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Abstract
A hollow-core anti-resonant fiber (HC-ARF) design based on hybrid silica/silicon cladding is proposed for single-polarization, single-mode and high birefringence. We show that by adding silicon layers in a semi-nested HC-ARF, one of the polarization states can be strongly suppressed while simultaneously maintaining low propagation loss for other polarization states, single-mode and high birefiringence. The optimized HC-ARF design exhibits propagation loss, high birefringence, and polarization-extinction ratio of 0.05 dB/m, 0.5 × 10−4, >300 respectively for y-polarization while the loss of x-polarization is >5 dB/m at 1064 nm. The fiber also has low bend-loss and thus can be coiled to a small bend radii of 5 cm having ≈0.06 dB/m bend loss.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Controlling and preserving the polarization state of light plays a crucial role in many polarization-sensitive optical fiber systems including but not limited to fiber sensors [1], fiber lasers [2], polarization maintaining optical amplifier [3], and fiber based gyroscopes [4]. Polarization maintaining fibers (PMFs) based on solid-core fiber designs have been studied and fabricated over the past few years in which the two polarization states have different propagation constants ($\beta$) [5,6]. In these fibers, high birefringence (Hi-Bi) is achieved by introducing either stress [5] or anisotropy in the core ($e.g.,$ by making the core elliptical) [6]. However, these high birefringent polarization maintaining fibers (Hi-Bi-PMFs) have some fundamental limitations which are: (i) polarization mode dispersion can not be completely eliminated due to the different $\beta$ of two polarization states, and (ii) have relatively low damage threshold and high non-linearity dictated by the core material [7,8]. The polarization mode dispersion can be eliminated by using single-polarization fibers (SPFs) as light propagates in only one polarization state, whereas the other polarization state experiences high loss and it is suppressed [8,9].
The fundamental limitations in the above mentioned fibers can be eliminated by introducing air in the core region in a so called hollow-core fiber (HCFs). Two types of HCFs have been studied in the past decade [10]. The first type is a hollow-core photonic bandgap fiber (HC-PBGF), which guides light in the air-core via photonic bandgap effect and offers limited transmission band [11]. In this regards, a 19-cell HC-PBGF was reported in [12], which has low loss of $<$10 dB/km at 1530 nm and a birefringence of $\approx 3\times 10^{-4}$, but with significantly limited bandwidth (BW) ($<$10 nm). The birefringence and higher-order-modes (HOMs) were controlled by using two 7-cell shunts surrounding the core region. A single-polarization HC-PBGF with an asymmetric cladding was studied in which the polarization dependent loss can reach as low as 800 dB/m at 1550 nm [13]. All of proposed HC-PBGF designs so far have limited BW.
The second type of HCF which is called hollow-core anti-resonant fiber (HC-ARF) guides light via inhibited coupling (IC) between the core and cladding modes (CMs) [14,15]. Recently, HC-ARFs have attracted massive interest owing to their outstanding optical properties [15–17]. In contrast to HC-PBGFs, HC-ARFs offer larger transmission BW [15,18–21], low power overlap with cladding elements [22], low anomalous dispersion [23] and extremely low loss [24–26], and they are considered ideal platform for nonlinear optics and lasers [27]. Since the light is guided in the air-core region and the light overlap with the glass is low, it is challenging to introduce Hi-Bi in HC-ARFs of the same order that can be achieved in solid-core PMFs ($\approx 10^{-4}$) [7]. Furthermore, there are only a few theoretical investigations for achieving Hi-Bi and PM based on HC-ARFs have been demonstrated to date [7,8,28–32]. The first demonstration of Hi-Bi and single-polarization HC-ARF was first proposed by Mousavi et al., [7,33], where multiple nested resonators with different wall thicknesses were used to achieve Hi-Bi with a polarization extinction ratio (PER) of 1000 at 1550 nm. A Hi-Bi HC-ARF was theoretically demonstrated in a hybrid transmission band regime in which multiple layers of resonators were used [31]. A birefringence of $10^{-5}$ and PER of 850 at 1550 nm were numerically obtained using six-tube HC-ARF with two nested tubes [30]. However, the single-mode and bend loss analysis of these HC-ARFs were not discussed. Recently, a double ring HC-ARF is proposed having two different wall thickness which offers PER of 17000 at 1550 nm wavelength [8]. The HC-ARFs proposed and investigated in [7,8,30,33] have considered different wall thicknesses of the cladding capillaries. Due to their different wall thicknesses, these designs have some limitations, for example, a very fine control of wall thickness is required because the Hi-Bi and single-polarization is quite sensitive to the change of wall thickness. More recently, a single-polarization HC-ARF is proposed in which silicon as a high index material used on the inner surface of one vertical cladding tube [29]. A birefringence of $4\times 10^{-4}$ and single-polarization in the wavelength range of 1512–1587 nm with a broad BW of 75 nm was obtained. However, single-modeness property of the HC-ARF was not investigated. Table 1 shows an overview of the reported birefringence, loss, PER, BW, and bend loss of different HCFs and provides a direct comparison on the performance of the proposed fiber designs.
Table 1. Summary of the optical properties (birefringence, loss, PER, BW, and bend loss) of different HCF designs.
In this work, we propose a semi-nested HC-ARF design consisting of a hybrid silica/silicon cladding for achieving single-polarization, single-mode transport, and Hi-Bi at 1064 nm. The choice of silicon in our investigation is based upon the fact that the silicon coated/filled fibers have been studied and experimentally demonstrated [34–37]. Our main motivation is to provide a systematic investigation on how the optical properties of HC-ARF based on silica/silicon cladding can be optimized toward novel PM fiber-based devices. We investigate the optical properties of our proposed waveguide structure to optimize the fiber parameters (e.g., silica/silicon tube thickness, gap separation, nested tube ratio, and refractive index of silicon). The performance of the fiber was directly compared with a regular HC-ARF. We found that the semi-nested HC-ARF outperforms the regular HC-ARF under bend condition whereas both fibers show a comparable performance under straight condition.
2. Fiber geometry
The HC-ARF designs considered in our investigations are shown in Fig. 1. Figure 1(a) shows a typical 6-tube non-touching nested HC-ARF design. The proposed fiber design having core diameter $D_{\textrm {c}}$, tube diameter $D$, wall thickness of silica tubes $t_{1}$, and a gap separation between the outer tubes, $g$ is shown in Fig. 1(b). A non-touching configuration is chosen as it usually offers better loss properties than the touching configuration [15,38]. To obtain single-polarization and Hi-Bi simultaneously, silicon tubes are placed [marked as orange color in Fig. 1(b)] inside the silica tubes. In addition, two of the nested tubes are removed in the horizontal direction (semi-nested tubes) because the polarization state (x-polarization in this case) is lossy compared to the low-loss of the other polarization state. The wall thickness of silicon tubes is indicated by $t_2$. For the regular HC-ARF, we removed the nested tubes from outer tubes keeping other parameters same as semi-nested HC-ARF which is shown in Fig. 1(c). In all our numerical calculations, we choose a core diameter of $D_{\textrm {c}}$ = 25 $\mu$m. However, we optimize the silica/silicon tube thickness ($t_1/t_2$), and gap separation, $g$ to ensure single-polarization, single-mode, and Hi-Bi. The outer diameter $D$ is related to the core diameter $D_{\textrm {c}}$, wall thickness $t_1$, and number to tubes $N$, which can be written as [39],
Fig. 1. HC-ARF geometries considered in the simulations. (a) Typical 6-tube nested HC-ARF; (b) proposed HC-ARF in which silicon tubes are inserted inside of two silica tubes. (c) regular HC-ARF in which the nested tubes are removed from the outer tubes. The magnification of one of the silica/silicon tube is shown in (d). All fibers have the same core diameter $D_{\textrm {c}}$ = 25 $\mu$m, $t_1$/$t_2$ = wall thickness of silica/silicon tubes, $d$ = nested tube diameter, $D$ = outer tube diameter, and $g$ = separation between the outer tubes.
In our all simulations, a small penetration of $t_1/2$ of all cladding tubes into the outer silica tube was considered which is the typical case in fabricated fibers [40].
3. Numerical results and discussions
A finite-element modeling (FEM) based on COMSOL software was used to perform the numerical simulations. To accurately model the fiber modal properties, a perfectly-matched layer (PML) boundary was placed outside the fiber domain [15,38]. The modeling of HC-ARF is crucial, particularly the selection of mesh size and PML boundary conditions are important [15]. We used extremely fine mesh sizes of $\lambda /6$ and $\lambda /4$ in the silica walls and air regions respectively and optimized the PML boundary conditions according to [15,38]. The choice of such mesh sizes and PML boundary conditions provide excellent agreement with the experimental results [24,41]. The propagation loss was calculated by adding the contributions from leakage loss, effective material loss (EML), and surface scattering loss (SSL). The details of calculating the SSL can be found in [15,42]. The EML was calculated by estimating the the power overlap with silica and silicon walls, and then added to other loss contributions. The material attenuation of silica and silicon were taken from [43] and [44] respectively. It is to be noted that the material attenuation of silicon is $>$4200 dB/m at 1064 nm which is far higher than silica. When the fiber is at anti-resonant, the power overlap with silica walls and silicon walls are $10^{-4}$ and $5\times 10^{-6}$ respectively.
3.1 Optimization of silica/silicon wall thickness ($t_1/t_2$)
We begin our analyses by optimizing the wall thickness of silica/silicon ($t_1/t_2$) for a fixed core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, gap separation, $g$ = 3 $\mu$m, and normalized nested tube ratio, $d/D$ = 0.5. The wall thickness plays vital role for achieving single-polarization, birefringecence, and low-loss guidance. Figure 2 displays the 2D surface plots of fundamental mode (FM) loss for the x- and y- polarization, PER which is the loss ratio of two polarization states, and phase birefringence, $|n_x-n_y|$ as a function of silica and silicon wall thickness. One of the important aspects of these 2D plots is that the region of low-loss, single-polarization, and phase birefringence can easily be found from these 2D maps. It can be seen from Fig. 2(b) that at $\lambda$ = 1064 nm the FM loss for y-polarization remains $<$0.3 dB/m over wide range of 340 nm $<t_1<$ 355 nm and 210 nm $<t_2<$ 217 nm with a minimum loss of 0.02 dB/m. Figure 2(a) shows that the FM loss for x-polarization can be 5 dB/m for $t_1>$360 nm and $t_2$ >215 nm. The loss of x-polarization is due to the strong coupling between the HOMs and CMs. The PER $>$150 can be made over a wide range of $t_1$ and $t_2$ which is shown in Fig. 2(c). The fiber offers Hi-Bi of $>$0.5$\times 10^{-4}$ for $t_1/t_2$ = 350/217 nm, and can be made even higher by sacrificing the high value of PER.
Fig. 2. Calculated propagation loss of (a) x-polarization, (b) y-polarization, (c) PER, and (d) phase birefringence as a function of silicon thickness, $t_2$ with different values of silica thickness, $t_1$. The propagation loss of x-polarization is defined as the lowest loss among x-pol.$^1$ and x-pol.$^2$. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, normalized tube ratio, $d/D$ = 0.5, and gap separation, $g$ = 3 $\mu$m. The simulations were performed at 1064 nm. To plot the 2D surface plots, silica wall thickness, $t_1$, and silicon wall thickness, $t_2$ were scanned with 30 and 25 data points respectively and between the data points are interpolated.
In order to get a better understanding of the effect of of silicon wall thickness, $t_2$ on loss and birefringence, we plot relative effective mode index $\Delta n_{\textrm {eff}}$ = 1–$n_{\textrm {eff}}$, phase birefringence, and PER as a function of $t_2$ for a fixed core diameter, $D_c$ = 25 $\mu$m, $g$ = 2 $\mu$m, $d/D$ = 0.5, and $t_1$ = 350 nm. The results are shown in Fig. 3. The relative effective mode index of y-polarization remains unchanged while it changes for x-polarization as a function of $t_2$. It is to be noted that for x-polarization there are two states of polarization exists and this is due to the mode coupling between x-polarization core mode and the mode present in the silicon tube region. In our PER calculations, the lowest loss between the two x-polarization states (x-polarization (x-pol.$^1$) and x-polarization (x-pol.$^2$)) were considered. The mode field profiles are shown in Fig. 3(b) in which it is clear that the light is well guided inside the core for y-polarization whereas the light leaks and interact with cladding for x-polarization which simultaneously provide PER and Hi-Bi.
Fig. 3. (a) Relative effective mode index, $\Delta n_{\textrm {eff}}$ = 1–$n_{\textrm {eff}}$, (b) phase birefringence, (c) propagation loss, and (d) PER as a function of silicon thickness, $t_2$. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica wall thickness, $t_1$ = 350 nm, gap separation, $g$ = 2 $\mu$m, and normalized tube ratio, $d/D$ = 0.5. Inset of (b) presents the normalized mode field profiles for different polarization states. The color bar shows the intensity distributions in a linear scale. The simulations were performed at 1064 nm.
3.2 Optimization of gap separation, $g$ and silicon wall thickness, $t_2$
In this section, we discuss the effect of gap separation, $g$, and silicon wall thickness, $t_2$ on FM loss of x- and y- polarization, PER, and phase birefringence while keeping the core diameter fixed to 25 $\mu$m, silica wall thickness, $t_1$ = 350 nm, and normalized nested tube ratio, $d/D$ = 0.5 for semi-nested HC-ARF (top panel) and regular HC-ARF (bottom panel). The low-loss, single-polarization, and Hi-Bi strongly depend on gap separation, $g$ for both semi-nested and regular HC-ARF which can be seen in Fig. 4. This is due to the hybrid nature of the silica/silicon tubes in which the light of FM for one of the polarization states can spread to the cladding while maintaining well confinement for the other polarization state when gap separation, $g$ changes. It can be seen from Fig. 4 that the fiber has low-loss guidance for y-polarization whereas low-loss for x-polarization over a wide range of $g$ and $t_2$. From the PER plot it is evident that the fiber shows high PER for small values of $g$. It is expected because for small gap separation, $g$ the light interacts more with both silica/silicon tubes which provides loss for x-polarization whereas the light for the y-polarization is well guided. A birefringence of $>$0.5$\times 10^{-4}$ is achieved. These 2D plots also provide a suitable range of $g$ and $t_2$ values that the fiber can be operated. It is interesting to note that both fiber designs have almost similar loss patterns, however semi-nested HC-ARF has better loss performance for y-polarization compared to regular structure (see the scale bar for the better comparison). The minimum FM loss for y-polarization of semi-nested and regular HC-ARF is $\approx$0.02 dB/m and $\approx$0.1 dB/m respectively.
Fig. 4. Calculated propagation loss of (a) x-polarization, (b) y-polarization, (c) PER, and (d) phase birefringence as a function of silicon thickness, $t_2$ with different values of gap separation, $g$ for semi-nested HC-ARF (top panel) and regular HC-ARF (bottom panel). The propagation loss of x-polarization is defined as the lowest loss among x-pol.$^1$ and x-pol.$^2$. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica wall thickness, $t_1$ = 350 nm, and normalized tube ratio, $d/D$ = 0.5. The simulations were performed at 1064 nm. To plot the 2D surface plots, gap separation, $g$, and silicon wall thickness, $t_2$ were scanned with 30 and 40 data points respectively and between the data points are interpolated.
The effect of gap separation, $g$ on the FM loss and birefringence for a fixed core diameter, $D_c$ = 25 $\mu$m, $t_1/t_2$ = 350/217 $\mu$m, and $d/D$ = 0.5 are shown in Fig. 5. The relative effective mode index of y-polarization remains unchanged as a function of $g$. Unlike in Fig. 3(b), the relative effective mode index of x-pol.$^1$ also remains constant which suggests that the birefringence and the single-polarization will have less impact while changing the gap separation, $g$. This is confirmed by the results presented in Fig. 5.
Fig. 5. (a) Relative effective mode index, $\Delta n_{\textrm {eff}}$ = 1–$n_{\textrm {eff}}$, (b) phase birefringence, (c) propagation loss, and (d) PER as a function of gap separation, $g$. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica wall thickness, $t_1$ = 350 nm, silicon wall thickness, $t_2$ = 217 nm, and normalized tube ratio, $d/D$ = 0.5. Inset of (b) presents the normalized mode field profiles for different polarization states. The color bar shows the intensity distributions in a linear scale. The simulations were performed at 1064 nm.
3.3 HOM suppression and effectively single-mode operation
In this section, we discuss how the HOMs can be effectively suppressed in our designs and obtaining thus effectively single-mode (ESM) operation. The HOMs suppression was achieved by optimizing the nested tube ratio, $d/D$, and gap separation, $g$. The 2D surface plots of fundamental mode (FM) and HOMs propagation loss as a function of $g$ and $d/D$ for $\lambda$ = 1064 nm is shown in Fig. 6. The low-loss FM and ESM operation regime can be identified from these 2D maps. Figure 6(a) depicts that the FM loss can be maintained $<$0.04 dB/m in the range of 0.5 $\mu$m $<g<$ 3.5 $\mu$m and 0.45 $<d/D<$0.8. It also shows that the FM loss weakly depends on a wide range of $d/D$, and $g$. However, the FM loss increases with the decrease of $d/D$. The loss of HOMs (lowest loss among the LP$_{11a}$, LP$_{11b}$, LP$_{21a}$, and LP$_{21b}$ modes) is shown in Fig. 6(b). Unlike FM, the HOMs can be strongly coupled with CMs giving loss values by suitably choosing $g$, and $d/D$ [15,19,38]. The FEM modeling predicts that the HOM loss can be made $>$10 dB/m for $g$ $\approx$0.5 $\mu$m and $d/D$ $\approx$0.48, while maintaining the FM loss slightly higher than 0.05 dB/m. It is worth to mention that the HOMs suppression follow a unique ‘V-shape’ pattern originating at $d/D$ $\approx$0.6, which is recently demonstrated in [45]. Such a loss value for HOMs indicates that the HOMs are strongly suppressed and the fiber will not be able to guide HOMs after propagating short piece of length leading to ESM fiber.
Fig. 6. Calculated y-polarization propagation loss of (a) LP$_{01}$-like FM and (b) HOMs as a function of $d/D$ with different values of gap separation, $g$. HOMs loss is defined as the lowest loss among the four LP modes (LP$_{11a}$, LP$_{11b}$, LP$_{21a}$, and LP$_{21b}$). All simulations are performed at 1064 nm. The fiber has a fixed core diameter, $D_{\textrm {c}}$ = 25 $\mu$m and silica/silicon wall thickness, $t_1/t_2$ = 350/215 nm. To plot the 2D surface plots, gap separation, $g$, and normalized nested tube ratio, $d/D$ were scanned with 25 and 30 data points respectively and between the data points are interpolated.
In order to get better insights of how the normalized tube ratio, $d/D$ affects the FM loss, HOMs loss, and HOMER, we investigate with a fixed core diameter, $D_c$ = 25 $\mu$m, $g$ = 1.5 $\mu$m, $t_1/t_2$ = 350/215 nm while we change $d/D$ from 0.3 to 0.8. In addition, this analysis also helps to understand the coupling phenomena between the HOMs and CMs. The effective refractive index, $n_{\textrm{eff}}$ of the first six core-guided modes (LP$_{01}$, LP$_{11a}$, LP$_{11b}$, LP$_{21a}$, LP$_{21b}$, LP$_{02}$) and CMs, propagation loss of the first five core-guided modes (LP$_{01}$, LP$_{11a}$, LP$_{11b}$, LP$_{21a}$, LP$_{21b}$), and HOMER as a function of $d/D$ are shown in Fig. 7. It can be seen from Fig. 7(a) that the effective refractive index of the FM-like mode (red dotted line) remains unchanged as a function of $d/D$. Moreover, effective refractive index of FM-like mode avoids any phase matching with CMs ensuring low-loss guidance. The refractive index of HOMs (LP$_{11a}$, LP$_{11b}$, LP$_{21a}$, LP$_{21b}$) significantly changes at various values of $d/D$. This is due to the fact that the HOMs strongly coupled with the CMs as the refractive index of CMs can be changed drastically with $d/D$ (black dotted lines). The strong coupling effect between the HOMs and CMs can be clearly seen form the mode field profiles. Due to the strong coupling of HOMs and CMs, HOMs are highly lossy. The HOMs loss of $>$5 dB/m can be achieved by suitably choosing $d/D$. We expect that the fiber will not guide HOMs after propagating few meters of length. The HOMER is shown in Fig. 7(c) (which is defined as the ratio between the propagation loss of the HOM having the lowest propagation loss and the propagation loss of the fundamental mode (FM)) which is an indication of effectively single-modeness of a fiber. The HOMER as as $>$50 can be made for a wide range of $d/D$ which ensures that the proposed fiber can be a effective single-mode fiber.
Fig. 7. Effect of changing normalized nested tube ratio, $d/D$ on (a) effective refractive index, (b) loss, and (c) HOMER with a fixed core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica/silicon wall thickness, $t_1/t_2$ = 350/215 nm, and gap separation, $g$ = 1.5 $\mu$m. The normalized tube ratio, $d/D$ were scanned from 0.3 to 0.8. The simulations were performed at 1064 nm. The electric field intensities of the first six core-guided modes and CMs are shown for $d/D$ $\approx$0.38 and $g$ = 1.5 $\mu$m on a linear color scale.
3.4 Bend loss analysis
Here the effect of gap separation, $g$ and silicon wall thickness, $t_2$ on the bend loss is presented for a fixed bend radius, $R_{\textrm {b}}$ of 5 cm. In order to calculate the bend loss, the bent structure is transformed into its equivalent straight structure with equivalent refractive index profile, $n_{\textrm{eq}}$ defined by [46] $n_{\textrm {eq}}=n(x,y)e^{(x,y)/R_{\textrm {b}}}$, where $R_{\textrm {b}}$ is the bending radius, $(x,y)$ is the transverse distance from the center of the fiber, and $n(x,y)$ is the refractive index profile of the straight fiber. The elasto-optic effect was neglected in the bend loss calculations [15]. The bend direction was chosen in the x-direction. Figure 8 shows the bend loss of x- and y- polarization, PER, and phase birefringence for semi-nested HC-ARF (top panel) and regular HC-ARF (bottom panel) as a function of $g$ and $t_2$. For semi-nested HC-ARF, the y-polarization FM loss remains $<$0.2 dB/m in the range of 0.5 $\mu$m $< g <$ 2.1 $\mu$m and 210 nm < $t_2$ < 217 nm with a minimum loss of $<$0.02 dB/m whereas the x-polarization FM loss is higher in this range as expected which offers PER and Hi-Bi. As a comparison, 2D surface plots for regular HC-ARF is shown in the bottom panel of Fig. 8. Almost similar FM loss for y-polarization is found for regular and semi-nested case. However, for regular case the coupling between FM and CMs is much stronger than semi-nested case and it occurs for several gap separations, $g$.
Fig. 8. Calculated bend loss of (a) x-polarization, (b) y-polarization, (c) PER, and (d) phase birefringence as a function of silicon thickness, $t_2$ with different values of gap separation, $g$ under 5 cm bend radius for semi-nested HC-ARF (top panel) and regular HC-ARF (bottom panel). The propagation loss of x-polarization is defined as the lowest loss among x-pol.$^1$ and x-pol.$^2$. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica wall thickness, $t_1$ = 350 nm, and normalized tube ratio, $d/D$ = 0.5. The simulations were performed at 1064 nm. To plot the 2D surface plots, gap separation, $g$, and silicon wall thickness, $t_2$ were scanned with 40 and 30 data points respectively and between the data points are interpolated.
The bend loss of a regular and semi-nested HC-ARF as a function of bend radius for x- and y-bend direction is shown in Fig. 9. The bend radius was scanned from 3 cm to 10 cm. It can be seen from Fig. 9 that the semi-nested design outperforms the regular design. For regular structure, there is a strong coupling between FM and CMs occur at small bend radius of 3.35 cm (x-dir.) and 4.95 cm (y-dir.) which can be seen from the mode field profiles in Fig. 9. In contrast to the regular structure, the semi-nested HC-ARF design does not couple to the semi-nested tubes [17]. The results indicate that that semi-nested HC-ARF can be bent with small bend radius.
Fig. 9. Calculated bend loss of y-polarization as a function of bend radius for semi-nested HC-ARF (orange) and regular HC-ARF (green) for both bend directions. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica/silicon wall thickness, $t_1/t_2$ = 350/215 nm, gap separation, and $g$ = 2 $\mu$m. The normalized tube ratio, $d/D$ = 0.5 was chosen for semi-nested HC-ARF. The simulations were performed at 1064 nm. The electric field intensities of regular (left) and semi-nested (right) HC-ARF design for bend radius of 3.35 cm (x-dir.) and 4.95 cm (y-dir.) is shown in the inset on a linear color scale.
3.5 Wavelength dependence on phase birefringence, FM loss and PER
The phase birefringence, FM loss of x- and y-polarization, and PER as a function of wavelength for a fixed core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica/silicon wall thickness, $t_1/t_2$ = 350/215 nm, and normalized nested tube ratio, $d/D$ = 0.5 is shown in Fig. 10. The wavelength is scanned from 1045 nm to 1140 nm. The phase birefringence is $>$0.25$\times 10^{-4}$ in the entire wavelength range which is shown in Fig. 10(b). Figure 10(d) shows the PER which is >100 in the 1053–1094 nm wavelength range with highest PER of >500 is found at 1080 nm.
Fig. 10. (a) Relative effective mode index, $\Delta n_{\textrm {eff}}$ = 1–$n_{\textrm {eff}}$, (b) phase birefringence, (c) propagation loss, and (d) PER as a function of wavelength. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica/silicon wall thickness, $t_1/t_2$ = 350/215 nm, gap separation, $g$ = 1.5 $\mu$m, and normalized tube ratio, $d/D$ = 0.5. Inset of (b) presents the normalized mode field profiles for different polarization states. The dotted line in (a-d) indicate the operating wavelength 1064 nm.
3.6 Effect of silicon index
In this work, to obtain single-polarization and Hi-Bi, silicon is used as high index material. In the literature, there are several kinds of silicon with slightly different refractive indices mentioned. In addition, the index of silicon depends on temperature [47]. For example, silicon index material of $\approx$3.6 was considered for amorphous hydrogenated silicon. The influence of silicon index on the propagation loss and phase birefringence is shown in Fig. 11. The index of silicon was varied from 3.4 to 3.7. In this work, we choose the refractive index of silicon to be 3.5548 at 1064 nm [44]. It can be seen from Fig. 11(a) that the relative index of y-polarization remains constant while the index of x-polarization changes rapidly with silicon index. Therefore, the phase birerringence increases with the increase of silicon index which is shown in Fig. 11(b). It can be seen from Fig. 11(c) that the propagation loss of y-polarization decreases with the increase of silicon index from 3.4 to 3.55, while the loss increases sharply beyond 3.59 and maximum loss occurs for silicon index of $\approx$3.6276 which is due to the coupling between the core guided mode and CMs (anti-crossing between FM y-polarization and CMs can be seen in Fig. 11(a) and mode-field profiles). Unlike y-polarization, the propagation loss of x-polarization increases with the increase of silicon’s refractive index because the mode field spreads to the cladding. Figure 11(d) shows that the PER can be maintained a high value over a wide range of silicon index. The FEM modeling suggests that the fiber offers single-polarization and Hi-Bi over a wide range of silicon index. It is worth to mention that the low-loss regime can be tuned by simply changing the wall thickness of silicon layer, $t_2$. Our proposed design can be applied to any other high-index materials by suitably choosing the fiber parameters.
Fig. 11. (a) Relative effective mode index, $\Delta n_{\textrm {eff}}$ = 1–$n_{\textrm {eff}}$, (b) phase birefringence, (c) propagation loss, and (d) PER as a function of silicon index. The HC-ARF has core diameter, $D_{\textrm {c}}$ = 25 $\mu$m, silica/silicon wall thickness, $t_1/t_2$ = 350/215 nm, gap separation, $g$ = 1.5 $\mu$m, and normalized tube ratio, $d/D$ = 0.5. The simulations were performed at 1064 nm. The inset of (a) shows the magnified relative mode index for silicon index of $\approx$3.6276. The dotted line in (c) shows the operating silicon index of 3.5548 used in this work. The high-loss region is shown by light green bar in (c).
4. Conclusion
In conclusion, in this work we present a novel fiber designs that have single-mode, single-polarization, and high birefringence properties based on a silica/silicon regular and semi-nested HC-ARF sturctures. The semi-nested tube HC-ARF design has better loss performance, single-mode and polarization extinction ratio compared to the regular HC-ARF structure. Moreover, the semi-nested HC-ARF presents low bend induced loss and can be coiled with a small bend radii. The semi-nested HC-ARF has a bend loss of $\approx$0.06 dB/m while regular HC-ARF shows a bend loss of >1 dB/m at 5 cm bend radius. Interestingly, we found that the choice of silicon wall thickness, $t_2$ and tube gap separation, $g$ are crucial in order to achieve Hi-Bi, and high PER, whereas the ESM can be achieved by suitably changing the nested tube ratio, $d/D$. The optimized semi-nested HC-ARF design has a propagation loss, Hi-Bi, and PER of 0.05 dB/m, $0.5\times 10^{-4}$, >300 respectively at 1064 nm. The proposed fiber design is generic and can find applications in the near to IR wavelength regime ($i.e.,$ 1.55 $\mu$m or 2 $\mu$m) by suitably scaling the fiber parameters (silica/silicon tube thickness). Therefore, we believe that the results presented in this work provide a new design approach for single-mode, single-polarization, and Hi-Bi HC-ARFs.
Funding
Villum Fonden (36063); Det Frie Forskningsråd (8022-00091B); Air Force Research Laboratory (FA86511820019); Air Force Office of Scientific Research (FA9550-15-10041); Army Research Office (W911NF-12-1-0450, W911NF-17-1-0501, W911NF1910426).
Acknowledgments
The authors would like to thank Francesco Poletti and Walter Belardi for useful discussions.
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.
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