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Abstract
A surface plasmon resonance based D-shaped tunable and broadband single-polarization filter is proposed. The complex propagation constants and the guided modes’ corresponding field profiles are calculated using the full-vector finite element method based COMSOL Multiphysics v5.0 software. From the simulation study, it is found that the confinement losses of the desired x-polarized mode are 0.17 dB/cm, and 0.30 dB/cm at the resonance wavelength of 1.31 µm, and 1.55 µm, respectively, while the losses for y-polarized unwanted guided mode are 736.30 dB/cm, and 573.32 dB/cm at the mentioned wavelengths, respectively. It is also evident that the proposed fiber offers a crosstalk of 625.10 dB at the wavelength of 1.31 µm, and 495.31 dB at the wavelength of 1.55 µm with the corresponding bandwidth of 490 nm and 485 nm over which the crosstalk is more than 20 dB for a fiber length of 1.0 mm. The study also underpins that the resonant peak shifts linearly with the change of lattice constant, radius of the gold-coated channel in the polished surface, and the size and the distance between the tuning air holes of the proposed filter. Hence, this filter can be a potential fit for micro-integration in optical systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
For the last few years, photonic crystals have become one of the most compelling research areas in the field of non-linear optics [1,2], micro-structured fiber based sensing and detection [3–5] and so on. Based on the properties of photonic crystals, photonic crystal fiber (PCF) is introduced, which is a micro-structured holey fiber with periodically modulated refractive index profile and photonic dispersion characteristics analogous to the electronic band structure in solid [6]. Normally, the periodically modulated refractive index profile results from two-dimensional periodically arranged hollow channels along the fiber length. By exploiting the advantages of this unique refractive index profile and dispersion characteristics, the guided modes’ propagation characteristics can be tailored in a way that was unprecedented with conventional optical fibers $e.g$. broadband dispersion compensation [7,8], supercontinuum generation [9,10] etc. In the perspective of applications, photonic crystal-based polarizing devices can play a pivotal role in polarization-sensitive systems like optical measurement instruments [11], biomedical applications [12] etc.
Traditional anisotropic absorption sheets, prism polarizers by refraction, Brewster-angle polarizers by reflection, etc., are not convenient for integrating dense optical systems due to the lack of compactness. An optical phenomenon named surface plasmon resonance (SPR) can offer a promising solution to this problem [13]. If the parallel wave vector of the incident wave matches the propagation constant of the surface wave in the interface of two mediums having an opposite sign of permittivity like a metal-dielectric interface, energy exchange happens between the incident photons and the oscillating electrons in that interface due to the formation of SPR [14]. Unlike prism-based setups, metal-coated or nano-wire inserted PCFs can offer small and compact setups for the phase matching of the electromagnetic wave and surface plasmon polariton (SPP). A careful design leads a particular orthogonal component of the HE$_{11}$ guided mode prone to more loss than the rest; it is the underlying physics of the PCF-based single-polarization filter.
Zhang et al. demonstrated the filtering characteristics of a selectively silver coated and filled micro-structured optical fiber (MOF) with broken C$_{5v}$ symmetry in 2007 [15]. In the year of 2011, Q. Bao et al. demonstrated a graphene-coated fiber polarizer that allows propagation of TE polarization and offers extinction co-efficient up to 27 dB within the communication bands [16]. Besides, J. Xue et al. proposed a gold-coated and purified water-filled PCF-based plasmonic polarizer that works at a wavelength of 1311 nm with a maximum loss of 508 dB/cm for $y$-polarization [17] in 2013. In addition, A. Khaleque et al. in the year of 2015 proposed a squeezed lattice plasmonic single-polarization filter. In that proposal, the $x$-polarized and $y$-polarized guided mode experience confinement loss of 1221 dB/cm and 1.6 dB/cm, respectively, at an operating wavelength of $1.31\: \mu m$ [18]. However, the authors did not report the essential performance parameters like crosstalk and bandwidth. Later on, An et al. proposed a gold nano-wire inserted $C_{2v}$ symmetric PCF based plasmonic polarization filter in 2016, which can filter out $x$-polarized light at a wavelength of 1.31 $\mu$m with resonance strength 231.60 dB/cm and $y$-polarized light at a wavelength of 1.55 $\mu$m with resonance strength 237.90 dB/cm just by adjusting the diameter of gold nano-wires [19].
Furthermore, in the year of 2017, M. Li et al. proposed a hexagonal single-polarization plasmonic filter that offers full width at half maxima (FWHM) of 13 nm and a fiber of 1.0 mm length can provide maximum operating bandwidth (crosstalk over 30 dB) of 70 nm and 235 nm covering second and third optical communication windows respectively [20]. Also, Boyao Li et al. proposed a duel wavelength single-polarization filter in the same year where $y$-polarized guided mode experiences loss of 265.04 dB/cm and 230.50 dB/cm at wavelengths of 1310 nm and 1550 nm, respectively, but the maximum available bandwidth is 440 nm with a comparatively large fiber length of 10 mm [21]. In the same year, Y. Guo et al. also proposed a D-shaped single-polarization broadband filter that operates at a wavelength of 1.55 $\mu$m with a loss of 326.70 dB/cm for $y$-polarization and the available bandwidth (crosstalk above 20 dB) is 480 nm when the fiber length is 1.0 mm [22]. Moreover, in the year of 2019, C. Liu et al. proposed a single polarization filter using gold and silver coating at the same time with liquid inclusion, and the losses of 544.30 dB/cm and 147.30 dB/cm at wavelength of 1310 nm and 1560 nm, respectively, are reported [23].
Recently, in the year 2020, M. M. Rahman et al. has proposed a dual-band single-polarization filter that offers an operating bandwidth of more than 800 nm when the fiber length was only 100 $\mu$m; however, the compactness of this fiber comes at the cost of fabrication difficulties due to the presence of elliptical air holes [24]. In 2021, Y. Wang et al. has proposed a side-leakage external gold-coated single-polarization filter in which phase matching occurs at a wavelength of 1.41 $\mu$m and 1.59 $\mu$m with a respective loss peak of 725.74 dB/cm and 1097.94 dB/cm [25]. X. Meng et al., in that same year, proposed a D-shaped single-polarization wavelength tunable filter covering wavelength ranges from 1250-1350 nm and 1550-1750 nm and provides operating bandwidth of 170 nm and 550 nm in the second and the third optical windows respectively [26].
From the literature survey, it is evident that single-polarization filters available in the prior literature typically incorporate tiny metal-coated and filled air holes; as a result, internal coating and filling of the small air openings often raises fabrication difficulties since filling micron-scale air holes with metal and maintaining the appropriate thickness and the roughness of the deposited metal film are somewhat challenging. Furthermore, the resonant peak of the undesired guided mode does not shift linearly with the design parameters in most of the single polarization filters. The linear shifting of the phase-matching point with design parameters would be a desirable feature for tuning the filter at an appropriate wavelength by altering those design parameters.
In this paper, a D-shaped SPR based broadband and tunable single-polarization filter is introduced with a gold-coated truncated channel in its flat surface, which is a plausible alternative to coating the whole flat surface of the filter with a metal film. Besides, the required area covered with metal film for the proposed filter is considerably small than the D-shaped or externally coated fibers. In addition, the resonant peak of the undesired orthogonal component of the guided mode shifts linearly with the radius of this channel. The proposed filter offers the coarse tuning of the resonant peak at the desired wavelength since redshift occurs linearly with the increase of lattice constant. Thus, the proposed fiber can filter out one of the orthogonal components of its linearly polarized mode in the second and third optical windows just by altering its lattice constant. Furthermore, increasing the radial dimension of the tuning air holes and the truncated channel on the flat surface offers an additional way to fine-tune the resonant wavelength. The tight confinement of the desired guided mode and leaky nature of its orthogonal counterpart results in high differential loss, consequently offering significantly high crosstalk over a broad wavelength band. Hence, this filter can play a pivotal role in the realm of micro-integration of the optical systems.
2. Schematic of the proposed filter
Figure 1 depicts the schematic diagram of the proposed D-shaped PCF filter. A two-dimensional triangular lattice arrangement with specific lattice constant $(\Lambda )$ and air hole diameter (d) modulates the refractive index profile of the proposed filter. The filling fraction of the proposed fiber $(d/\Lambda )$ is considered 0.7. Two comparatively large air holes induce asymmetry in the $x$-direction like the stress component of the polarization maintenance PANDA fiber [27]. The diameter of this stress component $(D)$ is considered $1.25\Lambda$. Besides, two tiny air holes are placed in the fiber core to fine-tune the resonant wavelength. The diameter $(d_1)$ and the center to center distance between the tuning air holes$(\Lambda _1)$ are considered as $d_1=0.35\Lambda$ and $\Lambda _1=0.42\Lambda$ respectively. The polished layer maintains a depth (h) such that h is equal to $2.18\Lambda$ from the center of the filter. The radius of the truncated metal-coated air opening in the polished layer is denoted by r and considered $r=0.93\:\mu m$. The coating thickness of the metal film is denoted by t and the optimum thickness is considered $50\;nm$. Fused silica is chosen as the background material due to its transparent board region, low absorption, and scattering losses in both infrared and ultra-violet spectrum [28]. Moreover, the hard and robust nature of the fused silica avails polishing convenience for the flat portion of the proposed filter. The wavelength-dependent refractive index profile of the fused silica is described by the following Sellmeier equation [29]:
Fig. 1. (a) Three-dimensional view of the proposed filter (b) Cross-sectional view of the proposed filter.
Table 1. Values of the fitting coefficients of the Sellmeier equation of fused silica
Table 2. Values of the parameters of the Drude-Lorentz model of gold
3. Methodology
The complex propagation constants and the corresponding field profiles of the guided modes are calculated using commercially available full-vector finite element method (FV-FEM) based software COMSOL Multiphysics v5.0. The total computational domain is discretized into 52,824 triangular elements and 2,836 edge elements for computational accuracy. A cylindrical perfectly matched layer (PML) is used as the absorbing boundary condition to avoid the unwanted reflection of the scattered field. In this simulation work, a PML of $1.2\: \mu m$ thickness is used since the imaginary part of the effective mode index of the guided modes remains almost constant for any values of thickness higher than 1.2 $\mu m$. The modal properties of the guided modes are investigated by solving the following eigenvalue equation for eigenvalue $\lambda =-j\beta$ [32,33]:
where $k_o$ denotes the free space wavenumber and $n$ denotes the refractive index. The electric field propagating in the z-direction is denoted by $E$ and can be expressed as $E(x,y,z,t)=E(x,y)\: e^{j(\omega t-\beta z)}$ where $\beta$ is the propagation constant and $\omega$ is the angular frequency. In plasmonic single-polarization filters, energy exchange occurs between one of the orthogonal components of the guided mode and the SPP mode when the guided mode is propagating across the fiber. This phenomenon can be described theoretically with couple-mode equations [34]:4. Synopsis of the transmission properties of the proposed filter
To investigate the filtering properties of the proposed fiber, the optimized value of the lattice constant of the proposed fiber, $\Lambda$, is considered $1.8\:\mu m$ initially. The effective mode indices $(n_{eff}=\beta /k_o)$ of the linearly polarized guided mode and SPP mode are extracted by solving the Eq. (3) using the numerical solver. It is evident from Fig. 2(a) that the effective mode index of both guided core mode and second-order SPP mode decreases smoothly with the increasing wavelength except for the presence of an S-knot at 1.31 $\mu m$ which indicates the anti-crossing of the mentioned two modes; thus phase matching occurs at 1.31 $\mu m$ with a complete mode coupling. Due to the mode coupling, the signal strength of $y$-polarized guided mode will be reduced significantly, and confinement loss is a reference parameter to indicate this optical power deterioration and resonance strength. The confinement loss $(\alpha )$ of the guided modes can be expressed mathematically by the following equation [35]:
Fig. 2. (a) Effective mode index and loss profile of the proposed filter for $\Lambda =1.8\:\mu m$ (b) Mode coupling mechanism between guided core mode and 2nd order SPP mode for $\Lambda =1.8\:\mu m$ with normalized electric field $(V/m)$ distribution around the fiber core, (c) Effective mode index and loss profile of the proposed filter for $\Lambda =2.0\:\mu m$ (d) Mode coupling mechanism between guided core mode and 2nd order SPP mode for $\Lambda =2.0\:\mu m$ with normalized electric field $(V/m)$ distribution around the fiber core
4.1 Tunable features of the proposed filter
It is evident that only by changing the lattice constant $(\Lambda )$ the proposed filter can be operated at a wavelength of 1.31 $\mu m$ and 1.55 $\mu m$. So, the wavelength at which the phase matching occurs can be tailored by changing the lattice constant $(\Lambda )$ of the proposed filter. At any specific wavelength, with the increase of lattice constant $(\Lambda )$, the effective mode index of both the $y$-polarized guided mode and second-order SPP mode increases. The effective mode index of both the $y$-polarized guided mode and SPP mode decreases with the operating wavelength keeping the lattice constant $(\Lambda )$ fixed. For lattice constant $(\Lambda )$ equal to 1.85 $\mu m$, phase matching occurs at wavelength 1.37 $\mu m$, and the confinement loss of the $y$-polarized guided mode at this wavelength is 684.98 dB/cm.
If the lattice constant is further increased, the phase matching point shifts further at larger wavelengths. For lattice constant equal to 1.9 $\mu m$ and 1.95 $\mu m$, the phase matching between the $y$-polarized guided mode and the SPP mode occurs at an operating wavelength of 1.43 $\mu m$ with resonance strength of 645.43 dB/cm and operating wavelength of 1.49 $\mu m$ with resonance strength of 605.36 dB/cm respectively. Therefore, by altering the lattice constant of the proposed fiber, it can be operated at different operating wavelengths. Fig. 3(b) portrays the effect of the fiber dimension in terms of lattice constant $(\Lambda )$ on the resonant wavelength of the proposed filter. The resonant wavelength shifts linearly with the change of lattice constant, which eventually assures the viability of the tunable property of the proposed filter. The confinement loss of the $x$-polarized guided mode increases with the operating wavelength though a small loss peak appears in the operating bandwidth due to a very weak coupling with SPP mode, as shown in Fig. 3(d). Thus, the confinement loss of the $x$-polarized mode at its peak due to coupling is considerably slight than the confinement loss of the $y$-polarized mode at the resonant wavelength of the proposed fiber.
Fig. 3. (a) Effective mode index of the $y$-polarized guided mode and 2nd order SPP mode as a function of operating wavelength for different lattice constant, (b) Resonant wavelength of the proposed filter as a function of lattice constant, (c) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different lattice constant, (d) Loss profile of the $x$-polarized guided mode as a function of operating wavelength for different lattice constant
The proposed fiber offers an additional way for fine-tuning the resonant wavelength by altering the size of the truncated gold-coated channel in the polished surface and the tuning air holes adjacent to the fiber core while variation in lattice constant $(\Lambda )$ of the crystal structure avails coarse tuning at the same time. To investigate the effect of the truncated air hole, numerical simulation is performed keeping $d_1/\Lambda =0.35$ and $\Lambda _1/\Lambda =0.42$ fixed for $\Lambda =1.8\:\mu m$ and $\Lambda =2.0\:\mu m$. For $\Lambda =1.8\:\mu m$, the resonant peak appears at a wavelength of 1.40 $\mu m$, 1.356 $\mu m$, 1.31 $\mu m$, 1.266 $\mu m$, and 1.23 $\mu m$ wavelength when $r$ is equal to 0.87 $\mu m$, 0.90 $\mu m$, 0.93 $\mu m$, 0.96 $\mu m$ and 0.99 $\mu m$ respectively. Similarly, the resonant peak appears at a wavelength of 1.646 $\mu m$, 1.598 $\mu m$, 1.55 $\mu m$, 1.49 $\mu m$, and 1.446 $\mu m$ wavelength when $r$ is equal to 0.87 $\mu m$, 0.90 $\mu m$, 0.93 $\mu m$, 0.96$\mu m$ and 0.99 $\mu m$ respectively for $\Lambda =2.0\: \mu m$. Therefore, it is evident from Fig. 4 that an increase in the radius of the truncated air hole in the polished surface results in a blue shift of the resonant peak.
Fig. 4. (a) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different values of $r$ where lattice constant, $\Lambda =1.8\:\mu m$ (b) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different values of $r$ where lattice constant, $\Lambda =2.0\:\mu m$, (c) Resonant wavelength of the proposed filter as a function of $r$
Fig. 5. (a) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different values of $d_1$ where $\Lambda _1/\Lambda =0.42$ and lattice constant, $\Lambda =1.8\:\mu m$ (b) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different values of $d_1$ where $\Lambda _1/\Lambda =0.42$ and lattice constant, $\Lambda =2.0\:\mu m$, (c) Resonant wavelength of the proposed filter as a function of $d_1/\Lambda$, (d) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different values of $\Lambda _1$ where $d_1/\Lambda =0.35$ and lattice constant, $\Lambda =1.8\:\mu m$, (e) Loss profile of the $y$-polarized guided mode as a function of operating wavelength for different values of $\Lambda _1$ where $d_1/\Lambda =0.35$ and lattice constant, $\Lambda =2.0\:\mu m$, (f) Resonant wavelength of the proposed filter as a function of $\Lambda _1/\Lambda$
To demonstrate the effect of diameter variation of the tuning air holes, the distance between the tuning air holes is kept fixed and considered $\Lambda _1/\Lambda$ equal to 0.42. It is evident from Fig. 5 that the resonant peak shifts proportionately with the size of tuning air holes for lattice constant $(\Lambda )$ equal to 1.8 $\mu m$ and 2.0 $\mu m$; approximately a change of 18 nm of the diameter of tuning air holes $(d_1)$ results in 12 nm shift of resonant peak from wavelength band 1287 nm to 1334 nm when $\Lambda =1.8\:\mu m$ and approximately a change of 20 nm of the diameter of tuning air holes $(d_1)$ results in 20 nm shift of resonant peak from wavelength band 1515 nm to 1586 nm when $\Lambda =2.0\:\mu m$. However, an increasing distance between the tuning air holes $(\Lambda _1)$ results in a blueshift of the resonant wavelength of the proposed filter. To investigate the influence of the distance between the tuning air holes $(\Lambda _1)$ on loss profile, $d_1/\Lambda$ is considered 0.35 for lattice constant, $\Lambda =1.8\:\mu m$ and $\Lambda =2.0\:\mu m$. It is evident that an approximately 18 nm increase in the distance between tuning air holes $(\Lambda _1)$ shifts the resonant peak 15 nm from wavelength band 1339 nm to 1284 nm for lattice constant, $\Lambda =1.8\:\mu m$ and for lattice constant, $\Lambda =2.0\: \mu m$, approximately an increment of 20 nm distance between the tuning air holes $(\Lambda _1)$ results in 22 nm shift of the resonant peak from wavelength band 1591 nm to 1511 nm.
4.2 Performance of the proposed filter in terms of crosstalk and insertion loss
In a single polarization filter, crosstalk (CT) is a performance parameter that indicates the extent of unwanted guided mode suppression and the available bandwidth of the proposed device. The wavelength spectrum over which the crosstalk is more than 20 dB is considered the operating bandwidth. The crosstalk (CT) of the proposed filter can be mathematically expressed as [36,37]:
Fig. 6. (a) Mapping of the crosstalk as a function of fiber length over the operating wavelength band for lattice constant, $\Lambda =1.8\:\mu m$, (b) Crosstalk as a function of operating wavelength for lattice constant, $\Lambda =1.8\:\mu m$, (c) Crosstalk and bandwidth of the proposed filter in terms of fiber length at the wavelength of 1.31 $\mu m$ for lattice constant, $\Lambda =1.8\:\mu m$, (d) Mapping of the crosstalk as a function of fiber length over the operating wavelength band for lattice constant, $\Lambda =2.0\:\mu m$, (e) Crosstalk as a function of operating wavelength for lattice constant, $\Lambda =2.0\:\mu m$, (f) Crosstalk and bandwidth of the proposed filter in terms of fiber length at the wavelength of 1.55 $\mu m$ for lattice constant, $\Lambda =2.0\:\mu m$.
Crosstalk can be tuned by altering the length of the fiber, which consequently affects the operating bandwidth of the proposed filter. For the convenience of understanding, the crosstalk of the proposed filter over operating wavelength band is portrayed in Fig. 6(b) and Fig. 6(e) for fixed fiber lengths of $250\:\mu m$, $500\:\mu m$, $750\:\mu m$ and $1000\:\mu m$. Crosstalk increases linearly with the increasing fiber length as depicted in Fig. 6(c) and Fig. 6(f); the proposed filter offers maximum crosstalk of 625.1 dB and 495.31 dB for a fiber length of 1000 $\mu m$ at resonant wavelength of 1.31 $\mu m$ and 1.55 $\mu m$, respectively. In addition, the proposed filter offers a reasonable bandwidth of 490 nm (covering telecom optical wavelength bands O, E, S, C, L, and U) and 485 nm (covering telecom optical wavelength bands E, S, C, L, and U) when the resonant peak is at wavelength 1.31 $\mu m$ and 1.55 $\mu m$, respectively.
Insertion loss refers to the attenuation of the signal power of the desired guided mode as the light pulse travels through the fiber. Insertion loss of the proposed filter is denoted by $IL$ and can be defined mathematically as [38]:
where $P_{out}$ and $P_{in}$ are the output power and the input power of the desired guided mode. Fig. 7(a) and Fig. 7(b) depict the insertion loss of the proposed filter for $\Lambda =1.8\:\mu m$ and $\Lambda =2.0\: \mu m$, respectively. Due to the weak coupling between the $x$-polarized guided mode and SPP mode, a small insertion loss peak appears at $1.09\:\mu m$ for $\Lambda =1.8\:\mu m$ and the filter experiences 0.0137 dB, 0.0273 dB, 0.0410 dB and 0.0547 dB insertion loss for a fiber length of $250\:\mu m$, $500\:\mu m$, $750\:\mu m$ and $1000\:\mu m$, respectively. Similarly, at a wavelength of $1.24\:\mu m$, the insertion loss peak appears for $\Lambda =2.0\:\mu m$ and the filter experiences insertion loss of 0.0497 dB, 0.0993 dB, 0.1490 dB, and 0.1986 dB for a fiber length of $250\:\mu m$, $500\:\mu m$, $750\:\mu m$ and $1000\:\mu m$ respectively. A point to be noted here is that the insertion loss is considerably low as 0.017 dB and 0.03 dB for a fiber length of $1000\:\mu m$ at the wavelength of $1.31\: \mu m$ and $1.55\:\mu m$, respectively, where the phase matching between the $y$-polarized guided mode and SPP mode occurs. The performance of the proposed single-polarization filter is compared with some previous works in Table 3.Fig. 7. (a) Insertion loss of the proposed filter as a function of wavelength for $\Lambda =1.8\:\mu m$ (b) Insertion loss of the proposed filter as a function of wavelength for $\Lambda =2.0\:\mu m$
Table 3. Comparison of the proposed filter with some previous works
4.3 Effect of other design parameters
In this section, we are going to illustrate the effect of the diameter of the big air holes $(D)$, thickness of the gold film $(t)$, and the polishing depth $(h)$ of the proposed filter on its loss profile. The two symmetric big air holes in the horizontal direction of the fiber geometry influence the tight confinement of the desired guided mode and assist in maintaining the polarization state. To observe the influence of big air holes, the loss profile of the $y$-polarized guided mode is analyzed for D equal to $1.10\:\Lambda$, $1.15\Lambda$, $1.20\Lambda$, $1.25\Lambda$, and $1.30\Lambda$. With the increase of the diameter of the big air holes, the resonant peak shifts slightly at higher wavelengths when $\Lambda =1.8\:\mu m$ as portrayed Fig. 8(a). A similar trend is observed for $\Lambda =2.0\:\mu m$ (Fig. 8(b)); however, the resonant peaks are a little bit spaced than when $\Lambda =1.8\:\mu m$. The optimum dimension of the big air holes is $D=1.25\Lambda$ because phase matching occurs at wavelength $1.31\:\mu m$ and $1.55\:\mu m$ for $\Lambda =1.8\:\mu m$ and $\Lambda =2.0\:\mu m$ respectively.
Fig. 8. (a) Loss profile of the $y$-polarized guided mode for different values of D when $\Lambda =1.8\:\mu m$ (b) Loss profile of the $y$-polarized guided mode for different values of D when $\Lambda =2.0\:\mu m$
Next, the effect of gold film thickness $(t)$ is analyzed for $t$ is equal 40 nm, 45 nm, 50 nm, 55 nm, and 60 nm. It is evident from Fig. 9(a) that the blue shift of the resonant peak occurs with the increase of the thickness of the metal film. In addition, the loss of the $y$-polarized guided mode at the resonant peak decreases with the increase of film thickness. For $t=40\:nm$, the resonance occurs at a wavelength of $1.345\:\mu m$, and the confinement loss is 817.77 dB/cm when $\Lambda =1.8\:\mu m$. The resonant peak shifts at $1.31\:\mu m$ wavelength with the reduction of resonance strength of 736.3 dB/cm when the metal film thickness is 50 nm. Further increase of the film thickness shifts the resonant peak at a lower wavelengths. However, the resonance strength does not alter significantly with the increase of film thickness above 50 nm. A similar phenomenon is observed for $\Lambda =2.0\:\mu m$; the phase matching occurs at $1.587\:\mu m$ wavelength, and the $y$-polarized guided mode experiences confinement loss of 642.93 dB/cm when $t=40\:nm$. The resonant peak shifts at $1.55\:\mu m$ wavelength when the metal film thickness is 50 nm. Further increase of the metal film does not significantly change the $y$-polarized guided mode’s confinement loss. Hence, the optimum thickness of the metal film is considered 50 nm for the proposed filter.
Fig. 9. (a) Loss profile of the $y$-polarized guided mode for different values of t (b) Mode field diameter and the splicing loss of the proposed filter for $x$-polarized guided mode
For the integration of the proposed single-polarization filter in the practical optical system, single-mode fibers (SMFs) are spliced at the ends of it, and as a result of mode field diameter (MFD) mismatch between the SMFs and the filter, splicing loss occurs. The MFD is the diameter that corresponds to the effective area of the fiber $(A_{eff})$ which can be expressed mathematically as $A_{eff}=\frac {\left ( \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } |E|^{2}dxdy\right )^{2}}{\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|E|^{4}dxdy}$ where $E$ is the electric field derived from Maxwell’s equation [32]. The MFD of the fiber is related to the effective area by the equation $A_{eff}=k_n\pi r^{2}$ where $k_n$ is the correction factor and $MFD=2r$ [43]. The splicing loss $(L_s)$ can be expressed by the well-known equation [44]:
where $w_{SMF}$ and $w_{PCF}$ are the mode field diameter of the SMF and the proposed filter respectively. The loss due to the splicing for $y$-polarized guided mode is outweighed by its confinement loss because of the formation of SPR induced mode coupling. Figure 9(b) portrays the MDF and the splicing loss of the $x$-polarized guided mode when the fiber is being spliced with a SMF whose MFD is assumed to be 10 $\mu m$. The correction factor $(k_n)$ is considered unity approximating that the electric field distribution is Gaussian in this analysis. It is evident that the MFD of the filter increases with the increase of operating wavelength which consequently reduces the mismatch between the MFDs of the proposed filter and the spliced SMF. Hence, the splicing loss decreases with the increase of operating wavelength.Maintaining the polishing depth of the proposed filter $(h)$ is one of the most challenging parts. The optimum polishing depth is considered $2.18\Lambda$. If the flat surface of the filter is polished more than the requirement, the resonant peak shifts at lower wavelengths and, conversely, shifts at the higher wavelengths when the flat surface is inadequately polished. The challenging part of the wheel polishing is to keep the variation of the polishing depth within the $\pm 4\%$ of the optimum depth. Because, beyond this limit, if the fabricated fiber is polished to a height less than the optimum polishing level, the uppermost horizontal array of the air holes are deformed; in addition, if the fabricated fiber is inadequately polished, the formation of a truncated channel in the polished surface is inhibited. The effect of the variation of the polishing depth is appended in Table 4 where $\Delta h$ is the percentage variation of polishing depth, $\lambda _{peak}$ is the resonant wavelength and Loss peak is the maximum loss of the $y$-polarized guided mode at resonant peak.
Table 4. Effect of the polishing depth $(h)$ on loss spectrum
5. Fabrication feasibility
For the practical realization of the proposed single-polarization filter, fabrication feasibility is one of the most crucial issues. Conventional stack and drawing method of fiber fabrication can be a potential choice for this type of structure. A preform similar to the actual cross-section of the proposed fiber is needed to be constructed using a combination of hollow fused silica capillaries and solid fused silica rods. Two comparatively thicker wall capillaries for the tuning air holes are required in the central region of the fiber preform. The upper portion of the preform would be constructed with solid fused silica rods since the upper portion of the drawn fiber have to be polished at the optimum level of polishing depth utilizing the wheel polishing method [45]. The real-time monitoring of the polishing depth can be possible in this polishing technique by observing the varying intensity of the light at the receiving end of the fiber as the amount of light leaks out from the fiber varies with the polishing depth. The proposed filter has defects in its triangular lattice due to the presence of air holes that had radius other than that of the air holes in the regular lattice. As a result, some small solid fused silica rods would be used for covering those free spaces. Avoiding the collapse of the tuning air holes is one of the mechanically challenging parts of the fiber drawing process. N. A. Wolchover et al. implemented a ’distance’ brush flaming approach where the fiber preform is heated with a butane torch keeping it close but never directly applied on the preform and the authors reported that the preform can be tapered to an extent where the dimension of the air holes are between 120 nm to 200 nm keeping the flame 2.0 cm apart from the preform [46]. In addition, G. S. Wiederhecker et al. fabricated a PCF using the stack and draw method and experimentally characterized the fiber which has a minimum pitch of 650 nm and an air hole in the fiber core with a diameter of 100 nm [47]. In our proposed work, the smallest diameter of the air hole is considered 630 nm for the tuning air holes. Hence, we expect that our proposed fiber can be fabricated successfully by implementing the existing stack and drawing crafts with the proper dimension of the truncated channel in the flat surface and avoiding the collapse of the tuning air holes. Magnetron sputtering could be a viable option for coating the truncated air opening in the polished surface with the metal film [48,49]. Another important issue is the splicing of the SMFs with the proposed filter. Conventional electric arc fusion splicing method can be implemented for fiber splicing. However, S. G. Leon Saval et al. reported a versatile and splicing free approach for interfacing of virtually any type of index guided silica PCFs for instance PCFs for highly non-linear applications, and multi-core PCFs and we expect that this approach can be successfully adapted for the integration of the proposed filter in practical optical systems [50].
6. Conclusion
In this article, a highly tunable and broadband single-polarization filter is proposed and analyzed using FEM based simulation tool. The proposed filter features ability to tune the resonant wavelength since the phase matching point shifts linearly with some of its specific design parameters. In addition, the loss of the $y$-polarized guided mode is 4331.17 and 1911.06 times of the $x$-polarized guided mode when the phase matching occurs at a wavelength of $1.31\;\mu m$, and $1.55\:\mu m$, respectively. Besides, at those resonant wavelengths, the proposed filter offers crosstalk of 625.10 dB and 495.31 dB for a fiber length of 1.0 mm with a corresponding bandwidth of 490 nm and 485 nm, respectively. The proposed single-polarization filter can be a good choice for dense integration of optical systems because of its compactness and high crosstalk in the optical communication bands.
Acknowledgments
The authors would like to cordially thank anonymous reviewers for their meaningful insights.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
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