Double phase-conjugate mirror: oscillation and amplification properties

I April 1995

OPTICS

COMMUNICATIONS ELSETIER

Optics Communications

1 IS (1995) 539-544

Double phase-conjugate mirror: oscillation and amplification properties N.A. Korneev ‘, S.L. Sochava A.F. loffe Physical Technical Institute. 194021 St.-Petersburg, Russia Received 3 March 1994; revised

version received 22 November 1994

Abstract We solve a simple approximate equation which describes the operation of a double phase-conjugate mirror for the case of a small angle between pump beams of finite size. The solution has the features of both an amplifier and oscillator. The model qualitatively agrees with data obtained for a photorefractive Bi,*TiO*,, conjugate mirror.

1. Introduction The double phase-conjugate mirror (DPCM) is a popular device producing two waves which are phaseconjugated to mutually incoherent pump beams (Fig. la) [ 1,2]. A recent resurgence of interest in this scheme was caused by a theoretical demonstration of the non-oscillatory character of its operation for the case of pumping by plane waves of finite size [ 3,4]. The reason for this is that an initial disturbance in DPCM with tilted pump beams propagates in space and finally goes out of the interaction region. On the other hand, experiments and recent theoretical investigations [5] demonstrate that the oscillator model describes well some features of the real device, in particular, the existence of clear threshold for conjugate beam appearance $ = 2, where y is the amplitude two-wave mixing gain, and d the crystal thickness. The exact solution proposed by Zozulya [ 31 is rather complicated which makes its application to a real experimental situation difficult. We propose a simple ’ Present address: INAOE, Apt. Postal 51 y 216, CP 72000, Puebla, Mexico. 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00005-4

E 1:.

B

lb) i_

__!

Fig. 1. Geometry of the phase-conjugate mirror. R, and R, are pump beams, A, and Az are phase-conjugate beams.

approximate solution for the case of a small angle between the pump beams. This case is realized in BSOtype crystals with applied external AC or DC electric field [ 6-81. The influence of the crystal parameters and the geometry on operation of DPCM is then easily analyzed, and it is possible to suggest a new criterion for the validity of the oscillator model [ 1 ] which is less

540

N.A. Komeev. XL. Sochava/Optics Communications 115 (1995) 539-544

rigorous than the one with the angle between the beams smaller than their diffraction-limited divergence as used in Ref. [ 31.

Tl-a4x0, z,>/at1 + 4x0, Z”) d/2

=- Y

2

JC

4x0, z) -alz-&.I I

-d/2

(6)

2. Equations and their solution The geometry is depicted in Fig. 1. We use the same equations as in Ref. [ 31, but amplitudes of pump beams are supposed to equal unity: aA,/&,

= v,

(1)

8A,lth,

= v,

(2)

~(dv/at)+

v=(y/2)(A,

+A2).

(3)

Here A, and A, are the amplitudes of the phase-conjugate waves, v is the amplitude of the refractive index grating, y is the amplitude two-wave mixing gain, and r is the grating relaxation time. The angle CYis supposed to be small ((Y e 1) and we also assume that the length of the beam overlap region (in thex-direction) is much larger than the crystal thickness: l/a Z+ d (here 1 is the aperture of the interacting beams). As boundary conditions for the conjugate waves we take: A,(z=d/2)

=A2(z=

-d/2)

=O.

(4)

Using Eq. ( 1) for the amplitude A, of the first wave in the point (xc), ze) (Fig. lb) we have:

A,(+ ~0)=

I

vdn,

.

(W

IX

A similar equation is valid for the second wave: A,(-G, ~0) =

udn,.

(5b)

I BC

Combining Eq. (3):

the two gives for the right-hand

(Y/~)(A, +A,) = (y/2)

4.~ z) dr ,

side of

(5c)

BCD

where dr is the length element along the line BCD. For the small angle between the beams we can approximate the integral in Eq. (5~) and, using Eq. (3):

where integration is performed now along the line EF (Fig. lb). To get the right-hand side of Eq. (6) only terms independent on (Y and proportional to (Y are retained in Eq. (5). The difference between dr and dz is proportional to a2, and the expression in the square brackets is equal to the first order of (Yto the value of v on the line DCB. The condition for the validity of this approximation is that v does not change considerably at a distance ad along the x-axis. The trial solution of the form: V(X, z, t) =exp(iot+ikn)

ch(pz)

(7)

fits Eq. (6) only if the two conditions connecting k with wand pare satisfied. This gives the dispersion relations. As, according to the nature of our approximation, we are interested only in solutions with kda K 1, these relations for small p (the lowest root for given k) can be written in a very simple form: ior= (pf)‘=

( yd/2 - 1) - ik yd2a/6 -2i

adk.

,

(8a) (8b)

As adk -=x 1, from Eq. (8b) it follows that pd GI 1, i.e. the solution is changing slowly with z. Note that the simplest solution for the case of the crystal infinite along the x-axis is independent on z. Eqs. (7) and (8a) mean that if at t = 0 the grating with amplitude V, is in the region from 0 to Ax along the x-axis, it will move as a whole in the positive direction with constant velocity u,,= ayd2/(67) and will grow with time as exp [ (yd/2 - 1) t] . So, there is a clearly defined threshold yd = 2, which determines whether initial disturbance will grow or subside. Limitation kda -=z 1 means that front and back edges of the grating will be distorted in the regions with the characteristic size ad. Above the threshold the operation of DPCM is determined by the amplitude of the grating caused by crystal imperfections or stray light at the point x = 0 since the disturbance born there undergoes the highest amplification before going out of the interaction region. Assuming model initial conditions

N.A. Korneev, S.L. Sochava /Optics Communications 1 I5 (1995j S39-S44

v(x=O,

t) = v(x, t=O) = v,

541

(9)

we have V(X, t) z v. exp[ (yd/2 - 1)6x/( y&d’) ] for t>6mlyad2, Y(.x, t) = v. exp[ (yd/2-

(loa) l)t/T]

for t<6mlyad2,

(lob)

Eqs. ( 10) can be obtained as a result of the time development of a set of the gratings occupying at the moment t = 0 regions [ 0, AX], [AX, FAX] , etc. In the point x = 0 the noise grating u. exists permanently. Note that Eq. ( 10a) is the steady-state solution which can be obtained from Eqs. (7), (8) by putting w= 0, and ( lob) is the oscillator solution for infinite beams. Eqs. ( 10) can also be obtained if we suppose from the beginning that v is independent on z and calculate the integral in Eq. (5) along the line GH (Fig. lb) which is somewhat lower than EF. For the correct result, the distance between the lines has to be c&/3. According to the condition of the slow change of the solution along the x axis, it is valid for the near-threshold region lyd/2-11


(II)

v as a function of x and t is shown in Fig. 2a.

3. Comparison

with previous solutions

The solution obtained combines features of “amplifier” and “oscillator” approaches. On the one hand, for any y there is a finite (for t + a) value of V, proportional to the crystal noise level ZY*The solution demonstrates a strong nonuniformity of v and accordingly, conjugate waves along the x-direction which was first predicted in Ref. [ 3 1 and confirmed experimentally in Ref. [ 41. On the other hand, there is clearly a defined threshold value yd/2 = 1, the same as in the oscillator model [ 21. Above this value conjugate waves appear above noise level and their intensities sharply (exponentially) grow with y. Such behavior was observed, for example in Refs. [7,8]. The comparison of the solution (Fig. 2a) with oscillator one (Fig. 2b) which is expressed by Eq. ( lob) for r> 0 demonstrates that if the saturation of pump

Fig. 2. Time and space development of initial grating with uniform amplitude vOfor the case of finite (a) and infinite (b) size of beams.

beams is taken into consideration, these two solutions may become similar. It occurs if the oscillator solution grows to a saturation level in the time t, which is less than the characteristic time of the amplifier 67-1/yad’, equal to the time for which the initial disturbance passes through the device (Eq. ( 10) ) . As A, cannot exceed pump level (supposed to be equal to unity), the maximal value of vcan be estimated from Eq. ( 1) as v__ = 2/d. The maximal value of the grating strength is obtained for t --, ~0 in the point x = t. Comparing this value with %Ax we have from Eq. (10a) the criterion of the equivalence of the two approaches:

542

N.A. Korneev, S.L. Sochava /Optics Communications 115 (1995) 539-544

yd ad --ln(2/dv,) yd/2-1 61

5 1.

(12)

Estimating ln( 2/d vo) as 10-30 which is a fairly wide range, we see that for typical parameters of the BTO crystal with the external electric field (Y= l/20, dz I, and for yd/2 - 1 z l/3 (which is the range of validity of Eq. (10)) DPCM is rather near to an oscillator regime. In other words, when DPCM works well (i.e. for high yd), both approaches are equivalent and insufficient because they do not consider saturation of gain due to the pump depletion.

t 2-

n n

n

1

t

0 C’ ,

0

4

2’

Applied

voltage

amplitude

d U,

kV

Fig. 3. Gain factor of two-wave mixing as function of the amplitude of AC electric field applied to the crystal for the angle LI= 0.05 rad.

4. Experimental

Intensity,

results

arb.units

I

The experiments were carried out using Bii2TiOZ0 crystal grown by Mr. V.V. Prokof’ev at the Department for Quantum Electronics of the A.F. Ioffe Physical Technical Institute. The sample was cut in the ( 110) crystallographic plane and oriented so that the [ 1711 axis was in the incidence plane. This orientation combined with a H-polarization of the interacting beams provides the most effective utilization of the linear electro-optic effect in cubic crystals [9]. The sample dimensions were 4.8 mm X 10 mm X 6 mm (the first size corresponds to the sample width and the last one corresponds to the sample thickness). An external square wave alternating field with an amplitude of up to 1.5kV/cm and a frequency of 70 Hz was applied to the crystal along the [ 1i 1] axis using silver paste electrodes. Two mutually incoherent beams from two He-Ne lasers (P, = 1 mW, P2 = 3 mW) interacted within the sample volume at an optimal angle (Y= 0.05 rad (inside the sample), which yields the maximum gain coefficient y for the crystal [ 61. Both Gaussian and speckle-like pump beams were used in our experiments. The nonstationary mechanism of a holographic recording in an external ac field allows the gain factor to be changed by varying the amplitude of the voltage applied to the crystal. Fig. 3 shows the experimental dependence of the amplitude gain factor y on the applied voltage U measured in a conventional experiment on two-wave amplification of a weak beam: (13)

08

‘I I

0.6

1 I 0

00

0

I

CJOOO

"b

0 0

I

0

I

2

3

x,

*

cl=3

k”

o

u=7

w

1

mm

Fig. 4. Spatial distributions of output beam intensity for near-threshold operation (Cl= 3 kV) and operation at high amplification ( (I= 7 kV). In the first case the distribution is strongly nonuniform. Here, Z, and Z, are the intensities

of the signal and the reference beams, respectively. It is seen that for voltages ranging from 1.5 to 4 kV, i.e. just in the region of the “threshold” in DPCM of interest, the dependence is linear and has a slope of = 1 kV - ‘. The studies of the spatial intensity distribution of excited waves A, and A2 along the x-axis (Fig. la) have confirmed the obtained theoretical results. For instance, in a fairly narrow range of voltages close to the “threshold”, when condition ( 12) is obviously not satisfied, the intensity distribution (as in Ref. [ 4 ] ) is strongly asymmetrical and its maximum is shifted to the crystal boundary x = I (Fig. 4, curve a). In view of the Gaussian intensity distribution over the cross section of the pump beam, this shape supports the conclusion (10a) about the exponential distribution of the grating amplitude along the x-axis. For higher gain factors of the medium the intensity distribution in the

N.A. Korneev, XL. Sochava /Optics

Inverse

characteristic

time

1/T.

1 /s

/ *

* * *

-J”

/

35

55

45

Applied

voltage

amplitude

U,

kV

Fig. 5. Inverse characteristic time of conjugate beam build-up as a function of applied voltage. The time grows drastically near the threshold.

Intensltv.

0

orb

,

units

2

3

Applied

,

Intensity 10000 ;

4

voltage

5

6

amplitude

7

B

U ,kV

I

F

x x a 0 3

a

1 D

I

2

Applied

s43

excited waves A, and A2 over the crystal aperture becomes more uniform due to the depletion of pump beams Rt and R, (Fig. 4, curve b). The data shown in Fig. 4 were obtained for a crystal pumped by plane waves. Similar distributions were also obtained in the experiment where one of the plane waves was replaced by a speckle-like one. As Eq. (lob) predicts, the dynamics of the phaseconjugate wave growth is initially exponential, as in the generation model, and the characteristic time T grows infinitely at yd + 2 ( T= T/ ( yd/2 - 1) ). Measurements on the crystal pumped by Gaussian beams focused in a vertical plane [7] agree with this result (Fig. 5). This behavior differs qualitatively from that one theoretically obtained for another geometry in Ref. [ 31 where the transient process proceed with a negative second derivative, as in a conventional amplifier, and the characteristic time has no pole at yd = 2. According to (lOa) the exponential factor in the V( -y) curve depends linearly on the Z/ad parameter in the vicinity of the “threshold” value of yd = 2. Fig. 6 shows the experimental dependencies of the conjugate wave intensity IA on the applied voltage U measured for different pump beam apertures (Fig. 6a) and for different angles of interaction (Fig. Sb), plotted on a logarithmic scale. The accuracy of the measurements in the vicinity of the “threshold” is not sufficient to make quantitative conclusions, but qualitatively the slope of the curves in the region where the conjugate wave just appear above the noise level is increased as the pump beam aperture is increased and as the angle of interaction is decreased.

@ @ *

5. Conclusion

0

x

115 (1995) 539-544

arb.units

(b)

x x x 000000

Communications

x

3

voltage

impli;ude

0.032

0

10

0 064

y

o

6

kV7

8

U,

Fig. 6. DPCM operation for different pump beam widths (a) and different angles n (0.032 and 0.064 rad) between them (b). In the second case the shift of threshold is due to the change of yas function of the applied voltage.

To summarize, we have suggested a fairly simple theoretical model of DPCM for a small angle between the pump beams which qualitatively describes the behavior of the device near the threshold for a BTO crystal. It has been shown that if the condition of strong amplification (12) is satisfied, the oscillator description is a good approximation to a more rigorous amplifier approach, but both describe only the initial stage of the conjugate wave development due to the saturation of gain. The experiments indicate the existence of a clearly defined threshold, the same as in the oscillator model. Due to the big effective interaction lengths in

544

N.A. Korneev, S.L. Sochava /Optics

the device with a small angle between beams, even a moderate shift from the generation threshold yd = 2 leads to a behavior with prominent generator features. But near the threshold the nonuniformity of the conjugate beams, as well as the dependence of their amplitude on the interaction angle and the beam width characteristic of the amplifier, are clearly seen.

Communications

I I5 (I 995) 539-544

121 S. Weiss, S. Stemklar and B. Fischer, Optics Lett. 12 (1987) 114. [3] A.A. Zozulya, OpticsLen.

16 (1991) 545.

141 N.V. Bogodaev. V.V. Eliseev, A.A. Zozulya, L.I. Ivleva, AS. Korshunov, S.S. Orlov and N.M. Polozkov, Kvantovaja Elektronika

19 ( 1992) 648.

151 M. Segev, D. Engin. A. Yariv and G.C. Valley, Optics Lett. I8 (1993) 1828. [6] M.P. Petrov, S.I. Stepanov and A.V. Khomenko, Photorefractive

Acknowledgements The authors thank S.I. Stepanov and M.P. Petrov for valuable discussions.

crystals

in coherent

[I I S. Sternklar, S. Weiss, M. Segev and B. Fischer, Optics Lett. 11 (1986) 528.

systems

(Springer,

171 M.P. Petrov, S.L. Sochava and S.I. Stepanov,

Heidelberg,

Optics Lett. I4

(1989) 284. [81 M.P. Petrov, J.D. Caulfield and E.V. Mokrushina, 1731 (1991)

References

optical

1992).

Proc. SPIE

18.

191 S.I. Stepanov, S.M. Shandarov Solid State 29 (1987) 1754.

and N.D. Hat’kov,

Sov. Phys.