SBS wave front reversal for the depolarized light — Theory and experiment

Volume 27, number 1

OPTICS COMMUNICATIONS

October 1978

SBS WAVE FRONT REVERSAL FOR THE DEPOLARIZED LIGHT - THEORY AND EXPERIMENT

V.N. BLASCHUK, V.N. KRASHENINNIKOV, N.A. MELNIKOV, N.F. PILIPETSKY, V.V. RAGULSKY, V.V. SHKUNOV and B.Ya. ZEL'DOVICH *

Institute of Mechanics Problems, USSR Academy of Sciences, 11726 Moscow, USSR Received 27 July 1978

Theoretical results concerning the wave front reversal (WFR) of spatially depolarized light in stimulated scattering are presented. The modes of scattering medium are found and the deterioration of reversal quality for the depolarized light is predicted. Experimental investigation of WFR in backward stimulated Brillouin scattering (SBS) in acetone was carried out when the light beam with h = 1.06 ~tm was depolarized by the etched calcite plate. The share of scattered energy which has exactly reversed configuration, was measured. Experimental results quantitatively coincide with the theoretical conclusions.

I. Introduction

The phenomenon of wave front reversal (WFR) in stimulated scattering of light revealed in ref. [1] (see also refs. [ 2 - 6 ] ) consists in the following. Under certain conditions of backward scattering of spatially nonuniform pump E(r, z), the field of scattered wave Es(r, z) appears to be complex conjugated with respect to pumping field:

Es(r, z) = const. E*(r, z)

(1)

The solution in the form (1) for Es(r, z) exp(-icot + iksz ) may be called "time-reversed" pump field and corresponds to wave front reversal. Interpretation of this phenomenon [ 1] is the following. When a local gain distribution is spatially non-uniform, g(r, z)= G IE(r, z)l 2, the most effective amplification has the scattered wave configuration Es(r, z) for which the local intensity maxima coincide with the pump ones. In the case of backward scattering, for such a coincidence throughout the whole interaction volume, in spite of both pump and Stokes waves diffraction, the Stokes wave has to correspond to time-reversed pump wave (1). Detailed theory [5] shows that the "reproducting" mode of scattered field of the form (1) has the doubled gain grep = 2 (71with respect to the gain * P.N. Lebedev Physical Institute, Moscow, USSR.

GI of the rest (unreproducting) modes. Here I = (IE 12>is the pump intensity averaged over the crosssection. Very high full amplification in stimulated scattering (typically exp {gz } ~ e 25 '~ 1011) results in the overwhelming intensity of the reproducting mode at the output of the cell. Recently very tempting perspectives of WFR applications for the correction of radiation divergency of powerful laser systems (see the experiment [2]) and for the autotargeting of laser beams (see [6] and first experiment [7]) were discussed. However Nd-glass powerful amplifiers introduce into a beam not only phase distortions, but spatially non-uniform variations of polarizations state too owing to the induced birefringence. But all previous theoretical and experimental works [1-7] treated fully polarized beams only.

2. Theory The results of a detailed theory of the WFR for a depolarized pump, developed in [8], are the following. Let us present the pump vector field E(r, z) in the form

E(r, z) = R 1 (r, z) e 1 + R2 (r, z) e 2 ,

(2)

where e 1 and e 2 are complex polarization orthogonalen, (el, e~) = 0, which were choosen so that the 137

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functions R 1 (r, z) and R 2 (r, z) became orthogonal (in the sense of integration over the cross-section) add statistically independent. Denote I 1 = ([R 112), 12 = (1R212),I=I1 +12,11 i>12. Then the pump polarization degree p (the magnitude of normalized Stokes vector) is given by the expression P = (I1 - I2)/(I1 + •2)"

(3)

It was found [8], that four reproducing modes of scattering medium M 1 - M4 exist with effective gains gl - g 4 (cm-1):

October 1978

presents the WFR for the more powerful (R 1 (r)) of two independent pump components R 1 and R 2 with reproduction (without conjugation) of polarization vector e I . Within the framework of linear theory considered, that is without taking into account pump saturation, just this mode M 3 (r) should be presented in scattered radiation (unreproducting modes again possess two times smaller gain). If the pump is entirely depolarized, p = 0, all the three modes M2, M 3 and M 4 have the same gain. They are excited by independent spontaneous sources and hence have fluctuatihg relative phases and magnitudes.

M 1 (r) = R~(r)e 2 - R~(r)e 1 = [E*(r) × e l , M2(r)=R~(r)e2(1 - p ) + R ~ ( r ) e l ( 1 M 3(r)= R ~ ( r ) e l , gl = O,

g2 = GI,

+p),

(4)

M4(r)=R~(r)e2, g3,4 = GI(1 +-p).

(5)

Besides that, there are a multitude of unreproducting modes with the polarization e 1 and the gain g5 = GI 1 = GI × ½(1 + p) as well as with the polarization e 2 and gain g6 = GI2 = GI × ½(1 - p). Fig. 1 shows the dependencies of gains g2 - g 6 versus the polarization degree p for the fixed total pump intensity. Ifp > O, i.e. for a partially polarized pump, the highest gain corresponds to the mode M 3 (r) which re26J ~3

.}sI 6I

I el

0

_7,-02

[ o~

qa

o,i

I

to

Fig. 1. Gain coefficients for reproducting (gl - g 4 ) and unreproducting (g5, g6) modes versus the pump polarization degree p. 138

3. Experiment The experimental scheme is shown on fig. 2. Ndglass laser linearly polarized radiation (X = 1.06 tam, pulse duration on halfwidth = 17 ns) with diffractional divergency was directed through the square 5 × 5 mm 2 aperture A (maximum intensity after the aperture was "~3.5 MW) on the device DP, which induced spatially non-uniform distribution of the field and its polarization state in the beam. After the depolarizer DP the radiation was focussed by the lens f = 17 cm in the cell with aceton, where backward SBS was excited. The depolarizer was made of a calcite plate etched in azotic acid, so that the optical axis was parallel to the plate surfaces. Etching produced holes with lateral dimension ~250/am and with depth "~8/am, what was sufficient to create path difference A ~> X for orthogonal polarizations (n o - n e = 0.16). The plate was immersed into a liquid with a refractive index n = 0.5(n o + ne) to obtain the same divergencies for both polarizations. In our experiment the optical axis of DP was directed vertically. To vary the pump polarization degree p within the cell, the input laser beam polarization had been turned with respect to the plate axis by the Faraday cell F and settled exactly by the polarizer P1 on the angle ~o (~0~<45°). The beam polarization degree p according to (3) equals to p = Icos 2~1. Backward scattered radiation passed through the depolarizer and was directed on the registering system. The energy and angular distribution were measured simultaneously for both polarizations (vertical and horizontal). This has been ensured by settling the polarizer P2 and Brewster plate B. Calorimeters C 1 and C2 with large aperture measured

October 1978

OPTICS COMMUNICATIONS

Volume 27, number 1

" '-"

........ !.-v~ • ....

.....

r

-

%

',~

®

b:___

/-

-IJ

c,

V'-, I ------=--~----~-'

) Fig. 2. The experimental set-up. the whole energy of backward scattered radiation for each polarization. Calorimeter C 3 registered the energy of incident light. In our experiments the energy of the scattered radiation was 15-20% of the incident light energy. The system of angular distribution registration included the calcite wedge W 1 and step-by-step attenuator W2 (see [3]), consisting of two mirrors with R = 70% turned at a small angle ~10 -2 rad to each other. On the photoplate placed in the focal plane of the lens I.2 with f = 80 cm, four series of gradually attenuated spots were registered, which corresponded to the angular distribution of the laser and scattered radiation for both polarizations. Fig. 3 shows the angular distributions for both pump polarizations before (3a, 3b) and after (3c, 3d) the depolarizer. Typical angular distributions of scattered radiation, which were reconstructed up to the diffractional quality by the backward passage through the uniform plate DP, for vertical and horizontal polarizations, are presented respectively on fig. 3e, f. It is well known [ 1], that such a a reconstruction by the passage through the phase object means wave front reversal (in this case for each polarization). It is convenient to introduce the "share of reproduction", i.e. the part of the scattered beam energy, that belongs to the component with exactly reversed wave front. A photographic method does not allow the reveal the halo with weak brightness around the reconstructed spot of diffraction quality in scattered

Fig. 3. Far-field angular distributions for the next beams: diffraction quality spots for the pump with vertical and horizontal polarizations before depolarizer DP - 3a, 3b; the same pump components after DP - 3c, 3d; reconstructed SBS radiation for the same polarizational components - 3e, f. radiation, although such halo could in principle contain considerable portion of scattered energy. Therefore to measure the share of reproduction quantitatively, the photographic data on the pump and recon139

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structed scattered field brightness were compared with calorimetric measurements for both polarizations. On fig. 4a the solid lines represent the "shares of reproduction" 771 and 7/2 defined as r/1 = e e,

r/2 = e r / e ,

(6)

where e is the total (over the angles and polarizations) scattered energy, er~ is the energy in the reconstructed diffraction quality spot for the vertical polarization, er is the same for the horizontal one. The pointed curve stands for r/3 = 1 - r/1 - r/2 and it characterizes

® i,o

October 1978

the part of the energy in the halo. To measure the threshold pump intensity corresponding to stationary gain, the oscillographic scheme with photocells S 1 and S 2 were assembled (see [3]). Photocell S 3 registered the shape of light pulse. Fig. 4b shows the inverse value of stationary threshold intensity Ith as a function of polarization degree p. To study specific features of the process with entirely depolarized pump (p = 0), the following additional experiment has been carried out. The optical axis of depolarizer DP was turned on the angle 45 ° with respect to the vertical, but the laser beam polarization was kept vertical. In this experiment we observed the spots of the scattered field, reconstructed up to diffractional quality, for both the horizontal and the vertical polarizations. However large fluctuations of intensity ratio for these two spots (up to 10 times) from pulse to pulse were observed.

4. Discussion of the results q~

o,z "'"'*',..

,~= q2

.........

,

, ...............

P

:

q~

q¢

~g

40

® ~

0, s

4".- q6 ~"

p o,~

,

,

i

i

qz

o,~

q~

q~,

;%0

Fig. 4. a. Reproduction share rh for the vertical polarization, versus polarization degree p; ~2 - the same for the horizontal polarization; beside pointed line denotes ha. b. Gain coefficient g ~ I ~ 1. The inverse threshold intensity for entirely polarized radiation is choosen as a unit. Solid line - theoretical result for g3140

Theoretical consideration of the passage of the modes M 2, M 3 and M 4 through the depolarizer DP gives the following. Modes M 3 and M 4 yield reversed waves of diffractional quality for vertical and horizontal polarizations respectively; mode M 2 after passing the depolarizer should give the halo in both polarizations without any spot. Due to the exponential dependence of scattered intensity on the gain g, the mode M 3 having the highest gain for depolarized pump, would be dominating for all p > 0. Besides that, for p = 0 all the three modes M 2 , M 3 , M 4 have the same gain and hence, according to the theory, should be presented in the scattered wave with equal intensities, 1 so that 171 = I12 = r / 3 = g. At the same time, the value of inverse threshold intensity, being proportional to the highest gain, should, according to the theory, depend on the polarization degree as (1 + p)/2, see eq. (5). Comparison of the theoretical predictions with the experimental data allowes to come to the following conclusions: 1) For partially spatially polarized radiation full spatial-polarizational WFR does not exist. For polarization degree p ~ 0.2 the scattered radiation is presented mainly (more then 60%) by the mode M 3 which reverses the stronger pump component of polarization, in conformity with theory. 2) The gain (or the inverse threshold intensity) depends on the polarization

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degree as (1 + p)/2 in conformity with theory. 3) For an entirely depolarized pump, experimentally measured values 7/1 , r/2 and halo energy r/3 are approximately equal to I/3, as predicts the theory. 4) The experiment with depolarizer axis turned on the angle 45 ° with respect to vertical axis requires the conclusion, that reversed mode M 2,M 3 and M 4 are not linked in phase. If such a relationship existed, then the ratio of intensities of the reversed diffraction quality spots would be constant throughout all the laser shots. There exist also a set of experimental facts (too early drop oft/1 and increase of r/2 when p ~ 0), which could not be explained within the framework of the theory presented. It is possible, that their explanation would require to take into account effects of saturation and unstationarity. Thus, in the present work the laws of WFR in stimulated scattering of a depolarized pump are established theoretically and experimentally.

References

October 1978

[2] O.Yu. Nosach, V.I. Popovichev, V.V. Ragulsky and F.S. FaizuUov, Pis'ma ZhETF 16 (1972) 617. [3] V.V. Ragulsky, Trudi FIAN, Vol. 85 (Nauka, Moscow, 1976) s. 8. [4] V.I. Bespalov, A.A. Betin and G.A. Pasmanik, Pis'ma ZhETF 3 (1977) 215; V.N. Blaschuk, B.Ya. Zel'dovich, N.A. Melnikov, N.F. Pilipetsky, V.I. Popovichev and V.V. Ragulsky, Pis'ma ZhETF 3 (1977) 211; V.I. Bespalov, A.A. Betin and G.A. Pasmanik, Izvestiya VUZov, Radiofizika 20 (1977) 791; B.Ya. Zel'dovich, N.A. Melnikov, N.F. Pilipetsky and V.V. Ragulsky, Pis'ma ZhETF 25 (1977) 41. [5] V.G. Sidorovich, ZhTF 46 (1976) 2168; I.M. Bel'djugin et al., Kvantovaya electronika 3 (1976) 2467; B.Ya. Zel'dovich and V.V. Shkunov, Kvantovaya electronika 4 (1977) 1090, 2353; 5 (1978) 36; N.B. Baranova, B.Ya. Zel'dovich, V.V. Shkunov, Kvantovaya electronika 5 (1978) 973. [6] V. Vang and G.R. Giuliano, Optics Lett. 2 (1978) 4; S.D. Zakharov, preprint FIAN, N 210 (1977); Yu.I. Kruzhilin, Kvantovaya electronika 5 (1978) 625. [7] N.F. Pilipetsky, V.I. Popovichev and V.V. Ragulsky, Pis'ma ZhETF 27 (1978) 619. [8] B.Ya. Zel'dovich and V.V. Shkunov, preprint FIAN, N 1 (1978); ZhETF 75, N 2 (8) (1978).

[1] B.Ya. Zel'dovich, V.I. Popovichev, V.V. Ragulsky add F.S. FaizuUov, Pisma ZhETF 15 (1972) 160.

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