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November 1981 / Vol. 6, No. 11 / OPTICS LETTERS 519

Amplified reflection, transmission, and self-oscillation in real-time holography

Baruch Fischer, Mark Cronin-Golomb, Jeffrey 0. White, and Amnon Yariv
California Institute of Technology, Pasadena, California 91125
Received July 20, 1961 A theory of phase conjugation in asymmetric materials that allowa phase shift between the grating and the lightinterference pattern is developed. We find that when this phase is nonzero, maximum phase-conjugate reflectivity occurs for unequal pump intensities. The conditions for self-oscillationare studied.

The application by Yariv and Pepper' of the formalism of nonlinear optics to the study of four-wave mixing
in phase-conjugate optics led to the prediction of a variety of phenomena, including amplified (phase-conjugate) reflection, amplified transmission, and mirrorless self-oscillation. These phenomena were later observed by a number of investigators. 2' 3 Also, the kinematic similarity between phase-conjugate optics and real-time holography has been noted,4 and a large number of experiments have been performed recently in photorefractive crystals. 5'6
There exists a fundamental difference between four-wave mixing in holographic, say photorefractive, media7 and in media, such as atomic vapors and anistropic molecular liquids, used in phase-conjugate optics; in the second case the nonlinearity that gives rise
to four-wave interaction is due to a local atomic, or
molecular (nonlinear), electronic response. A complex susceptibility Xijkl(-W, O, w, -w) in this case reflects a temporal displacement between the induced polarization and the product of optical fields that drive that polarization. This displacement is large when the frequency a' is near that of an atomic or molecular transition.8 In the holographic case the mixing is due to
spatial holograms written by the four interacting waves. Here the complex coupling constant reflects a spatial displacement between the interference pattern of a pair of beams and the resultant holographic grating. The physically distinct origin of the temporal phase and the spatial phase leads to fundamental and hence qualitative differences in the mechanisms of energy exchange
between the interacting waves in two cases. In this Letter we formulate the nonlinear four-wave
coupled-mode equations of real-time holography in a manner that is close to the spirit of phase-conjugate optics.1 We then proceed to solve these equations in a number of important special cases and obtain ex-
pressions for oscillation thresholds, amplified reflection, and amplified transmission.
The basic interaction geometry is illustrated in Fig.
1. Four waves of equal frequency a' and, for simplicity,
of the same polarization, are propagating through the nonlinear medium. Let the electric-field amplitude associated with the jth beam be

Ej = Aj(r) exp[i(kj - r - cot)] + c.c.

(1)

We solve the problem in steady state so that the A1 may

be taken to be time independent. The propagation
directions come in two oppositely directed pairs, k1 =
-k 2 and k3 =-k 4 , whereasthe relative direction ofk3
and k1 is arbitrary. It is the fringes in the time-independent part of the
light intensity that generate the hologram, whose fringes have the same periodicity as the light-interference pattern. In general, the holographic fringes of refractive index will have a spatial phase shift with respect to the light-interference pattern,7 so we can write the fundamental components of the intensity-induced
grating as

= no + nieiol(A,*A4 + A2A3*) exp(ik, . r) + c.c.

2

Io~1

+ntiei~ (A1 A3 * + A2*A4 ) exp(ikl-,r) + c.c.

2

So

+ niii ' (AlA2*)exp(ikii*,r) + c.c.

2

0o

+nlveiQIv(A3*A4) exp(ikiv. r) + c.c.,

(2)

2

10

where

Io= Z4 1A12,

(3)

i=1

I4,44I, 'kina, nd XIv are real, nI, nil, niii, and nIVare real and positive, k1 = k4 -k = k2- k3, kII= k- k3 = k4 - k2, ki, = 2k,, and kiv = 2k4. The complex constant
nleikI, as an example, characterizes the spatial hologram written by the stationary intensity-interference pattern

1
0 4

z 3 (P. C.)
-_Z4<CL 2

Fig. 1. Scheme of the four beams involved in phase conjugation.

0146-9592/81/110519-03$0.50/0

© 1981, Optical Society of America

520 OPTICS LETTERS / Vol. 6, No. 11 / November 1981

of beams 1 and 4 and also that of beams 3and 2. These two pairs of waves are characterized by the same constant because k4 - k, = k2- k3.
The expressions for njeionIi'ieion/niil,eill", and nIvei>lv are obtained by solving the specific physical process responsible for the hologram formation. In the
case of photorefractive crystals, such as Bi12 SiO2 0 and BaTiO3 as examples, the hologram is due to a refractive-index modulation produced by trapped charges that are excited by the intensity-interference pattern.
This charge distribution gives rise to a spatially alternating electric field, which in turn spatially modulates the index through the electro-optic effect. Expressions for nj and XI, for example, derived from two theories of the process are9g10

ni= no3Ep -reff flI

n3
=-ref 0

L (E+O2(E+dE+dEE2d))2]11/2 '

(4)

+E tan 'ki

Ed(~EdE+-_E_~Pp))+E

2 0

where reffis the relevant electro-optic coefficient,E 0 is an applied electric dc field directed along ki, and Ed and Ep are electric fields characteristic of diffusion and maximum space charge, respectively.
Ed = kBTkI/e and Ep = epd/ (ekj), where Pd is the density of traps in the material, kB is Boltzmann's constant, T is the temperature, e is the electron charge,
and Eis the permittivity of the material. Now, by using
expression (2) for n and the scalar-wave equation, we can derive, by the standard slowly varying field approximation, 1' the followingfour coupled-wave equa-
tions:

2c cos a,, dA=

co

dz

inje (AIA4 * + A2 *A3)A4 Io
_iniiell (AA 3* + A2*A4)A3 Io

inilei

(AlA 2 *)A 2 ,

(6)

Io

2c Cos

ddAz 22

inle-io' 10 (Ai*A 4 +A 2 A3*)A3

+ mnieikIo (A,*A3 + A2A4*)A4

+ mnIIo '0

(Ai*A 2 )Al,

(7)

2c os a2 dA3 = inle (AlA 4* + A2 *A3 )A2

a,

dz

Io

+ Io (A,*A3 + A2A4*)AI

+ i7?ive irkV (A 3A4 *)A 4 , Io

(8)

2c os
a,

dA4 _ine-i- (A,*A4 + A2 A*)A,

dz

Io

inlieitII (AA3* + A2*A4)A2

Io

-

(A3 *A4 )A3.

Io

(9)

When A3 and A2 are taken to be zero in the above equations, we recover the well-knownand analytically soluble theory of holographic two-beam coupling7. There too, the spatial phase difference 44between the light interference pattern and the grating plays an im-
portant role. Its sign determines the direction of energy transfer from one beam to the other. The effect of the phase is to shift the spatial pattern of refractive index toward one beam and away from the other. It intro-
duces an asymmetry that allows one beam to accept and the other to donate power. In the present analysis of phase conjugation, we will show that this leads to an asymmetry between the roles of the counterpropagating pump beams Al and A2. The problem may be simplified by making two assumptions. First we take only ni # 0. That is, we consider a holographic system whose spatial-frequency response is such that of all the grat-
ings present in the system, only one grating, in this case
the one created by the interference of beams 1 and 4 and
2 and 3, which is characterized by ni exp(ioj), gives rise to strong beam coupling. This predominance of one grating is common in many practical situations and is due to the directions, polarization, and coherence re-
lationships of the four beams relative to the crystal axes
and to the application, in some cases, of an electric field that enhances certain gratings.
Second, we use the nondepleted pump approximation, in which jA1j2, jA2j2 >> A3j 2, 1A42. In this case, the derivatives in Eqs. (6) and (7) are of the order of A42 or A3A4, and the approximation (dA,/dz) = (dA2/dz) = 0 becomes reasonable. Moreover, as can be seen from its definition, Eq. (3), the normalization factor 1obecomes constant. With these assumptions, the equa-
tions reduce to

2c

dA 3 nje [IA 212A 3 + (AlA 2)A 4 *],

a,

dz

Io

(10)

cos

dA4 = In [IA,1 2A4* + (AA 2)*A3].

(11)

With the boundary conditions A3(1) = 0 and A4*(0),the solutions of Eqs. (10)and (11) are

A3 (z) = A 4*(0) el/A12 + 1 (e'(4z 1 - 1),

(12)

A4*(z) = A4*(O)r'e-' + 1 (rleY(z-1) + 1), (13)

where r is the pump-beam intensity ratio,

A2 A2* I2
A1A1* I,

(14)

iwonleioi

y = =2c

coes ae2

-
E

iaeio,

(15)

with a positive and 0 real. The phase-conjugate reflectivity is thus

R- A3(0j2= | sinh)7

2

(16)

A4*(0) |cosh |a1+ Inr)

November 1981 / Vol. 6, No. 11 / OPTICS LETTERS 521

R

a=

50

100

Fig. 2. Phase-conjugate reflectivity as a function of pump-
intensity ratio for 4 = wr/2and different coupling strengths
al.

0~~~~~

4

Fig. 3. Phase-conjugate reflectivity for coupling strength at = 3.627and, peaking from left to right, 4)= 0, 7r/6, ir/3,and r/2. Oscillation here occurs for 4 = 7r/6 and r = 6.13.

the transmittivity of A4 is

_AS])

~e1712 cosh (inL 2~

T =

2

nAj)2 2

7)

A 4*(0)

cosh i- + I

2 2

and wesee that only the intensity-independent quantity
4yanld the pump-intensity ratio r enter the expressions
for R and T. The coupling constant in the second coupled-wave equation is the same as that in the first; it is not its complexconjugate,as is the case with the coupled-wave equations associated with four-wave mixing by third-order nonlinear susceptibility.4 This
is because the complex coupling constant in the holographic case represents a spatial phase shift, whereas,
in the case of a third-order nonlinearity, a complex coefficient is due to absorption; that is, it represents a
temporalphase shift. We find that the maximum phase-conjugate reflec-
tivity occurs when r = exp(al sin '):

R max =

2 (18)

If al sin 0 isnot small comparedto unity, the maximum reflectivity, Eq. (25) occurs at a value of a pumping ratio r that differs from unity by a large factor. The case in which ' = 7r/2 is of special interest, for it occurs in photorefractive materials that operate by diffusion only7 (Eo = 0). If we use symmetric pumps, then the reflectivity tanh2(al/2) never exceeds unity. Large reflectivities may, however, be reached simply by using asymmetric pumps. The maximum reflectivityobtains with r = exp(at) and is equal to sinh2(al/2). In Fig. 2 we show plots of reflectivity versus r for various values of the coupling constant. These values are large but nevertheless obtainable in certain crystals with large electro-optic coefficients, such as BaTiO3.' 2
The threshold conditions for self-oscillation(R = a) in holographic phase conjugation may be obtained from Eq. (16). We see that the phase-conjugate reflectance becomes infinite for
at cos 0 = 7r, r = exp(al sin'). (19)
In Fig. 3, we plot, for a given value of at, the phaseconjugate reflectivity as a function of r for several values of t. We find, for the value of at chosen, that self-oscillation occurs for 0 = 7r/6 and r c 6.
In order to achieve self-oscillation, it is necessary according to Eq. (18) that cos 0 # 0. In particular, we find that our model predicts that self-oscillation is impossible for X = 7r/2(the pure diffusion case), no matter how strong the coupling constant is.
When an ordinary mirror of amplitude reflectivity p is used to reflect the output AS]) in the direction of A3, we find that the oscillation threshold is simply RIp 12=
1.
This work was supported by the U.S. Air Force Office of Scientific Research and the U.S. Army Research Office, Durham, North Carolina. B. Fischer would like to acknowledge the support of the Weizmann postdoctoral fellowshipand M. Cronin-Golombthe support of the University of Sydney.
References
1. A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 (1977). 2. D. M. Bloom, P. F. Liao, and N. P. Economou, Opt. Lett.
2, 58 (1978). 3. D. M. Pepper, D. Fekete, and A. Yariv, Appl. Phys. Lett.
33, 41 (1978). 4. A. Yariv, Opt. Commun. 25, 23 (1978).
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6. J. P. Huignard, J. P. Herriau, G. Rivet, and P. GOinter, Opt. Lett. 5, 102 (1980).
7. For a review, see V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, Usp. Fiz. Nauk 129,113 (1979) [Sov. Phys. Usp. 22, 742 (1979)].
8. A. Yariv, IEEE J. Quantum Electron. QE-13, 943 (1977).
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