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Polarization dependent guiding in liquid crystal filled photonic crystal fibers
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Optics Communications 281 (2008) 1598–1606 www.elsevier.com/locate/optcom
Polarization dependent guiding in liquid crystal filled photonic crystal fibers Guobin Ren a,*, Ping Shum a, Xia Yu a, JuanJuan Hu a, Guanghui Wang a, Yandong Gong b a
Network Technology Research Centre, Nanyang Technological University, 50 Nanyang Drive, 637553 Singapore, Singapore b Institute for InfoComm Research, 21 Heng Mui Keng Terrace, 119613 Singapore, Singapore Received 11 October 2007; received in revised form 21 November 2007; accepted 21 November 2007
Abstract A theoretical study of nematic liquid crystal filled photonic crystal fibers (LCPCFs) is presented. Detailed investigations including the polarization dependent bandgap formation and the modal properties are given for LCPCFs, in which alignment of the molecules could be controlled by external static electric field. The polarization dependent bandgap splitting caused by the high index difference between the ordinary and the extraordinary dielectric index of nematic liquid crystals provides the possibility of single-mode single-polarization guiding. A polarization operation diagram is proposed to describe the guiding behavior of LCPCFs. The influence of rotation angle / of the director of liquid crystals on the modal properties is investigated. It is shown that the polarization axis of the guided mode is determined by the rotation angle /, which could be controlled by external electric field. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Fiber optics; Photonic bandgap fibers; Liquid crystal devices; Polarization-maintaining
1. Introduction Photonic crystal fibers (PCF) have attracted large scientific and commercial interest since they offer a wide range promising applications in communication and sensing [1,2]. Light guiding in a PCF is governed by one of two principal mechanisms responsible for light trapping within the core. The first is a simple mechanism based on the modified total internal reflection (TIR) phenomenon, which is well known within conventional fiber. The other is known as a photonic bandgap (PBG) effect, which relies on the coherent backscattering of light into the core. Propagation and polarization properties of PCF can be manipulated by filling fluid materials such as polymer, oil, or liquid crystal (LC) [3–5] into the air holes of PCF. Among these materials LC is of particular interest due to
*
Corresponding author. E-mail address: [email protected] (G. Ren).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.084
their electrically or thermally tunable refractive index, which could boost PCFs to a higher level of tunability. The thermal tunability of liquid crystal filled PCFs (LCPCFs) was achieved taking advantage of thermo-optic effect of LCs by using either the external heaters or the guided pump beam absorption in dye-doped LCs [6–8]. The first experimental demonstration of electrical tuning in hollow-core LCPCF was reported in Ref. [9]. By filling the holes of an air-core photonic bandgap fiber (PBGF), TIR guiding and electrically tunable optical switch effects are demonstrated. Haakestad et al. [10] reported electrically tunable PBG guiding of a nematic liquid crystal (NLC) filled PCF (NLCPCF), which was able to operate as an optical switch. In Ref. [11], an electrically controlled continuous tunable LCPCF that allows continuous control of the spectral position of the bandgaps is demonstrated. The continuous tunability is due to dual frequency LC is used to fill the holes of PCF. More recently, Alkeskjold et al. [12] demonstrated an electrically driven broadband LCPCF polarizer, which has potential applications for
G. Ren et al. / Optics Communications 281 (2008) 1598–1606
polarization monitoring and fiber-optic sensors. Polarizer axis control and polarization extinction ratio tuning are achieved by bipolar electrodes driving. Although some experimental works of LCPCFs have been reported in recent years, the theoretical investigation is relatively lagging behind especially taking the anisotropic property of LC into account. The LCPCFs for singlepolarization or high-birefringence guidance were theoretically modeled by Zografopoulos et al. [13]. Since the LCPCF proposed has a NLC core, the polarization characteristics of this fiber is mainly determined by the NLC, and the guiding mechanism is TIR in this case. More recently, Sun et al. [14] investigated the alignment effect of NLC on bandgap formation of the NLCPCF. A simple analytic model is proposed for analyzing the bandgap formation with the assumption of limited index difference. In this model, the coupling of transverse field is neglected. Modal properties of LCPCF are fundamentally interesting and important for LCPCF based device application. However, so far the detailed investigation on modal properties of LCPCF is not available. In this paper, we present a thorough theoretical analysis of LCPCFs including the polarization dependent bandgap formation and modal properties considering the alignment effect of NLC. The coupling terms of transverse field in wave equations are taken into account for the bandgap formation analysis; a general derivation is proposed for analyzing the bandgap formation more accurately. The polarization dependent bandgap is proposed to investigate the mode guiding. The modal properties including polarization splitting behavior, confinement loss, effective area, and the position of transmission windows are investigated. Furthermore, the influence of the alignment of the NLC molecules, which could be controlled by external static electric field, on the modal properties is investigated. 2. Polarization dependent bandgap formation Nematic liquid crystals (NLCs) are anisotropic materials consisting of rod-like molecules; the orientational order of molecules combined with their anisotropy lead to anisotropic optical properties. The local orientation of the NLCs is described by the director: a unit vector along the direction of the average orientation of the molecules, which can be aligned by application of proper boundary conditions to achieve a macroscopic alignment. Under the application of a static electric field the director’s orientation can be controlled, since the liquid crystal molecules tend to align their axis according to the applied field. The LCPCF investigated in this paper is shown in Fig. 1, K is the lattice spacing and d is diameter of hole, in which the NLCs are infiltrated. The director of a NLC is shown in left. In a practical point of view, for silica-based PCF, due to the strong boundary condition in silica surface, the director of NLCs is not homogeneous, especially in the proximity of silica surface. For simplified analysis, the inhomogeneity of the director is neglected and the
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Fig. 1. LCPCF structure with triangular lattices. The shaded region is infiltrated with liquid crystals. The left indicates the director of a nematic liquid crystal.
alignment of the director is assumed to be uniform within the holes in this paper. Generally LCs possess two kinds of dielectric indices. One is the ordinary dielectric index eo, and the other is the extraordinary dielectric index ee. Light waves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary refractive indices, respectively. When electric field is applied transversely (in x–y plane), the dielectric permittivity tensor takes the form of 2 3 exx exy 0 6 7 er ¼ 4 eyx eyy 0 5 ð1Þ 0
0
ezz
where exx ðrÞ ¼ eo ðrÞ sin2 / þ ee ðrÞ cos2 / exy ðrÞ ¼ eyx ðrÞ ¼ ½ee ðrÞ eo ðrÞ sin / cos / eyy ðrÞ ¼ eo ðrÞ cos2 / þ ee ðrÞ sin2 / ezz ðrÞ ¼ eo ðrÞ
ð2Þ
/ is the rotation angle of the director of the LCs and it can be controlled by the external static electric field. n = (cos/, sin/) is the director of the LC, as shown in Fig. 1. For bandgap formation analysis, we assume that the rotation angle of the director / is 0 or p/2, which give rise to a simplification of er = diag[eeeoeo]; for / = 0, and er = diag[eoeeeo] for / = p/2. We will justify this simplification in the last discussion session that the contribution of the rotation angle of the director / for bandgap calculation is small enough to be neglected. The eigenvalue equations in terms of transverse electric fields for / = p/2 is [15]: o2 E x o 1 o ðeo Ex Þ þ k 20 eo Ex þ ox eo ox oy 2 o2 E y o 1 o ðee Ey Þ ¼ b2 Ex þ oxoy ox eo oy ð3Þ o2 E y o 1 o 2 ðee Ey Þ þ k 0 ee Ey þ oy eo oy ox2 o2 E x o 1 o ðeo Ex Þ ¼ b2 Ey þ oxoy oy eo ox
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Eq. (4) is actually the eigenvalue equation of the isotropic systemer = eo. For y-polarization mode HE11y mode of LCPCF, we consider Ex Ey, therefore eoEx is replaced by eeEx in Eq. (3). We have: o2 Ex n2e o 1 o ðfEx Þ þ k 20 n2e fEx þ 2 oy 2 no ox f ox o2 Ey n2e o 1 o ðfEy Þ ¼ b2 Ex þ 2 oxoy no ox f oy ð5Þ 2 o Ey n2e o 1 o 2 2 ðfEy Þ þ k 0 ne fEy þ 2 ox2 no oy f oy o2 Ex n2e o 1 o ðfEx Þ ¼ b2 Ey þ oxoy n2o oy f ox Eq. (5) is the eigenvalue equation of the uniaxial anisotropic systemer = diag[eeeeeo]. Here the director of the anisotropic medium is along z-axis. The refractive index of the PCF background material (silica) is assumed to be 1.45. The 5CB type NLCs are considered in this paper, the ordinary and extraordinary refractive indices of are no = 1.522 and ne = 1.706, respec-
tively. Fig. 1 indicates the director n of a liquid crystal and the rotation angle / of the director to the x-axis. 5CB LC is a suitable choice for demonstration because it exhibits a nematic phase at room temperature and its nematic range is >10 °C. We assume that the operating temperature is at a constant room temperature and that the absorption loss is negligible. A full-vectorial plane wave method is used to calculate the bandgap of the LC cladding of the LCPCF shown in Fig. 1. Fig. 2a shows the polarization independent total bandgap map of the LCPCF. It should be noticed that this bandgap map is governed by Eq. (3) without approximation. The relative hole diameter is d/K = 0.4. The rotation angle of NLC is / = p/2. The shaded parts are the bandgaps in which no photonic state could be supported in fiber
a 1.65 1.6 1.55
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This equation determines the polarization independent total gaps. It is well known that the fundamental mode of the triangular lattice PCF contains two degenerate orthogonal modes, HE11x and HE11y mode [16]. The degeneracy would be split when the asymmetry of fiber structure or birefringent materials is introduced. When the LCs are filled into the air holes of PCF, the intrinsic anisotropy of LC would affect the photonic bands and also the guided modes of fiber, which are totally different from the isotropic situation. The polarization dependent photonic bandgaps would be open up and these bandgaps would guide modes with different polarization states. The formation of polarization dependent bandgap is similar to TE and TM bandgap formation of 2-D photonic crystals with anisotropic materials contained, but due to the hybrid nature of the fiber mode, the strict decoupling of Maxwell equation will not occur in this case. For x-polarization mode HE11x mode of LCPCF, we consider Ey Ex, therefore eeEy is replaced by eoEy in Eq. (3). The couplings of transverse fields are taken into account. With this approximation, when the elements of dielectric tensor take the form of ee ¼ n2e f ðx; yÞ, eo ¼ n2o f ðx; yÞ, f(x, y) is the refractive index profile, Eq. (3) turns out: o2 E x o 1 o ðfEx Þ þ k 20 n2o fEx þ ox f ox oy 2 o2 E y o 1 o ðfEy Þ ¼ b2 Ex þ oxoy ox f oy ð4Þ o2 E y o 1 o 2 2 ðfEy Þ þ k 0 no fEy þ oy f oy ox2 o2 E x o 1 o ðfEx Þ ¼ b2 Ey þ oxoy oy f ox
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Nomalized wavelength λ/Λ Fig. 2. Bandgap maps of LCPCF. (a) Polarization independent total bandgap map of LCPCF with rotation angle of the director / = p/2. (b) Polarization dependent bandgap maps for x-polarization mode HE11x and y-polarization mode HE11y. The intersection of the two polarization dependent bandgap maps constitutes the total polarization independent bandgap map.
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cladding. The core line neff = 1.45 represents the boundary between states that are propagating or evanescent in cladding. The total bandgap have only small section below the core line of LCPCF, which means that the fiber modes may be guided in these regions. But it is found that the polarization independent total bandgap map is not sufficient to describe the mode guiding in LCPCF. It is necessary to consider the polarization dependent properties of bandgaps for the modal properties analysis of LCPCF. We artificially defined the polarization dependent bandgaps, for which x-polarization bandgaps are determined by Eq. (4), while y-polarization bandgaps are determined by Eq. (5). Fig. 2b shows these two polarization dependent bandgap maps for x-polarization mode HE11x and y-polarization mode HE11y. Unlike all-solid bandgap fibers [17–20], the LCPCFs discussed here show splitting discrete bands of high transmission corresponding to the bandgap of the fiber cladding for x and y-polarization modes. For each polarization dependent bandgap map, it behaves like allsolid bandgap fibers, except for the photonic bands formed by the second order modes of high index rods (as shown in Fig. 2b). This is because of the splitting of almost degenerate TE01 and TM01 modes of high index rods in anisotropic system er = diag[eeeeeo]. The x-polarization bandgaps shift to short wavelength direction compared with y-polarization bandgaps, since the ordinary refractive index no = 1.522 is less than the extraordinary one ne = 1.706. This phenomenon gives rise to the most interesting characteristics of the polarization dependent bandgap splitting due to the anisotropic NLC. This characteristic provides the possible application of LCPCF for dynamic polarization control, single-mode single-polarization guidance, switching and polarizer. Furthermore, compare with Fig. 2a, the intersection of the two polarization dependent bandgap maps constitutes the total polarization independent bandgap map. This observation indicates the polarization dependent bandgaps determined by approximation Eqs. (4) and (5) agrees well with the exact solution provided by Eq. (3). 3. Modal properties Based on Eq. (3), after discretizing the partial differential operator and the waveguide dielectric constant, a finite-difference method (FDM) with PML boundary [21] is developed to calculate the guided modes of anisotropic optical waveguides. We consider a LCPCF with six rings of holes included in fiber cladding. When the rotation angle of NLC is / = p/2; the relative diameter of holes filled by NLCs (5CB type) is d/K = 0.4. The modes dispersion curves and the corresponding polarization dependent bandgaps are shown in top of Fig. 3. The bottom show the mode profiles and transverse electric filed distributions of HE11x at normalized wavelength k/K = 0.5, and HE11y at k/K = 0.35, k/K = 0.7. For clarity, only the first, second of y-polarization bandgaps, and first x-polarization bandgaps are graphed in the figure. It is shown that the
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x(y)-polarization modes are supported by corresponding x(y)-polarization bandgaps. Furthermore, within these polarization dependent bandgaps, no high-order modes are observed. This observation means that due to the anisotropy of NLC, although there is no polarization independent total bandgaps existing in these regions, the linear polarization modes still exist, and supported by corresponding polarization dependent bandgaps. For example, within the first bandgap of y-polarization bandgaps, there is only HE11y mode existing, which means single-mode single-polarization guidance in this bandgap (as shown in Fig. 3). This unique property can be utilized for applications of LCPCF for dynamic polarization control by applying external electric field. We can see the second bandgap of y-polarization is intersected with the first bandgap of x-polarization. So within this intersected wavelength region, the x-polarization mode HE11x and y-polarization mode HE11y exist simultaneously with high birefringence. Compared with index guiding LCPCF with a LC core (such as in Ref. [13]), the PBG guiding LCPCF offers splitting discrete high transmission windows, which may lead to potential applications for spectral filters. Fig. 4 shows the modal birefringence and the singlemode single-polarization operation wavelength regions of LCPCF. The shaded areas represent the wavelength regions where single-mode single-polarization guidance occurred. In the intersected region of x and y-polarization bandgap, the modal birefringence increases as the normalized wavelength increases. The LCPCF shows an ultrahigh birefringence (exceeding 0.01) around normalized wavelength k/K = 0.6. The confinement losses as a function of normalized wavelength k/K of the LCPCF are shown in Fig. 5. The confinement losses are normalized to aK (in dB), but may be scaled for any chosen value of the lattice spacing. It is evidently that the high-order bandgap provides lower confinement loss than low-order ones, but with the narrower transmission window. The confinement loss can be further reduced simply by increasing the number of the rings. The confinement loss increases rapidly at the edge of the bandgaps [17,22–24]. In addition, the confinement losses showed several high peaks within the bandgaps. For example, for x-polarization mode within the first bandgap, the peaks of confinement loss appears around the normalized wavelength k/K = 0.45. We noticed that the locations of these peaks are around the cut-off wavelength of the second modes of y-polarization bandgap map. It is believed that the appearance of high confinement loss peaks is due to the coupling between the x-polarization core mode and the second y-polarization band of LCPCF cladding. Fig. 6 shows the normalized effective areas (Aeff/K2) of linear polarization modes guided by polarization dependent bandgaps. Several characteristics are shown in Fig. 6. First, the effective area of guided mode in the high-order bandgap is less than that of the low-order ones. This is expected since generally the modal field is better confined in high-order bandgaps (short wavelength) [24].
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First y-polarization bandgap
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Fig. 3. Top: polarization dependent bandgap maps and modal dispersion curves for polarization modes HE11x and HE11y. Dot line represents the xpolarization mode HE11x, solid lines represent the y-polarization modes HE11y. Bottom: mode profiles and transverse electric field distributions at normalized wavelength k/K = 0.35 (HE11y), k/K = 0.5 (HE11x) and k/K = 0.7 (HE11y). 0. 02 0. 018
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Normalized wavelength λ/Λ Fig. 4. Birefringence (solid line) and single-mode single-polarization operation wavelength regions of LCPCF.
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Nomalized wavelength λ/Λ Fig. 5. Confinement losses as a function of normalized wavelength k/K for x(y)-polarization modes guided by polarization dependent bandgaps.
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Normalized wavelength λ/Λ Fig. 6. Normalized effective areas of guided mode Aeff/K2 as a function of normalized wavelength k/K for x(y)-polarization modes guided by polarization dependent bandgaps. The insets show mode profile and transverse electric field distribution of HE11x and HE11y at normalized wavelength k/K = 0.4325 and 0.5425, respectively.
Second, the minima wavelengths of effective area are generally not identical to that of confinement loss. Third, corresponding to the locations of high confinement loss peaks, the mode fields extend to fiber cladding at these regions as shown in insets of Fig. 6.
The width and position of x and y-polarization bandgaps are important to determine the transmission windows for x or y-polarization modes of LCPCF. As shown in Fig. 4, it determined modal birefringence and the singlemode single-polarization operation wavelength regions of
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LCPCF. But as shown in Fig. 5, at the long wavelength edge of the polarization dependent bandgap, the confinement loss of mode presents several orders larger than that at the middle bandgap. Due to the large differences of the confinement losses between the x-polarization and y-polarization mode, the LCPCF could be regarded as an effectively single-mode single-polarization fiber in these wavelength regions. So the position of the transmission window for different polarization modes is approximately defined as the wavelength region in between the intersections of the core line and the edges of x or y-polarization bandgap (as shown in Fig. 3). The single-mode singlepolarization guidance or high-birefringence guidance within certain wavelength regions could be achieved by adjusting the relative position of x and y-polarization transmission windows. It is found that the refractive index contrast of the high index inclusions and the relative size of these inclusions (corresponding to holes of PCF) would determine the positions of the transmission windows. Here we will consider the influence of the relative diameter of the holes of PCF. Fig. 7 illustrates the operation diagram of LCPCF, i.e. the transmission windows as a function of the relative diameter d/K. Only the first, second y-polarization bandgaps, and the first x-polarization bandgap are shown in figure. The darker regions are the single-mode single-polarization state for x and y-polarization mode respectively; the light region is the high birefringence state. This figure would help us design LCPCF for single-mode single-polarization operation within certain wavelength region. However, as shown in Fig. 5, the polarization guided modes are severely restricted by confinement loss.
So the confinement loss should be taken into account when the operation diagram is utilized. 4. Effect of the rotation angle / As mentioned earlier, the rotation angle / of the director n of the LCs can be controlled by the external static electric field. We will discuss the effect of the / on the mode properties in this section. It is found the bandgap map (shown in Fig. 2) keep approximately invariant when we vary the rotation angle / from 0 to p/2. This could be interpreted by ARROW (anti-resonant reflection optical waveguide) model [17,18]. The modes (including the leaky modes below cut-off) of individual high index LC inclusion couple together to form photonic bands; the width of these bands are determined by the coupling strength. The bandgaps form in between of these photonic bands [22,25]. For LCPCF discussed in this paper, the effect of rotation angle / on coupling strength is small enough to be neglected, thus the bandgap map of LCPCF can be considered approximately independent of the rotation angle. It also justifies our derivation of bandgap formation equations, i.e. Eq. (3) where we consider special cases of / equal to 0 or p/2. We show the mode index (in terms of neff – 1.41522084) of LCPCF as a function of rotation angle / from 0 to p/2 at normalized wavelength k/K = 0.75 in Fig. 8 (top). The LCPCF have four rings of holes with d/K = 0.4 included in fiber cladding. The mode profiles and transverse electric field distributions at rotation angle / = 0, p/12, p/6, p/4, p/3 and p/2 are shown in bottom. It should be noticed that
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Fig. 8. Top: mode index of LCPCF as a function of rotation angle / from 0 to p/2 at normalized wavelength k/K = 0.75. Bottom: mode profile and transverse electric field distribution evolution as the rotation angle / increases from 0 to p/2.
there is only one polarization mode exists at this wavelength (see Fig. 4). The mode index periodically varied with a period p/3 as rotation angle / increases. This observation is due to the C6m symmetry of fiber: since the PCF is p/3 rotation invariant, the LCPCF with rotation angle / is just itself with rotation angle / + p/3. We noticed that the difference for maximum and minimum mode index is about 4.4 107. Since this difference is so small, we further confirm it is not the computation error, because it is one order larger than the computation error 3 108. We can see from Fig. 8 (bottom), the polarization axis of the mode field is parallel with the director n. In addition, when we chose another wavelength, such as k/K = 0.48, the polarization axis will be perpendicular to the director n. For LCPCF, if the wavelength span is choose as single-mode single-polarization operation region as shown in Fig. 4, we could dynamically control the polarization axis of mode by adjusting the direction of external electric field. The
polarization axis of fiber mode would be parallel or perpendicular to the direction of external electric field, depending on the operation wavelength. 5. Conclusion In this paper, we investigated the polarization dependent bandgap formation and the mode properties of LCPCF in detail. Due to the strong anisotropy of NLC, the polarization independent total bandgap map is not enough to describe the guiding mechanism of LCPCF. The polarization dependent bandgap map is then proposed to investigate the mode guidance of the LCPCF, the x or y-polarization bandgaps are responsible for mode field confinement and guiding of the x or y-polarization modes. The intersection of the two polarization dependent bandgap maps constitutes the polarization independent total bandgap map. The polarization dependent bandgap splitting caused by the high
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index difference between the ordinary and the extraordinary dielectric index of NLC provides the possibility of singlemode single-polarization guiding for LCPCF. The mode properties including mode profile, confinement loss and effective area are also discussed. A polarization operation diagram for LCPCF is proposed to describe the mode guiding behavior. The influence of rotation angle / of the director n on the bandgap and mode index of LCPCF has been observed to be insignificant. However the polarization axis of the guided mode is determined by the rotation angle /, which could be controlled by external electric field. These results provide physical insight into the polarization tuning mechanism of LCPCFs, which is crucial for making compact and diverse polarization-manipulating devices. Acknowledgements This work is partially supported by Open Fund of Key Laboratory of Optical Communication and Lightwave Technologie, (Beijing University of Posts and Telecommunications), Ministry of Education, PR China. References [1] P.St.J. Russell, Science 299 (2003) 358. [2] T. Ritari, J. Tuominen, H. Ludvigsen, J.C. Petersen, T. Sørensen, T.P. Hansen, H.R. Simonsen, Opt. Express 12 (2004) 4080. [3] B. Eggleton, C. Kerbage, P. Westbrook, R. Windeler, A. Hale, Opt. Express 9 (2001) 698. [4] R.T. Bise, R.S. Windeler, K.S. Kranz, C. Kerbage, B.J. Eggleton, D.J. Trevor, Opt. Fiber Commun. Conf. Tech. Digest (2002) 466.
[5] T.T. Larsen, A. Bjarklev, D.S. Hermann, J. Broeng, Opt. Express 11 (2003) 2589. [6] C. Kerbage, B. Eggleton, Opt. Express 10 (2006) 246. [7] T.R. Wolin´ski, K. Szaniawska, S. Ertman, P. Lesiak, A.W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, J. Wojcik, Measur. Sci. Technol. 17 (2006) 985. [8] T.T. Alkeskjold, J. Laegsgaard, A. Bjarklev, D.S. Hermann, J. Broeng, J. Li, S.T. Wu, Opt. Express 12 (2004) 5857. [9] F. Du, Y.Q. Lu, S.T. Wu, Appl. Phys. Lett. 85 (2004) 2181. [10] M.W. Haakestad, T.T. Alkeskjold, M.D. Nielsen, L. Scolari, J. Riishede, H.E. Engan, A. Bjarklev, IEEE Photon. Technol. Lett. 17 (2006) 819. [11] L. Scolari, T.T. Alkeskjold, J. Riishede, A. Bjarklev, D. Hermann, A. Anawati, M. Nielsen, P. Bassi, Opt. Express 13 (2005) 7483. [12] T.T. Alkeskjold, A. Bjarklev, Opt. Lett. 32 (2007) 1707. [13] D.C. Zografopoulos, E.E. Kriezis, T.D. Tsiboukis, Opt. Express 14 (2006) 914. [14] J. Sun, C.C. Chan, Opt. Lett. 32 (2007) 1989. [15] C.L. Xu, W.-P. Huang, M. Stern, S.K. Chaudhuri, IEEE Proc. Pt. J: Optoelectron. 141 (1994) 281. [16] M.J. Steel, T.P. White, C. Martijn de Sterke, R.C. McPhedran, L.C. Botten, Opt. Lett. 26 (2001) 488. [17] T.P. White, R.C. McPhedran, C.M. de Sterke, N.M. Litchinitser, B.J. Eggleton, Opt. Lett. 27 (2002) 1977. [18] N.M. Litchinitser, S.C. Dunn, B. Usner, B.J. Eggleton, T.P. White, R.C. McPhedran, C.M. de Sterke, Opt. Express 11 (2003) 1243. [19] F. Luan, A.K. George, T.D. Hedley, G.J. Pearce, D.M. Bird, J.C. Knight, P.St.J. Russell, Opt. Lett. 29 (2004) 2369. [20] T.A. Birks, F. Luan, G.J. Pearce, A. Wang, J.C. Knight, D.M. Bird, Opt. Express 14 (2006) 5688. [21] S. Guo, F. Wu, S. Albin, H. Tai, R.S. Rogowski, Opt. Express 12 (2004) 3341. [22] J. Laegsgaard, J. Opt. A: Pure and Appl. Opt. 6 (2004) 798. [23] P. Steinvurzel, C. Martijn de Sterke, M.J. Steel, B.T. Kuhlmey, B.J. Eggleton, Opt. Express 14 (2006) 8797. [24] P. Steinvurzel, C. Martijn de Sterke, B.J. Eggleton, B.T. Kuhlmey, M.J. Steel, Opt. Commun. 263 (2) (2006) 207. [25] T.A. Birks, G.J. Pearce, D.M. Bird, Opt. Express 14 (2006) 9483.
Optics Communications 281 (2008) 1598–1606 www.elsevier.com/locate/optcom
Polarization dependent guiding in liquid crystal filled photonic crystal fibers Guobin Ren a,*, Ping Shum a, Xia Yu a, JuanJuan Hu a, Guanghui Wang a, Yandong Gong b a
Network Technology Research Centre, Nanyang Technological University, 50 Nanyang Drive, 637553 Singapore, Singapore b Institute for InfoComm Research, 21 Heng Mui Keng Terrace, 119613 Singapore, Singapore Received 11 October 2007; received in revised form 21 November 2007; accepted 21 November 2007
Abstract A theoretical study of nematic liquid crystal filled photonic crystal fibers (LCPCFs) is presented. Detailed investigations including the polarization dependent bandgap formation and the modal properties are given for LCPCFs, in which alignment of the molecules could be controlled by external static electric field. The polarization dependent bandgap splitting caused by the high index difference between the ordinary and the extraordinary dielectric index of nematic liquid crystals provides the possibility of single-mode single-polarization guiding. A polarization operation diagram is proposed to describe the guiding behavior of LCPCFs. The influence of rotation angle / of the director of liquid crystals on the modal properties is investigated. It is shown that the polarization axis of the guided mode is determined by the rotation angle /, which could be controlled by external electric field. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Fiber optics; Photonic bandgap fibers; Liquid crystal devices; Polarization-maintaining
1. Introduction Photonic crystal fibers (PCF) have attracted large scientific and commercial interest since they offer a wide range promising applications in communication and sensing [1,2]. Light guiding in a PCF is governed by one of two principal mechanisms responsible for light trapping within the core. The first is a simple mechanism based on the modified total internal reflection (TIR) phenomenon, which is well known within conventional fiber. The other is known as a photonic bandgap (PBG) effect, which relies on the coherent backscattering of light into the core. Propagation and polarization properties of PCF can be manipulated by filling fluid materials such as polymer, oil, or liquid crystal (LC) [3–5] into the air holes of PCF. Among these materials LC is of particular interest due to
*
Corresponding author. E-mail address: [email protected] (G. Ren).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.084
their electrically or thermally tunable refractive index, which could boost PCFs to a higher level of tunability. The thermal tunability of liquid crystal filled PCFs (LCPCFs) was achieved taking advantage of thermo-optic effect of LCs by using either the external heaters or the guided pump beam absorption in dye-doped LCs [6–8]. The first experimental demonstration of electrical tuning in hollow-core LCPCF was reported in Ref. [9]. By filling the holes of an air-core photonic bandgap fiber (PBGF), TIR guiding and electrically tunable optical switch effects are demonstrated. Haakestad et al. [10] reported electrically tunable PBG guiding of a nematic liquid crystal (NLC) filled PCF (NLCPCF), which was able to operate as an optical switch. In Ref. [11], an electrically controlled continuous tunable LCPCF that allows continuous control of the spectral position of the bandgaps is demonstrated. The continuous tunability is due to dual frequency LC is used to fill the holes of PCF. More recently, Alkeskjold et al. [12] demonstrated an electrically driven broadband LCPCF polarizer, which has potential applications for
G. Ren et al. / Optics Communications 281 (2008) 1598–1606
polarization monitoring and fiber-optic sensors. Polarizer axis control and polarization extinction ratio tuning are achieved by bipolar electrodes driving. Although some experimental works of LCPCFs have been reported in recent years, the theoretical investigation is relatively lagging behind especially taking the anisotropic property of LC into account. The LCPCFs for singlepolarization or high-birefringence guidance were theoretically modeled by Zografopoulos et al. [13]. Since the LCPCF proposed has a NLC core, the polarization characteristics of this fiber is mainly determined by the NLC, and the guiding mechanism is TIR in this case. More recently, Sun et al. [14] investigated the alignment effect of NLC on bandgap formation of the NLCPCF. A simple analytic model is proposed for analyzing the bandgap formation with the assumption of limited index difference. In this model, the coupling of transverse field is neglected. Modal properties of LCPCF are fundamentally interesting and important for LCPCF based device application. However, so far the detailed investigation on modal properties of LCPCF is not available. In this paper, we present a thorough theoretical analysis of LCPCFs including the polarization dependent bandgap formation and modal properties considering the alignment effect of NLC. The coupling terms of transverse field in wave equations are taken into account for the bandgap formation analysis; a general derivation is proposed for analyzing the bandgap formation more accurately. The polarization dependent bandgap is proposed to investigate the mode guiding. The modal properties including polarization splitting behavior, confinement loss, effective area, and the position of transmission windows are investigated. Furthermore, the influence of the alignment of the NLC molecules, which could be controlled by external static electric field, on the modal properties is investigated. 2. Polarization dependent bandgap formation Nematic liquid crystals (NLCs) are anisotropic materials consisting of rod-like molecules; the orientational order of molecules combined with their anisotropy lead to anisotropic optical properties. The local orientation of the NLCs is described by the director: a unit vector along the direction of the average orientation of the molecules, which can be aligned by application of proper boundary conditions to achieve a macroscopic alignment. Under the application of a static electric field the director’s orientation can be controlled, since the liquid crystal molecules tend to align their axis according to the applied field. The LCPCF investigated in this paper is shown in Fig. 1, K is the lattice spacing and d is diameter of hole, in which the NLCs are infiltrated. The director of a NLC is shown in left. In a practical point of view, for silica-based PCF, due to the strong boundary condition in silica surface, the director of NLCs is not homogeneous, especially in the proximity of silica surface. For simplified analysis, the inhomogeneity of the director is neglected and the
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Fig. 1. LCPCF structure with triangular lattices. The shaded region is infiltrated with liquid crystals. The left indicates the director of a nematic liquid crystal.
alignment of the director is assumed to be uniform within the holes in this paper. Generally LCs possess two kinds of dielectric indices. One is the ordinary dielectric index eo, and the other is the extraordinary dielectric index ee. Light waves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary refractive indices, respectively. When electric field is applied transversely (in x–y plane), the dielectric permittivity tensor takes the form of 2 3 exx exy 0 6 7 er ¼ 4 eyx eyy 0 5 ð1Þ 0
0
ezz
where exx ðrÞ ¼ eo ðrÞ sin2 / þ ee ðrÞ cos2 / exy ðrÞ ¼ eyx ðrÞ ¼ ½ee ðrÞ eo ðrÞ sin / cos / eyy ðrÞ ¼ eo ðrÞ cos2 / þ ee ðrÞ sin2 / ezz ðrÞ ¼ eo ðrÞ
ð2Þ
/ is the rotation angle of the director of the LCs and it can be controlled by the external static electric field. n = (cos/, sin/) is the director of the LC, as shown in Fig. 1. For bandgap formation analysis, we assume that the rotation angle of the director / is 0 or p/2, which give rise to a simplification of er = diag[eeeoeo]; for / = 0, and er = diag[eoeeeo] for / = p/2. We will justify this simplification in the last discussion session that the contribution of the rotation angle of the director / for bandgap calculation is small enough to be neglected. The eigenvalue equations in terms of transverse electric fields for / = p/2 is [15]: o2 E x o 1 o ðeo Ex Þ þ k 20 eo Ex þ ox eo ox oy 2 o2 E y o 1 o ðee Ey Þ ¼ b2 Ex þ oxoy ox eo oy ð3Þ o2 E y o 1 o 2 ðee Ey Þ þ k 0 ee Ey þ oy eo oy ox2 o2 E x o 1 o ðeo Ex Þ ¼ b2 Ey þ oxoy oy eo ox
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Eq. (4) is actually the eigenvalue equation of the isotropic systemer = eo. For y-polarization mode HE11y mode of LCPCF, we consider Ex Ey, therefore eoEx is replaced by eeEx in Eq. (3). We have: o2 Ex n2e o 1 o ðfEx Þ þ k 20 n2e fEx þ 2 oy 2 no ox f ox o2 Ey n2e o 1 o ðfEy Þ ¼ b2 Ex þ 2 oxoy no ox f oy ð5Þ 2 o Ey n2e o 1 o 2 2 ðfEy Þ þ k 0 ne fEy þ 2 ox2 no oy f oy o2 Ex n2e o 1 o ðfEx Þ ¼ b2 Ey þ oxoy n2o oy f ox Eq. (5) is the eigenvalue equation of the uniaxial anisotropic systemer = diag[eeeeeo]. Here the director of the anisotropic medium is along z-axis. The refractive index of the PCF background material (silica) is assumed to be 1.45. The 5CB type NLCs are considered in this paper, the ordinary and extraordinary refractive indices of are no = 1.522 and ne = 1.706, respec-
tively. Fig. 1 indicates the director n of a liquid crystal and the rotation angle / of the director to the x-axis. 5CB LC is a suitable choice for demonstration because it exhibits a nematic phase at room temperature and its nematic range is >10 °C. We assume that the operating temperature is at a constant room temperature and that the absorption loss is negligible. A full-vectorial plane wave method is used to calculate the bandgap of the LC cladding of the LCPCF shown in Fig. 1. Fig. 2a shows the polarization independent total bandgap map of the LCPCF. It should be noticed that this bandgap map is governed by Eq. (3) without approximation. The relative hole diameter is d/K = 0.4. The rotation angle of NLC is / = p/2. The shaded parts are the bandgaps in which no photonic state could be supported in fiber
a 1.65 1.6 1.55
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This equation determines the polarization independent total gaps. It is well known that the fundamental mode of the triangular lattice PCF contains two degenerate orthogonal modes, HE11x and HE11y mode [16]. The degeneracy would be split when the asymmetry of fiber structure or birefringent materials is introduced. When the LCs are filled into the air holes of PCF, the intrinsic anisotropy of LC would affect the photonic bands and also the guided modes of fiber, which are totally different from the isotropic situation. The polarization dependent photonic bandgaps would be open up and these bandgaps would guide modes with different polarization states. The formation of polarization dependent bandgap is similar to TE and TM bandgap formation of 2-D photonic crystals with anisotropic materials contained, but due to the hybrid nature of the fiber mode, the strict decoupling of Maxwell equation will not occur in this case. For x-polarization mode HE11x mode of LCPCF, we consider Ey Ex, therefore eeEy is replaced by eoEy in Eq. (3). The couplings of transverse fields are taken into account. With this approximation, when the elements of dielectric tensor take the form of ee ¼ n2e f ðx; yÞ, eo ¼ n2o f ðx; yÞ, f(x, y) is the refractive index profile, Eq. (3) turns out: o2 E x o 1 o ðfEx Þ þ k 20 n2o fEx þ ox f ox oy 2 o2 E y o 1 o ðfEy Þ ¼ b2 Ex þ oxoy ox f oy ð4Þ o2 E y o 1 o 2 2 ðfEy Þ þ k 0 no fEy þ oy f oy ox2 o2 E x o 1 o ðfEx Þ ¼ b2 Ey þ oxoy oy f ox
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Nomalized wavelength λ/Λ Fig. 2. Bandgap maps of LCPCF. (a) Polarization independent total bandgap map of LCPCF with rotation angle of the director / = p/2. (b) Polarization dependent bandgap maps for x-polarization mode HE11x and y-polarization mode HE11y. The intersection of the two polarization dependent bandgap maps constitutes the total polarization independent bandgap map.
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cladding. The core line neff = 1.45 represents the boundary between states that are propagating or evanescent in cladding. The total bandgap have only small section below the core line of LCPCF, which means that the fiber modes may be guided in these regions. But it is found that the polarization independent total bandgap map is not sufficient to describe the mode guiding in LCPCF. It is necessary to consider the polarization dependent properties of bandgaps for the modal properties analysis of LCPCF. We artificially defined the polarization dependent bandgaps, for which x-polarization bandgaps are determined by Eq. (4), while y-polarization bandgaps are determined by Eq. (5). Fig. 2b shows these two polarization dependent bandgap maps for x-polarization mode HE11x and y-polarization mode HE11y. Unlike all-solid bandgap fibers [17–20], the LCPCFs discussed here show splitting discrete bands of high transmission corresponding to the bandgap of the fiber cladding for x and y-polarization modes. For each polarization dependent bandgap map, it behaves like allsolid bandgap fibers, except for the photonic bands formed by the second order modes of high index rods (as shown in Fig. 2b). This is because of the splitting of almost degenerate TE01 and TM01 modes of high index rods in anisotropic system er = diag[eeeeeo]. The x-polarization bandgaps shift to short wavelength direction compared with y-polarization bandgaps, since the ordinary refractive index no = 1.522 is less than the extraordinary one ne = 1.706. This phenomenon gives rise to the most interesting characteristics of the polarization dependent bandgap splitting due to the anisotropic NLC. This characteristic provides the possible application of LCPCF for dynamic polarization control, single-mode single-polarization guidance, switching and polarizer. Furthermore, compare with Fig. 2a, the intersection of the two polarization dependent bandgap maps constitutes the total polarization independent bandgap map. This observation indicates the polarization dependent bandgaps determined by approximation Eqs. (4) and (5) agrees well with the exact solution provided by Eq. (3). 3. Modal properties Based on Eq. (3), after discretizing the partial differential operator and the waveguide dielectric constant, a finite-difference method (FDM) with PML boundary [21] is developed to calculate the guided modes of anisotropic optical waveguides. We consider a LCPCF with six rings of holes included in fiber cladding. When the rotation angle of NLC is / = p/2; the relative diameter of holes filled by NLCs (5CB type) is d/K = 0.4. The modes dispersion curves and the corresponding polarization dependent bandgaps are shown in top of Fig. 3. The bottom show the mode profiles and transverse electric filed distributions of HE11x at normalized wavelength k/K = 0.5, and HE11y at k/K = 0.35, k/K = 0.7. For clarity, only the first, second of y-polarization bandgaps, and first x-polarization bandgaps are graphed in the figure. It is shown that the
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x(y)-polarization modes are supported by corresponding x(y)-polarization bandgaps. Furthermore, within these polarization dependent bandgaps, no high-order modes are observed. This observation means that due to the anisotropy of NLC, although there is no polarization independent total bandgaps existing in these regions, the linear polarization modes still exist, and supported by corresponding polarization dependent bandgaps. For example, within the first bandgap of y-polarization bandgaps, there is only HE11y mode existing, which means single-mode single-polarization guidance in this bandgap (as shown in Fig. 3). This unique property can be utilized for applications of LCPCF for dynamic polarization control by applying external electric field. We can see the second bandgap of y-polarization is intersected with the first bandgap of x-polarization. So within this intersected wavelength region, the x-polarization mode HE11x and y-polarization mode HE11y exist simultaneously with high birefringence. Compared with index guiding LCPCF with a LC core (such as in Ref. [13]), the PBG guiding LCPCF offers splitting discrete high transmission windows, which may lead to potential applications for spectral filters. Fig. 4 shows the modal birefringence and the singlemode single-polarization operation wavelength regions of LCPCF. The shaded areas represent the wavelength regions where single-mode single-polarization guidance occurred. In the intersected region of x and y-polarization bandgap, the modal birefringence increases as the normalized wavelength increases. The LCPCF shows an ultrahigh birefringence (exceeding 0.01) around normalized wavelength k/K = 0.6. The confinement losses as a function of normalized wavelength k/K of the LCPCF are shown in Fig. 5. The confinement losses are normalized to aK (in dB), but may be scaled for any chosen value of the lattice spacing. It is evidently that the high-order bandgap provides lower confinement loss than low-order ones, but with the narrower transmission window. The confinement loss can be further reduced simply by increasing the number of the rings. The confinement loss increases rapidly at the edge of the bandgaps [17,22–24]. In addition, the confinement losses showed several high peaks within the bandgaps. For example, for x-polarization mode within the first bandgap, the peaks of confinement loss appears around the normalized wavelength k/K = 0.45. We noticed that the locations of these peaks are around the cut-off wavelength of the second modes of y-polarization bandgap map. It is believed that the appearance of high confinement loss peaks is due to the coupling between the x-polarization core mode and the second y-polarization band of LCPCF cladding. Fig. 6 shows the normalized effective areas (Aeff/K2) of linear polarization modes guided by polarization dependent bandgaps. Several characteristics are shown in Fig. 6. First, the effective area of guided mode in the high-order bandgap is less than that of the low-order ones. This is expected since generally the modal field is better confined in high-order bandgaps (short wavelength) [24].
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G. Ren et al. / Optics Communications 281 (2008) 1598–1606 1.5
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Fig. 3. Top: polarization dependent bandgap maps and modal dispersion curves for polarization modes HE11x and HE11y. Dot line represents the xpolarization mode HE11x, solid lines represent the y-polarization modes HE11y. Bottom: mode profiles and transverse electric field distributions at normalized wavelength k/K = 0.35 (HE11y), k/K = 0.5 (HE11x) and k/K = 0.7 (HE11y). 0. 02 0. 018
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Normalized wavelength λ/Λ Fig. 6. Normalized effective areas of guided mode Aeff/K2 as a function of normalized wavelength k/K for x(y)-polarization modes guided by polarization dependent bandgaps. The insets show mode profile and transverse electric field distribution of HE11x and HE11y at normalized wavelength k/K = 0.4325 and 0.5425, respectively.
Second, the minima wavelengths of effective area are generally not identical to that of confinement loss. Third, corresponding to the locations of high confinement loss peaks, the mode fields extend to fiber cladding at these regions as shown in insets of Fig. 6.
The width and position of x and y-polarization bandgaps are important to determine the transmission windows for x or y-polarization modes of LCPCF. As shown in Fig. 4, it determined modal birefringence and the singlemode single-polarization operation wavelength regions of
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LCPCF. But as shown in Fig. 5, at the long wavelength edge of the polarization dependent bandgap, the confinement loss of mode presents several orders larger than that at the middle bandgap. Due to the large differences of the confinement losses between the x-polarization and y-polarization mode, the LCPCF could be regarded as an effectively single-mode single-polarization fiber in these wavelength regions. So the position of the transmission window for different polarization modes is approximately defined as the wavelength region in between the intersections of the core line and the edges of x or y-polarization bandgap (as shown in Fig. 3). The single-mode singlepolarization guidance or high-birefringence guidance within certain wavelength regions could be achieved by adjusting the relative position of x and y-polarization transmission windows. It is found that the refractive index contrast of the high index inclusions and the relative size of these inclusions (corresponding to holes of PCF) would determine the positions of the transmission windows. Here we will consider the influence of the relative diameter of the holes of PCF. Fig. 7 illustrates the operation diagram of LCPCF, i.e. the transmission windows as a function of the relative diameter d/K. Only the first, second y-polarization bandgaps, and the first x-polarization bandgap are shown in figure. The darker regions are the single-mode single-polarization state for x and y-polarization mode respectively; the light region is the high birefringence state. This figure would help us design LCPCF for single-mode single-polarization operation within certain wavelength region. However, as shown in Fig. 5, the polarization guided modes are severely restricted by confinement loss.
So the confinement loss should be taken into account when the operation diagram is utilized. 4. Effect of the rotation angle / As mentioned earlier, the rotation angle / of the director n of the LCs can be controlled by the external static electric field. We will discuss the effect of the / on the mode properties in this section. It is found the bandgap map (shown in Fig. 2) keep approximately invariant when we vary the rotation angle / from 0 to p/2. This could be interpreted by ARROW (anti-resonant reflection optical waveguide) model [17,18]. The modes (including the leaky modes below cut-off) of individual high index LC inclusion couple together to form photonic bands; the width of these bands are determined by the coupling strength. The bandgaps form in between of these photonic bands [22,25]. For LCPCF discussed in this paper, the effect of rotation angle / on coupling strength is small enough to be neglected, thus the bandgap map of LCPCF can be considered approximately independent of the rotation angle. It also justifies our derivation of bandgap formation equations, i.e. Eq. (3) where we consider special cases of / equal to 0 or p/2. We show the mode index (in terms of neff – 1.41522084) of LCPCF as a function of rotation angle / from 0 to p/2 at normalized wavelength k/K = 0.75 in Fig. 8 (top). The LCPCF have four rings of holes with d/K = 0.4 included in fiber cladding. The mode profiles and transverse electric field distributions at rotation angle / = 0, p/12, p/6, p/4, p/3 and p/2 are shown in bottom. It should be noticed that
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φ=π/2
Fig. 8. Top: mode index of LCPCF as a function of rotation angle / from 0 to p/2 at normalized wavelength k/K = 0.75. Bottom: mode profile and transverse electric field distribution evolution as the rotation angle / increases from 0 to p/2.
there is only one polarization mode exists at this wavelength (see Fig. 4). The mode index periodically varied with a period p/3 as rotation angle / increases. This observation is due to the C6m symmetry of fiber: since the PCF is p/3 rotation invariant, the LCPCF with rotation angle / is just itself with rotation angle / + p/3. We noticed that the difference for maximum and minimum mode index is about 4.4 107. Since this difference is so small, we further confirm it is not the computation error, because it is one order larger than the computation error 3 108. We can see from Fig. 8 (bottom), the polarization axis of the mode field is parallel with the director n. In addition, when we chose another wavelength, such as k/K = 0.48, the polarization axis will be perpendicular to the director n. For LCPCF, if the wavelength span is choose as single-mode single-polarization operation region as shown in Fig. 4, we could dynamically control the polarization axis of mode by adjusting the direction of external electric field. The
polarization axis of fiber mode would be parallel or perpendicular to the direction of external electric field, depending on the operation wavelength. 5. Conclusion In this paper, we investigated the polarization dependent bandgap formation and the mode properties of LCPCF in detail. Due to the strong anisotropy of NLC, the polarization independent total bandgap map is not enough to describe the guiding mechanism of LCPCF. The polarization dependent bandgap map is then proposed to investigate the mode guidance of the LCPCF, the x or y-polarization bandgaps are responsible for mode field confinement and guiding of the x or y-polarization modes. The intersection of the two polarization dependent bandgap maps constitutes the polarization independent total bandgap map. The polarization dependent bandgap splitting caused by the high
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index difference between the ordinary and the extraordinary dielectric index of NLC provides the possibility of singlemode single-polarization guiding for LCPCF. The mode properties including mode profile, confinement loss and effective area are also discussed. A polarization operation diagram for LCPCF is proposed to describe the mode guiding behavior. The influence of rotation angle / of the director n on the bandgap and mode index of LCPCF has been observed to be insignificant. However the polarization axis of the guided mode is determined by the rotation angle /, which could be controlled by external electric field. These results provide physical insight into the polarization tuning mechanism of LCPCFs, which is crucial for making compact and diverse polarization-manipulating devices. Acknowledgements This work is partially supported by Open Fund of Key Laboratory of Optical Communication and Lightwave Technologie, (Beijing University of Posts and Telecommunications), Ministry of Education, PR China. References [1] P.St.J. Russell, Science 299 (2003) 358. [2] T. Ritari, J. Tuominen, H. Ludvigsen, J.C. Petersen, T. Sørensen, T.P. Hansen, H.R. Simonsen, Opt. Express 12 (2004) 4080. [3] B. Eggleton, C. Kerbage, P. Westbrook, R. Windeler, A. Hale, Opt. Express 9 (2001) 698. [4] R.T. Bise, R.S. Windeler, K.S. Kranz, C. Kerbage, B.J. Eggleton, D.J. Trevor, Opt. Fiber Commun. Conf. Tech. Digest (2002) 466.
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