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Rice University Physics 332

DIFFRACTION AND FOURIER

I. Introduction!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "# II. Theoretical considerations!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! $# III. Measurements !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%$# IV. Analysis !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%&#

Revised July 2011 I. Introduction

In this experiment, you will first examine some of the main features of the Huygens-Fresnel scalar theory of optical . This theory approximates the vector electric and magnetic fields with a single scalar function, and adopts a simplified representation of the interaction of an electromagnetic with matter. As you will see, the model accounts for a number of optical phenomena rather well. If one accepts the Huygens-Fresnel theory, it becomes possible to manipulate images by altering their spatial spectrum in much the same way that electronic circuits manipulate sound by altering the temporal frequency spectrum. In the second part of the experiment, you will see how this is done in some simple cases. Most of the information you will need for the data acquisition and analysis is contained in the following sections. You should probably read through the theoretical discussion before attempting the experiments. Corresponding parts of the measurement and analysis sections are best studied together as you proceed through the experimental work. Further details, particularly of the theory, can be found in your favorite optics textbooks. More extensive discussion of applications will be found in the literature on optical . We conclude this introduction with a short list of the topics which you should cover in your work: 1. Compare experimental and calculated plots of single and multiple-slit patterns. 2. Obtain plots of the pattern of a single slit as a function of distance. Compare your results with the calculated patterns, and discuss possible reasons for any differences you note. 3. Experimentally demonstrate the effect of removing various components from the images of a slit and of a grid. Show that the theory accounts for your observations. 4. Optional: Examine Fresnel patterns for other objects, look at the effect of filtering on other images, or carry out other related measurements and calculations as desired.

2 II. Theoretical considerations

A. Huygens-Fresnel scalar theory Imagine an opaque screen with a hole in it, illuminated by monochromatic light. Upon close examination, one finds an intricate pattern of light and dark behind the screen, not simply a geometric shadow as would be predicted by a particle theory of light. The first attempt to explain the observed pattern was made by Christiaan Huygens, in 1678, on the basis of his wave theory of light. In 1818, Augustin Fresnel combined Huygens' intuitive ideas with Young's principle of interference to produce a reasonably quantitative wave theory of optics. Briefly put, the model assumes that each point within the illuminated aperture of a screen is the source of a spherical wave. The amplitude of the optical field at any point beyond the screen is found by adding the spherical arriving from each fictitious point source. Additional assumptions are needed to insure that the point sources only radiate in the forward direction, and that the edge of the aperture makes no special contribution. Huygens and Fresnel did their work before Maxwell, so they did not really have a proper description of the electromagnetic field and were forced to resort to rather arbitrary assumptions. To the modern student of optics it seems more sensible to start with Maxwell's equations, appropriate boundary conditions at the screen, and a description of the source of electromagnetic ! ! waves. The E and B fields can then be calculated at all points in space, and the optical intensity ! 2 found from E . It turns out that this approach is nearly impossible to carry through to a successful conclusion for any reasonably interesting geometry. The most important difficulty is ! ! that E and B are coupled vector fields, so the basic equations are difficult to solve except for highly symmetric cases. Then, even when the geometry is simple, a proper description of the response of the screen material to the electromagnetic wave is difficult. As a result only a few simplified geometries have been calculated rigorously. Fortunately a number of scientists, including Kirchoff, Rayleigh and Sommerfeld, were able to develop a simplified theory of optical diffraction between about 1880 and 1900. The results of this theory agree well with experiment, and with complete calculations in those cases where the latter can be carried out. Accordingly, we will work with this simplified model in what follows. ! The first step in the simplification is to replace the vector for E with a scalar equation for one component, E(x,y,z,t):

2 2 1 " E ! E = 2 2 (1) c "t

3 By treating only one component of the field, we are assuming that interactions with the aperture cannot affect the polarization of the incident wave. Since we already know the explicit time dependence of the wave, we write E(x,y,z,t) = Re[Eexp(-i!t)] and substitute into (1) to obtain the desired scalar wave equation

2 2 ()! + k E =0 (2) where

! 2" k = = (3) c #

The solution of (2), with appropriate boundary conditions, will be our description of the diffracted wave. Equation 2 is to be solved for the geometry shown in Fig. 1, which depicts a planar aperture (the object) in an opaque screen illuminated from the left, and an image plane. We assume that the aperture acts as the source of a field Eo(xo,yo) which may be attenuated and shifted relative to the incident field. Eo is assumed to be identically zero beyond the opaque portions of the screen. These conditions cannot be exactly correct because the aperture must influence the fields for distances of the order of a few , but they are good approximations for large apertures. One can then show rather generally that Ei(xi,yi)is given by

1 exp()ikrio Ei = Eo cos(#) ds (4) i! "" rio

y o

y x i o r io x i

z

Fig. 1 Sketch of object and image plane coordinates showing the distance rio between image and object points. The line rio makes an angle " with the z-axis.

4 From our assumptions, Eo is zero except directly behind the aperture, so the surface integral effectively runs over the aperture in the screen. Although Eq. 4 appears complicated, it is really nothing more than a formal statement of the Huygens-Fresnel argument, which has now been more nearly justified. To see this, recall that a spherical wave of amplitude A diverging from a point source is described by

exp(ikr) E = A (5) r

The integral (4) is a sum of such waves, of amplitude Eo, originating from the object points (xo,yo) within the aperture. The cos(!) term corresponds to Fresnel's assumption that only the forward-propagating part of the spherical waves is to be retained. The screen is characterized by a complex transmission function To(xo,yo) which describes the phase shift and attenuation produced by the aperture. If, as is frequently the case, the barrier either totally transmits or totally blocks the incident beam, To(xo,yo) becomes identically 1 for (xo,yo) within the aperture and zero otherwise. The object amplitude Eo is found by multiplying the incident wave by To(xo,yo). The product is particularly simple for a point-source on the coordinate axis a distance zs from the aperture, provided zs is large compared to the dimensions of the aperture. In that limit we can replace r by zs in the denominator of Eq. 5. We must be more careful with the r in the exponential, because kr is likely to be a big number which we must determine to a fraction of 2". A simple expansion of r gives

) 2 2, 1 # xi " xo & 1 # yi " yo & r ! zs 1 + % ( + % ( (6) + 2 $ z ' 2 $ z ' . * s s - which can be inserted into (5) to yield

A ! ik $ E exp(ikz ) exp x2 y2 T x , y (7) = s # ()o + o & o()o o zs " 2zs %

Further simplification is obtained for plane wave illumination normal to the aperture, which is obtained by letting zs ! " holding A/zs ! A' finite. The varying phase factors then vanish, leaving Eo = A' To(xo,yo).

5 At this point, most of the physics is done and the content of the model has been presented. The remainder of our job is to make some geometrical approximations that will let us evaluate (4) in the cases of interest.

B. Fresnel and Fraunhofer diffraction Referring again to Fig. 1, we see that ! in Eq. 4 is determined by the distance z between aperture and image planes, and by the distance from the image point to the origin. If we restrict our attention to a finite region of the image plane, corresponding to ! less than a few degrees, it is adequate to take cos! ! 1. Exactly the same considerations allow us to replace rio in the denominator by z and by the expansion (6) in the exponential. Substituting into (4) we get

exp(ikz) $ ik 2 2 ' E = E exp% ()x # x + ()y # y ( dx dy (8) i i!z "" o & 2z[]i o i o ) o o

This is known as the Fresnel approximation to the scalar diffraction theory. It is useful when z is very large compared to a , but not necessarily much bigger than the linear dimensions of the aperture. (The expansion (6) seems to require z » (xi - xo), but most of the contribution to the integral (8) comes from regions where (xi - xo) ! 0. Outside of those regions the exponential oscillates rapidly, leading to a cancellation of positive and negative contributions.) A further rearrangement of the Fresnel diffraction expression will be computationally convenient. Expanding the quadratic terms in the exponential, we get

exp(ikz) ik E = exp" x2 + y2 $ i i!z # 2z ()i i % (9) ' " ik 2 2 $ * " ik $ & (E o exp xo + yo + exp . ()xi xo . yi yo dxodyo -- ) # 2z ()% , # z %

This tells us that the Fresnel diffraction pattern can be found, within phase factors, by computing 2 2 the of {Eo exp[(ik/2z)(xo +yo )]}. A very efficient algorithm, the Fast Fourier Transform or FFT, exists to do this computation. The physical significance of the transform is discussed in the topical notes. 2 2 If we move farther away from the aperture, so that z » k(xo +yo )max, the quadratic phase factor in (9) is approximately unity over the entire aperture. This infinite-distance limit is called the Fraunhofer regime, and is the case usually considered in elementary treatments. The diffraction pattern is explicitly given by

6 exp(ikz) ik ik E = exp" x2 + y2 $ E exp" ' ()x x + y y $ dx dy (10) i i!z # 2z ()i i % && o # z i o i o % o o which is simply the Fourier transform of the aperture illumination. We will usually want to know the optical intensity, which is proportional to |E|2, so the phase factor in front is irrelevant.

C. Thin and spatial filtering The geometrical-optics analysis of lenses is already familiar to you. Here we are concerned with the phase of the optical waves, in order to treat interference effects properly, so we must reexamine the effect of a on an incoming wave. We will eventually find that lenses do indeed form images, but with a phase shift which depends on position. Fortunately, we are usually concerned with intensity, not amplitude, so the phase shift is unimportant, and we can expect to use lenses as usual to form magnified images of objects. More surprisingly, we will also find that a lens can be used as an optical Fourier transformer. This will allow us to observe the Fraunhofer diffraction pattern at a finite distance from the diffracting object, and to do the Fourier transform required for spatial filtering experiments.

1. Lens geometry The type of lenses we need to understand consist of optically dense material bounded by spherical surfaces, as shown in Fig. 2a. We will treat only thin lenses, in the sense that a ray entering at (x,y) on one face emerges at nearly the same coordinate on the other face. A thin lens therefore serves only to delay the incident wavefront by an amount proportional to the thickness of the lens at each point. Denote the thickness at any point by !(x,y), and the maximum thickness by !0. The total phase delay between the input wave E! and the output wave E!' is

! (x, y) = kn"(x,y) + k[]"0 #"(x, y) (11) where n is the index of refraction. Rearranging, we get

E = E exp ik" exp ik(n #1)"(x, y) (12) ! ! ! []0 []

To calculate !(x,y), we imagine splitting the lens, as shown in Fig. 2b, and work out the geometry to obtain

7 E! E!' !(x,y)

-R 2

!0

R1

ab Fig. 2 Definition of the geometry for a thin lens.

) 2 2 1/ 2 , ) 2 2 1/ 2, # x + y & # x + y & ! = !0 " R1+1 " % 1 " ( . + R2 +1 " % 1 " ( . (13) R2 R2 * $ 1 ' - * $ 2 ' -

This simplifies enormously if we stay near the lens axis, since then we can expand the square roots to obtain:

x2 + y2 # 1 1 & !(x,y) = !0 " % + ( (14) 2 $ R1 R2 '

Eq. 14 limits us to paraxial rays, but that is the normal approximation in geometric optics anyway. Using the lens-maker's formula to relate the focal length f to the lens parameters, we substitute (14) into (12) to arrive at the desired equation for E!' :

# ik & E = E exp " x2 + y2 (15) ! ! ! % ()! ! ( $ 2 f '

(We have also suppressed a constant phase factor.) As a check on the derivation, you might wish to convince yourself that an incident plane wave E! will be transformed into a spherical wave converging on a point a distance f behind the lens, as you would expect.

8 do di

* oi!!'

d s Fig. 3 Object, lens and image planes for a thin lens. A point-source is located on the axis a distance ds from the lens.

2. Arbitrary do and di We now have all the machinery we need to handle the situation shown in Fig. 3. An object a distance do from the lens is illuminated by a point source. The object transmits a wave Eo(xo,yo) which is modified by the lens and then travels a distance di to the "image" plane, where it is described by Ei(xi,yi). To calculate Ei we start from the Huygens-Fresnel description of the object as a sum of point sources, each emitting a spherical wave of strength Eo(xo,yo). The superposition of all the spherical waves will give an expression for E! of the form of Eq. 8. This wave is transformed to E!' by the lens according to Eq. 15. If we consider E!' to be a new superposition we can describe Ei in the form of Eq. 8 again. Writing out these steps is messy but necessary. From (8), dropping the constant phase exp(ikz), we have

1 $ ik 2 2 ' E = E exp% ()x # x + ()y # y ( dx dy (16) ! i!d "" o 2d []! o ! o o o o & o )

We then get E!' from (15), and put it into (8) again to get

1 % ik 2 2 ( Ei = E! " exp& ()x! " $ xi + ()y! " $ yi ) dx! " dy! " (17) i!d ## 2d [] i ' i *

It is convenient to introduce a new function h(xi,yi;xo,yo) defined by

Ei = !! h(xi, yi ;x o, yo ) Eo dxodyo (18)

The explicit form of h is found from (15), (16), and (17):

9

1 " ik % " ik % h exp x2 y2 exp x2 y2 = 2 $ ()i + i ' $ ()o + o ' ! dodi # 2di & # 2do & " ik * 1 1 1 - % " * x x - % exp , / q2 exp ik, o i / q dq (19) ( $ + ) 1 ' $ ) + 1' 1 0 # 2 + di do f . & # + do di . & " ik * 1 1 1 - % " * y y - % ( exp , + ) / q2 exp )ik, o + i / q dq 0 $ 2 ' $ 2 ' 2 # 2 + di do f . & # + do di . &

Clearly, this equation is too messy to be useful, and we need to set about evaluating some special cases. (Before proceeding, note that we could have gotten h directly by finding the image of a point source located at (xo,yo) since the definition (18) is just the Huygens-Fresnel superposition integral again. Alternatively, an engineer would note that h is the transfer function of the system, and call (18) a integral.)

3. Image condition As a first case, let us look at the geometry for forming an ordinary image. Recall from that a real image is formed by a positive lens when (1/do) + (1/di) = 1/f. For that geometry the quadratic phase factors in (19) vanish leaving two integrals over exp(iqx), which are delta functions. Substituting into Eq. 18 we get, after some algebra,

1 % ik ! 1 2 2 ( ! x y E exp 1 # x y E i , i # (20) i = ' " + $ ()i + i * o " + + $ M & 2di M ) M M where M = di/do is the geometric magnification. Except for a phase shift, Ei is a scaled copy of 2 Eo. The phase factor will vanish when we take |Ei| to get the intensity at di, so we have shown that we do indeed get an image of the input object when we satisfy the geometrical-optics condition for imaging.

4. Fourier transform condition Although it is not obvious, a simple Fourier transform can describe the "image" in the plane at dt shown in Fig. 4. This is the plane conjugate to the source, that is (1/ds) + (1/dt ) = 1/f. When Eo is given by Eq. 7 and we set di = dt in Eq. 19 the integrand in Eq. 18 becomes

10 d d o i

oit

d t Fig. 4 The lens forms the Fourier transform at plane t and a geometric image at plane i. The source at distance ds is not shown.

iA d ! f # ik d ! f d ! f & hE = ()s exp ! ()o ()s x2 + y2 o 2 % 2 d d ()t t ( "fd()s ! do $ 2 f ()s ! o ' (21) # ik() ds ! f & T x , y exp x x y y ) o()o o % ! ()o t + o t ( $ fd()s ! do ' where (xt,yt) are coordinates in the plane at dt. This is almost the transform of the aperture transmission function. The quadratic phase factors can be eliminated by setting do = f and using plane-wave illumination for which ds ! " and dt = f. These conditions lead to

iA! % ik ( Et ()xt, yt = To ()xo, yo exp $ ()xoxt + yo yt dxodyo (22) "f ## &' f )* which indicates that under these conditions the image formed at dt is exactly the Fourier transform of the object transmission function. One sees from (21) and (22) that the image formed in the plane conjugate to the point source can have some interesting properties. Comparison with Eq. 10 shows that the intensity distribution in the transform plane is the same as in the Fraunhofer diffraction pattern of the object. This is true even though the transform plane is rather close to the object, a potential advantage in setting up experiments. The image is also closely related to the spatial frequency spectrum of the object, a point we now consider in detail.

5. Spatial filtering Equation 22 shows that the complex amplitude Et in the plane at dt is the spatial-frequency spectrum of the object transmission function To. Specifically, the observed amplitude at a point

11 (xt,yt) in the "transform plane" is proportional to the spectral amplitude at frequency (!x,!y) = (kxt/2"f,kyt/2"f). By inserting an appropriate barrier (filter) in the optical path at the transform plane, we can remove or phase shift any desired part of the spatial frequency spectrum. If we pick the filter correctly, the processed image may be modified in any of several well-defined ways. For example, a filter which removes the higher spatial (the parts of the transform farthest from the optic axis) will blur all sharp edges in the final image. Unfortunately, the condition do = f, required to completely eliminate the phase factors in the transform plane, leads to an image at infinity. This is not very satisfactory for experiments. Suppose instead that we consider the more general case of do > f to obtain a filtered image at a finite distance. Taking the limit ds ! " in (21) then yields an expression like (22) but with a 2 2 2 multiplicative phase factor exp[-(ik)/(2f )(do-f)(xt +yt )]. A filter in the transform plane t will modify this complex amplitude according to Et' = Tf(xt,yt)Et. The modified wave then propagates to the final image plane at di according to Eq. 9, with the result

# 2 2 & A! ik() xi + yi Ei = 2 exp%ikz + ( " fz 2z $ ' (23) # ik & ) Tf F{}To exp + ()xixt + yi yt dxtdyt ** $ z ' where

# ik & F{}To = To exp " ()xox f + yoy f dxodyo (24) !! $% f '( is the Fourier transform of To and z = di - f. Note that the quadratic phase factors in the transform plane have vanished, so we can retain all of our previous discussion based on Eq. 22. Also note that, when Tf = 1, Ei is the Fourier transform of the Fourier transform of To. Double- transforming a function results in an inverted version of the original function, so we recover the geometric image, consistent with Eq. 11, as we must.

12 III. Measurements

A. General setup The experiment will be done on an optical bench so that all the components can be accurately aligned. The light source is a small laser, and a photometer is provided to measure the light intensity in the various diffraction patterns. The output of the photometer is connected to a computer so that plots of intensity vs position can be directly recorded. The operation of these items is explained in this section, while the optical configurations for particular measurements are discussed in the following sections.

1. Alignment and beam expansion The laser beam is used to establish the optical axis, an imaginary horizontal line centered over the optical bench. Two adjustable mirrors deflect the laser beam so that it can be accurately aligned with the bench. Adjust the lower mirror to position the beam over the near end of the bench at a convenient height and then set the upper mirror to direct the beam parallel to the axis of the bench. A screen mounted in one of the holders will provide a convenient target which you can position at either end of the optical bench as needed. Once the laser beam is aligned with the bench, you can use it as a reference to place slits, lenses etc. on the optic axis. Simply put the desired component, in its holder, on the rail and adjust the holder to center the component in the beam. Fine adjustment of lenses is best accomplished by centering the broad spot of transmitted light on a centered target at the far end of the bench. After aligning each component you can carefully remove it, in its holder, and replace it without significant adjustment. For the most critical work, the component should be aligned near its working position since the bench may not be entirely straight. To better approximate an incident plane wave, the diameter of the laser beam should be expanded. Mount a 10X microscope objective and a 5 cm lens on the bench and slide the lens until the output beam has a constant diameter over the whole length of the bench. This will give more uniform illumination of the diffraction objects. (Use the short 10x objective for beam expansion. The longer one is needed later for spatial filtering.)

2. Photometer The operation of this device is mostly self-explanatory. A fiber-optic light guide conducts the light to a sensing device. The resulting electrical signal is read on a meter and is also available at terminals on the back of the instrument. The voltage signal is picked up by a computer interface for plotting and analysis by LoggerPro software. For the best resolution, set the sensitivity so that

13 the maximum signal matches the 10V range of the interface, even if this results in the panel meter going off scale. A mechanical slide is used to position the light guide along a line perpendicular to the axis of the optical bench. The relative position of the slide is also converted to an electrical signal, which is connected to CHAN1 of the interface device. The startup file FToptics.cmbl will provide a good starting point for obtaining intensity vs position plots. At the start of each session, calibrate voltage channel 1 to correspond to the actual carriage position, as the battery voltage will slowly decrease with use. The diameter of the fiber-optic input is inconveniently large relative to the diffraction patterns. A slit approximately 0.1 mm wide has therefore been installed in front of the input to restrict the area of the diffraction pattern which is seen by the instrument. Any feature smaller than the aperture width will be averaged out, so pattern size should be chosen accordingly.

B. Fraunhofer patterns The optical bench is long enough to reach the Fraunhofer regime with small diffracting objects. Glass slides with various sizes and numbers of slits are used as diffracting objects. They can be magnetically mounted to a holder which fits into one of the fixed carriers. The distance between the object and the screen should be chosen to give a pattern which can easily be measured within the range of the photometer motion. Small lateral adjustments will be needed to center the object in the expanded laser beam. To get a symmetric intensity profile, rotate the object about the optical axis until the diffraction pattern is horizontal at the photometer. Make plots of a typical single, a double and a multi-slit Fraunhofer pattern for later comparison with calculations. You may find it useful to expand the vertical scale on the graphs to see more detail between the main peaks, particularly with the multiple-slit objects.

C. Fresnel regime For reasonable slit widths the Fresnel regime is limited to positions within a few centimeters of the slit. In this region the pattern is not much wider than the slit, so some magnification is necessary for easy observation. The necessary arrangement is shown in Fig. 5. Note that the fixed slits have been replaced by an adjustable slit, and a lens has been added to project a approx. 1 m Laser photometer f=5cm slit f=5cm 10x Fig. 5 Optical bench arrangement for Fresnel diffraction. The distance from lens to photometer should be 1 - 1.5 m, depending on desired magnification.

14 magnified real image onto the photometer screen. The corresponding object for this image will be the optical disturbance Eo present at the position given by the lens law. Install the slit on the precision translation carriage, center the opening in the beam, and open the jaws to get a bright beam. Hold a piece of ground glass in the laser beam ahead of the slit and adjust the lens position until a focused image of the slit is formed on the screen. This determines the plane that the lens will image. Once you have found the focus, remove the diffuser. Note the position of the lens carriage on the bench scale so you can come back to the nominal focus if desired. Narrow the slit to a small fraction of a millimeter (this is not critical), and then move the lens away from the slit. You will see the image on the screen change from a simple image of the slit to a complicated Fresnel pattern and finally to something resembling the familiar Fraunhofer pattern. Particular patterns should reproduce if you return the lens carriage to the same position. Obtain two or three distinct patterns in the Fresnel regime, perhaps showing one, two and several dark lines within the pattern. You will probably find that the plotted intensities are asymmetric. This can be due to non- uniformities of the input beam and to shifts between the optical center line of the slit and the lens which introduce unwanted phase shifts. To minimize these effects, be sure that all optical components are centered as carefully as possible. Once you have the Fresnel pattern on the photometer screen you can make small lateral and rotational adjustments of the slit position to symmetrize the intensities. This can be done quite easily when the slit position is set to give a single dark line in the Fresnel diffraction image. With sufficient manual dexterity it is possible to get symmetry within about 10%.

D. Spatial filtering The optical arrangement is shown in Fig. 6. The 5 cm lens is now used to form the Fourier transform of the object transmission function at its focal plane and a magnified image at the photometer input.

approx. 1 m Laser photometer 10x f=5cm slitf=5cm 10x magnetic holder Fig. 6 Optical bench arrangement for spatial filtering. The 10X objective produces a magnified image of the transform plane for use while adjusting baffles.

15 Install the 5 cm lens at about the same position as you did for the Fresnel patterns, and lock it to the bench. Hold a card against the magnetic holder, slide it along the bench until the expanded laser beam is focused on it, and lock it down. This is the focal plane of the lens, where spatial filtering will be done. Now set the variable slit on the optical bench and move the slit carrier along the bench to get the best focus on the photometer screen. You should now see a very small Fraunhofer diffraction pattern at the position of the magnetic holder, and an image of the slit at the photometer. You may need to make small adjustments to get a symmetrical image, as before. You can now proceed to try spatial filtering. To understand what needs to be done, recall that the pattern in the transform plane is the Fourier transform of the object transmission function. The slit simply blocks part of the incident wave, so To is one over the open width of the slit and zero elsewhere, if we neglect the y variable for our long slit. The optical intensity in the 2 transform plane, |Et| , is therefore proportional to the square of the Fourier transform of the "spatial pulse" To(xo). This function peaks strongly at ! = 0 with weaker secondary peaks symmetrically located on either side, as will be evident when you look at the transform plane. The spatial frequency !x is related to position in the transform plane by (k/2"f)xt, so intensity close to the optical axis (small xt) represents low frequency information, while intensity far from the optical axis represents high frequency information. For example, if we allow only the central portion of the pattern to pass, we will have eliminated some of the high frequency information from the final image. Conversely, if we block the central portion of the pattern we will get an image with high frequency components but no low frequency components. The actual filtering is done by mounting opaque baffles on the magnetic holder. Single-edge razor blades can be used to cut off parts of the pattern, while a slide with (fragile) wires can be used to remove the central maximum. The pattern at the transform plane will be quite small, so it is helpful to insert a 10X objective, as shown, to magnify the pattern as you position the baffles. It may also be helpful to view the transform plane image onto a card placed at a convenient distance from the 10X objective, rather than the photometer screen. The magnifier and card must be removed before taking measurements, of course. Examine the effect of passing only the central peak and then the central peak plus successively more of the subsidiary peaks. What happens if you remove the central maximum while passing all the outer peaks? Record some of the resulting images, and pick simple enough filtering conditions that you can reproduce them later on the computer.

16 IV. Analysis

In principle, one could numerically fit the model equations to the intensity data. This is a rather difficult task because of the amount of computation needed and because of imperfections in the laser and lenses which lead to deviations from ideal diffraction patterns. Instead, we set ourselves the more modest goal of qualitatively reproducing the observed patterns. This limitation also avoids the need to normalize distances and amplitudes in the calculation to agree with the physical quantities. The analysis is based on the equations in Section II. The input object is represented by a one- dimensional array in which "ones" denote transparent regions. The array is multiplied by a complex phase factor if needed, and Fourier transformed to obtain the diffraction pattern. Spatial filtering is done by Fourier transforming the input array, multiplying by a filter function represented as an array of zeros and ones, and then Fourier transforming again. A suggested analysis using MATLAB is given below. It should also apply to other programs, such as Mathematica. Purists will note that most programs use the Fast Fourier Transform algorithm (FFT) to compute a discrete approximation to the Fourier transform. The discrete FFT of an N-point array un is defined to be

N !1 $ nm c u exp i2 & (25) m = " n % ! # ' n=0 N for m = 0, 1, ..., N-1. The algorithm requires that N be a power of 2. For a real input function un, * cN-m = cm, which has the effect of transposing the output spectrum about N/2. The folding can be removed with the MATLAB operator fftshift or a similar operator in other languages. Otherwise, the discrete transform is, for our purposes, equivalent to the ideal Fourier transform. Further discussion can be found in the topical notes on Fourier transforms.

Using MATLAB The array processing features of MATLAB make it relatively easy to program the necessary operations. We represent the slit(s) by a one-dimensional array containing ones where light can pass, and zeros otherwise. This has the effect of approximating To(xo,yo) by To(xo), which is reasonable for our long narrow slits. The size of this array must be a power of 2, with at least 2048 elements needed for adequate plotting resolution.

17 2 The Fraunhofer case is the simplest, since we only need to calculate the intensity |Ei| from Eq. 10. For plane-wave illumination, as produced by the laser, Eo = A'To so this simplifies to evaluating the power spectrum of To(xo). The MATLAB operator fft computes the complex Fourier transform, fftshift puts zero frequency at the center for convenience, and then the intensity is found by multiplying by the complex conjugate. You will probably find it convenient to normalize to the largest value before plotting. By constructing the proper input arrays you can do this for each of the slit patterns you measured in the lab. In the Fresnel regime Eq. 9 instructs us to calculate the power spectrum of 2 exp[(ik/2z)xo ]To(xo). The procedure is therefore the same as in the Fraunhofer situation, except that we must multiply by the exponential phase function before transforming and choose proper scales for x and z. Physically, we are interested in positions within a few slit widths of the screen, so it is convenient to choose z proportional to the slit width. The column number of To can be used directly as the xo distance, but this will move the transform off the center of the output array. The problem is cured if we use the distance from the center of the array in the phase factor. Once set up, simple variation of z should suffice to reproduce the observed patterns for the single slit. The extension to the double slit case is also possible, if you made those measurements. Spatial filtering is simulated by sequentially carrying out the steps required by Eq. 23. Use the FFT to compute the complex fourier transform of To at the transform plane. Select the parts of the transform that pass the filter, transform again, and plot the intensity of the result. You can do the filtering by setting part of the transform array to zero directly, or by multiplying by a filter function which contains zeros in the areas you want to eliminate. If you use the latter approach you can plot the filter function and the intensity in the transform plane on the same graph to verify that the filter is correctly placed in the transform plane.

18