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236 OPTICS LETTERS / Vol. 11, No. 4 / April 1986
Theory of self-frequency detuning of oscillations by wave
mixing in photorefractive crystals
Baruch Fischer
Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, 32000,
Israel
Received August 14, 1985; accepted January 27, 1986
We present a theory of the frequency-detuning properties of various oscillators formed by four-wave mixing in
photorefractive crystals. It is shown that the detuning originates from the self-induced grating dynamics in the
mixing
crystal, governed
by phase conditions of
the optical paths and several
other parameters, such as external and
internal electric fields in the mixing crystal.
Interesting experimental findings of self-frequency-
detuning effects of phase-conjugate mirrors (PCM's)
were reported recently. 1-3A dye laser in conjunction
with a passive (self-pumped) phase-conjugate mirror
(PPCM) caused an unexplained self-frequency scan-
ning of the laser.1'2Another system of a cavity formed
by two PCM's also demonstrated a nondegenerate os-
cillation. 3
Recently4we presented a first theory that explains
the self-frequency-shift properties of the ring PPCM
and discussed its meaning as a new type of active
interferometry with applications to optical sensors
such as gyroscopes. In this Letter we present a gener-
al study of the frequency-detuning properties of vari-
ous oscillators with photorefractive crystals. The re-
sults also suggest possible causes for detuning in the
PPCM used in self-scanning experiments
1'2but do not
establish the mechanism conclusively; this will be
done in further experimental work.
Figures 1(a)-i(d) describe the oscillators discussed
in this work. In the past5'6these oscillators were ana-
lyzed with the assumption of degenerate frequencies.
However, the consideration of complex amplitudes
with the phases of the beams in the cavity plus the
crystal dictates a detuning of the mixing beams. The
phase contribution of the induced moving gratings
written by these beams is an essential ingredient. It is
due to the detuning dependence of the complex cou-
pling constant or
of two mixing beams detuned by 6 in a
photorefractive crystal 4
'5:
7(5) = Ym/[1
+ i(r6)], (1)
where r is the time constant of the grating buildup.
The additional detuning-dependent phase from y(6)
is
added to the r/2 spatial phase shift, which exists be-
tween the gratings and interference fringes in diffu-
sion-dominated photorefractive crystals.
The mathematical treatment of the nondegenerate
four-wave mixing is straightforward in most systems
in which the phase-mismatch factor due to the detun-
ing is negligible, i.e., 61/c << 1. 1 is the effective width
of the mixing crystal and c is the speed of light in the
crystal. This assumption is valid for photorefractive
crystals since 1- 1-5 mm and 6 < 1hr 1-104
rad/sec.
Thus we will
use previous calculations for the degener-
ate case5simply by plugging in the complex y(
6) with
the same assumptions of plane waves, transmission
gratings, and negligible absorption in the crystal.
Consider the standard nondegenerate phase conju-
gation in which three of the four-wave mixing
beams-the two pumps and the signal-are externally
supplied. If the signal's frequency is w + 6 compared
with w of the two pumps, the reflected beam is down-
shifted to w -6. Although the magnitude of -y(6)
decreases for larger 6, the reflectivity may be highest
for 6 5d
0. This can be shown, for example, from the
expression for the small-signal reflectivity7
.[-y(6)l
1 y(6)l ln rl2
R = sinh[ 2 Osn 2 + 2] (2)
] 2 ]/ 1 2 2]l
where r is the pump's intensity ratio. Self-oscillation
(R = a) occurs for a nonzero 6
and a specific r such that
[ ii )]+ Inr = iq7r, (3)
where q is an odd integer. It might have been
thought8
'9that the preference for high gain is the cause
for the frequency shift of oscillators with photorefrac-
tive crystals. It will be shown, however, that for the
various oscillators discussed here, the frequency de-
tuning is dictated by phase considerations of the cavi-
ty and the crystal.
We use the notation and results of Ref. 5, recalculat-
ed for the complex amplitudes Ai(z) of
the waves rath-
er than their intensities. Thus we define m1=
A,(0)/(A*2(0), M2 = A*2(1)/(A1(1),
r1= A3(0)/(A*4(0),
and r2= 1/M
2.For boundary conditions where A3(1)
= 0, valid for all the configurations of Fig. 1 except for
Fig. (1d), we obtain 4
ml
= T+Q __
m2[(A
+ B)T +
Q]
rl =_ (A+1)T
m2[(AT + Q)]
(4)
(5)
where
Q = [A2+ (A + 1)2 1r
212]1/2, T = tanh[(yl/2)Q],
B
= (1 + A)1r
212, and A = [(I2 + I3) -(II + I4)]/10
is the
conserved intensity flux normalized by the total inten-
sity Io = E, Ii = F,1Ai12
The properties of the ring PPCM [Fig. (lb)] can be
obtained easily.4The ring's complex amplitude
0146-9592/86/040236-03$2.00/0 © 1986, Optical Society of America
April 1986 / Vol. 11, No. 4 / OPTICS LETTERS 237
(a)
(+8 \c/3 )
a... .. ( C)
( b)
/ 4
Fig. 1. The oscillators with photorefractive crystals C and
C' described in this Letter. (a) Semilinear and linear
(dashed lines) PPCM's. (b) Ring PPCM. (c) 2IR PPCM.
Usually the two regions C and C' are in one crystal of which
the faces are the mirrors. (d) Unidirectional and double-
directional (dashed arrows) ring oscillators.
transmissivities for the counterpropagating beams
provide the boundary conditions at the z = 0 surface of
the mixing crystal:
m = AlMIA30), m = A4(0)/A2(0).
Realizing that mi/ri = mmn* = MeiO, where 0 is a
nonreciprocal phase in the ring and equating this to
the ratio obtained from Eqs. (4) and (5) results in
(T +Q) (AT +Q)_ = _Men'. (7)
[(A
+ B)T + Q](A
+ 1)T
Here, A = (1 -M)/(1 + M) is known.4'5Equation (7)
gives the reflectivity Ir212 and the detuning 6 with a
wide linear region 4around r 0:
(,r) - at, (8)
where a = [(M/(M + 1)]sinh(yo1)/(yo1).
Note that any reciprocal phase in the ring is can-
celed out. The effect of the detuning 6 on 0 was
ignored in this calculation. It adds 5L/c to i, where L
is the cavity's length and c the speed of light in the
cavity. For long L, however, 0 must be renormalized. 4
Equation (4) provides a solution for a linear PPCM
[Fig. (la)], where (mlm2) = M'eiO' is known. It gives
6(tY)
and A, and Eq. (5) gives the reflectivity of this
device, rl. This is not exactly compatible with the
linear PPCM of Fig. 1(a), since the mirrors plus the
cavity provide information about Al/A2and not
A1/A*2, which is required for mi. Then O' and 6 will
not be specified by Eq. (4). However, for the semilin-
ear PPCM with only one mirror the solution is imme-
diate: ml = 0 in Eq. (4) results in
T+ Q = 0. (9)
Since A and Q are real, this implies 6 = 0 and a non-
shifted oscillation for any O'.
The two-interaction-region PPCM (2IR PPCM)
(Ref. 2) of Fig. 1(c) is a combination of a ring PPCM
with a double phase conjugator (at the region C') in its
feedback loop.
5The boundary conditions for the dou-
ble PCM (DPCM) with two vanishing beams at the
crystal's surfaces are similar to those for the semilin-
ear PPCM with one mirror, producing the same condi-
tions of Eq. (9) and stationary gratings, i.e., 61
= 62 = 6-
This still permits different frequencies for the two
couples of the writing beams in the region C' and
moving gratings in the first region C. The precise 6
will be determined by another property of the DPCM
obtained by the conserved constant5c = Al (z)A
2'(z) +
A3'(z)A4'(z) in the region C'. Since A34(=') Al (0) = 0
and c(0) = c(1'),
it follows that
(10)
This means that the complex amplitude transmissivi-
ties of the counterpropagating beams through the
DPCM (C') are the same, and the ring is reciprocal
Therefore, as for the ring PPCM, 6 = 0, all the beams
of the 2IR PPCM are degenerate, unless a nonrecipro-
cal phase 0 exists in the ring. The i dependence of 6 is
similar to that of the ring PPCM and is given by
relation (8). The apparent contradiction with the ex-
perimental findings is discussed below.
The unidirectional and double-directional ring os-
cillators of Fig. 1(d)6are different from the previous
rings. The existence of a feedback loop of the oscillat-
ing beams into themselves [A4(0) = mA4(l) and A3(1) =
mA3(0)] results in a dependence on the reciprocal
phases of the resonator paths (m and mi), whereas
reciprocal phases were canceled out in the previous
rings, which depend on mm*. We do not elaborate on
this configuration since an analysis has already been
published. 9.
Besides the explicit detuning dependence on the
crystal and cavity parameters such as a, M, and ,yo the
effect of an electric field in the crystal is interesting.
An applied dc field adds a phase source in the cavity
through its effect on the spatial phase between the
gratings and the fringes of the mixing beams in the
crystal. This affects the frequency shift of the oscilla-
tion. 10
The electric-field dependence of y is given by 5'6
f(E0)
,y(E
0, 6) = Yo
1 + i(r6) (11)
where
f(E0) Ep(Eo
+ iEd)
aE
+ (Ed+
E)
and a = (Ed + Ep)/(EpEd) normalizes f(Eo) such that
,Y(Eo = 6 = 0) = yo,
Ed = kBTk/e, Ep = epd/Ek, kB is
Boltzmann's constant, T is the temperature, E is the
dielectric constant, e is the electron charge, k is the
grating's wave number, and Pd is the trap's density.
Inserting y(Eo,
6)
into the equations that describe the
various oscillators gives the detuning dependence on
E0.(We neglect here the weak r dependencel1on E0.)
Applying Eq. (9) for the semilinear PPCM with one
Al,(0/4(0) = AXOVA241%
238 OPTICS LETTERS / Vol. 11, No. 4 / April 1986
mirror will dictate that -y(Eo,
6)
be real and
(r6)
= -E-E 0
Eo
2+ Ed
(Ed + Ep) (Ed(Ed
+
EP))E
= -OEo (12)
for E02<<
Ed(Ed + Ep).
A similar procedure for the ring PPCM with an
electric field, using Eq. (7), results in10
(T6) -t9 -f3Eo (13)
in the linear region, where
a = [M/(M + 1)] sinh(-y 01)/(-y
01), Ed(Ed + Ep)
For the 2IR PPCM, the detunings in the two regions
are determined by Eq. (12) and relation (13), giving
1(61+ 62)
= a9 -fl(E
0)1
,T2(1- 62)
=-f(Eo)2
(14)
where ri and (Eo)i are the time response and electric
fields, respectively, in the two regions.
Even in the absence of an applied electric field, an
internal field can activate a detuning. The bulk photo-
voltaic effect, for example, can cause a dc electric field in
the crystal and also influences the nonuniform space-
charge field. We carried out a detailed calculation to
this effect on y, assuming the photovoltaic current to be
of the form of11-
13
Jpu = vnpc, (15)
where n and p are the densities of the mobile electrons
or holes and the ionized donors or acceptors, c is a unit
vector along the crystal's c axis, and v is a constant.
Assumptions similar
10to those for the derivation of
expression (12) give
7
= -Yo'/[l + i(rb)], (16)
where
f aEp(E0+ Epv
+ iEd)
Eo + i(Ed + EP)
such that y(Eo = Epv = 6 = 0) = -yo,
Epu = v(c )
Pd/(e,), y is the mobility, and k is the wave vector of
the grating. The photovoltaic effect will modify Eqs.
(12)-(14) such that
-0E0 -3Eo -(l/Ed)Epu. (17)
The dc field E0in the crystal is determined by the
electrical circuitry.
Our analysis shows that an internal electric field
alone cannot satisfactorily explain the up-and-down
shift of the frequency in the same system. It must be
accompanied by some nonreciprocal phase in the ring.
Such nonreciprocity may originate from different
paths of the counterpropagating beams in the ring not
being exactly phase-conjugate waves or may result
from some noise or instability. We note that an ex-
perimental possibility exists of forming reflection
gratings, which may cause an additional reciprocal
phase dependence of the detuning, as in the unidirec-
tional and double-directional ring oscillators. The
reflection gratings may be particularly important in
the compact 2IR PPCM, since a limited coherence
length does not wash out these gratings.
In conclusion, we have presented a basic analysis of
the frequency-shift behavior of oscillators with pho-
torefractive crystals, which opens the way for resolving
the unexplained spontaneous detuning effects and a
systematic experimental evaluation of these oscilla-
tors.
Note added in proof: Following the submission of
this Letter, an experimental study of the detuning
properties of
various oscillators was carried out by the
author and his colleagues.
14
Results of this work were presented at the annual
Israel Physical Society meeting, April 1985.
References
1. W. B. Whitten and J. M. Ramsey, Opt. Lett. 9,44 (1984).
2. F. J. Jahoda, R. G. Weber, and J. Feinberg, Opt. Lett. 9,
362 (1984); J. Feinberg and G. D. Bacher, Opt. Lett. 9,
420 (1984).
3. M. Cronin-Golomb, B. Fischer, S. K. Kwong, J. 0.
White, and A. Yariv, Opt. Lett. 10, 353 (1985).
4. B. Fischer and S. Sternklar, Appl. Phys. Lett. 47, 1
(1985).
5. M. Cronin-Golomb, B. Fischer, J. 0. White, and A.
Yariv, IEEE, J. Quantum Electron. QE-20,12 (1984).
6. J. 0. White, M. Cronin-Golomb, B. Fischer, and A.
Yariv, Appl. Phys. Lett. 40,450 (1982).
7. B. Fischer, M. Cronin-Golomb, J. 0. White, and A.
Yariv, Opt. Lett. 6, 519 (1981).
8. H. Rajbenbach and J. P. Huignard, Opt. Lett. 10, 137
(1985).
9. After this Letter was completed and submitted, other
relevant work was published, among them a paper by K.
R. MacDonald and J. Feinberg [Phys.
Rev. Lett. 55,821
(1985)]
with an approach similar to that of
Ref.8. Other
papers, by A. Yariv and S. K. Kwong [Opt. Lett. 10, 359
(1985); Appl. Phys. Lett. 47, 460 (1985)] and M. D. Ew-
bank and P. Yeh [Opt. Lett. 10, 496 (1985)], describe
mainly oscillators by two-wave
mixing.
10. S. Sternklar, S. Weiss, and B. Fischer, Appl. Opt. 24,
3121 (1985).
11. V. M. Fridkin, Appl. Phys. 13, 357 (1977).
12. J. Lam, Appl. Phys. Lett. 46, 909 (1985).
13. A different photovoltaic current dependence, J cc ,
where I is the light intensity [A. M. Glass, D. Von der
Linde, and T. J. Negran, Appl. Phys. Lett. 25, 233
(1974)], gives similar conclusions.
14. Recent experimental findings confirm the results of the
present theory: S. Sternklar, S. Weiss, and B. Fischer,
Opt. Lett. 11, 165 (1986).
