international atomicenergy agency

theabdus salaminternational centre for theoretical physics

SMR 1302- 5

WINTER SCHOOL ON LASER SPECTROSCOPY AND APPLICATIONS

19 February - 2 March 2001

MODERN SPECTROSCOPY AND INSTRUMENTATION

Parti

Wolfgang DEMTRODERUniv. Kaiserslautern - Fachbereich Physik

67653 Kaiserslautern - Germany

These are preliminary lecture notes, intended only for distribution to participants.

strada costiera, I I - 34014 trieste italy - tel.+39 04022401 I I fax +39 040224163 - sci_info@ictp.trieste.it - www.ictp.trieste.it

VIA o'

Spec {AVISO

T'

ICTP

This chapter is devoted to a discussion of instruments and techniques whichare of fundamental importance for the measurements of wavelengths andline profiles, or for the sensitive detection of radiation. The optimum selec-tion of proper equipment or the application of a new technique is oftendecisive for the success of an experimental investigation. Since the develop-ment of spectroscopic instrumentation has shown great progress in recentyears it is most important for any spectroscopist to be informed about thestate-of-the-art regarding sensitivity, spectral resolving power, and signal-to-noise ratios attainable with modern equipment.

At first we discuss the basic properties of spectrographs and monochro-mators. Although for many experiments in laser spectroscopy these instru-ments can be replaced by monochromatic tunable lasers (Chaps. 5-6) theyare still indispensible for the solution of quite a number of problems inspectroscopy.

Probably the most important instruments in laser spectroscopy are in-terferometers^ applicable in various modifications to numerous problems.We therefore treat these devices in somewhat more detail. Recently newtechniques of measuring laser wavelengths with high accuracy have beendeveloped; they are mainly based on interferometric devices. Because oftheir relevance in laser spectroscopy they will be discussed in a separatesection.

Great progress has also been achieved in the field of low-level signaldetection. Apart from new photomultipliers with an extended spectral sen-sivity range and large quantum efficiencies, new detection instruments havebeen developed such as image intensifiers, infrared detectors, or opticalmultichannel analyzers, which could escape from classified military re-search into the open market. For many spectroscopic applications theyprove to be extremely useful.

4.1 Spectrographs and Monochromators

Spectrographs were the first instruments for measuring wavelengths andthey still hold their position in spectroscopic laboratories, particularly whenequipped with modern accessories such as computerized microdensitometersor optical multichannel analyzers. Spectrographs are optical instruments

99

Fig.4.1. Prism spectrograph

S2(X,)

which form images S2(A) of the entrance slit Sx; the images are laterallyseparated for different wavelengths A of the incident radiation (Fig.2.10).This lateral dispersion is due to either spectral dispersion in prisms or dif-fraction on plane or concave reflection gratings.

Figure 4.1 depicts the schematic arrangement of optical components ina prism spectrograph. The light source L illuminates the entrance slit Sxwhich is placed in the focal plane of the collimator lens hv Behind Lx theparallel light beam passes through the prism P where it is diffracted by anangle 0(A) depending on the wavelength A. The camera lens L2 forms animage S2(A) of the entrance slit Sx. The position x(A) of this image in thefocal plane of L2 is a function of the wavelength A. The linear dispersiondx/dA of the spectrograph depends on the spectral dispersion dh/dA of theprism material and on the focal length of L2.

When a reflecting diffraction grating is used to separate the spectrallines S2(A), the two lenses Lx and L2 are commonly replaced by two spheri-cal mirrors Mx and M2 which image the entrance slit onto the plane of ob-servation (Fig.4.2). Both systems can use either photographic or photoelec-tric recording. According to the kind of detection we distinguish betweenspectrographs and monochromators.

In spectrographs a photoplate or a CCD diode array is placed in thefocal plane of L2 or M2. The whole spectral range AA = A1(x1)-A2(x2) cov-ered by the lateral extension Ax = Xi-x2 of the photoplate can be simul-

Fig.4.2. Grating monochromator

100

taneously recorded. If the exposure of the plate remains within the linearpart of the photographic density range, the density Da(x) of the developedphotoplate at the position x(A)

Da(x) = C(A) | I(A)dt0I, (4.1)

is proportional to the spectral irradiance I(A), as received in the focal planeB, and integrated over the exposure time T. The sensitivity factor C(A) de-pends on the wavelength A, furthermore on the developing procedure andthe history of the photoplate (presensitizing). The photoplate can accumu-late the incident radiant power over long periods (up to 50h). In astrophy-sics, for instance, presensitized photographic plates are still the often-useddetectors for spectrally dispersed radiation from distant stars or galaxies.Photographic detection can be employed for both pulsed and CW lightsources. The spectral range is limited by the spectral sensitivity of availablephotoplates and covers the region between about 200 f 1000 nm. The devel-opment of sensitive CCD cameras and optical multichannel analysers whichcombine the advantages of fast photoelectric recording and signal integra-tion (Sect.4.6) has displaced photoplates for many applications.

Monochromators, on the other hand, use photoelectric recording of aselected small spectral interval. An exit slit S2, selecting an interval Ax2 inthe focal plane B, lets only the limited range AA through to the photoelec-tric detector. Different spectral ranges can be detected by shifting S2 in thex direction. A more convenient solution (which is also easier to construct)turns the prism or grating by a gear-box drive, which allows the differentspectral regions to be tuned across the fixed exit slit S2. Modern devicesuses a direct drive of the grating axis by step motors and measure the turn-ing angle by electronic angle decoders. This avoids backlush of the drivinggear. Unlike the spectrograph, different spectral regions are detected notsimultaneously but successively. The signal received by the detector is pro-portional to the product of the exit-slit area hAx2 with the spectral inten-sity Jl(A)dA, where the integration extends over the spectral range dispersedwithin the width Ax2 of S2.

Whereas the spectrograph allows the simultaneous measurement of alarge region with moderate time resolution, photoelectric detection allowshigh time resolution but permits, for a given spectral resolution, only asmall wavelength interval AA to be measured at a time. With integrationtimes below some minutes photoelectric recording shows a higher sensitiv-ity, while for very long detection times of several hours, photoplates maystill be more convenient, although nowadays cooled CCD-arrays allow in-tegration times up to one hour.

In spectroscopic literature the name spectrometer is often used for bothtypes of instruments.

We now discuss the basic properties of spectrometers, relevant for laserspectroscopy. For a more detailed treatment see for instance [4.1-8].

101

lens

L.S

Fig.4.3. Light-gathering power of a spectrometer

4.1.1 Basic Properties

The selection of the optimum type of spectrometer for a particular experi-ment is guided by some basic characteristics of spectrometers and theirrelevance to the particular application. The basic properties which are im-portant for all dispersive optical instruments may be listed as follows:

a) Speed of a Spectrometer

When the spectral intensity 1̂ within the solid angle df! = 1 sterad is in-cident on the entrance slit of area A, a spectrometer with an acceptanceangle H transmits the radiant flux within the spectral interval dA

IA*(A/As)T(A)HdA (4.2)

where Aa > A is the area of the source image at the entrance slit (Fig.4.3)and T(A) the transmission of the spectrometer.

The product U = AH is often named etendue. For the prism spectro-graph the maximum solid angle of acceptance, Q = F / ^ 2 , is limited by theeffective area F = hD of the prism, which represents the limiting aperturewith height h and width D for the light beam (Fig.4.1). For the gratingspectrometer the sizes of the grating and mirrors limit the acceptance solidangle 0.

Example 4.1For a prism with height h = 6 cm, D = 6 cm, fj = 30 cm —• D/f =1:5 and Q= 0.04 sterad. With an entrance slit of 5x0.1 mm2 the etendue is U =5-10"3-4-10"2 = 2-10"4 cm2 -sterad.

In order to utilize the optimum speed, it is advantageous to image thelight source onto the entrance slit in such a way that the acceptance angle Ois fully used (Fig.4.4). Although more radiant power from an extendedsource can pass the entrance slit by using a converging lens to reduce thesource image on the entrance slit, the divergence is increased. The radiationoutside the acceptance angle H cannot be detected but may increase thebackground by scattering from lens holders and spectrometer walls.

102

Fig.4.4. Optimized imaging of a light source onto the entrance slit of a spectrometer isachieved when the solid angle ft' of the incoming light matches the acceptance angle ft« (a/d)2 of the spectrometer

Often the wavelength of lasers is measured with a spectrometer. In thiscase it is not recommendable to direct the laser beam directly onto theentrance slit, because the prism or grating would be not uniformely illumi-nated. This decreases the spectral resolution. Furthermore, the symmetry ofthe optical path with respect to the spectrometer axis is not guaranteed withsuch an arrangement resulting in systematic errors of wavelengths measure-ments. It is better to illuminate a groundglass plate with the laser and to usethe incoherently scattered laser light as a secondary source, which is imagedin the usual way (Fig.4.5).

Fig.4.5. (a) Imaging of an extended light source onto the entrance slit of a spectrome-ter with ft* = ft. (b) Correct imaging optics for laser wavelength measurements with aspectrometer. The laser light, scattered by the ground glass forms the source which isimaged onto the entrance slit

103

0.4Bor.Sil.Glas3.5I 1

,0.15 Quartz 4.5 ,

.Q.U Sapphire 6.5.

0.11 Mg

0.12 Li F

,0.13 CaF 12.

,0.2 NaCl 26

0.2 Cs I

0.25 Diamond

•1.8 Germanium 23

80.

80,

1

0.8

0.6

OM

0.2

1

0.8

0.6

0A

0.2

MgF2Na Cl

0.2 0.3 0.4 0.5

Na Cl

/ j m

0.1 0.5 1 5 10 50 100 jum 2 U 6 8 10 jum

a) b)Fig.4.6. (a) Useful spectral ranges of different optical materials and (b) transmittanceof different materials with thicknesses 1 cm [4.5b]

b) Spectral Transmission

For prism spectrometers, the spectral transmission depends on the materialof the prism and the lenses. Using fused quartz the accessible spectral rangespans from about 180 to 3000 nm. Below 180 nm (vacuum-ultraviolet re-gion) the whole spectrograph has to be evacuated, and lithium fluoride orcalcium fluoride has to be used for the prism and the lenses, although mostVUV spectrometers are equipped with reflection gratings and mirrors.

In the infrared region, several materials (for example, CaF2 NaCl, andKBr crystals) are transparent up to 30 jum (Fig.4.6). However, because ofthe high reflectivity of metallic, coated mirrors and gratings in the infraredregion, grating spectrometers with mirrors are preferred rather than prismspectrographs.

Many vibrational-rotational transitions of molecules such as H2O orCO2 fall within the range 3rl0 /xm causing selective absorption of thetransmitted radiation. Infrared spectrometers therefore have to be eitherevacuated or filled with dry nitrogen. Because dispersion and absorption areclosely related, prism materials with low absorption losses also show lowdispersion, resulting in a limited resolving power (see below).

Since the ruling or holographic production of high-quality gratings hasnowadays reached a high technological standard, most spectrometers usedtoday are equipped with diffraction gratings rather than prisms. The spec-tral transmission of grating spectrometers reaches from the VUV region in-to the far infrared. The design and the coatings of the optical componentsand the geometry of the optical arrangement are optimized according to thespecified wavelength region.

104

•VA2/ tion of two nearly overlapping lines

c) Spectral Resolving Power

The spectral resolving power of any dispersing instrument is defined by theexpression

R = |A/AA| = |i//Ai/| (4.3)

where AA = Ax-A2 stands for the minimum separation of the central wave-lengths Xx and A2 of two closely spaced lines which are considered.to bejust resolved. It is possible to recognize that an intensity distribution iscomposed of two lines with the intensity profiles ^(A-Aj) and I2(A-A2) ifthe total intensity I(A) = ^(A-Aj) + I2(A-A2) shows a pronounced dip be-tween two maxima (Fig.4.7). The intensity distribution I(A) depends, ofcourse, on the ratio li/l2

anc* o n ^ e profiles of both components. There-fore, the minimum resolvable interval AA will differ for different profiles.

Lord Rayleigh has introduced a criterion of resolution for diffractionlimited line profiles, where two lines are considered to be just resolved ifthe central diffraction maximum of the profile ^(A-Aj) coincides with thefirst minimum of I2(A-A2) [4.3].

Let us consider the attainable spectral resolving power of a spectrome-ter. When passing the dispersing element (prism or grating), a parallel beamcomposed of two monochromatic waves with wavelengths A and A+AA issplit into two partial beams with the angular deviations 0 and 0+A0 fromtheir initial direction (Fig.4.8). The angular separation is

= (d0/dA)AA, (4.4)

dispersingelement

L2 Fig.4.8. Angular dispersion of a parallel beam

105

with focal length f2 images the entrance slit Sx into the plane B (hig.4.1)the distance Ax2 between the two images S2(A) and S2(A+AA) is, accordingto Fig.4.8,

Ax9 = f9 A0 = f9 -rrAA = -7T-2 2 2 dA dA(4.5)

The factor dx/dA is called linear dispersion of the instrument. It is generallymeasured in mm/A. In order to resolve two lines at A and A+AA, the sepa-ration Ax2 in (4.5) has to be at least the sum £x2(A) + 6x2(A+AA) of thewidths of the two slit images. Since the width 8x2 is related to the width 8xxof the entrance slit according to geometrical optics by

£x2'= (f2/f \)8x1 , (4.6)

the resolving power A/AA can be increased by decreasing 8xx. Unfortu-nately there is a theoretical limitation set by diffraction. Because of thefundamental importance of this resolution limit we discuss this point inmore detail.

When a parallel light beam passes a limiting aperture with diameter a,a Fraunhofer diffraction pattern is produced in the plane of the focussinglens L2 (Fig.4.9). The intensity distribution 1(0) as a function of the angle with the optical axis of the system is given by the well-known formula [4.3]

sin(a7rsin = 0. The central maximumcontains about 90% of the total intensity.

a) b)

Fig.4.9. (a) Diffraction in a spectrometer by the limiting aperture with diameter a. (b)Limitation of spectral resolution by diffraction

106

age ot wiatn

(4.8)

defined as the distance between the central diffraction maximum and thefirst minimum which is approximately equal to the FWHM of the centralmaximum.

According to the Rayleigh criterion two equally intense spectral lineswith wavelengths A and A+AA are just resolved if the central diffractionmaximum of S2(A) coincides with the first minimum of S2(A+AA) (seeabove). From (4.7) one can compute that in this case both lines partly over-lap with a dip of (8/7T2)I 0.8Imax between the two maxima. The dis-tance between the centers of the two slit images is from (4.8) (Fig.4.9b)

Ax2 = f2(A/a) . (4.9a)

With (4.5) one therefore obtains the fundamental limit on the resolvingpower,

|A/AA| < a(d0/dA) , (4.10)

which clearly depends only on the size a of the limiting aperture and on theangular dispersion of the instrument.

For a finite entrance slit with width b the separation Ax2 between thecentral peaks of the two images I(A-AX) and I(A-A2) must be

Ax9 > fo — + b 7— (4.9b)

in order to meet the Rayleigh criterion (Fig.4.10). With Ax2 = f2(d0/dA)AAthe smallest resolvable wavelength intervall AA is then

AA; (4.11)

Note, that the spectral resolution is limited, not by the diffraction due tothe entrance slit, but by the diffraction caused by the much larger aperturea, determined by the size of the prism or grating.

Although it does not influence the spectral resolution, the much largerdiffraction by the entrance slit imposes a limitation on the transmitted in-tensity at small slit widths. This can be seen as follows. When illuminatedwith parallel light, the entrance slit with width b produces a Fraunhoferdiffraction pattern analogous to (4.7) with a replaced by b. The central dif-fraction maximum extends between the angles 8 = ±A/b (Fig.4.11) and cancompletely pass the limiting aperture a only if 28 is smaller than the ac-

107

k/kfaAx2

Fig.4.10. Intensity profiles of two monochromatic lines measured with an entrance slitwidth b and a magnification factor f2/fx. Solid line: without diffraction; dashed line:with diffraction. The minimum resolvable distance between the line centers is Ax2 =f2(b/f1+A/a)

Fig.4.11. Diffraction by the entrance slit

ceptance angle a/fx of the spectrometer. This imposes a lower limit to theuseful width bm i n of the entrance slit,

bmin^Af i / a . (4.12)

In all practical cases, the incident light is divergent, which demands that thesum of divergence angle and diffraction angle has to be smaller than a/fand the minimum slit width b correspondingly larger.

Figure 4.12a illustrates the intensity distribution I(x) in the plane B fordifferent slit widths b. Figure 4.12b shows the dependence of the widthAx2(b) of the slit image S2 on the slit width b taking into account the dif-fraction caused by the aperture a. This demonstrates that the resolutioncannot be increased much by decreasing b below b ^ . The peak intensityI(b)x=0 is plotted in Fig.4.12c as a function of the slit width. According to

108

b) 5 vdiffraction

without

2f2 • X/ab/bm

Fig.4.12. (a) Diffraction limited intensity distribution I(x2) in the plane B for differ-ent widths b of the entrance slit, (b) The width £x2(b) of the entrance slit image S2(x2)including diffraction by the aperture a. (c) Intensity I(x2) as a function of entrance slitwidth for a spectral continuum and for a monochromatic spectral line (ni) with dif-fraction (solid curves) and without diffraction (dashed curves)

(4.2) the transmitted radiation flux (A) depends on the product U = AQ ofentrance slit area A and acceptance angle Q = (a/fx)

2. The flux in B wouldtherefore depend linearly on the slit width b, if diffraction were not pre-sent. This means that for monochromatic radiation the peak intensity[W/m2] in the plane B should then be constant (curve lm), while for aspectral continuum it should decrease linearly with decreasing slit width(curve lc). Because of the diffraction by Sj the intensity decreases with theslit width b both for monochromatic radiation (2m) and for a spectral con-tinuum (2c). Note the steep decrease for b < bmin.

Substituting b = bmin = 2fA/a into (4.11) yields the practical limit forAA imposed by diffraction by Sx and by the limiting aperture with width a

AA = 3f(A/a)dA/dx (4.13)

Instead of the theoretical limit (4.10) given by the diffraction through theaperture a, a smaller practically attainable resolving power is obtained from(4.13) which takes into account a finite minimum entrance slit width bminimposed by intensity consideration and which yields:

R = A/AA = (a/3)d0/dA . (4.14)

Example 4.2For a = 10 cm, A = 510"5 cm, f = 100 cm, dA/dx = 10 A/mm, with b = 10/im, —• AA = 0.15 A, with b = 5 /im, —>AA = 0.10 A. However, fromFig.4.12, one can see that the transmitted intensity with b = 5 /zm is only25% of that with b = 10 /mi.

Note: For photographic detection of line spectra, it is actually better to usethe lower limit bn^n for the width of the entrance slit, because the density

109

of the developed photographic layer depends only on the spectral irradiance[W/m2] rather than on the radiation power [W]. Increasing the slit widthbeyond the diffraction limit bm i n , in fact does not significantly increase thedensity contrast on the plate, but does decrease the spectral resolution.

Using photoelectric recording, the detected signal depends on the radi-ation power 0^dA transmitted through the spectrometer and therefore incre-ases with increasing slit width. In the case of completely resolved line spec-tra, this increase is proportional to the slit width b since x oc b. For contin-uous spectra it is even proportional to b2 because the transmitted spectralinterval dA also increases proportional to b and therefore f2 from M2. In this arrangement [4.9] the grating is illu-minated with slightly divergent light.

Fig.4.13. Curvature of the image of a straight entrance slit, due to astig-matic imaging errors

110

P(x2)Fig.4.14. Signal profile P(t) aP(x2) at the exit slit of a mono-chromator with b » bmin and^2 < (f2/fi)bi f° r monochro-matic incident light with uni-form turning of the grating

Image of S!

When the spectrometer is used as a monochromator with an entranceslit width bx and an exit slit width b2 , the power P(t) recorded as a func-tion of time while the grating is uniformly turned, has a trapezoidal shapefor bx » bm i n (Fig.4.14a) with a base line ( ^ / f ^ b j + b2. Optimum resolu-tion at maximum transmitted power is achieved for b2 = (fg/fjjbj. Theline profile P(t) = P(x2) then becomes a triangle.

d) Free Spectral Range

The free spectral range of a spectrometer is the wavelength interval 8X ofthe incident radiation for which a one-valued relation exists between A andthe position x(A) of the entrance-slit image. While for prism spectrometersthe free spectral range covers the whole region of normal dispersion of theprism material, for grating spectrometers 8X is determined by the diffrac-tion order m and decreases with increasing m (Sect.4.1.3).

Interferometers, which are generally used in very high orders (m =104-108), have a high spectral resolution but a small free spectral range.For unambiguous wavelength determination they need a preselector, whichallows one to measure the wavelength within 8X of the high-resolution in-strument (Sect.4.2.3).

4.1.2 Prism Spectrometer

When passing through a prism, a light ray is refracted by an angle $ whichdepends on the prism angle e, the angle of incidence al9 and the refractiveindex n of the prism material (Fig.4.15). The minimum deviation 0 is ob-tained when the ray passes the prism parallel to the base g (symmetricalarrangement with ax = a2 = a). In this case one can derive [4.5]

sin(0 + £) . , ...v—K——L = nsin(€/2) . (4.15)

From (4.15) the derivation d0/dn = (dn/dO)'1 is

d$ _ 2sin(€/2)dn cos[(0 + c)/2]

2sin(e/2)

- n2sin2sin2(6/2)(4.16)

111

prism at minimum deviation where= a9 = a and 0 = 2a-e

The angular dispersion dO/dX = (d0/dn)(dn/dA) is therefore

d$ 2sin(6/2) dn

- n2sin2(£/2)(4.17)

This shows that the angular dispersion increases with the prism angle e butdoes not depend on the size of the prism.

For the deviation of laser beams with small beam diameters smallprisms can therefore be used without losing angular dispersion. In a prismspectrometer, however, the size of the prism determines the limiting aper-ture a and therefore the diffraction; it has to be large in order to achieve alarge spectral, resolving power (see previous section). For a given angulardispersion, an equilateral prism with e = 60° uses the smallest quantity ofprism material (which might be quite expensive). Because sin30° = 1/2,(4.17) then reduces to

dn/dA (4.18)dA Vl - (n/2)2

The resolving power A/AA is according to (4.10)

A/AA < a(d0/dA) .

The diameter a of the limiting aperture in a prism spectrometer is (Fig.4.16)

gcosa= 2^72) •

Substituting d0/dA from (4.17) gives

A/AA dnV\ - n2 sin2 (6/2) d A

(4.19)

(4.20)

At minimum deviation, (4.15) gives nsin(€/2) = sin(0+€)/2 = sinax andtherefore (4.20) reduces to

A/AA = g(dn/dA) . (4.21a)

112

According to (4.21a), the theoretical maximum resolving power dependssolely on the base length g and on the spectral dispersion of the prism mat-erial. Because of the finite slit width b > bm i n the resolution, reached inpractice, is somewhat lower and the corresponding resolving power can bederived from (4.12) to be at most

(4.21b)

The spectral dispersion dn/dA is a function of prism material andwavelength A. Figure 4.17 shows dispersion curves n(A) for some materialscommonly used for prisms. Since the refractive index increases rapidly inthe vicinity of absorption lines, glass has a larger disperison in the visibleand near-ultraviolet regions than quartz which, on the other hand, can beused advantageously in the UV down to 180 nm. In the vacuum-ultravioletrange CaF, MgF, or LiF prisms are sufficiently transparent. Table 4.1 givesa summary of the optical characteristics and useful spectral ranges of someprism materials.

If achromatic lenses (which are expensive in the infrared and ultravi-olet region) are not employed, the focal length of the two lenses decreaseswith the wavelength. This can be partly compensated by inclining the plane

n2

1.8

1.6

U

1.2

heaviest flint

fluorite

15 200 300 400 500 600 700 800 \LnmJ

Fig.4.17. Refractive index n(A) for some prism materials

113

Table 4.1. Refractive index and dispersion ot some materials usea in prism spectro-meters

Material Useful spectralrange [/im]

Refractiveindex n

Dispersion-dn/dA [nm-

Glass (BK7)

Heavy flint

Fused quartz

NaCl .

LiF

0.35

0.4-

0.15

0.2-

0.12

- 3.5

•2

4.5

26

- 9

.516

.53

.755

.81

.458

.470

.79

.38

.44

.09

4.6-10-51.110-41.410-44.4-10-43.4-10-51.110-46.3-10"3

1.710-56.6 10-48.6-10-5

atatatatatatatatatat

589 nm400 nm589 nm400 nm589 nm400 nm200 nm20 /im

200 nm10/im

B against the principal axis in order to bring it at least approximately intothe focal plane of L2 for a large wavelength range (Fig.4.1).

In Summary: The advantage of a prism spectrometer is- the unambiguousassignment of wavelengths, since the position S2(A) is a monotonic functionof A. Its drawback is the moderate spectral resolution. It is mostly used forsurvey scans of extended spectral regions.

Example 4.3a) Suprasil (fused quartz) has a refractive index n = 1.47 at A = 400 nm anddn/dA = 1100 cm"1. This gives d0/dA = 1.6-10'4 rad/nm.b) For heavy flint glass at 400 nm n = 1.81 and dn/dA = 4400 cm"1, giving69/d\ = 1.0-10"3 rad/nm. This is about 6 times larger than that for quartz.With a focal length f = 100 cm for the camera lens one achieves a lineardispersion dx/dA = 0.1 mm/A with a flint prism, but only 0.015 mm/A witha quartz prism.

4.1.3 Grating Spectrometer

In a grating spectrometer (Fig.4.2) the collimating lens Lx is replaced by aspherical mirror Mx with the entrance slit Sx in the focal plane of Mx. Thecollimated, parallel light is reflected by Mx onto a reflection grating con-sisting of many straight grooves (about 105) parallel to the entrance slit. Thegrooves have been ruled onto an optically smooth glass substrate or havebeen produced by holographic techniques [4.10-15]. The whole grating sur-

114

A r « A . / d

incidentbeam

grating normal

d • sin

A s = d (sin a- sin p)a) b)

Fig.4.18. (a) Reflection of incident light from a single groove into the diffractionangle A/d around the specular reflection angle r = i. (b) Illustration of the gratingequation (4.22) B

face is coated with a highly reflecting layer (metal or dielectric film). Thelight reflected from the grating is focussed by the spherical mirror M2 ontothe exit slit S2 or onto a photographic plate in the focal plane of M2.

The many grooves, which are illuminated coherently, can be regardedas small radiation sources, each of them diffracting the light incident ontothis small groove with a width of about one wavelength A, into a largerange of angles r around the direction of geometrical reflection (Fig.4.18a).The total reflected light consists of a coherent superposition of these manypartial contributions. Only in those directions where all partial waves, emit-ted from the different grooves, are in phase will constructive interferenceresult in a large total intensity, while in all other directions the differentcontributions cancel by destructive interference.

Figure 4.18b depicts a parallel light beam incident onto two adjacentgrooves. At an angle of incidence a to the grating normal (which is normalto the grating surface, but not necessary to the grooves) one obtains con-structive interference for those directions fi of the reflected light for which

d(sina ± sin/9) = mA , (4.22)

the plus sign has to be taken if fi and a are on the same side of the gratingnormal; otherwise the minus sign, which is the case shown in Fig.4.18b.

The reflectivity R(/?,0) of a ruled grating depends on the diffractionangle fi and on the blaze 6 of the grating, which is the angle between thegroove normal and the grating normal (Fig.4.19). If the diffraction angle ficoincides with the angle r of specular reflection from the groove surfaces,R(fi,0) reaches its optimum value Ro, which depends on the reflectivity ofthe groove coating. From Fig.4.19 one infers i = a-6 and r = 6+p whichyields for specular reflection i = r the condition for the optimum blazeangle 0

= (a- fi)/2 . (4.23)

115

groovenormal

grating normal

A s = 2d sin a= m Xa) b)

Fig.4.20. (a) Littrow mount of a grating with ft = a. (b) Illustration of blaze angle fora Littrow grating

Because of the diffraction of each partial wave into a large angular rangethe reflectivity R(P) will not have a sharp maximum at p = a-29 but willrather show a broad distribution around this optimum angle. The angle ofincidence a is determined by the particular construction of the spectrometerand the angle p for which constructive interference occurs depends on thewavelength A. Therefore the blaze angle 0 has to be specified for the desiredspectral range and the spectrometer type.

In laser-spectroscopic applications the case a = p often occurs, whichmeans that the light is reflected back into the direction of the incident light.For such an arrangement, called a Littrow-grating mount (shown in Fig.4.20), the grating equation (4.22) for constructive interference reduces to

2dsina = mA . (4.22a)

Maximum reflectivity of the Littrow grating is achieved for i = r = 0 —> 6 =a (Fig.4.20b). The Littrow grating acts as a wavelength-selective reflectorbecause light is only reflected if the incident wavelength satisfies the con-dition (4.22a).

116

when a monochromatic plane wave is incident onto an arbitrary grating.The path difference between partial waves reflected by adjacent

grooves is As = d(sina±sin/?) and the corresponding phase difference is

27T 27T

(/> = -T~As = — d(sina ± sin/?) . (4.24)

The superposition of the amplitudes reflected from all N grooves in thedirection p gives the total reflected amplitude

N-l

AR,VR (4.25)m=0

where R(p) is the reflectivity of the grating, which depends on the reflec-tion angle p, and Ag is the amplitude of the partial wave incident onto eachgroove. Because the intensity of the reflected wave is related to its ampli-tude by IR = eocARAR, see (2.30c), we find from (4.25)

siri'W/2)C60 A g A g (4.26)

This intensity distribution is plotted in Fig.4.21 for two differentvalues of the total groove number N. The principal maxima occur for (/> =2m7T, which is, according to (4.24), equivalent to the grating equation (4.22)and means that at a fixed angle a the path difference between partial beamsfrom adjacent grooves is for certain angles Pm an integer multiple of thewavelength, the integer m being called the order of the interference. Thefunction (4.26) has (N- l ) minima with IR = 0 between to successive princi-pal maxima. These minima occur at values of $ for which N0/2 = &r, t = 1,2..., N - l , and mean that for each groove of the grating another one can befound which emits light into the direction p with a phase shift TT, such thatall pairs of partial waves just cancel.

The line profile I(/3) of the principal maximum of order m around thediffraction angle Pm can be derived from (4.26) by substituting p = /?m+€.Because for large N, l(P) is very sharply centered around Pm, we can as-sume e « pm. With the relation

sin(/?m + e) = sin/?mcos€ + cos/?msine ~ sin/9m + £cos/?m ,

we obtain from (4.24) because of (27rd/A)(sino: + sin/5m) = 2m?r

cf)(p) = 2m7r + 27r(d/A)6COS/?m = 2m?r + 81 with 6X « 1 (4.27)

and (4.26) can be written as

117

16

N = 20

AI I IAAAAAAAAAI I IAAAAAAAAAI I IA / ^

-2TC 0

| 4 I R

IAAAI

N = 5

IAAAI+ 2 71

Jt/(Nd)

Fig.4.21. Intensity distribution I(/?) for two different numbers N of illuminatedgrooves. Note the different scale of the ordinate!

Ht - R I 0 " R 1 0 Nsin2(N51/2)

( N V 2 ) 2 '(4.28)

with 6X = 27r(d/A)ecos/3m.The first two minima on both sides of the central maximum at fim are

at

= ± 2TT±A

Ndcos/9n(4.29)

The central maximum of m th order therefore has a line profile (4.28)with a base halfwidth A/3 = A/(Ndcos/3m). This corresponds to a diffractionpattern produced by an aperture with width b = Nd cos/3m, which is justthe size of the whole grating projected onto a direction normal to /3m (Fig.4.18).

Note that according to (4.28) the angular halfwidth A/? = 2e of the interfer-ence maxima decreases as 1/N while the peak intensity increases oc N2 withan increasing number N of illuminated grooves. The area under the mainmaxima is therefore proportional to N which is due to the increasing con-centration of light into the directions /3m.

The intensity of the N-2 small side maxima, which are caused by in-complete destructive interference, decreases proportional to 1/N with in-

118

20. For gratings used in practical spectroscopy, with groove numbers ofabout 105, the reflected intensity IR(A) at a given wavelength A has verysharply defined maxima only in those directions pm, as defined by (4.22).The small side maxima are completely negligible at such large values of N,provided the distance d between the grooves is exactly constant over thewhole grating area.

Differentiating the grating equation (4.22) with respect to A we obtainat a given angle a the angular dispersion

d/3 = mdA d cos/9 '

Substituting from (4.22) (m/d) = (sina±sin/3)/A, we find

d/3 _ sina ± sin/3dA "

(4.30a)

(4.30b)

This illustrates that the angular dispersion is determined solely by the anglesa and fi and not by the number of grooves* For the Littrow mount with a =/?, we obtain

dp/'dA = 2 tana/A . (4.30c)

The resolving power can be immediately derived from (4.30a) and the basewidth A£ = 2e = 2A/(Ndcos/3) of the principal diffraction maximum (4.29)if we apply the Rayleigh criterion (see above) that -two lines A and A+AAare just resolved when the maximum of I(A) fallsiinto the adjacent mini-mum for I(A+AA). This is equivalent to the condition

or(d£/dA)AA = A/(Nd cos/9) ,

A Nd(sina ± sin/3)AA " A

which reduces with (4.22) to

(4.31)

AAA (4.32)

The theoretical spectral resolving power is the product of the diffractionorder m with the total number N of illuminated grooves. If the finite slitwidth bj and the diffraction at limiting aperatures are taken into account,the practically achievable resolving power is according to (4.14) about 3times lower.

119

2) which increases the spectral resolution by a factor 2 without losing muchintensity, if the blaze angle 9 is correctly choosen to satisfy (4.22 and 23)with m = 2.

Example 4.4A grating with a ruled area of 10x10 cm2 and 103 grooves/mm allows insecond order (m=2) a theoretical spectral resolution of R = 2-105. Thismeans that at A = 500 nm two lines which are separated by AA = 2.5-10~3

nm = 0.025 A should be resolvable. The dispersion is for a = p = 30° and afocal length f = 1 m: dx/dA = fd/?/dA = 0.2 mm/A. With a slit width bx =b2 = 50 /xm a spectral resolution of AA = 0.25 A can be achieved. Linesaround A = 1 /xm in the spectrum would appear in 1st order at the sameangles p. They have to be suppressed by filters.

A special design is the so-called echelle grating, which has very widelyspaced grooves forming right-angled steps (Fig.4.22). The light is incidentnormal to the small side of the grooves. The path difference between tworeflected partial beams incident on two adjacent grooves with an angle ofincidence a = 90° -9 is As = 2dcos0 and the grating equation (4.22) gives forthe angle p of the m th diffraction order

d(cos0 + sin/3) = mA , (4.33)

where p is close to a = 90° -0.With d » A the grating is used in a very high order (m ~ 10T100) and

the resolving power is very high according to (4.32). Because of the largerdistance d between the grooves, the relative ruling accuracy is higher andlarge gratings (up to 30 cm) can be ruled. The disadvantage of the echelle isthe small free spectral range 8\ = A/m between successive diffractionorders.

Example 4.5N = 3 104, d = 10 /xm, 9 = 30°, A = 500 nm, m = 34. The spectral resolvingpower is R = 106, but the free spectral range is only SX = 15 nm. Thismeans that the wavelengths A and A+5A overlap in the same direction p.

grating normal

reflectivecoating

Fig.4.22. Echelle grating

120

by inaccuracies during the ruling process, may result in constructive in-terference from parts of the grating for "wrong" wavelengths. Such un-wanted maxima, which occur for a given angle of incidence a into "wrong"directions p, are called grating ghosts. Although the intensity of theseghosts is generally very small, intense incident radiation at a wavelength X{may cause ghosts with intensities comparable to those of other weak lines inthe spectrum. This problem is particularly serious in laser spectroscopywhen the intense light at the laser wavelength, which is scattered by cellwalls or windows, reaches the entrance slit of the monochromator.

In order to illustrate the problematics of achieving that ruling accuracywhich is required to avoid these ghosts, let us assume that the carriage ofthe ruling engine expands by only 0.1 /xm during the ruling of a 10x10 cm2

grating, e.g., due to temperature drifts. The groove distance d in the secondhalf of the grating differs then from that of the first half by 10"5d. With N= 105 grooves the waves from the second half are then completely out ofphase with those from the first half. The condition (4.22) is then fulfilledfor different wavelengths in both parts of the grating, giving rise to un-wanted wavelengths at the wrong positions p. Such ghosts are particularlytroublesome in laser Raman spectroscopy (Chap.8) or low-level fluores-cence spectroscopy, where very weak lines have to be detected in the pres-ence of extremely strong excitation lines. The ghosts from these excitationlines may overlap with the fluorescence or Raman lines and complicate theassignment of the spectrum.

Although modern ruling techniques with interferometric length controlhave greatly improved the quality of ruled gratings [4.10-12] the most satis-factory way of producing completely ghost-free gratings is with hologra-phy. The production of holographic gratings proceeds as follows. A photo-sensitive layer on the grating's blank surface is illuminated by two coherentplane waves with the wave vectors kx and k2 (\kx\ = |k2 | ) which form theangles a and p against the surface normal (Fig.4.23). The intensity distribu-tion of the superposition in the plane z = 0 of the photolayer consists ofparallel dark and bright fringes imprinting an ideal grating into the layerwhich becomes visible after developing the photo-emulsion. The gratingconstant depends on the wavelength A = 2TT/ | k | and on the angles a and p.Such holographic gratings are essentially free of ghosts. Their reflectivityR, however, is lower than that of ruled gratings and is furthermore strongly

Z = 0Fig.4.23. Photographic production of a holo-graphic grating

121

that holographically produced grooves are no longer plane, but have a sin-ussodial surface and the "blaze-angle" 6 varies across each groove [4.14].

For Littrow gratings, used as wavelength-selective reflectors, it is de-sirable to have a high reflectivity in a selected order m and low reflectionsfor all other orders. This can be achieved by selecting the width of thegrooves and the blaze angle correctly. Due to diffraction by each groovewith a width d light can only reach angles p within the intervall /3Q±\/d(Fig.4.18a).

Example 4.6With a blaze angle 0 = a = p = 30° and a step height h = A the grating canbe used in 2nd order , while the 3. order appears at ft = ^0+37°. With d =A/tan0 = 2A the central diffraction lobe extends only to /30±30° and the in-tensity in the 3rd order is very small.

Summarizing the considerations above we find that the grating acts asa wavelength-selective mirror, reflecting light of a given wavelength onlyinto definite directions /?m, called the m

fch diffraction orders, which are de-fined by (4.22). The intensity profile of a diffraction order corresponds tothe diffraction profile of a slit with width b = Ndcos/?m representing thesize of the whole grating projection as seen in the direction /3m. The spec-tral resolution A/AA = mN = Nd(sina+sin/?)/A is therefore limited by theeffective size of the grating measured in units of the wavelength.

For a more detailed discussion of special designs of grating monochro-mators such as the concave gratings used in VUV spectroscopy, the readeris referred to the special literature on this subject [4.10-15]. An excellentaccount of the production and design of ruled gratings can be found in[4.10].

4.2 Interferometers

For the investigation of the various line profiles discussed in Chap.3, in-terferometers are preferentially used because, with respect to the spectralresolving power, they are superior even to large spectrometers. In laserspectroscopy the different types of interferometers not only serve to meas-ure emission - or absorption - line profiles, but they are also essential de-vices for narrowing the spectral width of lasers, monitoring the laser line-width, and controlling and stabilizing the wavelength of single-mode lasers(Chap. 5).

In this section we discuss some basic properties of interferometers withthe aid of some illustrating examples. The characteristics of the differenttypes of interferometer that are essential for spectroscopic applications are

122

lo~Ao2

\

Z S i IT~(A,(s,) • A2(s2) • A 3 ( s 3 ) )2

Fig.4.24. Schematic illustration of the basic principle for all interferometers

discussed in more detail. Since laser technology is inconceivable withoutdielectric coatings for mirrors, interferometers, and filters, an extra sectiondeals with such dielectric multilayers. The extensive literature on interfer-ometers [4.16-19] informs about special designs and applications.

4.2.1 Basic Concepts

The basic principle of all interferometers may be summarized as follows(Fig.4.24). The indicent lightwave with intensity Io is divided into two ormore partial beams with amplitudes Ak, which pass different optical pathlengths sk = nxk (n: refractive index) before they are again superimposed atthe exit of the interferometer. Since all partial beams come from the samesource, they are coherent as long as the maximum path difference does notexceed the coherence length (Sect.2.7). The total amplitude of the transmit-ted wave, which is the superposition of all partial waves, depends on theamplitudes Ak and on the phases k = 0+27rsk/A of the partial waves. // istherefore sensitively dependent on the wavelength A.

The maximum transmitted intensity is obtained when all partial wavesinterfere constructively. This gives the condition for the optical path differ-ence Asik = s r s k , namely

Asik = mA , m =1,2,3 (4.34)

The condition (4.34) for maximum transmission of the interferometerapplies not only to a single wavelength A but to all Am for which

A = A s / m (m= 1,2, 3, . . . ) .

The wavelength interval

As 2A=

m + 1 m2 + m 2m + 1

(4.35a)

where A = ±(Am + Am+1), is called the free spectral range of the interfer-ometer. It is more conveniently expressed in terms of frequency. With v =c/A, (4.34) yields As = mc/Vm and the free spectral frequency range

123

= "m+1 = C / A s (4.35b)

becomes independent of the order m.It is important to realize that from one interferometric measurement

alone one can only determine A modulo m-8X because all wavelengths A =A0+m5A are equivalent with respect to the transmission of the interferome-ter. One therefore has at first to measure A within one free spectral rangeusing other techniques, before the absolute wavelength can be obtainedwith an interferometer.

Examples of devices in which only two partial beams interfere, are theMichelson interferometer and the Mach-Zehnder interferometer. Multi-/?/e-beam interference is used, for instance, in the grating spectrometer, theFabry-Perot interferometer and in multilayer dielectric coatings of highlyreflecting mirrors.

Some interferometers utilize the optical birefringence of specific cry-stals to produce two partial waves with mutually orthogonal polarization.The phase difference between the two waves is generated by the differentrefractive index for the two polarizations. An example of such a "polariza-tion interferometer" is the Lyot filter [4.20] used in dye lasers to narrow thespectral linewidth (Sect.4.2.9).

4.2.2 Michelson Interferometer

The basic principle of the Michelson interferometer (M.I.) is illustrated inFig.4.25. The incident plane wave

is split by the beam splitter S (with reflectivity R and transmittance T) intotwo waves

Ej = Axexp[i(a;t - kx + 0X)] and E2 = A2exp[i(u;t - ky + 02)] •

If the beam splitter has negligible absorption (R+T = 1), the amplitudes Axand A2 are determined by Ax = VR AO with A0

2 = A ^ 2

-HiinniiFig.4.25. Two-beam interference in aMichelson-interferometer

124

After being reflected at the plane mirrors Mx and M2 the two wavesare superposed in the plane of observation B. In order to compensate forthe dispersion which beam 1 suffers by passing twice through the glassplate of beam splitter S, often an appropriate compensation plate P is placedin one side arm of the interferometer. The amplitudes of the two waves inthe plane B are VTRA0 , because each wave has been transmitted and re-flected once at the beam-splitter surface S. The phase difference 0 betweenthe two waves is

= ^2(SM 1 -SM 2 ) (4.36)

where A0 accounts for additional phase shifts which may be caused by re-flection. The total complex field amplitude in the plane B is then

(4.37)

The detector in B cannot follow the rapid oscillations with frequencybut measures the time-averaged intensity I, which is according to (2.30c)

1= ^C£0A02RT(l+e i = C6OAO2RT(1 + cos0)

= i Io (1 + for R = T = I and Io = ± ceQ A o 2 (4.38)

If mirror M2 (which is mounted on a carriage) moves along a distance Ay,the optical path difference changes by As = 2nAy (n is the refractive indexbetween S and M2) and the phase difference (/> changes by 2TTAS/A. Figure4.26 shows the intensity 1^(0) in the plane B as a function of for a mono-chromatic incident plane wave. For the maxima at = 2m?r (m = 0,1,2,...)the transmitted intensity IT becomes equal to the incident intensity Io,which means that the transmission of the interferometer is Tj = 1 for (j> =2m7r. In the minima for (j> = (2m+l)7r the transmitted intensity IT is zero!The incident plane wave is being reflected back into the source.

1.0

0.5

0 ft 2ft 3ft $

Fig.4.26. Intensity transmitted through the Michelson interferometer in dependence onthe phase difference ^ between the two interfering beams for R = T = 0.5

125

B

Fig.4.27. Circular fringe pattern producedby the MI with divergent incident light

This illustrates that the M.I. can be regarded either as a wavelength-dependent filter for the transmitted light, or as a wavelength-selective re-flector. In the latter function it is often used for mode selection in lasers(Fox-Smith selector, Sect.5.4.3).

For divergent incident light the path difference between the two wavesdepends on the inclination angle (Fig.4.27). In the plane B an interferencepattern of circular fringes, concentric to the symmetry axis of the system, isproduced. Moving the mirror M2 causes the ring diameter to change. Theintensity behind a small aperture still follows approximately the function1(0) in Fig.4.26. With parallel incident light, but slightly tilted mirrors Mxor M2, the interference pattern consists of parallel fringes which move intoa direction perpendicular to the fringes when As is changed.

The M.I. can be used for absolute wavelength measurements by count-ing the number N of maxima in B when the mirror M2 is moved along aknown distance Ay. The wavelength A is then obtained from

A = 2nAy/N .

This technique has been applied to very precise determinations of laserwavelengths (Sect.4.4).

The M.I. may be described in another, equivalent way, which is quiteinstructive. Assume that the mirror M2 in Fig.4.25 moves with a constantvelocity v = Ay/At. A wave with frequency w and wave vector k incidentperpendicularly on the moving mirror suffers a Doppler shift

Aw = w - w' = 2k-v = (4TT/A)V (4.39)

on reflection.Inserting the path difference As = Aso+2vt and the corresponding

phase difference = (27r/A)As into (4.38) gives, with (4.39) and As0 = 0,

I = - I 0 ( l + cosAwt) with Aw = 2wv/c (4.40)

We recognize (4.40) as the time-averaged beat signal, obtained from the su-perposition of two waves with frequencies w and w' = a;-Aw, giving theaveraged intensity of

126

1-I = I0(l + cosAwt)cos

2[(a/ + w)t/2]x = ^I 0 ( l + cosAwt) .

Note that the frequency w = (c/v)Aw/2 of the incoming wave can bemeasured from the beat frequency Aw, provided the velocity v of themoving mirror is known. The M.I. with uniformly moving mirror M2 canbe therefore regarded as a device which transforms the high frequency w(1014 - 1015 s"1) into an easily accessible audio range (v/c)w.

Example 4.6v = 3 cm/s -> (v/c) = 10"10. The frequency w = 3-1015 Hz (A = 0.6/zm) istransformed to Aw = 6-105 Hz ~ Ai/ ~ 100 kHz.

The maximum path difference As which still gives interference fringesin the plane B is limited by the coherence length of the incident radiation(Sect.2.7). Using spectral lamps the coherence length is limited by theDoppler width of the spectral lines and is typically a few cm. With stabil-ized single-mode lasers, however, coherence lengths of several kilometerscan be achieved. In this case the maximum path difference in the M.I. is, ingeneral, not restricted by the source but by technical limits imposed by lab-oratory facilities.

The attainable path difference As can be considerably increased by anoptical delay line, placed in one arm of the interferometer (Fig.4.28). Itconsists of a pair of mirrors, M3, M4, which reflect the light back and forthmany times. In order to keep diffraction losses small, spherical mirrors arepreferable which compensate by collimation the divergence of the beamcaused by diffraction. With a stable mounting of the whole interferometer,optical path differences up to 350 m could be realized [4.21], allowing aspectral resolution of v/Av ~ 1011. This was demonstrated by measuring thelinewidth of a HeNe laser oscillating at v = 5 1014 Hz as a function of dis-charge current. The accuracy obtained was better than 5 kHz.

For gravitational-wave detection [4.22] a M.I. with side arms of about1 km length is being built where the optical path difference can be in-

M M4 M2

b)Fig.4.28. Michelson interferometer with optical delay line allowing a large pathdifference between the two interfering beams (a) schematic arrangement (b) spot sizesof the reflected beams on mirror M3

127

IU LAS > IUU Kin uy using icuc^uvc niniuio anu an

ultrastable argon laser with a coherence length of Asc » As [4.23,24].When the incoming radiation is composed of several components with

frequencies wk, the total amplitude in the plane B of the detector is the sumof all interference amplitudes (4.37),

(4.41)

A detector with a time constant large compared with the maximum periodl/(wi-wk) does not follow the rapid oscillations of the amplitude at fre-quencies uk or at the difference frequencies (wj-o^) but gives a signal pro-portional to the sum of the intensities Ik in (4.38). We therefore obtain forthe time-dependent total intensity

cosAu,kt) , (4.42)

where the audio frequencies Awk = 2wk v/c are determined by the frequen-cies wk of the components and by the velocity v of the moving mirror.Measurements of these frequencies Aa;k allows one to reconstruct the spec-tral components of the incoming wave with frequencies u)k (Fourier-transform spectroscopy [4.25]).

For example, when the incoming wave consists of two componentswith frequencies u)x and o;2, the interference pattern will vary with time ac-cording to

+cos2o;1(v/c)t]+ | cos(2w2(v/c)t]1(0= ;jli

where we have assumed I10 = I20 = Io. This is a beat signal, where the amp-litude of the interference signal at (o;1+a;2)(v/c) is modulated at the differ-ence frequency (o;1-o;2)v/c (Fig.4.29).

The spectral resolution can roughly be estimated as follows: If Ay isthe path difference travelled by the moving mirror, the number of interfer-ence maxima which are counted by the detector is Nx = 2Ay/Ax for an in-cident wave with the wavelength Aa, and N2 = 2Ay/A2 for A2 < Xv The twowavelengths can be clearly distinguished when N2 > Nx + 1. This yields withX1 = A2+AA and AA « A for the spectral resolving power

AAwith A = (A!+A2)/2 and N (4.43a)

128

1-

0.5-

Fig.4.29. Interference signal behind the MI with uniformly moving mirror M2 whenthe incident wave consists of two components with frequencies u)x and u)2

The equivalent consideration in the frequency domain is as follows:In order to determine the two frequencies OJX and w2, one has to meas-

ure at least over one modulation period

c 2TTV UJi - ( J ,

1

The frequency difference which can be resolved is then

c c c Ai/ 1 v KTA " = v T = A l = N A ^ ^ 7 A = N ° r A i J = N -

(4.43b)

Example 4.7a) Ay = 5 cm, A = 10 /zm -> N = 104,b) Ay = 100 cm, A = 0.5 \im -* N = 4-106

where the latter example can be realized only with lasers which have a suf-ficiently large coherence length (Sect.4.4).c) Xx = 10 /xm, A2 = 9.8 /xm -> (^ 2 -^ ) = 610

1 1 Hz; with v = 1 cm/s -> T =50 ms.

4.2.3 Mach-Zehnder Interferometer

Analogous to the Michelson interferometer, the Mach-Zehnder interfer-ometer is based on the two-beam interference by amplitude splitting of theincoming wave. The two waves travel along different paths (Fig.4.30a). In-serting a transparent object into one arm of the interferometer alters theoptical path difference between the two beams. This results in a change ofthe interference pattern, which allows a very accurate determination of therefractive index of the sample and its local variation. The Mach-Zehnderinterferometer may be regarded therefore as a sensitive refractometer.

129

/ — - - - - - - • /

2-Q'CoscC

a) b)Fig.4.30. Mach-Zehnder interferometer (a) schematic arrangement (b) path differencebetween the two parallel beams

If the beam splitters Bj, B2 and the mirrors M2, M2 are all strictlyparallel, the path difference between the two split beams does not dependon the angle of incidence a because the path difference Ax = B J M J =2acosa between the beam 1 and 3 is exactly compensated by the sampe pathlength between M2 and B2 (Fig.4.30b). This means that the interferingwaves in the symmetric interferometer (without sample) experience thesame path difference on the solid path as on the dashed path in Fig.4.30a.Without the sample the total path difference is therefore zero, and it is As =(n- l )L with the sample having the refractive index n in one arm of the in-terferometer.

Expanding the beam on path 3 gives an extended interference-fringepattern, which reflects the local variation of the refractive index. Using alaser as a light source with a large coherence length, the path lengths in thetwo interferometer arms can be made different without losing the contrastof the interference pattern (Fig.4.31). With a beam expander (lenses Lx andL2), the laser beam can be expanded up to 10r20 cm and large objects canbe tested. The interference pattern can either be photographed or may beviewed directly with the naked eye or with a television camera [4.26]. Sucha laser interferometer has the advantage that the laser-beam diameter canbe kept small everywhere in the interferometer, except between the two

i t ! ! }

Plane ofobservation

Fig.4.31. Laser interferometer for sensitive measurements of local variations of theindex of refraction in extended samples

130

not deviate from an ideal plane by more than A/10 in order to obtain goodinterferograms, smaller beam diameters are advantageous.

The Mach-Zehnder interferometer has found a wide range of applica-tions. Density variations in laminar or turbulent gas flows can be seen withthis technique and the optical quality or mirror substrates of interferometerplates can be tested with high sensitivity [4.26,27].

In order to get quantitative information of the local variation of theoptical path through the sample, it is useful to generate a fringe pattern forcalibration purposes by slightly tilting the plates B l s Mx and B2, M2 in Fig.4.31, which makes the interferometer slightly asymmetric. Assume that Bxand Mx are tilted clockwise around the z direction by a small angle p andthe pair B2M2 is tilted counterclockwise by the same angle p. The opticalpath between B2 and Mx is then Ax = 2acos(a+^), whereas B2M2 = A2 =2acos(a-/?). After being recombined, the two beams therefore have the pathdifference

A = A2 - = 2a[cos(a-/3) - cos(a+/?)] = 4asinasin/? (4.44)

which depends on the angle of incidence a. In the plane of observation, aninterference pattern of parallel fringes is observed with an angular separa-tion Ae between the fringes m and m+1 given by Ae = (sinam-sinam+1) =

A sample in path 3 introduces an additional path difference As(/5) =(n-l)L/cos/? depending on the local refractive index n and the path lengththrough the sample. The resulting phase difference shifts the interferencepattern by an angle 7 =As/(4a/3). Using a lens with a focal length f, whichimages the interference pattern onto the plane O, the linear shift is Ay=f As/(4a/?). Figure 4.32 shows for illustration the interferogram of the con-

Fig.4.32. Interferogram of the den-sity profile in the convection zoneabove a candle flame [4.26a]

131

Fringes

Fig.4.33. Combination of Mach-Zehnder interferometer and spectrograph used for thehook method

vection zone of hot air above a candle flame, placed below one arm of thelaser interferometer in Fig.4.31. It can be seen that the optical path throughthis zone changes by many wavelengths.

The Mach-Zehnder interferometer has been used in spectroscopy tomeasure the refractive index of atomic vapors in the vicinity of spectrallines (Sect. 3.1). The experimental arrangement (Fig.4.33) consists of a com-bination of a spectrograph and a interferometer, where the plates ^X^A\and B2 ,M2 are tilted in such a direction that without the sample the parallelinterference fringes with the separation Ay(A) = Af/(4a/?) are perpendicularto the entrance slit. The spectrograph disperses Ay(A) in the x direction.Because of the wavelength-dependent refractive index n(A) of the atomicvapor (Sect.3.1.3). The fringe shift follows a disperions curve in the vicinityof the spectral line (Fig.4.34). The dispersed fringes look like hooks aroundan absorption line, which gave this technique the name hook method. Tocompensate for background shifts caused by the windows of the absorptioncell, a compensating plate is inserted into the second arm. For more detailsof the Hook method, see [4.27,28].

3944009* 396I523AFig.4.34. Position of fringes as a function of wavelength around the absorption linedoublet of aluminum atoms, as observed behind the spectrograph [4.28]

132

plane parallel partially reflecting surfaces

4.2.4 Multiple-Beam Interference

In a grating spectrometer the interfering partial waves, emitted from thedifferent groves of the grating, have all the same amplitude. In contrast, inmultiple-beam interferometers these partial waves are produced by multiplereflection at plane or curved surfaces and their amplitude decreases withincreasing number of reflections. The resultant total intensity will thereforediffer from (4.26).

Assume that a plane wave E = Aoexp[i(u;t-kx)] is incident at the anglea on a plane transparent plate with two parallel, partially reflecting surfaces(Fig.4.35). At each surface the amplitude A{ is split into a reflected com-ponent AR = AjVft and a refracted component AT = Aj\ / l -R, neglectingabsorption. The reflectivity R = IR/IJ depends on the angle of incidence aand on the polarization of the incident wave. Provided the refractive indexn is known, R can be calculated from Fresnel's formulas [4.3]. From Fig.4.35, the following relations are obtained for the amplitudes AA of wavesreflected at the upper surface, Bj of refracted waves, Cl of waves reflectedat the lower surface, and DA of transmitted waves

(4.45)j | =(1 - R ) V R | A 0 | ,

| D 2 | = R ( l - R ) | A 0 | ;|C2 | = R\/R(1 - R ) | A 0 | ;

|A3 | = v ^ R |C2 | = R3/2 (1 - R) |Ao | ... .

This scheme can be generalized to the equations

(4.46a)

(4.46b)

133

n=1

n>1

between two beams being reflected fromthe two surfaces of a plane parallel plate

= 2 n a - b s i n

Two successively reflected partial waves E4 and E i+1 have the optical pathdifference (Fig.4.36)

As = (2nd/cos/3) - 2dtan/?sina .

Because sina = nsin/?, this can be reduced to

As = 2ndcos/3 = 2dn\/l - sin2/? (4.47a)

if the refractive index within the plane-parallel plate is n > 1 and outsidethe plate n = 1. This path difference causes a corresponding phase differ-ence

2TTAS/A (4.47b)

where A takes into account possible phase changes caused by the reflec-tions. For instance, the incident wave with amplitude Ax suffers the phasejump A = n while being reflected at the medium with n > 1. Including thisphase jump, we can write

Ax = \/RAoexp(i7r) = - \ /RA0 .

The total amplitude A of the reflected wave is obtained by summation overall partial amplitudes Ax taking into account the different phase shifts,

m=l

= - \/RAr

1 - ( l -

p~2

m=0

l (4.48)

For vertical incidence (a = 0), or for an infinitely extended plate we havean infinite number of reflections. The geometrical series in (4.48) has for p-> oo the limit (1 -Re1^) '1 and we obtain for the total amplitude

134

A = - >/R Ao - i -1 - Re**

The intensity I = 2c€0AA* of the reflected wave is then

4sin2(0/2)+4Rsin2(0/2) '

(4.49)

(4.50a)

In an analogous way, we find for the total transmitted amplitude

D = V Dmei(m-l)^ = (1 _ R)A0 V

m=l 0

which gives the total transmitted intensity

(1 - R)2

4Rsin2(0/2) '(4.51a)

Equations (4.50,51) are called the Airy formulas. Since we have neglectedabsorption, we should have I R + I T = Io, as can easily be verified from (4.50,51).

The abbreviation F = 4R/(1-R)2 is often used, which allows the Airyequations to be written in the form

TXR

I T -

J

Io

F1 +

1 +

Fsin2(0/2) '

1Fsin2(0/2) '

(4.50b)

(4.51b)

Figure 4.37 illustrates (4.51) for different values of the reflectivity R. Themaximum transmittance is T = 1 for = 2m7r. At these maxima IT = Io, andthe reflected intensity IR is zero. The frequency range &v between twomaxima is the free spectral range of the interferometer. With = 27rAs/Aand A = c/y, we obtain from (4.47a)

As 2d>/n2 - sin2 a

For vertical incidence (a = 0) the free spectral range becomes

2nd

(4.52a)

(4.52b)

135

0.5

• c ( 2 n d )

F * = 1

F*=50

2mTT 2(m*1)'TT

Fig.4.37. Transmittance of an absorption-free multiple beam interferometer as afunction of the phase difference for different values of the finesse F*

The full halfwidth e = l ^ - ^ l with I(

Fig.4.38. Transmitted intensity lT{v) tor twoclosely spaced spectral lines at the limit ofspectral resolution

quency separation Av is larger than 8u/¥*, which means that their peakseparation should be larger than their full halfwidth.

Quantitatively this can be seen as follows: Assume the incident radia-tion consists of two components with the intensity profiles I1(i>'-i>'1) andh(u'u2) a n d ec*ual P e a k intensities I ^ x ) = ^(^2) = *o- F o r a P e a k s e P a r a "tion V2-V\ = Su/¥* = 28v/nVF the total transmitted intensity l(u) =Ix(u)-¥l2M is obtained from (4.51a) as

- io1 1

8v/¥*)/8h(4.55)

where the phase shift $ = 2TTASI//C = 2w/6v has been replaced by the freespectral range 8v. The function I(^) is plotted in Fig.4.38 around the fre-quency v = (y1+v2)/2. For v = vx = mc/2nd the first term in (4.55) becomes1 and the second term can be derived with sin[7r(i/1+&//F*)/&/) = sin?r/F ~TT/F* and F(TT/F*)2 = 4 to become 0.2. Inserting this into (4.55) yields:l(u=ux) = 1.2 Io; l(i/=(i/1+i/2)/2) a Io and I(y=i/2) = 1.2I0. This just corre-sponds to the Rayleigh criterion for the resolution of two spectral lines. Thespectral resolving power of the interferometer is therefore

v/Av = (v/8v)¥* -> Av = 6v/¥* (4.56)

This can be also expressed by the optical path differences As between twosuccessive partial waves

u A_ • AsAi /~ A A " A *

(4.57)

The resolving power of an interferometer is the product of finesse F* andoptical path difference As/X in units of the wavelength A.

A comparison with the resolving power v/Av = mN = NAs/A in agrating spectrometer with N grooves show that the finesse F* can indeed be

138

regarded as the ettective number ot interfering partial waves and F^As asthe maximum path difference between these waves.

Example 4.9d = 1 cm, n = 1.5, R = 0.98, A = 500 nm. An interferometer with negligiblewedge and high-quality surfaces, where the finesse is mainly determined bythe reflectivity, achieves with F* = 7n/R/(l-R) = 155 a resolving power ofA/AA = IO7. This means that the instrument's line width is about AA ~5-10"5 nm or, in frequency units Av = 60 MHz.

Taking into account the absorption A = (1-R-T) of each reflectivesurface, (4.51) has to be modified to

T 2 10 (A + T)2 [1 +Fsin2(S/2)] '

(4.58a)

where T2 = T2T2 is the product of the transmittance of the two reflectingsurfaces. The absorption causes three effects:

(i) The maximum transmittance is decreased by the factor

IT T 2 T2

Io (A + T)2 ( 1 - R ) 2

< 1 . (4.58b)

Note that already a small absorption of each reflecting surface results in adrastic reduction of the total transmittance. For A = 0.05, R = 0.9 -+ T =0.05 and T2/( 1-R)2 = 0.25.

(ii) For a given transmission factor T the reflectivity R = 1 -A-T decreaseswith increasing absorption. The quantity

F = 4R4(1 - T - A)

(1 - R)2 (T +A)2(4.58c)

decreases with increasing A. This makes the transmission peaks bro-ader. The physical reasons for this is the decreasing number of in-terfering partial waves. The contrast

Tinax1 T

ymin= 1 + F (4.58d)

of the transmitted intensity also decreases.

139

(iii) The absorption causes a phase shift A at each reflection, which de-pends on the wavelength A, the polarization, and the angle of incidencea [4.3]. This effect causes a wavelength-dependent shift of themaxima.

4.2.5 Plane Fabry-Perot Interferometer

A practical realization of the multiple beam-interference discussed in thissection may use either a solid plane-parallel glass or fused quartz plate withtwo coated reflecting surfaces (Fabry-Perot etalon, Fig.4.39a) or two separ-ate plates, where one surface of each plate is coated with a reflection layer.The two reflecting surfaces oppose each other and are aligned to be as par-allel as achievable (Fabry-Perot Interferometer, FPI, Fig.4.39b). The outersurfaces are coated with antireflection layers in order to avoid reflectionsfrom these surfaces which might overlap the interference pattern. Further-more, they have a slight angle against the inner surfaces (wedge).

Both devices can be used for parallel as well as for divergent incidentlight. We now discuss them in some more detail where, at first, consideringtheir illumination with parallel light.

a) The Plane FPI as a Transmission Filter

In laser spectroscopy etalons are mainly used as wavelength-selective trans-mission filters within the laser resonator to narrow the laser bandwidth(Sect. 5.4). The wavelength Am or frequency vm for the transmission maxi-mum of m th order, where the optical path between successive beams is As= mA, can be deduced from (4.47a) and Fig.4.36 to be

Ar =2dm

2ndVn2 - sin2 a = cos/3

me2nd cos/9

m(4.59a)

(4.59b)

reflectioncoatings

Quartz

a)

reflectioncoatings

1

antireflectioncoatings

Fig.4.39. Two realizations of aFabry-Perot interferometer (a) solidetalon (b) air-spaced plane parallelreflecting surfaces

140

For all wavelengths A = Am (m = 0,1,2 ...) in the incident light, the phasedifference between the transmitted partial waves becomes 8 = 2m7r and thetransmitted intensity is according to (4.58)

IT =T 2 T 2

(1 - R ) 2 u (A + T)2In (4.60)

where A = 1-T-R is the absorption of the etalon (substrate absorption plusabsorption of one reflecting surface). The reflected waves interfere destruc-tively for A = Am and the reflected intensity becomes zero.

Note, however, that this is only true for A « 1 and infinitely ex-tended plane waves where the different reflected partial waves completelyoverlap. If the incident wave is a laser beam with the finite diameter D, thedifferent reflected partial beams do not completely overlap because they arelaterally shifted by b = 2dtan/3cosa (Fig.4.40). For a rectangular intensityprofile of the laser beam the fraction b/D of the reflected partial ampli-tudes does not overlap and cannot interfere destructively. This means thateven for maximum transmission the reflected intensity is not zero but abackground reflection remains, which is missing in the transmitted light.For small angles a one obtains for the intensity loss per transit due to re-flection [4.30a] for a rectangular beam profile

4RIo (1 - R )

2 i n D J(4.61a)

For a Gaussian beam profile the calculation is more difficult, and the solu-tion can only be obtained numerically. The result for a Gaussian beam withthe radius w (Sect.5.3) is [4.30b]

IR _ 8R p d a ] 2

Io " (1 - R)2 Inw J

(4.61b)

A parallel light beam with the diameter D passing a plane parallel platewith the angle of incidence a therefore suffers, besides eventual absorptionlosses, reflection losses which increase with a2 and which are proportionalto the ratio (d/D)2 of the etalon thickness d and the beam diameter D(walk-off losses).

Fig.4.40. Incomplete interference of tworeflected beams with finite diameter D,causing a decrease of the maximumtransmitted intensity

141

Example 4.10d = 1 cm, D = 0.2 cm, n = 1.5, R = 0.3, awhich means 5% walk-off losses.

1° =0.017 rad — IR / I 0 = 0.05,

The transmission peak Am of the etalon can be shifted by tilting theetalon. According to (4.59) the wavelength Am decreases with increasingangle of incidence a. The walk-off losses limit the tuning range of tiltedetalons within a laser resonator. With increasing angle a the losses may be-come intolerably large.

b) Illumination with Divergent Light

Illuminating the FPI with divergent monochromatic light (e.g., from an ex-tended source or from a laser beam, behind a diverging lens), a continuousrange of incident angles a is offered to the FPI, which transmits those di-rections am that obey (4.59a), In the transmitted light we then observe aninterference pattern of bright rings (Fig.4.41). Since the reflected intensityIR = I 0 - I T is complementary to the transmitted one, a corresponding systemof dark rings appears in the reflected light at the same angles of incidence

When p is the angle of inclination to the interferometer axis inside theFPI, the transmitted intensity is maximum, according to (4.59), for

mA = 2nd cos/5 , (4.62a)

where n is the refractive index between the reflecting planes. Let us num-ber the rings by the integer p, beginning with p = 0 for the central ring.With m = m o -p we can rewrite (4.63) for small angles fiP as

(mo-p)A = 2ndcos/9P ~ 2nd(l - /?p2/2) = 2nd (4.63)

where n0 is the refractive index of air, and Snell's law sina ~ a = nfi hasbeen used (Fig.4.42).

Fig.4.41. The interference ring system of the transmitted intensity may be regarded aswavelength selective imaging of corresponding ring areas of an extended light source

142

lens Fig.4.42. Illustration of (4.64)

D/2

When the interference pattern is imaged by a lens with the focal lengthf into the plane of the photoplate, we obtain for the ring diameters D =2f ap the relations

(m0 - p)A = 2nd[l - (no /n)2Dp

2 /(8f2)] ,

(m0 - p - 1)A = 2nd[l - (no /n)2Dp + 1

2 /(8f2)] .

Subtracting the second equation from the first one yields

4nf2

(4.64)

" Dp2 nn2d

A. (4.65a)

For the smallest ring with p = 0 (4.63) becomes

m0A = 2nd(l - /?02/2)

which can be written as

(m0 + 6) A = 2nd . (4.62b)

The "excess" e < 1, also called fractional interference order, can be obtainedfrom a comparison of (4.62a and b) as

Inserting e into (4.65) yields the relation

~ o 4nf2 . / vD P 2 = ~ T T A ( P + O8n2f2

no2(mo

-(p + (4.65b)

A linear fit of the squares Dp2 of the measured ring diameters versus the

ring number P yields the excess e and therefore from (4.63b) the wave-length A, provided the refractive index n and the value of d is known froma previous calibration of the interferometer. However, the wavelength is

143

imageintensifier

Fabry - Perot

Mono-chrornator

photomultiplier

nuorescence-ceil

calibrationlamp

topump

Laserbeam

Fabry - PerotRingsystem

Fabry - Perot-plate

Mono-chromatorentranceslit

imagingplane

m • X. = 2n • d cos 6Invar - Ring

lightsource

Fig.4.43. Combination of FPI and spectrograph for the unambiguous determination ofthe integral order m0

determined by (4.65) only modulo a free spectral range 2nd. This meansthat all wavelengths Am differing by m free spectral ranges produce thesame ring systems. For an absolute determination of A the integer order m0has to be known.

The experimental scheme for the determination of A utilizes a com-bination of FPI and spectrograph in a so-called crossed arrangement (Fig.4.43) where the ring system of the FPI is imaged onto the entrance slit of aspectrograph. The spectrograph disperses the slit images S(A) with amedium dispersion in the x direction (Sect.4.1) and the FPI provides highdispersion in the y direction. The resolution of the spectrograph must onlybe sufficiently high to separate the images of two wavelengths differing byone free spectral range of the FPI. Figure 4.44 shows, for illustration, a sec-tion of the Na2 fluorescence spectrum excited by an argon-laser line. Theordinate corresponds to the FPI dispersion and the abscissa to the spectro-graph dispersion [4.31].

The angular dispersion of the FPI can be deduced from (4.63)

dAdAV1 m 1

2ndsin/?with A = 2nd/m . (4.66)

Equation (4.66) shows that the angular dispersion becomes infinite for /3->0.

Fig 4 44. Section of the argon laser-excited fluorescence spectrum of Na, obtainedwith the arrangement of crossed FPI and spectrograph, shown in Fig.4.43

The linear dispersion of the ring system on the photoplate is

dd = f d / ? = _ _ f _dA dA Am sin/9 ' (4.67)

Example 4.11

n w2! °m2 A = ° 5 Mm- A t a d i s t a n c e o f ' m m f r o m t h e "ng center is B =0.1/50 and we obtain a linear dispersion of dd/dA = 50 mm/A This is atleast one order of magnitude larger than the disperison of a large spectro-

c) The Air-Spaced FPI

Different from the solid etalon, which is a plane-parallel plate coated onboth sides w.th reflecting layers, the plane FPI consists of two wedged^ ^ n u f J 1 ^ 1 " 8 ° n e h i8h- r e f |ection and one antireflection coating(Fig.4.39b). The finesse of the FPI critically depends, apart from the re-lectivity R and the optical surface quality, on the parallel alignment of the

two reflecting surfaces. The advantage of the FPI, that any desired freespectral range can be realized by choosing the corresponding plate separa-tion d, must be paid for by the inconvenience of careful alignment. Insteadof changing the angle of incidence a, wavelength tuning can be achievedtor a = 0 by variation of the optical path difference As = 2nd either bvchanging d with piezoelectric tuning of the plate separation or by alteringtte refractive index by a pressure change in the container enclosing the

The pressure-tunable FPI is used for high-resolution spectroscopy ofline profiles. The transmitted intensity IT(p) as a function of pressure isgiven by the convolution picture is

144145

alignment

LSP2

detector

gas inlet

Fig.4.45. Use of a plane FPI for photoelectric recording of the spectrally resolvedtransmitted intensity IT(nd,A) emitted from a point source

where T(nd,A) = T(0) can be obtained from (4.51).With photoelectric recording (Fig.4.45) the large dispersion at the ring

center can be utilized. The light source is imaged onto a small pinhole PIwhich serves as a point source in the focal plane of LI. The parallel lightbeam passes the FPI, and the transmitted intensity is imaged b< L2 ontoanother pinhole P2 in front of the detector. All light rays around ft = 0,where fi is the angle against the interferometer axis (4.59) contribute to thecentral fringe. If the optical path length nd is tuned, the different transmis-sion orders with m = m0, mo+l, mo+2 ... are successively transmitted for awavelength A according to mA = 2nd. Light sources which come close to apoint source, can be realized when a focussed laser beam crosses a samplecell and the laser-induced fluorescence emitted from a small section of thebeam length is imaged through the FPI onto the entrance slit of a mono-chromator, which is tuned to the desired wavelength interval AA around Am

^m-th order|(m+1)th order-of laserline

nL- d (pressure)

Fig.4.46. Photoelectric recording of a Doppler broadened laser-excited fluorescenceline of Na2 molecules in a vapor cell and the Doppler-free scattered laser line. Thepressure scan Ap = a corresponds to one free spectral range of the FPI

146

[D1detectorfor total

fluorescence

LaserL3

excitationspectrum

collimatedmolecular beam

-A

L1 FPI

mono-chromator

photo- [D2multiplier

fluorescencespectrum

Fig.4.47. Experimental arrangement for photoelectric recording of high resolution flu-orescence lines excited by a single mode laser in a collimated molecular beam andobserved through FPI plus monochromator

(Fig.4.43). If the spectral interval AA resolved by the monochromator issmaller then the free spectral range 6X of the FPI, an unambigious determi-nation of A is possible. For illustration Fig.4.46 shows a Doppler-broadenedfluorescence line of Na2 molecules excited by a single-mode argon laser atA = 488 nm, together with the narrow line profile of the scattered laserlight. The pressure change Ap = 2dAnL = a corresponds to one free spectralrange of the FPI, i.e., 2dAnL = A.

For Doppler-free resolution of fluorescence lines (Chap.9) the laser-induced fluorescence of molecules in a collimated molecular beam can beimaged through a FPI onto the entrance slit of the monochromator (Fig.4.47). In this case the crossing point the laser and molecular beams, indded,represents a point source.

4.2.6 Confocal Fabry-Perot Interferometer

A confocal interferometer, sometimes called incorrectly a spherical FPI\consists of two spherical mirrors Mx, M2 with equal curvatures (radius r)which oppose each other at a distance d = r (Fig.4.48a) [4.32-36]. These in-terferometers have gained great importance in laser physics: firstly, ashigh-resolution spectrum analyzers for detecting the mode structure and thelinewidth of lasers [4.34-36]; and secondly, in the nearly confocal form, aslaser resonators (Sect.5.2).

Neglecting spherical aberration, all light rays entering the interferome-ter parallel to its axis would pass through the focal point F and would reachthe entrance point PI again after having passed the confocal FPI four times.Figure 4.48b,c illustrate the general case of a ray which enters the confocalFPI at a small inclination 6 and passes the successive points PI, A, B, C, PIshown in Fig.4.48d in a projection. 0 is the skew angle of the entering ray.

Because of spherical aberration rays with different distances px fromthe axis will not all go through F but will intersect the axis at different pos-itions F depending on px and 0. Also each ray will not exactly reach the

147

c) d)

Fig.4.48. Trajectories of rays in a confocal FPI (a) incident beam parallel to the FPIaxis (b) inclined incident beam (c) perspective view for illustrating the skew angle (d)projection of the skewed rays onto the mirror surfaces

entrance point ?x after four passages through the confocal FPI since it isslightly shifted at successive passages. However, it can be shown [4.32,35]that for sufficiently small angles 0, all rays intersect in the vicinity of thetwo points P and P\ located in the central plane of the confocal FPI at adistance p(p1j) from the axis (Fig.4.48b).

The optical path difference As between two successive rays passingthrough P can be calculated from geometrical optics. For px « r and 0 « 1one obtains for the near confocal case d « r [4.35]

As = 4d + Px2p22cos20/r3 + higher-order terms . (4.68)

An incident light beam with diameter D = 2px therefore produces, in thecentral plane of a confocal FPI, an interference pattern of concentric rings.Analogous of the treatment in Sect.4.2.5, the intensity I(p,A) is obtained byadding all amplitudes with their correct phases 8 = 60+(2TT/A)AS. Accordingto (4.51) we get

4Rsin2[(7r/A)As](4.69)

where T = 1-R-A is the transmission of Mx. The intensity has maxima for8 = 2m7T, which is equivalent to

148

4d + p4 / r3 = mA (4.70)

when we neglect the higher-order terms in (4.68) and set 0 = 0, and p2 =

The free spectral range 8v, i.e., the frequency separation between suc-cessive interference maxima, is for the near-confocal FPI with p « d

4d + p4 / r3 '(4.71)

which is different from the expression 8v = c/2d for the plane FPI.The radius pm of the m

th order interference ring is obtained from(4.70),

= [(mA-4d)r3]i/4 (4.72)

which reveals that pm depends critically on the separation d of the sphericalmirrors. Changing d by a small amount e from d = r to d = r+e changes thepath difference to

As = 4(r + e) + p4/(r + 6)3 ~ 4(r + e) + p4 /r3 . (4.73)

For a given wavelength A the value of e can be chosen such that 4(r+e) =m0A. In this case the radius of the central ring becomes zero. We can num-ber the outer rings by the integer p and obtain with m = mo+p for theradius of the p t h ring the expression

(4.74)

(4.75)

The radial dispersion deduced from (4.72),

mr3/4dA - 4d)r3]3/4 '

becomes infinite for mA = 4d, which occurs according to (4.72) at thecenter with p = 0.

This large disperison can be used for high-resolution spectroscopy ofnarrow line profiles with a scanning confocal FPI and photoelectric record-ing (Fig.4.49).

If the central plane of the near-confocal FPI is imaged by a lens onto acircular aperture with sufficiently small radius b < (Ar3)1/4 only the centralinterference order is transmitted to the detector while all other orders arestopped. Because of the large radial dispersion for small p one obtains ahigh spectral resolving power. With this arrangement not only spectral lineprofiles but also the instrumental bandwidth can be measured, when an in-cident monochromatic wave (from a stabilized single-mode laser) is used.

149

d = r r /2

Fig.4.49. Photoelectric recording of the transmitted power of a scanning confocal FPI

The mirror separation d = r+6 is varied by the small amount e and thepower

P(A,b,6) = 27T pl(pJp=O

(4.76)

transmitted through the aperture, is measured as a function of £, at fixedvalues of A and b.

The integrand I(p,A,6) can be obtained from (4.69), where the phasedifference 8(e) = 2TTAS/A is deduced from (4.73).

The optimum choice for the radius of the aperture is based on a com-promise between spectral resolution and transmitted intensity. When the in-terferometer has the finesse F*, the spectral half width of the transmissionpeak is 8v/F*, see (4.54b), and the maximum spectral resolving power be-comes F* As/A (4.57). For the radius b = (r3A/F*)1/4 of the aperture, whichis just (F*)1/4 times the radius px of a fringe with p = 1 in (4.74), the spec-tral resolving power is reduced to about 70% of its maximum value. Thiscan be verified by inserting this value of b into (4.76) and calculating thehalf width of the transmission peak P(A1,F*,£).

The total finesse of the confocal FPI is, in general, higher than that ofa plane FPI for the following reasons:• The alignment of spherical mirrors is by far less critical than that of

plane mirrors, because tilting of the spherical mirrors does not change(to a first approximation) the optical path length 4r through the confo-cal FPI, which remains approximately the same for all incident rays(Fig.4.50). For the plane FPI, however, the path length increases forrays below the interferometer axis, but decreases for rays above theaxis.

• Spherical mirrors can be polished to a higher precision than plane mir-rors. This means that the deviations from an ideal sphere are less forspherical mirrors than those from an ideal plane for plane mirrors.Furthermore, such deviations do not wa