SECTION 3

## Design and Numerical Results

The cross section of the proposed NLC-PCF coupler is shown in Fig. 1. The cladding holes have been infiltrated with a NLC of type E7 and arranged in a soft glass background. The infiltrated holes of diameter d are arranged in a triangular lattice with hole pitch Λ. The separation between the centers of the two cores, A and B, shown in Fig. 1 is equal to . The NLCs considered in the proposed structure are anisotropic materials consisting of rod-like molecules which are characterized by ordinary index n_{o} and extraordinary index n_{e}. Moreover, the local orientation of the NLCs as shown in Fig. 1 is described by its director, which is a unit vector n along the direction of the average orientation of the molecules.

Having applied 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. Therefore, the fiber is placed between two pairs of electrodes [16] allowing for the arbitrary control of the alignment of the NLC director via an external voltage, as schematically shown in Fig. 1. Additionally, two silica rods with appropriate diameters are used to control the spacing between the electrodes and the fiber is surrounded by silicone oil, which has higher dielectric strength than air. Therefore, the external electric field as reported by Haakestad *et al.* [16] will be uniform across the fiber cross section which results in good alignment of the director of the NLC with constant rotation angle φ. Moreover, the nonuniform electric field region will be only at the edges far away from the cores where the light will be propagating. As a result, the proposed coupler overall performance will avoid the nonuniform field distribution. Other layouts, such as those described in [17] and [18], can also be used to ensure better field distribution uniformity over the fiber cross section.

The practical techniques that have been utilized in the manufacturing of the nonsilica PCF are capillary stacking [19], drilling [20], build-in casting [21], and extrusion [22], [23], [24]. However, extrusion mechanism offers a controlled and reproducible approach for fabricating complex structured PCFs with a considerable surface quality. Furthermore, extrusion can be used to produce structures that could not be created with the capillary stacking approaches. Therefore, in the literature, most of the fabricated nonsilica PCFs are based on extrusion. Recently, the extrusion approach has been extended to construct the soft glasses such as lead silicate (SF57 glass) [22], [23], [24] and tellurite [25]. The soft glass of type SF57 has low processing temperature of sim520 °C [26], while the softening temperature for silica glass is 1500–1600 °C. Therefore, it is possible to extrude the PCF preform directly from the bulk glass. Also, lead silicate glasses [22] offer the highest thermal and crystallization stability making them particularly attractive for PCF fabrication.

The filling of PCFs with liquid or liquid-crystalline materials has already been demonstrated in the literature [16], [18], [27], [28], [29]. Arc-fusion techniques have been successfully implemented for the infiltration of central defect cores [28], while extensive control in the infiltration process of either core or cladding capillaries can be achieved by using UV curable polymers [29]. In [16], all the cladding holes of the PCF are filled with NLC using capillary forces and an electrically tunable photonic bandgap guidance is reported. In addition, tunable light switch using PCF whose central defect and cladding holes are filled with NLC is studied by Fang *et al.* [18].

The ordinary n_{o} and extraordinary n_{e} refractive indices of the E7 material were measured previously by J. Li *et al.* [30] at different visible wavelengths in the temperature range from 15 °C to 50 °C with a step of 5 °C. Then, the Cauchy model was adopted to fit the measured n_{o} and n_{e}, which can be described as follows [30]:
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$$\eqalignno{{\rm n}_{\rm e} =&\, {\rm A}_{\rm e} + ({\rm B}_{\rm e}/\lambda^{2}) + ({\rm C}_{\rm e}/\lambda^{4})&\hbox{(6)}\cr\noalign{\vskip1pt} {\rm n}_{\rm o} =&\, {\rm A}_{\rm o} + ({\rm B}_{\rm o}/\lambda^{2}) + ({\rm C}_{\rm o}/\lambda^{4})&\hbox{(7)}}$$ where A_{e}, B_{e}, C_{e}, A_{o}, B_{o} and C_{o} are the coefficients of the Cauchy model. The Cauchy coefficients at T = 25 °C are A_{e} = 1.6933, B_{e} = 0.0078 μm^{2}, C_{e} = 0.0028 μm^{4}, A_{o} = 1.4994, B_{o} = < > < > 0.0070 μm^{2}, and C_{o} = 0.0004 μm^{4}. Fig. 2 shows the wavelength dependence n_{o} and n_{e} of the E7 material at different temperatures T from 15 °C to 50 °C with a step of 5 °C. It is evident from this figure that n_{e} is greater than n_{o} at the measured temperature values within the reported wavelength range. In the proposed design, the relative permittivity tensor ε_{r} of the E7 material is taken as [14]
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$$\varepsilon_{\rm r} = \left(\matrix{{\rm n}_{\rm o}^{2}\sin^{2}\varphi + {\rm n}_{\rm e}^{2}\cos^{2}\varphi & ({\rm n}_{\rm e}^{2} - {\rm n}_{\rm o}^{2})\cos \varphi \sin \varphi & 0\cr\noalign{\vskip-1pt} ({\rm n}_{\rm e}^{2} - {\rm n}_{\rm o}^{2})\cos \varphi \sin \varphi & {\rm n}_{\rm o}^{2}\cos^{2}\varphi + {\rm n}_{\rm e}^{2}\sin^{2}\varphi & 0\cr\noalign{\vskip-1pt} 0 & 0 & {\rm n}_{\rm o}^{2}}\right)\eqno{\hbox{(8)}}$$ where φ is the rotation angle of the director of the NLC, as shown in Fig. 1. The proposed in-plane alignment of the NLC can be exhibited under the influence of an appropriate homeotropic anchoring conditions [14], [31]. Haakestad *et al.* [16] demonstrated experimentally that in the strong field limit, the NLC of type E7 is aligned in plane in *y*-direction in capillaries of diameter 5 μm. Moreover, Alkeskjold and Bjarklev [32] presented in-plane alignment with three different rotation angles, 0°, 45°, and 90° in solid core PCF with hole diameter of 3 μm infiltrated with NLC of type E7 using two sets of electrodes.

The background material of the reported NLC-PCF coupler is a soft glass of type SF57 (lead silica). Fig. 2 shows the wavelength-dependent refractive index of the SF57 material. As revealed from this figure, the refractive index of the SF57 material is larger than n_{o} and n_{e} of the E7 material. Therefore, the average refractive index of the soft glass core is greater than that of the infiltrated NLC cladding region. Consequently, the propagation through the suggested design has been taken place by the modified total internal reflection The Sellmeier equation of the soft glass of type SF57 [22] is given by
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$$n_{SF57}^{2} = A_{o} + A_{1}\lambda^{2} + {A_{2} \over \lambda^{2}} + {A_{3} \over \lambda^{4}} + {A_{4} \over \lambda^{6}} + {A_{5} \over \lambda ^{8}}\eqno{\hbox{(9)}}$$ where n_{SF57} is the refractive index of the SF57 material, A_{o} = 3.24748, A_{1} = −0.00954782 μm^{−2}, A_{2} = 0.0493626 μm^{2}, A_{3} = 0.00294294 μm^{4}, A_{4} = −1.48144 × 10^{−4} μm^{6}, and A_{5} = 2.78427 × < > < > 10^{−5} μm^{8} [22].

In the proposed design, the cladding holes have the same diameter d and are arranged with a hole pitch Λ = 2.0 μm. In addition, n_{o}, n_{e}, n_{SF57} are taken as 1.5024, 1.6970, and 1.802, respectively, at the operating wavelength λ = 1.55 μm. Moreover, the rotation angle of the director of the NLC and the temperature are taken as 90° and 25 °C, respectively. The effective indices of the even mode n_{effe} and odd mode n_{effo} of the NLC-PCF coupler are evaluated by the FVFDM [8] with perfect matched layer boundary conditions. The coupling length, defined as the minimum longitudinal distance at which maximum power is transferred from one core to another, is one of the important characteristics of the directional couplers. The coupling length L_{C} can be obtained using the operating wavelength λ, n_{effe} and n_{effo} as follows:
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$${\rm L}_{\rm C} = {\lambda \over 2 ({\rm n}_{{\rm eff}\_{\rm e}} - {\rm n}_{{\rm eff}\_{\rm o}})}.\eqno{\hbox{(10)}}$$ Fig. 3 shows the variation of the coupling lengths of the NLC-PCF coupler for the quasi TE and TM modes with the *d*/Λ ratio at the operating wavelength of 1.55 μm while the hole pitch, is fixed to 2 μm. It is observed from this figure that the coupling lengths for the two polarized modes increase with increasing the *d*/Λ ratio at constant hole pitch Λ. As the *d*/Λ ratio increases at constant Λ, the soft glass bridge between the two cores of the NLC-PCF coupler decreases. Therefore, the distance taken by the modes to transfer between the two cores and, hence, the coupling lengths of the two polarized modes increases. As the *d*/Λ ratio increases from 0.6 to 0.85, the coupling lengths of the quasi TE and TM modes increase from 429 μm and 241 μm to 1914 μm and 454 μm, respectively. It is also evident from Fig. 3 that the coupling length of the quasi TE mode at φ = 90° is longer than that of the quasi TM mode. This can be explained by analyzing the dominant field components of the quasi TE and TM modes and the direction of the director of the NLC. The dominant field components of the quasi TE and TM modes are E_{x} and E_{y}, respectively. At φ = 90°, the director of the NLC is parallel to E_{y} while it is perpendicular to E_{x}, and the relative permittivity tensor ε_{r} of the E7 material has the diagonal form [n^{2}_{o},n^{2}_{e},n^{2}_{o}]. In this case, ε_{yy} is greater than ε_{xx}, and therefore, the index contrast seen by the quasi TE mode is greater than that seen by the quasi TM mode. Consequently, the quasi TE modes are more confined in the core regions than the quasi TM modes. As a result, the quasi TE modes take longer distance than the quasi TM modes to transfer from one core to another and hence the coupling lengths of the quasi TE modes are longer than those of the quasi TM modes.

It is also revealed from Fig. 3 that the NLC-PCF coupler has strong polarization dependence. This is due to the infiltration of the NLC which increases the birefringence between the two fundamental polarized modes in the proposed coupler. Therefore, the NLC-PCF coupler can be easily designed as a polarization splitter and its polarization dependence is stronger than those splitters presented in [6], [33], and [34]. However, the conventional silica PCF coupler [2] has low birefringence and the high birefringence can be realized by adjusting the size of the air holes around the two core regions [6], [33], [34], which enlarges the difference between the coupling lengths for the two polarized modes. In [33], the two identical cores are formed by combination of large and small air holes which makes the two cores birefringent. Zhang and Yang [34] reported a polarization splitter based on two nonidentical cores with also a combination of large and small air holes. However, in [6], two elliptical cores are used to improve the polarization dependence of the conventional silica PCF coupler.

The fiber coupler can separate the two polarized states, i.e., quasi TE and TM modes, at a given wavelength if the coupling lengths L_{CTE} and L_{CTM} of the quasi TE and TM modes, respectively, satisfy the coupling ratio [5]
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$${\vskip-12pt}\gamma = {\rm L}_{\rm CTE}: {\rm L}_{\rm CTM} = {\rm i}: {\rm j}\eqno{\hbox{(11)}}{\vskip12pt}$$ where i and j are two integers of different parities. In this case, the length of the coupler is equal to L_{f} = L_{CTM} × i/j. Therefore, to achieve the shortest splitter, the optimal value of γ should be 2. Fig. 4 shows the coupling length ratio between the coupling lengths of the quasi TE and TM modes as a function of the *d*/Λ ratio at two different hole pitch values: 3.7 μm and 3.9 μm. In this study, the operating wavelength, rotation angle of the director of the NLC and temperature are taken as 1.55 μm, 90°, and 25 °C, respectively. It is found that the coupling length ratio γ increases with increasing the *d*/Λ ratio at a given hole pitch while the crosstalk decreases as revealed from Figs. 4 and 5. The crosstalk is a measure of the unwanted power, remaining at the end of the NLC-PCF coupler. Fig. 5 shows the variation of the crosstalk for the quasi TE and TM modes with the *d*/Λ ratio at the operating wavelength λ = 1.55 μm at two different hole pitch values: 3.7 μm and 3.9 μm. As shown from this figure, the crosstalk of the quasi TE is lower than that of the quasi TM mode due to the better confinement of the quasi TE modes through the core regions than the quasi TM mode at φ = 90°. As can be seen from Fig. 4, the coupling length ratio equals 2.013 at *d*/Λ = 0.7 and hole pitch of 3.9 μm. The coupling lengths calculated by the FVFDM are 8252 μm and 4099 μm for the quasi TE and TM modes, respectively at the operating wavelength λ = 1.55 μm.

In order to confirm the polarization splitter based on the NLC-PCF coupler, the FVFD-BPM is used to study the propagation along its axial direction. Initially, at z = 0, the fundamental components H_{y} and H_{x} of the quasi TE and TM modes, respectively of soft glass PCF with air holes obtained using the FVFDM [8] at λ = 1.55 μm are launched into the left core A of the NLC-PCF coupler. These input fields, in turn, start to transfer to the right core of the coupler, and at the corresponding coupling lengths, the fields are completely transferred to the right core. The coupling lengths calculated by the FVFD-BPM are 8253 μm and 4100 μm for the quasi TE and TM modes, respectively, which are in excellent agreement with those obtained by the FVFDM. The ratio between the coupling lengths L_{CTE} to L_{CTM} is slightly larger than 2.0. Therefore, the length of the proposed coupler is L_{f} = (8253 + 2 ∗ 4100)/2.0 = 8227 μm at which the two polarized states are separated well. Fig. 6 shows the power transfer normalized to the input power for the quasi TE and TM modes at the operating wavelength of 1.55 μm in the left core of the NLC-PCF coupler. It is evident from Fig. 6 that the two polarized modes are separated well after a propagation distance equals to L_{f} = 8227 μm. The normalized powers of the quasi TE mode in the left and right cores of the coupler are 0.000435 and 0.9995, respectively, at z = 8227 μm. However, the normalized powers of the quasi TM mode in the left and right cores of the coupler are 0.9987 and 0.0013, respectively. Therefore, when launched from one end of the splitter, the signals of the quasi TE mode will exit at the other core B, while the signals of the quasi TM mode will exit at the same core A.

The field distributions of the dominant field component H_{y} and H_{x} of the quasi TE and TM modes, respectively at λ = 1.55 μm are shown in Fig. 7 at different waveguide sections z, 0, 4100 μm, and 8227 μm. It is evident from this figure that, at z = 0, the input fields are launched into the left core, and as the propagation distance increases, the normalized power in the right core increases, and that in the left core decreases. At z = 4100 μm, which is equal to the coupling length of the quasi TM mode, the normalized power of quasi TM mode is approximately completely transferred to the right core. The normalized powers of the quasi TM mode in the left and right cores of the coupler are 0.00096 and 0.99904, respectively. However, the normalized powers of the quasi TE mode in the left and right cores of the coupler are 0.5051 and 0.4949, respectively, at z = 4100 μm. Finally, the two polarized modes are separated after a propagation distance L_{f} = 8227 μm.

The crosstalk CT around the operating wavelength λ = 1.55 μm for the quasi TE and TM modes are shown in Fig. 8(a) and (b), respectively. The crosstalk in decibel [3] for the desired quasi TE mode at the right core B is defined such that
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$$\hbox{CT}_{\rm TE} = 10\log_{10}({\rm P}_{{\rm B},{\rm uTM}}/{\rm P}_{{\rm B}, {\rm dTE}})\eqno{\hbox{(12)}}$$ where P_{B,dTE} and P_{B,uTM} are the normalized power of the desired quasi TE and undesired quasi TM modes, respectively, at core B. However, the crosstalk [3] of the desired quasi TM mode at the left core A is given by
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$$\hbox{CT}_{\rm TM} = 10\log_{10}({\rm P}_{{\rm A},{\rm uTE}}/{\rm P}_{{\rm A}, {\rm dTM}})\eqno{\hbox{(13)}}$$ where P_{A,uTE} and P_{A,dTM} are the normalized power of the undesired quasi TE and desired quasi TM modes, respectively, at core A. It is revealed form Fig. 8 that the proposed splitter has large BWs of 30 nm and 75 nm for the quasi TE and TM modes, respectively, at which the crosstalks are better than −20 dB. In addition, the BWs are 15 nm and 42 nm for the quasi TE and TM modes, respectively, for the crosstalks that are lower than −25 dB. Therefore, the proposed polarization splitter is less sensitive to the perturbation introduced during the fabrication process due to the low-level crosstalks with wide wavelength ranges. The BWs of the NLCPCF splitter are much larger than those reported in [3] and [4]. The BW of the quasi TE mode in [3] is 2.7 nm around λ = 1.55 μm, while the BW in [4] is 2.0 nm. In addition, the proposed splitter is shorter than those reported in [3] and [4] of lengths 15.4 mm and 9.08 mm, respectively. Moreover, the NLC-PCF splitter has wide wavelength range larger than the reported splitter by Chen *et al.* [5] of BW 25.4 nm around λ = 1.55 μm for the quasi TE mode. Furthermore, the splitter in [5] has longer length of 10.69 mm than that of the NLC-PCF splitter.

It is also shown from Fig. 8, that the BW of the quasi TE mode is less than that of the quasi TM mode. At φ = 90°, the quasi TE mode is more confined through the core region than the quasi TM mode. Therefore, the quasi TM mode is more affected by the wavelength variation around λ = 1.55 μm than the quasi TE mode. As a result, the undesired normalized power of the quasi TM mode at core B at the device length of 8227 μm increases with the wavelength variation around λ = 1.55 μm more than the undesired normalized power of the quasi TE mode at core A. Therefore, the BW of the quasi TE mode at core B is less than that of the quasi TM mode at core A.

The tolerances of the fiber length, rotation angle φ of the director of the NLC and temperature are also investigated. It is worth noting that the tolerance of a specific parameter is calculated while the other parameters of the proposed design are kept constant. It is found that the fiber length and rotation angle φ allow a tolerance of ±3% and ± 5°, respectively, at which the crosstalks are still better than −20 dB.

As shown in Fig. 2, the ordinary n_{o} and extraordinary n_{e} refractive indices of the E7 material are influenced by the temperature variation which affects the coupling characteristics of the NLC-PCF coupler. It can be observed from Fig. 2 that n_{e} of the E7 material is more dependent on the temperature than n_{o}. As the temperature T increases from 15 °C to 50 °C, n_{e} of the E7 material decreases from 1.7096 to 1.6438 at the operating wavelength λ = 1.55 μm. However, n_{o} of the E7 material decreases slightly from 1.5034 to 1.5017 when the temperature changes from 15 °C to 35 °C, while n_{o} increases from 1.5017 to 1.5089 when T increases from 35 °C to 50 °C. Therefore, the tolerance of the temperature is the next parameter to be studied while the other parameters of the reported design are not modified. It is found that the crosstalk of the quasi TM mode is better than −30 dB over a temperature range from 15 °C to 50 °C. However, the crosstalk of the quasi TE mode has a tolerance of ± 5 °C at which the crosstalk is better than −14 dB. This can be explained as follows. At φ = 90°, the relative permittivity tensor ε_{r} of the E7 material has the diagonal form [n^{2}_{o},n^{2}_{e},n^{2}_{o}]. Therefore, as the temperature increases, ε_{yy} decreases while ε_{xx} is nearly invariant. As a result, the index contrast seen by the quasi TM mode increases by increasing the temperature which increases the confinement of the quasi TM mode through the core regions. Thus, the coupling length of the quasi TM mode increases with increasing the temperature. However, the index contrast seen by the quasi TE mode and, hence, its coupling length is nearly constant with the temperature variation. Therefore, the undesired normalized power of the quasi TM mode at core B at the device length of 8227 μm increases, which increases the crosstalk of the quasi TE mode. On the other hand, the undesired power of the quasi TE modes at core A is approximately invariant, which has little effect on the crosstalk of the quasi TM mode.