cx% . . _- __ EB 15 February 1997

OPTICS COMMUNICATIONS

ELSEVIER Optics Communications 135 (1997) 337-341

Stabilization of an externally-pumped phase conjugator by the control of mean phases of incident beams

Atsushi Nakamura, Tsutomu Shimura, Kazuo Kuroda * Institute of Industrial Science, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan

Received 9 May 1996; revised version received 22 October 1996; accepted 23 October 1996

Abstract

The performance of an externally-pumped photorefractive phase conjugator is greatly improved, especially in stability and response to signal change, by the phase control of incident beams. Using piezoelectrically-driven mirrors in all incident optical paths, the backward pump is made incoherent with the other input beams and the fluctuation of mutual mean phases between forward pump and signal beams is canceled.

Keywords: Phase conjugation; Photorefractive material; Barium titanate; Stabilization; Phase modulation

1. Introduction

Phase conjugation is one of the promising applications of nonlinear optical materials [l]. Photorefractive phase conjugators, either externally- or self-pumped (including mutually-pumped), have already been investigated widely and deeply. However, some shortcomings that should be overcome remain. Among them, the response speed and stability are the most important issues. The speed is limited by the response rate of photorefractive materials, whereas it is possible to improve the stability.

Instability in externally-pumped phase conjugators is mainly caused, or at least triggered, by the fluctuation of mutual phases between incident waves. We have to distin- guish the mean phase (constant phase over the beam cross section) from the relative phase distribution (the deviation from the mean). The mean phase only indicates the direc- tion of propagation of the light beam. On the other hand, the relative phase distribution determines the wave front of the beam. The mean phase has no significance in an ideal phase conjugator. Nothing should occur when only the mean phase changes. However, in a phase conjugator based on the four wave mixing, the change of mean phase

* Corresponding author.

induces the shift of interference fringes, which leads to rewriting the index gratings in photorefractive materials. This rewriting process sometimes triggers large fluctuation of phase conjugation. By canceling the variation of the mean phases of incident waves, we can avoid an unwanted rewriting of the gratings. The stabilization of photorefrac- tive nonlinear interaction has been carried out in two-wave mixing schemes by Shimura, Miao, Itoh, and Kuroda [2] and by Bian and Frejlich [3]. A similar technique can be applied to four wave mixing.

Another cause of the instability is the coexistence of different gratings. In the degenerate four-wave mixing, three different gratings are formed simultaneously, i.e., (1) transmission, (2) reflection and (3) 2k gratings if all the incident beams are mutually coherent. Competition be- tween these gratings makes the phase conjugator dynami- cally unstable. Furthermore, the phase conjugate wave is the coherent superposition of three beams of different paths, i.e., (1) the wave originated from the backward

pump diffracted by the transmission grating, (2) that from the forward one diffracted by the reflection grating, and (3) that from the signal beam diffracted by 2 k gratings. There- fore, the fluctuation of mean phases between incident beams directly results in the change of the phase conjugate power because of the interference. These difficulties are easily overcome by breaking the coherence between the input beams. De La Cruz, MacCormack, Feinberg, He,

0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-40 18(96)0067 l-2

338 A. Nakamura et al./ Optics Communicutions 135 (1997) 337-341

PD1

0

BS HP PBS PZM,

Fig. 1. Experimental setnp. PZM: piezoelectrically driven mirror; PD: photodetector; PBS: polarizing beam splitter; BS: beam split- ter; HP: half wave plate.

Liu, and Yeh used the phase modulation technique to control the coherence, and they investigated the influence of partial coherence on the reflectivity of mutually-pumped phase conjugators [4]. This technique is applicable to the externally-pumped phase conjugator too. If we make the backward pump incoherent with the other two incident waves, only the transmission grating is formed and the phase conjugate wave is only generated by the diffraction of the backward pump beam. As a result, we can eliminate the competition between gratings and the fluctuation due to the interference.

0.5 L ._............_......; ._.....................~ ._____._____________...~ . . . . . . . . i . . . .._.. L

O~.“~‘~““‘~“~‘.“““.~ 0 5 10 15 20 25

Time (set)

Fig. 2. Phase conjugate reflectivity when the backward pump is coherent (lower trace) and incoherent (upper trace) with the other input beams.

In this paper we demonstrate how to improve the performance of an externally-pumped phase conjugator.

2. Two-wave mixing between phase modulated signals

Before describing the experiments, here we briefly summarize the two-wave mixing between phase modulated signals in photorefractive materials. If the modulation fre- quency is much faster than the response rate of the pho- torefractive material, a stationary index grating is formed responding to the time average of interference fringes, and the oscillating component of index grating is negligible. Therefore, we can treat the two wave mixing as if the index grating is fixed.

Let a phase modulated input signal be A,(t) =

A, exp[ip sin(On)] and an unmodulated signal be Be, where fi is the modulation frequency and p is the depth

,,“I ..“I - “‘I”“]““:

. .._............... + . . . . . . . . . . . . . . . . . . . . . . . . _....________ _ __.......; . . . . j . . L

3

x 2.5 c .z

H 2

5 a , 5 . . . . . . .._.... i . .._....________._______! .__._................. .-

4 . . . . . . . . . . . . . . . . . . . . . i ..,.................... i . . . .._..._._.... i..... ._._._.___.________~ . ..__ J . . . . . . . . . . . . . . . . .._.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .._....._..... _..___. _ __.__.____.____ i . .

” ‘.I’. ” ” ” ” “.I”” 0 10 20 30 40 50

Time (set)

(b) vow

60

40

t

______..___...___...~ _..._________________ ..I __._______..._....._____~ _....___..____.____..... f ..,....___.___ _ ___.__

-901 ’ ’ , ’ ’ ’ . 1 0 10 20 30 40 50

Time (set)

Fig. 3. Traces of (a) the phase conjugate reflectivity and (b) the mutual mean phase before and after the stabilization feedback loop is turned on.

A. Nakamura et al./ Optics Communications 135 (1997) 337-341 339

of phase modulation. The output signals A,(t) and B,(t) are the superposition of two input signals. In the transmis- sion geometry, they are expressed in the form of scattering matrix [5]

A,(t) = cAO(t) + sB,, (1)

B,(t) = -s*A,(r) +c*B,.

If the absorption is negligible, then 1 c I* + 1 s I* = 1, be- cause of the conservation of energy. The output intensity of the beam A is given by

IAl= IcA,j*+ IsB,l*

+ 2 Re{ cA,s*B,* exp[ip sin( Or)]}

=&a+Inc+2~~ i J,(p)cos(nfit+4) n= --m

= Lo + 4~ + 2 J,( P)~~COS 9

- 4J,( p)dmsin 4 sin(tif) + . . . , (2)

where IA0 = I CA, I ’ is the output intensity of the beam A

just after the input beam B is blocked, Zaa = I sB, I * is the

t....,....,....,....i 0 5 10 15 20

Time[sec]

t....,,...,....,....i 0 5 10 15 20

Time[sec]

- 0 10 15 20 Time[sec] _-

output intensity of the beam B just after the input beam A is blocked, I#J is the phase difference between CA, and sB,, and J,,(p) is the Bessel function of order n which comes from the Bessel expansion of phase modulated signal. We can obtain the relative phase 4 by measuring the sin( 0 t) component. Note that Eq. (1) is general for the diffraction by the transmission grating. Even in the four- wave mixing, if only the transmission grating is formed, Eqs. (1) and (2) are still valid, although in this case, the coefficients c and s are the sum of contributions from the interference between the signal and the forward pump, and that between the backward pump and the phase conjuga- tion.

The coefficients c and s depend on the material param- eters, such as photorefractive coupling coefficient, as well as the input signals [6,7]. We assume that the phase shift between the interference fringe and the index grating due to the nonlocality is r/2. This is true when the migration of charges are mainly caused by the diffusion, and the contribution of the drift due to external field or photo- voltaic effect is negligible, such as in barium titanate. In

this case, the coupling coefficient r is real. Under this

0 5 10 15 20 Time[sec]

5 10 15 20 Time[sec]

0 5 10 15 20 Time[sec]

Fig. 4. Recovery of the phase conjugate power when the input signal is suddenly changed when the stabilization feedback loop is off.

Experiments are repeated 6 times.

of phase, and the two wave coupling vanishes. In other words, the phase modulation of p = 2.4 makes the two beams ‘incoherent’.

340 A. Nakamura et al. / Optics Communications 135 (1997) 337-341

assumption, the explicit expressions of c and s are given

by [61

1 + 6 exp( $qL)

‘= JiX?dl + 6 exp(qL) ’

2Ao%Jo( p)[exp( %) - 11 3. Elimination of reflection gratings

‘= (F+I IBo12- I A,1*1)di?+l++ exP(&) ’

(3) where

The experimental setup is shown in Fig. 1. Rh doped

BaTiO, was used for a photorefractive phase conjugator. The laser source was a frequency-doubled Nd:YAG laser. The powers of signal, forward-, and backward-pumps are denoted by I,, 12, and Ia, respectively. The power ratio is I, : I, : Z3 = 1 : 4.5 : 160 in the following experiments. Mean phases of all incident waves are modulated or controlled by mirrors attached on piezoelectric transducers PZM,- PZM,.

F-lIB,l*- l&,1*1 ‘= F+i IB,l*- I A,l*] ’ (4)

‘= lAoI*+ lB,I*’

and L is the interaction length. Since s is proportional to J,(p), at the zero points of J,(p), (p= 2.4, . ..>. the index grating is completely swept away by the modulation

“‘[ .,,,,,,..,...,,.,,, j 3 U1 a I

0 5 10 15 20 0 5 10 15 20

Time[sec] Time[sec]

3,....,....,....,....,

t....,..,.,....,....i 0 5 10 15 20 0 5 10 15 20

Time[sec] Timetsec]

In the first experiment the reflection and 2k gratings

are eliminated. We drive PZM,, located in the backward- pump path, with the sinusoidal signal of the frequency (0s = 100 Hz) to which the photorefractive BaTiO, can- not respond. The amplitude of the vibration of PZM, is adjusted to make the backward pump 1s incoherent with I, and I,. Fig. 2 shows the conjugate reflectivity in both

3 U1 ck 1

t....,....,....,....i 0 5 10 15 20 0 5 10 15 20

Time[sec] Time[sec]

Fig. 5. The phase conjugate power for the same experiment as Fig. 4 when the stabilization feedback loop is on.

A. Nakamura et al./ Optics Communications 135 (1997) 337-341 341

coherent and incoherent cases. The reflectivity increases almost 3 times when Z3 becomes incoherent with I, and Z2. This indicates that the wave from Z2 diffracted by the reflection grating interferes destructively with that from Z3 by the transmission grating. The fluctuation is also sup- pressed slightly. The standard deviation of the fluctuation is 3.9% in the free running, and is reduced to 1.9% by breaking the coherence.

4. Compensation of phase fluctuation

In the next experiment we control the mutual mean phase 4 between the signal and the forward pump. To monitor the phase 4, we vibrate PZM, with small ampli- tude ( p = 0.24) and high modulation frequency (0 = 2.26 kHz). The intensity of the signal beam transmitted through the photorefractive material was detected by the detector PD,. The sin(0f) component of the intensity is detected by a lock-in amplifier and the relative phase I$ is calcu- lated by a personal computer. Then we drive PZM, to compensate the fluctuation of mutual mean phase. The frequency of this feedback loop is 40 Hz, which is fast enough for slow response of barium titanate. In this experi- ment, PZM, is still working to sweep away the reflection grating. The result is shown in Fig. 3. We observed drastic decrease of fluctuation after turning on the stabilization feedback loop. The standard deviation decreases from 3.9% to 0.58%. It should be noted that the fluctuation increases from 1.9% to 3.9% by vibrating PZM,, which induces additional fluctuation to the system.

5. Response to sudden change of input image

Finally we measure the response to sudden change of input signal. When the input signal is suddenly changed, both mean phase and phase distribution change. Then the rewriting process starts in the photorefractive material in order to adapt the index grating to the new input signal. This rewriting process should respond to only the change of phase distribution. Although the change of mean phase affects the rewriting of the grating, the wavefront of phase conjugated wave is not involved with the mean phase. If the feedback loop is working, the change of mutual mean phase is compensated by controlling the phase of forward pump, which minimizes the rewriting of the grating.

In the experiment we used a photographic film as an input signal, and we rapidly changed it to a new one. In

this experiment, we make the backward pump incoherent with the other input beams by vibrating PZM,. Since the transmission of the second image is lower than that of the first one in this experiment, the steady state output power reduces after the input is changed to the new image. We

repeatedly measured the phase conjugate power for the sudden change of image. When the stabilization feedback loop is off, transient behavior of the phase conjugate power differs for every experiment, and it takes a longer time to converge to a steady state (Fig. 4). This is partly because of the large fluctuation of the phase conjugation without feedback, but also because the change of mean phase is not the same from one experiment to another. When the feedback is on, the phase conjugation is rapidly transferred to a new steady state (Fig. 5). The change of mean phase is compensated and the phase conjugation is always recovered with minimum delay.

6. Conclusions

In conclusion, we have achieved substantial improve- ment of the performance of an externally pumped phase conjugator by controlling both coherence and mean phases of incident waves. At least, one of the difficulties of photorefractive phase conjugators can be overcome by equipping them with a stabilization feedback loop. Since the response of barium titanate is slow, fast electronics are not required for the feedback loop. If we use a nonlinear optical material of fast response, this technique is limited by the availability of much faster feedback loop. This technique would be useful for the application of phase conjugators to precise measurements where the fluctuation of the reflectivity may add unnecessary noise to the sys- tem.

References

[l] See, for example, G. Gower and D. Proch, eds., Optical Phase

Conjugation (Springer, Berlin, 1994).

[2] T. Shimura, H.Y. Miao, M. Itoh and K. Ku&a, Optics

Comm. 87 (1992) 171. [3] S. Bian and J. Frejlich, J. Opt. Sot. Am. B 12 (199.5) 2060.

[4] S. De La Cruz, S. MacCormack, J. Feinberg, Q.B. He, H.-K.

Liu and P. Yeh, J. Opt. Sot. Am. B 12 (1995) 1363.

[5] C. Gu and P. Yeh, Optics Lett. 16 (1991) 455.

[6] C. Xie, M. Itoh, K. Kuroda and I. Ogura, Optics Comm. 82

(1991) 544.

[7] K. Tei and H. Yokota, Optics Comm. 107 (1994) 133.