Created by the technical staff of Newport Corporationwith the assistance of Dr. Donald C. O‘Shea of the School of Physics

at the Georgia Institute of Technology.We gratefully acknowledge J. Wiley and Sons, publishers of

The Elements of Modern Optical Design by Donald C. O‘Sheafor use of copyrighted material in the Optics Primer section.

Projectsin

OpticsApplications Workbook

P/N 16291-01, Rev. G

Table of Contents

Page

Preface .................................................................................................... 1An Optics Primer ................................................................................... 3

0.1 Geometrical Optics .......................................................................... 30.2 Thin Lens Equation.......................................................................... 60.3 Diffraction ......................................................................................... 90.4 Interference .................................................................................... 130.5 Component Assemblies ................................................................ 160.6 Lasers .............................................................................................. 220.7 The Abbe Theory of Imaging ........................................................ 300.8 References ...................................................................................... 35

Component Assemblies ....................................................................... 36Projects Section .................................................................................... 45

1.0 Project 1: The Laws of Geometrical Optics ................................ 452.0 Project 2: The Thin Lens Equation .............................................. 513.0 Project 3: Expanding Laser Beams .............................................. 554.0 Project 4: Diffraction of Circular Apertures ............................... 595.0 Project 5: Single Slit Diffraction and Double Slit Interference.. 636.0 Project 6: The Michelson Interferometer ................................... 677.0 Project 7: Lasers and Coherence ................................................. 718.0 Project 8: Polarization of Light..................................................... 759.0 Project 9: Birefringence of Materials ........................................... 79

10.0 Project 10: The Abbe Theory of Imaging .................................... 82

1

Projects In Optics

PrefaceThe Projects in Optics Kit is a set of laboratory equip-ment containing all of the optics and optomechanicalcomponents needed to complete a series of experi-ments that will provide students with a basic back-ground in optics and practical hands-on experience inlaboratory techniques. The projects cover a wide rangeof topics from basic lens theory through interferometryand the theory of imaging. The Project in OpticsHandbook has been developed by the technical staff ofNewport Corporation and Prof. D. C. O’Shea, in order toprovide educators with a convenient means of stimulat-ing their students’ interest and creativity.

This handbook begins with a description of severalmechanical assemblies that will be used in variouscombinations for each experiment. In addition, thesecomponents can be assembled in many other configura-tions that will allow more complex experiments to bedesigned and executed. One of the benefits fromconstructing these experiments using an optical bench(sometimes called an optical breadboard) plus stan-dard components, is that the student can see that thecomponents are used in a variety of different circum-stances to solve the particular experimental problem,rather than being presented with an item that willperform only one task in one way.

A short Optics Primer relates a number of opticalphenomena to the ten projects in this handbook. Eachproject description contains a statement of purposethat outlines the phenomena to be measured, theoptical principle is being studied, a brief look at therelevant equations governing the experiment or refer-ences to the appropriate section of the Primer, a list ofall necessary equipment, and a complete step-by-stepinstruction set which will to guide the student throughthe laboratory exercise. After the detailed experimentdescription is a list of somewhat more elaborateexperiments that will extend the basic conceptsexplored in the experiment. The ease with which theseadditional experiments can be done will depend bothon the resources at hand and the inventiveness of theinstructor and the student.

The equipment list for the individual experiments isgiven in terms of the components assemblies, plusitems that are part of the project kits. There are acertain number of required items that are to be sup-plied by the instructor. Items such as metersticks andtape measures are easily obtainable. Others, for the

2

elaborate experiments, may be somewhat more diffi-cult, but many are found in most undergraduateprograms. Note that along with lasers and adjustablemirror mounts, index cards and tape is used to acquirethe data. The student should understand that thepurpose of the equipment is get reliable data, usingwhatever is required. The student should be allowedsome ingenuity in solving some of the problems, but ifhis or her choices will materially affect their data aninstructor should be prepared to intervene.

These experiments are intended to be used by instruc-tors at the sophomore/junior level for college engineer-ing and physical science students or in an advancedhigh school physics laboratory course. The projectsfollow the general study outline found in most opticaltext books, although some of the material on lasers andimaging departs from the standard curriculum at thepresent time. They should find their greatest applicabil-ity as instructional aids to reinforcing the conceptstaught in these texts.

Acknowledgement: A large part of the text and many ofthe figures of “An Optics Primer” are based on ChapterOne of Elements of Modern Optical Design by Donald C.O‘Shea, published by J. Wiley and Sons, Inc., New York©1985. They are reprinted with permission of JohnWiley & Sons, Inc.

3

0.0 An Optics Primer

The field of optics is a fascinating area of study. In manyareas of science and engineering, the understanding ofthe concepts and effects in that field can be difficultbecause the results are not easy to display. But inoptics, you can see exactly what is happening and youcan vary the conditions and see the results. This primeris intended to provide an introduction to the 10 opticsdemonstrations and projects contained in this Projectsin Optics manual. A list of references that can provideadditional background is given at the end of thisprimer.

0.1 Geometrical OpticsThere is no need to convince anyone that light travelsin straight lines. When we see rays of sunlight pouringbetween the leaves of a tree in a light morning fog, wetrust our sight. The idea of light rays traveling instraight lines through space is accurate as long as thewavelength of the radiation is much smaller than thewindows, passages, and holes that can restrict the pathof the light. When this is not true, the phenomenon ofdiffraction must be considered, and its effect upon thedirection and pattern of the radiation must be calcu-lated. However, to a first approximation, when diffrac-tion can be ignored, we can consider that the progressof light through an optical system may be traced byfollowing the straight line paths or rays of light throughthe system. This is the domain of geometrical optics.

Part of the beauty of optics, as it is for any good game,is that the rules are so simple, yet the consequences sovaried and, at times, elaborate, that one never tires ofplaying. Geometrical optics can be expressed as a set ofthree laws:

1. The Law of Transmission.

In a region of constant refractive index, lighttravels in a straight line.

2. Law of Reflection.

Light incident on a plane surface at an angle θi

with respect to the normal to the surface isreflected through an angle θr equal to the incidentangle (Fig. 0.1).

θi = θr (0.1)

Figure 0.1 Reflection and refraction of light at aninterface.

ni

r

nt

n < ni t

θ

tθ

iθ

4

3. Law of Refraction (Snell’s Law).

Light in a medium of refractive index ni incidenton a plane surface at an angle θi with respect tothe normal is refracted at an angle θt in a mediumof refractive index nt as (Fig. 0.1),

ni sinθi = nt sinθt (0-2)

A corollary to these three rules is that the incident,reflected, and transmitted rays, and the normal to theinterface all lie in the same plane, called the plane ofincidence, which is defined as the plane containing thesurface normal and the direction of the incident ray.

Note that the third of these equations is not written as aratio of sines, as you may have seen it from your earlierstudies, but rather as a product of n sinθ. This isbecause the equation is unambiguous as to whichrefractive index corresponds to which angle. If youremember it in this form, you will never have anydifficulty trying to determine which index goes where insolving for angles. Project #1 will permit you to verifythe laws of reflection and refraction.

A special case must be considered if the refractiveindex of the incident medium is greater than that of thetransmitting medium, (ni >nt ). Solving for θt, we get

sinθt = (ni /nt) sinθi (0-3)

In this case, ni /nt> 1, and sinθi can range from 0 to 1.Thus, for large angles of θi it would seem that we couldhave sinθt > 1. But sinθt must also be less than one, sothere is a critical angle θi = θc, where sin θc = nt /ni andsinθt = 1. This means the transmitted ray is travelingperpendicular to the normal (i.e., parallel to the inter-face), as shown by ray #2 in Fig. 0.2. For incidentangles θi greater than θc no light is transmitted.Instead the light is totally reflected back into theincident medium (see ray #3, Fig. 0.2). This effect iscalled total internal reflection and will be used inProject #1 to measure the refractive index of a prism.

As illustrated in Fig. 0.3, prisms can provide highlyreflecting non-absorbing mirrors by exploiting totalinternal reflection.

Total internal reflection is the basis for the transmis-sion of light through many optical fibers. We do notcover the design of optical fiber systems in this manualbecause the application has become highly specializedand more closely linked with modern communicationstheory than geometrical optics. A separate manual andseries of experiments on fiber optics is available fromNewport in our Projects in Fiber Optics.

Figure 0.3. Total internal reflection from prisms.

Figure 0.2. Three rays incident at angles near or at thecritical angle.

ni nt

n < nt i

1

1

2

23

3

cθ

5

0.1.1. Lenses

In most optical designs, the imaging components — thelenses and curved mirrors — are symmetric about aline, called the optical axis. The curved surfaces of alens each have a center of curvature. A line drawnbetween the centers of curvatures of the two surfacesof the lens establishes the optical axis of the lens, asshown in Fig.0.4. In most cases, it is assumed that theoptical axes of all components are the same. This lineestablishes a reference line for the optical system.

By drawing rays through a series of lenses, one candetermine how and where images occur. There areconventions for tracing rays; although not universallyaccepted, these conventions have sufficient usage thatit is convenient to adopt them for sketches and calcula-tions.

1. An object is placed to the left of the opticalsystem. Light is traced through the system fromleft to right until a reflective component altersthe general direction.

Although one could draw some recognizableobject to be imaged by the system, the simplestobject is a vertical arrow. (The arrow, imaged bythe optical system, indicates if subsequentimages are erect or inverted with respect to theoriginal object and other images.) If we assumelight from the object is sent in all directions, wecan draw a sunburst of rays from any point onthe arrow. An image is formed where all the raysfrom the point, that are redirected by the opticalsystem, again converge to a point.

A positive lens is one of the simplest image-forming devices. If the object is placed very faraway (“at infinity” is the usual term), the raysfrom the object are parallel to the optic axis andproduce an image at the focal point of the lens, adistance f from the lens (the distance f is thefocal length of the lens), as shown in Fig. 0.5(a).A negative lens also has a focal point, as shown inFig. 0.5(b). However, in this case, the parallelrays do not converge to a point, but insteadappear to diverge from a point a distance f infront of the lens.

2. A light ray parallel to the optic axis of a lens willpass, after refraction, through the focal point, adistance f from the vertex of the lens.

3. Light rays which pass through the focal point ofa lens will be refracted parallel to the optic axis.

4. A light ray directed through the center of the lensis undeviated.

Figure 0.4 Optical axis of a lens.

R

Center ofcurvatureof surface 2

Center ofcurvature

of surface 1

1R2

Optical AxisOptical Axis

f

f

a.

b.

Figure 0.5. Focusing of parallel light by positive andnegative lenses.

6

The formation of an image by a positive lens isshown in Fig. 0.6. Notice that the rays cross at apoint in space. If you were to put a screen at thatpoint you would see the image in focus there.Because the image can be found at an accessibleplane in space, it is called a real image. For anegative lens, the rays from an object do notcross after transmission, as shown in Fig. 0.7, butappear to come from some point behind the lens.This image, which cannot be observed on ascreen at some point in space, is called a virtualimage. Another example of a virtual image is theimage you see in the bathroom mirror in themorning. One can also produce a virtual imagewith a positive lens, if the object is locatedbetween the vertex and focal point. The labels,“real” and “virtual”, do not imply that one type ofimage is useful and the other is not. They simplyindicate whether or not the rays redirected bythe optical system actually cross.

Most optical systems contain more than one lensor mirror. Combinations of elements are notdifficult to handle according to the following rule:

5. The image of the original object produced by thefirst element becomes the object for the secondelement. The object of each additional element isthe image from the previous element.

More elaborate systems can be handled in asimilar manner. In many cases the elaboratesystems can be broken down into simplersystems that can be handled separately, at first,then joined together later.

0.2 Thin Lens EquationThus far we have not put any numbers with the ex-amples we have shown. While there are graphicalmethods for assessing an optical system, sketching raysis only used as a design shorthand. It is throughcalculation that we can determine if the system will dowhat we want it to. And it is only through these calcula-tions that we can specify the necessary components,modify the initial values, and understand the limitationsof the design.

Rays traced close to the optical axis of a system, thosethat have a small angle with respect to the axis, aremost easily calculated because some simple approxima-tions can be made in this region. This approximation iscalled the paraxial approximation, and the rays arecalled paraxial rays.

Figure 0.6. Imaging of an object point by a positivelens. A real inverted image with respect to the objectis formed by the lens.

Figure 0.7. Imaging of an object point by a negativelens. A virtual erect image with respect to the objectis formed by the lens.

ff

f f

7

Before proceeding, a set of sign conventions should beset down for the thin lens calculations to be considerednext. The conventions used here are those used in mosthigh school and college physics texts. They are not theconventions used by most optical engineers. This isunfortunate, but it is one of the difficulties that is foundin many fields of technology. We use a standard right-handed coordinate system with light propagatinggenerally along the z-axis.

1. Light initially travels from left to right in apositive direction.

2. Focal lengths of converging elements are positive;diverging elements have negative focal lengths.

3. Object distances are positive if the object islocated to the left of a lens and negative if locatedto the right of a lens.

4. Image distances are positive if the image is foundto the right of a lens and negative if located to theleft of a lens.

We can derive the object-image relationship for a lens.With reference to Fig. 0.8 let us use two rays from anoff-axis object point, one parallel to the axis, and onethrough the front focal point. When the rays are traced,they form a set of similar triangles ABC and BCD. InABC,

h hs

hf

o i

o

i+ = (0-4a)

and in BCD

h hs

hf

o i

i

o+ = (0-4b)

Adding these two equations and dividing through byho + hi we obtain the thin lens equation

1 1 1f s si o

= + (0-5)

Solving equations 0-4a and 0-4b for ho + hi , you canshow that the transverse magnification or lateralmagnification, M, of a thin lens, the ratio of the imageheight hi to the object height h0, is simply the ratio ofthe image distance over the object distance:

Mhh

ss

i

o

i

o

= = − (0-6)

With the inclusion of the negative sign in the equation,not only does this ratio give the size of the final image,its sign also indicates the orientation of the image

Figure 0.8. Geometry for a derivation of the thin lensequation.

B

C D

A

h i hih0

h0

+

f s i

s0

f

8

relative to the object. A negative sign indicates thatthe image is inverted with respect to the object. Theaxial or longitudinal magnification, the magnification ofa distance between two points on the axis, can beshown to be the square of the lateral or transversemagnification.

M Ml = 2 (0-7)

In referring to transverse magnification, an unsub-scripted M will be used.

The relationship of an image to an object for a positivefocal length lens is the same for all lenses. If we startwith an object at infinity we find from Eq. 0-5 that for apositive lens a real image is located at the focal point ofthe lens ( l/so = 0, therefore si = f ), and as the objectapproaches the lens the image distance increases untilit reaches a point 2f on the other side of the lens. At thispoint the object and images are the same size and thesame distance from the lens. As the object is movedfrom 2f to f, the image moves from 2f to infinity. Anobject placed between a positive lens and its focal pointforms a virtual, magnified image that decreases inmagnification as the object approaches the lens. For anegative lens, the situation is simpler: starting with anobject at infinity, a virtual image, demagnified, appearsto be at the focal point on the same side of the lens asthe object. As the object moves closer to the lens sodoes the image, until the image and object are equal insize at the lens. These relationships will be explored indetail in Project #2.

The calculation for a combination of lenses is not muchharder than that for a single lens. As indicated earlierwith ray sketching, the image of the preceding lensbecomes the object of the succeeding lens.

One particular situation that is analyzed in Project #2 isdetermining the focal length of a negative lens. The ideais to refocus the virtual image created by the negativelens with a positive lens to create a real image. In Fig.0.9 a virtual image created by a negative lens of un-known focal length f1 is reimaged by a positive lens ofknown focal length f2. The power of the positive lens issufficient to create a real image at a distance s3 from it.By determining what the object distance s2 should befor this focal length and image distance, the location ofthe image distance for the negative lens can be foundbased upon rule 5 in Sec. 0.1: the image of one lensserves as the object for a succeeding lens. The imagedistance s1 for the negative lens is the separationbetween lenses t1 minus the object distance s2 of thepositive lens. Since the original object distance s0 andthe image distance s1 have been found, the focal length

Figure 0.9 Determination of the focal length of anegative lens with the use of a positive lens of knownfocal length.

0

f1

t1

f2

s2 s

ObjectVirtualImage

Image onScreen

ss

1

3

9

of the negative lens can be found from the thin lensequation.

In many optical designs several lenses are used to-gether to produce an improved image. The effectivefocal length of the combination of lenses can be calcu-lated by ray tracing methods. In the case of two thinlenses in contact, the effective focal length of thecombination is given by

1 1 1

1 2f f f= + (0-8)

0.3 DiffractionAlthough the previous two sections treated light as rayspropagating in straight lines, this picture does not fullydescribe the range of optical phenomena that can beinvestigated within the experiments in Projects inOptics. There are a number of additional concepts thatare needed to explain certain limitations of ray opticsand to describe some of the techniques that allow us toanalyze images and control the amplitude and directionof light. This section is a brief review of two importantphenomena in physical optics, interference and diffrac-tion. For a complete discussion of these and relatedsubjects, the reader should consult one or more of thereferences.

0.3.1 Huygen’s Principle

Light is an electromagnetic wave made up of manydifferent wavelengths. Since light from any source (evena laser!) consists of fields of different wavelength, itwould seem that it would be difficult to analyze theirresultant effect. But the effects of light made up of manycolors can be understood by determining what happensfor a monochromatic wave (one of a single wavelength)then adding the fields of all the colors present. Thus byanalysis of these effects for monochromatic light, weare able to calculate what would happen in non-monochromatic cases. Although it is possible toexpress an electromagnetic wave mathematically, wewill describe light waves graphically and then use thesegraphic depictions to provide insight to several opticalphenomena. In many cases it is all that is needed to getgoing.

An electromagnetic field can be pictured as a combina-tion of electric (E) and magnetic (H) fields whosedirections are perpendicular to the direction of propa-gation of the wave (k), as shown in Fig. 0.10. Becausethe electric and magnetic fields are proportional toeach other, only one of the fields need to be describedto understand what is happening in a light wave. In

Figure 0.11. Monochromatic plane wave propagat-ing along the z-axis. For a plane wave, the electricfield is constant in an x-y plane. The solid lines anddashed lines indicate maximum positive andnegative field amplitudes.

Figure 0.10. Monochromatic plane wave propagatingalong the z axis. For a plane wave, the electric field isconstant in an x-y plane. The vector k is in thedirection of propagation.

y

x z

z

λ/2

A

-A

E

x

E

H

y z

k

10

most cases, a light wave is described in terms of theelectric field. The diagram in Fig 0.10 represents thefield at one point in space and time. It is the arrangementof the electric and magnetic fields in space that deter-mines how the light field progresses.

One way of thinking about light fields is to use theconcept of wavefront. If we plot the electric fields as afunction of time along the direction of propagation, thereare places on the wave where the field is a maximum inone direction and other places where it is zero, andother places where the field is a maximum in the oppo-site direction, as shown in Fig. 0.11. These representdifferent phases of the wave. Of course, the phase of thewave changes continuously along the direction ofpropagation. To follow the progress of a wave, however,we will concentrate on one particular point on thephase, usually at a point where the electric field ampli-tude is a maximum. If all the points in the neighborhoodhave this same amplitude, they form a surface of con-stant phase, or wavefront. In general, the wavefrontsfrom a light source can have any shape, but some of thesimpler wavefront shapes are of use in describing anumber of optical phenomena.

A plane wave is a light field made up of plane surfaces ofconstant phase perpendicular to the direction of propa-gation. In the direction of propagation, the electric fieldvaries sinusoidally such that it repeats every wave-length. To represent this wave, we have drawn theplanes of maximum electric field strength, as shown inFig. 0.11, where the solid lines represent planes in whichthe electric field vector is pointing in the positive y-direction and the dashed lines represent plane in whichthe electric field vector is pointing in the negative y-direction. The solid planes are separated by one wave-length, as are the dashed planes.

Another useful waveform for the analysis of light wavesis the spherical wave. A point source, a fictitious sourceof infinitely small dimensions, emits a wavefront thattravels outward in all directions producing wavefrontsconsisting of spherical shells centered about the pointsource. These spherical waves propagate outward fromthe point source with radii equal to the distance be-tween the wavefront and the point source, as shownschematically in Fig. 0.12. Far away from the pointsource, the radius of the wavefront is so large that thewavefronts approximate plane waves. Another way tocreate spherical waves is to focus a plane wave. Figure0.13 shows the spherical waves collapsing to a point andthen expanding. The waves never collapse to a truepoint because of diffraction (next Section). There aremany other possible forms of wave fields, but these twoare all that is needed for our discussion of interference.

Figure 0.12. Spherical waves propagating outwardfrom the point source. Far from the point source, theradius of the wavefront is large and the wavefrontsapproximate plane waves.

Figure 0.13. Generation of spherical waves by focus-ing plane waves to a point. Diffraction prevents thewaves from focusing to a point.

PointSource

11

What we have described are single wavefronts. Whathappens when two or more wavefronts are present inthe same region? Electromagnetic theory shows that wecan apply the principle of superposition: where wavesoverlap in the same region of space, the resultant fieldat that point in space and time is found by adding theelectric fields of the individual waves at a point. For thepresent we are assuming that the electric fields of allthe waves have the same polarization (direction of theelectric field) and they can be added as scalars. If thedirections of the fields are not the same, then the fieldsmust be added as vectors. Neither our eyes nor anylight detector “sees” the electric field of a light wave. Alldetectors measure the square of the time averagedelectric field over some area. This is the irradiance ofthe light given in terms of watts/square meter (w/m2 ) orsimilar units of power per unit area.

Given some resultant wavefront in space, how do wepredict its behavior as it propagates? This is done byinvoking Huygen’s Principle. Or, in terms of thegraphical descriptions we have just defined, Huygen’sConstruction (see Fig. 0.14): Given a wavefront ofarbitrary shape, locate an array of point sources on thewavefront, so that the strength of each point source isproportional to the amplitude of the wave at that point.Allow the point sources to propagate for a time t, sothat their radii are equal to ct (c is the speed of light)and add the resulting sources. The resultant envelopeof the point sources is the wavefront at a time t after theinitial wavefront. This principle can be used to analyzewave phenomena of considerable complexity.

0.3.2 Fresnel and Fraunhofer Diffraction

Diffraction of light arises from the effects of aperturesand interface boundaries on the propagation of light. Inits simplest form, edges of lenses, apertures, and otheroptical components cause the light passing through theoptical system to be directed out of the paths indicatedby ray optics. While certain diffraction effects proveuseful, ultimately all optical performance is limited bydiffraction, if there is sufficient signal, and by electricalor optical “noise”, if the signal is small.

When a plane wave illuminates a slit, the resulting wavepattern that passes the slit can be constructed usingHuygens’ Principle by representing the wavefront in theslit as a collection of point sources all emitting in phase.The form of the irradiance pattern that is observeddepends on the distance from the diffraction aperture,the size of the aperture and the wavelength of theillumination. If the diffracted light is examined close tothe aperture, the pattern will resemble the aperturewith a few surprising variations (such as finding a point

Wavefrontafter time

PointSource

InitialWavefront

t

Figure. 0.14. Huygen’s Construction of a propagatingwavefront of arbitrary shape.

12

of light in the shadow of circular mask!). This form ofdiffraction is called Fresnel (Freh-nell) diffraction andis somewhat difficult to calculate.

At a distance from the aperture the pattern changesinto a Fraunhofer diffraction pattern. This type ofdiffraction is easy to calculate and determines in mostcases, the optical limitations of most precision opticalsystems. The simplest diffraction pattern is that due toa long slit aperture. Because of the length of the slitrelative to its width, the strongest effect is that due tothe narrowest width. The resulting diffraction pattern ofa slit on a distant screen contains maxima and minima,as shown in Fig. 0.15(a). The light is diffracted stronglyin the direction perpendicular to the slit edges. Ameasure of the amount of diffraction is the spacingbetween the strong central maximum and the first darkfringe in the diffraction pattern. The differences inFraunhofer and Fresnel diffraction patterns will beexplored in Project #4.

At distances far from the slit, the Fraunhofer diffractionpattern does not change in shape, but only in size. Thefringe separation is expressed in terms of the sine of theangular separation between the central maximum andthe center of the first dark fringe,

sinθ λ=w

(0-9)

where w is the slit width and λ is the wavelength of thelight illuminating the slit. Note that as the width of theslit becomes smaller, the diffraction angle becomeslarger. If the slit width is not too small, the sine can bereplaced by its argument,

θ λ=w

(0-10)

If the wavelength of the light illuminating the slit isknown, the diffraction angle can be measured and thewidth of the diffracting slit determined. In Project #5you will be able to do exactly this.

In the case of circular apertures, the diffraction patternis also circular, as indicated in Fig. 0.15(b), and theangular separation between the central maximum andthe first dark ring is given by

sin .θ λ=1 22D

or for large D,

θ λ=1 22.D

(0-11)

1st dark ring

θ=1.22 λD

λ

Lightwavelength

D

ω

λ

Lightwavelength

θ= λω

Central Maximum

1st Dark Fringes

(a)

(b)

Figure 0.15. Diffraction of light by apertures. (a)Single slit. (b) Circular aperture.

13

where D is the diameter of the aperture. As in the caseof the slit, for small values of λ /D, the sine can bereplaced by its angle. The measurement of the diameterof different size pinholes is part of Project #4.

One good approximation of a point source is a brightstar. A pair of stars close to one another can give ameasure of the diffraction limits of a system. If the starshave the same brightness, the resolution of the systemcan be determined by the smallest angular separationbetween such sources that would still allow them to beresolved. This is provided that the aberrations of theoptical system are sufficiently small and diffraction isthe only limitation to resolving the images of these twopoint sources. Although it is somewhat artificial, a limitof resolution that has been used in many instances isthat two point sources are just resolvable if the maxi-mum of the diffraction pattern of one point source fallson the first dark ring of the pattern of the second pointsource, as illustrated in Fig. 0.16, then

θ λR D

=1 22. (0-12)

This condition for resolution is called the Rayleighcriterion. It is used in other fields of optical design,such as specifying the resolution of a optical systems.

0.4 InterferenceWhile diffraction provides the limits that tells us howfar an optical technique can be extended, interferenceis responsible for some of the most useful effects in thefield of optics — from diffraction gratings to hologra-phy. As we shall see, an interference pattern is oftenconnected with some simple geometry. Once thegeometry is discovered, the interference is easilyunderstood and analyzed.

0.4.1. Young’s Experiment

In Fig. 0.17 the geometry and wave pattern for one ofthe simplest interference experiments, Young’s experi-ment, is shown. Two small pinholes, separated by adistance d, are illuminated by a plane wave, producingtwo point sources that create overlapping sphericalwaves. The figure shows a cross-sectional view of thewavefronts from both sources in a plane containing thepinholes. Notice that at points along a line equidistantfrom both pinholes, the waves from the two sources arealways in phase. Thus, along the line marked C theelectric fields always add in phase to give a field that istwice that of a single field; the irradiance at a point

Figure 0.16. Rayleigh criterion. The plot of the inten-sity along a line between the centers of the twodiffraction patterns is shown below a photo of twosources just resolved as specified by the Rayleighcriterion. (Photo by Vincent Mallette)

Figure 0.17. Young’s Experiment. Light diffractedthrough two pinholes in screen S1 spreads out towardscreen S2. Interference of the two spherical wavesproduces a variation in irradiance (interferencefringes) on S2 that is plotted to the right of the screen.

CC

CC

DD

DD

DD

DD

ll

S

CC

1

S2

d

θ=1.22 λD

14

along the line, which is proportional to the square ofthe electric field, will be four times that due to a singlepinhole. When electric fields add together to give alarger value it is referred to as constructive interfer-ence. There are other directions, such as those alongthe dotted lines marked D, in which the waves from thetwo sources are always 180° out of phase. That is, whenone source has a maximum positive electric field, theother has the same negative value so the fields alwayscancel and no light is detected along these lines markedD, as long as both sources are present. This conditionof canceling electric fields is called destructive interfer-ence. Between the two extremes of maximum construc-tive and destructive interference, the irradiance variesbetween four times the single pinhole irradiance andzero. It can be shown that the total energy falling on thesurface of a screen placed in the interference pattern isneither more nor less than twice that of a single pointsource; it is just that interference causes the lightdistribution to be arranged differently!

Some simple calculations will show that the differencein distances traveled from pinholes to a point on thescreen is

∆r = d sinθ. (0-13)

In the case of constructive interference, the wavefrontsarrive at the screen in phase. This means that there iseither one or two or some integral number of wave-length difference between the two paths traveled by thelight to the point of a bright fringe. Thus, the angles atwhich the bright fringes occur are given by

∆r = d sinθ = n λ (n = 1, 2, 3, . . .). (0-14)

If the above equation is solved for the angles θn atwhich the bright fringes are found and one applies theapproximation that for small angles the sine can bereplaced by its angle in radians, one obtains:

θn ≅ n λ/d (n = 1, 2, 3, . . .). (0-15)

The angular separation by neighboring fringes is thenthe difference between θn+1 and θn:

∆θ = λ/d. (0-16)

It is this angular separation between fringes that will bemeasured in Project #5 to determine the separationbetween two slits.

0.4.2 The Michelson Interferometer

Another interference geometry that will be investigatedin Project #6 and used to measure an important pa-rameter for a laser in Project #7 is shown in Fig. 0.18.This is a Michelson interferometer, which is con-structed from a beamsplitter and two mirrors. (This

Figure 0.18. Michelson interferometer. By reflectingthe mirror M1 about the plane of the beamsplitter BSto location M’1, one can see that a ray reflecting offmirror M2 travels an additional distance 2(L2 - L1) overa ray reflecting off M1.

M 1

M 2M'1

L1

BS

L1

L 2

PointSource

Lens

Screen

15

device is sometimes called a Twyman-Green interfer-ometer when it is used with a monochromatic source,such as a laser, to test optical components.) Thebeamsplitter is a partially reflecting mirror that sepa-rates the light incident upon it into two beams of equalstrength. After reflecting off the mirrors, the two beamsare recombined so that they both travel in the samedirection when they reach the screen. If the two mirrorsare the same distance (Ll = L2 in Fig. 0.18) from thebeamsplitter, then the two beams are always in phaseonce they are recombined, just as is the case along theline of constructive interference in Young’s experiment.Now the condition of constructive and destructiveinterference depends on the difference between thepaths traveled by the two beams. Since each beam musttravel the distance from the beamsplitter to its respec-tive mirror and back, the distance traveled by the beamis 2L. If the path-length difference, 2L1 - 2L2, is equal toan integral number of wavelengths, mλλλλλ, where m is aninteger, then the two waves are in phase and theinterference at the screen will be constructive.

L1 - L2 = m λλλλλ /2 (m = . . ., - 1, 0, 1, 2, . . .). (0-17)

If the path-length difference is an integral number ofwavelengths plus a half wavelength, the interference onthe screen will be destructive. This can be expressed as

L1 - L2 = m λλλλλ /4 (m = odd integers). (0-18)

In most cases the wavefronts of the two beams whenthey are recombined are not planar, but are sphericalwavefronts with long radii of curvature. The interfer-ence pattern for two wavefronts of different curvatureis a series of bright and dark rings. However, the abovediscussion still holds for any point on the screen.Usually, however, the center of the pattern is the pointused for calculations.

In the above discussion, it was assumed that themedium between the beamsplitter and the mirrors isundisturbed air. If, however, we allow for the possibilitythat the refractive index in those regions could bedifferent, then the equation for the bright fringes shouldbe written as

n1L1 - n2L2 = m λ/2 (m = . . . - 1, 0, 1, 2, . . .). (0-17a)

Thus, any change in the refractive index in the regionscan also contribute to the interference pattern as youwill see in Project #6.

In optical system design, interferometers such as theMichelson interferometer can be used to measure verysmall distances. For example, a movement of one of themirrors by only one quarter wavelength (corresponding

sin

d

d

Planewave Grating

Light diffractedat

dθ

d∆ θx=d

dθ

dθ

Figure 0.19. Diffraction of light by a diffractiongrating.

16

to a path-length change of one half wavelength) changesthe detected irradiance at the screen from a maximum toa minimum. Thus, devices containing interferometerscan be used to measure movements of a fraction of awavelength. One application of interference that hasdeveloped since the invention of the laser is holography.This fascinating subject is explored in a separate set ofexperiments in Newport’s Projects in Holography.

0.4.3. The Diffraction Grating

It is a somewhat confusing use of the term to call theitem under discussion a diffraction grating. Althoughdiffraction does indeed create the spreading of lightfrom a regular array of closely spaced narrow slits, it isthe combined interference of multiple beams thatpermits a diffraction grating to deflect and separate thelight. In Fig. 0.19 a series of narrow slits, each separatedfrom its neighboring slits by distance d, are illuminatedby a plane wave. Each slit is then a point (actually a line)source in phase with all other slits. At some angle θd tothe grating normal, the path-length difference betweenneighboring slits will be (see inset to Fig. 0.19)

∆x = d sin(θd),

Constructive interference will occur at that angle if thepath-length difference ∆x is equal to an integral numberof wavelengths:

m λ = d sin(θd) (m = an integer). (0-19)

This equation, called the grating equation, holds for anywavelength. Since any grating has a constant slit separa-tion d, light of different wavelengths will be diffracted atdifferent angles. This is why a diffraction grating can beused in place of a prism to separate light into its colors.Because a number of integers can satisfy the gratingequation, there are a number of angles into whichmonochromatic light will be diffracted. This will beexamined in Project #5. Therefore, when a grating isilluminated with white light, the light will be dispersedinto a number of spectra corresponding to the integersm = . . ., ±1, ±2, . . ., as illustrated in Fig. 0.20(a). Byinserting a lens after the grating, the spectra can bedisplayed on a screen one focal length from the lens, Fig.0.20(b). These are called the orders of the grating andare labeled by the value of m.

0.5. PolarizationSince electric and magnetic fields are vector quantities,both their magnitude and direction must be specified.But, because these two field directions are alwaysperpendicular to one another in non-absorbing media,

Figure 0.20. Orders of diffraction from a gratingilluminated by white light. (a) Rays denoting theupper and lower bounds of diffracted beams for thered (R) and blue (B) ends of the spectrum; (b) spectraproduced by focusing each collimated beam ofwavelengths to a point in the focal plane.

(a)

(b)

1

1

WhiteLight

Grating

R

B

1

1

R

B

-1

-1

R

B

0W

0W

-1

-1

RB

-2-2

RB

-2-2

RB

WhiteLight

Grating

R

B

R

B

R

B

R

Bf

f

2nd Order

-2nd Order

1st Order

0th Order

-1st Order

17

Figure 0.21. Three special polarization orientations:(a) linear, along a coordinate axis; (b) linear, compo-nents along coordinate axes are in phase (∆Φ∆Φ∆Φ∆Φ∆Φ = 0) andthus produce linear polarization; (c) same compo-nents, 90° out of phase, produce elliptical polariza-tion.

the direction of the electric field of a light wave is usedto specify the direction of polarization of the light. Thekind and amount of polarization can be determined andmodified to other types of polarization. If you under-stand the polarization properties of light, you cancontrol the amount and direction of light through theuse of its polarization properties.

0.5.1. Types of Polarization

The form of polarization of light can be quite complex.However, for most design situations there are a limitednumber of types that are needed to describe thepolarization of light in an optical system. Fig. 0.21shows the path traced by the electric field during onefull cycle of oscillation of the wave (T = 1/ν) for anumber of different types of polarization, where ν is thefrequency of the light. Fig. 0.21(a) shows linearpolarization, where orientation of the electric fieldvector of the wave does not change with time as thefield amplitude oscillates from a maximum value in onedirection to a maximum value in the opposite direction.The orientation of the electric field is referenced tosome axis perpendicular to the direction of propaga-tion. In some cases, it may be a direction in the labora-tory or optical system, and it is specified as horizontallyor vertically polarized or polarized at some angle to acoordinate axis.

Because the electric field is a vector quantity, electricfields add as vectors. For example, two fields, Ex and Ey,linearly polarized at right angles to each other andoscillating in phase (maxima for both waves occur atthe same time), will combine to give another linearlypolarized wave, shown in Fig. 0.21(b), whose direction(tanθ = Ey /Ex) and amplitude (√Ex

2+Ey2) are found by

addition of the two components. If these fields are 90°out of phase (the maximum in one field occurs whenthe other field is zero), the electric field of the com-bined fields traces out an ellipse during one cycle, asshown in Fig.0.21(c). The result is called ellipticallypolarized light. The eccentricity of the ellipse is theratio of the amplitudes of the two components. If thetwo components are equal, the trace is a circle. Thispolarization is called circularly polarized. Since thedirection of rotation of the vector depends on therelative phases of the two components, this type ofpolarization has a handedness to be specified. If theelectric field coming from a source toward the observerrotates counterclockwise, the polarization is said to beleft handed. Right-handed polarization has theopposite sense, clockwise. This nomenclature appliesto elliptical as well as circular polarization. Light whosedirection of polarization does not follow a simplepattern such as the ones described here is sometimes

y

x

(a)

y

x

Ey

Ex

∆φ = 90°

(b)

y

x

Ey

Ex

∆φ = 90°

(c)

18

referred to as unpolarized light. This can be somewhatmisleading because the field has an instantaneousdirection of polarization at all times, but it may not beeasy to discover what the pattern is. A more descriptiveterm is randomly polarized light.

Light from most natural sources tends to be randomlypolarized. While there are a number of methods ofconverting it to linear polarization, only those that arecommonly used in optical design will be covered. Onemethod is reflection, since the amount of light reflectedoff a tilted surface is dependent on the orientation ofthe incident polarization and the normal to the surface.A geometry of particular interest is one in which thepropagation direction of reflected and refracted rays atan interface are perpendicular to each other, as shownin Fig. 0.22. In this orientation the component of lightpolarized parallel to the plane of incidence (the planecontaining the incident propagation vector and thesurface normal, i.e., the plane of the page for Fig. 0.22)is 100% transmitted. There is no reflection for thispolarization in this geometry. For the component oflight perpendicular to the plane of incidence, there issome light reflected and the rest is transmitted. Theangle of incidence at which this occurs is calledBrewster’s angle, θB, and is given by:

tanθB = ntrans/nincident (0-20 )

As an example, for a crown glass, n = 1.523, and theBrewster angle is 56.7°. Measurement of Brewster'sangle is part of Project #8.

Sometimes only a small amount of polarized light isneeded, and the light reflected off of a single surfacetilted at Brewster’s angle may be enough to do the job.If nearly complete polarization of a beam is needed, onecan construct a linear polarizer by stacking a number ofglass slides (e.g., clean microscope slides) at Brewster’sangle to the beam direction. As indicated in Fig. 0.23,each interface rejects a small amount of light polarizedperpendicular to the plane of incidence.

The “pile of plates” polarizer just described is some-what bulky and tends to get dirty, reducing its effi-ciency. Plastic polarizing films are easier to use andmount. These films selectively absorb more of onepolarization component and transmit more of the other.The source of this polarization selection is the alignedlinear chains of a polymer to which light-absorbingiodine molecules are attached. Light that is polarizedparallel to the chains is easily absorbed, whereas lightpolarized perpendicular to the chains is mostly trans-mitted. The sheet polarizers made by Polaroid Corpora-tion are labeled by their type and transmission. Three

Figure 0.22. Geometry for the Brewster angle.

Figure 0.23. A “Pile of Plates” polarizer. This deviceworking at Brewster angle, reflects some portion ofthe perpendicular polarization (here depicted as adot, indicating an electric field vector perpendicularto the page) and transmits all parallel polarization.After a number of transmissions most of the perpen-dicular polarization has been reflected away leaving ahighly polarized parallel component.

Parallel

Perpendicular

No reflection of parallelpolarization

θ B

θ B

θB

θB

19

common linear polarizers are HN-22, HN-32, and HN-38, where the number following the HN indicates thepercentage of incident unpolarized light that is trans-mitted through the polarizer as polarized light.

When you look through a crystal of calcite (calciumcarbonate) at some writing on a page, you see a doubleimage. If you rotate the calcite, keeping its surface onthe page, one of the images rotates with the crystalwhile the other remains fixed. This phenomenon isknown as double refraction. (Doubly refracting is theEnglish equivalent for the Latin birefringent.) If weexamine these images through a sheet polarizer, wefind that each image has a definite polarization, andthese polarizations are perpendicular to each other.

Calcite crystal is one of a whole class of birefringentcrystals that exhibit double refraction. The physicalbasis for this phenomenon is described in detail in mostoptics texts. For our purposes it is sufficient to knowthat the crystal has a refractive index that varies withthe direction of propagation in the crystal and thedirection of polarization. The optic axis of the crystal(no connection to the optical axis of a lens or a system)is a direction in the crystal to which all other directionsare referenced. Light whose component of the polariza-tion is perpendicular to the optic axis travels throughthe crystal as if it were an ordinary piece of glass with asingle refraction index, n0. Light of this polarization iscalled an ordinary ray. Light polarized parallel to aplane containing the optic axis has a refractive indexthat varies between n0 and a different value, ne. Thematerial exhibits a refractive index ne where the fieldcomponent is parallel to the optic axis and the directionof light propagation is perpendicular to the optic axis.Light of this polarization is called an extraordinary ray.The action of the crystal upon light of these twoorthogonal polarization components provides thedouble images and the polarization of light by transmis-sion through the crystals. If one of these componentscould be blocked or diverted while the other compo-nent is transmitted by the crystal, a high degree ofpolarization can be achieved.

In many cases polarizers are used to provide informa-tion about a material that produces, in some manner, achange in the form of polarized light passing through it.The standard configuration, shown in Fig. 0.24, consistsof a light source S, a polarizer P, the material M, anotherpolarizer, called an analyzer A, and a detector D.Usually the polarizer is a linear polarizer, as is theanalyzer. Sometimes, however, polarizers that produceother types of polarization are used.

The amount of light transmitted by a polarizer dependson the polarization of the incident beam and the quality

Figure 0.24. Analysis of polarized light. Randomlypolarized light from source S is linearly polarizedafter passage through the polarizer P with irradianceI0. After passage through optically active material M,the polarization vector has been rotated through anangle θθθθθ. (The dashed line of both polarizers A and Pdenote the transmission axes; the arrow indicates thepolarization of the light.) The light is analyzed bypolarizer A, transmitting an amount I0cos2θθθθθ that isdetected by detector D.

cos

SP

M

A

D

θI02

θ

20

of the polarizer. Let us take, for example, a perfectpolarizer — one that transmits all of the light for onepolarization and rejects (by absorption or reflection) allof the light of the other polarization. The direction ofpolarization of the transmitted light is the polarizationaxis, or simply the axis of the polarizer. Since randomlypolarized light has no preferred polarization, therewould be equal amounts of incident light for twoorthogonal polarization directions. Thus, a perfectlinear polarizer would have a Polaroid designation ofHN-50, since it would pass half of the incident radiationand absorb the other half. The source in Fig. 0.24 israndomly polarized, and the polarizer passes linearlypolarized light of irradiance Io. If the material M changesthe incident polarization by rotating it through an angleθ, what is the amount of light transmitted through ananalyzer whose transmission axis is oriented parallel tothe axis of the first polarizer? Since the electric field is avector, we can decompose it into two components, oneparallel to the axis of the analyzer, the other perpen-dicular to this axis. That is

E = E0 cosθ ê|| + E0 sinθ ê⊥ (0-21)

(Note that the parallel and perpendicular componentshere refer to their orientation with respect to the axis ofthe analyzer and not to the plane of incidence as in thecase of the Brewster angle.) The transmitted field is theparallel component, and the transmitted irradiance Itrans

is the time average square of the electric field

Itrans = ⟨E02

cos2θ⟩ = ⟨E02⟩ cos2θ

or

Itrans = I0 cos2θ (0-22)

This equation, which relates the irradiance of polarizedlight transmitted through a perfect polarizer to theirradiance of incident polarized light, is called the Lawof Malus, after its discoverer, Etienne Malus, an engi-neer in the French army. For a nonperfect polarizer, I0

must be replaced by α I0, where α is the fraction of thepreferred polarization transmitted by the polarizer.

0.5.2. Polarization Modifiers

Besides serving as linear polarizers, birefringentcrystals can be used to change the type of polarizationof a light beam. We shall describe the effect that thesepolarization modifiers have on the beam and leave theexplanation of their operation to a physical optics text.

In a birefringent crystal, light whose polarization isparallel to the optic axis travels at a speed of c /n||; for apolarization perpendicular to that, the speed is c /n ⊥. Incalcite n ⊥ > n||, and therefore the speed of light polar-ized parallel to the optic axis, v||, is greater than v⊥.

21

Thus, for calcite, the optic axis is called the fast axisand a perpendicular axis is the slow axis. (In othercrystals n|| may be greater than n⊥ and the fast-slowdesignation would have to be reversed.)

The first device to be described is a quarter-waveplate. The plate consists of a birefringent crystal of aspecific thickness d, cut so that the optic axis is parallelto the plane of the plate and perpendicular to the edge,as shown in Fig.0.25. The plate is oriented so that itsplane is perpendicular to the beam direction and its fastand slow axes are at 45° to the polarized direction ofthe incident linearly polarized light. Because of this 45°geometry, the incident light is split into slow and fastcomponents of equal amplitude traveling through thecrystal. The plate is cut so that the components, whichwere in phase at the entrance to the crystal, travel atdifferent speeds through it and exit at the point whenthey are 90°, or a quarter wave, out of phase. Thisoutput of equal amplitude components, 90° out ofphase, is then circularly polarized. It can be shown thatwhen circularly polarized light is passed through thesame plate, linearly polarized light results. Also, itshould be noted that if the 45° input geometry is notmaintained, the output is elliptically polarized. Theangle between the input polarization direction and theoptic axis determines the eccentricity of the ellipse.

If a crystal is cut that has twice the thickness of thequarter-wave plate, one has a half-wave plate. In thiscase, linearly polarized light at any angle θ with respectto the optic axis provides two perpendicular compo-nents which end up 180° out of phase upon passagethrough the crystal. This means that relative to one ofthe polarizations, the other polarization is 180° from itsoriginal direction. These components can be combined,as shown in Fig. 0.26, to give a resultant whose direc-tion has been rotated 2θ from the original polarization.Sometimes a half-wave plate is called a polarizationrotator. It also changes the “handedness” of circularpolarization from left to right or the reverse. Thisdiscussion of wave plates assumes that the crystalthickness d is correct only for the wavelength of theincident radiation. In practice, there is a range ofwavelengths about the correct value for which thesepolarization modifiers work fairly well.

Waveplates provide good examples of the use ofpolarization to control light. One specific demonstra-tion that you will perform as part of Project #9 con-cerns reflection reduction. Randomly polarized light issent through a polarizer and then through a quarterwave plate to create circularly polarized light, as notedabove. When circularly polarized light is reflected off a

Figure 0.26. Half-wave plate. The plate produces a180° phase lag between the E|| and E⊥⊥⊥⊥⊥ components ofthe incident linearly polarized light. If the originalpolarization direction is at an angle θθθθθ to the optic axis,the transmitted polarization is rotated through 2θθθθθfrom the original.

Figure 0.25. Quarter-wave plate. Incident linearlypolarized light is oriented at 45° to the optic axis sothat equal E|| and E⊥⊥⊥⊥⊥ components are produced. Thethickness of the plate is designed to produce a phaseretardation of 90° of one component relative to theother. This produces circularly polarized light. Atany other orientation elliptically polarized light isproduced.

90° outof phase

d

Circularlypolarized

Optic Axis

E

E⊥

E E⊥

E total

Linearlypolarized

In phase

E

E⊥

E

θθ

180° outof phasewith originalE•fieldcomponents

d

E⊥

E

θ

θ

Optic Axis

E

22

surface, its handedness is reversed (right to left or left toright). When the light passes through the quarter waveplate a second time, the circularly polarized light of theopposite handedness is turned into linearly polarizedlight, but rotated 90° with respect to the incident polariza-tion. Upon passage through the linear polarizer a secondtime, the light is absorbed. However, light emanating frombehind a reflective surface (computer monitor, forexample) will not be subject to this absorption and alarge portion will be transmitted by the polarizer. Acomputer anti-reflection screen is an application of thesedevices. Light from the room must undergo passagethrough the polarizer-waveplate combination twice andis, therefore suppressed, whereas light from the com-puter screen is transmitted through the combination butonce and is only reduced in brightness. Thus, the con-trast of the image on the computer screen is enhancedsignificantly using this polarization technique.

0.6 LasersThe output of a laser is very different than most otherlight sources. After a description of the simplest type ofbeam, the TEM00 mode Gaussian beam and its parameters,we look at means of collimating the beam. The effect of alaser’s construction on its output and a method by whichthis output can be measured will be discussed.

0.6.1. Characteristics of a Gaussian Beam

The term Gaussian describes the variation in the irradi-ance along a line perpendicular to the direction ofpropagation and through the center of the beam, asshown in Fig. 0.27. The irradiance is symmetric about thebeam axis and varies radially outward from this axis withthe form

I r I e r r( ) /= −0

2 212

(0-23)

or in terms of a beam diameter

I d I e d d( ) /= −0

2 212

where rl and dl are the quantities that define the radialextent of the beam. These values are, by definition, theradius and diameter of the beam where the irradiance is1/e2 of the value on the beam axis, I0.

0.6.1.1. Beam Waist and Beam Divergence

Figure 0.27 shows a beam of parallel rays. In reality, aGaussian beam either diverges from a region where thebeam is smallest, called the beam waist, or converges toone, as shown in Fig. 0.28. The amount of divergence orconvergence is measured by the full angle beam diver-gence θ, which is the angle subtended by the 1/e2 diam-

Figure 0.27. Gaussian beam profile. Plot of irradi-ance versus radial distance from the beam axis.[Elements of Modern Optical Design, Donald C.O’Shea, copyright ©, J. Wiley & Sons, 1985. Re-printed by permission of John Wiley & Sons, Inc.]

r

r

d1 r1 I 0e-2

l0 I

23

eter points for distances far from the beam waist asshown in Fig. 0.28. In some laser texts and articles, theangle is measured from the beam axis to the 1/e2 asymp-tote, a half angle. However, it is the full angle divergence,as defined here, that is usually given in the specificationsheets for most lasers. Because of symmetry on eitherside of the beam waist, the convergence angle is equal tothe divergence angle. We will refer to the latter in bothcases.

Under the laws of geometrical optics a Gaussian beamconverging at an angle of θ should collapse to a point.Because of diffraction, this, does not occur. However, atthe intersection of the asymptotes that define θ, thebeam does reach a minimum value d0 the beam waistdiameter. It can be shown that for a TEM00 mode d0

depends on the beam divergence angle as:

d04= λπθ (0-24)

where λ is the wavelength of the radiation. Note that fora Gaussian beam of a particular wavelength, the productd0θ is constant. Therefore for a very small beam waist thedivergence must be large, for a highly collimated beam(small θ), the beam waist must be large.

The variation of the beam diameter in the vicinity of thebeam waist is shown in Fig. 0.28 and given as

d 2=d 02+θ 2z 2 (0-25)

where d is the diameter at a distance ±z from the waistalong the beam axis.

0.6.1.2. The Rayleigh Range

It is useful to characterize the extent of the beam waistregion with a parameter called the Rayleigh range. (Inother descriptions of Gaussian beams this extent issometimes characterized by the confocal beam parame-ter and is equal to twice the Rayleigh range.) RewritingEq. 0.25 as

d d z d= +0 021 ( / )θ (0-26)

we define the Rayleigh range as the distance from the

beam waist where the diameter has increased to d0 2 .Obviously this occurs when the second term under theradical is unity, that is, when

z=zR=d0/θ (0-27)

Although the definition of zR might seem rather arbitrary,this particular choice offers more than just convenience.Figure 0.29 shows a plot of the radius of curvature of thewavefronts in a Gaussian beam as a function of z. Forlarge distances from the beam waist the wavefronts are

Figure 0.28. Variation of Gaussian beam diameter inthe vicinity of the beam waist. The size of the beam atits smallest point is d0; the full angle beam diver-gence, defined by the smallest asymptotes for the l/e2

points at a large distance from the waist is θθθθθ.

θd0

z1

2

(z )1d

line describing the 1/epoints

24

nearly planar, giving values tending toward infinity. Atthe beam waist the wavefronts are also planar, and,therefore, the absolute value of the radius of curvatureof the wavefronts must go from infinity at large dis-tances through a minimum and return to infinity at thebeam waist. This is also true on the other side of thebeam waist but with the opposite sign. It can be shownthat the minimum in the absolute value of the radius ofcurvature occurs at z = ± zR, that is, at a distance oneRayleigh range either side of the beam waist. From Fig.0.29, the “collimated” region of Gaussian beam waistcan be taken as 2zR.

The Rayleigh range can be expressed in a number ofways:

(0-28)

From this we see that all three characteristics of aGaussian beam are dependent on each other. Given anyof the three quantities, d0 θ , zR, and the wavelength ofthe radiation, the behavior of the beam is completelydescribed. Here, for example, if a helium-neon laser(λ=633 nm) has a specified TEM 00 beam diameter of1mm, then

θ = 4λ/πd0 = (1.27 x 6.33 x 10-7m)/(1 x 10-3m) = 0.8 mrad

and

zR = d0/θ = (l x 10-3m)/(0.8 x 10-3rad) = l.25 m.

The Rayleigh range of a typical helium-neon laser isconsiderable.

0.6.2 Collimation of a Laser Beam

Through the use of lenses the divergence, beam waist,and Rayleigh range of the Gaussian beam can bechanged. However, from the above discussion it is clearthat the relations between the various beam parame-ters cannot be changed. Thus, to increase the collima-tion of a beam by reducing the divergence requires thatthe beam waist diameter be increased, since the beamwaist diameter-divergence product is constant. This isdone by first creating a beam with a strong divergenceand small beam waist and then putting the beam waistat the focal point of a long focal length lens. What thisamounts to is putting the beam through a telescope —backward. The laser beam goes in the eyepiece lens andcomes out the objective lens.

There are two ways of accomplishing this. One uses aGalilean telescope, which consists of a negative eye-piece lens and a positive objective lens, as shown inFig. 0.30(a). The light is diverged by the negative lensproducing a virtual beam waist and the objective lens ispositioned at a separation equal to the algebraic sum of

Figure 0.29. Plot of radius of curvature (R) versusdistance (z) from the beam waist. The absolute valueof the radius is a minimum at the Rayleigh rangepoint, zR. In the limit of geometrical optics, the radiusof curvature of the wavefronts follows the dashedline.

zd d

R = = =02

024

4θλ

πθπ

λz

R

z2

-z zR

R

R

25

the focal lengths of the lenses to produce a morecollimated beam. It can be shown that the decrease inthe divergence is equal to the original divergencedivided by the magnification of the telescope. Themagnification of the telescope is equal to the ratio ofthe focal lengths of the objective divided by the eye-piece The second method uses a Keplerian telescope(Fig. 0.30(b)). The eyepiece lens is a positive lens so thebeam comes to a focus and then diverges to be colli-mated by the objective lens.

Project #3 will demonstrate the design of these twotypes of laser beam expanders. Each has distinctadvantages. The advantage of the Galilean type of beamexpander occurs for high power or pulsed laser sys-tems. Since the beam does not come to a focus any-where inside of the beam expander’s optical path, thepower density of the beam decreases. Thus, if thelenses and environment can survive the initial beam,they can survive the beam anywhere in the opticalpath. Although the Keplerian beam expander can givesimilar ratios of beam expansion, the power density atthe focus of the first lens is very large. In fact, with ahigh power, pulsed laser it is possible to cause abreakdown of the air in the space between the lenses.This breakdown is caused by the very strong electricalfield that results from focusing the beam to a smalldiameter creating miniature lightning bolts. (Manyresearchers have been unpleasantly surprised whenthese “miniature” lightning bolts destroyed some veryexpensive optics!)

The primary advantage for the Keplerian beam ex-pander is that a pinhole of an optimum diameter can beplaced at the focus of the first lens to “clean” up thelaser beam by rejecting the part of the laser energy thatis outside of the pinhole diameter. This concept of“spatial filtering” will be explored in Project #10.

0.6.3 Axial Modes of a Laser

The properties of laser light include monochromaticity,low divergence (already explored in the previoussections), and a high degree of coherence, whichencompasses both of these properties. This section is adiscussion of the coherence of the laser and a historicalexperiment that illustrates one of the concepts using amodern device.

A complete discussion of the principle of laser actionwould take a substantial amount of space and readingtime. For an explanation of the concept we refer you tothe references. The basis of lasers is a physical processcalled stimulated emission. It appears as the third andfourth letters of the acronym, LASER (Light Amplifica-tion by Stimulated Emission of Radiation). Amplification

Figure 0.30. Gaussian beam collimation. (a) Galileantelescope. (b) Keplerian telescope. Eyepiece focallength, fe; objective focal length, fo.

(a)

(b)

fo

fe

fo

fe

26

is only the beginning of the process in most lasers,since the increase in light as it passes through anamplifying volume is usually quite modest. If theradiation was only amplified during a single passthrough the volume, it would be only marginally useful.However, when mirrors are placed at both ends of theamplifying medium, the light is returned to the mediumfor additional amplification (Fig. 0.31). The usefuloutput from the laser comes through one of the mir-rors, which reflects most of the light, but transmits asmall fraction of the light, usually on the order of 5%(up to 40% for high power lasers). The other mirror istotally reflecting. But the laser mirrors do more thanconfine most of the light. They also determine thedistribution of wavelengths that can support amplifica-tion in the laser.

The mirrors serve as a simple, but effective, interfer-ometer and for only certain wavelengths, just as in thecase of the Michelson interferometer, will there beconstructive interference. The mirrors form a resonantstructure that stores or supports only certain frequen-cies. It is best compared to the resonances of a guitarstring in which the note that the string produces whenplucked is determined by the length of the string. Bychanging the location of the finger on a guitar string adifferent note is played. The note (really, notes) isdetermined by the amount of tension the guitarist hasput on the string and the length of the string. Anyfundamental physics book will show that the conditionsimposed upon the string of length L will produce a notewhose wavelength is such that an integral number ofhalf wavelengths is equal L,

q λ /2 = L. (0-29)

In Fig. 0.32 a standing wave with three half wave-lengths is shown. In most lasers, unless special precau-tions are taken, a number of wavelengths will satisfythis resonance condition. These wavelengths arereferred to as the axial modes of the laser. Since L=q λ /2, where q is an integer, the wavelengths supported bythe laser are

λq = 2L/q (0-29a)

The frequencies of these modes are given by ν = c /λ,where c is the speed of light.

Inserting the expression for the wavelengths, thefrequencies of the resonant modes are given by

νq = q (c /2L),

where q is an integer. The frequency separationbetween these axial modes equals the differencebetween modes whose integers differ by one:

Figure 0.32. Standing wave picture.

Figure 0.31. The laser cavity. The distance betweenmirrors is an important parameter in the output of alaser.

L

Output Beam

100% ReflectingMirror

95% Reflecting -5% Transmitting Mirror

3λ/2

27

∆n = νq+1 - νq = (q +1)c /2L - q c /2L = c /2L, (0-30)

so the separation between neighboring modes of a laseris constant and dependent only on the distance be-tween the mirrors in the laser, as shown in Fig. 0.33.Since the amount of power obtained from small helium-neon lasers, such as those used for the projects de-scribed in this manual, is related to the length of thelaser, the separation between mirrors is set by the lasermanufacturers to produce the required power for thelaser. But the band of wavelengths that can maintainstimulated emission is determined by the atomicphysics of the lasing medium, in this case, neon. Thatband does not change radically for most helium-neonlaser tubes. Therefore, the number of axial modes ismainly dependent on the distance between the mirrors,L. The farther apart the mirrors are, the closer are theaxial mode frequencies. Thus, long high power helium-neon lasers have a large number of axial modes,whereas, the modest power lasers used in this Projectsin Optics kit produce only a small number (usuallythree) of axial modes.

One of the other relations between neighboring lasermodes, beside their separation, is that their polariza-tion is orthogonal (crossed) to that of their neighbors(Fig. 0.34). Thus, if we examined a three-mode laserwith the appropriate tools, we would expect to find thattwo of the modes would have one polarization and theother would have a perpendicular polarization. Thismeans that, while axial modes are separated in fre-quency by c /2L, modes of the same polarization areseparated by c /L.

Looking through a diffraction grating at the output of athree-mode laser, we see a single color. High resolutioninterferometers must be employed to display the axialmodes of a laser. However, it is also possible to use aMichelson interferometer to investigate the modeswithout resorting to high resolution devices. Thistechnique has special applications in the infraredregion of the spectrum.

0.6.4 Coherence of a Laser

If we speak of something as being “coherent” in every-day life, we usually mean that it, a painting, a work ofmusic, a plan of action, “makes sense.” It “hangstogether.” There is in this concept the idea of consis-tency and predictability. The judgement of what iscoherent, however, is one of individual taste. What oneperson may find coherent in heavy metal rock music,another person would hear in rhythm and blues … orelevator music, perhaps. This concept of coherence asa predictable, consistent form of some idea or work ofart has much the same meaning when applied to light

Figure 0.34. Output from a three mode laser. Therelative polarization of each mode is indicated at itsbase.

Figure 0.33. Laser mode distribution. Plot of powerin laser output as a function of frequency.

c L2

c L2

c L2

28

sources. How consistent is a light field from one pointto another? How do you make the comparison? Theinterference of the light beam with itself does thecomparing. If there is a constant relation between onepoint on a laser beam and another point, then theinterference of waves separated by that distance shouldproduce a stable interference pattern. If, however, theamplitude or phase or wavelength changes betweenthese two points, the interference, while it is still thereat all times, will constantly vary with time. This un-stable interference pattern may still exhibit fringes, butthe fringes will be washed out. This loss of visibility offringes as a function of the distance between the pointsof comparison is measure of the coherence of the light.This visibility can be measured by the contrast of theinterference fringes. The contrast is defined by

CI II I

= −+

max min

max min(0-31)

where Imax is the irradiance of the bright interferencefringes and Imin is the irradiance of the dark interferencefringes (Fig. 0.35). This contrast is determined bypassing the light from the source through a Michelsoninterferometer with unequal arms. By changing thepath length difference between the arm in the interfer-ometer, the visibility of the fringes as a function of thisdifference can be recorded. From these observations,the measurement of the coherence of a source can bedone using a Michelson interferometer.

If a source were absolutely monochromatic, therewould be no frequency spread in its spectrum. That is,its frequency bandwidth would be zero. For this to betrue, all parts of the wave exhibit the same sinusoidaldependence from one end of the wave to the other.Thus, a truly monochromatic wave would never showany lack of contrast in the fringes, no matter how largeof a path length difference was made. But all sources,even laser sources contain a distribution of wave-lengths. Therefore, as the path length difference isincreased, the wavefront at one point on the beam getsout of phase with another point on the beam. A meas-ure of the distance at which this occurs is the coher-ence length lc of a laser. It is related to the frequencybandwidth of a laser by

∆ν = c /lc (0-32)

Any measurement of the coherence length of a lightsource by observation of the visibility of fringes from aMichelson interferometer will yield information on thebandwidth of that source and, therefore, its coherence.For example, suppose the source is a laser with somebroadening. As the length of the one of the arms in a

Figure 0.35. Contrast.

Figure 0.36. Visibility function.

I I

I

max

min

100% Contrast

I

I

I

max

min

50% Contrast

∆ x

AB

C

Contrast

A B C

29

Michelson interferometer, as shown in Fig. 0.36,becomes unequal (mirror moved from A to B), the onepart of a wave will interfere with another part that isdelayed by a time equal to the difference in path lengthdivided by the speed of light. Eventually the wavesbegin to get out of step and the fringe contrast beginsto fall because the phase relations between the twowaves is varying slightly due to the spread in frequen-cies in the light. The greater the broadening, the morerapidly the visibility of the fringes will go to zero.

One particularly interesting case consists of a sourcewith only a few modes present as is the case for thethree-mode helium-neon laser discussed above. Be-cause only light of the same polarization can interfere,there will be two modes (λ1, λ3) in the laser that caninterfere with each other. The third mode (λ2) withorthogonal polarization is usually eliminated by passingthe output of the laser through a polarizer. With theinterferometer mirrors set at equal path length thereare two sets of fringes, one from each mode. Since thepath length difference is zero, these two sets of highcontrast fringes overlap each other. But as the pathlength increases, the fringes begin to get out of step.Until, finally, the interference maximum of one set offringes overlaps the interference minimum of the otherset of fringes and the fringe contrast goes to zero. Thecalculation of this condition is fairly simple. Thecondition for an interference maximum is given by

L1 - L2 = mλ/2 m = an integer (0-33)

and for an interference minimum by

L1 - L2 = mλ/4 m = odd integers (0-34)

If we assume that the change in path length is from zeropath length to the point where the visibility first goes tozero, then for one wavelength, λ1,

L1 - L2 = mλ1/2 m = an integer (0-35)

and for the other mode with the same polarization,there is a minimum.

L1 - L2 = mλ3/2 + λ3/4 (0-35)

Equating these two expressions and rearranging terms,gives

mλ1/2 - mλ3/2 = m(λ1-λ3)/2 = λ3/4. (0-35)

or

m∆λ = λ3/2

Wavelength separation can be expressed as a frequencyseparation by ∆ν

∆λ = λ∆ν/ν (0-36)

30

where λ and ν are the average values in the intervals ∆λand ∆ν. Inserting this expression for ∆λ, we obtain

∆ν = ν /2m . (0-37)

The integer m is an extremely large number in mostcases and is not easily determined, but it is related tothe average wavelength of the source by L1 - L2 = mλ /2.If we set ∆L = L1 - L2, solve for m and insert in theexpression for ∆ν,

∆ν = ν /2m = λν /4∆L = c /4∆L, (0-38)

since λν = c.

Thus the frequency separation between modes can bemeasured by determining the path length differencewhen the two interference fringe patterns are out ofstep with one another, causing the visibility to go tozero, as depicted in Fig. 0.37. It can also be demon-strated that there are additional minima in the visibilityat ∆ν =3c /4∆L, 5c /4∆L, etc. Visibility maxima occurhalfway between these minima as the two fringespatterns get back into step. In Project#7, this effect willenable you to determine the mode separations for thelaser used in these projects. What has been derivedhere is a simple case of a much more involved applica-tion of this technique. It is possible to measure thefringe contrast as a function of mirror position (calledan interferogram) and store it in the memory of acomputer. It has been shown that a mathematicaltransformation (the same Fourier Transform that willbe discussed in the next section) of the visibilityfunction yields the frequency spectrum of the source.

While it might be considered difficult, the advent ofpowerful computers has reduced the cost and en-hanced the utility of this technique, particularly in thefar infrared part of the spectrum. These devices areknown as Fourier transform spectrometers.

0.7 Abbe Theory of ImagingThe earlier discussion of imaging depended upontracing a series of rays to determine the location andsize of the image. It was shown that only a few rayswere needed. This approach ignores the possibilitiesthat the source could be monochromatic and suffi-ciently coherent that diffraction and interference effectscould play a part in the formation of an image. What wewill describe and then demonstrate in Project #10, isthat after the light that will form an image has traversedthe lens, we can intervene and change the image in veryspecial ways. This approach to imaging has found usein a number of applications in modern optics. To beginto understand this concept, we need to review brieflythe diffraction grating discussed in Section 0.4.3, since

Figure 0.37. Visibility function for two mode system.

Figure 0.38. Diffraction orders.

IrradianceEnvelopeof fringeIrradianceFringes

Contrast

r∆

MonochromaticBeam Lens

DiffractionGrating

Screen

= +2

= +1

= 0

= -1

= -2

m

m

m

m

m

31

the grating is one of the simplest illustrations of thisnew way of thinking about imaging. Consider a diffrac-tion grating consisting of a series of equally spaced,narrow absorbing and transmitting (black and white)bands. It is possible to determine mathematically notonly the directions of the diffracted orders

sinθm = m λ /d m = an integer, (0-39)

but also the relative irradiances of the diffracted spotsto one another. If we insert a lens after the diffractiongrating, we can relocate the orders of the diffractiongrating from infinity to the back focal plane of the lens(Fig. 0.38). We will see how this can be used to under-stand imaging.

0.7.1 Spatial Frequencies

We are used to the idea of repetitions in time. Electricaland audio sources of signals with single frequencies,particularly as they relate to sound are used to testequipment for their response. A good high fidelitysystem will reproduce a wide range of frequenciesranging from the deep bass around 20 Hz (cycles persecond), that is as much felt as it is heard, to the nearlyimpossible to hear 15,000 Hz, depending on how wellyou have treated your ears during life. As noted earlier,the frequency of the electromagnetic field determineswhether the radiation is visible to the eye. Again, thisperiodic variation in the electric field takes place intime. Just as it is possible to speak about variation ofelectrical waves and sound with time, in optics, varia-tions in space can be expressed as spatial frequencies.These are usually expressed in cycles/mm (or mm-1).They indicate the rapidity with which an object orimage varies in space instead of time. An example thatshows a number of spatial frequencies is given in Fig.0.39.

As in the case of many sounds and electrical signals,most spatially repetitive patterns do not consist of asingle frequency, but as a musical chord, are made upof some fundamental frequency plus its overtones, orhigher harmonics. The discussion of spatial frequenciesin optics is based on some interesting, but relativelycomplicated mathematics. You may want to read thissection once to get the general ideas, then come backlater after you have done Project #10. Certainly, here isa case where hands-on work will improve your under-standing of the discussion of the subject.

An example of an object with a few spatial frequenciesis the diffraction grating. If the grating just discussedconsisted of a sinusoidal variation, as shown in Fig.0.40(a), there would only be a zero order and the first Figure 0.40. Sinusoidal grating versus black and

white grating (Fourier analysis).

Figure 0.39. Spatial frequencies in an object.

Fourier Coefficientsfor Square Grating

f

f0 f f f f f f f2 3 4 5 6 7 8 9f

Fourier Coefficientsfor Sine Grating

f

f0 f f f f f f f2 3 4 5 6 7 8 9f(a)

(b)

32

orders (m = ±1). As repetitive patterns depart fromsinusoidal, additional diffraction orders appear and inthe case of the black and white grating, a whole seriesof diffraction orders are present (Fig. 0.40(b)).

All of this can be expressed mathematically in terms ofFourier (Four-ee-ay) Theory. We will not go into themathematical expression of the theory, but onlygraphically express the result as simply as possible.

Any periodic (repeating) function can be expressed as aseries of sine and cosine functions consisting of thefundamental periodic frequency (f) and its higherharmonics (those frequencies that are multiples of thefundamental frequency, f (2f, 3f, 4f, …) The amount thateach frequency contributes to the original function canbe calculated using some standard integral calculusexpressions. The decomposition of the periodic patterninto its harmonics is referred to as Fourier Analysis.This analysis determines the amplitude of each har-monic contribution to the original function and itsphase relative to the fundamental (in phase or 180° outof phase).

The procedure can be, in a sense, reversed. If a patternat the fundamental frequency is combined with theappropriate amounts of the higher harmonics, it ispossible to approximate any function with a repetitionfrequency of the fundamental. This is referred to asFourier Synthesis. To completely synthesize a functionsuch as our example of an alternating black and whitegrating, an infinite number of harmonics would beneeded. If only frequencies up to some specific valueare used, the synthesized function will resemble thefunction, but it will have edges that are not as sharp asthe original. A simple example (Fig. 0.41) using only afundamental and two harmonics shows the beginning ofthe synthesis of a square wave function, similar to ourblack and white grating. What you will be investigatingin Project #10 are optical techniques that use Fourieranalysis and synthesis in creating images.

0.7.2 Image Formation

If the black and white grating is illuminated with planewaves of monochromatic light, a number of diffractionorders will be generated by the grating. These planewave beams diffracted at different angles given by Eq. 0-39, can be focused with a lens located behind thediffraction grating, as shown in Fig. 0.42. The focusedspots have intensities that are proportional to thesquare of the amplitudes that we could calculate forthis diffraction grating. In effect, the laser plus lenscombination serves as an optical Fourier analyzer for adiffractive object.

Figure 0.41. Fourier synthesis.

sin x

sin 3x

sin 5x

sin x +sin 3x +sin 5x

33

When a laser beam illuminates a grating, the light willdiffract and the spectrum of spatial frequencies will bedisplayed in the back focal plane of the lens. It turns outthat even if the object is not a grating or a series of lineswith a number of repetitive spacings, the light patternin the back focal plane still describes the content ofspatial frequencies found in the object. Objects that arelarge and smoothly varying in their shading representlow spatial frequencies and will not diffract the beammuch. Their contributions, therefore, lay close to theoptical axis of the lens. Objects that are small or havefine detail and sharp edges will cause substantialdiffraction and their contributions will be found furtherfrom the optical axis in the back focal plane of the lens.

Suppose as an object we use a fine, square mesh wirescreen. This is a pair of crossed gratings only coarserthan the diffraction grating just discussed earlier (Fig.0.42). When this square grating is illuminated by a laserbeam, the diffraction orders are focused to a series ofspots at the back of the focal plane of the lens and thespatial frequencies form a two-dimensional grid ofpoints. The separation between the points is deter-mined by the distance between neighboring wires,representing the grating constant for this coarsegrating.

If the lens is more than one focal length from thegrating, then somewhere beyond the Fourier transformplane, a real image of the grating will be formed. Thegeometry is shown in Fig. 0.43. The interesting point ofthis arrangement is that the image can be understoodas a light distribution that arises out of the interferenceof the light from the spatial frequency components ofthe object located at the back focal plane of the lens. Inother words, the image is a Fourier synthesis of the thespatial frequencies in the grating.

This imaging can be thought of as a two step process:Fourier analysis followed by Fourier synthesis. Thisapproach to analyzing images was first proposed byErnst Abbe, a lecturer at the University of Jena, whowas hired by Carl Zeiss to understand and designmicroscope lenses. After some study one sees that thelarger the collection angle of an imaging lens, the higherthe possible resolution, since the higher spatial fre-quency components appear farther from the axis.Although very little light is collected near the edge ofthe lens, it is this light that contributes to the fine detailin high resolution images.

Figure 0.42. Optical Fourier analysis.

Diffractio

n

grating

Focal

plane

Lens

monochromaticbeam

Ordersrepresentingfrequencycomponentsin the grating

34

0.7.3 Spatial Filtering

Since the light in the Fourier transform plane (Fig 0.43)is arranged according to increasing spatial frequencywith radius, then any intervention in that plane in theform of a mask will change the distribution of spatialfrequencies in the plane. It will also change the contentof the image, but in a very predictable way.

The procedure of modifying an image by “changing”the spatial frequencies contained in it is called spatialfiltering. One example of such a procedure is thespatial filtering of a transparency of a picture of atelevision screen. The Fourier transform of the pictureis a rather raggedy-looking patch at the center of thepicture and a series of equally spaced dots arranged ina vertical line. These dots represent a periodic, grating-like feature in the picture. This grating is due to theseries of parallel lines, called a raster, that is used tobuild up an image on the TV screen. The electron beamthat writes on the face of the tube in a TV set does so asa series of parallel lines. By turning the beam on and offas it is swept across the screen and dropping down alittle on each sweep, the circuitry builds up a picture onthe tube. If you look at a TV screen up close you can seethe raster. If the dots represent the raster, where is therest of the image? It resides in the raggedy-lookingpatch at the center of the beam. This analysis andsynthesis process of imagery will be demonstrated asone of the experiments in Project #10.

There are a number of applications that are based onthis approach to imaging. One of these “cleans up” alaser beam. The irradiance distribution of the beam inmany lasers is Gaussian (Section 0.6.1) as it exits theoutput mirror of the laser. However, dust and smallimperfections in the lenses, windows, and surfaces thatit traverses or reflects from can produce irregularitiesin the irradiance pattern. The Gaussian distributionrepresents a low frequency spatial variation in thebeam, whereas the irregularities contain higher spatialfrequencies. When the laser beam is focused with amicroscope objective, as shown in Fig. 0.44, thesevariations are arranged according to their spatialfrequencies. If a small pinhole, whose diameter issufficiently large to pass the low frequency Gaussianportion of the beam and block the high frequency part,the irregularities will be removed from emerging beamand a “clean” laser beam will result.

Laseroutput

“Dirty”beam

Spatialfrequenciesin the beam

Pinholein Fouriertransform

plane

Microscopelens

Imperfectionsand dust oncomponents

“Clean”beam

Figure 0.44. Spatial filtering.

Figure 0.43. Abbe theory of imaging.

Lens

Mono-chromatic

beam

Grating Transformplane

Imageplane

2

1

0

1

2

2

1

0

1

2

35

Another application involves the use of spatial frequen-cies for object recognition. In some areas of photo-graphic surveys, the amount of data to be analyzed isenormous. Provided a feature of interest has someparticular set of spatial frequencies connected with it(spacing between ties in the case of railroads, forexample), the laser and lens combination can be usedto recognize the possible presence of these features inthe photograph. Other applications include inspectionof products, such as the tips of hypodermic needles.The average spatial frequency pattern for a largenumber of good needles is stored in a computermemory. Then the pattern of each new tip is comparedto it. Those needles that do not fit the stored pattern towithin certain criteria are then rejected. Since theactual position of the tip does not affect the spatialfrequency pattern, the test is insensitive to locationerrors, whereas a direct inspection of the image of theneedle tip would have to locate it with a high degree ofprecision.

0.8 ReferencesElements of Modern Optical Design, Donald C. O'Shea

J. Wiley & Sons, Inc.,1985,

Optics (2nd Edition), E. HechtAddison Wesley,

Fundamentals of Optics , F. Jenkins and H. WhiteMcGraw Hill, 1957,

Contemporary Optics for Scientists and Engineers ,NussbaumPrentice-Hall, inc. 1976

The Optics Problem Solver,Research and Education Association, 1986

36

Component Assemblies

All ten experiments use a number of similar componentassemblies. In order to simplify the experimental set upprocedure we have included a section on building theseassemblies. This components section contains drawingsof each assembly and easy to understand instructions.

The parts are labeled with a single letter in the instruc-tions. The Newport catalog number that appears on someof the items is listed next to the part label. The catalognumbers in parentheses denote metric versions of thesame item.

Alignment of Components

All assemblies, except the lens chuck assembly (LCA),are intended to be screwed directly into the rectangularlab bench (sometimes referred to as an optical bread-board, since it can be used to set up optical systems onan experimental basis). Alignment of many of the compo-nents is made simpler by directing the optical pathsalong the screw holes on the surface. To adjust theheight or alignment of a component, rotate the assemblyin the post holder and reposition the post within theholder.

In a number of the later experiments, you will notice thatthere are some standard assemblies in the same loca-tions. One particular geometry that is used for all experi-ments after Project #3, has a U-shaped form: One armconsists of a laser assembly (LA) and beam steeringassembly (BSA-I). The side of the “U” is between this BSA-I and a second BSA-1. In some experiments a laser beamexpander, described and constructed in Project #3, islocated within the arm between the two beam steeringassemblies. The third arm consists of any number ofcomponents that are specific to the particular experi-ment. If the experiments are being done in order, then itshould be possible to construct the basic U-shapedgeometry, LA plus two BSA-I’s, and conduct the balanceof the experiments without having to tear down aftereach set of experiments.

A Note on Handling OpticsWhen handling optical components treat theoptical surfaces of lenses and mirrors with care.Never touch those surfaces. If there are lenstissues or finger cots available, use them toprotect the surfaces from finger oils and con-tamination. If not, handle the components bytheir edges. Assemble components over a tablesurface. (Do not lift components any higher offthe table than necessary)

37

Beam Steering Assembly: (BSA-I)

This assembly consists of:

Part Cat # (Metric #) Qty Description

A 10D20ER.1 1 Mirror 1 inchB UPA1 1 Mirror holderC P100-A (M-P100-A) 1 Mirror mount,adjD COR-1 1 Center of rotation barE SP-3 (M-SP-3) 1 Post, 3 inchesF VPH-2 (M-VHP-2) 1 Post holder, 2 inchesAA 8-32 (M4) 2 Socket hd screwsBB 1⁄4-20 (M6) 1 Socket hd screwsCC 8-32 (M4) 1 Set screw

1. Place the 1 inch mirror (A) into its holder (B). Ifthe mirror seems to be stuck, place severalsheets of lens tissue on the table and gently laythe holder (B) and mirror (A) on the table withthe mirror is facing down. Push straight downuntil the mirror bottoms out in the holder.

Do not move the mirror side to side or severedamage to the mirror coating could result.

2. Put the mirror holder (B) into the adjustablemirror mount (C). Secure the mirror holder (B)into the mount (C) using the set screw (CC) in itsside. Do not tighten more than finger tight.

3. Use one of the two 8-32 socket head cap screws(AA) to bolt the bar (D) that locates the edge ofthe mirror over the center of the post (E). Youmay need to remove the 8-32 set screw thatcomes shipped with the post (E) first. Save thisscrew.

4. Use the second 8-32 socket head cap screw (AA)to bolt the mirror assembly (A-B-C) to the otherhole in the bar (D).

5. Insert a 1⁄4-20 socket head cap screw (BB) into thepost holder (F) from the inside, place the holderover the hole in the optical breadboard wherethe assembly is to be located and tighten untilthe screw bottoms out in the holder (F).

6. Place the post assembly (A-B-C-D-E) in the holder(F) and tighten the side screw sufficiently to holdthe unit together.

NOTE: For some assemblies, the post holder (F) isnot attached directly to the breadboard. Insteadthe cap screw (BB) is inserted through a baseplate (K) from the bottom. This gives the assem-bly a base and allows it to be moved around inthe table.

Optional Base Set-up

(F)

(K) Optional

(A)

(B)

(C)

(CC)

(D)

(AA)

(AA)

(E)

(F)

38

Modified Beam Steering Assembly: (BSA-II)

This assembly consists of:

Part Cat # (Metric #) Qty Description

C P100-P (M-P100-1) 1 Mirror mountE SP-3 (M-SP-3) 1 Post, 3 inchesF VPH-2 (M-VHP-2) 1 Post holder, 2 inchesR RSP-1T 1 Rotation StageBB 1⁄4-20 (M6) 1 Socket head screwsCC 8-32 (M4) 1 Set screw

This modified version of this beam steering assembly(BSA-II) consists of an adjustable mirror mount (C)whose surface (adjustment knob side) is mountedperpendicular to the post (E) with a 8-32 set screw (CC)as shown in the inset. The post holder is attached tothe center plug in the rotation stage (R).

(C)

(E)

(F)

(BB)

(R)

39

Modified Beam Steering Assembly: (BSA-III)

This assembly consists of:

Part Cat # (Metric #) Qty Description

A 10D20ER.1 1 Mirror 1 inchB UPA1 1 Mirror holderC P100-P (M-P100-P) 1 Mirror mount,adjE SP-3 (M-SP-3) 2 Post, 3 inchesF VPH-2 (M-VHP-2) 1 Post holder, 2 inchesH CA-2 1 Variable angle holderBB 1⁄4-20 (M6) 1 Socket hd screwsCC 8-32 (M4) 1 Set screwDD 8-32 (M4) 1 Set screw

1. Place the 1 inch mirror (A) into its holder (B). Ifthe mirror seems to be stuck, place severalsheets of lens tissue on the table and gently laythe holder (B) and mirror (A) on the table withthe mirror is facing down. Push straight downuntil the mirror bottoms out in the holder.

Do not move the mirror side to side or severedamage to the mirror coating could result.

2. Put the mirror holder (B) into the adjustablemirror mount (C). Secure the mirror holder (B)into the mount (C) using the set screw (CC) in itsside. Do not tighten more than finger tight.

3. Insert the 8-32 set screw (DD) into one end of thepost (E). Screw the mirror mount (C) onto this 8-32 set screw.

4. Insert this mirror mount-post combination (E-A-B-C) into the variable angle holder (H) in the holefarthest from the tightening knob. Insert a secondpost (E) into the remaining hole in the variableangle holder (H). Lightly tighten the knob to holdthe two posts in place.

5. Insert a 1⁄4-20 socket head cap screw (BB) into thepost holder (F) from the inside, place the holderover the hole in the optical breadboard wherethe assembly is to be located and tighten untilthe screw bottoms out in the holder (F).

6. Insert the post assembly (E-A-B-C-H-E) into thepost holder (F) and lightly tighten the thumbscrew to hold the assembly together.

This version of the beam steering assembly is intendedto be located in one place and target movement is doneby changing the length and direction of the posts (E) inthe variable angle holder (H).

(C)

(CC)

(H)

(E)(A)

(B)

(F)

(E)

40

Target Assembly (TA-I)

This assembly consists of:

Part Cat # (Metric #) Qty Description

E SP-3 (M-SP-3) 2 Post, 3 inchesF VPH-2 (M-VPH-2) 1 Post holder, 2 inchesG FC-1 (M-FC-1) 1 Filter holderH CA-2 1 Variable angle holderBB 1⁄4-20 (M6) 1 Socket head screwDD 8-32 (M4) 1 Set screw

1. Insert the 8-32 set screw (DD) into one end of thepost (E). Screw the filter holder (G) onto this 8-32set screw.

2. Insert this filter holder-post combination (E-G)into the variable angle holder (H) in the holefarthest from the tightening knob. Insert a secondpost (E) into the remaining hole in the variableangle holder (H). Lightly tighten the knob to holdthe two posts in place.

3. Insert a 1⁄4-20 socket head cap screw (BB) into thepost holder (F) from the inside, place the holderover the hole in the optical breadboard wherethe assembly is to be located and tighten untilthe screw bottoms out in the holder (F).

4. Insert the post assembly (E-G-H-E) into the postholder (F) and lightly tighten the thumb screw tohold the assembly together.

This version of the target assembly is intended to belocated in one place and target movement is done bychanging the length and direction of the posts (E) in thevariable angle holder (H).

(G)

(E)

(E)

(F)

(H)

41

Modified Target Assembly (TA-II)

This assembly consists of:

Part Cat # (Metric #) Qty Description

E SP-3 (M-SP-3) 1 Post, 3 inchesF VPH-2 (M-VPH-2) 1 Post holder, 2 inchesG FC-1 (M-FC-1) 1 Filter holderK B-2SA 1 Base plateBB 1⁄4-20 (M6) 1 Socket head screwDD 1⁄4-20 (M6) 1 Set screw

A modified assembly does not use the variable angleholder (H) and consists of the filter-post combination(E-G) mounted directly into the post holder (F) andbase plate (K) with the 1⁄4-20 socket head screw (BB).This assembly is not intended to be screwed into theoptical bench, but moved around on the base plate (K).

(G)

(E)

(F)

(K)

42

Lens Chuck Assembly (LCA)

This assembly consists of:

Part Cat # (Metric #) Qty Description

E SP-3 (M-SP-3) 1 Post, 3 inchesF VPH-2 (M-VPH-2) 1 Post holder, 2 inchesJ AC-1A 1 Lens chuckK B-2SA 1 Base plateBB 1⁄4-20 (M6) 1 Socket head screwCC 8-32 (M4) 1 Set screw

1. Insert the set screw (CC) into the base of the lenschuck (J).

2. Screw the post (E) onto the set screw (CC) in thebase of the lens chuck (J). Tighten the postfirmly.

3. Place the holder (F) over the center hole in thebase plate (K), insert a 1⁄4-20 screw through thebaseplate (K) and tighten until the screw bottomsin the holder (F).

4. Insert the lens chuck-post combination (E-J) intothe post holder (F) and tighten the thumb screwto hold the assembly together.

This assembly is not intended to be bolted into theoptical bench, but moved around on the table.

(J)

(CC)

(E)

(F)

(K)

43

Rotational Stage Assembly (RSA-I)

This assembly consists of:

Part Cat # (Metric #) Qty Description

E SP-3 (M-SP-3) 1 Post, 3 inchesF VPH-2 (M-VPH-2) 1 Post holder, 2 inchesR RSP-1T 1 Rotation stageBB 1⁄4-20 (M6) 1 Socket head screwCC 8-32 (M4) 1 Set screw

This assembly will be used in two slightly differentversions. The type RSA-I will be have the rotation stage(R) mounted such that the 1 inch clear aperture isperpendicular to the table surface.

1. Insert the set screw (CC) into the post (E). Leaveapproximately 1⁄4 inch protruding from the post.

2. Into one of the holes in the center of the narrowside, screw the rotation stage (R) onto the setscrew (CC) in the post (E).

3. Insert a 1⁄4-20 socket head cap screw (BB) into thepost holder (F) from the inside, place the holderover the hole in the optical breadboard wherethe assembly is to be located and tighten untilthe screw bottoms out in the holder (F).

4. Insert the rotation stage-post assembly (E-R) intothe post holder (F) and tighten the thumb screwto hold the assembly in place.

(E)

(R)

(F)

44

Laser Assembly (LA)

This assembly consists of:

Part Cat # (Metric #) Qty Description

M 41 (M-41) 1 7 inch rodN 340-RC (M-340-RC) 1 Rod clampP ULM 1 1-3⁄4 in laser mountQ R-31005 1 HeNe laserBB 1⁄4-20 (M6) 2 Socket head screw

WarningThe HeNe laser can cause permanent damageto your vision. Never look directly into thelaser tube or at a reflection from a specularsurface. Do not wear rings or other shinyjewelry when working with lasers. Use onlydiffuse (white 3x5 card) reflectors for viewingHeNe laser beam. Protect fellow co-workersfrom accidental exposure to the laser beam.Always block laser beam close to the laserwhen the experiment is left unattended.

1. Mount the rod (M) to the breadboard by placingthe socket driver into the rod and mounting therod over one of the tapped holes. Tighten thebolt with the socket driver until the rod is firmlyattached to the table.

2. Screw the laser clamp (P) onto the rod clamp (N)with the two 1⁄4-20 socket screws (BB).

3. Slide the laser mount-rod clamp combination (N-P) onto the rod (M) and tighten the large knurledknob on the rod clamp (N).

4. Insert the laser (Q) into the laser mount (P) suchthat approximately 3 inches protrudes fromeither side of the clamp. Rotate the laser (Q) untilthe alignment mark on the front of the laser is upand gently tighten the set screw in the laserclamp to hold the laser. Do not over tighten orthe laser could be damaged.

4. The laser supplied with the Newport OpticsEducation Kit is equipped with an internalshutter. The shutter is opened and closed byinserting a flat-bladed screw driver into the slotabove the laser output aperture. Rotate the screwdriver clockwise to close and counterclockwiseto open (the laser is shipped with the shutter inthe closed position).

(M)

(N)

(Q)

(P) (BB)

(BB)

LASER

45

Project #1The Law of

Geometrical Optics:

All optical designs are based upon two very simple lawsof optics: the laws of reflection and refraction. Anyanalysis of an optical system, no matter how elaborate,is done using these two laws to simulate the passage oflight rays through the lenses and windows and off themirrors that make up the optical device. So, the basisof almost everything you will do in optics begins withthese two simple laws. It is, therefore, appropriate thatthe first experiment in this manual is a demonstrationof these laws.

The first thing that you will do is to verify the law ofreflection: “the angle of reflection equals the angle ofincidence”. Then you will verify the law of refraction,also known as Snell’s Law, which states that “theproduct of the refractive index of a medium and thesine of the angle of incidence of a ray on one side of aninterface between two optical media is equal to theproduct of the refractive index times the sine of thetransmitted ray on the other side of the interface.” Thiscan be stated mathematically by,

ni sinθi = nt.sinθt (1-1)

where ni is the refractive index in the incident medium,θi is the angle between the local normal to the interfaceand direction of the incident ray, nt is the refractiveindex in the transmitted medium, and θt is the anglebetween the local normal and direction of the transmit-ted ray. These angles are measured between the rayand the normal to the surface where the ray hits theinterface. The direction of the normal changes oncurved surfaces, such as those on a lens or curvedmirror, so the normal is sometimes called the localnormal, since it applies only at that point on thesurface and not to neighboring points.

In addition to verifying the basic laws of geometricaloptics, this first experiment will also familiarize thestudent with the “tools of the trade”, the componentsthat make up the experimental set up. The labelsplaced on the items are the same as those used in thecomponent assembly section.

46

1.2 The Law of Reflection

A Note on Taking DataThe measurement and recording of data forthese projects are as important as the effectsyou will be exploring. It is only by measuringthe size of the effect and checking it with theexpressions given in the text, that the subjectunder discussion can be truly understood.Until the data are analyzed the project is a nicedemonstration of an optical effect and not anexperiment in optics.

Data should be taken in a standard, boundlaboratory record book, if it is available. Re-cording should be as neat as possible. If some-thing is recorded incorrectly, it should belined out with a single line and the correctvalue recorded next to it. Do not erase anydata that you record. When there is sufficientdata and a reasonable range of data, it shouldbe plotted in the most useful manner. Yourinstructor can help you determine this.

The law of reflection will be verified by showing thatthe angle through which a beam reflected by a mirror(angle of incidence plus angle of reflection) is twice theangle made by the beam incident and the normal to themirror surface (angle of incidence). By scanning theangle of incidence at which the He-Ne laser beamstrikes the mirror you will be able to show that the totalangle through which the beam is reflected is twice thatangle.

1.3 Experimental Set Up1. The optical breadboard should be located on a

table near a wall or the side of a cabinet. Tape asheet of paper on the wall or cabinet at the sameheight at which you will set the laser in the nextstep.

2. Mount a He-Ne laser as described in the LaserAssembly (LA) section of the Component Assem-blies Section and place it at the rear of thebreadboard.

3. Mount a Beam Steering Assembly (BSA-I) onto aRotation Stage (R). The Rotation Stage (R) isthen attached to the table with 1⁄4-20 screws. TheBSA-I should be located such that the incidentlaser beam is parallel to a nearby wall, as shownin Fig. 1-1.

Figure 1-1. Schematic view of Project #1, Law ofReflection.

BSA-I on RLA

Table 1.1 - Required Equipment

Newport Equipment Required:

Part Qty Description

LA 1 Laser AssemblyBSA-I 2 Beam Steering AssyBSA-II 1 Beam Steering AssyBSA-III 1 Beam Steering AssyTA-I 1 Target AssemblyRSP-1T 1 Rotation stage05BR08 1 Prism16569-01 1 Clear plastic tank

Additional Equipment Required:

Part Qty Description

QI 1 Index cardQW 1 Metric ruler or meterstick

47

y

θ

θ

∆θLaser Beam

∆θ=θ−θ0

Wall

r

r

Figure 1-2. Geometry for calculating incident andbeam deflection angles.

4. Adjust the beam steering mirror to reflect thelaser beam back onto itself. Record the positionof the rotation stage θo in your laboratory note-book or lab sheet.

EXTREME CARE SHOULD BE USED TO AVOIDACCIDENTAL EXPOSURE OF CO-WORKERS WHENDIRECTING THE LASER BEAM OUTSIDE OF THEOPTICAL TABLE AREA.

5. Scan the angle of the mirror by turning (R) suchthat the laser beam is reflected onto the piece ofpaper on the wall. Record the new angle θ in yournotebook and on the sheet of paper on the walljust above the mark locating the center of thebeam. Do this for a number of different rotationstage angles that produce beam positionsseparated by an inch or more on the wall.

6. Measure the perpendicular distance Y from themirror to the wall and the distance X from thatpoint on the wall to the marks on the wall asshown in Fig. 1-2. Use your knowledge of thedefinitions of the trigonometric functions and acalculator to determine the angle between thelaser beam direction and the reflected beamdirection using the distance measurements youhave just made. Record your calculations andthese angles next to those for the incidenceangles.

7. Compare the total reflected angles θr to theincident angles (θ -θo). You should find that thetotal beam deviation angle is twice the angle ofincidence, confirming the law of reflection.

A Note on Mirror MountsAll of the beam movement in this part of theexperiment was done using the rotation stage (R).However, you can also move the beam using theknobs on the adjustable mount (C). Note that oneknob moves the beam horizontally and the othervertically. Not all adjustable mounts move thebeam in two directions perpendicular to eachother. Those mounts that do are called orthogo-nal mounts, because the movements are at rightangles, or orthogonal, to each other. It is mucheasier to locate a beam with such adjustments.Examine the mount and see if you can under-stand how its design provides this feature.

48

The Law of RefractionThe verification of the law of refraction will be shownby measuring the incident and transmitted angles of aHe-Ne laser beam incident upon an air-water interface.

Experimental Set Up

1. Mount a laser assembly (LA) along the edge ofthe breadboard with the beam parallel to theedge of the board, as shown in Fig. 1.3. Set thelaser using the adjustable rod clamp (S) at themaximum height above the breadboard surface.

2. Mount two beam steering assemblies (BSA-1) atthe corners of the optical breadboard, as shownin Fig 1.3.

3. Mount a Modified Beam Steering Assembly(BSA-III) described in the Components AssemblySection. Locate the post holder (F) off the beamaxis, so that the mirror is in line with the laserbeam. Rotate the mirror to direct the laser beamapproximately at an angle of 45 degrees to thebreadboard surface.

4. Measure the perpendicular distance from wherethe laser intersects the mirror (H1 in Fig. 1.4) tothe bottom of the plastic box. Measure thedistance from this point to where the laser beamstrikes the bottom of the plastic box (V1 in Fig.1.4). Fill the plastic box with clear water towithin 1 cm of the top.

5. In the same manner as in step #4 measure theperpendicular distance from the point where thelaser beam enters the water’s surface to thebottom of the box (H2 in Fig. 1.4) and from thispoint on the bottom of the plastic box to wherethe laser beam strikes the bottom of the plasticbox. (V2 in Fig. 1.4)

6. You can now calculate the incident and refractedangles of the beam in water. Using Eq. 1-1 givenabove and the fact that the refractive index of airis 1.0, find the sines of the angles and determinethe refractive index of water. Your value shouldbe close to 1.33. If not, you might want to checkyour measurements and calculations. Be sure thedistances that you measured are the distancesdescribed above.

Additional ExperimentsIf there are other liquids available that can fill the tank,you can measure their refractive index also. You cancheck your answer in reference tables of refractiveindices in standard handbooks.

H 1

H 2

V2

V1

θi

θr

Figure 1-4. Measurements to determine the refractiveindex of a liquid.

Figure 1-3. Schematic view of Project #1, Law ofrefraction.

BSA-III

TANKBSA-I

BSA-I

LA

49

#1 – Measurement of Refractive Index of a Transpar-ent Solid using Total Internal Reflection

The phenomenon of total internal reflection (TIR)discussed in the Primer will be used to determine therefractive index of a prism. The geometry is somewhattricky, but it permits the determination of the refractiveindex of a standard 45°-45°-90° prism without resorting toany damage to the prism.

As pointed out in the Primer, the angle of incidence atwhich an interface switches from transmitting some lightand then to total internal reflection is called the criticalangle, θc. At this angle where the transmitted ray istraveling along the boundary of an air-glass interface, thetransmitted angle is 90°. The critical angle is related tothe refractive index of the material n as

sin θc = 1/n.

In the case of the experiment you are about to do, thingsare a little more complicated, but not much. Instead of asingle interface to worry about there are two. The first ofthe interfaces does not involve TIR. It is by rotating aprism until TIR occurs at the second interface andmeasuring the angle β between an unrefracted beam andthe beam at the critical angle that we can determine therefractive index of the prism from an equation whoseproof is left as a challenge to the student.

n2 = sin2θ0 + (√2+sinθ0)2 (1-2)

Where θ0 is related to the angle β through which thebeam is deviated by

θ0 = β - 45°. (1-3)

Experimental set up1. Mount the laser assembly (LA) along the rear edge

of the breadboard with the output toward a nearbywall. Mount a beam steering assembly that hasbeen modified to place the mirror mount parallel tothe table surface (BSA-II in the Component Assem-bly Section) onto the center of a rotation stage (R).The laser beam should be 4 to 5 mm higher thanthe BSA-II and parallel to the table surface. Tape apiece of paper on the wall.

Place a 1 inch round mirror flat against the walland adjust the angle of the laser such that thebeam is retro-reflected on itself. This assures thebeam is at a right angle to the wall for our calcula-tions. Remove the mirror and mark the position ofthe laser beam on the paper taped to the wall. Thisrepresents the undeviated beam.

50

θ

BSA-II

LA0

β

X

Y

Location of beamat critical angle

Location of beamwith prism removed

Figure 1-5. Top view refractive index experiment.

2. Place the prism such that the laser beam isincident to one of the short sides as shown in Fig.1-5. Retro-reflect the laser beam back to the laserhead. You may need to adjust the angle screws ofthe mount on which the prism is sitting. Thecenter of the prism hypotenuse should be over thecenter of rotation. Measure the distance from thecenter of rotation of the prism to the wall.

3. Rotate the prism clock-wise by turning the rotationstage (R) until the laser beam just exits thehypotenuse of the prism and strikes the wall. Dothis several times until you feel that you are at thetransition at which light just begins to be transmit-ted out the hypotenuse. The measurement of θ0 isgiven by tan(θ0 + 45°) = Y/X, where X is the dis-tance from the interface to the wall and Y is thedistance from the location of the beam at thecritical angle to the point where the beam hits thewall before the prism is placed in the beam.

4. Substitute this calculation into the formula givenabove and compare this value (1.517) with thepublished index of BK7 glass at the wavelength ofthe helium-neon laser.

This experiment has provided you with the opportunityto use a type of laboratory equipment available in mostcompanies. Later in some projects, the angles anddistances are determined and the modular equipmentyou used here can be replaced by specific opticalcomponents and holders machined to the specificationsobtained from the experiment. But as long as a project isin an experimental phase, the flexibility of the equipmentused here enables engineers to setup rapidly and revisetheir optical systems.

#2 – Index Gradients

This particular experiment must be prepared ahead oftime. Fill the tank with water and add several tablespoons of salt to it. Let the tank stand undisturbedovernight. Direct a laser beam along the length of thetank underneath and parallel to the surface. Try this adifferent heights within the tank. Note that the beamemerges from the tank at a different height than it enters.This is because there is a refractive index gradient in thetank. Index gradients bend light in various locations.They are responsible for mirages and the “wet” appear-ance of a distant spot on a hot road. Optical technologynow depends on small optical components that haveindex gradients designed into them so that they will actas lenses. They are referred to as GRIN lenses, where thefirst two letters are taken from “gradient” and the secondtwo are taken from “index.”

51

Figure 2-1. Definition of lens parameters.

ff

s i

so

Project #2The Thin Lens Equation:

While the idea of creating images with lenses is easy tograsp, understanding of location, magnification andorientation of an image usually comes from workingwith lenses. This project is more than a verification ofthe thin lens equation. It is also a study in the sizes andorientations of images and in the effect of combinationof lenses and their equivalent focal length.

In this experiment you will apply the Gaussian form ofthe thin lens equation:

1/f =1/so+1/si (2-1)

to determine the focal length of a lens or a combinationof lenses. By careful measurements of object distances(so) and image distances (si) (Fig. 2-1) it is possible tocalculate the focal length (f ) of an unknown lens to lessthan a percent of the true focal length. To use thisequation requires that the thickness of the lens besmall relative to its focal length. If the lens is “toothick”, the lens equation breaks down and a morecomplicated calculation is required to determine theimage distance and magnification (see references).

Since a negative lens alone cannot produce a realimage, a combination of a positive lens and a negativelens is used to determine the focal length of the nega-tive lens.

The relation between object and image location is takenfor granted once you are given the Gaussian form of thethin lens equation. But it is usually forgotten that someapproximations have gone into deriving the equation.As a method of rapidly laying out the location and focallengths of various lenses, it has some use, but as ameans of doing optical engineering, it cannot providethe precision required for serious calculations ofoptical performance. Still, as a means of exploring thenature of imaging, experiments involving the thin lensequation can be most useful.

52

Experimental Set Up:1. Construct a target (QT) on an index card by ruling

a square grid of lines 5 mm apart. This will serveas your object. You can compare the size of theimages you generate to this object to determinethe magnification. Add some arrows to enable youto determine if the images are upside down orright side up. Mount this in a target assembly (TA-I) close to the edge of the breadboard (Fig. 2-2).

2. Unroll the tape measure along the breadboardedge so that the start of the tape is directly underthe target. Place a high intensity lamp approxi-mately 2 inches behind and at the same height asthe target.

Positive Lens

3. Read the note on handling lenses in the compo-nents assembly section if you have not done soalready. Take a positive 100 mm focal length lensfrom the lens kit and mount it in a lens chuckcomponent assembly (LCA).

4. Place the lens about 125 mm from the target andrecord in your notebook, the exact distancebetween the lens and the target. This is the firstobject distance.

5. Mount the white card in a second target assemblymount (TA-II). Place the TA-II at the end of thebreadboard and slowly move it toward the lensuntil an image is seen. Continue moving the TA-IIuntil the image starts to become blurred. Move itaway from the lens until the image is again seen.Move the TA-II to produce the sharpest image andmark this position. The distance from the lens tothe card is the image distance. Record this valuealong with object distance.

6. Mark on the white card two points on the imagethat represent the distance between a specificnumber of grid lines and, either now or later,measure the distance between these points.Measure the distance between the correspondingpoints on the object grid. Record the values inyour notebook along with the image and objectdistances.

7. Redo steps 4 through 6 with the object distanceset close to the values of 150, 200, 400, and 600mm. Record the object and image distances andthe distance between two points on the image.This will give you sufficient data to make a gooddetermination of the focal length of the lens.

Newport Equipment Required:Part Cat # Qty Description

TA-I 1 Target AssemblyTA-II 1 Target AssemblyLCA 2 Lens Chuck AssemblyLKIT-2 1 Lens kit, as noted belowLP1 KPX094 1 100 mm focal length lensLP2 KPX106 1 200 mm focal length lensLP3 KPX076 1 25.4 mm focal length lensLN1 KPC043 1 -25.4 mm focal length lens

Additional Equipment Required:Part Qty Description

QW 1 Meterstick or tape measureQI 1 Index cardQT 1 TargetQQ 1 High intensity lamp

Table 2.1 - Required Equipment

TA-II

TA-I

LCAQQ

Figure 2-2. Project #2, Positive lens set up.

53

Figure 2-3. Project #2, Negative lens set-up.

LCA

TA-I

LCAQQ TA-II

8. Using the relationship for lens focal length, objectdistance and image distance, 1/f = 1/so + 1/si, calcu-late the focal length of the lens using each of thesets of data. Find the average of the results andcompare this value to that specified in the lenskit. Also calculate the magnification of the imagefor each object distance from the distancesmarked and measured on the white card dividedby the corresponding distance on QT. Comparethese to the ratio of the image distance dividedby the object distance. (See Eq. 0-6 in thePrimer.)

9. Return to the first lens placement with an objectdistance of 125 mm. Verify that the image islocated at the point that you recorded it earlierby moving the TA-II into the correct position.Now, keeping the TA-II fixed, move the lenstoward it until you get an image on the indexcard. Measure the new image and object dis-tances and compute the magnification. Do theybear any relation to the earlier measurements?

Negative Lens

Bi-concave lenses have negative focal lengths and theimage they form is virtual. Since only image distancesfor real images can be directly measured, the techniquethat uses an auxiliary positive lens of known focallength, described in Section 0.2 of the Primer, is usedto determine the focal length of a negative lens.

10. Place a negative focal length lens (LN1) in an LCAwith its concave side facing the object andmeasure the distance from the object to the lens.

11. Next place the positive lens whose focal lengthyou have just measured more than 100 mmbeyond the negative lens. Obtain a sharp imageand measure the image distance from the positivelens.

12. Since you know the focal length of the positivelens and you have measured the image distance,you can calculate the object distance that wouldbe required for this image distance if only apositive lens were present. The image of thenegative lens is the object for the positive lens(Rule #5 in Section 0.1). Subtract the calculatedobject distance for the positive lens from thespacing between the two lenses. This is the imagedistance for the negative lens and will be nega-tive. See Fig 0-9 in the Primer to help you visual-ize this. Recalculate the focal length from thelens formula above (remember to use the correctsigns on the image and object distances). Com-pare this to the value in the lens kit guide.

54

Additional experiments:Combinations of lenses

1. Using a combination of lenses from the lens kit(for example, a 100 mm EFL (LP1) and a 200 mmEFL (LP2)), mount the two lenses next to oneanother. You might tape the two lenses togetherat several points near the edges. Do not attachthe tape near the center of the lens. Measure thefocal length of the combination of lenses as aboveto find the effective focal length. Compare theseresults to that calculated using Eq. 0-8 in thePrimer.

2. Repeat the previous step using one positive lensand one negative lens. To assure that you will geta real image you can use a negative lens whoseabsolute value of the focal length is greater thanthe focal length of the positive lens. Why is thisnecessary?

3. Using two LCA’s, put the 100mm EFL (LP1) in oneand the 25.4 mm EFL (LP3) in the other. LocateLP1 about 200 mm from the object and record theobject distance. Locate and record the imagedistance and orientation. Place LP3 about 60 mmbeyond the image location and move the TA-II tofind the image for the combination. Record theseparation between the lenses and the magnifica-tion and orientation of the final image. Note thatwhereas the first image was inverted, the secondimage is erect with respect to the original object.Verify your measurements by applying the thinlens equation twice to calculate the locations andmagnifications of the intermediate and finalimages.

55

Figure 3-1. Gaussian beam collimation. (a) Galileantelescope. (b) Keplerian telescope. Eyepiece focallength, fe; objective focal length, fo.

(a)

(b)

fo

fe

fo

fe

Project #3Expanding Laser Beams:

Many times when a laser is used in an optical system,there is a requirement for either a larger beam or abeam that has a small divergence (doesn’t change sizeover the length of the experiment). In some cases thesize of the beam becomes critical, for example; whenmeasuring the distance from the Earth to the Moon, abeam one meter in diameter travels to the Moon whereit has expanded to several hundreds of meters indiameter and when the return beam intersects theEarth’s surface it is several kilometers in diameter. Thesignal returned from this expansion is millions of timessmaller than the original signal, so that the divergenceof a laser beam must be reduced to produce strong,detectable signals. Even in the case of earthboundexperiments, higher degrees of collimation are requiredfor many applications including some of the projects inthis manual.

As was pointed out in the Primer, the product of thebeam waist and the divergence of a lens is a constant:

(3-1)

Therefore, if we want a more collimated beam, thedivergence θ must be reduced and that can only bedone by increasing the beam waist. This process cannotbe easily done by a single lens. First, the beam must bediverged with a short focal length lens and then thediverging beam is recollimated with a large beam waistand smaller divergence. The arrangement of the lensesare essentially those of an inverted telescope. It isinverted since the light goes in the eyepiece lens (theshorter focal length lens) and comes out the objectivelens. The amount of beam expansion, and thereforedivergence reduction, is equal to the power of thetelescope, which is simply the ratio of the focal lengthsof the telescope lenses. Therefore after passagethrough a beam expander, the divergence should beequal to the old divergence divided by the power of thetelescope.

This experiment will demonstrate the design of twotypes of laser beam expanders — the Galilean and theKeplerian. Each has distinct advantages. From theseexperiments you will gain experience in the alignmentof laser beams and components and learn some simpletechniques that make the process of alignment mucheasier.

doθλ

π= 4

56

The set up that you will be constructing in this experi-ment will be used for a number of other experiments(#4, 6, 7, and 10) that require expanded laser beamillumination. It is worthwhile to write down in your notebook anything that helps you to rapidly set up and alignthe beam expander, since you will be doing this again.

Notes on Alignment of Laser BeamsTape a card with a hole slightly larger than thelaser beam to the output end of the laser, sothat the beam will pass through and backreflections from components can be easilyseen.

For each lens there are two reflections, onefrom each surface. When the centers of thetwo reflections are at the height of the laserbeam, the height of the lens is properly ad-justed. When they are overlapping, the beamis at the center of the lens. And when they arecentered about the laser output, the lens is nottilted with respect to the beam.

In some cases, if the return beam is too strong(as in the case of this experiment), the laserwill give an erratic output because vibrationsfrom the outside world can be coupled into thelaser. However, in the case of items such asbeam expanders, where you do not try to sendall the light back into the laser, the smallreflections from the components have no meas-urable effect on the projects described in thismanual.

Experimental Set Up:

1. Mount a laser assembly (LA) to the rear of thebreadboard. Adjust the position of the laser suchthat the beam is parallel to the edge and on top ofa line of tapped holes in the breadboard top.Tape an index card with a small (about 2 mm)hole in it to the front of the laser, so that the laserbeam can pass through it. This card will be usedas a screen to monitor the reflections from thecomponents as they are inserted in the beam.These reflections, when they are centered aboutthe beam output, indicate that lens is centered inthe beam with its optic axis parallel to the beam.

Newport Equipment Required:Part Catalog # Qty Description

LA 1 Laser AssemblyBSA-I 2 Beam Steering AssyLCA 2 Lens Chuck AssemblyTA-I 1 Target AssemblyLKIT-2 1 Lens kit, specificallyLP2 KPX106 1 200 mm focal len. lensLP3 KPX076 1 25.4 mm focal len. lensLN1 KPC043 1 -25 mm focal len. lens

Additional Equipment Required:Part Qty Description

QI 1 Index card1 Tape1 Non-shiny, non-metallic ruler or meterstick

Table 3.1 - Required Equipment

57

Figure 3-2. Schematic view of Galilean beamexpander experiment.

BSA-ILA

TA-ILCA

LCA

BSA-I

2. Mount a beam steering assembly (BSA-I) approxi-mately 4 inches in from the far corner of thebreadboard (Fig. 3-2). Adjust the height of themirror mount until the beam intersects the centerof the mirror. Then rotate the post in the postholder until from the laser beam is parallel to theleft edge and the surface of the optical bread-board.

3. Place a second beam steering assembly (BSA-I) inline with the laser beam at the lower left cornerof the optical breadboard, (Fig. 3-2). Adjust themirror mount until the laser beam is parallel tothe front edge and the surface of the opticalbreadboard.

4. Use a meter stick or ruler to measure the beam atseveral distances from the output of the laser.The distances should be up to 10 m, if the roomallows it. You will have to estimate the beam size,since, as was discussed in the Primer, the beamirradiance falls off smoothly from the center.Record the beam sizes at various distances abouta meter apart. Calculate a divergence for the laserbeam. Record the beam sizes at various distancesabout a meter apart. As indicated in the Primer,the beam diameter varies as

d 2(z)=d02+θ2z2. (0-25)

Assume that d0 is the beam diameter measuredclose to the laser and that z is the distance fromthe laser, use the above equation to determinethe value for θ based upon the measurement atvarious values of z. You will find the values of θare more accurate for larger values of z. Theaverage of measured values should be in theneighborhood of 1 milliradian.

The Galilean Beam Expander

5. Insert a short focal length (-25.4 mm) negativelens (LN1) into a lens chuck assembly (LCA) andmount it five inches from the first beam steeringassembly. Align the lens by raising or loweringthe post in the post holder and sliding the LCA sothat the diverging beam is centered on the mirrorof the second beam steering assembly. You canalso use the alignment hints given previously.

6. Insert a longer positive focal length (200 mm)lens (LP2) into an LCA and place it about 175 mm(the sum of the focal lengths of the two lenses,remembering the first lens is a negative lens)from the first lens in the diverging laser beam.

58

Note:Either of these beam expanders can be used in Projects#4, 6, 7, and 10. The beam expander you constructdepends more on the required expansion ratio thananything else.

Figure 3-3. Schematic view of Keplerian beam ex-pander experiment.

BSA-ILA

TA-ILCA

LCA

BSA-I

Again, center the beam on the mirror of thesecond BSA and use the reflections off the lens toassist in aligning the beam.

7. Rotate the second BSA such that the beamreturns back through the two lenses just to eitherside of the laser output aperture. (If the returnbeam enters the output aperture the laser mayexhibit intensity fluctuations and you cannotdetermine the size of the return beam).

8. Carefully adjust the position of the last lens bymoving it back and forth along the beam until thereturning beam is the same size as the outputbeam.

9. Rotate the second BSA and shine the laser beamto the end of the room and measure the diameterjust after the beam expander and at a number ofplaces along the beam (at least a meter apart).Depending upon the accuracy of your alignmentand distance available, it may be difficult to seeany divergence at all.

10. As was discussed above, the divergence de-creases with increasing beam waist diameter. Thebeam expander increases the diameter of thebeam and as a result decreases the divergence ofthe beam in the same ratio as the beam expan-sion. Although it is difficult to be accurate inmeasuring the divergence, compare the estimateof the divergence of the undiverged beam dividedby the power of the telescope. You may wish totry other combinations of lenses. Be sure thatyou do not choose a lens combination that, at thecorrect separation, has a beam which overfillsthe second lens and causes diffraction (SeeProject #4).

The Keplerian Beam Expander:

1. Replace the negative lens (LN1) with a short focallength positive lens (25.4 mm.) (LP3) and use thesame adjustments to center the beams in thelenses and the mirror of the second BSA. Adjustthe distance between the two lenses to be thesum of their focal lengths (Fig. 3-2).

2. Adjust the last beam steering mirror again for thecondition of equal spot size at the laser output.

3. Repeat steps 7, 8, and 9 in the Galilean BeamExpander including an estimate of the collimation

59

Figure 4-1. Diffraction from a circular aperture.

of the laser beam for this geometry. You areencouraged to try other lens combinations.

Project #4

Diffraction ofCircular Apertures:

Most of the optical systems that you will work with aremade up of components whose apertures are circular.They can be mirrors, lenses, or holes in the structuresthat contain the components. While they do permitlight to be transmitted, they also restrict the amount oflight in an optical system and cause a basic limitation tothe resolution of the optical system.

In this experiment you will measure the diffractioneffects of circular apertures (Fig. 4-1). The diffractionassociated with the size of the aperture determines theresolving power of all optical instruments from theelectron microscope to the giant radiotelescope dishes.In addition, you will discover that a solid object notonly casts a shadow but that it is possible for a brightspot to appear in the center of the shadow!

The diffraction patterns you will examine are locatedboth close to the diffraction aperture and far away. Thefirst is called Fresnel (Freh - NEL) diffraction; thesecond is Fraunhofer (FRAWN - hoffer) diffraction.

WARNING.To be able to see some of the diffractionpatterns, this experiment will be performed ina darkened room. Extreme care should betaken concerning the He-Ne laser beam. Yourpupils will be expanded and will let in 60 timesmore light than in a lighted room. DO NOTLOOK AT A DIRECT SPECULAR REFLECTIONOR THE DIRECT LASER BEAM. BE AWARE OFYOUR SURROUNDINGS AND FELLOW CO-WORKERS.

1st dark ring

θ=1.22 λD

λ

Lightwavelength

D

60

Newport Equipment Required:Part Catalog # Qty Description

LA 1 Laser AssemblyBSA-I 2 Beam Steering AssyLCA 2 Lens Chuck AssyTA-I 1 Target AssyTA-II 1 Target AssyLP4 KPX100 1 150mm focal length lensLN1 KPC043 1 -25mm focal length lensTP1 1 Target, pinhole, 0.001" diam.TP2 1 Target, pinhole, 0.002" diam.TP3 1 Target, pinhole, 0.080" diamTF 1 Target, 3 zone Fresnel

Additional Equipment Required:Part Qty Description

QI 1 Index cardQW 1 Metric ruler or meterstick

Table 4.1 - Required Equipment

Figure 4-2. Schematic view of Fraunhofer diffractionexperiment using TP1.

BSA-ILA

LCA

BSA-ITA-II

WALKING IN A DARKENED ROOM CAN BEHAZARDOUS!

Experimental Set Up:1. Mount a laser assembly (LA) at the rear of the

breadboard. Adjust the position of the laser suchthat the beam is parallel to the edge and on top ofa line of tapped holes in the breadboard. Tape anindex card with a small (about 2 mm) hole in it tothe front of the laser, so that the laser beam canpass through it. This card will be used as ascreen to monitor the reflections from thecomponents as they are inserted in the beam. Seethe note in Project #3 on the alignment of laserbeams.

2. Mount a beam steering assembly (BSA-I) approxi-mately 4 inches in from the far corner of thebreadboard (Fig. 4-2). Adjust the height of themirror mount until the beam intersects the centerof the mirror. Then rotate the post in the postholder until the laser beam is parallel to the leftedge and the surface of the optical breadboard.

3. Place a second beam steering assembly (BSA-I) inline with the laser beam at the lower left cornerof the optical breadboard, (Fig. 4-2). Rotate andadjust the mirror mount until the laser beam isparallel to the front edge and the surface of theoptical breadboard.

4. Place an index card in a modified target holderassembly (TA-II) and set it at the end of thebreadboard so that the beam hits the center ofthe card.

5. Mount a lens chuck assembly (LCA) five inches tothe right of the last beam steering mirror anddirectly in line with the laser beam. This will bethe aperture holder.

Fraunhofer Diffraction of a Circular Mask

6. Carefully place the pinhole target (TP1) into theLCA. Adjust the mount such that the laser beamstrikes the target approximately in the center.WARNING — The target will reflect a largepercentage of the beam.

7. Adjust the last beam steering mirror such thatthe laser beam fills the pinhole. This can best beaccomplished by viewing the back side of thetarget (beyond the laser) from 45° and looking fora bright red glow. This will occur when the laserbeam (or part of the laser beam) is illuminatingthe aperture.

61

Figure 4-2. Schematic view of Fraunhofer diffractionexperiment using TP2.

BSA-ILA

BSA-ITA-II

TA-I

8. Watch the white card. Carefully adjust the lastbeam steering mirror to produce the brightestimage. You should see a bright central circlesurrounded by dark and light circular bands. Thisis the Airy disc pattern. Measure the distancefrom TP1 to the index card in TA-II. Mark andthen measure the diameter of the first darkcircular band around the bright central circle.This is a measure of the amount of diffractioncaused by the pinhole.

9. As was pointed out in the Primer, the angularsubtense of the dark band is related to thewavelength and diameter of the pinhole by

sinθ = 1.22 λ /D (0-11)

where D is the diameter and λ is the wavelength.Since the diffraction angle is small, the sine andthe tangent of the angle are equal. The tangent isfound from dividing the radius of the dark bandby the distance from the pinhole to the indexcard that was recorded in step #8. Inserting thewavelength of the He-Ne laser (λ = 633 nm) intothe equation, calculate the diameter of thepinhole.

10. All circular apertures will exhibit an Airy pattern.Replace the TP1 with TP2. You will need to use atarget holder mount (TA-I) approximately fourinches in from the breadboard edge and in linewith the position of the lens chuck assembly.Measure the diameter of the first dark band andthe distance between the pinhole and the indexcard. Calculate the pinhole diameter based onthis data.

This series of rings from a circular aperturecauses objects that are close together to overlapat the focal plane of the observing instrument andwill limit the resolving power of large aperturetelescopes.

11. Assemble a 6:1 beam expander, using the tech-niques of Project #3, between the first andsecond beam steering mounts (BSA-I) as shown inFig. #4-3. Replace TP2 with TP3. Because theaperature is so large, replace the card mount TA-II with a third BSA-I and direct the beam to a wallmore than 10 feet away. Measure the diameter ofthe first dark band and estimate the distancebetween the pinhole and the wall. Calculate thepinhole diameter.

62

BSA-ILA

TA-ILCA

LCA

BSA-IBSA-I

Figure 4-3. Schematic view of Fresnel diffractionexperiment.

In the Primer we discussed that at far field theFraunhofer diffraction pattern does not change inshape, but only in size. Using the index card, lookat the diffraction pattern starting at the pinholeand moving away toward the wall. At a distanceof about 2 feet from the pinhole you will see thecenter bright spot become a small dark spot.Depending on how well the beam expander is set,this small dark spot may be difficult to resolve.However, the center spot changing from bright todark and then to bright again is Fresnel diffrac-tion.

Fresnel Diffraction of Circular Mask

12. Replace TP3 with the Fresnel target (TF). Look atthe diffraction pattern on the target screen. Notethat the center of the image has several brightand dark rings. This is also Fresnel diffraction.Depending on the distance of TA-II from TF, thecenter of the pattern may be bright or dark.Although the Fresnel target (TF) has a centralabsorbing circle, note there is still light at thecenter of the pattern. The bright spot at thecenter is sometimes called the Poisson spot orthe spot of Arago.

13. Examine the shadows of other objects put in theexpanded laser beam. Pencil points, wires, andsmall beads on a string are good objects that giveinteresting Fresnel patterns. Note how thepatterns change as you move the objects alongthe beam direction. Sketch some of the moreinteresting patterns in your notebook.

A detailed description of the results can be found inThe Optics Problem Solver by The Research andEducation Association.

63

Figure 5-1. Diffraction from a single slit.

Figure 5-2. Schematic view of diffraction experiments.

BSA-ILA

TA-I

BSA-ITA-II

ω

λ

Lightwavelength

θ= λω

Central Maximum

1st Dark Fringes

Project #5

Single Slit Diffraction andDouble Slit Interference:

Diffraction of light occurs whenever light illuminates anaperture that has dimensions that are on the order ofthe wavelength of light being used. In the case of a slitwhich has a narrow opening that is “infinitely” tall, thediffraction takes place in the direction perpendicular tothe small dimension.

In addition light from one slit will interfere with the lightfrom a second nearby slit to produce an interferencepattern that combines the interference properties ofthe single slit with the interference pattern of twonearby sources. (Fig. 5-1)

Experimental Set Up:1. Mount a laser assembly (LA) to the rear of the

breadboard (Fig 5-2). Adjust the position of thelaser such that the beam is parallel to the edgeand on top of a line of tapped holes in the bread-board top. Tape an index card with a small(about 2 mm) hole in it to the front of the laser,so that the laser beam can pass through it. Thiscard will be used as a screen to monitor thereflections from the components as they areinserted in the beam. See the note in Project #3on the alignment of laser beams.

2. Mount a beam steering assembly (BSA-I) approxi-mately 4 inches in from the far corner of thebreadboard (Fig. 5-2). Adjust the height of themirror mount until the beam intersects the centerof the mirror. Then rotate the post in the postholder until the laser beam is parallel to the leftedge and the surface of the optical breadboard.

3. Place a second beam steering assembly (BSA-I) inline with the laser beam at the lower left cornerof the optical breadboard, (Fig. 5-2). Rotate andadjust the mirror mount until the laser beam isparallel to the front edge and the surface of theoptical breadboard.

4. Place an index card in a modified target holderassembly (TA-II) and mount it at the end of thebreadboard so that the beam hits the center ofthe card.

64

Newport Equipment Required:Part Cat # Qty Description

LA 1 Laser AssemblyBSA-1 2 Beam Steering AssemblyTA-1 1 Target AssemblyTA-1I 1 Target AssemblyTSS 1 Target, single slitTDS 1 Target, dual slitDG 1 Diffraction grating

Additional Equipment Required:Part Qty Description

QI 1 Index cardQW 1 Metric ruler or meterstick

Table 5.1 - Required Equipment

5. Mount a TA-I five inches to the right of the lastbeam steering mirror and four inches in from thelaser beam. This will be the holder for parts TSSand TDS.

Single Slit Diffraction

6. Carefully place the TSS (single slit – 0.002 in.wide) into the TA-I. Adjust the mount such thatthe laser beam strikes the target approximately inthe center.

WARNINGThe target will reflect a large percentage of thebeam.

7. Adjust the last beam steering mirror such thatthe laser beam fills the slit. This can best beaccomplished by viewing the back side of thetarget from 45° and looking for a bright red glow.This will occur when the laser beam is illuminat-ing the slit.

8. Watch the white card. Carefully adjust the lastbeam steering mirror to produce the brightestimage. You should see a bright central band withseveral dimmer bands on each side. This is thesingle slit diffraction pattern. Mark on the indexcard the locations of as many dark bands as youcan easily see. Note that the central band islarger than the sidebands. Measure the distancebetween the center of the dark bands on bothsides of the central band and the distance fromthe slit to the index card. Calculate the angle atthe slit between the central peak and the firstdark band. Remember the distance between thefirst dark bands is twice the distance between thecentral part and the first dark band. Based onexpressions given in the Primer, the angularsubtense of the center band from central peak tofirst dark band is given by

sinθ = mλ/ω (0-9)

If the wavelength of the He-Ne laser is 633 nm,determine the width of the slit from your calcula-tions.

65

Figure 5-3. Schematic view of Young’s double slitexperiment.

Figure 5-4. Schematic view of diffraction gratingexperiment.

BSA-ILA

TA-I

BSA-ITA-II

BSA-ILA

TA-I

BSA-ITA-II

Young’s Double Slit Experiment

1. A second and related experiment can be per-formed with the previous setup. Replace the TSSwith a TDS (dual slit – 0.002 in. wide with 0.008 inspacing) target and the index card used as theobservation screen. Measure the slit to indexcard distance R.

2. The pattern will now have a series of maxima andminima spaced within the envelope of the originalsingle slit pattern. These fringes are the interfer-ence pattern of the double slit. Mark the locationsof the minima x1,x2. . . of these closely spacedfringes. Calculate the average separations ∆x=x1-x2. Also mark the location of the minimum of thelarge band (the position where the interferencefringes disappear).

3. Calculate the fringe separation from

∆θ=∆x/R (5-2)

and record it in your notebook. From this value of∆θ and the wavelength of the laser (λ=633nm) youcan calculate the slit separation using Eq. 0–16 inthe primer.

4. Take a card or the edge of a ruler and carefullyinsert it in front of one of the two slits. This takesa little practice. If you do it right you will see theinterference pattern disappear and a single slitdiffraction pattern remain. Note that when youtake away the item blocking the light from any ofthe slits, you introduce dark fringes. Of coursethe light at the bright fringes is brighter. You areusing light to push light around!

Additional experiments:Diffraction grating

Higher order diffraction patterns exist for 3,4,5 ...equally spaced slits. Eventually, the number of slitsbecomes very large and the results approaches thediffraction grating described in Section 0.4.3 in thePrimer. Most high resolution instruments for determin-ing the transmission or reflection characteristics ofoptical materials and coatings use some form of agrating.

1. Mount a diffraction grating (DG) in the TA-1 andilluminate it with the laser beam. Mount a newindex card in the TA-II and locate it behind DG, sothat a number of diffraction orders can be seen.

66

2. Move the TA-II away from DG until only a few dotsremain on the screen and their separations areeasily measured. Mark the locations of thediffraction orders on the card and label each withthe order (0 for the undiffracted beam). Measurethe distance from DG to the screen.

3. Calculate the diffraction angles from the measure-ments. Note the angles are large enough that youcannot use any small angle approximation. Youmust use the inverse tangent to arrive at theangle.

4. The groove separation, or grating constant, forthis grating is found by taking the reciprocal ofthe grating frequency, which is 13,400 groovesper inch. From the groove separation and theangular measurements for several orders,determine the wavelength of the helium-neonlaser. Compare it to 633 nm.

The spectra of other sources can be studied using thisdiffraction grating, but additional components arerequired. Since most will not be lasers with sharplydefined beams, the sources illuminate a slit. The lightfrom the slit is then collimated to provide a constantangle of incidence to the grating. The diffracted beamsare then refocused to a series of slit images that areseparated into the colors in the spectrum of the sourcelocated in the focal plane of the focus lens. See any ofthe major optics references for description of a simpletransmission grating spectrometer.

In most commercial spectrometers, the diffractiongrating is a reflection, rather than a transmission type.This is because the instrument is more compact andthe gratings tend to be more efficient in this mode.

67

Project #6

The MichelsonInterferometer:

In this experiment you will build a Michelson interfer-ometer similar to the one described in the Primer anduse it as a means to observe small displacements andrefractive index changes. When this arrangement ofcomponents is used to test optical components inmonochromatic light it is called a Twyman-Greeninterferometer. The Twyman-Green interferometer iswidely used for testing optics and optical systems, andprovides a means for measuring the amount of aberra-tions present in these optical systems. Rather thanmake such a distinction here, the device will be referredto as a Michelson interferometer throughout thismanual.

Experimental Set Up:1. Mount a laser assembly (LA) to the rear of the

breadboard (Fig 6-1). Adjust the position of thelaser such that the beam is parallel to the edgeand on top of a line of tapped holes in the bread-board top. Tape an index card with a small(about 2 mm) hole in it to the front of the laser,so that the laser beam can pass through it. Thiscard will be used as a screen to monitor thereflections from the components as they areinserted in the beam. See the note in Project #3on the alignment of laser beams.

2. Mount a beam steering assembly (BSA-I) approxi-mately 4 inches in from the far corner of thebreadboard (Fig. 6-1). Adjust the height of themirror mount until the beam intersects the centerof the mirror. Then rotate the post in the postholder until the laser beam is parallel to the leftedge and the surface of the breadboard.

3. Place a second beam steering assembly (BSA-I) inline with the laser beam at the lower left cornerof the breadboard, (Fig. 6-1).

4. Set up a beam expander between the first twoBSA-I’s as explained in Project #3. This beamexpander generates an expanded plane wave thatis needed to build the interferometer.

Figure 6-1. Schematic view of Michelson Interferome-ter experiment.

LABSA-I

BSA-I

LCA

LCA

TA-II

BSA-I

BSA-ILCA

Newport Equipment Required:Part Cat # Qty Description

LA 1 Laser AssemblyBSA-I 4 Beam Steering AssemblyLCA 3 Lens Chuck AssemblyTA-II* 1 Target Assembly without B-2 baseLKIT-2 1 Lens kitFK-BS 1 Beam splitter

Additional Equipment Required:Part Qty Description

QI 1 Index cardQS 1 Soldering iron

Table 6.1 - Required Equipment

68

5. Place a lens chuck assembly (LCA) approximatelyfive inches to the right of the last BSA-I mount.Mount a 50/50 beam splitter in the LCA and rotatethe assembly 45 degrees to the optical path. Thebeam reflected off of the first surface should beperpendicular to and going in the direction of thefront edge of the breadboard. The beamsplitterdivides the incoming laser beam into two equalcomponents for the two arms of the interferom-eter.

6. Place a BSA-I with its mirror centered about thepath of the reflected beam and about five inchesfrom the beamsplitter such that the beam isretro-reflected back onto the index card taped tothe laser. This mirror will be called the referencemirror.

7. Place a second BSA-I with its mirror centeredabout the path of the transmitted beam fiveinches beyond the beam splitter to intercept thetransmitted beam (Fig. 6-1). Adjust the mirroruntil its beam is directed back to the index cardtaped to the laser. This mirror will be called thetest mirror.

8. Use a TA-II* (no base) with an index card (QI) asan observation screen on the other side of thebeamsplitter from the reference mirror (See Step6). Adjust the mirrors in the two arms of theinterferometer until the two beams overlap onthe screen. There should be combined reflectionsat the observation screen and on the card at thefront of the laser. Aligning the two beams on thecard at the front of the laser is good for a quickrough alignment.

9. As the two beams are brought into coincidence, aseries of bright and dark fringes should appearrepresenting the interference pattern betweenthe two wavefronts. The orientation and separa-tion of the fringes can be controlled by adjustingthe reference and test mirrors. Usually it is bestto use one mirror for adjustment. Adjust themirror so that approximately five fringes appearacross the beam on the card. The number offringes can be varied in a particular direction bytilting the reference mirror in a direction perpen-dicular to the direction of the fringes. YourMichelson interferometer is now completed.

10. Any curvature present in the fringes representphase differences between the waves that havetraversed the two arms of the interferometer, i.e.the reference mirror arm and the test mirror arm.If the reference mirror is assumed to be perfectly

69

flat, then the curvature in the fringes may be dueto the fact that the mirror under test is not flat,but has a long radius of curvature or aberrations.These aberrations cause the plane wave, gener-ated by the beam expander, in the test arm todepart from a plane wave. The interference of theplane wave reference wavefront with the testmirror wavefront will create a pattern of curvedfringes with varying separation. The amount ofthe departure of a curved fringe from a straightline, represents the phase shift introduced by thecomponent under test. This departure, measuredin number of fringes, gives twice the departure ofthe test wavefront from the reference wavefrontin wavelength of laser light. The amount of testmirror aberration (W ) may be calculated asfollows:

W = (fringe shift)/2 (6-1)

where W is expressed in units of the wavelengthof the laser used (in this case the laser wave-length is 633 nm), and fringe shift is the height ofthe fringe expressed in units of the average fringeseparation distance in the interference pattern.The factor of two arises from the fact thatreflection doubles the amount of aberration.

11. Move the second lens of the beam expanderslowly toward the first. The expanded beam nowdiverges, causing the wavefront to be sphericalinstead of planar. The fringes will becomecircular and if you further adjust the beam co-incidence, a bull’s-eye pattern can be seen.

12. Turn the soldering iron (QS) on, and after itwarms up place it in the path of the light in thetest arm. Observe the changes in the fringesaround the tip of the soldering iron. The shift inthe fringes is due to the extra phase shift intro-duced by the hot air surrounding the iron tip. Hotair has a different density and refractive indexthan cold air and consequently the two armshave a different optical path-lengths.

13. Insert your finger partway into one of the arms ofthe interferometer, so that its shadow can beseen on the screen. Notice the variations in thefringes just due to the heat of your finger. Alsohold your hand palm up just below one of theinterferometer arms.

14. Push on the test mirror and note that very littleforce leads to tiny deflections of the test mirror.These deflections are measurable as indicated bythe shift in the fringes. For each fringe that

70

moves past a point in the center of the patternthe mirror has moved one half wavelength alongthe beam direction. Try to devise a means ofslowly moving one mirror. If the motion is slowenough you can count the number of fringes anddetermine the amount of mirror displacement.

15. Another means of changing the optical pathinside the interferometer is to insert a transpar-ent material, such as a microscope slide or otherflat material, in one of the arms. The departure ofthe fringe pattern from the undisturbed system isa measure of the refractive index and thicknessvariation in the material.

Additional experiments:Distance measurement.

Time dependence of the fringes

1. After setting up the Michelson interferometer,monitor the change in the fringe pattern as afunction of time. Usually thermal changes willcause small expansion and contractions in thecomponent distances and this will result in a shiftof the fringes with time.

Vibration dependence of the fringes

2. Tap on the surface of the table and monitor thechanges in the fringes. How long does it take thevibration to damp out? Do you detect any vibra-tional motion when you slam a door, walk acrossthe room, or jump up and down? Some tableshave air cushions or springs to isolate an opticalsystem such as a Michelson Interferometer fromthe vibrations of the outside world.

Motion detector

3. This is a somewhat more elaborate experimentand requires a light detector such as a photocellor phototransistor. *Replace the observationscreen with the detector and a large pinhole thatallows only one fringe to pass. When the fringepattern moves the light on detector will createalternately strong and weak signals. If the detec-tor is hooked up to an audio amplifier andspeaker, the alternating signal will provide anaudible sound. The frequency of the sound willdepend upon the number of fringes per secondthat sweep past the pinhole. Since each fringerepresents one half wavelength mirror move-ment, the pitch of the sound wave represents thespeed of motion of the mirror.

*The circuit for a simple photocell detector is given at the end ofProject #8.

71

LABSA-I

BSA-I

LCA

LCA

TA-II

BSA-I

BSA-ILCA

Figure 7-1. Schematic view of coherence experiment.

Project #7Coherence and Lasers:

This project is a departure from the classical opticsexperiments described up to this point in this manual.In this project you will examine one of the characteris-tics of lasers with a Michelson interferometer todetermine the frequency separation between the axialmodes of a laser (Section 0.6.3). The standard He-Nelaser supplied in the kit produces three wavelengthsseparated in frequency by c/2L, where c is the speed oflight and L is the distance between the ends of the lasermirrors (approximately the length of the laser tube).

The technique used for this project was described inSection 0.6.4 in the Primer. By observing the visibilityof the fringes of the Michelson interferometer you willbe able to measure the frequency between axial modesin the laser. The Michelson interferometer is describedin Section 0.4.2 in the Primer and was constructed inProject #6.

Experimental Set Up

NOTE:If you have already built a Michelson interfer-ometer in Project #6, the set up is almost com-plete. The only modification you will have tomake is to change the reference mirror (Step 5)from a fixed base assembly that screws into theoptical table to a movable assembly by attachinga B-2 base from one of the LCA assemblies to theBSA-I assembly (Fig 7-1). Also, adjust the beamexpander for a planar wavefront; the small re-flected beams on the card at the front of the laserwill make coincident alignment much quicker.Once this is done you can start at Step #10.Because the reference mirror assembly is nolonger fixed, you will have to readjust to find thefringes every time you move or bump this assem-bly. Patience!

1. Mount a laser assembly (LA) to the far side of thebreadboard (Fig 7.1). Adjust the position of thelaser such that the beam is parallel to the edgeand in line with a line of tapped holes in thebreadboard top. Tape an index card (QI) with asmall (about 2 mm) hole in it to the front of thelaser, so that the laser beam can pass through it.

Newport Equipment Required:Part Qty Description

LA 1 Laser AssemblyBSA-I 3 Beam Steering AssemblyBSA-I* 1 Beam Steering Assembly

with B-2 BaseLCA 3 Lens Chuck AssemblyTA-II* 1 Target Assembly

without B-2 baseLKIT-2 1 Lens KitFK-BS 1 Beam Splitter

Additional Equipment Required:Part Qty Description

QI 1 Index CardQW 1 Measuring Tape

Table 7.1 - Equipment Required

72

This card will be used as a screen to monitor thereflections from the components as they areinserted in the beam. See the note in Project #3on the alignment of laser beams.

2. Mount a beam steering assembly (BSA-I) approxi-mately 4 inches in from the far corner of thebreadboard (Fig. 7-1). Adjust the height of themirror mount until the beam intersects the centerof the mirror. Then rotate the post in the postholder until from the laser beam is parallel to theleft edge and the surface of the optical bread-board.

3. Place a second beam steering assembly (BSA-I) inline with the laser beam at the lower left cornerof the optical breadboard (Fig. 7-1).

4. Set up a beam expander between the first twoBSA-I’s as explained in Experiment #3. Mountthe first lens to the optical table without a B-2base (Fig 7-1). This beam expander generates anexpanded plane wave that is needed to build theinterferometer.

5. Place a lens chuck assembly (LCA) approximatelyfive inches to the right of the last BSA-I mount.Mount a 50/50 beam splitter in the LCA and rotatethe assembly 45 degrees to the optical path. Thebeam reflected off of the first surface should beperpendicular to and going in the direction of thefront edge of the breadboard. The beamsplitterdivides the incoming laser beam into two equalcomponents for the two arms of the interferom-eter.

6. Place a BSA-I with its mirror centered about thepath of the reflected beam five inches from thebeam splitter to intercept the reflected beam(Fig. 7-1). Adjust the mirror until its beam isdirected back to the index card taped to thelaser. This mirror will be called the fixed mirror.

7. Place a BSA-I* (modified with a B-2 base, so that itmay be moved on the optical table) with itsmirror centered about the path of the transmittedbeam and about five inches beyond the beamsplitter such that the beam is retro-reflected backto the laser. This mirror will be called themoveable mirror.

8. Use a TA-II* (no base) with an index card (QI) asan observation screen on the other side of thebeamsplitter from the fixed mirror (see Step 6).Adjust the mirrors in the two arms of the interfer-

73

ometer until the two beams overlap on thescreen. There should be combined reflections atthe observation screen and on the card at thefront of the laser.

9. As the two beams are brought into coincidence, aseries of bright and dark fringes should appearrepresenting the interference pattern betweenthe two wavefronts. The orientation and separa-tion of the fringes can be controlled by adjustingthe moveable and fixed mirrors. Usually it is bestto use one mirror for the adjustment mirror.Adjust the mirror so that approximately fivefringes appear across the beam on the card. Thenumber of fringes can be varied in a particulardirection by tilting the reference mirror in adirection perpendicular to the direction of thefringes. Your Michelson interferometer is nowadjusted.

10. Adjust the mirror position until the path differ-ence between the two arms of the Michelsoninterferometer are equal. In this case, however,the two paths are not quite equal since the lightthat has to pass through the glass to reflect offthe beamsplitter travels an additional distance,since the path inside the glass must be multipliedby the refractive index of the glass. In the case ofthe beamsplitter (FK-BS), the additional opticalpath length amounts to approximately 2√!2 timesthe thickness of the beamsplitter.

11. Place the measuring tape on the breadboard withsome major division on it even with the center ofthe beamsplitter. Record the value in you note-book.

12. Translate the moveable mirror away from thebeamsplitter by 1⁄2 inch increments. Each time themirror is moved readjust the mirror tilt until fourto six fringes are observed. There will be aposition in path difference where the fringesseem to fade in and out. Carefully move themirror about this position until the fringes cannotbe made to appear. Record this position.

13. Continue to move the mirror away from thebeamsplitter and observe that the contrastincreases. Using the same technique as above,look for the mirror positions that give the stron-gest possible contrast. You may have to gobeyond that position and return several times inorder to verify your judgement. Record the valuefor the contrast maximum.

74

14. Attempt to find additional contrast maxima andminima. If you are able to do so, record thesepositions also. The distance between successivemaxima and minima should be the same. If youdo have more than one value, take the average ofthe maximum to minimum distances.

15. From the Primer (Section 0.6.4) it was shownthat if the fringe visibility went from a maximumto a minimum in a distance ∆L, the frequencyseparation between two outputs of a laser thatcaused this contrast variation is

∆v = c/2L = λv/4∆L = c/4∆L . (7-1)

Calculate the frequency difference based on thevalues you have measured for ∆L.

16. It was shown that the frequency separationbetween the neighboring modes of a laser wasc/2L. Based on the frequency separation justdetermined in Step #15, find the distance be-tween mirrors in the laser tube you are using. Isthis value reasonable based on the exteriordimensions of the laser?

Additional experiments:If a polarizer is added to the system at the output of thelaser while the mirror is at the location of minimumcontrast and the polarizer is rotated parallel to thedouble mode output, the contrast of the fringes will bemaximized. The modes contributing to the majorcontrast variation will now be separated by c/L. Sincethe modes are spaced twice as far apart as in theprevious experiment, the contrast minimum will occurat one half the value of ∆L found in Step #15. Rotate thepolarizer by 90° and observe the contrast variation asyou move the mirror. Don’t expect anything as dramaticas the effects you have just been measuring, since youare now observing the interference of a single modelaser. You would need the length of a football fielddifference in one of the arms in order to reduce thecontrast. Thus the coherence length of this laser with asingle mode selected by polarization is of the order ofhundreds of meters. This can be compared to thecoherence length of a standard laboratory gradesodium light source (0.045 nm half width @ λ = 550 nm)which has a coherence length of about 2 mm.

75

Project #8

Polarization of Light:

While the idea of polarization is fairly simple (Section0.5 in the Primer), it remains somewhat abstract untilyou can work with light and its various forms of polari-zation. The object of this project is to give you someexperience in the orientation and generation of polar-ized light.

As was pointed out in the Primer (Section 0.6.3) andexplored in Project #7, the output of the laser used inthe Project in Optics Kit has three modes with two ofthe modes polarized orthogonally to the third mode.Because the laser has no special stabilization circuitry,the modes of the laser will tend to drift in frequency, sothat one of the modes of one polarization may drop outand a mode of the orthogonal polarization pop in. Sowhat had been the single mode polarization becomesthe two mode polarization and vice versa! This phe-nomenon is referred to as mode sweeping. Its effect onthese experiments is that the output of the laser in aparticular polarization will change slowly over time.

So, as you take measurements during this experiment,be aware that some of the power variation may not bedue to your efforts to change a variable, but maycaused by mode sweeping effects. Two ways to mini-mize these effects are; (1) to let the laser warm up byturning it on as soon as you enter the lab and (2) takethe data series more than once to account for anysource variation. If there is time, you may want tomonitor and record the power output of the laser forsome period of time when you are not performing theexperiments. These “baseline” measurements are usefulto assess this power variation phenomenon.

NOTE: Part of this experiment requires a measurementof optical power levels. It is not possible for you to relyon your eye to perform this task. Its very constructionwith a diaphragm that closes down when the light getstoo bright makes it a great image detector but a poorpower meter. Therefore, some sort of optical detectormust be used or constructed. If you do not have aNewport 615 or equivalent Optical Detector, you willneed to obtain a standard laboratory voltmeter andconstruct a fairly simple detector. While there are anumber of devices that will do the job, the instructionsat the end of this Project should be sufficient to con-struct a simple photodetector circuit.

E

E⊥

E

θθ

180° outof phasewith originalE•fieldcomponents

d

E

θ

θ

Optic AxisOptic Axis

E

E⊥

Figure 8-1. Half-wave plate. The plate produces a180° phase lag between the E|| and E⊥ components ofincident linearly polarized light. If the originalpolarization direction is at an angle θ to the optic axis,the transmitted polarization is rotated through 2θfrom the original.

76

Newport Equipment Required:

Part Qty Description

LA 1 Laser AssyBSA-I 2 Beam Steering AssyRSP-I 1 Rotation Stage AssyLCA 2 Lens Chuck AssemblyTA-II 1 Modified Target AssyR (RSP-1T) 1 Rotation Stage and Plug

2 Linear Polarizer

Additional Equipment Required:

Part Qty Description

QI 1 Index CardQV 1 VoltmeterQD 1 Photodetector or Solar CellQM 1 Microscope Slide

Experimental Set Up:1. Mount a laser assembly (LA) to the rear of the

breadboard. Adjust the position of the laser suchthat the beam is parallel to the edge and on to ofa line of tapped holes in the breadboard top.Tape an index card with a small (about 2 mm)hole in it to the front of the laser, so that the laserbeam can pass through it. This card will be usedas a screen to monitor the reflections from thecomponents as they are inserted in the beam. Seethe note in Project #3 on the alignment of laserbeams.

2. Mount a beam steering assembly (BSA-I) approxi-mately 4 inches in from the far corner of thebreadboard (Fig. 8-2). Adjust the height of themirror mount until the beam intersects the centerof the mirror. Then rotate the post in the postholder until the laser beam is parallel to the leftedge and the surface of the optical breadboard.

3. Place a second beam steering assembly (BSA-I) inline with the laser beam at the lower left cornerof the optical breadboard, (Fig. 8-2). Rotate andadjust the mirror mount until the laser beam isparallel to the front edge and the surface of theoptical breadboard.

4. Place the detector in a lens chuck assembly(LCA) and mount it well beyond the second BSA-Iso that the beam hits the center of the detector.

5. Mount a polarizer in an LCA assembly with thenotch in the disk facing up. Tape a secondpolarizer by the edges to a rotation stage assem-bly (RSA-I) such that the notch is vertical whenthe rotation stage is set at 360°. Place both ofthese assemblies directly in line with the laserbeam between the second BSA-I and the detector.The output of the device will be proportional tothe irradiance of the light (Watts/m2). Thisquantity is proportional to the square of theamplitude of the electric field, as discussed inSection 0.5.1 of the Primer. Rotate the secondpolarizer by 10° increments between 0° and 180°,recording the angle and the output of the detec-tor as measured by the voltmeter.

6. Plot the results of your measurements andcompare them to the Law of Malus.

Itrans = I0 cos2θ (0-18)

You will have to scale your comparison plot tothe I0, which is the maximum value recorded.

BSA-1

LA

RSA-ILCA/QDLCA

BSA-1

Table 8.1 - Equipment Required

Figure 8-2 Schematic view of polarization of lightexperiment.

77

Figure 8-3. Schematic view of Brewster angleexperiment.

BSA-I

LA

LCA/RSA

TA-II

LCA

BSA-I

Also, you may have missed the orientation of oneof the polarizers by some amount and thereforethe two curves may be shifted along the angleaxis. You may have to adjust your plots to makethe comparison, but you should justify anyadjustment in your notebook.

7. Remove the RSA-I from the bench and demountthe Rotation Stage. Remove the polarizer andinsert a plug in the 1 inch hole. Mount a LensChuck Assembly (LCA) without a base onto thisstage by using the hole in the center of the plug.The Rotation Stage (R) is then attached to thetable with 1/4-20 screws. Place the LCA such thatthe laser beam passes through the center of thelens holder. Tape a microscope slide (QM) to thelens holder so that the slide is held firmly in placeand the beam does not pass through the tape (Fig.8-3).

8. Set the Rotation Stage to 0°. Rotate the lens chuckin its holder so that the beam reflected from theslide is sent back along the input beam. You maywant to use the index card with the hole in it toset the beam. Tighten the screw on the post, sothat the lens chuck is fixed in the holder.

9. Rotate the polarizer so that the transmissiondirection is horizontal to the table (notch up-ward). This means that the polarization vector isnow horizontal and in a plane formed by the laserbeam direction and the normal to the surface ofthe microscope slide. The plane defined by thesetwo directions is called the plane of incidence.

10. Replace the detector in the LCA with a modifiedtarget assembly (TA-II) since you are going tohave to follow the beam on the table.

11. Rotate the Rotation Stage (R) away from 0° andobserve the reflection from the slide on the indexcard mounted in TA-II. At some point in thisprocess the reflection from the slide will becomevery dim. Scan past the point of minimum reflec-tion and observe that the reflected beam irradi-ance increases. By successive approximations,bring the stage to produce the minimum reflec-tion. (You may find that you can improve theminimum by slightly tweaking the input polariza-tion angle by a small amount.) Record the angle ofthe Rotation Stage and determine the anglebetween the beam and the normal to the surface.Compare this angle to that of Brewster’s angle,discussed in Section 0.5.1 in the Primer, and thevalue given there.

78

Q1

10K 1%

2

1 V/100 µa

9V+–

Light

3 41

8+–

U1 Out

9V

+ –

DVM

9V

Q1

10K 1%10 MEG

1 V/100 µa

Light

Alternate Circuit

Photo Transistor Test Circuit

Q1 IR Photo TransistorRadio Shack P/N 276-145 SDP 8403-301

U1 JFET OP AMPRadio Shack P/N 276-1715

12. Rotate the input polarization of the beam to theorthogonal polarization and observe that no suchreduction in the reflected beam irradiance occursat any angle.

Additional Exercises:If your photodetector is sensitive enough, it is possibleto measure the reflected power of the beam for bothpolarizations. A plot of the recorded powers as afunction of the angle of incidence can be compared tothe theoretical curves in most college optics texts inthe sections of polarization of light by reflection.

Another use of polarized light is to measure the sugarcontent of various syrups in the candy making industry.The basis for this measurement is the fact that sugarsare optically active materials, causing the plane ofpolarization of light passing through the liquid torotate. The amount of rotation is dependent on thesugar concentration and the thickness of the samplethrough which the light travels. Thus by constructing adevice with a standard sample length and knowing therotation constant of the sugar, the concentration of anunknown sugared liquid can be determined.

For example, colorless corn syrup (Karo is standardbrand) has a rotation constant for visible light that iseasily measured. Using a clear plastic tank such as theone in Project #1, fill the bottom of the tank with syrupto height sufficient to pass a beam through and place iton a box or other object to raise the liquid to thecorrect beam height. Without the tank in the beam, setup two crossed polarizers, then insert the tank of syrupbetween them. Because of the optical activity of thesyrup the light will be rotated and light will be transmit-ted by the analyzer. By rotating the analyzer to extin-guish the beam, the angle of rotation can be deter-mined. The rotation constant is equal to that angledivided by the distance the beam traveled through thesyrup.

When you are finished, empty the tank and store thesyrup in a covered jar. Rinse the tank and dry it toprevent a sticky mess the next time it is used.

Circuit used to measure light in polarized lightexperiment.

Alternate circuit is for low impedance measuringequipment. Ground pins 5 and 6 if not used.

79

Table 9.1 - Equipment Required

Figure 9-1. Schematic view of Birefringence ofMaterials experiment.

BSA-I

BSA-IBeam Stop

TA-II

TA-IILCA

LCA

LA

LCARSA-I

Project #9Birefringence of Materials:

The polarization of light can be used to control thepassage of light through an optical system and toimpress information on a light wave by changing(modulating) the amount of light birefringent material.

In this project the birefringence of a material will beused to change the polarization of the laser. Using aquarter-wave plate and a polarizer you will build anoptical isolator. When you add a second quarter-waveplate, a polarization rotator results.

Experimental Set Up:1. Mount a laser assembly (LA) to the rear of the

breadboard. Adjust the position of the lasersuch that the beam is parallel to the edge and inline with a line of tapped holes in the bread-board top. Tape an index card with a small(about 2 mm) hole in it to the front of the laser,so that the laser beam can pass through it. Thiscard will be used as a screen to monitor thereflections from the components as they areinserted in the beam. See the note in Project #3on the alignment of laser beams.

2 Mount a beam steering assembly (BSA-I) ap-proximately 4 inches in from the far corner ofthe breadboard. Adjust the height of the mirrormount until the beam intersects the center ofthe mirror. Then rotate the post in the postholder until from the laser beam is parallel tothe left edge and the surface of the breadboard.

3. Place a second beam steering assembly (BSA-I)in line with the laser beam about two thirds ofthe way along the left side of the breadboard.Rotate and adjust the mirror mount until thelaser beam is parallel to the front edge and thesurface of the breadboard.

4. Mount the beamsplitter in a lens chuck assembly(LCA) and place the unit four inches to the rightfrom the last beam steering mirror. Adjust thebeamsplitter such that the beamsplitter surfaceis oriented at 45° to the last beam steeringmirror.

Additional Equipment Required:

Part Catalog #. Qty Description

QI 2 Index CardTape

Newport Equipment Required:

Part Catalog #. Qty Description

LA 1 Laser AssyBSA-I 3 Beam Steering AssyRSA-I 1 Rotation Stage AssyLCA 1 Lens Chuck AssyLCA* 2 Lens Chuck Assy

without B-2 BaseTA-II 2 Modified Target AssyFK-B2 1 Beam Splitter

2 Linear Polarizer2 Quarter-wave plates

80

5. Place a piece of cardboard or some other objectat the edge of the optical breadboard to blockreflected light from the back surface of thebeamsplitter (the back surface of a beamsplitteris always the surface opposite the beam splittingsurface). There will be at least two beams ofunequal intensity.

6. Place a modified target assembly (TA-II) on theopposite side of the beamsplitter. Insert a whitecard in the holder at the beam height. This will beused to monitor the amount of reflected light.

7. Mount a polarizer into a lens chuck assembly(LCA) without a B-2 base and mount it to theoptical breadboard in line with the laser beamand several inches to the right of the beamsplit-ter. Set the polarizer with the indicating notch up.Rotate the lens chuck mount slightly so that thereflection off this polarizer can be seen on theindex card.

8. Mount a second polarizer into a lens chuckassembly (LCA) without a B-2 base and mount it 7inches to the right of the first polarizer. Place asecond target assembly TA-II to the right of thesecond polarizer. Rotate the second polarizeruntil it completely blocks the light from the firstpolarizer (i.e. polarization axes crossed). Rotatethe lens chuck mount slightly so that the reflec-tion off this second polarizer can be recognizedas a separate beam.

9. Insert the quarter-wave plate into a rotation stageassembly (RSA-I). Place the assembly betweenthe two polarizers. Slowly rotate the quarter-wave plate until the output through the secondpolarizer is a maximum.

10. Loosen the second polarizer in its lens chuck androtate. Note that the output does not changesince the light is now circularly polarized andthere is an equal component in any direction ofthe polarization orientation. The quarter-waveplate has converted the linear input beam to acircularly polarized beam.

81

11. Having verified that the plate has producedcircularly polarized light, replace the secondpolarizer with a mirror on a beam steeringassembly, BSA-I (Fig. 9-2). Observe the lightreflections on the index card. There will besurface reflections off the surfaces of thepolarizer and the quarter-wave plate, but therewill be no strong reflection from the mirrorbecause as was pointed out in Section 0.5.2 inthe Primer, the mirror reverses the circularlypolarized light and on second passage throughthe quarter-wave plate, the beam is againlinearly polarized but at right angles to theoriginal polarization. When the reflected beamhits the polarizer again it is absorbed. You cantest this by either removing or rotating thequarter-wave plate or rotating the input polar-izer somewhat. In either case, the light on themirror is no longer circularly polarized. This isequivalent to saying that the outgoing beam hasbeen “isolated” from reflections after thequarter-wave plate.

12. Starting with the arrangement completed in Step#9 with a single quarter-wave plate in the RSA-I,hold a second quarter-wave plate between thecrossed polarizers without disturbing theorientation of the first. Rotate the secondquarter-wave plate until the light passingthrough the second polarizer is a maximum.Carefully tape this second quarter-wave plate tothe RSA-I in which the first quarter-wave plate ismounted. You have now created a half-waveplate which rotates the input polarization by 90°.To check that this is indeed so, rotate thesecond polarizer to produce the minimumtransmission. You will find that you will have torotate through 90° and that the axes of the twopolarizers are now parallel.

13. Finally, rotate the first polarizer by somespecific angle, say 10°, and then note the amountby which the second polarizer must be rotatedto extinguish the beam. You will find that theanalyzer must rotate through twice the angle ofthe initial polarizer.

BSA-I

BSA-IBeam Stop

TA-II

LCA

LCA

LA

RSA-I BSA-I

Figure 9-2.

82

Newport Equipment Required:

Part Cat. # Qty Description

LA 1 Laser AssyBSA-I 2 Beam Steering AssyLCA 1 Lens Chuck AssyLCA* 2 Lens Chuck Assy

without B-2 BaseTA-II 2 Modified Target Assy

2 Lenses for Beam ExpanderTL KPX100 1 Transform Lens (150 mm EFL)

Object Transparencies

Additional Equipment Required:

Part Qty Description

QI 1 Index cardQM 1 Microscope slide

Toothpicks, ink, other masking materials

Project #10Abbe Theory of Imaging:

This experiment touches on the subject of spatialfrequency content of objects and how they could beused to control the shape and quality of an image. Thissubject is similar to finding the frequency harmoniccontent of a waveform such as that produced by amusical instrument. For example, a musical instrumentmay produce both a low pitch tone and a high pitchtone. We can control the quality of the sound byfiltering out one of the two frequency harmonics with alow pass or a high pass filter. A discussion of the basictheory was given in Section 0.7 in the Primer.

As noted in the Primer, objects have certain intensityprofiles which translate into a corresponding spatialfrequency distribution. In this project the frequencydistribution of an object illuminated with a laser beamwill be examined with a single lens placed after thepicture or slide. The light distribution formed at thatfocal plane tells us the frequency content of the object,and by manipulating the light in that plane we gaincontrol of the quality and content of the image to bedisplayed.

The laser beam we are using has a smooth profile, i.e.the intensity distribution does not have any wiggles,and when it is focused it produces a single small spot,i.e. the original beam contains only low spatial frequen-cies. On the other hand, if we pass this beam through agrating or screen which introduces many variations onthe laser profile, then in the focal plane of the lens wewill see several spots indicating that additional spatialfrequency components have been added. Let us build aset up that allows us to examine this feature.

1. Mount a laser assembly (LA) to the far side ofthe breadboard (Fig. 10-1). Adjust the positionof the laser such that the beam is parallel to theedge and in line with a line of tapped holes inthe breadboard top. Tape an index card with asmall (about 2 mm) hole in it to the front of thelaser, so that the laser beam can pass throughit. This card will be used as a screen to monitorthe reflections from the components as they areinserted in the beam. See the note in Project #3on the alignment of laser beams.

LABSA-I

BSA-I

LCA

LCA

LCATA-II TA-II

Figure 10-1. Schematic view of imaging experiment.

Table 10.1 - Equipment Required

83

2. Mount a beam steering assembly (BSA-I)approximately 4 inches in from the far corner ofthe breadboard. Adjust the height of the mirrormount until the beam intersects the center ofthe mirror. Then rotate the post in the postholder until from the laser beam is parallel tothe left edge and the surface of the opticalbreadboard.

3. Place a second beam steering assembly (BSA-I)in line with the laser beam at the lower leftcorner of the optical breadboard. Rotate andadjust the mirror mount until the laser beam isparallel to the front edge and the surface of theoptical breadboard.

4. Set up a beam expander between the first twoBSA-I’s as explained in Project #3. Mount thefirst lense to the optical table without a B-2 base(Fig 10-1).

5. Mount the 150mm EFL transform lens in an LCAwithout a base 12 inches from the second BSA-Iand center it in the path of the beam.

6. Set up a modified target assembly (TA-II) withan index card and place it at the beam focus,150 mm from the transform lens. The plane ofthe card represents the back focal plane of thelens. For some of the experiments the indexcard will be replaced by a microscope slide.

7. Mount a second modified target assembly (TA-II) without a base 225 mm before the transformlens. This assembly will be used to hold thepicture slides before the transform lens in thepath of the laser light. This assembly will becalled the slide holder.

8. The set up is now ready for the examination ofslides. The image will be observed on an indexcard in another modified target assembly. Thelocation of the observation plane will be about450 mm after the last lens and will give an imageabout twice the size as the object slide when thecard at the back focal point of the lens isremoved.

9. Remove any slides attached to the slide holder.At the back focal plane we see a single focalspot. The position of the spot locates the “dclevel” of illumination of the beam entering thelens. Any other spots appearing on the cardindicate that other spatial frequencies arepresent. Remove the card from the back focalplane and you will see uniform (or dc level)illumination at the observation place.

84

10. Place the target containing the square mesh inthe slide holder and replace the index card to theback focal plane of the lens. There you will see asquare grid pattern of dots representing thefrequency content of the mesh in both horizontal(or x) and the vertical (or y) axes. Mark on thecard with a pencil the location of these axes. Thedots on the x (or y) axis represent frequenciespresent in that direction in the slide.

11. Remove the card at the back focal plane andmove the TA-II in the image plane to achieve thesharpest image. This image can be manipulatedby eliminating certain frequencies, much in thesame way a high fidelity audio filter controls thetone of a musical instrument. To illustrate this,cut out a narrow vertical slit in a section of theindex card such that only those dots on thehorizontal axis are passed through the cutout.Place the card in the TA-II at the back focal planeso that the rest of the dots are blocked by thecard. Examine the image on the index card in theobservation plane and record what you see. Youwill note that the image consists of only horizon-tal lines. When you remove the slit in the focalplane, the image will resemble the original object.

12. Rotate the card in Step #11 above such that thedots on the y-axis are passed. Record what yousee.

13. Make another cut out such that only the centralspot is transmitted. Note that only uniformillumination is present at the observation plane,i.e. you have filtered out all higher frequenciesand all that is left is the low frequency (i.e.relatively uniform illumination). This is theprinciple of the spatial filter. It “cleans” opticalbeams by removing the high frequencies byfocussing the beam through a pinhole, therebyobstructing the unwanted harmonics.

14. Other cutouts can be inserted at the back focalplane. For example, if you make a filter in theform of a larger hole that passes only the centraland the two adjacent spots. Record what you see.Note that this removes the sharp edges of theimage and produces a soft picture. Alternately,you can make a special filter that will obstructonly the central spot by putting a dot of ink on amicroscope and locating it at the center spot inthe back focal plane. Record your observations.This will remove the uniform illumination fromthe image and leave the edges enhanced.

85

15. Replace the square mesh slide at the slide holderwith any picture slide. Notice that this picturehas superimposed on it many horizontal lines. Atthe back focal plane of the lens the light distribu-tion consists of an irregular distribution of lightrepresenting the many spatial frequenciespresent in the picture. Superimposed on thisdistribution is a set of faint spots aligned alongthe vertical axis and passing through the centralspot. This line of spots represent the frequencycontent of the horizontal lines in the pictureslide.

16. Attach with tape to the microscope slide twopins (or toothpicks or thin objects) so thatwhen it is placed back in the back focal planethe objects will obstruct this vertical line ofspots. The central spot should not be ob-structed. Record your observations. Removeand then replace this “needle” filter.

17. Other cutouts may be made and used at theback focal plane. For example, cut a hole in theindex card so that it obstructs the outer spots.These spots contain the high frequency informa-tion. Record your observations. Obstructingthem leads to a softer, more fuzzy, picture as isseen at the observation plane.