- Email: [email protected]
Light-wave polarization bistability in a gyrotropic magnetic medium
Applied Surface Science 142 Ž1999. 272–275
Light-wave polarization bistability in a gyrotropic magnetic medium M. Wierzbicki ) , J. Kocinski ´ Institute of Physics, Warsaw UniÕersity of Technology, Koszykowa 75, 00-662 Warsaw, Poland
Abstract A linearly polarized light wave enters a Fabry–Perot ´ cavity containing a MnSerZnTe superlattice. The helical axis is perpendicular to the cavity walls, and an external magnetic field is parallel to that axis. The coherence matrix is calculated and the polarization state of the outgoing beam is determined. It is shown that the ratio of the semi-axes of the polarization ellipse and the spatial orientation of the ellipse major axis as functions of the incoming light intensity exhibit bistable behaviour. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Light wave; Polarization bistability; Gyrotropic
1. Introduction Optical dispersive bistability in media with forced gyrotropy was theoretically investigated in Ref. w1x, where significant effects of polarization bistability were calculated. Polarization bistability was experimentally measured in Ref. w2x and optical properties of ZnSerZnTe superlattices in Ref. w3x. The investigation w1x will be extended to the case of light propagation in gyrotropic magnetic superlattices exhibiting antiferromagnetic helical spin structure, characterized by a fourth-rank optical tensor connected with magnetic order.
2. The nonlinear polarization The appearance of the third-order electric polar™ ization P Ž3., which is basic for the effect of polariza-
tion bistability, is conditioned by the fourth-rank optical tensor x i jk l 4 . We assume that this tensor is connected with the antiferromagnetic helical spin structure of the MnSerZnTe superlattice. For the calculation of the spatial modulation and symmetry properties of that tensor involving group-theoretical methods, we refer to Ref. w4x. The tensor x i jk l 4 has 16 nonzero components of which five are linearly independent. There appear two kinds of spatial modulation of these components, depending on 2 a or 4a , where a is the turning angle between two adjacent antiferromagnetic planes of the helix. In this paper, we consider the 4a modulation. Utilizing the calculated expression for the relevant tensor components and passing from™the Cartesian to circularly ™ polarized electric field, E, and polarization, P Ž3., by means of the formulae: E1,2 s Ž E x " iE y .r2 and Ž3. P1,2 s Ž PxŽ3. " iPyŽ3. .r2, we obtain the left and right circular-polarization components in the form: P1Ž3. s 4g E22 E1U eyi 4 a
)
Corresponding author. Tel.: q48-22-6605230; Fax: q48-26282171; E-mail: [email protected]
P2Ž3. s 4g E12 E2U e i4 a ,
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 6 8 9 - 8
Ž 1.
M. Wierzbicki, J. Kocinskir Applied Surface Science 142 (1999) 272–275 ´
where g denotes the amplitude of the fourth-rank optical tensor connected with the presence of the antiferromagnetic spiral, and E1 and E2 are the respective electric field circular components.
3. The wave equation for circularly polarized electric fields Without currents and charges, Maxwell’s equations yield the following wave equation:
E
™
= = = = E s ymr m 0
2
Et
2
™
D,
Ž 2.
where mr denotes the isotropic linear magnetic permeability of the MnSerZnTe superlattice for the paramagnetic phase of the ™MnSe layers, m 0 is the ™ ™ vacuum permeability and D s ee ˆ 0 E q PNL , with eˆ denoting the mean electric permeability and e 0 the vacuum permeability. The fourth-rank tensor xˆ appears only in the MnSe layers of the superlattice. The average value of the tensor in the superlattice therefore is required. Since the spiral pitch is much larger than the thickness of a single MnSe layer, a single layer is approximately spatially homogeneous. The method of evaluation of average values of tensors in magnetic superlattices was elaborated in Refs. w5,6x. In our case, this method leads to an effective tensor to which the magnetic and non-magnetic layers contribute with appropriate weights, proportional to the thickness of the layers. Consequently, the magnitude of the tensor is diminished in comparison with that in a single magnetic layer by a constant factor which can be absorbed by g . We assume that the second-rank tensor character of the linear permeability is caused by the presence of a weak external magnetic field B directed along the spiral axis, i.e., z-axis. The magnetic field brings about an anisotropic contribution D e to the isotropic permeability e of the form: 0 D e s ih B 1 0
ž
y1 0 0
0 0 , 0
/
Ž 3.
where h is a constant depending on the magnitude of the magnetic field. The linear Faraday effect depends on D e . When circular components E1 and E2 of the
273
light wave are introduced, the tensor D eˆ is diagonalized. We assume a monochromatic incoming wave of frequency v depending only on the z-coordinate, along the spiral axis. We introduce the circularly ™ polarized fields E1 and E2 with the wave vectors k 1 ™ and k 2 in the linear approximation determined by:
ž
2 k 1,2 s k 02 1 .
hB er
/
,
Ž 4.
(
where k 0 s vry and y s 1r mr m 0 e r e 0 is the light velocity in a linear medium. With E1 and E2 fields, Eq. Ž2. takes the form: d2
k 02
dz
er e 0
E q k m2 Em s y 2 m
PmNL ,
where m s 1,2.
Ž 5. The fields E1 and E2 are represented by forward Žf. and backward Žb. moving waves: Em Ž z . s Emf Ž z . q Emb Ž z . s A m Ž z . eyi k m z q Bm Ž z . e i k m z ,
Ž 6.
where A mŽ z . and BmŽ z . are the complex amplitudes, m s 1,2. We utilize the approximation of slowly varying amplitude w7x.
4. The nonlinear equations for the wave amplitudes and phases We assume that the wave vector ™ q of the spiral spin structure is much smaller than the wave vectors of the light waves E and E2 , and at the same time ™ ™1 that: 2 < ™ q < f < k 1 < y < k 2 <. The latter difference is determined by the magnitude of the external magnetic field and hence, it can be adjusted to fit the length of the spiral pitch. Multiplying both sides of Eq. Ž5. by expŽ"ik m z . and retaining only the terms containing the factor expw"4iŽ q y Ž k 1 y k 2 . z .x, we obtain the equations for the complex amplitudes A1 , A 2 and B1 , B2 : d Am
s yGm AUm A22ym ,
d Bm
s 0, Ž 7. dz where Gm s y2 ig k 02rŽ e 0 e r k m ., m s 1,2. From Eq. Ž7., it follows that the spiral structure acts like a dz
M. Wierzbicki, J. Kocinskir Applied Surface Science 142 (1999) 272–275 ´
274
tions to only two equations for b and d , from which we deduce the second conservation law in the form:
Ž b 2 y C12 . cos d s C2 s const.
Ž 11 .
Expressing d in terms of b , C1 and C2 , we obtain a single equation for b with the solution:
b Ž z . s b 2 sn Ž b 1 z q u 0 , k . ,
Fig. 1. The ratio ar b of the semi-axes of the polarization ellipse of the transmitted wave as a function of the incoming wave intensity I0 , measured in the units of the quantity G 1 L, which appears in Eq. Ž14.. The difference G 2 y G 1 is proportional to the intensity of external magnetic field and was assumed to be equal 2% in the numerical calculations.
nonlinear optical filter in which one of the two waves moving in opposite directions undergoes a nonlinear interaction, while the other propagates without an interaction. From Eq. Ž7., we obtain the energy conservation law in the form:
G 2 A1 AU1 q G 1 A 2 AU2 s C1 s const.
Ž 12 .
2 where k s b 2rb 1 , with b 1,2 s C12 " < C2 < is the modulus of the elliptic function. The constant u 0 is determined by the value of b at z s 0. From Eq. Ž8., 2 s C1 " b . Introducing these expreswe find that a1,2 sions into Eq. Ž10. for the phases f 1 and f 2 and utilizing Eq. Ž11. for cos d and Eq. Ž12. for b Ž z ., we obtain the expressions for the phase differences:
D f 1,2 s f 1,2 Ž L . y f 1,2 Ž 0 . s
C2 2 b 1 C1
du
u L , Husu Ž0 . 1 " a sn u Ž .
Ž 13 .
where uŽ z . s b 1 q u 0 and a s b 2rC1 , Ž0 F a F 1.. The right-hand side of Eq. Ž13. represents an incomplete elliptic integral of the third kind w8x. The calculation of the phase difference is based on the assumption that the change of the amplitudes a1 and a 2 in the Fabry–Perot ´ cavity is negligible and only
Ž 8.
To obtain the second conservation law which will be indispensable for the integration of Eq. Ž7., we express the complex amplitudes A1 and A 2 through the real amplitudes a1 and a 2 and phases f 1 and f2: A m Ž z . s am Ž z . e i f mŽ z . ,
m s 1,2,
Ž 9.
obtaining the equations: d am dz d fm dz
s Ž y1 .
mq1
Gm a m a22ym sin d ,
2 s yGm a 2ym cos d ,
Ž 10 .
with d s 2Ž f 1 y f 2 .. Introducing the parameter: b s G 2 a12 y G 1 a 22 , we can reduce the above four equa-
Fig. 2. The inclination angle c , with respect to the x-axis of the longer semi-axis of the polarization ellipse of the transmitted wave as a function of the incoming wave intensity I0 , measured in the units of the quantity G 1 L, which appears in Eq. Ž14.. The angle c is measured in degrees.
M. Wierzbicki, J. Kocinskir Applied Surface Science 142 (1999) 272–275 ´
the phases f 1 and f 2 are essentially altered. We obtain the expressions: D fm s y
Gm L T
Imout cos d ,
m s 1,2,
Ž 14 .
where I0 T 2
out I1,2 s 2
Ž 1 y R . q 4 R sin
2
ž
1 2
. D f i y kL
Ž 15 .
/
From the boundary conditions of a light wave in the cavity, we obtain an additional relation between the initial phases and the nonlinear phase differences in the form: tan fm Ž 0 . s
R sin Ž D fm y 2 kL .
,
1 y R cos Ž D fm y 2 kL .
m s 1,2.
275
6. Conclusions The polarization bistability discussed in this paper is conditioned by two circumstances: Ž1. the existence of third-order electric polarization and Ž2. the presence of external magnetic field parallel to the helical axis which implies the anisotropy of the refractive index for the circularly polarized waves. Only the combination of these two conditions leads to the polarization bistability. In the present constructions of optical logical devices, only the effect of light wave intensity bistability is utilized. The here examined two forms of polarization bistability which can be controlled by a weak external magnetic field open new possibilities in the field of construction of optical logical devices.
Ž 16 . Eqs. Ž14. – Ž16. allow to determine numerically the outgoing-waves intensities I1out , I2out and cos d as functions of the incoming intensity I0 .
5. The polarization bistability The polarization state of the wave transmitted through the cavity is determined by the coherence matrix w9x. The components J x x , J y y and J x y of that ™ matrix for the transmitted wave E out are expressed through the circularly polarized components E1out and E2out. Consequently, for the ratio of the semi-axes of the polarization ellipse of the transmitted wave, we obtain the formula: a b
s tan
1 2
arc sin
ž
I1out y I2out I1out q I2out
/
,
Ž 17 .
and for the angle c of inclination of the longer semi-axis with respect to the x-axis the formula: U
1
c s arc tan 2
U
i Ž E1out . E2out y i Ž E2out . E1out U
Ž E1out .
U
E2out q Ž E2out . E1out
,
Ž 18 .
The results of numerical calculations based on Eqs. Ž17. and Ž18. are represented in Figs. 1 and 2, respectively.
Acknowledgements This work was sponsored by a grant of the Warsaw University of Technology and by grant no. 2 PO3B 105 14 of the State Committee for Scientific Research ŽPoland.. We thank Prof. Jan Petykiewicz for inspiring discussions concerning problems of optical bistability.
References w1x J. Petykiewicz, D. Strojewski, Opt. Quantum Electron. 22 Ž1990. 131. w2x K. Panajotov, T. Tenev, G. Zartov, M. Pelt, J. Danckaert, H. Thienpont, I. Veretennicoff, Int. J. Nonlinear Opt. Phys. 5 Ž1996. 351. w3x R.C. Tu, Y.K. Su, H.J. Chen, Y.S. Huang, S.T. Chou, W.H. Lan, S.L. Tu, J. Appl. Phys. 84 Ž1998. 2866. w4x M. Wierzbicki, J. Kocinski, Surface Science 377–379 Ž1997. ´ 418. w5x N.S. Almeida, D.L. Mills, Phys. Rev. B 38 Ž1988. 6698. w6x N. Raj, D.R. Tilley, Phys. Rev. B 36 Ž1987. 7003. w7x J. Petykiewicz, Wave Optics, PWN, Warszawa, 1992. w8x P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971. w9x M. Born, E. Wolf, Principles of Optics, Pergamon, London, 1965.
Light-wave polarization bistability in a gyrotropic magnetic medium M. Wierzbicki ) , J. Kocinski ´ Institute of Physics, Warsaw UniÕersity of Technology, Koszykowa 75, 00-662 Warsaw, Poland
Abstract A linearly polarized light wave enters a Fabry–Perot ´ cavity containing a MnSerZnTe superlattice. The helical axis is perpendicular to the cavity walls, and an external magnetic field is parallel to that axis. The coherence matrix is calculated and the polarization state of the outgoing beam is determined. It is shown that the ratio of the semi-axes of the polarization ellipse and the spatial orientation of the ellipse major axis as functions of the incoming light intensity exhibit bistable behaviour. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Light wave; Polarization bistability; Gyrotropic
1. Introduction Optical dispersive bistability in media with forced gyrotropy was theoretically investigated in Ref. w1x, where significant effects of polarization bistability were calculated. Polarization bistability was experimentally measured in Ref. w2x and optical properties of ZnSerZnTe superlattices in Ref. w3x. The investigation w1x will be extended to the case of light propagation in gyrotropic magnetic superlattices exhibiting antiferromagnetic helical spin structure, characterized by a fourth-rank optical tensor connected with magnetic order.
2. The nonlinear polarization The appearance of the third-order electric polar™ ization P Ž3., which is basic for the effect of polariza-
tion bistability, is conditioned by the fourth-rank optical tensor x i jk l 4 . We assume that this tensor is connected with the antiferromagnetic helical spin structure of the MnSerZnTe superlattice. For the calculation of the spatial modulation and symmetry properties of that tensor involving group-theoretical methods, we refer to Ref. w4x. The tensor x i jk l 4 has 16 nonzero components of which five are linearly independent. There appear two kinds of spatial modulation of these components, depending on 2 a or 4a , where a is the turning angle between two adjacent antiferromagnetic planes of the helix. In this paper, we consider the 4a modulation. Utilizing the calculated expression for the relevant tensor components and passing from™the Cartesian to circularly ™ polarized electric field, E, and polarization, P Ž3., by means of the formulae: E1,2 s Ž E x " iE y .r2 and Ž3. P1,2 s Ž PxŽ3. " iPyŽ3. .r2, we obtain the left and right circular-polarization components in the form: P1Ž3. s 4g E22 E1U eyi 4 a
)
Corresponding author. Tel.: q48-22-6605230; Fax: q48-26282171; E-mail: [email protected]
P2Ž3. s 4g E12 E2U e i4 a ,
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 6 8 9 - 8
Ž 1.
M. Wierzbicki, J. Kocinskir Applied Surface Science 142 (1999) 272–275 ´
where g denotes the amplitude of the fourth-rank optical tensor connected with the presence of the antiferromagnetic spiral, and E1 and E2 are the respective electric field circular components.
3. The wave equation for circularly polarized electric fields Without currents and charges, Maxwell’s equations yield the following wave equation:
E
™
= = = = E s ymr m 0
2
Et
2
™
D,
Ž 2.
where mr denotes the isotropic linear magnetic permeability of the MnSerZnTe superlattice for the paramagnetic phase of the ™MnSe layers, m 0 is the ™ ™ vacuum permeability and D s ee ˆ 0 E q PNL , with eˆ denoting the mean electric permeability and e 0 the vacuum permeability. The fourth-rank tensor xˆ appears only in the MnSe layers of the superlattice. The average value of the tensor in the superlattice therefore is required. Since the spiral pitch is much larger than the thickness of a single MnSe layer, a single layer is approximately spatially homogeneous. The method of evaluation of average values of tensors in magnetic superlattices was elaborated in Refs. w5,6x. In our case, this method leads to an effective tensor to which the magnetic and non-magnetic layers contribute with appropriate weights, proportional to the thickness of the layers. Consequently, the magnitude of the tensor is diminished in comparison with that in a single magnetic layer by a constant factor which can be absorbed by g . We assume that the second-rank tensor character of the linear permeability is caused by the presence of a weak external magnetic field B directed along the spiral axis, i.e., z-axis. The magnetic field brings about an anisotropic contribution D e to the isotropic permeability e of the form: 0 D e s ih B 1 0
ž
y1 0 0
0 0 , 0
/
Ž 3.
where h is a constant depending on the magnitude of the magnetic field. The linear Faraday effect depends on D e . When circular components E1 and E2 of the
273
light wave are introduced, the tensor D eˆ is diagonalized. We assume a monochromatic incoming wave of frequency v depending only on the z-coordinate, along the spiral axis. We introduce the circularly ™ polarized fields E1 and E2 with the wave vectors k 1 ™ and k 2 in the linear approximation determined by:
ž
2 k 1,2 s k 02 1 .
hB er
/
,
Ž 4.
(
where k 0 s vry and y s 1r mr m 0 e r e 0 is the light velocity in a linear medium. With E1 and E2 fields, Eq. Ž2. takes the form: d2
k 02
dz
er e 0
E q k m2 Em s y 2 m
PmNL ,
where m s 1,2.
Ž 5. The fields E1 and E2 are represented by forward Žf. and backward Žb. moving waves: Em Ž z . s Emf Ž z . q Emb Ž z . s A m Ž z . eyi k m z q Bm Ž z . e i k m z ,
Ž 6.
where A mŽ z . and BmŽ z . are the complex amplitudes, m s 1,2. We utilize the approximation of slowly varying amplitude w7x.
4. The nonlinear equations for the wave amplitudes and phases We assume that the wave vector ™ q of the spiral spin structure is much smaller than the wave vectors of the light waves E and E2 , and at the same time ™ ™1 that: 2 < ™ q < f < k 1 < y < k 2 <. The latter difference is determined by the magnitude of the external magnetic field and hence, it can be adjusted to fit the length of the spiral pitch. Multiplying both sides of Eq. Ž5. by expŽ"ik m z . and retaining only the terms containing the factor expw"4iŽ q y Ž k 1 y k 2 . z .x, we obtain the equations for the complex amplitudes A1 , A 2 and B1 , B2 : d Am
s yGm AUm A22ym ,
d Bm
s 0, Ž 7. dz where Gm s y2 ig k 02rŽ e 0 e r k m ., m s 1,2. From Eq. Ž7., it follows that the spiral structure acts like a dz
M. Wierzbicki, J. Kocinskir Applied Surface Science 142 (1999) 272–275 ´
274
tions to only two equations for b and d , from which we deduce the second conservation law in the form:
Ž b 2 y C12 . cos d s C2 s const.
Ž 11 .
Expressing d in terms of b , C1 and C2 , we obtain a single equation for b with the solution:
b Ž z . s b 2 sn Ž b 1 z q u 0 , k . ,
Fig. 1. The ratio ar b of the semi-axes of the polarization ellipse of the transmitted wave as a function of the incoming wave intensity I0 , measured in the units of the quantity G 1 L, which appears in Eq. Ž14.. The difference G 2 y G 1 is proportional to the intensity of external magnetic field and was assumed to be equal 2% in the numerical calculations.
nonlinear optical filter in which one of the two waves moving in opposite directions undergoes a nonlinear interaction, while the other propagates without an interaction. From Eq. Ž7., we obtain the energy conservation law in the form:
G 2 A1 AU1 q G 1 A 2 AU2 s C1 s const.
Ž 12 .
2 where k s b 2rb 1 , with b 1,2 s C12 " < C2 < is the modulus of the elliptic function. The constant u 0 is determined by the value of b at z s 0. From Eq. Ž8., 2 s C1 " b . Introducing these expreswe find that a1,2 sions into Eq. Ž10. for the phases f 1 and f 2 and utilizing Eq. Ž11. for cos d and Eq. Ž12. for b Ž z ., we obtain the expressions for the phase differences:
D f 1,2 s f 1,2 Ž L . y f 1,2 Ž 0 . s
C2 2 b 1 C1
du
u L , Husu Ž0 . 1 " a sn u Ž .
Ž 13 .
where uŽ z . s b 1 q u 0 and a s b 2rC1 , Ž0 F a F 1.. The right-hand side of Eq. Ž13. represents an incomplete elliptic integral of the third kind w8x. The calculation of the phase difference is based on the assumption that the change of the amplitudes a1 and a 2 in the Fabry–Perot ´ cavity is negligible and only
Ž 8.
To obtain the second conservation law which will be indispensable for the integration of Eq. Ž7., we express the complex amplitudes A1 and A 2 through the real amplitudes a1 and a 2 and phases f 1 and f2: A m Ž z . s am Ž z . e i f mŽ z . ,
m s 1,2,
Ž 9.
obtaining the equations: d am dz d fm dz
s Ž y1 .
mq1
Gm a m a22ym sin d ,
2 s yGm a 2ym cos d ,
Ž 10 .
with d s 2Ž f 1 y f 2 .. Introducing the parameter: b s G 2 a12 y G 1 a 22 , we can reduce the above four equa-
Fig. 2. The inclination angle c , with respect to the x-axis of the longer semi-axis of the polarization ellipse of the transmitted wave as a function of the incoming wave intensity I0 , measured in the units of the quantity G 1 L, which appears in Eq. Ž14.. The angle c is measured in degrees.
M. Wierzbicki, J. Kocinskir Applied Surface Science 142 (1999) 272–275 ´
the phases f 1 and f 2 are essentially altered. We obtain the expressions: D fm s y
Gm L T
Imout cos d ,
m s 1,2,
Ž 14 .
where I0 T 2
out I1,2 s 2
Ž 1 y R . q 4 R sin
2
ž
1 2
. D f i y kL
Ž 15 .
/
From the boundary conditions of a light wave in the cavity, we obtain an additional relation between the initial phases and the nonlinear phase differences in the form: tan fm Ž 0 . s
R sin Ž D fm y 2 kL .
,
1 y R cos Ž D fm y 2 kL .
m s 1,2.
275
6. Conclusions The polarization bistability discussed in this paper is conditioned by two circumstances: Ž1. the existence of third-order electric polarization and Ž2. the presence of external magnetic field parallel to the helical axis which implies the anisotropy of the refractive index for the circularly polarized waves. Only the combination of these two conditions leads to the polarization bistability. In the present constructions of optical logical devices, only the effect of light wave intensity bistability is utilized. The here examined two forms of polarization bistability which can be controlled by a weak external magnetic field open new possibilities in the field of construction of optical logical devices.
Ž 16 . Eqs. Ž14. – Ž16. allow to determine numerically the outgoing-waves intensities I1out , I2out and cos d as functions of the incoming intensity I0 .
5. The polarization bistability The polarization state of the wave transmitted through the cavity is determined by the coherence matrix w9x. The components J x x , J y y and J x y of that ™ matrix for the transmitted wave E out are expressed through the circularly polarized components E1out and E2out. Consequently, for the ratio of the semi-axes of the polarization ellipse of the transmitted wave, we obtain the formula: a b
s tan
1 2
arc sin
ž
I1out y I2out I1out q I2out
/
,
Ž 17 .
and for the angle c of inclination of the longer semi-axis with respect to the x-axis the formula: U
1
c s arc tan 2
U
i Ž E1out . E2out y i Ž E2out . E1out U
Ž E1out .
U
E2out q Ž E2out . E1out
,
Ž 18 .
The results of numerical calculations based on Eqs. Ž17. and Ž18. are represented in Figs. 1 and 2, respectively.
Acknowledgements This work was sponsored by a grant of the Warsaw University of Technology and by grant no. 2 PO3B 105 14 of the State Committee for Scientific Research ŽPoland.. We thank Prof. Jan Petykiewicz for inspiring discussions concerning problems of optical bistability.
References w1x J. Petykiewicz, D. Strojewski, Opt. Quantum Electron. 22 Ž1990. 131. w2x K. Panajotov, T. Tenev, G. Zartov, M. Pelt, J. Danckaert, H. Thienpont, I. Veretennicoff, Int. J. Nonlinear Opt. Phys. 5 Ž1996. 351. w3x R.C. Tu, Y.K. Su, H.J. Chen, Y.S. Huang, S.T. Chou, W.H. Lan, S.L. Tu, J. Appl. Phys. 84 Ž1998. 2866. w4x M. Wierzbicki, J. Kocinski, Surface Science 377–379 Ž1997. ´ 418. w5x N.S. Almeida, D.L. Mills, Phys. Rev. B 38 Ž1988. 6698. w6x N. Raj, D.R. Tilley, Phys. Rev. B 36 Ž1987. 7003. w7x J. Petykiewicz, Wave Optics, PWN, Warszawa, 1992. w8x P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971. w9x M. Born, E. Wolf, Principles of Optics, Pergamon, London, 1965.