

It is convenient to employ the so-called Glanprism as the above-mentioned additional element. TheGlan prism (Fig. 74) comprises two identical rectangularprisms cut out of an Iceland spar crystal in suchfashion that the principal axis of the crystal is parallelto the edge of the angle ~ (i. e. normal to the plane ofthe drawing); ~ = 500. The Glan prism has a veryinteresting property: it lets through, and withoutchanging the direction of propagation, only theplane-polarized light wave with the plane of polarizationnormal to the plane of the drawing, while the lightwave polarized in this plane is reflected away. Thereflection takes place on the interface of therectangular prisms.

Let us place a nonlinear crystal and a Glan prismalong the optical axis 00 inside the oscillator cavity asshown in Fig. 75 (AA is the principal axis of thenonlinear crystal, and the hatched plane is that of thecardinal section). The waves with frequencies v and VI

Nonlinear Optics 186 18. Principle of Operation

are extraordinary, so that their polarization planecoincides with the cardinal section plane. These waveswill therefore be transmitted by the Glan prism. The(v - vl)-wave is, on the contrary, an ordinary wave; itis polarized normal to the cardinal section plane andso will be eliminated by the prism.

A Glan prism in the oscillator cavity can thereforesubstantially affect the process of parametric generationof secondary light waves: the (v - vd-wave will not begenerated at all and the pumping wave energy will beconverted only to the wave with frequency VI' Oneimportant fact is that the Glan prism continues tofunction even if frequency is varied since the characterof polarization of the interacting waves is obviouslyunaffected by the tuning operations.

Parametric light oscillator and optically pumpedlaser. The parametric light oscillator and opticallypumped laser seem to be very similar instruments, atleast in a number of characteristics. Both generatecoherent light, and both employ an optical resonator

187

Fig. 75

A

Nonlinear crystal

o

(cavity). The similarity becomes more obvious if wetake into account that a laser may be pumped bya coherent (laser) source and that the frequency oflaser emissi.on ~an. be varied continuously (for example,III lase~s. wIth lIqUId lasants). If a laser is pumped byan auxIlIary laser, then the functional diagrams of thetwo sources (the laser and the parametric oscillator) arenearly identical.

Despite the superficial similarity outlined above,t~e two .sources differ in one essential point. TheIIght-to-lIght transformation in the active medium ofthe laser proceeds via incoherent processes: thepumping wave is first absorbed by the matter andexcites the active centres, and only then do the activec.entres emit light, thereby generating the secondarylIght wave. As to the parametric oscillator, its "activemediu~" is a scene of coherent processes: the pumpingwave IS "transformed" by a nonlinear crystal intosecondary light waves, and no changes can be detectedin the state of the crystal. The difference between theoptically pumped laser and the parametric oscillatorlies therefore in the difference between incoherent andcoherent processes of light-to-light transformation.

The same factors determine the difference between"active mediums" in the sources in question. Thepa.rametric. oscillator uses a nonlinear crystal speciallyonented wIth respect to the resonator's optical axis'this "active medium" contains no active centres. od thecontrary, lasers need no nonlinear mediums but thepresence of active centres in them is essential (thesystem of energy levels, concentration of centres thenature of their interaction with other particles ~f thelasant, etc.). In the case of the parametric oscillator themedium as a whole participates in the process, while in

Nonlinear Optics 188 19. Nonlinear Optics and Progress in Laser Technology

the laser the "participants" are active centres, withspecific quantum transitions occurring in each.

In contrast to the parametric oscillator, coherenceof the pumping wave of a laser is not a necessarycondition. Normally optical pumping involvesincoherent light. Furthermore, pumping is in manycases not optical at all. For instance, excitation ofactive centres of gas lasers is often realized by meansof inelastic collisions of particles in the plasma of gasdischarge. Obviously, in this case even a superficialsimilarity of lasers and parametric oscillatorsvanishes.

19. Nonlinear Opticsand Progressin Laser TechnologyThe best debt is the one paid back. Nonlinear

optics and laser technology are closely interdependentfields. On one hand, nonlinear optics could developonly on the basis of laser successes. On the otherhand, it is nonlinear optics that at present determinesto a great extent further potentials of laser technology."Born of the laser", nonlinear optics now opens newhorizons to the laser applications. Indeed, the best debtis the one paid back.

Phototropic gate discussed in Sec. 14 gives anexample of the effect of nonlinear optics on the laserfield. Several years ago giant light pulses were obtainedby various optical shutters of revolving type. Various"vanes" could be found in the laboratories - rotatingbeam choppers, rotating resonator mirrors or prisms,

189

Nonlinear Optics

and so on. Now such gates are replaced almostcompletely by more efficient devices, and among themby phototropic gates employing the effect of inducedtransparency of the medium. Advantages of suchdevices (and first of all their automatic operation) weredescribed in Sec. 14.

190 19. Nonlinear Optics and Progress in Laser Technology

Nonlinear optics serves to develop new types ofsources of coherent light. As we mentioned in Section 5,an important problem is the expansion of the rangealready "mastered" by coherent-light generators. One ofthe most promising methods of solving this problem isfound in making use of frequency converters ofnonlinear optics. Among them are

generators of optical harmonics, and first of all thesecond-harmonic generators (frequency doublers);

parametric light oscillators;oscillators for various operations of frequency

mixing (summation. subtraction. combining. and so on).All these devices are based on nonlmear optical

phenomena, that is on various coherent proce~"es oflight-to-light transformation.

As an example, let us consider two diagrams ofpractically employed methods of successive frequencyconversion. The first of them is shown in Fig. 76.A laser emits a coherent light wave with frequency v.This wave is used to pump the parametric oscillatorwhich emit~ radiation that can be continuously tunedin ~0me Irl"qJCncy range, from VI - ~v to VI + ~v.

The output of the parametric oscillator ~erves as thepumping wave for the second-harmonic oscillator; thefinal result i, a coherent ramation with frequencycontinuously tunable from 2Vl - 2~v to 2vI + 2~v.

Nonlinear Optics

The second diagram is shown in Fig. 77. Laseremits a wave with frequency v. After passing the wavethrough two second-harmonic oscillators we obtaina wave with frequency 4v, that is the fourth opticalharmonic. Mixing this harmonic with the initial signal(i. e. summing up 4v + v = 5v) we obtain the fifthoptical harmonic.

It must not be overlooked, however, thatconversion at each stage of the system is far fromtotal. Energy is inevitably lost at each of them, andthese losses determine practical value of each specificconfiguration of the system. Normally a system isconsidered acceptable if from 10 to 20% of theinitial-wave energy is converted to the secondary waveat each stage. In some cases the conversion coefficientwas substantially improved, up to 0.3-0.5 (and evenhigher).

The feature common for schematics of Figs. 76and 77 is that all instruments function independentlyand start operating on a sequential basis: first one,then the other, then the third. The laser as the sourceof the initial pumping wave is at the beginning of this"chain".

Although this principle of organization of theconversion appears to be quite logical, it is by nomeans optimal. If, for instance, the chain involves threesuccessive stages of conversion, then even for theconversion coefficient at each stage equal to 0.5 theultimate conversion coefficient will only be(0.5)3 = 0.125. The following question is thus in order:is it possible to drop the principle of sequential(successive) frequency conversion? Is there an essentially

192

,

19. Nonlinear Optics and Progress in Laser Technology

?..,.

?

L /) ~~~ ijl'l'----,I'~"0E l:

J::: "Eoil .----IL...&..---,

~

Fruitful idea. An essentially different approach doesexist. In fact, it has been described in this book in atleast three places. We want to remind about all three,as it is justified by the importance of theproblem.

Case 1. Utilization of a prism inside the lasercavity was analyzed in Sec. 4. In order to suppress theunwanted wave with frequency V2, it could be possibleto place the prism in the path of the beampropagating outside the cavity, and thus spatiallyseparate the waves with frequencies VI and V2' Thiswould mean, however, wasting the energy consumed forgeneration of an unwanted wave. It is expedienttherefore to place the prism inside the cavity and thusinfluence the process of light generation, that is makethe appearance of the wave with frequency V2

impossible.Case 2. It was explained in Sec. 4 why the end

faces of the laser tube are oriented at the Brewsterangle. It could be possible, in order to suppress theunwanted wave polarized normal to the "generationplane", to place a plane-parallel plate oriented at theBrewster angle across the beam emitted from theresonator. The plate would let through only the wavepolarized in the "generation plane", but the energyspent on generation of the unwanted component wouldbe lost. Therefore it is expedient to introduce the platetilted at the Brewster angle into the cavity and at thesame time make it the outlet window of thegas-discharge tube. This intrusion into the process of

What is the "internal generation of the secondharmonic"? The above idea may prove very fruitful forfrequency conversion of coherent light. This means thatharmonic generation). Then this wave is used to pumpharmonic generation, should be placed inside theresonator.

Let us have a look at Fig. 78. The figure comparestwo qualitatively different situations: a-secondharmonic generation with laser pumping, andh -- internal SHG.

In the case of Fig. 78a, first a coherent light waveat frequency V is obtained from the laser (firstthe nonlinear crystal meant, for instance, for the secondthe second harmonic oscillator, whereby part of thepumping energy is converted into a wave at frequency

generation damps out the excitation of a wave withpolarization normal to the "generation plane".

Case 3. The possibility of generating only onesecondary light wave was discussed in Sec. 18. Anunwanted component with frequency V - VI could beremoved by placing a Glan prism in the path of thelight beam leaving the resonator. But again this wouldmean the loss of energy spent on generation of theunwanted component. This is why the Glan prism isplaced inside the cavity which means intrusion into theprocess of parametric excitation and sUfJpression of thegeneration of a wave with frequency V - VI'

All the three examples provide a conclusivedemonstration of the basic idea: it is much better toinfluence the process of generation inside the resonatorthan to operate with the beam already out of theresonator.

19519. Nonlinear Optics and Progress in Laser Technology194

new approach substantial1y increasing the efficiency ofnonlinear frequency converters?

Nonlinear Optics

13*

Nonlinear Optics

2v, that is into the second harmonic. If we denote theconversion coefficient by k, then the second harmonicintensity will be kI where I stands for the firstharmonic intensity.

Figure 78b shows the case of internal generation ofthe second harmonic. In this configuration the wavewith frequency v may be totally absent. Photons withenergy hv are almost immediately involved into thethree-photon coherent processes giving rise to emissionof secondary photons with energy 2hv. Studiesdemonstrate that the intensity of the second harmonicemission may reach the level of the intensity I of thefirst harmonic which would be emitted if the nonlinearcrystal were removed from the laser resonator.

The next step is possible.... One could go evenfurther. Why stop at placing a lasant and a nonlinearcrystal in the same resonator and not to try and

Fig. 78

active nonlinear

resonat01tfmedium ~rew",,",~ B-frrystalresonatormIrror mlITor mIrror------- - - -

t t t(al I I

active nonlinearresonator medium crystal resonatormirror .....---......-=--.., mirror

(b)

196

I

II

i

19. Nonlinear Optics and Progress in Laser Technology

combine the two mediums into one? We meana nonlinear uniaxial crystal with specially added activecentres. Such a step may prove a logical extension ofthe previous development. It is too early yet to assessthe consequencies of such a step. Still, activatednonlinear crystals are being grown already.

We find therefore that nonlinear optics greatlyinfluences the laser field and opens ways of developingnew types of lasers. At the same time, the "laserfrequency range" is widened, the intensity of coherentemission enhanced, and its coherent propertiesimproved.

197

Historical Background

We are near the end of the story about the laserage in optics. To bring this story to a close, let usscan the history of the field: some more importantdates and names, the steps leading to the developmentof the laser, optical holography, and nonlinear optics.

The history of the laser. The word "laser" is anacronym of the words Light Amplification byStimulated Emission of Radiation. The very termreflects, therefore, the fundamental role of thestimulated emission in generators and amplifiers ofcoherent light. Our outline of the history must logicallystart with 1917 when A. Einstein introduced theconcept of stimulated emission.

This was the first step on the way to the laser.The next step was made by V. Fabrikant in the USSRwho indicated in 1939 that stimulated emission couldbe used for amplification of the electromagneticradiation passing through the medium. Fabrikant's ideaconsisted in using microscopic systems with inverted


population. After the end of World War ll, Fabrikantreturned to this idea and in 1951 applied (togetherwith M. Vudynsky and F. Butayeva) for the inventor'scertificate of a method of amplification ofelectromagnetic radiation by making use of thestimulated emission. The subject of invention asformulated in the certificate was: "A method ofamplification of electromagnetic radiation (in theultraviolet, visual, infrared, and radio frequency ranges)having as its distinct feature the transmission of theradiation to be amplified through the medium in whichan excessive, compared to the equilibrium,concentration of atoms or other particles and theirsystems at upper energy levels corresponding to theexcited states of the said medium is produced by anadditional radiation or by other means".

Originally, this method of amplification wasrealized in the radio frequency range (to be precise, inthe microwave range). The Soviet physicists N. Basovand A. Prokhorov presented a report to the USSRConference on Microwave Spectroscopy about thepossibility of developing, in principle, a microwaveamplifier. Basov and Prokhorov suggested the term"molecular generator" (a beam of ammonia moleculeswas to be employed). The suggestion to use stimulatedemission to amplify and generate microwave radiationcame at practically the same time from the Americanphysicist Ch. Townes of the Columbia University.A molecular generator later termed maser (MicrowaveAmplification by Stimulated Emission of Radiation) wasrealized in 1954. It was developed simultaneously andindependently in two points of the globe: in theLebedev Physics Institute of the USSR Academy ofSciences (by N. Basov and A. Prokhorov and their

199

co-workers) and in the Columbia University in theUSA (by Ch. Townes and his co-workers).

The term "laser" is a later derivation from "maser"[substitution of M (for Microwave) in the acronym byL (for Light)]. Both the maser and the laser are basedon the same principle formulated by Fabrikant in 1951.

The advent of the maser manifested that a newdirection is gaining strength in today's science andtechnology. The field was first called quantumradiophysics, and then quantum electronics.

Ten years later, in 1964, A. Prokhorov said on theceremony of presentation of Nobel Prize thateverybody anticipated the appearance of the quantumlight generators soon after masers had been developed.The prediction proved wrong, it took five to six yearsto develop the first laser. The reason for this lag, saidProkhorov, was two-fold: first, resonators (cavities) forthe optical frequency range were not yet proposed, andsecond, there were no realistic systems and techniquesin the optical range for realizing inverted population.

The five-to-six year span mentioned by Prokhorovwas indeed consumed by the research which in thelong run made it possible to bridge the gap betweenthe maser and the laser. In 1955 Basov and Prokhorovsubstantiated the application of optical pumping toachieve population inversion. In 1957 Basov suggestedthat semiconductors be used to develop quantum lightsources; he suggested, moreover, that speciallymachined surfaces of the semiconductor sample itself beused as the resonator walls. The same year Fabrikantand Butayeva recorded quantum amplification in theoptical range in experiments with electric discharge ina mixture of mercury vapor with small amounts ofhydrogen or helium. The next year Prokhorov, and

Historical Background 200

I

(

tI

Historical Background 201

quite independently A. Shavlov and Ch. Townes, gavea theoretical basis for the utilization of the stimulatedemission in the optical range. In 1959 N. Basav.B. Vul, and Yu. Popov published a paper theoreticallysubstantiating the idea of developing semiconductorquantum light sources (semiconductor lasers) andanalyzing a number of necessary conditions. Finally,N. Basov, O. Krokhin, and Yu. Popov publisheda lengthy paper in 1960 in which they discussed indetail the principles of developing quantum generatorsand amplifiers in the IR and visible ranges. Theauthors concluded: "There being no principalrestrictions, one can hope that generators andamplifiers in the IR and visible ranges will bedeveloped in the nearest future".

The intensive theoretical and experimental researcheffort in the USSR and USA has brought the scientiststo the very "brink" of the laser development by the~nd of the 50's. The American physicist T. Maimanwas the first to report the success. In August andSeptember 1960 he published communications in twoBritish journals on the achieved emission in the visiblerange. The world was thus informed about the birth ofthe first "optical maser", the ruby laser. The first laserlooked quite modest: a small ruby cube (1 x 1 x 1 cmin dimensions) with two opposite faces coated withsilver film (acting as cavity mirrors). The crystal wasperiodically illuminated with green light ofa high-power flash lamp which was wound, likea snake, into a helix around the ruby cube. Red lightpulses of the emitted radiation were let out througha small hole in the silver coating of one of the faces.

The same year the American physicists A. Javan,W. Bennett, and D. Herriot succeeded in achieving

Historical Back~rollnd 202 Historical Back~rollnd203

generation of optical radiation from the gas discharge(the active medium was a mixture of helium and neongases). This was the first gas laser whose appearancewas in fact prepared by the 1957 experimental resultsof Fabrikant and Butayeva.

Beginning with 1961, the laser (a solid state ora gas laser) becomes a permanent piece of equipmentin optics laboratories. New lasants are being developedand the laser design technology is improving. Firstsemiconductor lasers appear simultaneously in theUSSR and in the USA in 1962-63.

This was the beginning of the "laser age" in optics.

To the history of optical holography. It issignificant that the main ideas, principles, andtechniques of holography were suggested long beforethe laser was developed. The advent of the laser in the60's immediately provided a solid foundation to theholographic ideas and techniques; a new direction inoptics, optical holography, became a reality.

In principle, the idea of the holographic method ofimage formation was suggested and verifiedexperimentally in 1920 by the Polish physicistM. Wolfke. In his paper, "On the Possibility ofObtaining an Optical Image of a Molecular Lattice" hedemonstrated that X-ray diffraction patterns of a crystalcan be used to reconstruct an optical image of thiscrystal. Unfortunately, Wolfke's publication did not findany response from his contemporaries in physics.Wolfke remains a forerunner of holography who wasneither understood nor supported while he was active.

The ideas and principles of holography wereformulated anew in 1948 by the British physicist

1

D. Gabor (future Nobel Prize winner for hiscontribution) who at the time was not aware ofWolfke's results. Gabor's contribution to holograph~ isexceptionally large; the term holography was also hiSsuggestion. Incidentally, Gabor came to holographywhile working at a quite practical problem, ~amely, hewas developing methods of impr~ving resolutiOn. of theelectron microscope. Gabor mentiOned that the idea ofthe holographic method in electron ~icr?scopy asa two-stage process, in which an object is re~orded byan electron beam and is reconstructed by a hght beam,has appeared as a modification of an idea ?f thewell-known British physicist W Bragg. In hiS paper"X-Ray Microscope" Bragg presented in 1942 ~ met~odof visualizing the crystal lattice by means o~ diffractiOnof light on a diffraction pattern reco~ded With X-rays.

With the invention of the laser m 1960, theprinciples of holography were soon. used to developa serious and promising field of SCience. In 1961 theAmerican physicists E. Leith and J. Upatnieksdeveloped the widely used method of the two-?eamholography in which the reference laser beam IS usedtogether with the object bea~: .

In 1962 the Soviet physIcist Yu. Denzsyuk suggestedthe idea of the three-dimensional holograms based onthick photoemulsion layers. He also developed.a method of recording these holograms by usmg. the .back scattered object beam. Denisyuk's three-dImenSionalholograms reconstructed in the ordinary sun light area result of many years of research started long beforethe invention of the laser as well as before thefundamental work of Gabor. Denisyuk mentions thatfor him the starting point was the method of colourphotography developed as early as 1892 by the French

Historical Back/<round204 Historical Back/<round 205

physicist G. Lippmann. Later, in 1970, the sequence ofDenisyuk's papers under the general heading"Holography with Recording in a Three-dimensionalMedium" was conferred the Lenin Prize for Science.

Almost simultaneously a number of laboratories inthe USSR, USA, and England began to studyholographic interferometry and immediatelydemonstrated its high practical significance and greatpotentialities. Holography was maturing.

We want to emphasize that although theholographic principles were formulated long before thelaser age, the optical holography grew into a seriousscientific and technological field only owing to progressin the laser science. As Denisyuk notes, "Holographywithout lasers would remain an interesting principlewhich possibly could be applied to some specialproblems. The laser gave holography a new life, openedfor it numerous inroads into practical applications".

To the history of nonlinear optics. Two Sovietphysicists, S. Vavilov and V. Levshin, conducted in 1925a very interesting experiment. They observed inexperiments with high-intensity light (high-density sparkdischarge was used as the light source) a decrease inthe uranium glass absorption coefficient by 1.5%, withthe average experimental error of ± 0.3%. In fact, itwas the first ever observation of a nonlinear effect moptics. It was the observation of the light-inducedtransparency of the medium.

At that time, however, these experiments had nofuture. The appropriate equipment, and first of allhigh-intensity light sources, was lacking. The pre-laseroptics simply had nothing of the sort.

1

Despite this obstacle, Vavilov s~ent mu~h timepondering the possibiliti~s of an~lyzmg ~onl.mea.r ..phenomena in optics. HIs e~ceptlOnal sCientific mtmtIon,his ability to see far ahead mto the future weredemonstrated to the full in the book "Microstructure ofLight" published in 1950. The main problems ?f thenew field of optics and the approaches to so~vmg theseproblems were outlined in this monograph with .surprising insight and completenes~. The n~me given byVavilov to the new field was nonlmear optICS.

We shall not be very wrong to say that originallynonlinear optics was born "at the tip of a pen", or inthe scientist's mind. Vavilov did not live to see thelaser-age progress in nonlinear optics; he died in 1951,ten years before the laser was invented. .

The advent of the laser resulted in the rebirth ofnonlinear optics. P. Franken, the American physicis~,

observed in 1961 generation of the second harmomc o~

the ruby laser radiation in a quartz c~y~tal. An analysIsof his results enabled two Soviet physIcists,R. Khokhlov and S. Akhmanov, to formulate !n 1962the conditions of maximum efficiency for vanousnonlinear phenomena in optics (including ge?eration ofoptical harmonics); they h.ave ad.vanced the Idea of theparametric generation of lIght. Simultaneously, theAmerican physicists G. Giordmaine and R. Terhuneanalyzed the possibility of satisfying the wavesynchronism conditions in uniaxial crystals. .

In 1961 the Soviet physicist G. Askaryan predictedthe self-focusing effect; this prediction was confirm~d bythe experiments of N. Pilipetsky and S. Rustamov ~n

the USSR who were the first to observe self-focusmg oflight (in 1963). In 1964 the American physicist


M. Hercher was the first to report the superthin"filaments" in self-focused beams.

Fundamental theoretical work on nonlinear opticswas carried out in the period from 1961 to 1963 in theUSSR by R. Khokhlov and others, and in the USA byN. Bloembergen and his co-workers. In 1965 AkhmaRovand Khokhlov have published a fundamentalmonograph "Problems in Nonlinear Optics".

By 1965 nonlinear optics was established as anindependent powerful branch of the contemporaryoptics. Sufficiently effective generators of opticalharmonics and tunable parametric light oscillators werealready available.

206

TO THE READER

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Basic Concepts of Quantum Mechanics

by L. TARASOV, Cando Sc.

This book gives a detailed and systematic expositionof the fundamentals of non-relativistic quantummechanics for those who are not acquainted with thesubject. The character of the physics of micro-objectsand the problems of the physics of microprocesses(interference of amplifudes, the principleof superposition, the specific nature of measuringprocesses, casuality in quantum mechanics) areconsidered on the basis of concepts about probabilityamplitudes. Besides, the quantum mechanical systemsmicro-objects with two basic states are analyzedin detail. The apparatus of quantum mechanics isconsidered as a synthesis of concepts about physicsand the theory of linear operators. A numberof specially worked out problems and examples havebeen included in order to demonstrate the workingof the apparatus.This book is meant for use by students of engineeringand teachers-training institutes. It may also be usedby engineers of different profiles.

tG7 ? 00

Ex Libris

Amit Dhakulkar

Klan Nosferatu

Date:~ Iff Iz.ooo

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Laser Age in Optics [L._v_tarasov]