The cross talk between wavelength-multiplexed holograms is analyzed. The signal-to-noise ratio (SNR) is calculated for a recording schedule that places the center of each image at the null (in wavelength space) of the adjacent hologram. An asymptotic closed-form expression for the minimum SNR is derived in a general reflection geometry. The reflection geometry with counterpropagating signal and reference beams is shown to have the best SNR

Curtis, Kevin

Gu, Claire

Psaltis, Demetri

English

Caltech Authors - Main

June 15, 1993 / Vol. 18, No. 12 / OPTICS LETTERS 1001
Cross talk in wavelength-multiplexed holographic memories
Kevin Curtis
Northrop B-2 Division, Pico Rivera, California 90660, and Department of Electrical Engineering, California Institute of Technology,
MS 116-81, Pasadena, California 91125
Claire Gu
Department of Electrical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802
Demetri Psaltis
Department of Electrical Engineering, California Institute of Technology, MS 116-81, Pasadena, California 91125
Received December 22, 1992
The cross talk between wavelength-multiplexed holograms is analyzed. The signal-to-noise ratio (SNR) is
calculated for a recording schedule that places the center of each image at the null (in wavelength space) of
the adjacent hologram. An asymptotic closed-form expression for the minimum SNR is derived in a general
reflection geometry. The reflection geometry with counterpropagating signal and reference beams is shown to
have the best SNR.
One can record multiple holograms in a material
by changing either the reference-beam angle'1
(angle multiplexing) or the recording wavelength14'5
(wavelength multiplexing). These volume holo-
graphic memories have potentially high storage
capacity and fast (page-at-a-time) readout. In
the past, most research concentrated on angle
multiplexing, primarily because convenient fre-
quency-tunable light sources were not available.
With the advent of semiconductor and solid-state
tunable lasers, there has been renewed interest
in wavelength multiplexing. Rakuljic et al. 5 also
M
Ae E Rm*Sm + c.c.,
m=-M
where c.c. is the complex-conjugate term.
ence wave is given by
Rm = exp(ikmr),
(1)
The refer-
(2)
where km refers to the wave vector coming from
the reference-point source with wavelength Am. In
addition, with standard Fourier-optics analysis,8 the
mth signal beam can be expressed as
(3)
argued that wavelength-multiplexed memories have
low cross-talk noise.
The information storage capacity of volume-
holographic memories is limited by geometry and
material dynamic range. The signal-to-noise ratio
(SNR) is a convenient measure used to assess the
storage capability of holographic memories.6 7 In
this Letter we calculate the SNR that results from
cross talk between holograms that are purely due to
geometric constraints.
Fourier-transform holograms are wavelength mul-
tiplexed in a volume holographic medium with the
setup shown in Fig. 1. A plane wave Rm is generated
at wavelength Am and angle 0 with respect to the z
axis by collimating light from a point source. This
interferes inside the holographic medium with Sm,
the Fourier-transform of the mth object image, also of
wavelength Am. When we label the holograms m =
-M, -(M - 1),...,O,... ,M, the presence of these
N = 2M + 1 holograms modulates the permittivity
of the material so that the change in the permittivity
of the medium can be written as
where fm(Xo,yo) is the mth object image; x, y, and z
are the coordinates at the back focal plane of lens
L,; and F is the focal length of all three lenses, Li,
L2, and L3. By substituting relations (2) and (3) into
relation (1), one can write Ae explicitly.
The recorded medium is illuminated with one of the
reference plane waves. Ideally, this reconstructs the
Input I
Yo
x0 &
F
Reference
Point Source
'lane L 2 o
L 1 4L
itput plane
Y2
-b x X2
F F F
Fig. 1. Recording and readout geometry for wavelength
multiplexing.
0146-9592/93/121001-03$6.00/0 © 1993 Optical Society of America
Sm(r) exp(i ZA ,) ff dxodyofm(xo~yo)exp-i A F (xxo + yyo)]exP[-i AmF2 (x. 2 +
1002 OPTICS LETTERS / Vol. 18, No. 12 / June 15, 1993
hologram associated with that particular reference.
In what follows we calculate the deviations of the
reconstructed wave from the desired image. The dif-
fracted plane-wave component E(r), with wave vector
kd,j and wavelength Aj, can be derived from standard
scalar diffraction theory.9 The following expression
assumes that the Born and paraxial approximations
are valid:
E(r) ko exp(ikr) C41T r f dr'exp(-iKr')Ae(r'),
NSR = E sinc2[-t(+ - lcos 0 - t(& -
mj Am ~ ~A Am A
t(xf2 + Y2) ( Am 1 I+ 2F2 AV2 A__ )
tAmY 2 1
FAj \AAm
+ tAm ( 1 _
2 A.
(4)
where K = kd,j - kj and kj is the illuminating ref-
erence beam with wavelength Ap. The stored per-
turbation A e contains a grating-vector component
K that will give the desired response as a strong
contribution to the integral in relation (4). However,
the finite volume over which the integration is carried
out causes frequency components other than K to
Ialso yield nonzero contributions to the above integral.
Therefore relation (4) describes both the diffracted
light caused by the desired Sj and the noise caused
by the diffracted light from the other holograms.
Using relations (1)-(4), we can explicitly state the
diffracted light resulting from the presence of the
permittivity perturbation. This result can then be
related to the detected pattern by using the paraxial
approximation to describe how lens L3 maps kdJ to a
point x2, y2 at the output plane:
27X2 27rY[ 2, /kdi = _, _,-V 1 -
X2 _ Y2 i l
2F 2 2F2 J-
I sin 6
Aj )
)21
(7)
To minimize the NSR, the spacing between the
holograms in frequency (or wavelength) is chosen to
place the center pixel of each hologram at the zero
of the sinc function of the two adjacent holograms:
AP = c/[t(l + cos 6)] =_ A. To calculate NSR with
this frequency schedule, we assume that the individ-
ual frequencies are given in terms of P'm = mA + Po.
Rewriting Eq. (7) in terms of frequency and substitut-
ing in the expression for the individual frequencies
results in
NSR = E sinc2[ -(m - j)
moj
(x22 + y22)(jA + o) ( _*)
2F 2 (1 + cos 0) (mA + v -j)
(jA + Vo y2(m - j) sin
ImA + P'o F(1 + cos 0)
Making use of the assumption that the transverse
dimensions of the medium are much larger than the
spatial bandwidth of the images and carrying out the
integration over the volume of the medium, we can
write the electric field at the output plane as
___ A,,_A_. Am FAin
E(X2, Y2) ( E m Y f ( x2 -- AKmix, A-Y2 -m=-M Aj 2 n; Aj 27r
2t(mAV + o) )(1 + cos 6)2 sin2
(8)
The NSR is a function of the hologram number j (or
equivalently the reconstructing Aj) and the location
A\Km sn2 -~ jz ( Aj
2Am
Am,,
± (x22 + Y 2) ( Am
F 2 Aj2
- ) + AF (AKmjx2 + AKmyjy2) + Am (AK j"2 + AKmjy2)1 ,
mjx mj
where AKm,,j = km - kj is the difference between the
mth reference wave vector km and the illuminating
beam's wave vector kj and AKmja, is the component
of AKmj in the a direction. In relation (6), t is the
thickness of the material in the z direction. Note
that, when km = kj, Ej(x2,y2) = fj(-x 2 , -Y2), which
is the mirror image of the desired stored image.
The cross-talk noise arises from the m # j terms
in relation (6). We first write explicit expressions
for the Bragg-mismatch terms AK mix = 0, AKmjy =
-(21T/Am - 2ir/Aj)sin 0, and AKmjz = -(27r/Am -
217-/Aj)cos 6, where 6 is the angle between the ref-
erence beam and the optical axis of the output plane.
To estimate the noise-to-signal ratio (NSR), we di-
vide the total average noise power by the magnitude
squared of the signal. We assume that each pixel of
the stored images is an independent random variable,
taking the values 0 and 1 with equal probability. In
this case,
at the output plane (x 2,y 2). Given t, PO, F, 0, and
N, the above expression can be numerically evalu-
ated. The NSR on the output plane for the hologram
stored at the center frequency (j = 0) was calculated
by numerically evaluating Eq. (8) with F = 30 cm,
t = 1 cm, X2max = Y2max = 1.5 cm, v0 = 6 X 1014 Hz,
and N = 401 holograms. Figure 2 shows the output
Fig. 2. NSR at the output plane for the j = 0 hologram
with 0 = 0.0O.
(6)
June 15, 1993 / Vol. 18, No. 12 / OPTICS LETTERS 1003
We can drop the last term in the argument of the sinc
function in Eq. (8) because it becomes comparable
with the other terms only when mi is large. When
m is large, the sinc function itself adds a negligible
contribution to the summation in Eq. (8). Assuming
that vo/(m\A + Po) = 1 and using the identities sin(x +
y) = sin x cos y + cos x sin y and sin2 x = 1/2 -
1/2 cos 2x, one can write Eq. (8) as
Fig. 3. NSR at the output plane for the j
with a = 15.00.
1600.0
1400.0
1200.0
1000.0
600.0
0=00600.0 --- 0=156
400.0
200.0 0 ---- -- -___----
0.0 . I .0 0 . . 40.. 
-6000-4000-2000 0 2000 4000 6000
j - Hologram Number
Fig. 4. SNR versus hologram number (j) for N = 10,001
holograms.
1200.0
1000.0
z
800.0
600.0
400.0
200.0
0.0 . .
0.0 1000.0 2000.0 3000.0 4000.0
N - total number of holograms
Fig. 5. SNR versus the total number of holograms.
plane for 6 = 0.00, and Fig. 3 shows the output plane
for 0 = 15.00. In both cases, the maximum NSR is
at the edges of the output plane. Setting x2 and Y2
to their maximum values, we can plot the worst-case
SNR versus hologram number j. Figure 4 shows the
results for 0 = 0.00 and 0 = 15.00 for N = 10,001
holograms. Note that the last j = ±5000 holograms
for both cases have SNR's that are twice as large as
the middle (j = 0) hologram. This is because the
j = +M holograms have holograms only on one side.
Therefore the minimum SNR hologram is always the
middle hologram in the frequency schedule. How-
ever, even this worst-case SNR is >800 for 6 = 0.00,
which is excellent.
From Fig. 4 we know that the middle (j = 0) holo-
gram has the minimum SNR and that the noise is
symmetric around this hologram. Therefore we can
set j = 0 in Eq. (8) and use it to find a closed-form
solution for the maximum NSR. Let us first define
a as
(X2
2 + co 2) + F( T) sin 0Y) (9Ya2F2(l + 6O ) F(l±co 6) si . 9
= 0 hologram
NSRmax =7r2(1 + a)2 [ M2
E cos(2.nram)i1
m=1 M
2 ]
(10)
This can be further simplified, if a << 1 (true for
typical lens systems), into
=:S _ (x a 2
2 + y22)
NSRmax a 2F2(1 ± COS 6) + Y2 T)sin60 F(1 + cos)
(11)
as the number of holograms stored becomes very
large. Note that the best SNR is achieved in the
0 = 0° geometry. In this case the SNR becomes
4F 2
S~mm =(x 2
2
+ y2
2 ) (12)
Figure 5 is a plot of the SNR at the worst point
in the output plane (x2 = Y2 = 1.5 cm) versus the
total number of holograms stored with the same
parameters as given for Figs. 2 and 3. Relation (11)
predicts that the SNR goes to 800 at 6 = 00 and -127
at 0 = 15.0°. The plots show that, at N = 4000 holo-
grams, the SNR is 837 and 131, respectively, showing
that the asymptotic expression that we derived is in
good agreement with the actual values for typical
parameters.
We thank the Defense Advanced Research Projects
Agency and Northrop Corporation for support and
Geoffrey Burr for helpful discussions.
References
1. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
2. J. J. Amodei and D. L. Staebler, RCA Rev. 33, 71
(1972).
3. F. H. Mok, M. C. Tackitt, and H. M. Stoll, Opt. Lett.
16, 605 (1991).
4. F. T. S. Yu, S. Wu, A. W. Mayers, and S. Rajan, Opt.
Commun. 81, 343 (1991).
5. G. A. Rakuljic, V. Leyva, and A. Yariv, Opt. Lett. 17,
1471 (1992).
6. C. Gu, J. Hong, I. McMichael, R. Saxena, and F. H.
Mok, J. Opt. Soc. Am. A 9, 1978 (1992).
7. C. Gu and J. Hong, in Annual Meeting, Vol. 23 of
1992 OSA Technical Digest Series (Optical Society of
America, Washington, D.C., 1992), paper WT2, p. 111.
8. J. W. Goodman, Introduction to Fourier Optics
(McGraw-Hill, New York, 1968), p. 57.
9. J. D. Jackson, Classical Electrodynamics (Wiley, New
York, 1975), p. 427.
10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,
Series, and Products (Academic, London, 1980), p. 39.
11. M. Abramowitz and I. E. Stegun, Handbook of Mathe-
matical Functions With Formulas, Graphs, and Math-
ematical Tables (National Bureau of Standards, Wash-
ington, D.C., 1972), p. 807.
I
- 0=0__ __ __
-a' - '
- __1 - __ -
a:
of


Cross talk in wavelength multiplexed holographic memories

June 15, 1993 / Vol. 18, No. 12 / OPTICS LETTERS 1001
Cross talk in wavelength-multiplexed holographic memories
Kevin Curtis
Northrop B-2 Division, Pico Rivera, California 90660, and Department of Electrical Engineering, California Institute of Technology,
MS 116-81, Pasadena, California 91125
Claire Gu
Department of Electrical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802
Demetri Psaltis
Department of Electrical Engineering, California Institute of Technology, MS 116-81, Pasadena, California 91125
Received December 22, 1992
The cross talk between wavelength-multiplexed holograms is analyzed. The signal-to-noise ratio (SNR) is
calculated for a recording schedule that places the center of each image at the null (in wavelength space) of
the adjacent hologram. An asymptotic closed-form expression for the minimum SNR is derived in a general
reflection geometry. The reflection geometry with counterpropagating signal and reference beams is shown to
have the best SNR.
One can record multiple holograms in a material
by changing either the reference-beam angle'1
(angle multiplexing) or the recording wavelength14'5
(wavelength multiplexing). These volume holo-
graphic memories have potentially high storage
capacity and fast (page-at-a-time) readout. In
the past, most research concentrated on angle
multiplexing, primarily because convenient fre-
quency-tunable light sources were not available.
With the advent of semiconductor and solid-state
tunable lasers, there has been renewed interest
in wavelength multiplexing. Rakuljic et al. 5 also
M
Ae E Rm*Sm + c.c.,
m=-M
where c.c. is the complex-conjugate term.
ence wave is given by
Rm = exp(ikmr),
(1)
The refer-
(2)
where km refers to the wave vector coming from
the reference-point source with wavelength Am. In
addition, with standard Fourier-optics analysis,8 the
mth signal beam can be expressed as
(3)
argued that wavelength-multiplexed memories have
low cross-talk noise.
The information storage capacity of volume-
holographic memories is limited by geometry and
material dynamic range. The signal-to-noise ratio
(SNR) is a convenient measure used to assess the
storage capability of holographic memories.6 7 In
this Letter we calculate the SNR that results from
cross talk between holograms that are purely due to
geometric constraints.
Fourier-transform holograms are wavelength mul-
tiplexed in a volume holographic medium with the
setup shown in Fig. 1. A plane wave Rm is generated
at wavelength Am and angle 0 with respect to the z
axis by collimating light from a point source. This
interferes inside the holographic medium with Sm,
the Fourier-transform of the mth object image, also of
wavelength Am. When we label the holograms m =
-M, -(M - 1),...,O,... ,M, the presence of these
N = 2M + 1 holograms modulates the permittivity
of the material so that the change in the permittivity
of the medium can be written as
where fm(Xo,yo) is the mth object image; x, y, and z
are the coordinates at the back focal plane of lens
L,; and F is the focal length of all three lenses, Li,
L2, and L3. By substituting relations (2) and (3) into
relation (1), one can write Ae explicitly.
The recorded medium is illuminated with one of the
reference plane waves. Ideally, this reconstructs the
Input I
Yo
x0 &
F
Reference
Point Source
'lane L 2 o
L 1 4L
itput plane
Y2
-b x X2
F F F
Fig. 1. Recording and readout geometry for wavelength
multiplexing.
0146-9592/93/121001-03$6.00/0 © 1993 Optical Society of America
Sm(r) exp(i ZA ,) ff dxodyofm(xo~yo)exp-i A F (xxo + yyo)]exP[-i AmF2 (x. 2 +
1002 OPTICS LETTERS / Vol. 18, No. 12 / June 15, 1993
hologram associated with that particular reference.
In what follows we calculate the deviations of the
reconstructed wave from the desired image. The dif-
fracted plane-wave component E(r), with wave vector
kd,j and wavelength Aj, can be derived from standard
scalar diffraction theory.9 The following expression
assumes that the Born and paraxial approximations
are valid:
E(r) ko exp(ikr) C41T r f dr'exp(-iKr')Ae(r'),
NSR = E sinc2[-t(+ - lcos 0 - t(& -
mj Am ~ ~A Am A
t(xf2 + Y2) ( Am 1 I+ 2F2 AV2 A__ )
tAmY 2 1
FAj \AAm
+ tAm ( 1 _
2 A.
(4)
where K = kd,j - kj and kj is the illuminating ref-
erence beam with wavelength Ap. The stored per-
turbation A e contains a grating-vector component
K that will give the desired response as a strong
contribution to the integral in relation (4). However,
the finite volume over which the integration is carried
out causes frequency components other than K to
Ialso yield nonzero contributions to the above integral.
Therefore relation (4) describes both the diffracted
light caused by the desired Sj and the noise caused
by the diffracted light from the other holograms.
Using relations (1)-(4), we can explicitly state the
diffracted light resulting from the presence of the
permittivity perturbation. This result can then be
related to the detected pattern by using the paraxial
approximation to describe how lens L3 maps kdJ to a
point x2, y2 at the output plane:
27X2 27rY[ 2, /kdi = _, _,-V 1 - X2 _ Y2 i l2F 2 2F2 J-
I sin 6
Aj )
)21 (7)
To minimize the NSR, the spacing between the
holograms in frequency (or wavelength) is chosen to
place the center pixel of each hologram at the zero
of the sinc function of the two adjacent holograms:
AP = c/[t(l + cos 6)] =_ A. To calculate NSR with
this frequency schedule, we assume that the individ-
ual frequencies are given in terms of P'm = mA + Po.
Rewriting Eq. (7) in terms of frequency and substitut-
ing in the expression for the individual frequencies
results in
NSR = E sinc2[ -(m - j)
moj
(x22 + y22)(jA + o) ( _*)
2F 2 (1 + cos 0) (mA + v -j)
(jA + Vo y2(m - j) sin
ImA + P'o F(1 + cos 0)
Making use of the assumption that the transverse
dimensions of the medium are much larger than the
spatial bandwidth of the images and carrying out the
integration over the volume of the medium, we can
write the electric field at the output plane as
___ A,,_A_. Am FAinE(X2, Y2) ( E m Y f ( x2 -- AKmix, A-Y2 -
m=-M Aj 2 n; Aj 27r
2t(mAV + o) )(1 + cos 6)2 sin2 (8)
The NSR is a function of the hologram number j (or
equivalently the reconstructing Aj) and the location
A\Km sn2 -~ jz ( Aj 2AmAm,,
± (x22 + Y 2) ( Am
F 2 Aj2 - ) + AF (AKmjx2 + AKmyjy2) + Am (AK j"2 + AKmjy2)1 ,mjx mj
where AKm,,j = km - kj is the difference between the
mth reference wave vector km and the illuminating
beam's wave vector kj and AKmja, is the component
of AKmj in the a direction. In relation (6), t is the
thickness of the material in the z direction. Note
that, when km = kj, Ej(x2,y2) = fj(-x 2 , -Y2), which
is the mirror image of the desired stored image.
The cross-talk noise arises from the m # j terms
in relation (6). We first write explicit expressions
for the Bragg-mismatch terms AK mix = 0, AKmjy =
-(21T/Am - 2ir/Aj)sin 0, and AKmjz = -(27r/Am -
217-/Aj)cos 6, where 6 is the angle between the ref-
erence beam and the optical axis of the output plane.
To estimate the noise-to-signal ratio (NSR), we di-
vide the total average noise power by the magnitude
squared of the signal. We assume that each pixel of
the stored images is an independent random variable,
taking the values 0 and 1 with equal probability. In
this case,
at the output plane (x 2,y 2). Given t, PO, F, 0, and
N, the above expression can be numerically evalu-
ated. The NSR on the output plane for the hologram
stored at the center frequency (j = 0) was calculated
by numerically evaluating Eq. (8) with F = 30 cm,
t = 1 cm, X2max = Y2max = 1.5 cm, v0 = 6 X 1014 Hz,
and N = 401 holograms. Figure 2 shows the output
Fig. 2. NSR at the output plane for the j = 0 hologram
with 0 = 0.0O.
(6)
June 15, 1993 / Vol. 18, No. 12 / OPTICS LETTERS 1003
We can drop the last term in the argument of the sinc
function in Eq. (8) because it becomes comparable
with the other terms only when mi is large. When
m is large, the sinc function itself adds a negligible
contribution to the summation in Eq. (8). Assuming
that vo/(m\A + Po) = 1 and using the identities sin(x +
y) = sin x cos y + cos x sin y and sin2 x = 1/2 -
1/2 cos 2x, one can write Eq. (8) as
Fig. 3. NSR at the output plane for the j
with a = 15.00.
1600.0
1400.0
1200.0
1000.0
600.0
0=00600.0 --- 0=156
400.0
200.0 0 ---- -- -___----
0.0 . I .0 0 . . 40.. 
-6000-4000-2000 0 2000 4000 6000
j - Hologram Number
Fig. 4. SNR versus hologram number (j) for N = 10,001
holograms.
1200.0
1000.0
z
800.0
600.0
400.0
200.0
0.0 . .
0.0 1000.0 2000.0 3000.0 4000.0
N - total number of holograms
Fig. 5. SNR versus the total number of holograms.
plane for 6 = 0.00, and Fig. 3 shows the output plane
for 0 = 15.00. In both cases, the maximum NSR is
at the edges of the output plane. Setting x2 and Y2
to their maximum values, we can plot the worst-case
SNR versus hologram number j. Figure 4 shows the
results for 0 = 0.00 and 0 = 15.00 for N = 10,001
holograms. Note that the last j = ±5000 holograms
for both cases have SNR's that are twice as large as
the middle (j = 0) hologram. This is because the
j = +M holograms have holograms only on one side.
Therefore the minimum SNR hologram is always the
middle hologram in the frequency schedule. How-
ever, even this worst-case SNR is >800 for 6 = 0.00,
which is excellent.
From Fig. 4 we know that the middle (j = 0) holo-
gram has the minimum SNR and that the noise is
symmetric around this hologram. Therefore we can
set j = 0 in Eq. (8) and use it to find a closed-form
solution for the maximum NSR. Let us first define
a as
(X22 + co 2) + F( T) sin 0Y) (9Ya2F2(l + 6O ) F(l±co 6) si . 9
= 0 hologram
NSRmax =7r2(1 + a)2 [ M2
E cos(2.nram)i1
m=1 M 2 ]
(10)
This can be further simplified, if a << 1 (true for
typical lens systems), into
=:S _ (x a 2 2 + y22)NSRmax a 2F2(1 ± COS 6) + Y2 T)sin60 F(1 + cos)
(11)
as the number of holograms stored becomes very
large. Note that the best SNR is achieved in the
0 = 0° geometry. In this case the SNR becomes
4F 2
S~mm =(x 2 2 + y2 2 ) (12)
Figure 5 is a plot of the SNR at the worst point
in the output plane (x2 = Y2 = 1.5 cm) versus the
total number of holograms stored with the same
parameters as given for Figs. 2 and 3. Relation (11)
predicts that the SNR goes to 800 at 6 = 00 and -127
at 0 = 15.0°. The plots show that, at N = 4000 holo-
grams, the SNR is 837 and 131, respectively, showing
that the asymptotic expression that we derived is in
good agreement with the actual values for typical
parameters.
We thank the Defense Advanced Research Projects
Agency and Northrop Corporation for support and
Geoffrey Burr for helpful discussions.
References
1. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
2. J. J. Amodei and D. L. Staebler, RCA Rev. 33, 71
(1972).
3. F. H. Mok, M. C. Tackitt, and H. M. Stoll, Opt. Lett.
16, 605 (1991).
4. F. T. S. Yu, S. Wu, A. W. Mayers, and S. Rajan, Opt.
Commun. 81, 343 (1991).
5. G. A. Rakuljic, V. Leyva, and A. Yariv, Opt. Lett. 17,
1471 (1992).
6. C. Gu, J. Hong, I. McMichael, R. Saxena, and F. H.
Mok, J. Opt. Soc. Am. A 9, 1978 (1992).
7. C. Gu and J. Hong, in Annual Meeting, Vol. 23 of
1992 OSA Technical Digest Series (Optical Society of
America, Washington, D.C., 1992), paper WT2, p. 111.
8. J. W. Goodman, Introduction to Fourier Optics
(McGraw-Hill, New York, 1968), p. 57.
9. J. D. Jackson, Classical Electrodynamics (Wiley, New
York, 1975), p. 427.
10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,
Series, and Products (Academic, London, 1980), p. 39.
11. M. Abramowitz and I. E. Stegun, Handbook of Mathe-
matical Functions With Formulas, Graphs, and Math-
ematical Tables (National Bureau of Standards, Wash-
ington, D.C., 1972), p. 807.
I
- 0=0__ __ __
-a' - '
- __1 - __ -
a:
of


A 9,

Classical Electrodynamics

Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards,

Introduction to Fourier Optics (McGraw-Hill,

paper WT2,

https://authors.library.caltech.edu/2962/1/CURol93.pdf

Cross talk in wavelength multiplexed holographic memories

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Caltech Authors - Main

Caltech Authors - Main