Diffusion recording in photorefractive sillenite crystals: an analytical approach for engineering purposes

1 June 2000

Optics Communications 180 Ž2000. 183–190 www.elsevier.comrlocateroptcom

Diffusion recording in photorefractive sillenite crystals: an analytical approach for engineering purposes E. Shamonina a , Yi Hu a , V.P. Kamenov a , K.H. Ringhofer a,) , V.Ya. Gayvoronsky b, S.F. Nichiporko c , A.E. Zagorskiy c , N.N. Egorov c , V.V. Shepelevich c a

c

Physics Department, UniÕersity of Osnabruck, ¨ D-49069 Osnabruck, ¨ Germany b Institute of Physics, National Academy of Sciences, 252650 KieÕ, Ukraine Mozyr State Pedagogical Institute, Laboratory of Coherent Optics & Holography, 247760 Mozyr, Belarus Received 18 February 2000; received in revised form 7 April 2000; accepted 15 April 2000

Abstract Simple formulae for the two-wave mixing ŽTWM. gain and diffraction efficiency obtained in the approximation of weak coupling and strong optical activity allow us to tune the experimental setup for interferometric applications. We prove that two invariant parameters, trace and determinant of the coupling tensor, are sufficient to optimize the experimental setup. Both practically important crystal cuts, Ž110. and Ž111., can be treated in a unified manner. Fundamental TWM properties Ž908-jumps of the optimum polarization, apparent doubling of the local maxima of the diffraction efficiency in comparison with the gain, drastic changes between regimes of small and large polarization rotation, the crucial role of elastooptic properties. are systematized. The guidelines for the design of an optimum experimental setup for interferometric applications including anisotropic self-diffraction using sillenite crystals are presented. Quantitative agreement with previous experimental data is demonstrated. q 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Sillenites; Photorefractive; Piezoelectric; Elastooptic; Optical activity

In the past, numerous studies have shown that cubic sillenite crystals are important for interferometric applications w1x. These photorefractive crystals are optically active and piezoelectric. Crucial characteristics for interferometry are gain and diffraction efficiency which are to be calculated by solving the coupled wave equations and material

) Corresponding author: Tel.: q49 541 969 2685, fax: q49 541 969 2351, e-mail: [email protected]

equations simultaneously w2x. It is laborious to solve the entire set of equations numerically but it is equally laborious to derive the complicated analytical solutions when taking all conceivable effects into account. For the practical aims of optical engineering both approaches are of little use; a set of simple analytical expressions is imperative. We fill in this gap and derive easy-to-use expressions both for gain and diffraction efficiency in dependence on the experimental parameters Žcrystal cut and thickness, grating vector orientation, light polarization and in-

0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 7 1 5 - X

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

184

tensity ratio, etc.. and on the crystal properties Žoptical activity, elastooptic parameters, etc... Gain and diffraction efficiency depend on the six elements of the coupling tensor w3x. We prove, however, that knowledge of only two invariant parameters, trace and determinant of the coupling tensor, is sufficient to optimize the experimental setup. Both practically important crystal cuts, Ž110. and Ž111., can be treated in the same unified manner. Fundamental TWM properties Ž908-jumps of the optimum polarization, apparent doubling of the local maxima of the diffraction efficiency in comparison with the gain, drastic differences between the regimes of small and large polarization rotation, the crucial role of elastooptic properties. are systematized. The guidelines for designing an optimum experimental setup for interferometric applications of sillenite crystals are presented. In TWM experiments the gain, G, andror the diffraction efficiency, h , is of interest w4x. Both are to be derived from the same system of coupled-wave equations Žsee Appendix A. and are defined as G s IsrIs0 y 1, h s IsrIp0 . Here Is is the signal intensity 0 in the presence of the pump, Is,p the signal Žpump. intensity in the absence of the pump Žsignal., all values being measured behind the crystal so that the optical absorption does not influence w5x G and h 1. We aim at interferometric applications using diffusion recording. The solution of the TWM problem is obtained within the framework of two standard approximations justified under diffusion recording: Ža. strong optical activity and Žb. weak coupling Žsee Appendix A.. For linearly polarized light we get GsG d

2b

bq1

hs Ž G d.

2

Ž b q 1.

2

H02 q t 2 Ž H12 q H32 .

q2t H0 Ž H1 Sw q H3 Cw . .

G max s G d

2b

bq1

h max s Ž G d .

H0 q
b

2

Ž b q 1.

Ž 1.

Here t s sinŽ D d .rD d, H0 s Ž H x x q H y y .r2, H1 s H x y , H3 s Ž H x x y H y y .r2, Sw s sinŽ2 w y D d ., Cw s cosŽ2 w y D d . with initial polarization angle w

< H0 < q
2

Ž 2.

(

H s H12 q H32 . The corresponding polarization angles are given by

sin Ž 2 wGmax y D d . s

H1

sin Ž 2 whmax y D d . s

sign Ž t . ,

H H3

cos Ž 2 wGmax y D d . s

H H1 H

sign Ž t . ,

sign Ž t H0 . ,

H3 H

sign Ž t H0 . ,

Ž 3.

except for the special cases t s 0, where both wGmax and whmax are arbitrary, and H0 s 0, where whmax is arbitrary. For circularly Žleft- or right-handed. polarized light we obtain

G circ s G d 1 Note that the similarity of the definitions of G and h is 0 deceptive. The initial conditions for the gain are Is,p / 0, while for the diffraction efficiency Is0 s 0.

2

with

cos Ž 2 whmax y D d . s

H0 q t Ž H1 Sw q H3 Cw . ,

b

defined as the angle between the vector e x and the polarization plane. The tensor Hˆ with dimensionless components characterizes the photo-induced change of the dielectric permittivity, Deˆ Žsee Appendix A.. Obvious optimization with respect to the initial polarization angle w leads to

2b

bq1

h circ s Ž G d .

2

H0 ,

b

Ž b q 1.

2

H02 q t 2 H 2 .

Ž 4.

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

Note that H0 and H are built from the trace, Tr Hˆ 4 , and the determinant, D Hˆ 4 , of the horizontal part of the tensor Hˆ Žwith indices x and y only., 2 H0 s Tr  Hˆ 4 ,

(

2 H s Tr 2  Hˆ 4 y 4D  Hˆ 4 .

Ž 5.

Having obtained the main results, let us discuss the properties of Eqs. Ž1. – Ž4. and illustrate them by examples. The following discussion is general and applicable to any crystal cut Ždefined by the propagation axis z .. We shall analyze the role of crystal thickness d and grating vector orientation. Without loss of generality the latter is defined by the angle j between e x and K. The reader is referred to the prominent publications w6–9x in order to convince him- or herself that our formulae are indeed most simple. Furthermore, our expressions include only experimentally relevant optical and material parameters Žlight polarization and intensity ratio, crystal cut and thickness, grating vector orientation, rotatory power, trap concentration, elastooptic parameters.. Shamonina et al. w10x have used a similar solution for the gain only and only for Ž110.-cut crystals. Recently Kamenov et al. w5x pointed out that the form of this solution for the gain is valid for any crystal cut. The salient feature of this paper is the introduction of the parameters H0 and H. Historically, the 6 matrix elements of the coupling matrix Hˆ w11x are the starting point for the optimization procedure. Within the paraxial approximation Žthe light waves propagates under small angles to the z axis. only three of the coupling tensor elements, H x x , H x y , and H y y , survive w9x. These three tensor elements depend not only on the propagation direction and on the orientation of the grating vector; for a given propagation direction Ž z axis., the choice of the x and y axis also affect the values of Hi j w9x. This non-physical and artificial feature disappears if we utilize H0 and H instead. Both are invariant with respect to the choice of the x and y axis Žthat is with respect to rotation around the propagation direction z .. This makes our formulae conveniently applicable to any experimental situation because any convenient choice of the coordinate system is allowed. After these general remarks, let us compare the behaviour of the gain and the diffraction efficiency.

185

The special case t s 0. Gain as well as diffraction efficiency become independent of the initial light polarization for crystal thicknesses such that t s 0; and t becomes zero whenever the polarization plane performs a half-integer number of rotations inside the crystal Ž D d s 1808,3608, . . . .. In this case, all possible polarization states contribute to the coupling strength so that the total result does not depend on

Fig. 1. Ž111.-cut. Ža. h ma x Ž j . and Žb. G ma x Ž j . in polar coordinates in BSO for ds 2.1 mm Ždashed lines. and ds8.2 mm Žsolid lines.. Experimental data Žsquares. for ds 2.1 mm from Ref. w24x are fitted with Nt s1.3P10 21 my3 , K s 4 mmy1 , b s11. The other parameters are given in Table 1.

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

186

the initial polarization. One finds that D d s 1808 in BSO at d , 5 mm Ž l s 515 nm. or at d , 8 mm Ž l s 633 nm. and in BTO at d , 16 mm Ž l s 515 nm. or at d , 29 mm Ž l s 633 nm.. Such a situation is quite possible for bulk BSO crystals, but is hardly achievable even in fiber-like BTO crystals w9x. The special case H0 s 0. The diffraction efficiency becomes independent of the initial polarization for such grating orientations that H0 s 0. It turns out that for both experimentally relevant crystal cuts, Ž110. and Ž111., H0 becomes zero for the grating vector K I w110x and for crystallographically equivalent orientations Že.g. K I w101x etc... It is worth mentioning that orientations with H0 s 0 provide anisotropic diffraction Žrotation of the polarization plane during diffraction by 908.. For such orientations h A sin2D d. For any given crystal cut, after optimizing with respect to the polarization angle w , we obtain the optimized functions G max and h max of the crystal thickness d and of the grating vector orientation Ždefined by the angle j between the x-axis and K . which can be thought of as surfaces in three dimensions. Whenever h Žor G . is independent of the initial polarization the surface for the maximum h max Žor G max . must touch the corresponding surface for the minimum h min Žor G min .. The optimum polarization undergoes a 908-jump at such points. We emphasize once again that for orientations K I w110x the 908-jumps of whmax take place, while for thicknesses D d s 1808,3608, . . . the 908-jumps of both whmax and wGmax take place. Thus Eqs. Ž1. – Ž3. summarize and explain sporadic observations of 908-jumps for optimum polarization by change of the thickness or crystal orientation w10,12x. At first glance at Eqs. 2 one might conclude that, for any given crystal thickness, there are twice as many maxima for the optimum diffraction efficiency

as for the optimum gain, since the expression for the diffraction efficiency contains only the absolute value of H0 . This property has even been emphasized in an earlier Ref. w12x. Recalling the definitions for G and h one can, however, hardly explain why G should only have half the number of maxima and, of course, it does not. The simple explanation is that, by definition, for the gain we have chosen a preference direction for the energy transfer, namely from the pump to the signal. For orientations with H0 ) 0 this makes sense. But for H0 - 0 the energy transfer is more effective in the opposite direction, from the signal to the pump. Consequently, we should look not for the maximum gain but for the minimum gain, which is then negative. It can be seen that H0 Ž K . s yH0 ŽyK . and therefore the solution for the minimum gain can be obtained from that for the maximum gain by changing the sign of the gain and of the grating vector and by letting the polarization undergo a 908-jump: G min Ž K . s yG max ŽyK ., wGmin Ž K . s wGmax Ž K . q 908. Eqs. Ž2. prompt to distinguish between two limiting cases: small polarization rotation D d < 1 Žt 1. and large polarization rotation D d 1808 Žt 0.. For small D d, the optimum gain and the optimum diffraction efficiency are defined by both H0 and H, while for D d s 1808 only H0 contributes. At least for the cuts Ž110. and Ž111. and for all known sets of elastooptic and electrooptic parameters of BSO and BTO crystals, H G 0.5, while < H0 < F 0.7. Therefore, the orientation dependence h max Ž j . is much less pronounced for D d < 1 than for D d s 1808. Also the orientation dependence G max Ž j . is much less pronounced for D d < 1 than for D d s 1808 for orientations with H0 ) 0. The elastooptic and piezoelectric parameters strongly affect the coupling coefficients w3,6,9, 10,13–20x. For sillenites, there are at least eight

™™

™

Table 1 E4 E4 , p1,2,34 '  p11 , p12 q p13 , p44 . Note that Optical and material parameters of BTO and BSO Ž l s 515 nm. w4,7,13x. c1,2,34 '  c11 ,c12 ,c 44 plotting the figures in this paper we put p12 s p13 for simplicity. While H0 is independent of p12 y p13 , H could modify if p12 / p13 . Up to now only the sum p12 q p13 is available from holographic measurements on Ž110.-cut crystals.

BTO-1 BTO-2 BSO

n0

e

D Ž8rmm.

S r41 ŽpmrV.

c1,2,34 Ž10 10 Nrm2 .

p1,2,34

e14 ŽCrm2 .

2.58 2.58 2.54

47 47 56

6.3 6.3 22

4.75 4.75 5.0

12.5, 2.75, 2.42 13.7, 2.8, 2.6 12.96, 2.99, 2.45

y0.055, 0.295, 0.0035 0.173, y0.003, y0.005 0.16, 0.25, 0.015

1.1 1.1 1.12

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

parameters which must be known additionally to the electrooptic constant: the elastooptic constants e14 , E E Ž p 11 , p 12 , p 13 , p44 , c11 , c12 , c 44 see Appendix A.. Since at least two contradictory sets of pi j for BTO can be found in the literature w7,13x, we use both to analyze the role of the elastooptic contribution in the wave coupling. To illustrate the above discussion we restrict ourselves to the less investigated Ž111.-cut which presently attracts the attention of the researchers w21–24x. The role of the elastooptic parameters which has been recognized in the past few years becomes especially important for this cut. Shepelevich et al. w24x were the first to demonstrate that for this cut the orientation dependence of the optimum gain is solely due to the elastooptic contribution. Fig. 1a shows the dependence of the optimum diffraction efficiency on the grating vector orientation h max Ž j . plotted in polar coordinates Žangle j versus radius h .. Without loss of generality j is defined as the angle between the w112x-axis and K. Fig. 1b shows G max Ž j . plotted in polar coordinates Žangle j versus radius G .. BSO parameters Žsee Appendix A. and two crystal thicknesses, d s 2.1 and 8.2 mm, are used. The dashed lines correspond to the limiting case of small polarization rotation Ž d s 2.1 mm, t , 0.9. and the solid lines correspond to the case of 1808-rotation of the polarization Ž d s 8.2 mm, t s 0.. Only positive values of the optimum gain are shown in Fig. 1b. Experimental points in Fig. 1b taken from Ref. w24x are well fitted by the theoretical curve for d s 2.1 mm. The apparent doubling of the maxima for h compared with G is now readily understood: Fig. 1a reflects an apparent ‘6fold’ symmetry of the w111x-axis; but there is the 3-fold symmetry for the energy transmission from pump to signal ŽFig. 1b. and the 3-fold symmetry for the opposite energy transmission. The maximum diffraction efficiency is achieved for K I w112x and equivalent orientations, the minimum corresponds to K I w110x and its equivalents Ž H0 s 0.. For d s 8.2 mm, the diffraction efficiency is much more sensitive to the grating vector orientation than for d s 2.1 mm. This might be undesirable for interferometric applications. Fig. 2 shows h max Ž j . for two sets of BTO parameters. The huge difference between Fig. 1a and Fig. 1b illustrates the drastic influence of the al-

187

Fig. 2. Ž111.-cut. h ma x Ž j . for ds 2,4,6 and 8 mm in Ža. BTO-1, Žb. BTO-2. Nt s10 22 my3 , Kr D s1, b s10, the other parameters are given in Table 1. Note that without the elastooptic contributions h ma x and G ma x are independent of the K-orientation: ‘flowers’ with 6 Žor 3. petals would transform here and in Fig. 1 into circles!

legedly small elastooptic contribution in BTO and proves that it is necessary to know the correct set of elastooptic parameters. Switching off the elastooptic contribution would take away any orientational dependence of G max and h max . The ‘flowers’ with 6 Žor 3. petals in Figs. 1 and 2 would transform then into circles! The reason

188

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

is that, without the elastooptic contribution, the invariants H0 and H are constant: H0 ' 0 and H ' 2r3 , 0.816 Žsee also Appendix A.. The analysis of the expressions for circularly polarized light runs along the same lines. In particular, for such orientations that H0 s 0 the gain for circularly polarized light vanishes and the diffraction efficiency oscillates with the thickness, h A sin2D d. For such thickness that D d s 1808 neither G nor h depend on the initial polarization and the expressions for circular and linear polarizations coincide. A special experimental situation may be achieved in BSO crystals of thickness d s 5 mm Žat l s 515 nm. or d s 8.2 mm Žat l s 633 nm.. For the grating vector K I w110x Žor any other crystallographically equivalent orientation. both the gain and the diffraction efficiency vanish at the crystal output although there is energy transfer inside the crystal. On the other hand, for half of this thickness the anisotropic diffraction reaches its maximum, h max s Ž H GrD . 2 brŽ b q 1. 2 . This is of interest for interferometric applications. In conclusion, simple formulae for h and G using linear and circular light polarization allow us to optimize the experimental setup for a sillenite crystal of any cut and thickness and explain fundamental properties of two-wave mixing in such a crystal.

'

Under diffusion recording the coupled wave equations for slowly varying amplitudes in the stationary state have the form: d Es dz d Ep dz

s i Dsˆ 2 Es q i g ) Hˆ Ž n . Ep , s i Dsˆ 2 Ep q i g Hˆ Ž n . Es .

Ž A.1 .

Here D is the rotatory power, sˆj the j component of the vector of Pauli matrices, Hˆ the 2 = 2 coupling tensor, and n s KrK the direction of the grating vector. The coupling constant, g, is proportional to the space-charge field, E sc : gsi G

m sy

S p n30 r41

2

l

S p n30 r41

Kk B Tre

E sc

Ž A.2 .

with

Gs

l

E sc s yi

1 q Ž Kr D . Kk B Tre

2

,

ms

2 Es) P Ep < Es < 2 q < Ep < 2

Es) P Ep

2 2 2 1 q Ž Kr D . < Es < q < Ep <

,

.

S Here n 0 is the refractive index, r41 the clamped electrooptic coefficient, l the wavelength, r D s

Acknowledgements We thank M. Shamonin for helpful discussions and constructive criticism. We acknowledge the financial support by the Deutsche Forschungsgemeinschaft ŽEmmy-Noether-Programm, Graduiertenkolleg ‘Mikrostruktur oxidischer Kristalle’, Sonderforschungsbereich 225..

Appendix A We describe the interaction of a pump and a signal wave in a sillenite crystal. In the framework of the paraxial approximation both waves are supposed to propagate at small angles to the z-direction so that the wave amplitudes Ep and Es have essentially only x- and y-components.

Fig. 3. Ž111.-cut. H0 Ž j . and H Ž j . for BSO Žsolid line., BTO-1 Ždashed., BTO-2 Ždashed-dotted. and without the elastooptic contribution Ždotted.. Only one period w0,1208x of the 3-fold Ž111.-axis is shown. j is the angle between the w112x-axis and K. Parameters are given in Table 1.

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

189

Table 2 Relationship between j and n s KrK for the Ž111.-cut

j n

08 w112x

308 w011x

608 w121x

908 w110x

1208 w211x

1508 w011x

Ž e e 0 k B TrNt e 2 . 1 r 2 the Debye radius, k B Boltzmann’s constant, T the temperature, e the electron charge, ee the dielectric permittivity, and Nt the effective trap density. We use two standard approximations w9x, that of strong optical activity Ž gr ˜ D < 1. and that of a weak coupling Ž gd ˜ < 1., where g˜ s bGrŽ b q 1. with b s < Ep < 2r< Es < 2 being the beam intensity ratio. The dimensionless components of the coupling tensor Hˆ are expressed by the components of the photoinduced change of the inverse tensor of the dielectric permittivity Deˆy1 :

1808 w112x

2108 w101x

2408 w121x

2708 w110x

3008 w211x

3308 w101x

The invariants of the coupling tensor important for optimization of the experimental setup, H0 and H s < H < are shown in Fig. 3 for the Ž111.-cut using four sets of parameters: the standard set of BSO parameters, two contradictory sets of BTO parameters, and another set without the elastooptic contribution. Note that without the elastooptic contribution H0 ' 0 and H ' 2r3 . Table 2 gives an overview of both types of crystallographically equivalent orientations of K for the Ž111.-cut. < H0 < is maximum for n I w112x and its equivalents. H0 s 0 for n I w110x and its equivalents.

'

S < sc < Hˆ Ž n . s Deˆy1 r Ž r41 E ..

In crystallographic coordinates the tensor Deˆy1 has the form w3,13x: S E < sc < , Dey1 m n s Ž r m n p q pm n k l g k i e p i j n l n j . n p E

Ž A.3 .

where rmS n p , pmE n k l , and e p i j are the components of the linear electrooptic, photoelastic, and piezoelectric tensors, respectively. The g k i are the components of the tensor inverse to the tensor with components Gi k s CiEjk l n j n l , where the CiEjk l are the elastic modules and n j are the components of the unit vector KrK. For a better understanding of the physical meaning of the coupling tensor invariants H0 and H defined by Eq. Ž5., Hˆ Žlike any 2 = 2 matrix. may be expressed w9x by the unit matrix 1ˆ and by the vectorial operator sˆ , Hˆ s H0 1ˆ q H P sˆ ,

Ž A.4 .

with the components sˆj being the Pauli matrices,

sˆ s Ž sˆ 1 , sˆ 2 , sˆ 3 . s

0 1

1 , 0

žž / ž

0 i

yi , 0

/ ž

1 0

0 y1

//

.

Ž A.5 .

References w1x M.P. Georges, P.L. Lemaire, Appl. Phys. B 68 Ž1999. 1073, and references cited therein. w2x L. Solymar, D.J. Webb, A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Clarendon Press, Oxford, 1996. w3x A.A. Izvanov, A.E. Mandel, N.D. Khat’kov, S.M. Shandarov, Optoelectronics 2 Ž1986. 80. w4x M.P. Petrov, S.I. Stepanov, A.V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Springer Series in Optical Sciences, Springer, Heidelberg, 1991. w5x V.P. Kamenov, Y. Hu, E. Shamonina, K.H. Ringhofer, Phys. Rev. E Ž1999., to be published. w6x V.V. Shepelevich, S.M. Shandarov, A.E. Mandel, Ferroelectrics 110 Ž1990. 235. w7x S.M. Shandarov, A. Reshet’ko, A.A. Emelyanov, O. Kobozev, M. Krause, Y.F. Kargin, V.V. Volkov, Nonlinear Optics of Low-Dimensional Structures, New Materials, SPIE Proc. 2969 Ž1996. 202. w8x V.V. Shepelevich, Opt. Spectrosc. 83 Ž1997. 161. w9x B.I. Sturman, E.V. Podivilov, K.H. Ringhofer, E. Shamonina, V.P. Kamenov, E. Nippolainen, V.V. Prokofiev, A.A. Kamshilin, Phys. Rev. E 60 Ž1999. 3332. w10x E. Shamonina, V. Kamenov, K.H. Ringhofer, G. Cedilnik, A. Kießling, R. Kowarschik, J. Opt. Soc. Am. B 15 Ž1998. 2552. w11x S.M. Shandarov, V.V. Shepelevich, N.D. Khat’kov, Opt. Spectrosc. 70 Ž1991. 627.

190

E. Shamonina et al.r Optics Communications 180 (2000) 183–190

w12x V.V. Shepelevich, N.N. Egorov, P.I. Ropot, P.P. Khomutovskiy, Optical Organic, Semiconductor Inorganic Materials, SPIE Proc. 2968 Ž1996. 301. w13x S. Stepanov, S.M. Shandarov, N.D. Khat’kov, Sov. Phys. Sol. State 29 Ž1987. 1754. w14x P. Gunter, M. Zgonik, Opt. Lett. 16 Ž1991. 1826. ¨ w15x G. Pauliat, P. Mathey, G. Roosen, J. Opt. Soc. Am. B 8 Ž1991. 1942. w16x N.V. Kukhtarev, T.I. Semenec, P. Hribek, Ferroelectrics Lett. 13 Ž1991. 29. w17x S. Shandarov, Appl. Phys. A 55 Ž1992. 91. w18x E. Anastassakis, IEEE J. Quant. El. 29 Ž1993. 2239.

w19x G. Montemezzani, A.A. Zozulya, L. Czaia, D.Z. Anderson, M. Zgonik, P. Gunter, Phys. Rev. A 52 Ž1995. 1791. ¨ w20x H.C. Ellin, L. Solymar, Opt. Commun. 130 Ž1996. 85. w21x N. Kukhtarev, B.S. Chen, P. Venkateswarlu, G. Salamo, M. Klein, Opt. Commun. 104 Ž1993. 23. w22x B. Sugg, F. Kahmann, R.A. Rupp, P. Delaye, G. Roosen, Opt. Commun. 102 Ž1993. 6. w23x H.J. Eichler, Y. Ding, B. Smandek, Phys. Rev. A 52 Ž1995. 2411. w24x V.V. Shepelevich, S.F. Nichiporko, A.E. Zagorskiy, N.N. Egorov, Y. Hu, K.H. Ringhofer, E. Shamonina, OSA TOPS 27 Ž1999. 353.