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Visualization of light propagation
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
VISUALIZATION OF LIGHT PROPAGATION H.O. BARTELT, S.K. CASE * and A.W. LOHMANN Physikalisches Institut, Universitiit Erlangen-Niirnberg 8520 Erlangen, Fed. Rep. Germany Received 24 April 1979
Abramson has demonstrated that by using the finite coherence properties of light, holograms can be constructed that allow one to visualize the path that a simulated light pulse takes through an optical system. We have demonstrated that using an alternate hologram construction and readout geometry, light with a very long coherence length can also be used to visualize light propagation. Experimental results showing well resolved time-sequence photographs of reflection, refraction, and diffraction in an optical system are given.
1. Introduction
It would often be desirable to view a pulse of light traveling through an optical system in order to study the system or the interaction of light with the system. Of course, because of the speeds involved, one cannot directly view or film the propagation of a light pulse. Abramson I1 ] has recently shown, however, that using a time exposure and a proper coding technique, a hologram can be recorded which can later be decoded to visualize the path that a simulated light pulse would take through an optical system. *1 His technique used the finite coherence length of his light source to map points on his object to given points on the hologram plate. By reconstructing the image from only a certain position on the hologram plate, he could see the position o f a simulated light pulse in his optical system at a certain "time". Thus a pseudo "time of flight" was coded as a position on his holographic plate. With his method, light o f very short coherence length must be used to provide good time resolution 4:2. Abramson has also proposed, that similar experiments could be performed with genuine short pulses from a picosecond laser.
* Fulbright Fellow on leave from the University of Minnesota, Dept. of Electrical Engineering, Minneapolis, Minnesota 55455 USA.
In this paper, we will show that a light source with very long coherence length can also be used to simulate the propagation of a light pulse through an optical system. With our method, "time" is coded as spatial frequency on the hologram. Subsequent spatial filtering to select a certain spatial frequency allows us to look at the position of a simulated light pulse at a certain time. We will first analyze our construction and readout method, calculate the position and time resolution of our system, discuss practical considerations involved in using this technique, and give experimental results which show reflection, refraction, and diffraction within an optical system. Finally, we will compare the two recording techniques.
2. Coding technique Fig. 1 shows a side view of the basic recording geometry (without any test objects) used in our experiment. Using light of long coherence length and with a wavelength X, we record the interference between a plane reference wave and a cylindrical object wave. ,1 The differences between a true light pulse and these simulated pulses are discussed at the end of section 5. ,2 Caulfield had previously used the f'mite coherence length of a light source to provide holographic depth selection [2]. 13
Volume 30, number 1
OPTICS COMMUNICATIONS
LiNE SOURCE PLANE REFERENCEWAVE
July 1979
v = X/X(X 2 + Z2) 1/2.
(4)
This could also be calculated using the angle 0 between a local object and reference ray. To introduce "time" into our experiment, we note that if a light pulse were emitted from the line source at time t = O, it would arrive at the film at a time t = (x 2 + z20)1/2/c
(S)
where c is the speed of light. Combining eqs. (4) and (5) yields HOLOGRAPHIC PLATE
t = (Zo/C)(1 - X2v2) -1/2. Fig. 1. Hologram recording geometry
Thus we can see that there is a unique mapping between time and local spatial frequency on the hologram plate. We now introduce an object into our experimental setup as shown in fig. 2. For the present analysis, the object must have no z dependence so that a cylindrical lens has been chosen for our example. We have illustrated how different "sheets" of light emerghag from the line source at different angles 0 arrive at the hologram plate. The light in a particular sheet can be thought of as the coding mechanism for a particular "time" on our hologram. Concerning any given ray in the object wave, the important point is that if the object has no z dependence, then the angle 0 between the object wave and the z axis (hence the reference wave) will not be changed by the object. Thus the
A: the hologram plate, the object wave, O, is 0 = A ( x , z O) exp (i(21r/X)(x 2 + z2)1/2 },
(1)
while the unit amplitude reference wave, R, is R = exp (-i(2zr/?,)z} = 1
(at z = 0).
(2)
The resulting hologram will have an amplitude transmittance proportional to T " 1 +A 2 + 2,4 cos(2rr/XXx 2 +z2) 1/2.
(3)
Neglecting the slow variation in A, the local spatial frequency, v, on the hologram can be found by taking the derivative of the argument of the cosine and is
LINE SOURCE
PLANE REFERENCE WAVE
I
I
I
I
HOLOGRAPHIC PLATE
Fig. 2. Recording geometry with a cylindrical lens used as a test object.
14
(6)
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OPTICS COMMUNICATIONS
two lines)that a slit served as the filter. The width of the output image (in the propagation direction) is a measure of the time resolution of our process and is a function of the filter slit width. If the slit is too wide, too many spatial frequencies will be passed producing an excessively wide output image. If the slit width is too small, diffraction from the slit will broaden the output image. An estimate of the optimum slit width can be made. Working the problem in one dimension, a slit of width W will pass a band of spatial frequencies
mapping between time and spatial frequency given by eq. (6) will still be accurate although we now mean v = (Vx 2 + Uy2)1/2. We might note that ifz 0 is small, such that z 0 ~ x , then the object wave will illuminate the test lens at nearly normal incidence, as the lens would usually be used in an optical system. With the above assumption, eq. (5) goes to the limit (7)
t = x/c
July 1979
so that our "time" would be that normally associated with a light pulse travelling in the x direction through the lens.
(8)
~v = w/~f,
where f i s the focal length of our Fourier transform lens. If the output image has width &x in the propagation direction and is centered at x, we can define the right and left edge locations, x R and x L, of the output image as
3. Spatial filtering Using the setup in fig. 2, we record a hologram which maps time into spatial frequency. As shown in fig. 3, the developed hologram could be illuminated with a plane wave so that the Fourier spectrum of the hologram appears in the filter plane. A filter passes a small band of frequencies which are again Fourier transformed to form the output image. Because of the mapping between time and spatial frequency, a given filter position selects a particular "time slice" to be seen at the output. By changing the filter we view different times at the output and hence can watch a simulated time pulse propagate through our optical element. The filter required would in general be an annulus. For most of the objects we tested, however, the frequency spectrum was sufficiently simple (e.g. only
xR=x+Ax/2
and
xL =x-Ao¢/2.
Using eq. (4), we relate the image width to the frequency band Av: AV = VR -- VL
xR x(4 +zob1/2
xL (lO) X(x +zob1/2
Assuming (see discussions in sections 4 and 5) that
6x
(11)
we can combine eqs. (8)-(11) to obtain ~x
=
Ax F
=
(W/f)(x 2
+
z2) 1/2,
(l 2)
where we have written zSxF to indicate the output
v LENS
X3
i
LENS
I
_[ HOLOGRAM PLANE
f
(9)
_l_
f
_W
FI LTER PLANE
L>Z
f_ OUTPUT PLANE
Fig.3. Spatialftlteringsetupfor "time"selection. 15
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OPTICS COMMUNICATIONS
image width based on the band of spatial frequencies passed. The width of the output image due to diffraction by a Fourier plane slit of width W is simply given by ,2ix = z2ix,D = 2 ~ , f / W .
(13)
The output image width depends on both of the effects used to calculate eqs. (12) and (1 3) such that we would like to simultaneously minimize both Ax F and Ax D. We can make an estimate of the optimum slit width by setting Ax F = z2ocD to get 141 = f ( 2 ~ , ) 1 / E ( x 2
+ z2) - 1/4.
(1 4)
Inserting eq. (14) back into eq. (12) gives the output resolution z ~ --- (2)01/2(x2 + z2) 1/4.
(15)
Using eq. (15) with the derivative of eq. (5), we obtain the time resolution A t = ( 2 ) O 1 / 2 ( x / c ) ( x 2 + z2) - 1/4.
(16)
4. Recording geometry Because time is coded as spatial frequency by our process, the best time resolution is obtained when we have a large change in spatial frequency with position. We would also like this frequency change to be relatively uniform across the hologram so that our resolution does not change with position. These goals can be achieved by the proper choices o f x and z 0. Looking at the limits, if z 0 ~ x, then the test objects will be illuminated in the manner in which we are used to looking at objects as described at the end of section 2. However there will be very little variation in spatial frequency across the hologram and hence poor time resolution in our filtered image. I f z 0 >>x, we will have a larger range of spatial frequencies for better time resolution (as long as z 0 does not become so large that our object wave becomes a plane wave). The objection to this type of illumination, however, is that we are not used to viewing objects with such a large incidence angle and it would cause unusual effects such as producing curved rays behind the lens and changing the appearant focal length of the lens that we are using as the test object. The configuration we chose, therefore, was 16
July 1979
between these two extremes. We used z 0 = 13 cm and the exposed area of the hologram was between x = 14 and 20 cm. The change in spatial frequency with position was large and relatively uniform. Therefore the optimum slit width and hence resolution varied by less than 12% across the field of view. The light transmitted by the objects also did not have a distorted appearance so that this was an adequate solution. The last remaining problem had to do with the magnitude of the spatial frequencies recorded on the hologram. For our construction geometry, the average frequency was v = 1600 ~/mm which means that if the hologram were read out as in fig. 3 the diffracted light would propagate at an angle greater than 50 ° with respect to the z axis. If these high frequencies existed only in one direction, we could simply bend our optical system and put our lenses and filters off axis. However our object will in general deflect the incident light at fairly large angles in the y direction (see fig. 2) so that the v x and Vy components of spatial frequency can be comparable in size. The lenses in fig. 3, therefore, would have to have very large diameters and low f numbers which means that this type of filtering is impractical. To solve the problem, we had to maintain our construction geometry yet lower the magnitude of the spatial frequencies on the hologram. This could conveniently be accomplished by making double exposure holograms at two slightly different wavelengths (Xl and X2) and then using the difference frequency (Moire) between the two recordings to do the spatial filtering. Using this technique, eq. (4) becomes
.
= _
(x 2 + z02)v2 ' 1
x
XE (x 2 +z2)1/2 '
(18)
where XE = K)taverage is the effective recording wavelength. Using X1 = 5145 A and X2 = 4880 A for exposure reduced the spatial frequencies by a factor of approximately K = 19 so that they could easily be filtered. A second technique for frequency reduction was to move the line source slightly between exposures to record slightly different spatial frequencies. This can also alleviate problems with testing objects that
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
are strongly wavelength dependent. Finally, one need not actually make double exposures as this usually leads to low diffraction efficiency. We most often used real time interferometry by exposing with X1, developing and replacing the plate and re-illuminating the object with X2. The diffracted light from the hologram enters the filtering system in fig. 3 and has the same angular spectrum as would the hologram described by eq. (17).
5. Experimental results Using the above techniques, we demonstrated light propagation through a number of objects. To begin, we wanted to show propagation of a plane wave with no other objects present so that our hologram was exposed with just the plane reference and cylindrical object waves. Fig. 4a schematically shows how our light beam with finite cross-section travels from left to right. Fig. 4b is the "time integrated image" present at the output which shows the total path swept out by our light beam. It is obtained by using no filters in the Fourier plane. Figs. 4e-4g show a sequence of photographs in which a single light pulse travels from left to right. A slit filter was used and moved by a given amount between photographs. For this test a slightly different geometry was used than for all others. Here we used z 0 >>x so that eq. (4) was approximately v ~ x / k z O.
(19)
The frequency increased linearly across the plate. Thus by replacing a few regularly spaced slits in the Fourier plane, we were able to show a wave packet in fig. 4c. By placing many regularly spaced slits (Ronchi ruling) in the Fourier plane, we showed a periodic wave in fig. 4d. Fig. 5a shows a plane wave traveling from the left and reflecting off of a mirror. The time integrated photograph is shown in fig. 5b. In fig~. 5c-5f, we show a light pulse entering from the left, reflecting from the mirror and then propagating toward the upper right. The Fourier spectrum for this object consisted of just two lines so that a slit filter of width W = 0.23 mm (see eq. (14) with X replaced by X/K) was used. The resolution in the direction of propagation was Ax = 2.6 mm on the hologram plate (see eq. (15) with X replaced by KX)while the cross-sectional
Fig. 4. Propagation of plane wave. Schematic diagram (a). "Time integrated" light path (b). Wavepacket (c). Periodic w a v e (d). Propagation of single pulse (e-g) width of the pulse was 12 mm. The time resolution from eq. (16) (with X replaced by/CA) is then 5.3 ps and the pictures are spaced approximately 20 ps apart in time. In fig. 6 we demonstrate refraction by a prism. A plane wave comes from the left and half of it is refracted toward the lower right by the prism. Figs. 6 c 6f show time-sequence photographs of the propagation of a pulse through the prism. Fig. 7 also demonstrates refraction and shows photographs of a plane wave being brought to focus by a cylindrical lens. Finally in fig. 8, we demonstrate diffraction by a grating. The object is a phase grating with a spatial frequency of approximately 600 ~/mm and a modulation such that the two first orders were approximately equal in strength to the zero order. An interesting point in these photographs is that since we look at the light at one particular time, the "wavefront" of the diffracted light in figs. 8 c - 8 f is not perpendicular to the direction of propagation. Thus this recording technique distinguishes between the processes of refraction and diffraction. The photographs also illustrate the difference between the propagation of a true light pulse and the simulated light 17
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OPTICS COMMUNICATIONS
/ I
t
/
Fig. 5. Reflection from a mirror,
Fig. 6. Refraction by a prism.
pulses in our experiments. Because the holograms were actually recorded with light of long coherence length, we had the constructive interference necessary to produce diffracted plane waves at well defined angles. Thus at readout, our simulated pulses follow these well defined paths even though this would not happen for an extremely short pulse.
6. Discussion
We have shown that by recording a coded hologram and subsequent spatial filtering, it is possible to visualize the path that a light pulse would take through an optical system. This was also done by Abramson so it is worthwhile to compare the two methods. In terms of resolution, Abramson used an argon laser without an etalon whose coherence length was approximately 6 cm This was therefore the spatial resolution in his experi18
July 1979
c
C
(l
d
e
/
Fig. 7. Focussing by a lens.
Fig. 8. Diffraction by a grating.
ment so that only large objects (~30 cm or larger) could be reasonably tested. Our method gave a factor of more than 20 finer resolution so that we could conveniently test optical elements of the size that are typically found in the laboratory. As Abramson pointed out, however, by using a light source with even shorter coherence length, his resolution could be substantially improved. Finding the limit of improvement would require an analysis which included the dimensions of the construction and readout setup, light coherence length, and the width of the slit placed on the hologram for time selection. The second consideration is light efficiency. With our method, almost all of the light that is incident on the test object reaches the holographic plate and is used for hologram formation (with little bias) so that we produce holograms with high diffraction efficiency. With Abramson's method, a considerable amount of light is lost because light must be scattered from the
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OPTICS COMMUNICATIONS
test surface toward the film. In addition, light from all parts of the test surface is scattered toward all parts on the film. Thus at a given point on the holographic film, most of the incident light acts as a bias exposure while a small fraction records the interferogram. We can estimate the signal-to-bias ratio for the object beam to be 1/N where N is the number of resolvable positions in the object field. Thus for large N, which one would try to achieve by using light of short coherence length, we would expect low diffraction efficiency from the hologram. This problem coulc be reduced by using a bandpass recording material such as thermoplastic [3] which would not respond to the incoherent bias.
July 1979
Aside from these practical considerations we concur with N. Abramson's enthusiasm about the value of these experiments for pedagogical purposes and as a tool for ultra high speed photography. The authors thank R. Mueller for comments on the manuscript.
References [1] N. Abramson, Opt. Lett. 3 (1978) 121. [2] H.J. Caulfield, J. Opt. Soc. Am. 58 (1968) 276. [3] L.H. Lin and H.L. Beauehamp, Appl. Opt. 9 (1970) 2088.
19
OPTICS COMMUNICATIONS
July 1979
VISUALIZATION OF LIGHT PROPAGATION H.O. BARTELT, S.K. CASE * and A.W. LOHMANN Physikalisches Institut, Universitiit Erlangen-Niirnberg 8520 Erlangen, Fed. Rep. Germany Received 24 April 1979
Abramson has demonstrated that by using the finite coherence properties of light, holograms can be constructed that allow one to visualize the path that a simulated light pulse takes through an optical system. We have demonstrated that using an alternate hologram construction and readout geometry, light with a very long coherence length can also be used to visualize light propagation. Experimental results showing well resolved time-sequence photographs of reflection, refraction, and diffraction in an optical system are given.
1. Introduction
It would often be desirable to view a pulse of light traveling through an optical system in order to study the system or the interaction of light with the system. Of course, because of the speeds involved, one cannot directly view or film the propagation of a light pulse. Abramson I1 ] has recently shown, however, that using a time exposure and a proper coding technique, a hologram can be recorded which can later be decoded to visualize the path that a simulated light pulse would take through an optical system. *1 His technique used the finite coherence length of his light source to map points on his object to given points on the hologram plate. By reconstructing the image from only a certain position on the hologram plate, he could see the position o f a simulated light pulse in his optical system at a certain "time". Thus a pseudo "time of flight" was coded as a position on his holographic plate. With his method, light o f very short coherence length must be used to provide good time resolution 4:2. Abramson has also proposed, that similar experiments could be performed with genuine short pulses from a picosecond laser.
* Fulbright Fellow on leave from the University of Minnesota, Dept. of Electrical Engineering, Minneapolis, Minnesota 55455 USA.
In this paper, we will show that a light source with very long coherence length can also be used to simulate the propagation of a light pulse through an optical system. With our method, "time" is coded as spatial frequency on the hologram. Subsequent spatial filtering to select a certain spatial frequency allows us to look at the position of a simulated light pulse at a certain time. We will first analyze our construction and readout method, calculate the position and time resolution of our system, discuss practical considerations involved in using this technique, and give experimental results which show reflection, refraction, and diffraction within an optical system. Finally, we will compare the two recording techniques.
2. Coding technique Fig. 1 shows a side view of the basic recording geometry (without any test objects) used in our experiment. Using light of long coherence length and with a wavelength X, we record the interference between a plane reference wave and a cylindrical object wave. ,1 The differences between a true light pulse and these simulated pulses are discussed at the end of section 5. ,2 Caulfield had previously used the f'mite coherence length of a light source to provide holographic depth selection [2]. 13
Volume 30, number 1
OPTICS COMMUNICATIONS
LiNE SOURCE PLANE REFERENCEWAVE
July 1979
v = X/X(X 2 + Z2) 1/2.
(4)
This could also be calculated using the angle 0 between a local object and reference ray. To introduce "time" into our experiment, we note that if a light pulse were emitted from the line source at time t = O, it would arrive at the film at a time t = (x 2 + z20)1/2/c
(S)
where c is the speed of light. Combining eqs. (4) and (5) yields HOLOGRAPHIC PLATE
t = (Zo/C)(1 - X2v2) -1/2. Fig. 1. Hologram recording geometry
Thus we can see that there is a unique mapping between time and local spatial frequency on the hologram plate. We now introduce an object into our experimental setup as shown in fig. 2. For the present analysis, the object must have no z dependence so that a cylindrical lens has been chosen for our example. We have illustrated how different "sheets" of light emerghag from the line source at different angles 0 arrive at the hologram plate. The light in a particular sheet can be thought of as the coding mechanism for a particular "time" on our hologram. Concerning any given ray in the object wave, the important point is that if the object has no z dependence, then the angle 0 between the object wave and the z axis (hence the reference wave) will not be changed by the object. Thus the
A: the hologram plate, the object wave, O, is 0 = A ( x , z O) exp (i(21r/X)(x 2 + z2)1/2 },
(1)
while the unit amplitude reference wave, R, is R = exp (-i(2zr/?,)z} = 1
(at z = 0).
(2)
The resulting hologram will have an amplitude transmittance proportional to T " 1 +A 2 + 2,4 cos(2rr/XXx 2 +z2) 1/2.
(3)
Neglecting the slow variation in A, the local spatial frequency, v, on the hologram can be found by taking the derivative of the argument of the cosine and is
LINE SOURCE
PLANE REFERENCE WAVE
I
I
I
I
HOLOGRAPHIC PLATE
Fig. 2. Recording geometry with a cylindrical lens used as a test object.
14
(6)
Volume 30, number 1
OPTICS COMMUNICATIONS
two lines)that a slit served as the filter. The width of the output image (in the propagation direction) is a measure of the time resolution of our process and is a function of the filter slit width. If the slit is too wide, too many spatial frequencies will be passed producing an excessively wide output image. If the slit width is too small, diffraction from the slit will broaden the output image. An estimate of the optimum slit width can be made. Working the problem in one dimension, a slit of width W will pass a band of spatial frequencies
mapping between time and spatial frequency given by eq. (6) will still be accurate although we now mean v = (Vx 2 + Uy2)1/2. We might note that ifz 0 is small, such that z 0 ~ x , then the object wave will illuminate the test lens at nearly normal incidence, as the lens would usually be used in an optical system. With the above assumption, eq. (5) goes to the limit (7)
t = x/c
July 1979
so that our "time" would be that normally associated with a light pulse travelling in the x direction through the lens.
(8)
~v = w/~f,
where f i s the focal length of our Fourier transform lens. If the output image has width &x in the propagation direction and is centered at x, we can define the right and left edge locations, x R and x L, of the output image as
3. Spatial filtering Using the setup in fig. 2, we record a hologram which maps time into spatial frequency. As shown in fig. 3, the developed hologram could be illuminated with a plane wave so that the Fourier spectrum of the hologram appears in the filter plane. A filter passes a small band of frequencies which are again Fourier transformed to form the output image. Because of the mapping between time and spatial frequency, a given filter position selects a particular "time slice" to be seen at the output. By changing the filter we view different times at the output and hence can watch a simulated time pulse propagate through our optical element. The filter required would in general be an annulus. For most of the objects we tested, however, the frequency spectrum was sufficiently simple (e.g. only
xR=x+Ax/2
and
xL =x-Ao¢/2.
Using eq. (4), we relate the image width to the frequency band Av: AV = VR -- VL
xR x(4 +zob1/2
xL (lO) X(x +zob1/2
Assuming (see discussions in sections 4 and 5) that
6x
(11)
we can combine eqs. (8)-(11) to obtain ~x
=
Ax F
=
(W/f)(x 2
+
z2) 1/2,
(l 2)
where we have written zSxF to indicate the output
v LENS
X3
i
LENS
I
_[ HOLOGRAM PLANE
f
(9)
_l_
f
_W
FI LTER PLANE
L>Z
f_ OUTPUT PLANE
Fig.3. Spatialftlteringsetupfor "time"selection. 15
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OPTICS COMMUNICATIONS
image width based on the band of spatial frequencies passed. The width of the output image due to diffraction by a Fourier plane slit of width W is simply given by ,2ix = z2ix,D = 2 ~ , f / W .
(13)
The output image width depends on both of the effects used to calculate eqs. (12) and (1 3) such that we would like to simultaneously minimize both Ax F and Ax D. We can make an estimate of the optimum slit width by setting Ax F = z2ocD to get 141 = f ( 2 ~ , ) 1 / E ( x 2
+ z2) - 1/4.
(1 4)
Inserting eq. (14) back into eq. (12) gives the output resolution z ~ --- (2)01/2(x2 + z2) 1/4.
(15)
Using eq. (15) with the derivative of eq. (5), we obtain the time resolution A t = ( 2 ) O 1 / 2 ( x / c ) ( x 2 + z2) - 1/4.
(16)
4. Recording geometry Because time is coded as spatial frequency by our process, the best time resolution is obtained when we have a large change in spatial frequency with position. We would also like this frequency change to be relatively uniform across the hologram so that our resolution does not change with position. These goals can be achieved by the proper choices o f x and z 0. Looking at the limits, if z 0 ~ x, then the test objects will be illuminated in the manner in which we are used to looking at objects as described at the end of section 2. However there will be very little variation in spatial frequency across the hologram and hence poor time resolution in our filtered image. I f z 0 >>x, we will have a larger range of spatial frequencies for better time resolution (as long as z 0 does not become so large that our object wave becomes a plane wave). The objection to this type of illumination, however, is that we are not used to viewing objects with such a large incidence angle and it would cause unusual effects such as producing curved rays behind the lens and changing the appearant focal length of the lens that we are using as the test object. The configuration we chose, therefore, was 16
July 1979
between these two extremes. We used z 0 = 13 cm and the exposed area of the hologram was between x = 14 and 20 cm. The change in spatial frequency with position was large and relatively uniform. Therefore the optimum slit width and hence resolution varied by less than 12% across the field of view. The light transmitted by the objects also did not have a distorted appearance so that this was an adequate solution. The last remaining problem had to do with the magnitude of the spatial frequencies recorded on the hologram. For our construction geometry, the average frequency was v = 1600 ~/mm which means that if the hologram were read out as in fig. 3 the diffracted light would propagate at an angle greater than 50 ° with respect to the z axis. If these high frequencies existed only in one direction, we could simply bend our optical system and put our lenses and filters off axis. However our object will in general deflect the incident light at fairly large angles in the y direction (see fig. 2) so that the v x and Vy components of spatial frequency can be comparable in size. The lenses in fig. 3, therefore, would have to have very large diameters and low f numbers which means that this type of filtering is impractical. To solve the problem, we had to maintain our construction geometry yet lower the magnitude of the spatial frequencies on the hologram. This could conveniently be accomplished by making double exposure holograms at two slightly different wavelengths (Xl and X2) and then using the difference frequency (Moire) between the two recordings to do the spatial filtering. Using this technique, eq. (4) becomes
.
= _
(x 2 + z02)v2 ' 1
x
XE (x 2 +z2)1/2 '
(18)
where XE = K)taverage is the effective recording wavelength. Using X1 = 5145 A and X2 = 4880 A for exposure reduced the spatial frequencies by a factor of approximately K = 19 so that they could easily be filtered. A second technique for frequency reduction was to move the line source slightly between exposures to record slightly different spatial frequencies. This can also alleviate problems with testing objects that
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
are strongly wavelength dependent. Finally, one need not actually make double exposures as this usually leads to low diffraction efficiency. We most often used real time interferometry by exposing with X1, developing and replacing the plate and re-illuminating the object with X2. The diffracted light from the hologram enters the filtering system in fig. 3 and has the same angular spectrum as would the hologram described by eq. (17).
5. Experimental results Using the above techniques, we demonstrated light propagation through a number of objects. To begin, we wanted to show propagation of a plane wave with no other objects present so that our hologram was exposed with just the plane reference and cylindrical object waves. Fig. 4a schematically shows how our light beam with finite cross-section travels from left to right. Fig. 4b is the "time integrated image" present at the output which shows the total path swept out by our light beam. It is obtained by using no filters in the Fourier plane. Figs. 4e-4g show a sequence of photographs in which a single light pulse travels from left to right. A slit filter was used and moved by a given amount between photographs. For this test a slightly different geometry was used than for all others. Here we used z 0 >>x so that eq. (4) was approximately v ~ x / k z O.
(19)
The frequency increased linearly across the plate. Thus by replacing a few regularly spaced slits in the Fourier plane, we were able to show a wave packet in fig. 4c. By placing many regularly spaced slits (Ronchi ruling) in the Fourier plane, we showed a periodic wave in fig. 4d. Fig. 5a shows a plane wave traveling from the left and reflecting off of a mirror. The time integrated photograph is shown in fig. 5b. In fig~. 5c-5f, we show a light pulse entering from the left, reflecting from the mirror and then propagating toward the upper right. The Fourier spectrum for this object consisted of just two lines so that a slit filter of width W = 0.23 mm (see eq. (14) with X replaced by X/K) was used. The resolution in the direction of propagation was Ax = 2.6 mm on the hologram plate (see eq. (15) with X replaced by KX)while the cross-sectional
Fig. 4. Propagation of plane wave. Schematic diagram (a). "Time integrated" light path (b). Wavepacket (c). Periodic w a v e (d). Propagation of single pulse (e-g) width of the pulse was 12 mm. The time resolution from eq. (16) (with X replaced by/CA) is then 5.3 ps and the pictures are spaced approximately 20 ps apart in time. In fig. 6 we demonstrate refraction by a prism. A plane wave comes from the left and half of it is refracted toward the lower right by the prism. Figs. 6 c 6f show time-sequence photographs of the propagation of a pulse through the prism. Fig. 7 also demonstrates refraction and shows photographs of a plane wave being brought to focus by a cylindrical lens. Finally in fig. 8, we demonstrate diffraction by a grating. The object is a phase grating with a spatial frequency of approximately 600 ~/mm and a modulation such that the two first orders were approximately equal in strength to the zero order. An interesting point in these photographs is that since we look at the light at one particular time, the "wavefront" of the diffracted light in figs. 8 c - 8 f is not perpendicular to the direction of propagation. Thus this recording technique distinguishes between the processes of refraction and diffraction. The photographs also illustrate the difference between the propagation of a true light pulse and the simulated light 17
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Fig. 5. Reflection from a mirror,
Fig. 6. Refraction by a prism.
pulses in our experiments. Because the holograms were actually recorded with light of long coherence length, we had the constructive interference necessary to produce diffracted plane waves at well defined angles. Thus at readout, our simulated pulses follow these well defined paths even though this would not happen for an extremely short pulse.
6. Discussion
We have shown that by recording a coded hologram and subsequent spatial filtering, it is possible to visualize the path that a light pulse would take through an optical system. This was also done by Abramson so it is worthwhile to compare the two methods. In terms of resolution, Abramson used an argon laser without an etalon whose coherence length was approximately 6 cm This was therefore the spatial resolution in his experi18
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Fig. 7. Focussing by a lens.
Fig. 8. Diffraction by a grating.
ment so that only large objects (~30 cm or larger) could be reasonably tested. Our method gave a factor of more than 20 finer resolution so that we could conveniently test optical elements of the size that are typically found in the laboratory. As Abramson pointed out, however, by using a light source with even shorter coherence length, his resolution could be substantially improved. Finding the limit of improvement would require an analysis which included the dimensions of the construction and readout setup, light coherence length, and the width of the slit placed on the hologram for time selection. The second consideration is light efficiency. With our method, almost all of the light that is incident on the test object reaches the holographic plate and is used for hologram formation (with little bias) so that we produce holograms with high diffraction efficiency. With Abramson's method, a considerable amount of light is lost because light must be scattered from the
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test surface toward the film. In addition, light from all parts of the test surface is scattered toward all parts on the film. Thus at a given point on the holographic film, most of the incident light acts as a bias exposure while a small fraction records the interferogram. We can estimate the signal-to-bias ratio for the object beam to be 1/N where N is the number of resolvable positions in the object field. Thus for large N, which one would try to achieve by using light of short coherence length, we would expect low diffraction efficiency from the hologram. This problem coulc be reduced by using a bandpass recording material such as thermoplastic [3] which would not respond to the incoherent bias.
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Aside from these practical considerations we concur with N. Abramson's enthusiasm about the value of these experiments for pedagogical purposes and as a tool for ultra high speed photography. The authors thank R. Mueller for comments on the manuscript.
References [1] N. Abramson, Opt. Lett. 3 (1978) 121. [2] H.J. Caulfield, J. Opt. Soc. Am. 58 (1968) 276. [3] L.H. Lin and H.L. Beauehamp, Appl. Opt. 9 (1970) 2088.
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