An exact solution of a nonlinear model of holographic four-wave mixing is derived. An expression for the reflectivity of a phase-conjugate mirror with depleted pumps is presented. We find that such a phase-conjugate mirror may exhibit bistability

Cronin-Golomb, Mark

Fischer, Baruch

White, Jeffrey O.

Yariv, Amnon

English

Caltech Authors - Main

July 1982 / Vol. 7, No. 7 / OPTICS LETTERS 313
Exact solution of a nonlinear model of four-wave mixing and
phase conjugation
Mark Cronin-Golomb, Jeffrey 0. White, Baruch Fischer, and Amnon Yariv
California Institute of Technology, Pasadena, California 91125
Received March 29, 1982
An exact solution of a nonlinear model of holographic four-wave mixing is derived. An expression for the reflectivi-
ty of a phase-conjugate mirror with depleted pumps is presented. We find that such a phase-conjugate mirror may
exhibit bistability.
Theories of the interactions of light in media possess-
ing optical nonlinearities are themselves strongly non-
linear; most theoretical approaches have depended on
such linearizing assumptions as the undepleted-pumps
approximation.1' 2 Between the early 1960's, when
second- and third-harmonic generations and Raman
pump conversion to first Stokes line4 were solved, and
the present, little work was done that analytically re-
tained the nonlinear nature of the problem. In 1979,
a model of degenerate four-wave mixing in isotropic
media was solved for the case in which all four beams are
collinear.5 As a result of recent progress in the search
for nonlinear media that are efficient in coupling light
at low power levels, four-wave mixing and phase-con-
jugation experiments in which the pumping beams
cannot be considered undepleted are becoming more
numerous.6-8 There is thus a more urgent need for
solutions of theories of four-wave mixing that do not rely
on the constancy of the pumps.
We present here such a solution, which is applicable
in the case of degenerate four-wave mixing by real-time
holography.9 This solution does not suffer from the
collinearity restriction and is applicable for any phase
shift between the refractive-index grating and the
light-interference pattern. The basic interaction ge-
ometry is illustrated in Fig. 1. Four waves of equal
frequency a, and of the same polarization, are propa-
gating through the nonlinear medium. Let the elec-
tric-field amplitude associated with the jth beam be
Ej = Aj(r)exp[i(kj - r -Cot)] + c.c. (1)
We solve the problem in the steady state, so that Aj may
be taken to be time independent. The propagation
directions come in two oppositely directed pairs, k, =
-k 2 and k3 = -k4 , and the angles between k, and the
crystal normal and k2 and the crystal normal are equal
in magnitude but opposite in sign. A set of four non-
linear coupled-wave equations can be derived by using
the standard slowly varying field approximations We
simplify the problem by considering a holographic
system whose response is such that, of all the gratings
present in the system, only one grating, in our case the
one created by the interference of beam 1 with beam
4 and beam 2 with 3, gives rise to strong beam coupling.
This predominance of one grating is common in many
laboratory situations and is due to the directions, po-
larizations, and coherence relationships of the four
beams relative to the nonlinear medium and to the ap-
plication, in some cases, of an electric field that en-
hances certain gratings. We do not use the unde-
pleted-pumps approximation. That is, the variation
of all four beams is retained, and the solution is valid
even when all four beams are comparable in amplitude.
The applicable coupled-wave equations are10
dA1 =-7 (A1A4* + A2*A3)A4,
dz Io
dA 2* =y (AiA4 * + A2*A3 )A3 *,dz Io
dA3 = " (A1A4 * + A2 *A3 )A2,dz Io
dA4* = "' (A1A4* + A2*A3)A,*,
dz Io
(2)
(3)
(4)
(5)
where y is a complex coupling constant, which is a ma-
terial parameter of the nonlinear medium.'1 When -y
is real, the phase shift between grating and fringes is
7r/2. When it is purely imaginary, the grating is in phase
with the fringes. Jo is the total average light inten-
sity.
I0 = Il + I2 + I3 + 14,
where Ij = I Aj 2.
We observe the following conservation laws:
A1A2 + A3 A4 = c,
Il + I4 = di,
12 + I3 = d2 ,
(7)
(8)
(9)
1 ~~~~~~3
4 S- W 2
Fig. 1. Scheme of the four beams involved in nonlinear in-
teraction.
0146-9592/82/070313-03$1.00/0 © 1982, Optical Society of America
314 OPTICS LETTERS / Vol. 7, No. 7 / July 1982
where c, d1 , and d2 are constants of integration.
With the help of these conservation laws, Eqs. (2) and
(3) can be decoupled from Eqs. (4) and (5):
dAl -"' [Aid, -A1 (I1 + 2) + A2*c,
dA2* = -8 [A1c*-A2*(Ii + 12) + A2*d2],dz 10
dA3 = B' [AAd - A3(I3 + 14) + A 4*c],dz =I
dA4* = .1 [A 3 C* - A 4* (13 + 14) + A4*dl].
dz Io
(10)
(11)
(12)
I-
l
U I
Or
(13)
By eliminating the term in I, + 12 between Eqs. (10)
and (11), and the term in I3 + 14 between Eqs. (12) and
(13), we find the following expressions for A3 /A 4 * and
Al/A 2*:
d IA 1 'j -- y (A 1 \ (*A1 P2
dz (A2 *) Io [c + (d - d2 )
d(AijdY +ddi)s (AV
dz A4 _) Io [4 (2 AA4 *)Jj
(15)
Noting that Io is constant because of the conservation
laws, we see that these equations are immediately in-
tegrable:
Al
A2*
A3=
A4*
COUPLING STRENGTH tY.1
Fig. 2. Reflectivity of a phase-conjugate mirror versus cou-
pling strength llyl. The incident pump beams are of equal
intensity, the intensity of the incident probe beam is 20% of
the total incident pumping intensity, and the phase shift be-
tween the index grating and interference fringes is 5°.
This solution, of course, reduces to the known unde-
pleted-pumps approximation theory,' 0 in the limit of
small I4(0). In Fig. 2 we plot the reflectivity of a
phase-conjugate mirror as a function of the coupling
strength Iy1I for the case in which the phase shift be-
|[[A - (A2+ 2 (De-zD ) + 4}c2)2D-le (16)
lA -(A 2 ±41c1) 2 c*(De-Az - D-lez) j [Ad - (A\2 + 41C1 2)1/21Ee -z-[A + (A2 + 41C1 2)1/21E-1e~z1 . 17)
I 2c*(Eez - E-legz) J 
where A is the power flux d2 -di
y(A2 + 41ci2)1/2
and D and E are constants of integration.
At this point, the problem has been transformed from
the set of nonlinear differential equations [Eqs. (2)-(5)]
to another set of equations [Eqs. (7)-(9), (16), and (17)1,
which may be solved by fitting boundary conditions.
To illustrate this solution in a particular set of
boundary conditions, we consider the reflectivity of a
phase-conjugate mirror. Let beams 1 and 2 be the
pumping beams, beam 4 be the probe beam, and beam
3 be the phase-conjugate beam. Il(O), I(0), and I2(1)
are given as boundary conditions, together with the
phase-conjugation condition N3(O) = 0. After elemen-
tary manipulations of the solution equations and the
boundary conditions, we find that the intensity reflec-
tivity R is given by
R = ~41el12J TJ2 (9
IAT+ (A2 + 41c12)1/2 2 (19)
where T = tanh(gl) and IC12 is given by the equation
[ le 2 - 11(0)12(1}]1 AT + (As + 41 cl 2)1121 2
+ 41c1 21 TI 214(0)I2(1)
+ 21C122I(O)(A2 + 41c12)1/2 (T+ T*) = 0. (20)
tween the grating and the interference fringes is 50.
The intensities of the two pumping beams I1(O) and
I2(0) have been taken to be equal, and the probe ratio
q, defined as
14(0)
q Ii(O) + I2(1) (21)
has been taken to be 0.2. The top of the graph (R = 2.5)
corresponds to the reflectivity that would result if all the
power of beam 2 were transferred to beam 3. This is the
maximum reflectivity consistent with the conservation
laws [Eqs. (7)-(9)]. The peaks in the curve correspond
to the poles in the reflectivity of a phase-conjugate
mirror with no pump depletion [R = Itanh(-yl/2)12 for
no phase shift between grating and interference fringes].
In the graph, we have set the phase shift slightly nonzero
to demonstrate the resultant damping of the oscillatory
behavior. The peaks bend toward the right, probably
because depletion causes high reflectivities to demand
higher coupling strengths than are required for the same
behavior in the undepleted phase-conjugate mirror.
The bending of the peaks can even be sufficient for
bistability, as can be seen in the first two peaks of Fig.
2.
In conclusion, then, we have solved a model of holo-
graphic four-wave mixing without using the unde-
J_. 
July 1982 / Vol. 7, No. 7 / OPTICS LETTERS 315
pleted-pumps approximation; the quantitative theo-
retical study of experiments involving pump depletion
has thus been made possible without using numerical
solutions of the coupled-wave differential equations.
The authors are grateful for valuable discussions with
F. Vachss. M. Cronin-Golomb would like to acknowl-
edge the support of the University of Sydney and B.
Fischer the support of the Weizmann postdoctoral fel-
lowship. This work was supported by the U.S. Air
Force Office of Scientific Research and the U.S. Army
Research Office, Durham, North Carolina.
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11. In photorefractive media, for example, y depends on
parameters such as crystal orientation, grating period,
charge-carrier concentration, and the angle of incidence
a.


Exact solution of a nonlinear model of four-wave mixing and phase conjugation

In photorefractive media, for example, y depends on parameters such as crystal orientation, grating period, charge-carrier concentration, and the angle of incidence a.

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Exact solution of a nonlinear model of four-wave mixing and phase conjugation

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