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Optical activity in photorefractive Bi12TiO20
15 January 1998
Optics Communications 146 Ž1998. 62–68
Optical activity in photorefractive Bi 12TiO 20 E. Shamonina a , V.P. Kamenov a , K.H. Ringhofer a,1, G. Cedilnik b, A. Kießling b, R. Kowarschik b, D.J. Webb c a Fachbereich Physik der UniÕersitat, ¨ 49069 Osnabruck, ¨ Germany Institut fur ¨ Angewandte Optik, Friedrich-Schiller-UniÕersitat, ¨ 07743 Jena, Germany Applied Optics Group, Physics Laboratory, The UniÕersity, Canterbury, Kent CT2 7NR, UK b
c
Received 17 April 1997; revised 15 August 1997; accepted 8 September 1997
Abstract The influence of optical activity on two-wave mixing ŽTWM. in photorefractive BTO and BSO crystals in the absence of an applied field is studied both theoretically and experimentally. For the conventinal orientations of the grating vector, K 5 w001x and K H w001x, the piezoelectric and photoelastic effects are either zero or negligible. This makes an analytical treatment of the TWM problem possible. We obtain an analytical solution for the coupled wave equations of TWM valid for arbitrary optical activity. This result is of special importance for BTO crystals. In these crystals under the condition of maximum energy transfer Ž< K < r D s 1, where r D is the Debye radius. neither the approximation of small optical activity nor the one of dominating optical activity is applicable and our analytical solution becomes essential. Our experimental setup uses beams with a trapezoidal overlap that allow us to study the thickness-dependence of the gain in a single measurement. Experimental and theoretical results for a BTO crystal are compared with those for a BSO crystal and are explained in the framework of the model used. q 1998 Elsevier Science B.V. PACS: 42.40.Ht; 42.40.Lx; 42.40.Pa; 42.65.HW Keywords: Sillenites; Photorefractive; Two-wave mixing; Optical activity
1. Introduction It is known that the natural optical activity of sillenites affects the photorefractive two-wave mixing ŽTWM. in these crystals strongly w1x. Optical activity leads to rotation of the polarization planes of the interacting waves. In different layers inside the crystal the components of the effective electrooptic tensor w2x Žwhich are due to both the electrooptic and photoelastic effects. contribute differently to the rate of energy transfer between the waves. In other words, the efficiency of the energy transfer varies with the crystal depth. The investigation of the role of optical activity is particularly interesting in view of recent progress
1 Corresponding uni-osnabrueck.de.
author.
E-m ail:
ringhofe@ m ail.rz.
in the understanding of photorefractive soliton propagation w3,4x and of TWM in fiber-like crystals w5x. In the conventional orientations of the grating vector, K 5 w001x and K H w001x w6–8x, the piezoelectric and photoelastic properties of the sillenite crystals are not pronounced w9x and the coupled wave equations have a simple structure. In these orientations it is worth looking for an analytical solution of the TWM problem that makes analysis of the photorefractive process and the determination of crystal parameters easier. In the current literature two main theoretical approaches are used. In the first approach the optical activity is neglected. This is justified for the case of a strong grating w10x. This method is frequently used to obtain estimates for the optimization of the TWM process with a moving grating technique with an external dc field. However, the optical activity cannot be neglected for diffusion recording which is of large importance for inter-
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 5 1 0 - 5
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
ferometric applications in particular w1x. The second method assumes that the optical activity is dominating w8,11,12x. The propagation of light waves in the optically active medium is first calculated from optical activity alone and is then disturbed by the weak interaction between the light waves via the photorefractive grating. This approximation works sufficiently well for BSO and BGO crystals for diffusion recording. It is also applicable to BTO for almost parallel propagation of waves when, owing to the small spatial frequency, the recorded diffusion grating is weak and the optical activity prevails. In the practically important case of maximizing the TWM gain Ž< K < r D s 1., however, a discrepancy between this theory and experiment is observed even for BSO crystals. In particular, the theory predicts that there is no interaction between two waves with the same circular polarization state for K H w001x although such an interaction was observed experimentally w12x. Furthermore, the theory suggests that the dependence of the gain on the azimuth of linear polarization, w , is of cosine type, i.e. the polarization dependence is a periodic function oscillating around the zero gain value with positive and negative extrema having equal absolute value. Deviations from such a type of polarization dependence can even be seen in BSO crystals and they are much larger in BTO crystals. The structure of the paper is as follows. In Section 2 we review shortly the coupled wave equations for TWM and show that in ‘‘pure’’ electrooptic orientations K 5 w001x and K H w001x piezoelectric and photoelastic effects do not need to be taken into account. After discussing the methods described above in Section 3, we obtain, in Section 4, an analytic solution of TWM which is valid for arbitrary optical activity and both for linearly and circularly polarized light. Our results explain the experimentally observed peculiarities of beam coupling in BTO crystals. Our experimental setup uses finite aperture beams which are partly overlapping in the crystal. This allows us to observe the thickness dependence of the TWM gain in a single measurement. In Section 5 we compare the experimental and numerical results for BTO and BSO crystals and explain the observed characteristic features.
2. Coupled wave equations We consider the Ž110.-cut crystal and choose the following basis vectors: 0 exs 0 , 1
ž/
eys
1
'2
y1 y1 , 0
ž /
ez s
1
'2
1 y1 . 0
ž /
two waves is described by the following equations for the slowly varying amplitudes: EsX s i Ds 2 Es q ig ) S Ž K . Ep , EXp s i Ds 2 Ep q ig S Ž K . Es ,
Ž2.
where D is the rotatory power, g the coupling constant, and sj the j component of the vector of Pauli matrices. S Ž K . is a coupling tensor with dimensionless components which, in general, include the electrooptic, photoelastic and piezoelectric contributions w13x. For diffusion recording gsiG
ms
m
,
2
Gs
p n3 r41
Kk BTre
l
1 q Ž Kr D .
2 Es) P Ep < Es < 2 q < Ep < 2
,
2
,
Ž3.
where m is the light modulation depth, r D s Ž ee 0 k BTrNt e 2 .1r2 the Debye radius, n the refractive index, r41 the electrooptic coefficient, l the wavelength, k B Boltzmann’s constant, T the temperature, e the electron charge, ee the static dielectric permittivity, and Nt the effective trap density. For diffusion recording the coupling constant g is purely imaginary. In Eqs. Ž2. the optical absorption is neglected because it has no influence on the gain, G w12x, defined in the following way: Gs
< Es < 2 y < Es0 < 2 < Es0 < 2
,
where Es0 is the signal beam amplitude in the absence and Es in the presence of the pump beam, both values being measured behind the crystal. For the orientations K 5 w001x and K H w001x the matrix S Ž K . is 5-case:
Ss
Iˆy s 3 2
,
ž
S s s1 1 q
H -case:
e14 p44 r41 c44
/
.
Here Iˆ is the unity matrix, e14 the piezoelectric coefficient, c 44 and p44 the component of the elasticity modules and photoelastic tensor, respectively. For K 5 w001x the photoelastic effects do not contribute to the tensor S at all. For K H w001x the photoelastic effects lead to an introduction of the new effective electrooptic coefficient only:
Ž1.
The pump wave, Ep expw iŽ k p P r y v t .x, and the signal wave, Es expw iŽ k s P r y v t .x, are supposed to propagate at small angles to the z-direction, so that in our coordinate system Ž1. the amplitudes Ep and Es have only x- and y-components. In the stationary state the interaction of the
63
ž
X r41 s r41 1 q
e14 p44 r41 c 44
/
without changing the structure of the tensor S . For the parameters given in Ref. w13x the relative changes in r41 for K H w001x are 4.5% for BTO and 13.7% for BSO. Therefore we conclude that for both these orthogonal
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
64
orientations of the grating the photoelastic effects can be neglected Žsee also Ref. w9x..
3. The basic approximations and their shortcomings for the orientations K 5 [001] and K H [001] We need to estimate the competition of the two terms in Eqs. Ž2. which are due to the optical activity and the electrooptic effect only. As we mentioned above, we distinguish two cases when one of the parameters g˜ or D prevails. Here
b g˜ s
bq1
3.2. The optical actiÕity dominates: D 4 g˜ Here the equations describing the light propagation in an optically active medium Ž g s 0 in Eqs. Ž2.. are first to be solved. The solutions represent circularly polarized waves e "s Ž1r '2 .Ž1," i . w15x. The photorefractive energy exchange acts here as a small perturbation. We assume additionally that the light modulation depth can be considered as approximately constant. This is fullfilled for crystals with gd ˜ < 1 w16x. We obtain for K 5 w001x and K H w001x for circularly polarized ŽCP. and linearly polarized ŽLP. waves:
G,
with b being the beam intensity ratio. We discuss the properties of both approximations and the regions of their applicability. We assume that the two beams enter the crystal with identical polarization. For diffusion recording there is no phase coupling between the interacting waves so that initially linearly polarized light waves maintain their linear polarization during the propagation through the sample.
5-case, CP waves:
Gs
5-case, LP waves:
Gs
bG d bq1
,
Ž6.
bG d bq1 = 1y
sin D d
Dd
cos Ž 2 w y D d . ,
Ž7.
3.1. The photorefractiÕe mechanism dominates: g˜ 4 D In this case we neglect optical activity. Eqs. Ž2. contain now only the term describing the energy exchange and the optical activity does not influence the polarization state. The interacting beams change, however, their polarizations because the structure of the coupling tensor g S ensures the coupling of different components of the polarization vectors. There are, however, a few special cases when the polarization of the propagating light waves remains constant: w s 0 and w s pr2 for K 5 w001x and w s "pr4 for K H w001x, where w is the angle between the polarization vector and the x-axis. For these cases the exact solutions of the TWM problem are known w10x. They give for the two orientations: 5-case:
G Ž w s pr2 . s
1
Ž b q 1. e G d
2 e G d q b ey G d
y1
Ž4.
G Ž w s 0. s 0 H -case:
G Ž w s "pr4. s
Ž b q 1. e " G d e" G d q b e. G d
y 1.
Ž5.
In general, the polarization angle determines the efficiency and, for K H w001x, also the direction of the energy exchange. The numerical analysis of the coupled equations shows that the special solutions given above in both orientations correspond to the maximum and minimum gain. It follows from Eqs. Ž4., Ž5. that in the undepleted pump approximation Ž m < 1. the gain grows Žor decays. exponentially with the crystal thickness. We note that in practice Eqs. Ž4., Ž5. can be used if g˜ 4 D and also D d < 1.
H -case, CP waves:
G s 0,
H -case, LP waves:
Gs
Ž8.
2 bG d sin D d
bq1
Dd
sin Ž 2 w y D d . .
Ž9. These results for CP waves were first obtained in Ref. w12x and for LP waves in Ref. w8x. One can see that for CP waves the gain depends linearly on the thickness d in the 5-case and is non-existant in the H -case. For LP waves the maximum gain in the 5-case and H -case is achieved for initial polarizations w s D dr2 and D dr2 q pr4, respectively. This corresponds to the polarization state in the middle of the crystal thickness, w Ž dr2. s 0 and pr4, respectively. For typical parameters of BSO Ž Kr D s 1 for Nt s 10 22 my3 , l s 514.5 nm. we have D f 628 my1 and g˜ f 35 my1 Ž b s 1. or f 70 my1 Ž b ™ `.. The ratio gr ˜ D is 0.06–0.12 and one can rely on the approximation of dominating optical activity. At the same time, for BTO Ž Kr D s 1 for Nt s 10 22 my3 , l s 514.5 nm. we obtain D f 200 my1 and g˜ f 47 my1 Ž b s 1. or f 94 my1 Ž b ™ `.. The ratio gr ˜ D is 0.24–0.48. It is clear that the optical activity cannot be neglected in BTO, but at the same time the ratio gr ˜ D is not small enough to justify the application of the approximation of dominating optical activity. In Figs. 1a and 1b the results obtained in two approximations of small and large optical activity are compared with the numerical solutions of Eqs. Ž2. for BSO and BTO,
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
65
K 5 w001x and K H w001x for circularly and linearly polarized waves: 5-case, CP waves:
bG d
Gs
bq1 q
1 b Ž b y 1. 4 Ž b q 1.
2
G
ž / k
2
sin2k d,
Ž 10. 5-case, LP waves:
bG d
Gs
bq1
1y
sin k d
kd
D
ž
= cos k dcos2 w q
q
1 b Ž b y 1. 4 Ž b q 1.
2
H -case, CP waves:
k
/
sin k dsin2 w 2
G
sin2k d,
ž / k
Gs
Ž 11.
b Ž b y 1.
Ž b q 1. 2
G
ž / k
2
sin2k d,
Ž 12. Fig. 1. Dependence of the gain on the crystal Ža. and BTO Žb.: negligible optical activity perturbation theory for large optical activity numerical solution Žsolid line..
thickness for BSO Ždash-dotted line., Ždashed line., and
respectively. The orientation K H w001x and LP waves with w "s "pr4 are chosen. It is seen that for both crystals the monotonic solutions obtained by neglecting of the optical activity Žcurves ‘‘ g˜ 4 D ’’. become unapplicable for even small thicknesses, d f 1 mm in the case of BSO ŽFig. 1a. and d f 4 mm in the case of BTO ŽFig. 1b.. The approximation of large optical activity leads to solutions strictly periodic and symmetric with respect to the x-axis. For BSO the agreement of this approximation with the precise numerical solution ŽFig. 1a. is satisfactory for any crystal thickness. For BTO the agreement is satisfactory for small d only ŽFig. 1b. and the discrepancy varies strongly with the thickness. At d s 5 mm the deviation of the approximate solutions from the precise ones for w s "pr4 are 0.24 and 0.15 Žabsolute values. and 36% and 56% Žrelative values normalized to the value of gain from the exact solution..
4. A solution for arbitrary optical activity Obviously the theory developed for large D must be modified to describe BTO crystals as well. For this purpose Eqs. Ž2. have to be solved without assuming anything about the strength of the optical activity. Using the methods developed in Ref. w17x, including the approximation of the constant light modulation depth Ž gd ˜ < 1., we find for
H -case, LP waves:
Gs
2 bG d sin k d
bq1
kd
D
ž
= cos k dsin2 w y
q
b Ž b y 1.
Ž b q 1. 2
G
ž / k
k
sin k dcos2 w
/
2
sin2k d.
Ž 13.
Here k s Ž< g < 2 q D 2 .1r2 . The terms present in Eqs. Ž6. – Ž9. above are now modified by the replacement of D by k everywhere and by the appearance of an additional factor Drk in the brackets in Eqs. Ž11., Ž13. for LP waves. In addition, new terms appear which are periodic and positive functions of the crystal thickness and are proportional to the square of the parameter Grk . These terms do not depend on the light polarization, but only on the grating vector orientation: for K 5 w001x they are 4 times larger than for K H w001x. They are most pronounced when the optical activity is not large as in BTO and they become negligible for large optical activity as in BSO. These new terms are of more importance for CP waves. For K H w001x, for example, the energy exchange is solely due to them. The ‘‘large optical activity approximation’’ predicts that there should be no interaction at all in this case. Such an interaction of CP waves for K H w001x was observed, however, in Ref. w12x. Since this effect is small in BSO, it was difficult to identify it. Strong effects of this type are observed in BTO as we show in Section 5. It is interesting that the new terms vanish for equal beam intensities, m s 1, and the perturbation theory Žapproximation of large optical activity. described in the previous subsection is justified even for small D . For m f 1, however, the basic equations Ž2. should no longer
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
66
Table 1 Parameters for BSO and BTO Žwavelength l s 515 nm. Material parameters
BSO
BTO
Refractive index n Electro-optic coefficient r w10y1 2 m Vy1 x Static dielectric constant ´ s t Trap density Nt w10 21 my3 x Rotatory power D w8 mmy1 x Crystal thickness d wmmx Angle of incidence u w8x Light intensity I0 wmW cmy2 x Beam ratio b
2.615 4.50 56 3.5 45.0 2 23 1.6 70
2.660 5.17 47 60 11.5 10 23 750 25
Fig. 2. Comparison of the analytical solutions: large optical activity Ždashed line., arbitrary optical activity Ždash-dotted line., and numerical solution Žsolid line..
be used since the higher spatial harmonics of the spacecharge field have to be included w18,19x. The above solution was obtained under the assumption that the changes in the light modulation depth are small. Let us compare now the approximate solutions for large optical activity and for arbitrary optical activity with the exact numerical solution. As an example we select BTO with the orientation K H w001x and LP waves with w s ypr4. Fig. 2 shows the corresponding dependences of the gain on the crystal thickness. We can see that the new approximation fits the exact solution much better than the former one. However, depending on the crystal thickness and the input polarization, even the deviation of this better approximation from the exact solution can reach 5–7%. The reason for this deviation is of course the change of the light modulation depth. We note, however, that the assumption of constant m is still quite well justified for BTO. Therefore for a wide class of experimental parameters it is better to use the new approximate solution. Utilization of the two simple approximations described above Ž g˜ 4 D and D 4 g˜ . clearly leads to larger errors in the determination of the crystal parameters.
ment using finite beams with plane wave fronts and constant amplitude distribution. We used a trapezoidal beamoverlap geometry Žsee Fig. 3. and the experimental setup described earlier in Ref. w14x. Essentially, this method allows us to measure in a single experiment the gain inside the crystal as a function of the crystal thickness by measuring the gain as a function of the detector transverse coordinate. The results for BSO were already published in Ref. w14x. Some of them are presented here for comparison with the new results for BTO. The main parameters of our BSO and BTO crystals are given in Table 1. All of them, with the exception of Nt , are known from the literature or defined by our experimental conditions. The values of Nt used correspond to the best fit of our experimental data. This was the only fit parameter. An external field was not applied. The pump and signal beams were linearly polarized in an identical way. In both orientations, K 5 w001x and K H w001x, measurements were
5. Experimental results To compare the influence of the optical activity on TWM in BSO and BTO crystals, we performed an experi-
Fig. 3. Trapezoidal overlap geometry.
Fig. 4. Gain distribution for a BSO crystal: experimental results Žsolid line. and numerical solution Ždashed line.. The different initial linear polarizations of the light are indicated by arrows: w sp r4 and 3p r4 Ža.; w s 0 and p r2 Žb..
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
67
light modulation depth different from 1, and vanishing gain for m s 1. In the experiment, for b s 10 the thickness dependence of the gain corresponds qualitatively to the theoretical predictions. The maximum gain is G s 0.2 " 0.1 for lefthand polarization and G s 0.5 " 0.1 for right-hand polarization. From our analytical theory we expect a value of 0.2. The corresponding numerical value is 0.23. For b s 1 the gain is G s y0.1 " 0.1 for left-hand polarization and G s 0.0 " 0.1 for right-hand polarization. In both cases, the gain remains independ of the crystal thickness. From both the analytical theory and the numerical simulations we expect no gain at all. As one can see the experiments demonstrate a slightly more complicated behaviour which cannot be fully explained at the present time. Our analysis is, however, in general accordance with the experimental data obtained. Fig. 5. Gain distribution for a BTO crystal: experimental results Žsolid line. and numerical solution Ždashed line.. The different initial linear polarizations of the light are indicated by arrows: w sp r4 and 3p r4 Ža.; w s 0 and p r2 Žb..
performed for different input light polarizations Ž w s 0, pr4, pr2, 3pr4.. The gain observed for BSO and BTO for K H w001x versus the normalized detector coordinate is shown in Fig. 4 and Fig. 5, respectively. The polarization rotation D d was close to pr2 for both cases Žsee Table 1., therefore the results for the two crystals can be easily compared. The solid lines represent the experimental data, the dashed lines the theoretical values corresponding to the exact numerical solution of Eqs. Ž2. with the finite beam geometry taken into account. The spread of the experimental data may be related to beam and crystal inhomogeneities. The small arrows indicate the input state of polarization. The saturation of the gain distribution observed at the right-hand side of the figures is related to the finite beam-overlap geometry. It indicates that the effective interaction length of the beams is constant for this part of the signal beam. As shown above, for K H w001x, the approximation of large optical activity gives a periodic gain dependence on the crystal depth. Furthermore, from Eq. Ž9. we conclude that for initial polarizations differing by pr2 the corresponding gains, considered as functions of the crystal depth, are equal but simply have an opposite sign. The experimental results for BSO show exactly this type of behaviour Žsee Figs. 4.. The appearance of an asymmetry in Fig. 5 indicates that the approximation D 4 g˜ is not fulfilled well for BTO. At the same time the presence of some periodicity shows that the optical activity cannot be neglected. Next we consider our results for CP waves. The approximation of large optical activity predicts zero gain for circularly polarized light waves. Our theory modified for arbitrary optical activity predicts some non-zero gain for a
6. Conclusions We investigated the role of optical activity in photorefractive TWM in cubic crystals both theoretically and experimentally and demonstrated that the interpretation of the data obtained for BTO crystals is considerably more complicated than that for BSO crystals. We compared different theoretical approaches of TWM for the case of diffusion recording. The approximation of small optical activity can be used only for very thin crystals. The opposite approximation of very large optical activity works well in BSO and BGO crystals but fails for BTO crystals in the case when Kr D s 1 Žwhen the diffusion TWM gain is maximal.. The reason is that in this case optical activity and photorefractive wave coupling are of comparable strength. An analytical solution of TWM for arbitrary optical activity is presented under the assumption of a constant light modulation depth. The new results explain the main experimental peculiarities of BTO crystals Žnon-zero gain for circularly polarized beams and the appearance of an asymmetry in the thickness-dependence of the gain for linearly polarized beams.. We compared the results of the theoretical analysis with the experimental data obtained for BSO and BTO crystals. In summary, the results obtained provide an understanding of the role of optical activity in BTO crystals which has been largely underestimated up to now.
Acknowledgements We thank M. Shamonin, S. Stepanov and B. Sturman for helpful discussions and constructive criticism. We acknowledge the financial support by the Sonderforschungsbereich 225 ŽGermany..
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References w1x M.P. Petrov, S.I. Stepanov, A.V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Springer Series in Optical Sciences, Springer, Heidelberg, 1991, pp. 233–239. w2x A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley Series in Pure and Appl. Optics, Wiley, New York, 1984, pp. 223–237. w3x W. Krolikowski, N. Akhmediev, D.R. Andersen, B. LutherDavies, Optics Comm. 132 Ž1996. 179. w4x S.R. Singh, D.N. Christodoulides, J. Opt. Soc. Am. B 13 Ž1996. 719. w5x E. Raita, A.A. Kamshilin, T. Jaaskelainen, Optics Lett. 21 Ž1996. 1897. w6x A. Marrakchi, R.V. Johnson, J.A.R. Tanguay, J. Opt. Soc. Am. B 3 Ž1986. 321. w7x A. Marrakchi, R.V. Johnson, J.A.R. Tanguay, IEEE J. Quantum Electron. 23 Ž1987. 2142. w8x S. Mallick, D. Rouede, ` A.G. Apostolidis, J. Opt. Soc. Am. B 4 Ž1987. 1247. w9x V.V. Shepelevich, N.N. Egorov, V. Shepelevich, J. Opt. Soc. Am. B 11 Ž1994. 1394.
w10x L. Solymar, D.J. Webb, A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Clarendon Press, Oxford, 1996. w11x A.A. Kamshilin, Optics Comm. 93 Ž1992. 350. w12x D.J. Webb, A. Kießling, B.I. Sturman, E. Shamonina, K.H. Ringhofer, Optics Comm. 108 Ž1994. 31. w13x S. Stepanov, S.M. Shandarov, N.D. Khat’kov, Sov. Phys. Sol. State 29 Ž1987. 1754. w14x E. Shamonina, M. Mann, K.H. Ringhofer, A. Kiessling, R. Kowarschik, Opt. Quantum Electron. 28 Ž1996. 25. w15x L. Solymar, D.J. Cooke, Volume Holography and Volume Gratings, Academic Press, London, 1981, pp. 164–207. w16x E. Shamonina, G. Cedilnik, M. Mann, A. Kiessling, D.J. Webb, R. Kowarschik, K.H. Ringhofer, Appl. Phys. B 64 Ž1997. 49. w17x B.I. Sturman, D.J. Webb, R. Kowarschik, E. Shamonina, K.H. Ringhofer, J. Opt. Soc. Am. B 11 Ž1994. 1813. w18x G.A. Brost, Optics Comm. 96 Ž1993. 113. w19x T.E. McClelland, D.J. Webb, B.I. Sturman, E. Shamonina, M. Mann, K.H. Ringhofer, Optics Comm. 131 Ž1996. 315.
Optics Communications 146 Ž1998. 62–68
Optical activity in photorefractive Bi 12TiO 20 E. Shamonina a , V.P. Kamenov a , K.H. Ringhofer a,1, G. Cedilnik b, A. Kießling b, R. Kowarschik b, D.J. Webb c a Fachbereich Physik der UniÕersitat, ¨ 49069 Osnabruck, ¨ Germany Institut fur ¨ Angewandte Optik, Friedrich-Schiller-UniÕersitat, ¨ 07743 Jena, Germany Applied Optics Group, Physics Laboratory, The UniÕersity, Canterbury, Kent CT2 7NR, UK b
c
Received 17 April 1997; revised 15 August 1997; accepted 8 September 1997
Abstract The influence of optical activity on two-wave mixing ŽTWM. in photorefractive BTO and BSO crystals in the absence of an applied field is studied both theoretically and experimentally. For the conventinal orientations of the grating vector, K 5 w001x and K H w001x, the piezoelectric and photoelastic effects are either zero or negligible. This makes an analytical treatment of the TWM problem possible. We obtain an analytical solution for the coupled wave equations of TWM valid for arbitrary optical activity. This result is of special importance for BTO crystals. In these crystals under the condition of maximum energy transfer Ž< K < r D s 1, where r D is the Debye radius. neither the approximation of small optical activity nor the one of dominating optical activity is applicable and our analytical solution becomes essential. Our experimental setup uses beams with a trapezoidal overlap that allow us to study the thickness-dependence of the gain in a single measurement. Experimental and theoretical results for a BTO crystal are compared with those for a BSO crystal and are explained in the framework of the model used. q 1998 Elsevier Science B.V. PACS: 42.40.Ht; 42.40.Lx; 42.40.Pa; 42.65.HW Keywords: Sillenites; Photorefractive; Two-wave mixing; Optical activity
1. Introduction It is known that the natural optical activity of sillenites affects the photorefractive two-wave mixing ŽTWM. in these crystals strongly w1x. Optical activity leads to rotation of the polarization planes of the interacting waves. In different layers inside the crystal the components of the effective electrooptic tensor w2x Žwhich are due to both the electrooptic and photoelastic effects. contribute differently to the rate of energy transfer between the waves. In other words, the efficiency of the energy transfer varies with the crystal depth. The investigation of the role of optical activity is particularly interesting in view of recent progress
1 Corresponding uni-osnabrueck.de.
author.
E-m ail:
ringhofe@ m ail.rz.
in the understanding of photorefractive soliton propagation w3,4x and of TWM in fiber-like crystals w5x. In the conventional orientations of the grating vector, K 5 w001x and K H w001x w6–8x, the piezoelectric and photoelastic properties of the sillenite crystals are not pronounced w9x and the coupled wave equations have a simple structure. In these orientations it is worth looking for an analytical solution of the TWM problem that makes analysis of the photorefractive process and the determination of crystal parameters easier. In the current literature two main theoretical approaches are used. In the first approach the optical activity is neglected. This is justified for the case of a strong grating w10x. This method is frequently used to obtain estimates for the optimization of the TWM process with a moving grating technique with an external dc field. However, the optical activity cannot be neglected for diffusion recording which is of large importance for inter-
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 5 1 0 - 5
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
ferometric applications in particular w1x. The second method assumes that the optical activity is dominating w8,11,12x. The propagation of light waves in the optically active medium is first calculated from optical activity alone and is then disturbed by the weak interaction between the light waves via the photorefractive grating. This approximation works sufficiently well for BSO and BGO crystals for diffusion recording. It is also applicable to BTO for almost parallel propagation of waves when, owing to the small spatial frequency, the recorded diffusion grating is weak and the optical activity prevails. In the practically important case of maximizing the TWM gain Ž< K < r D s 1., however, a discrepancy between this theory and experiment is observed even for BSO crystals. In particular, the theory predicts that there is no interaction between two waves with the same circular polarization state for K H w001x although such an interaction was observed experimentally w12x. Furthermore, the theory suggests that the dependence of the gain on the azimuth of linear polarization, w , is of cosine type, i.e. the polarization dependence is a periodic function oscillating around the zero gain value with positive and negative extrema having equal absolute value. Deviations from such a type of polarization dependence can even be seen in BSO crystals and they are much larger in BTO crystals. The structure of the paper is as follows. In Section 2 we review shortly the coupled wave equations for TWM and show that in ‘‘pure’’ electrooptic orientations K 5 w001x and K H w001x piezoelectric and photoelastic effects do not need to be taken into account. After discussing the methods described above in Section 3, we obtain, in Section 4, an analytic solution of TWM which is valid for arbitrary optical activity and both for linearly and circularly polarized light. Our results explain the experimentally observed peculiarities of beam coupling in BTO crystals. Our experimental setup uses finite aperture beams which are partly overlapping in the crystal. This allows us to observe the thickness dependence of the TWM gain in a single measurement. In Section 5 we compare the experimental and numerical results for BTO and BSO crystals and explain the observed characteristic features.
2. Coupled wave equations We consider the Ž110.-cut crystal and choose the following basis vectors: 0 exs 0 , 1
ž/
eys
1
'2
y1 y1 , 0
ž /
ez s
1
'2
1 y1 . 0
ž /
two waves is described by the following equations for the slowly varying amplitudes: EsX s i Ds 2 Es q ig ) S Ž K . Ep , EXp s i Ds 2 Ep q ig S Ž K . Es ,
Ž2.
where D is the rotatory power, g the coupling constant, and sj the j component of the vector of Pauli matrices. S Ž K . is a coupling tensor with dimensionless components which, in general, include the electrooptic, photoelastic and piezoelectric contributions w13x. For diffusion recording gsiG
ms
m
,
2
Gs
p n3 r41
Kk BTre
l
1 q Ž Kr D .
2 Es) P Ep < Es < 2 q < Ep < 2
,
2
,
Ž3.
where m is the light modulation depth, r D s Ž ee 0 k BTrNt e 2 .1r2 the Debye radius, n the refractive index, r41 the electrooptic coefficient, l the wavelength, k B Boltzmann’s constant, T the temperature, e the electron charge, ee the static dielectric permittivity, and Nt the effective trap density. For diffusion recording the coupling constant g is purely imaginary. In Eqs. Ž2. the optical absorption is neglected because it has no influence on the gain, G w12x, defined in the following way: Gs
< Es < 2 y < Es0 < 2 < Es0 < 2
,
where Es0 is the signal beam amplitude in the absence and Es in the presence of the pump beam, both values being measured behind the crystal. For the orientations K 5 w001x and K H w001x the matrix S Ž K . is 5-case:
Ss
Iˆy s 3 2
,
ž
S s s1 1 q
H -case:
e14 p44 r41 c44
/
.
Here Iˆ is the unity matrix, e14 the piezoelectric coefficient, c 44 and p44 the component of the elasticity modules and photoelastic tensor, respectively. For K 5 w001x the photoelastic effects do not contribute to the tensor S at all. For K H w001x the photoelastic effects lead to an introduction of the new effective electrooptic coefficient only:
Ž1.
The pump wave, Ep expw iŽ k p P r y v t .x, and the signal wave, Es expw iŽ k s P r y v t .x, are supposed to propagate at small angles to the z-direction, so that in our coordinate system Ž1. the amplitudes Ep and Es have only x- and y-components. In the stationary state the interaction of the
63
ž
X r41 s r41 1 q
e14 p44 r41 c 44
/
without changing the structure of the tensor S . For the parameters given in Ref. w13x the relative changes in r41 for K H w001x are 4.5% for BTO and 13.7% for BSO. Therefore we conclude that for both these orthogonal
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
64
orientations of the grating the photoelastic effects can be neglected Žsee also Ref. w9x..
3. The basic approximations and their shortcomings for the orientations K 5 [001] and K H [001] We need to estimate the competition of the two terms in Eqs. Ž2. which are due to the optical activity and the electrooptic effect only. As we mentioned above, we distinguish two cases when one of the parameters g˜ or D prevails. Here
b g˜ s
bq1
3.2. The optical actiÕity dominates: D 4 g˜ Here the equations describing the light propagation in an optically active medium Ž g s 0 in Eqs. Ž2.. are first to be solved. The solutions represent circularly polarized waves e "s Ž1r '2 .Ž1," i . w15x. The photorefractive energy exchange acts here as a small perturbation. We assume additionally that the light modulation depth can be considered as approximately constant. This is fullfilled for crystals with gd ˜ < 1 w16x. We obtain for K 5 w001x and K H w001x for circularly polarized ŽCP. and linearly polarized ŽLP. waves:
G,
with b being the beam intensity ratio. We discuss the properties of both approximations and the regions of their applicability. We assume that the two beams enter the crystal with identical polarization. For diffusion recording there is no phase coupling between the interacting waves so that initially linearly polarized light waves maintain their linear polarization during the propagation through the sample.
5-case, CP waves:
Gs
5-case, LP waves:
Gs
bG d bq1
,
Ž6.
bG d bq1 = 1y
sin D d
Dd
cos Ž 2 w y D d . ,
Ž7.
3.1. The photorefractiÕe mechanism dominates: g˜ 4 D In this case we neglect optical activity. Eqs. Ž2. contain now only the term describing the energy exchange and the optical activity does not influence the polarization state. The interacting beams change, however, their polarizations because the structure of the coupling tensor g S ensures the coupling of different components of the polarization vectors. There are, however, a few special cases when the polarization of the propagating light waves remains constant: w s 0 and w s pr2 for K 5 w001x and w s "pr4 for K H w001x, where w is the angle between the polarization vector and the x-axis. For these cases the exact solutions of the TWM problem are known w10x. They give for the two orientations: 5-case:
G Ž w s pr2 . s
1
Ž b q 1. e G d
2 e G d q b ey G d
y1
Ž4.
G Ž w s 0. s 0 H -case:
G Ž w s "pr4. s
Ž b q 1. e " G d e" G d q b e. G d
y 1.
Ž5.
In general, the polarization angle determines the efficiency and, for K H w001x, also the direction of the energy exchange. The numerical analysis of the coupled equations shows that the special solutions given above in both orientations correspond to the maximum and minimum gain. It follows from Eqs. Ž4., Ž5. that in the undepleted pump approximation Ž m < 1. the gain grows Žor decays. exponentially with the crystal thickness. We note that in practice Eqs. Ž4., Ž5. can be used if g˜ 4 D and also D d < 1.
H -case, CP waves:
G s 0,
H -case, LP waves:
Gs
Ž8.
2 bG d sin D d
bq1
Dd
sin Ž 2 w y D d . .
Ž9. These results for CP waves were first obtained in Ref. w12x and for LP waves in Ref. w8x. One can see that for CP waves the gain depends linearly on the thickness d in the 5-case and is non-existant in the H -case. For LP waves the maximum gain in the 5-case and H -case is achieved for initial polarizations w s D dr2 and D dr2 q pr4, respectively. This corresponds to the polarization state in the middle of the crystal thickness, w Ž dr2. s 0 and pr4, respectively. For typical parameters of BSO Ž Kr D s 1 for Nt s 10 22 my3 , l s 514.5 nm. we have D f 628 my1 and g˜ f 35 my1 Ž b s 1. or f 70 my1 Ž b ™ `.. The ratio gr ˜ D is 0.06–0.12 and one can rely on the approximation of dominating optical activity. At the same time, for BTO Ž Kr D s 1 for Nt s 10 22 my3 , l s 514.5 nm. we obtain D f 200 my1 and g˜ f 47 my1 Ž b s 1. or f 94 my1 Ž b ™ `.. The ratio gr ˜ D is 0.24–0.48. It is clear that the optical activity cannot be neglected in BTO, but at the same time the ratio gr ˜ D is not small enough to justify the application of the approximation of dominating optical activity. In Figs. 1a and 1b the results obtained in two approximations of small and large optical activity are compared with the numerical solutions of Eqs. Ž2. for BSO and BTO,
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
65
K 5 w001x and K H w001x for circularly and linearly polarized waves: 5-case, CP waves:
bG d
Gs
bq1 q
1 b Ž b y 1. 4 Ž b q 1.
2
G
ž / k
2
sin2k d,
Ž 10. 5-case, LP waves:
bG d
Gs
bq1
1y
sin k d
kd
D
ž
= cos k dcos2 w q
q
1 b Ž b y 1. 4 Ž b q 1.
2
H -case, CP waves:
k
/
sin k dsin2 w 2
G
sin2k d,
ž / k
Gs
Ž 11.
b Ž b y 1.
Ž b q 1. 2
G
ž / k
2
sin2k d,
Ž 12. Fig. 1. Dependence of the gain on the crystal Ža. and BTO Žb.: negligible optical activity perturbation theory for large optical activity numerical solution Žsolid line..
thickness for BSO Ždash-dotted line., Ždashed line., and
respectively. The orientation K H w001x and LP waves with w "s "pr4 are chosen. It is seen that for both crystals the monotonic solutions obtained by neglecting of the optical activity Žcurves ‘‘ g˜ 4 D ’’. become unapplicable for even small thicknesses, d f 1 mm in the case of BSO ŽFig. 1a. and d f 4 mm in the case of BTO ŽFig. 1b.. The approximation of large optical activity leads to solutions strictly periodic and symmetric with respect to the x-axis. For BSO the agreement of this approximation with the precise numerical solution ŽFig. 1a. is satisfactory for any crystal thickness. For BTO the agreement is satisfactory for small d only ŽFig. 1b. and the discrepancy varies strongly with the thickness. At d s 5 mm the deviation of the approximate solutions from the precise ones for w s "pr4 are 0.24 and 0.15 Žabsolute values. and 36% and 56% Žrelative values normalized to the value of gain from the exact solution..
4. A solution for arbitrary optical activity Obviously the theory developed for large D must be modified to describe BTO crystals as well. For this purpose Eqs. Ž2. have to be solved without assuming anything about the strength of the optical activity. Using the methods developed in Ref. w17x, including the approximation of the constant light modulation depth Ž gd ˜ < 1., we find for
H -case, LP waves:
Gs
2 bG d sin k d
bq1
kd
D
ž
= cos k dsin2 w y
q
b Ž b y 1.
Ž b q 1. 2
G
ž / k
k
sin k dcos2 w
/
2
sin2k d.
Ž 13.
Here k s Ž< g < 2 q D 2 .1r2 . The terms present in Eqs. Ž6. – Ž9. above are now modified by the replacement of D by k everywhere and by the appearance of an additional factor Drk in the brackets in Eqs. Ž11., Ž13. for LP waves. In addition, new terms appear which are periodic and positive functions of the crystal thickness and are proportional to the square of the parameter Grk . These terms do not depend on the light polarization, but only on the grating vector orientation: for K 5 w001x they are 4 times larger than for K H w001x. They are most pronounced when the optical activity is not large as in BTO and they become negligible for large optical activity as in BSO. These new terms are of more importance for CP waves. For K H w001x, for example, the energy exchange is solely due to them. The ‘‘large optical activity approximation’’ predicts that there should be no interaction at all in this case. Such an interaction of CP waves for K H w001x was observed, however, in Ref. w12x. Since this effect is small in BSO, it was difficult to identify it. Strong effects of this type are observed in BTO as we show in Section 5. It is interesting that the new terms vanish for equal beam intensities, m s 1, and the perturbation theory Žapproximation of large optical activity. described in the previous subsection is justified even for small D . For m f 1, however, the basic equations Ž2. should no longer
E. Shamonina et al.r Optics Communications 146 (1998) 62–68
66
Table 1 Parameters for BSO and BTO Žwavelength l s 515 nm. Material parameters
BSO
BTO
Refractive index n Electro-optic coefficient r w10y1 2 m Vy1 x Static dielectric constant ´ s t Trap density Nt w10 21 my3 x Rotatory power D w8 mmy1 x Crystal thickness d wmmx Angle of incidence u w8x Light intensity I0 wmW cmy2 x Beam ratio b
2.615 4.50 56 3.5 45.0 2 23 1.6 70
2.660 5.17 47 60 11.5 10 23 750 25
Fig. 2. Comparison of the analytical solutions: large optical activity Ždashed line., arbitrary optical activity Ždash-dotted line., and numerical solution Žsolid line..
be used since the higher spatial harmonics of the spacecharge field have to be included w18,19x. The above solution was obtained under the assumption that the changes in the light modulation depth are small. Let us compare now the approximate solutions for large optical activity and for arbitrary optical activity with the exact numerical solution. As an example we select BTO with the orientation K H w001x and LP waves with w s ypr4. Fig. 2 shows the corresponding dependences of the gain on the crystal thickness. We can see that the new approximation fits the exact solution much better than the former one. However, depending on the crystal thickness and the input polarization, even the deviation of this better approximation from the exact solution can reach 5–7%. The reason for this deviation is of course the change of the light modulation depth. We note, however, that the assumption of constant m is still quite well justified for BTO. Therefore for a wide class of experimental parameters it is better to use the new approximate solution. Utilization of the two simple approximations described above Ž g˜ 4 D and D 4 g˜ . clearly leads to larger errors in the determination of the crystal parameters.
ment using finite beams with plane wave fronts and constant amplitude distribution. We used a trapezoidal beamoverlap geometry Žsee Fig. 3. and the experimental setup described earlier in Ref. w14x. Essentially, this method allows us to measure in a single experiment the gain inside the crystal as a function of the crystal thickness by measuring the gain as a function of the detector transverse coordinate. The results for BSO were already published in Ref. w14x. Some of them are presented here for comparison with the new results for BTO. The main parameters of our BSO and BTO crystals are given in Table 1. All of them, with the exception of Nt , are known from the literature or defined by our experimental conditions. The values of Nt used correspond to the best fit of our experimental data. This was the only fit parameter. An external field was not applied. The pump and signal beams were linearly polarized in an identical way. In both orientations, K 5 w001x and K H w001x, measurements were
5. Experimental results To compare the influence of the optical activity on TWM in BSO and BTO crystals, we performed an experi-
Fig. 3. Trapezoidal overlap geometry.
Fig. 4. Gain distribution for a BSO crystal: experimental results Žsolid line. and numerical solution Ždashed line.. The different initial linear polarizations of the light are indicated by arrows: w sp r4 and 3p r4 Ža.; w s 0 and p r2 Žb..
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67
light modulation depth different from 1, and vanishing gain for m s 1. In the experiment, for b s 10 the thickness dependence of the gain corresponds qualitatively to the theoretical predictions. The maximum gain is G s 0.2 " 0.1 for lefthand polarization and G s 0.5 " 0.1 for right-hand polarization. From our analytical theory we expect a value of 0.2. The corresponding numerical value is 0.23. For b s 1 the gain is G s y0.1 " 0.1 for left-hand polarization and G s 0.0 " 0.1 for right-hand polarization. In both cases, the gain remains independ of the crystal thickness. From both the analytical theory and the numerical simulations we expect no gain at all. As one can see the experiments demonstrate a slightly more complicated behaviour which cannot be fully explained at the present time. Our analysis is, however, in general accordance with the experimental data obtained. Fig. 5. Gain distribution for a BTO crystal: experimental results Žsolid line. and numerical solution Ždashed line.. The different initial linear polarizations of the light are indicated by arrows: w sp r4 and 3p r4 Ža.; w s 0 and p r2 Žb..
performed for different input light polarizations Ž w s 0, pr4, pr2, 3pr4.. The gain observed for BSO and BTO for K H w001x versus the normalized detector coordinate is shown in Fig. 4 and Fig. 5, respectively. The polarization rotation D d was close to pr2 for both cases Žsee Table 1., therefore the results for the two crystals can be easily compared. The solid lines represent the experimental data, the dashed lines the theoretical values corresponding to the exact numerical solution of Eqs. Ž2. with the finite beam geometry taken into account. The spread of the experimental data may be related to beam and crystal inhomogeneities. The small arrows indicate the input state of polarization. The saturation of the gain distribution observed at the right-hand side of the figures is related to the finite beam-overlap geometry. It indicates that the effective interaction length of the beams is constant for this part of the signal beam. As shown above, for K H w001x, the approximation of large optical activity gives a periodic gain dependence on the crystal depth. Furthermore, from Eq. Ž9. we conclude that for initial polarizations differing by pr2 the corresponding gains, considered as functions of the crystal depth, are equal but simply have an opposite sign. The experimental results for BSO show exactly this type of behaviour Žsee Figs. 4.. The appearance of an asymmetry in Fig. 5 indicates that the approximation D 4 g˜ is not fulfilled well for BTO. At the same time the presence of some periodicity shows that the optical activity cannot be neglected. Next we consider our results for CP waves. The approximation of large optical activity predicts zero gain for circularly polarized light waves. Our theory modified for arbitrary optical activity predicts some non-zero gain for a
6. Conclusions We investigated the role of optical activity in photorefractive TWM in cubic crystals both theoretically and experimentally and demonstrated that the interpretation of the data obtained for BTO crystals is considerably more complicated than that for BSO crystals. We compared different theoretical approaches of TWM for the case of diffusion recording. The approximation of small optical activity can be used only for very thin crystals. The opposite approximation of very large optical activity works well in BSO and BGO crystals but fails for BTO crystals in the case when Kr D s 1 Žwhen the diffusion TWM gain is maximal.. The reason is that in this case optical activity and photorefractive wave coupling are of comparable strength. An analytical solution of TWM for arbitrary optical activity is presented under the assumption of a constant light modulation depth. The new results explain the main experimental peculiarities of BTO crystals Žnon-zero gain for circularly polarized beams and the appearance of an asymmetry in the thickness-dependence of the gain for linearly polarized beams.. We compared the results of the theoretical analysis with the experimental data obtained for BSO and BTO crystals. In summary, the results obtained provide an understanding of the role of optical activity in BTO crystals which has been largely underestimated up to now.
Acknowledgements We thank M. Shamonin, S. Stepanov and B. Sturman for helpful discussions and constructive criticism. We acknowledge the financial support by the Sonderforschungsbereich 225 ŽGermany..
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