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Polarization-independent triangular-groove fused-silica gratings with high efficiency at a wavelength of 1550 nm
Optics Communications 283 (2010) 4271–4273
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Polarization-independent triangular-groove fused-silica gratings with high efficiency at a wavelength of 1550 nm Hongchao Cao, Changhe Zhou ⁎, Jijun Feng, Peng Lv, Jianyong Ma Information Optics Lab, Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211, Shanghai 201800, China
a r t i c l e
i n f o
Article history: Received 6 April 2010 Received in revised form 17 June 2010 Accepted 17 June 2010
a b s t r a c t We describe polarization-independent triangular-groove fused-silica gratings illuminated by incident lights in the C + L bands as (de)multiplexers for dense wavelength division multiplexing (DWDM) application. The physical mechanisms of the grating can be shown clearly by using the simplified modal method with consideration of the corresponding accumulated average phase difference of two excited propagating grating modes, which illustrates that the grating structure depends mainly on the ratio of the average effective indices difference to the incident wavelength. Exact grating profile is optimized by using the rigorous coupled-wave analysis (RCWA). With the optimized grating parameters, the grating exhibits diffraction efficiencies of greater than 90% under TE- and TM-polarized incident lights for 101 nm spectral bandwidths (1500–1601 nm) and it can reach an efficiency of more than 99% for both polarizations at a wavelength of 1550 nm. Without loss of metal absorption, coating of dielectric film layers, the designed triangular-groove fused-silica grating should be of great interest for DWDM application. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The advent of the dense wavelength division multiplexing (DWDM) technology greatly reduces the channel spacing and considerably enhances the number of communication channels [1]. Therefore, the bandwidth of a normal fiber can be enlarged enormously. DWDM technology can be realized through thin-film filters, arrayed waveguide gratings, fiber Bragg gratings (FBGs), and free-space diffraction gratings [2].Generally, free-space diffraction gratings have advantages over other devices [3,4]. They can (de) multiplex optical signals in a parallel fashion with high dispersion; they are athermal devices under all environmental conditions; and, most important, they have relatively low polarization-dependent losses. Among different types of free-space diffraction gratings, metallic reflection gratings can achieve a diffraction efficiency of ∼85% through high reflectivity of a metal film [5]. However, the power absorption of a metal film is an impediment to further efficiency improvement. A dielectric film can be an alternative to the metal film because of its lower power absorption rate. However, it is fairly difficult to fabricate a highly reflecting dielectric stack below a surface-relief grating to realize high diffraction efficiency [6]. The combination of the metallic and the dielectric gratings, i.e., metallo-dielectric gratings, can offer very high reflective efficiencies with a reduced number of dielectric films [7,8]. Most reported high efficiencies gratings are designed for one polariza⁎ Corresponding author. E-mail address: [email protected] (C. Zhou). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.06.060
tion, that is to say, polarization-dependent. However, a polarizationinsensitive or polarization-independent device would be more favorable for practical application [9–12]. To our knowledge, a polarizationindependent triangular-groove fused-silica grating used in the C + L band as (de)multiplexers for DWDM has not been reported before. In this paper, we present the design of a polarization-independent high efficiency triangular-groove transmission fused-silica grating used in the C + L band as (de)multiplexers for DWDM with the simplified modal method [13–18]. Unlike the rectangular-groove grating, the average phase difference which was firstly proposed by Zheng et al. [17] and accumulated between even and odd grating modes should be considered in the triangular-groove grating designs. Optimum grating parameters can be determined by using the rigorous coupled-wave analysis (RCWA) [19,20]. Note that, the RCWA is not the best method when dealing with lamellar gratings [21]. However, it is sufficient in our calculations since only the dielectric material is involved. We also mention that the obtained ratio of grating period to incident wavelength can be used as guideline for grating designs at other wavelength. Finally some discussions and conclusions are given. 2. Grating design As shown in Fig. 1, the grating with period Λ and groove depth h is illuminated under Littrow mounting (incident angle θin = arcsin (λ/2/Λ)) at a central wavelength λ of 1550 nm (the index of fusedsilica substrate n2 = 1.44462). For triangular-groove grating it can be approximated by a stack of lamellar gratings. In each layer there will be a certain amount of phase difference accumulated between the
4272
H. Cao et al. / Optics Communications 283 (2010) 4271–4273
even and the odd grating modes excited by the incident wave [17] (the definition of odd and even modes of the grating in this paper accord with that in Ref. 17). As Zheng et al. has pointed out that the exchange of energy can only take place among the even and the odd modes separately in the adjacent layers. According to the modal method the diffraction efficiency can be mainly determined by the P accumulated average phase difference Δφ of the two modes [17]: 2
η−1 = sin
P
Δφ ; 2
ð1Þ
with P
P
Δφ = Δneff k0 h;
ð2Þ
P
Δneff = ∑ðni;even −ni;odd Þhi = h i
ð3Þ
where ηp is the diffraction efficiency of the pth (p = −1) order, k0 is the wave number in vacuum (k0 = 2π/λ), hi is the depth of the ith layer, ni,even and ni,odd are corresponding even and odd mode effective P indices, and Δneff is the average difference of mode indices. If P Δφ = ð2l−1Þπ,(l = 1, 2 …) for TE and TM polarizations, a high diffraction efficiency grating can be achieved. So, the guideline for the design of a polarization-independent device for the −1st order high efficiency should be [18] P Δneff XTM ΔφTM 2l−1 ; = P = P 2m−1 ΔφTE Δneff XTE P
ð4Þ
Eqs. (5) and (6) can be derived by solving the Helmholtz equation in the grating region with the corresponding boundary and pseudoperiodic conditions [22–24]. From Eqs. (5) and (6), we can see that the grating effective index neff varies with the duty cycle f and the ratio of wavelength to grating period λ/Λ. For triangular-groove gratings the duty cycle changes from 0 to 1. So the average difference of the effective grating modes is dependent only on the ratio of the wavelength to P grating period. The plot of average effective indices differences Δneff for TE and TM polarizations versus grating period Λ and the ratio of corresponding effective indices difference as a function of ratio of grating period to wavelength Λ/λ are shown in Fig. 2 (a) and (b), respectively. And different ratio corresponds to different l an m in Eq. (4). For ease of fabrication, the lowest groove depth is needed. As can be seen from Fig. 2 (b), the ratio can be chosen as 1/3. And the corresponding ratio of incident wavelength to grating period is 1.43. P P In this case, ΔφTM = π and ΔφTE = 3π, and the grating period Λ is P 1080 nm. The average effective indices difference is ΔneffPTM = 0:075 P and ΔneffPTE = 0:23. Thus the groove depth is calculated to be 10,300 nm. Fig. (3) shows the optimum minimum diffraction efficiencies between TE and TM polarizations in the −1st diffractive order versus grating period and groove depth. As can be seen from Fig. 3, there is a window (10,200 nmb h b10,400 nm, and 1070 nmb Λ b 1095 nm) within which the minus-first order diffraction efficiencies for both TE- and TM-polarized waves can exhibit a value of greater than 99%. So, a high-efficient polarization-independent transmission grating can be obtained. Fig. 4 shows the minus-first order diffraction efficiencies versus wavelength for TE and TM polarizations calculated by using the RCWA method, and the incident angle is Littrow
with l and m being arbitrary positive integers. The corresponding effective mode indices can be found by solving the following eigenfunction equation for TE polarization (electric field vector perpendicular to the plane of incidence) [22]: cos½k1 ð1−f ÞΛ cosðk2 f ΛÞ−
k21 + k22 sin½k1 ð1−f ÞΛ sinðk2 f ΛÞ = cosðαΛÞ; 2k1 k2
ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with ki = k0 n2i −n2eff and α = k0sinθin, under Littrow mounting, cosαΛ = − 1. For TM polarization (magnetic field vector perpendicular to the plane of incidence), the equation can be expressed as cos½k1 ð1−f ÞΛ cosðk2 f ΛÞ−
n42 k21 + k22 sin½k1 ð1−f ÞΛ sinðk2 f ΛÞ = cosðαΛÞ: 2n22 k1 k2 ð6Þ
Fig. 1. Schematic illustration of a triangular-groove transmission grating (n1 and n2: refractive indices of air and fused-silica, respectively, Λ: grating period, h: groove depth, θ− 1: diffraction angles of the − 1st order).
P
Fig. 2. Plot of the average effective indices difference Δneff versus grating period for (a). (b) Plot of the ratio of effective indices differences for TM polarization to that for TE polarization versus ratio of wavelength to period.
H. Cao et al. / Optics Communications 283 (2010) 4271–4273
Fig. 3. Contour of the minimum − 1st order diffraction efficiencies between TE and TM polarizations versus groove depth and grating period.
configuration for 1550 nm wavelength. We can see clearly that the designed grating exhibits diffraction efficiencies of greater than 90% under TE- and TM-polarized incident lights for 101 nm spectral bandwidths (1500–1601 nm) and it can reach an efficiency of more than 99% for both polarizations at a wavelength of 1550 nm, which would be useful in optical communication system. 3. Conclusions In conclusion, we have designed a deep-etched polarizationindependent triangular-groove fused-silica grating used as (de) multiplexers in the C + L bands. Such gratings can realize transmission efficiencies of nearly 100% with large spectral bandwidth (1540–1560) for TE- and TM-polarized incident light without coating metallic or dielectric layers. With optimized grating parameters, a triangular-groove fused-silica grating can realize efficiencies of greater than 99% at a wavelength of 1550 nm for both polarizations, realizing low polarization-dependent losses. This work provides an important guideline for the use of high efficiency triangular-groove gratings in DWDM application. Acknowledgements The authors acknowledge the support of National Natural Science Foundation of China (60878035), Shanghai Science and Technology Committee (09520709300, 10ZR1433500) and 863 programm (2008AA03Z406).
4273
Fig. 4. Plot of the − 1st order diffraction efficiencies versus incident wavelength with grating period Λ = 1080 nm, h = 10300 nm and θi = 45.85° (Littrrow angle for 1550 nm wavelength of incidence).
References [1] J.-P. Laude, DWDM Fundamentals, Components, and Applications, Artech House Optoelectronics Library, 2002. [2] J. Qiao, F. Zhao, R.T. Chen, J.W. Horwitz, W.W. Morey, Appl. Opt. 41 (2002) 6567. [3] A. Sappey, Photonics Spectra 36 (2002) 78. [4] A. Sappey, P. Huang, WDM Solutions R&D Rev. 3 (2001) 39. [5] http://www.agilent.com. [6] L. Li, J. Hirsh, Opt. Lett. 20 (1995) 1349. [7] N. Destouches, A.V. Tishchenko, J.C. Pommier, S. Reynaud, O. Parriaux, Opt. Express 13 (2005) 3230. [8] N. Bonod, J. Néauport, Opt. Commun. 260 (2006) 649. [9] N. Bonod, E. Popov, S. Enoch, J. Néauport, J. Eur. Opt. Soc. 1 (2006) 06029. [10] J.R. Marciante, J.I. Hirsh, D.H. Rauin, E.T. Prince, J. Opt, Soc. Am. A 22 (2005) 299. [11] Y. Zhang, C. Zhou, J. Opt, Soc. Am. A 22 (2005) 331. [12] S. Wang, C. Zhou, Y. Zhang, H. Ru, Appl. Opt. 45 (2006) 2567. [13] A.V. Tishchenko, Opt. Quantum Electron. 37 (2005) 309. [14] T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A.V. Tishchenko, O. Parriaux, Opt. Express 13 (2005) 10448. [15] T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, O. Parriaux, Appl. Opt. 46 (2007) 819. [16] T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, O. Parriaux, Opt. Express 16 (2008) 5577. [17] J. Zheng, C. Zhou, J. Feng, B. Wang, Opt. Lett. 33 (2008) 1554. [18] J. Feng, C. Zhou, J. Zheng, B. Wang, Opt. Commun. 281 (2008) 5298. [19] M.G. Moharam, D.A. Pommet, E.B. Grann, J. Opt, Soc. Am. A 12 (1995) 1077. [20] P. Lalanne, G.M. Morris, J. Opt. Soc. Am. A 13 (1996) 779. [21] L. Li, C.W. Haggans, J. Opt, Soc. Am. A 10 (1993) 1184. [22] L.C. Botten, M.S. Craig, R.C. Mcphedran, J.L. Adams, J.l.R. Andrewartha, Opt. Acta 28 (1981) 413. [23] P. Sheng, R.S. Stepleman, P.N. Sanda, Phys. Rev. B 26 (1982) 2907. [24] S.M. Rytov, Sov. Phys. JETP 2 (1956) 466.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Polarization-independent triangular-groove fused-silica gratings with high efficiency at a wavelength of 1550 nm Hongchao Cao, Changhe Zhou ⁎, Jijun Feng, Peng Lv, Jianyong Ma Information Optics Lab, Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211, Shanghai 201800, China
a r t i c l e
i n f o
Article history: Received 6 April 2010 Received in revised form 17 June 2010 Accepted 17 June 2010
a b s t r a c t We describe polarization-independent triangular-groove fused-silica gratings illuminated by incident lights in the C + L bands as (de)multiplexers for dense wavelength division multiplexing (DWDM) application. The physical mechanisms of the grating can be shown clearly by using the simplified modal method with consideration of the corresponding accumulated average phase difference of two excited propagating grating modes, which illustrates that the grating structure depends mainly on the ratio of the average effective indices difference to the incident wavelength. Exact grating profile is optimized by using the rigorous coupled-wave analysis (RCWA). With the optimized grating parameters, the grating exhibits diffraction efficiencies of greater than 90% under TE- and TM-polarized incident lights for 101 nm spectral bandwidths (1500–1601 nm) and it can reach an efficiency of more than 99% for both polarizations at a wavelength of 1550 nm. Without loss of metal absorption, coating of dielectric film layers, the designed triangular-groove fused-silica grating should be of great interest for DWDM application. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The advent of the dense wavelength division multiplexing (DWDM) technology greatly reduces the channel spacing and considerably enhances the number of communication channels [1]. Therefore, the bandwidth of a normal fiber can be enlarged enormously. DWDM technology can be realized through thin-film filters, arrayed waveguide gratings, fiber Bragg gratings (FBGs), and free-space diffraction gratings [2].Generally, free-space diffraction gratings have advantages over other devices [3,4]. They can (de) multiplex optical signals in a parallel fashion with high dispersion; they are athermal devices under all environmental conditions; and, most important, they have relatively low polarization-dependent losses. Among different types of free-space diffraction gratings, metallic reflection gratings can achieve a diffraction efficiency of ∼85% through high reflectivity of a metal film [5]. However, the power absorption of a metal film is an impediment to further efficiency improvement. A dielectric film can be an alternative to the metal film because of its lower power absorption rate. However, it is fairly difficult to fabricate a highly reflecting dielectric stack below a surface-relief grating to realize high diffraction efficiency [6]. The combination of the metallic and the dielectric gratings, i.e., metallo-dielectric gratings, can offer very high reflective efficiencies with a reduced number of dielectric films [7,8]. Most reported high efficiencies gratings are designed for one polariza⁎ Corresponding author. E-mail address: [email protected] (C. Zhou). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.06.060
tion, that is to say, polarization-dependent. However, a polarizationinsensitive or polarization-independent device would be more favorable for practical application [9–12]. To our knowledge, a polarizationindependent triangular-groove fused-silica grating used in the C + L band as (de)multiplexers for DWDM has not been reported before. In this paper, we present the design of a polarization-independent high efficiency triangular-groove transmission fused-silica grating used in the C + L band as (de)multiplexers for DWDM with the simplified modal method [13–18]. Unlike the rectangular-groove grating, the average phase difference which was firstly proposed by Zheng et al. [17] and accumulated between even and odd grating modes should be considered in the triangular-groove grating designs. Optimum grating parameters can be determined by using the rigorous coupled-wave analysis (RCWA) [19,20]. Note that, the RCWA is not the best method when dealing with lamellar gratings [21]. However, it is sufficient in our calculations since only the dielectric material is involved. We also mention that the obtained ratio of grating period to incident wavelength can be used as guideline for grating designs at other wavelength. Finally some discussions and conclusions are given. 2. Grating design As shown in Fig. 1, the grating with period Λ and groove depth h is illuminated under Littrow mounting (incident angle θin = arcsin (λ/2/Λ)) at a central wavelength λ of 1550 nm (the index of fusedsilica substrate n2 = 1.44462). For triangular-groove grating it can be approximated by a stack of lamellar gratings. In each layer there will be a certain amount of phase difference accumulated between the
4272
H. Cao et al. / Optics Communications 283 (2010) 4271–4273
even and the odd grating modes excited by the incident wave [17] (the definition of odd and even modes of the grating in this paper accord with that in Ref. 17). As Zheng et al. has pointed out that the exchange of energy can only take place among the even and the odd modes separately in the adjacent layers. According to the modal method the diffraction efficiency can be mainly determined by the P accumulated average phase difference Δφ of the two modes [17]: 2
η−1 = sin
P
Δφ ; 2
ð1Þ
with P
P
Δφ = Δneff k0 h;
ð2Þ
P
Δneff = ∑ðni;even −ni;odd Þhi = h i
ð3Þ
where ηp is the diffraction efficiency of the pth (p = −1) order, k0 is the wave number in vacuum (k0 = 2π/λ), hi is the depth of the ith layer, ni,even and ni,odd are corresponding even and odd mode effective P indices, and Δneff is the average difference of mode indices. If P Δφ = ð2l−1Þπ,(l = 1, 2 …) for TE and TM polarizations, a high diffraction efficiency grating can be achieved. So, the guideline for the design of a polarization-independent device for the −1st order high efficiency should be [18] P Δneff XTM ΔφTM 2l−1 ; = P = P 2m−1 ΔφTE Δneff XTE P
ð4Þ
Eqs. (5) and (6) can be derived by solving the Helmholtz equation in the grating region with the corresponding boundary and pseudoperiodic conditions [22–24]. From Eqs. (5) and (6), we can see that the grating effective index neff varies with the duty cycle f and the ratio of wavelength to grating period λ/Λ. For triangular-groove gratings the duty cycle changes from 0 to 1. So the average difference of the effective grating modes is dependent only on the ratio of the wavelength to P grating period. The plot of average effective indices differences Δneff for TE and TM polarizations versus grating period Λ and the ratio of corresponding effective indices difference as a function of ratio of grating period to wavelength Λ/λ are shown in Fig. 2 (a) and (b), respectively. And different ratio corresponds to different l an m in Eq. (4). For ease of fabrication, the lowest groove depth is needed. As can be seen from Fig. 2 (b), the ratio can be chosen as 1/3. And the corresponding ratio of incident wavelength to grating period is 1.43. P P In this case, ΔφTM = π and ΔφTE = 3π, and the grating period Λ is P 1080 nm. The average effective indices difference is ΔneffPTM = 0:075 P and ΔneffPTE = 0:23. Thus the groove depth is calculated to be 10,300 nm. Fig. (3) shows the optimum minimum diffraction efficiencies between TE and TM polarizations in the −1st diffractive order versus grating period and groove depth. As can be seen from Fig. 3, there is a window (10,200 nmb h b10,400 nm, and 1070 nmb Λ b 1095 nm) within which the minus-first order diffraction efficiencies for both TE- and TM-polarized waves can exhibit a value of greater than 99%. So, a high-efficient polarization-independent transmission grating can be obtained. Fig. 4 shows the minus-first order diffraction efficiencies versus wavelength for TE and TM polarizations calculated by using the RCWA method, and the incident angle is Littrow
with l and m being arbitrary positive integers. The corresponding effective mode indices can be found by solving the following eigenfunction equation for TE polarization (electric field vector perpendicular to the plane of incidence) [22]: cos½k1 ð1−f ÞΛ cosðk2 f ΛÞ−
k21 + k22 sin½k1 ð1−f ÞΛ sinðk2 f ΛÞ = cosðαΛÞ; 2k1 k2
ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with ki = k0 n2i −n2eff and α = k0sinθin, under Littrow mounting, cosαΛ = − 1. For TM polarization (magnetic field vector perpendicular to the plane of incidence), the equation can be expressed as cos½k1 ð1−f ÞΛ cosðk2 f ΛÞ−
n42 k21 + k22 sin½k1 ð1−f ÞΛ sinðk2 f ΛÞ = cosðαΛÞ: 2n22 k1 k2 ð6Þ
Fig. 1. Schematic illustration of a triangular-groove transmission grating (n1 and n2: refractive indices of air and fused-silica, respectively, Λ: grating period, h: groove depth, θ− 1: diffraction angles of the − 1st order).
P
Fig. 2. Plot of the average effective indices difference Δneff versus grating period for (a). (b) Plot of the ratio of effective indices differences for TM polarization to that for TE polarization versus ratio of wavelength to period.
H. Cao et al. / Optics Communications 283 (2010) 4271–4273
Fig. 3. Contour of the minimum − 1st order diffraction efficiencies between TE and TM polarizations versus groove depth and grating period.
configuration for 1550 nm wavelength. We can see clearly that the designed grating exhibits diffraction efficiencies of greater than 90% under TE- and TM-polarized incident lights for 101 nm spectral bandwidths (1500–1601 nm) and it can reach an efficiency of more than 99% for both polarizations at a wavelength of 1550 nm, which would be useful in optical communication system. 3. Conclusions In conclusion, we have designed a deep-etched polarizationindependent triangular-groove fused-silica grating used as (de) multiplexers in the C + L bands. Such gratings can realize transmission efficiencies of nearly 100% with large spectral bandwidth (1540–1560) for TE- and TM-polarized incident light without coating metallic or dielectric layers. With optimized grating parameters, a triangular-groove fused-silica grating can realize efficiencies of greater than 99% at a wavelength of 1550 nm for both polarizations, realizing low polarization-dependent losses. This work provides an important guideline for the use of high efficiency triangular-groove gratings in DWDM application. Acknowledgements The authors acknowledge the support of National Natural Science Foundation of China (60878035), Shanghai Science and Technology Committee (09520709300, 10ZR1433500) and 863 programm (2008AA03Z406).
4273
Fig. 4. Plot of the − 1st order diffraction efficiencies versus incident wavelength with grating period Λ = 1080 nm, h = 10300 nm and θi = 45.85° (Littrrow angle for 1550 nm wavelength of incidence).
References [1] J.-P. Laude, DWDM Fundamentals, Components, and Applications, Artech House Optoelectronics Library, 2002. [2] J. Qiao, F. Zhao, R.T. Chen, J.W. Horwitz, W.W. Morey, Appl. Opt. 41 (2002) 6567. [3] A. Sappey, Photonics Spectra 36 (2002) 78. [4] A. Sappey, P. Huang, WDM Solutions R&D Rev. 3 (2001) 39. [5] http://www.agilent.com. [6] L. Li, J. Hirsh, Opt. Lett. 20 (1995) 1349. [7] N. Destouches, A.V. Tishchenko, J.C. Pommier, S. Reynaud, O. Parriaux, Opt. Express 13 (2005) 3230. [8] N. Bonod, J. Néauport, Opt. Commun. 260 (2006) 649. [9] N. Bonod, E. Popov, S. Enoch, J. Néauport, J. Eur. Opt. Soc. 1 (2006) 06029. [10] J.R. Marciante, J.I. Hirsh, D.H. Rauin, E.T. Prince, J. Opt, Soc. Am. A 22 (2005) 299. [11] Y. Zhang, C. Zhou, J. Opt, Soc. Am. A 22 (2005) 331. [12] S. Wang, C. Zhou, Y. Zhang, H. Ru, Appl. Opt. 45 (2006) 2567. [13] A.V. Tishchenko, Opt. Quantum Electron. 37 (2005) 309. [14] T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A.V. Tishchenko, O. Parriaux, Opt. Express 13 (2005) 10448. [15] T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, O. Parriaux, Appl. Opt. 46 (2007) 819. [16] T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, O. Parriaux, Opt. Express 16 (2008) 5577. [17] J. Zheng, C. Zhou, J. Feng, B. Wang, Opt. Lett. 33 (2008) 1554. [18] J. Feng, C. Zhou, J. Zheng, B. Wang, Opt. Commun. 281 (2008) 5298. [19] M.G. Moharam, D.A. Pommet, E.B. Grann, J. Opt, Soc. Am. A 12 (1995) 1077. [20] P. Lalanne, G.M. Morris, J. Opt. Soc. Am. A 13 (1996) 779. [21] L. Li, C.W. Haggans, J. Opt, Soc. Am. A 10 (1993) 1184. [22] L.C. Botten, M.S. Craig, R.C. Mcphedran, J.L. Adams, J.l.R. Andrewartha, Opt. Acta 28 (1981) 413. [23] P. Sheng, R.S. Stepleman, P.N. Sanda, Phys. Rev. B 26 (1982) 2907. [24] S.M. Rytov, Sov. Phys. JETP 2 (1956) 466.