Download A simple approach to phase holography

A simple approach to phase holography
Vincent Toala)
Centre for Industrial and Engineering Optics, School of Physics, Dublin Institute of Technology,
Kevin Street, Dublin 8, Ireland
共Received 14 January 2003; accepted 9 May 2003兲
A simple mathematical description of thick hologram recording and playback is given using a basic
wave front representation. Results are obtained for amplitude and phase holography. This approach
is very accessible to students and predicts the important results that are obtained in early and some
more advanced holography experiments. The approach is also quite general and applies to
transmission and reflection holography. © 2003 American Association of Physics Teachers.
关DOI: 10.1119/1.1587702兴
I. INTRODUCTION
I 共 x,y 兲 ⫽
It is quite common in the undergraduate laboratory to prepare phase holograms by bleaching silver halide1 or through
the use of photopolymers.2 Students find the experiments interesting and rewarding not least because of the remarkably
life-like three-dimensional 共3-D兲 images that can be obtained. Holography is also very useful from the teaching
viewpoint, because it combines the phenomena of interference and diffraction as well as requiring careful laboratory
technique for worthwhile results.
In holography texts, the recording of an amplitude hologram and reconstruction of the 3-D image are discussed using simple theory,3 but the more common phase hologram is
neglected. For thick holograms, coupled wave theory4,5 provides a comprehensive treatment of both amplitude and
phase holography in transmission and reflection geometry,
but this approach is readily accessible only to more advanced
or graduate students. Furthermore in some presentations of
coupled wave theory, it is not obvious that the object wave is
reconstructed because plane waves are assumed throughout.
Therefore there is a need for a simple approach to the theory
of thick phase holograms that will be helpful to students in
the early stages of understanding and doing holography. The
approach taken here is to use simple wave front representation.
I 共 x,y 兲 ⫽
共4兲
Assume that the amplitude transmissivity of the recorded
hologram is proportional to the exposure, which in turn is
proportional to I(x,y). It is also assumed that ␪ ⫺ ␪ ⬘ at 共x,y兲
remains constant during recording. This implies that the two
waves are mutually coherent and that the recording setup is
interferometrically stable for the duration of the exposure.
Then if we illuminate the hologram using only the reference
wave we obtain transmitted waves by multiplication as follows:
a 2 ⫹b 2
a cos共 ky sin ␪ ⫺ ␻ t 兲
2
⫹a 2 b cos兵 ky 共 sin ␪ ⫺sin ␪ ⬘ 兲 其 cos共 ky sin ␪ ⫺ ␻ t 兲
a 2 ⫹b 2
a 2b
a cos共 ky sin ␪ ⫺ ␻ t 兲 ⫹
cos共 2ky sin ␪
2
2
a 2b
cos共 ky sin ␪ ⬘ ⫺ ␻ t 兲 .
2
共5兲
The first term in Eq. 共5兲 represents the zero order wave. The
second is a phase conjugated version of the object wave but
its general direction is now at an angle sin⫺1(2 sin ␪) to the
hologram plane. If a phase conjugate version of the reference
wave, a cos(⫺ky sin ␪⫺␻t), is used to illuminate the hologram, then it is easy to show that the general direction of
travel of this wave is that of the original object wave but in
the opposite sense and a pseudoscopic image of the object is
formed. The third term is a replica of the original object
wave 关Eq. 共2兲兴 and represents the familiar 3-D image.
共1兲
IV. PHASE HOLOGRAPHY
and the object wave is
共2兲
The total intensity, I(x,y), is given by the time average of
the square of the sum of 共1兲 and 共2兲,
Am. J. Phys. 71 共9兲, September 2003
a 2 ⫹b 2
⫹ab cos兵 ky sin共 ␪ ⫺ ␪ ⬘ 兲 其 .
2
⫺ky sin ␪ ⬘ ⫺ ␻ t 兲 ⫹
Assume that the reference wave is plane and has uniform
amplitude a. Its wave normal is at an angle ␪ to the hologram
front surface, which lies in the x,y plane. The object wave
front has amplitude b(x,y) and its normal is at angle ␪ ⬘ (x,y)
to the x,y plane. The x,y dependence of b indicates that the
light from the object wave varies spatially in brightness and
the x,y dependence of ␪ ⬘ indicates an object of complex
shape so that the wave normal direction will vary with position x,y on the hologram 共Fig. 1兲. The angular wave number
is k⫽2 ␲ /␭, the angular frequency is ␻, and t is time.
Thus, the reference wave is
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共3兲
III. AMPLITUDE HOLOGRAPHY
II. PROCEDURE
b cos共 ky sin ␪ ⬘ ⫺ ␻ t 兲 .
冔
where angular brackets denote time average.
Carrying out the time average yields the result
⫽
a cos共 ky sin ␪ ⫺ ␻ t 兲
冓
a 2 cos2 共 ky sin ␪ ⫺ ␻ t 兲 ⫹b 2 cos2 共 ky sin ␪ ⬘ ⫺ ␻ t 兲
,
⫹2ab cos共 ky sin ␪ ⫺ ␻ t 兲 cos共 ky sin ␪ ⬘ ⫺ ␻ t 兲
We now assume that the effect of the object and reference
wave illumination of the photosensitive layer is to modulate
the refractive index of the layer. Bleaching a silver halide
emulsion can do this or it can occur as a result of the pro-
http://ojps.aip.org/ajp/
© 2003 American Association of Physics Teachers
948
abc cos兵 ky 共 sin ␪ ⫺sin ␪ ⬘ 兲 其 ⫽ ␾ .
共9兲
If we assume that the refractive index modulation is low, as
is usually the case, ␾ is small and Eq. 共8兲 becomes
a cos共 ky sin ␪ ⫺ ␻ t 兲 cos ␾ ⫺a sin共 ky sin ␪ ⫺ ␻ t 兲 sin ␾
⫽a cos共 ky sin ␪ ⫺ ␻ t 兲 ⫺a sin共 ky sin ␪ ⫺ ␻ t 兲 ␾ .
共10兲
The first term in this last expression is simply the reference wave 关Eq. 共1兲兴. We rewrite the second term as
⫺a 2 bc sin共 ky sin ␪ ⫺ ␻ t 兲 cos兵 ky 共 sin ␪ ⫺sin ␪ ⬘ 兲 其
⫽⫺
Fig. 1. The expression a cos(ky sin ␪⫺␻t) represents a plane reference wave
and b(x,y)cos(ky sin ␪⬘⫺␻t), an object wave. Arrows represent the local
direction of the wave front normal.
duction of polymer in illuminated regions combined with
concentration driven diffusion of monomer into those
regions.6
The spatially varying refractive index modulation
⌬n(x,y) is assumed to be proportional to the exposure and
hence the intensity I(x,y).
If the effective thickness of the photosensitive layer is d, a
constant for a collimated reference beam, then the optical
path modulation is ⌬n(x,y)d and the phase modulation is
2 ␲ ⌬n 共 x,y 兲 d/␭,
which can be written as
cI⫽c
再
冎
a 2 ⫹b 2
⫹ab cos兵 ky 共 sin ␪ ⫺sin ␪ ⬘ 兲 其 ,
2
⫺
a 2 bc
sin共 2ky sin ␪ ⫺ky sin ␪ ⬘ ⫺ ␻ t 兲
2
a 2 bc
sin共 ky sin ␪ ⬘ ⫺ ␻ t 兲 .
2
共11兲
The first term in this last expression is again a phase conjugate version of the object wave traveling in the general
direction sin⫺1(2 sin ␪) to the hologram plane. The second
term is again a replica of the original object wave 共2兲 and is
in phase quadrature, i.e., 90° out of phase, with the illuminating reference wave. The latter result can also be obtained
from coupled wave theory7 in which it expresses the fact that
light is coupled back and forth between the two waves.
In conclusion it is seen that coupled wave theory is not
required when seeking to derive some of the important results in the theory of thick transmission and reflection holograms. An approach based on simple wave front representation can be used instead. Amplitude and phase holograms
have been treated separately for clarity. A plane reference
wave is also assumed but even this is not strictly necessary
using the approach adopted here.
共6兲
where c is a constant given by
c⫽2 ␲ d/␭.
共7兲
If the amplitude of the object wave is small compared with
the reference wave or if the object wave is of spatially uniform intensity, then the effect of the first term (a 2 ⫹b 2 )/2 in
Eq. 共6兲 is to impose a spatially uniform phase shift, which we
will ignore.
Thus, when the reference wave illuminates the hologram it
is phase modulated and we find the transmitted waves by
adding cI to the phase of the reference wave
a cos共 ky sin ␪ ⫺ ␻ t⫹cI 兲
⫽a cos兵 ky sin ␪ ⫺ ␻ t⫹abc cos兵 ky 共 sin ␪ ⫺sin ␪ ⬘ 兲 其其 .
共8兲
Let the second term
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Am. J. Phys., Vol. 71, No. 9, September 2003
ACKNOWLEDGMENT
The author is indebted to the original reviewer of this
paper for many helpful comments and suggestions.
a兲
Electronic mail: [email protected]
H. J. Bjelkhagen, Silver-Halide Recording Materials for Holography and
Their Processing, 2nd ed. 共Springer-Verlag, Berlin, 1995兲.
2
S. Piazzolla and B. K. Jenkins, ‘‘Holographic grating formation in photopolymers,’’ Opt. Lett. 21, 1075–1077 共1996兲.
3
Howard M. Smith, in Principles of Holography 共Wiley, New York, 1969兲.
4
P. Hariharan, Optical Holography, Principles, Techniques and Applications 共Cambridge U.P., New York, 1986兲.
5
H. Kogelnik, ‘‘Coupled wave theory for thick holograms,’’ Bell Syst. Tech.
J. 48, 2909–2947 共1969兲.
6
S. Martin, P. Leclere, Y. Renotte, V. Toal, and Y. Lion, ‘‘Characterisation
of an acrylamide-based dry photopolymer holographic recording material,’’ Opt. Eng. 33 共12兲, 3942–3946 共1995兲.
7
L. Solymar and D. J. Cooke, Volume Holography and Volume Grating
共Academic, London, 1981兲.
1
Vincent Toal
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