IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 48, NO. 8, AUGUST 2012
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Design Consideration of Polarization Converter Based on Silica Photonic Crystal Fiber Mohamed Farhat O. Hameed and Salah S. A. Obayya, Senior Member, IEEE
Abstract— In this paper, the performance of a novel design of passive polarization rotator (PR) based on silica photonic crystal fiber is studied and analyzed using the full vectorial finite difference method along with the full vectorial finite difference beam propagation method. The proposed design has a rectangular core region with a slanted sidewall. The influence of the different structure geometrical parameters and operating wavelengths on the PR performance is investigated. At a wavelength of 1.55 µm, nearly 100% polarization conversion ratio is obtained, with a device length of 2839 µm. In addition, it is expected that over the 1.5 to 1.6-µm wavelength range, polarization conversion would be more than 99%. Index Terms— Beam propagation method, finite difference method, photonic crystal fibers, polarization rotation (PR).
I. I NTRODUCTION
P
OLARIZATION ROTATORS (PRs) play an important role in modern optoelectronic and communication systems. Therefore, PRs have attracted the interest of many researchers in recent years. The PRs are used to control the polarization states in the communication systems such as polarization switches [1], [2] and polarization modulators [2]. Initially, polarization rotation in multi-sections asymmetric periodic loaded rib waveguides [3] was proposed. The multi-sections PR suffers from its longer device length (several mms) and large transition losses at the interface between alternating sections (several dBs). However, Obayya et al. [4] reported a complete polarization rotation at a moderate device length around 0.74 mm, and with minimal radiation loss as low as 0.13 dB by a careful adjustment of the waveguide width and/or the waveguide materials of an asymmetrically periodic loaded rib waveguides. Due to the radiation losses associated with multi-sections PR, great efforts have been directed toward the design of a single section polarization converter which contains no transition losses with shorter device length. The single section PR examples included curved optical waveguides [5], [6] and slanted sidewall waveguides [7], [8]. Polarization rotation in deeply etched semiconductor optical waveguide bends was reported in [5] and a very compact PR of length 60 μm,
Manuscript received September 1, 2011; revised May 1, 2012; accepted May 26, 2012. Date of publication June 4, 2012; date of current version June 11, 2012. M. F. O. Hameed is with the Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt (e-mail:
[email protected]). S. S. A. Obayya is with the Department of Electronics and Communications Engineering, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt (e-mail:
[email protected]). Digital Object Identifier 10.1109/JQE.2012.2202372
with extremely low overall loss, and nearly 100% polarization conversion ratio were achieved. Another PR based on cascaded configuration of curved waveguide sections each with an alternative curvature direction was presented in [6]. In this configuration [6], the polarization conversion builds up as the wave propagates along that cascaded arrangement of the curved waveguides. In addition, an improved design of a polarization rotator, based on a single asymmetrical slanted rib waveguide, was investigated in [7]. Moreover, Obayya et al. [8] reported a new design of polarization converter based on the use of a deeply etched semiconductor waveguide with double-slanted sidewalls. However, PRs with slanted sidewall require a complex fabrication process including dry and wet etching techniques. Also, compact PRs based on the curved optical waveguides suffer from radiation losses and rely on a very small radius of curvature which will not be easy for fabrication. The photonic bandgap silica photonic crystal fiber (PCF) filled with liquid crystal (LC) has been experimentally shown to have potential for polarization conversion [9], [10]. Scolari et al. [9] demonstrated a PR by using a silica core PCF infiltrated with a dual frequency LC while in [10] the PCF was filled with a negative LC. In addition, high tunable PR based on index guiding soft glass PCF whose cladding holes have been infiltrated with nematic liquid crystal (NLC-PCF) was reported by Hameed and Obayya [11]. The NLC-PCF PR [11] provides a strong polarization conversion ratio of 99.81% with a device length of 1072 μm. Moreover, the authors have recently investigated a novel design of single section high tunable PR based on air holes soft glass PCF with liquid crystal core (SGLC-PCF) [12]. The SGLC-PCF PR offers polarization conversion ratio of 99.67% with low crosstalk of −24.86 dB and a device length of 558 μm. Therefore, the SGLC-PCF PR [12] is shorter and easier for fabrication than the NLCPCF PR [11] with filling all the cladding holes. Most of the PCF PRs reported in the literature are based on the use of silica PCF infiltrated with anisotropic material such as LC, in order to enforce the coupling between the two fundamental polarization states through the off-diagonal elements of the LC permittivity tensor [9]–[12]. In this paper, we report for the first time (to the best of our knowledge) the design consideration of a novel design of single section passive silica PCF (PS-PCF) PR without adding any additive dopants to the structure. The new design relies on using PCF with rectangular core region with slanted sidewall which offers almost 100% polarization conversion ratio. The reported structure is easier for fabrication than the LC-PCF [9]–[12]
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with strong polarization conversion ratio. In addition, the slanted side wall of the core region can be simply achieved by using extra hole in the corner of the core region. Therefore, it does not require a complex fabrication process like the semiconductor waveguide with slanted sidewalls [7], [8]. In this investigation, the performance of the suggested design is studied in detail and the effects of the structure geometrical parameters and operating wavelength on the device length and polarization efficiency are studied thoroughly. The reported PS-PCF PR offers polarization conversion ratio of 99.95% with low crosstalk of −32.98 dB and a device length of 2839 μm. It is also expected that over the 1.5–1.6 μm wavelength range, polarization conversion would be more than 99% and polarization crosstalk would be better than −20 dB. The numerical results are evaluated using the full vectorial finite difference method (FVFDM) [13] and the full vectorial finite difference beam propagation method (FVFD-BPM) [14]. In the next section, the numerical methods are described briefly. Following that, in the results section, the design and simulated results obtained from the suggested single-section passive polarization rotator are presented. Then conclusions are drawn. II. N UMERICAL A PPROACHES Modal solutions are very useful for the characterization and design of PRs. To calculate accurately vector modal field profiles of the input waveguide, a full vectorial modal solution approach is necessary. Many modal solution techniques have been proposed in the past few years for modal analysis of PCFs such as the finite element method (FEM) [15], and finite difference method (FDM) [13]. The FDM [13] renders itself to have a number of numerical advantages such as; easier to implement, rigorously in accounting for polarization dependence of polarization coupling, accurate truncation of the computational domain using perfect matched layer (PML) boundary conditions [16]. In addition, the accuracy of meshing distribution of the FDM can be greatly enhanced through the use of graded mesh [17]. Alternatively, the FEM is superior to the FDM and it can provide high accuracy by means of flexible triangular and curvilinear meshes to represent the waveguide cross section. However, this accuracy results in an algorithm that is complex to implement. In addition, the FDM can produce much accurate results by increasing the mesh density around the curved holes of the PCF In this study, the FVFDM [13] with PML boundary conditions [16] is used to calculate the full vectorial quasi transverse electric (TE) and quasi transverse magnetic (TM) modes for the PCF waveguides. It should be noted that a series of mesh with resolutions of x = y = = 0.05 μm and less is tested. It has been noticed that the resolutions are very stable and convergent for ≤ 0.035 μm. Therefore, the grid sizes x and y in x and y directions, respectively, are taken as 0.03 μm through all simulations in this work. To study the propagation through the PR waveguide section, a more suitable approach such as beam propagation method (BPM) [14], [18] is required. The BPM has been considered in the last two decades as one of the most popular
Fig. 1. Variation of the real part of the complex effective index of the quasi TE mode of the triangular lattice PCF with the wavelength at two different d/ ratios, 0.4 and 0.5, while the hole pitch is fixed to 2.3 μm.
methods used for the simulation of wave propagation in various photonic devices. A full vectorial approach is particularly necessary to calculate the polarization conversion in the optical guided-wave devices or systems. Due to its numerical efficiency and versatility, some full vectorial BPM algorithms have been formulated based on the FEM [18]. In addition, many full vectorial BPM approaches based on the popular FDM have been reported [14]. In this study, the FVFD-BPM [14] is applied to study the polarization conversion based on the reported structure. Through all simulations, the transverse step sizes are also fixed to x = y = 0.03 μm while the longitudinal step size z is taken as 1.0 μm. In addition, the reference index n0 which is used to satisfy the slowly varying envelope approximation of the FVFD-BPM [14] is taken as the effective index of the fundamental mode launched at the input waveguide. Moreover, α is chosen within the range, 0.5 ≤ α ≤ 1.0, at which the FVFD-BPM [14] is unconditionally stable. The α parameter is responsible for controlling the scheme which is used to solve the finite difference equations. III. D ESIGN AND N UMERICAL R ESULTS In order to validate the simulation results, initially the conventional triangular lattice silica PCF shown in the inset of Fig. 1 [19] has been considered. In this study, all the holes have the same diameter d and the hole pitch is fixed to 2.3μm. In addition, the refractive index of the silica background is taken as 1.45 at the operating wavelength λ = 1.55μm. Fig. 1 shows the wavelength dependence of the real part of the complex effective index of the quasi TE mode of the conventional PCF at two different d/ ratios, 0.4 and 0.5. As revealed from this figure there is an excellent agreement between the FVFDM results and those reported in the literature by the FEM [19]. Fig. 2(a) shows cross section of the conventional rectangular lattice PCF. All the holes have the same diameter d1 and are arranged with hole pitches x and y in x and y directions, respectively. In the proposed design, an extra hole of diameter d2 is added to the right upper corner of the rectangular core region of the conventional rectangular lattice PCF. Therefore, the core region of the suggested PR has a slanted sidewall as shown by dotted lines in Fig. 2(b) and (c). The three holes in
HAMEED AND OBAYYA: DESIGN CONSIDERATION OF POLARIZATION CONVERTER
Fig. 2. Cross section of the (a) conventional rectangular lattice PCF, (b) and (c) proposed PS-PCF PRs.
the slanted side can have different diameters of d2 , d3 , and d4 as shown in Fig. 2(b). In the design of PR optical waveguides, it is necessary to increase the magnitude of the nondominant field components of the fundamental TE and TM polarized modes at the outset. As a result, their profiles will be modified to increase the overlap with the dominant field components. Therefore, to achieve polarization rotation in a single section device it is important to design a highly hybridness PR waveguide that can support the quasi-TE and quasi TM modes, with nearly equal amplitude of Hx and H y field components. The hybridness will peak up when the polarization states make ±45° with the horizontal and vertical axes. Working on this principle, if a nearly pure TE mode is introduced from a standard input waveguide as shown in Fig. 2(a) with very little hybrid modes, this incident mode excites both the quasi TE and quasi TM modes of almost similar modal amplitudes. However, as the two modes propagate along the PR waveguide, they are out of phase and their combined modal fields produce a nearly pure TM mode at the conversion length Lπ . According to the coordinate system defined in Fig. 2, the y fundamental quasi TE mode refers to the fundamental H11 or x E11 modes, while the fundamental quasi TM mode refers to x or E y modes. The minimum longitudinal the fundamental H11 11 distance at which maximum polarization conversion occurs is called the conversion length or the half-beat length Lπ which can be calculated using π Lπ = (1) β H y − β H11x 11
y
and β are the propagation constants of the H11 where β x (quasi TM) mode, respectively. For a (quasi TE) mode and H11 single section PR, the length of the PS-PCF PR with the hybrid mode should be equal to the half-beat length Lπ to reverse the polarization state. In this study, the modal hybridness can be obtained using the following overlap integral [20] S = E x (T E, 1)E x (T M, 2)+ E y (T E, 1)E y (T M, 2) d x d y y H11
x H11
(2) where E(TE,1) and E(TM,2), are the vector fields for TE mode in the input section and for TM mode in the proposed PS-PCF PR section, respectively.
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Fig. 3. Variation of the modal hybridness with hole diameter for three different cases.
The reported structure is based on rectangular core region PCF with a slanted sidewall. It has been shown earlier [7], [8], [21], and [22], that for a waveguide with a slanted sidewall, and more particularly without geometric symmetry, the hybrid nature of the vector modal field is significantly enhanced. Therefore, the slanted sidewall of the reported PR plays the fundamental role in achieving peak hybridness leading to possibility of designing such PR design. Hence, non-symmetrical PCF PR waveguide with only one slanted sidewall is considered for evaluation, as shown in Fig. 2(b) and (c). In addition, the suggested PR could be easily fabricated by using an extra hole in the corner of the core region. Therefore, it does not require a complex fabrication process like the semiconductor waveguide with slanted sidewalls [7], [8]. In the proposed structure, all the cladding air holes are arranged with hole pitches x = 1.5 μm and y = 2.15 μm. In addition, silica refractive index is taken as 1.45 at the operating wavelength 1.55 μm. The effective index contrast between the core and the cladding regions ensures the index guiding through the suggested PS-PCF PR. The effects of the hole diameters, d1 , d2 , d3 , d4 are first investigated. Fig. 3 shows the variation of the modal hybridness with the hole diameter for three different cases. First, all the hole diameters, d1 , d2 , d3 , and d4 are equal and their effects on the modal hybridness are studied. In the second case, d1 is fixed to 1.4 μm while the hole diameters of the slanted side, d2 , d3 , and d4 are equal and are changed from 0.8 μm to 1.5 μm. However, in the final case, d1 , d3 , and d4 are equal and are taken as 1.4 μm as shown in Fig. 2(c) while d2 is variant from 0.8μm to 1.5μm. It is evident from Fig. 3 that the final case when d2 is only variant, achieves the highest modal hybridness of 0.9165 at d2 = 1.5 μm. As d2 is increased from 0.8 μm to 1.5 μm, the modal hybridness is increased from 0.6893 to 0.9165. Therefore, complete polarization conversion can occur at d2 = 1.5 μm. The variation of the conversion length with the hole diameters for the three cases are shown in Fig. 4. It is revealed from this figure, that the conversion lengths decrease with increasing the hole diameter in all cases. Therefore, the suggested PS-PCF PR shown in Fig. 2(c) with d2 = 1.5 μm, and d1 = d3 = d4 = 1.4 μm can offer complete polarization conversion with short conversion length. Thus, this design
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Fig. 4. Variation of the conversion length with hole diameter for three different cases.
Fig. 5. Contour plot of the (a) nondominant Hx and (b) dominant H y y field profiles of the fundamental H11 (quasi-TE) mode for the conventional rectangular lattice PCF. However, Fig. 5(c) and (d) shows the contour plot of the nondominant H x and dominant H y field profiles of the quasi TE mode, respectively for the reported PS-PCF PR.
will be considered as the appropriate device in the subsequent simulations. In this case, all the cladding holes have the same diameter d1 except the extra air holes of diameter d2 in the right upper corner of the core region as shown in Fig. 2(c). Fig. 5(a) and Fig. 5(b) show the non-dominant Hx and y dominant H y field components of the H11 (quasi-TE) mode of the conventional rectangular lattice PCF with hole diameter of d1 = 1.4 μm, x = 1.5 μm and y = 2.15 μm. It is observed from these figures that the field profile of H y is symmetric in nature as the PCF structure itself is symmetric. In addition, the maximum magnitude of Hx is only 0.02, normalized to the maximum value of H y . The non-dominant H y field profile of x (quasi TM) mode is not shown here, but this profile is the H11 y similar to the non-dominant Hx field profile of the H11 (quasi TE) mode. Only a small amount of mode conversion can take place in the conventional rectangular lattice PCF with air holes because the dominant and non-dominant field profiles of the two polarized modes are not of very unequal amplitudes. This type of PCF with very little hybridization can be used as an input or output waveguide. The non-dominant Hx and dominant H y field profiles of y the H11 (quasi TE) mode of the suggested PS-PCF PR with d1 = 1.4 μm, d2 = 1.5 μm, x = 1.5 μm and y = 2.15 μm are shown in Fig. 5(c) and Fig. 5(d), respectively. It is observed
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 48, NO. 8, AUGUST 2012
Fig. 6. Evolution of the TM powers for the TE excitation along the propagation direction at different extra hole diameters d2 .
that the field profiles of the dominant and non-dominant y components of the H11 (quasi TE) mode are very similar. The maximum value of Hx is 0.99631, normalized to the maximum value of the dominant H y component, which is 50 times higher than that of the conventional rectangular lattice PCF. This ensures that the proposed PCF with slanted sidewall core region supports highly hybrid modes which can be used for designing polarization conversion devices. In addition, the expected overlap between the y x vector field components of the H11 (quasi-TE) and H11 (quasi TM) modes can improve the polarization conversion through the reported PS-PCF PR. When a TE polarized mode obtained from the conventional rectangular lattice PCF is launched directly into the proposed PS-PCF PR with x = 1.5 μm, y = 2.15 μm, d1 = 1.4 μm, and d2 = 1.5 μm, the input power excites two hybrid modes along the PR waveguide. These two modes become out of phase at a distance equals to Lπ from the beginning of the PR section. Therefore, H y component will be cancelled while the Hx component will be added which produces a nearly pure TM mode as shown in Fig. 6. The calculated Lπ using the FVFD-BPM is 2839 μm which is in an excellent agreement with 2837.6 μm calculated by the FVFDM. Fig. 6 shows the variations of the TM power Px for the TE input along the axial direction, at different d2 values, 0.8 μm, 1.1 μm, and 1.5 μm while the other parameters are fixed to x = 1.5 μm, y = 2.15 μm, and d1 = 1.4 μm. The polarization power factors, P y and Px , are defined as the power carried by the H y and Hx field components, respectively, over the PR waveguide cross section, normalized to the total power. It can be observed from this figure that for the TE excitation, initially Px is zero and it slowly increases to a maximum value at z = Lπ and if the PR section is not terminated at this position the optical power Px starts decreasing. It is also evident from Fig. 6 that the conversion ratio increases with increasing d2 which is very compatible with the behavior of the modal hybridness, shown in Fig. 3. It should also be noted that if the bigger hole of diameter d2 is placed at any other corner of the rectangular core region, the same modal hybridness of 0.9165 and conversion length of 2839 μm are obtained. This is due to the slanted side wall of the core region resulted
HAMEED AND OBAYYA: DESIGN CONSIDERATION OF POLARIZATION CONVERTER
Fig. 7. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with extra hole diameter d2 .
from the extra hole in any corner of the rectangular core region. Consequently, the proposed PR design has a very little sensitivity to the location of the bigger hole in any corner rendering itself to overcome potential fabrication errors. Hence it is of great importance to consider the fabrication tolerances and their effects on the polarization converter performance. The variation of the hole diameter d2 is the first parameter to be considered. To study the fabrication tolerances of the device length, the power conversion and the corresponding crosstalk for the TE excitation at specified longitudinal positions have been considered for various d2 values while the other parameters are taken as x = 1.5 μm, y = 2.15 μm, d1 = 1.4 μm and λ = 1.55 μm. Fig. 7 shows the variation of the TM power, Px , and the crosstalk values, in dB, with d2 , in the range from 0.8 μm to 1.5 μmat both the exact value of Lπ and the specified device length of 2839 μm. It is found that the behavior of the Px variation with d2 is very compatible with the behavior of the modal hybridness, shown in Fig. 3. As it has been shown before that the modal hybridness is maximum at d2 = 1.5 μm, the value of Px reaches its maximum value at d2 = 1.5 μm and also the corresponding crosstalk is a minimum value of −32.98 dB. It is also evident from Fig. 7 that changing d2 yields large variation in the TM power, Px as well as the corresponding crosstalk. This is due to the large effect of d2 on the modal hybridness of the two polarized modes and conversion length as shown in Fig. 3 and Fig. 4. It should be noted that when fabricating the device within the range of the d2 from 1.45 μm to 1.5 μm, the maximum Px power and the crosstalk at the designed length will be always better than 0.983 and −20 dB, respectively. Next, the effect of the deformation of the extra hole into elliptical hole on the PR performance is investigated. Here, a2 and b2 are the radii of the extra hole in x and y directions, respectively as shown in the inset of Fig. 8. In this study, the radius in y direction b2 , is fixed to 0.75 μm. In addition, d1 , hole pitches, x , and y are fixed to 1.4 μm, 1.5 μm, 2.15 μm, respectively. Fig. 8 shows the variation of the conversion length LC and modal hybridness, with the radius in x direction a2 of the extra hole. It may be shown from this figure that as a2 increases from 0.4 to 0.75, the conversion
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Fig. 8. Variation of the modal hybridness and the conversion length with the extra hole radius a2 in x direction.
Fig. 9. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with the extra hole radius a2 in x direction.
length decreases from 4123 μm to 2839 μm while the modal hybridness increases from 0.6836 to 0.9165. In order to investigate the tolerance in the hole radius a2 of the extra hole over the polarization rotation, the converted TM power Px for the TE excitation, at z = Lπ and z = 2839 μm and the crosstalk are calculated as shown in Fig. 9. It is revealed from this figure that fabricating the device within the range of a2 from 0.7 μm to 0.75 μm, the crosstalk at the designed length will be always better than −18 dB. The effect of the extra hole radius in y direction b2 at constant a2 of 0.75 μm on the PR performance is also considered. It is found that the effect of b2 at constant a2 has the same effect of a2 at constant b2 on the PR performance. The influence of shifting the position of the extra hole on the PR performance will be next studied. Here the shift ratio is defined as (S/x ) where S is the shift distance in +x or −x directions. The positive and negative shift ratio indicates the shift in positive and negative x directions, respectively. In this study the other parameters are kept constant at x = 1.5 μm, y = 2.15 μm, d1 = 1.4 μm, d2 = 1.5 μm and λ = 1.55 μm. The numerical results reveal that the conversion length increases with increasing the shift ratio in +x direction. However, the conversion length decreases with increasing the shift ratio in −x direction as shown in Fig. 10. Fig. 10 shows the variation of the conversion length LC and modal
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Fig. 10. Variation of the modal hybridness and the conversion length with the shift ratio in x direction (S/ x ).
Fig. 12. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with the cladding hole diameter d1 .
Fig. 11. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with the shift ratio in x direction (S/ x ).
Fig. 13. Variation of the modal hybridness and the conversion length with the hole pitch in x direction x .
hybridness, with the shift ratio (S/x ). It is also evident from this figure that the modal hybridness decreases with increasing the (S/x ) in both +x and −x directions. In order to investigate the tolerance in the position of the extra hole over the polarization rotation, the converted TM power Px for the TE excitation, at z = Lπ and z = 2839 μm and the crosstalk are calculated as shown in Fig. 11. It is observed from this figure that the crosstalk at exact Lπ increases from −32.98 dB to −14.86 dB and −23.56 dB with increasing the (S/x ) ratio from 0 to −0.2 and 0.1 in the −x and +x directions, respectively. However, the crosstalk at z = 2839 μm increases from −32.98 dB to −8.7 dB and −15.45 dB as (S/x ) ratio increases from 0 to −0.2 and 0.1 in −x and +x directions, respectively. In addition, there is a difference between Px at z = Lπ and at z = 2839 μm due to the effect of the (S/x ) ratio on the conversion length as shown in Fig. 10. The effect of the shift in ±y direction on the performance of the PR has also been performed. It is found that fabricating the device for a tolerance of shift ratio of 0.07 in ±x or ±y direction, the crosstalk at the designed length will be always better than −20 dB. The effect of the cladding hole diameter d1 , is also studied. The fabrication tolerances of the device length, the power conversion and the corresponding crosstalk at specified longitudinal positions have been considered for various cladding hole radius d1 while the hole pitches x , y and d2 are fixed
to 1.5 μm, 2.15 μm, and 1.5 μm respectively. In addition, the operating wavelength is fixed to 1.55 μm. For this purpose, the cladding hole diameter d1 has been varied from 0.8 μm to 1.5 μm, while the optimum device length has been fixed at 2839 μm, as it corresponds to the ideal in design conditions, x = 1.5 μm, y = 2.15 μm, d1 = 1.4 μm, d2 = 1.5 μm and λ = 1.55 μm. As shown in Fig. 12, the value of Px for the TE excitation at the corresponding values of Lπ increases from 0.9845 to 0.9995 as d1 increases from 0.8 μm to 1.4 μm. On the other hand, if the power conversion is to be calculated at a fixed device length of 2839 μm, the Px values will be much lower than those at the exact values of Lπ , except at a hole diameter of 1.4 μm, where the exact value of Lπ is itself 2839 μm. This is due to the effect of d1 on the conversion length. As d1 increases from 0.8 μm to 1.5 μm, the conversion length decreases from 4570 μm to 2608 μm. It is also evident from Fig. 12 that the crosstalk measured at the specified device length of 2839 μm decreases from −3.18 dB to a value of −32.98 dB as the cladding hole diameter d1 is increased from 0.8 μm to 1.4 μm. Then the crosstalk increases to a value of −16.35 dB as d1 is increased to 1.5 μm. Also, it is important to mention that for the cladding hole diameter range from 1.3 μm to 1.5 μm, the crosstalk value will be still less than −16 dB. The effect of the hole pitch x in x direction on the PR performance will be the next parameter to be considered.
HAMEED AND OBAYYA: DESIGN CONSIDERATION OF POLARIZATION CONVERTER
Fig. 14. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with the hole pitch x .
Fig. 15. Variation of the modal hybridness and the conversion length with the hole pitch in y direction y .
In this study, the other parameters are still invariant at y = 2.15 μm, d1 = 1.4 μm, d2 = 1.5 μm and λ = 1.55 μm. Fig. 13 shows the variation of the conversion length LC and modal hybridness, with the hole pitch x . It is evident from this figure that the modal hybridness decreases with increasing x . As x is increased from 1.5μm to 2.0μm, the modal hybridness is decreased from 0.9165 to 0.4723. Therefore, complete polarization conversion can occur at x = 1.5 μm. In addition, Fig. 13 reveals that the conversion length increases with increasing the hole pitch x . The LC increases from 2839μm to 4692μm with increasing x from 1.5 μm to 2 μm. In order to investigate the tolerance in the hole pitch x over the polarization rotation, the converted TM power Px for the TE excitation, at z = Lπ and z = 2839 μm and the crosstalk are obtained within the x range from 1.45 μm to 2.0 μm as shown in Fig. 14. It is observed from this figure that the crosstalk at z = 2839 μm decreases from −13.24 dB to −32.98 dB with increasing x from 1.45 μm to 1.5 μm. As x increases to 2.0 μm, the crosstalk increases to 3.68 dB. In addition, the power Px at exact Lπ and at z = 2839 μm decreases from 0.9995 to 0.4607 and 0.3002, respectively with increasing x from 1.5 μm to 2.0 μm. This is due to the large effect of x on the modal hybridness as shown in Fig. 13. It can also be seen from Fig. 14 that fabricating the device for a tolerance of x = 1.5 μm ±0.03 μm, the crosstalk at the designed length will be always better than −19 dB.
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Fig. 16. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with the hole y .
Fig. 17. Variation of the converted power Px for the TE excitation and the crosstalk at z = L π and z = 2839 μm with the operating wavelength.
The effect of the hole pitch in y direction y on the PR performance is also evaluated. In this study, the other parameters are taken as x = 1.5 μm, d1 = 1.4 μm, d2 = 1.5 μm and λ = 1.55 μm. Fig. 15 shows the variation of the conversion length LC and modal hybridness, with the hole pitch y . It is evident from this figure that the conversion length increases by increasing y . In addition, the modal hybridness increases with increasing y from 1.9 μm to 2.15 μm. If y is further increased the modal hybridness is decreased. Fig. 16 shows the converted TM power Px for the TE excitation and the crosstalk at z = Lπ and z = 2839 μm within the y range from 1.9 μm to 2.25 μm. It is revealed from the numerical results that fabricating the device for a tolerance of y = 2.15 μm ± 0.05 μm, the crosstalk at the designed length will be always better than −19 dB. The effect of rotating the reported PR structure by 90° is also studied. In this case, the bigger hole of diameter d2 is moved to the right lower corner. In addition, the hole pitch in x direction x is increased to 2.15 μm while the hole pitch in y direction y is reduced to 1.5 μm. The numerical results reveal that the modal hybridness of the 90° rotated trapezoid decreases to 0.1255 which is compatible with the reported results in Figs. 13 and 15. Figs. 13 and 15 show that the modal hybridness decreases by increasing x and by
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decreasing y . Therefore, small conversion ratio is achieved at conversion length of 1184 μm with crosstalk of 13.739dB. Also, for this single-section passive PR, the effect of varying the operating wavelength, λ, on the performance of the PR is studied in the range from 1.5μm to 1.63μm. In this study, the change of the refractive index of the composing materials with the wavelength is taken into account. Fig. 17 shows the variation of the TM power, Px for the TE excitation and the corresponding crosstalk at the exact values of Lπ and the design length of 2839 μm. At the given device length of 2839 μm, if the wavelength varies in the range of 1.55 μm ± 0.05 μm, the crosstalk is better than −21 dB. IV. C ONCLUSION A single section passive polarization converter based on silica PCF has been presented and analyzed. The numerical results demonstrate that the proposed design has the potential in providing a strong polarization conversion. The reported converter offers complete polarization conversion with almost 100% polarization conversion ratio with a device length of 2839 μm. In addition, the suggested PS-PCF PR is easier for fabrication than the LC PCF PR. The results shown here demonstrate a reasonably stable performance with slight variations in the fabrication parameters or the operating wavelength as might be expected. R EFERENCES [1] I. Morita, K. Tanka, N. Edagawa, and M. Suzuki, “40 Gb/s single-channel soliton transmission over transoceanic distances by reducing Gordon-Haus timing jitter and soliton-soliton interaction,” J. Lightw. Technol., vol. 17, no. 12, pp. 2506–2511, Dec. 1999. [2] S. Obayya, Computational Photonics, 1st ed. New York: Wiley, 2011, pp. 50–63. [3] Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett., vol. 59, no. 11, pp. 1278–1280, Sep. 1991. [4] S. S. A. Obayya, B. M. A. Rahmann, and H. A. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photon. Technol. Lett., vol. 12, no. 10, pp. 1346–1348, Oct. 2000. [5] S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Beam propagation modeling of polarization rotation in deeply etched semiconductor bent waveguides,” IEEE Photon. Technol. Lett., vol. 13, no. 7, pp. 681–683, Jul. 2001. [6] S. S. A. Obayya, N. Somasiri, B. M. A. Rahman, and K. T. V. Grattan, “Full vectorial finite element modeling of novel polarization rotators,” Opt. Quantum Electron., vol. 35, nos. 4–5, pp. 297–312, 2003. [7] B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikati, “Design and characterization of compact single-section passive polarization rotator,” J. Lightw. Technol., vol. 19, no. 4, pp. 512–519, Apr. 2001. [8] S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Improved design of a polarization converter based on semiconductor optical waveguide bends,” Appl. Opt., vol. 40, no. 30, pp. 5395–5401, 2001. [9] L. Scolari, T. Alkeskjold, J. Riishede, A. Bjarklev, D. Hermann, A. Anawati, M. Nielsen, and P. Bassi, “Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers,” Opt. Express, vol. 13, no. 19, pp. 7483–7496, 2005. [10] L. Wei, L. Eskildsen, J. Weirich, L. Scolari, T. Alkeskjold, and A. Bjarklev, “Continuously tunable all-in-fiber devices based on thermal and electrical control of negative dielectric anisotropy liquid crystal photonic bandgap fibers,” Appl. Opt., vol. 48, no. 3, pp. 497–503, 2009.
[11] M. F. O. Hameed and S. S. A. Obayya, “Analysis of polarization rotator based on nematic liquid crystal photonic crystal fiber,” J. Lightw. Technol., vol. 28, no. 5, pp. 806–815, 2010. [12] M. F. O. Hameed and S. S. A. Obayya, “Polarization rotator based on soft glass photonic crystal fiber with liquid crystal core,” J. Lightw. Technol., vol. 29, no. 18, pp. 2725–2731, Sep. 2011. [13] A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightw. Technol., vol. 26, no. 11, pp. 1423–1431, Jun. 2008. [14] W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron., vol. 29, no. 10, pp. 2639–2649, Oct. 1993. [15] S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, “Accurate finite element modal solution of photonic crystal fibres,” IEE Proc. Optoelectron., vol. 152, no. 5, pp. 241–246, Oct. 2005. [16] W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett., vol. 15, no. 6, pp. 363–369, 1997. [17] V. Hill, O. Farle, P. Ingelström, and R. Dyczij-Edlinger, “Efficient implementation of nonuniform refinement levels in a geometric multigrid finite-element method for electromagnetic waves,” IEEE Trans. Magn., vol. 43, no. 4, pp. 1521–1524, Apr. 2007. [18] S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, “New full-vectorial numerically efficient propagation algorithm based on the finite element method,” J. Lightw. Technol., vol. 18, no. 3, pp. 409–415, Mar. 2000. [19] S. Haxha and H. Ademgil, “Novel design of photonic crystal fibers with low confinement losses, nearly zero ultraflatted chromatic dispersion, negative chromatic dispersion and improved effective mode area,” J. Opt. Commun., vol. 281, no. 2, pp. 278–286, 2008. [20] C. M. Weinert and H. Heidrich, “Vectorial simulation of passive TE/TM mode converter devices on InP,” IEEE Photon. Technol. Lett., vol. 5, no. 3, pp. 324–326, Mar. 1993. [21] J. J. G. M. van der Tol, F. Hakimzadeh, J. W. Pedersen, D. Li, and H. van Brug, “A new short and low-loss passive polarization converter on InP,” IEEE Photon. Technol. Lett., vol. 7, no. 1, pp. 32–34, Jan. 1995. [22] B. M. A. Rahman, N. Somasiri, and M. Rajarajan, “Compact passive polarization converter using slanted semiconductor rib waveguides,” in Proc. Integr. Photon. Res. Conf., Quebec City, QC, Canada, Jul. 2000, pp. 60–62.
Mohamed Farhat O. Hameed was born in Egypt in 1979. He received the B.Sc. degree in electronics and control engineering and the M.Sc. degree in engineering physics from Mansoura University, Mansoura, Egypt, in 2000 and 2005, respectively, and the Ph.D. degree in photonics through a channel scheme from Mansoura University, and Glamorgan University, Wales, U.K., in 2010. His current research interests include the numerical modeling of photonic crystal fibers, directional couplers, splitters, and polarization converters.
Salah S. A. Obayya was born in Egypt in 1969. He received the B.Sc. degree in electronics and communications engineering from Mansoura University, Mansoura, Egypt, in 1991, and the Ph.D. degree from the Department of Electrical, Electronic, and Information Engineering, City University London, London, U.K. He was a Full Professor in 2008 and the Chair in photonics with the Photonics Research Group, Glamorgan University, Wales, U.K. He is currently with the Department of Electronics and Communications Engineering, Faculty of Engineering, Mansoura University. His current research interests include frequency-selective surfaces, and linear, nonlinear, passive, and active photonic devices.
