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Abstract
In this paper, a highly birefringent polarization maintaining low losses and a single mode antiresonant hollow core fiber is proposed and analyzed, that is able to exhibit better performances compared to the recent related structures. The usage of bi-thickness cladding tubes with additional high refractive index layers on our geometrically optimized structure improves birefringence nearly by one order: the highest birefringence is 4.7 × 10−4 at 1.51 µm and sustains > 1 × 10−4 for a wide bandwidth of 100 nm with a larger core diameter of 26 µm. Elliptical nesting on our proposed structure lowers the confinement loss to 0.007 dB/m at 1.51 µm and maintains a loss of < 1 dB/m for a wide range of 210 nm. A polarization extinction ratio of 300 and higher order mode extinction ratio of 63, for our fiber, ensure a single polarization and single mode operation at 1.51 µm. Moreover, the proposed fiber exhibits a bend robust performance with a very low bend loss of 0.009 dB/m at a small bend radius of 6 cm and sustains a bend loss of < 0.01 dB/m from a bend radius of 4 cm and above. Hence, our presented fiber, containing the above excellent characteristics, may be fruitful for designing polarization-controlled devices (fiber optic sensors, fiber optic amplifiers, fiber optic gyroscope, etc.) in the field of optical communication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The polarization state of light is required to be preserved and regulated for numerous polarization dependent applications in optical fiber networks, including gyroscope based on fiber optic [1], polarization maintaining (PM) optical amplifiers [2], atom spectroscopy [3], polarization splitters [4,5], fiber sensors [6,7], and so on [8,9]. This can be maintained by introducing a disproportionate loss or a significant birefringence between the two fundamental modes (FMs) which are polarized orthogonally. Throughout the previous few years, several investigations on solid core fibers (SCFs) have been performed for achieving high birefringence (Hi-Bi) by applying stress [10] or introducing anisotropy to the core of solid fiber (e.g., by using elliptical core) [11]. However, these high birefringent PM-SCFs have several inherent limitations [12], such as optical nonlinearity, low damage threshold, material absorption, etc.
The aforementioned elemental constraints can be overcome by using hollow core fibers (HCFs), in which light is trapped in an air core surrounded by dielectric layer cladding components [13]. Light guiding strategy leads HCFs to the categorization of two different types: photonic bandgap HCF (PBG-HCF) and antiresonant HCF (AR-HCF) [14]. In PBG-HCF, the photonic bandgap effect is applied to direct light in an air core, resulting in minimal loss with a narrow transmission bandwidth (BW) and significant dispersion [15–17]. Regarding this, a PBG-HCF having 19 cells was reported in [16], with a high birefringence of ∼ 3 × 10−4 and a minimal loss of 10 dB/km at 1.53 µm, but covering a substantially narrow bandwidth (<10 nm). On the contrary, AR-HCF concentrates light in air core area by inhibiting the interaction between the modes of core and cladding [18], and has gathered a lot of interest due to its impressive optical characteristics such as pretty low loss [19], superior group velocity dispersion, wider transmission BW [20], etc. In AR-HCFs, it is difficult to introduce Hi-Bi of the similar order like PM-SCFs (∼ 10−4) [12], because light is steered in the air-core, and the light interaction with the fiber material is limited.
Nowadays, researchers are focusing on incorporating birefringence in AR-HCFs, and only a few theoretical and experimental investigations have been conducted to date for obtaining Hi-Bi, which leads to PM characteristics on AR-HCFs [12,21–29]. Mousavi et al. [12,21] were the first to demonstrate Hi-Bi and single polarization on AR-HCF and obtained a Hi-Bi and polarization extinction ratio (PER) of 1.5 × 10−4 and 1000, respectively, at 1.55 µm with a confinement loss (CL) of 0.043 dB/m. Still, the structure was comprised of four cladding tubes with multiple nested resonator layers and relatively small core dimension of 14 µm. Besides that, S. Yerolatsitis and co-authors [25] fabricated a bi-thickness six tubes AR-HCF and acquired a relatively lower birefringence of 2.5 × 10−5 with a CL of 0.46 dB/m at 1.55 µm. In 2020, under static conditions, Taranta and co-workers [26] demonstrated that through launching light into a regular non-birefringent AR-HCF with a lower birefringence of 10−7, polarization purity has been sustained with PER > 40 dB, but the PM capability of the suggested fiber exhibited resistance to temperature variation only. Hence, in 2021, Habib and colleagues [27] propounded a silica/silicon hybrid cladding based AR-HCF at 1.064 µm to obtain Hi-Bi, henceforth, attained a loss of 0.05 dB/m and a birefringence of 5 × 10−5 that is yet away from solid core counterpart. In the same year, a double trigonal symmetrical AR-HCF having two different thicknesses of cladding tubes with different refractive indices was proposed with a lower birefringence of 1.38 × 10−5 [28]. Very recently in 2022, Hong and co-workers [29] proposed and fabricated a fourfold bi-thickness semi tube AR-HCF with a birefringence of 9.1 × 10−5, which is close to its solid core counterpart having a relatively high CL of 0.185 dB/m at 1.589 µm. On the basis of the aforementioned literatures, it is very difficult to produce a high birefringence (at the level of solid core fiber) covering wide bandwidth and low losses in a simpler AR-HCF structure with a large core dimension and lesser number of resonator layers.
In this paper, we propound and conduct an investigation into a silica/high refractive index layer based bi-thickness AR-HCF, having a larger core diameter of 26 µm and two antiresonant layers, to accomplish the aforementioned research gaps. Using cladding components of different thicknesses result into more interaction of light with the cladding area for one FM polarization whereas the other one is well guided through the core; thus, higher birefringence is created in our proposed AR-HCF. Additionally, the insertion of a layer of high refractive index material only alongside the cladding tubes of one axis further enhances the birefringence to a very high value of 4.7 × 10−4 at 1.51 µm by maintaining a birefringence value of > 1 × 10−4 for a wide bandwidth of 100 nm from 1.48 µm to 1.58 µm. Selection of properly optimized nested tubes of elliptical shape not only reduces the number of layers, compared to the three layers structures, but also provides low losses (leakage and bending) and maintains a larger BW compared to other types of nesting elements. The fiber also obtains a PER of 300 at 1.51 µm and offers a single polarization characteristic having a bandwidth of 25 nm ranging from 1.50 µm to 1.525 µm. It also maintains an effective single mode (ESM) behaviour in the entire highly birefringent region and exhibits a very bend robust performance which makes it more suitable for practical applications.
2. Fiber layout
The two-dimensional representation of our suggested AR-HCF is illustrated in Fig. 1. The proposed structure contains six antiresonant tubes (ARTs) which are non-touching and serve as cladding components surrounding the core, having nested tubes of elliptical shape in all the cladding components. As Non-touching cladding tubes configuration of AR-HCF offers better performance in terms of loss characteristics than the touching one, it is more preferable [13]. The diameter of the hollow core region is mentioned as Dc. Cladding tubes having different thicknesses are chosen for the proposed highly birefringent AR-HCF: t1 and t3 denote the thicknesses of cladding components along y and x axis, respectively, referring to the center of the core. Silicon layer with a thickness of t2 is added only to the cladding components alongside x axis. Here, choice of silicon is made based upon the fact that silicon has a high refractive index (RI) of 3.48 at 1.55 µm [30] and theoretical investigation along with experimental validation of silicon coated/filled AR-HCF has been performed already [31,32]. Optical loss of silicon is not included in simulation of this work [24,27]. Elliptical nested tube with a thickness of t1, is placed which lessens the leaky behavior of light from the hollow core region [20]. Silica is taken as the material of cladding tubes and optical properties of silica are characterized by Sellmeier's equation [14]. The semi-major axis and semi-minor axis of the elliptical nested tube are indicated as Rb and Ra, respectively, and the ellipticity is defined as $\alpha = {R_b}/{R_a}$. The gap between two cladding tubes and the distance between the elliptical nested and outer cladding tube are denoted by g and Z, respectively. All the mentioned structural parameters are optimized to obtain highly birefringent, low losses, single polarization, and single mode behavior of our proposed fiber as discussed later. In addition, core diameter (Dc) is related to the thickness of cladding tube (t1), the outer cladding tube diameter (Dt), separation between cladding (g), number of cladding components (n), and is defined as [33],
Fig. 1. Cross sectional view (two dimensional) of the proposed AR-HCF with the diameter of core (Dc), diameter of cladding tube (Dt), wall thickness of cladding tube alongside x axis (t3), wall thickness of y axis cladding tube as well as nested elliptical tube (t1), layer thickness of high index material (t2), gap separation between two cladding tubes (g), distance between cladding tubes and nested tubes (Z), and semi major and minor axis of the elliptical nested tubes of Rb and Ra, respectively. Silicon is chosen as high index layer in this structure.
In all the simulations of this work, the outer layer of silica tube is marginally penetrated to the outer cladding tubes and also the outer cladding tube is marginally penetrated by the nested tubes, as this is the usual case in manufactured fibers [27,34,35].
3. Results and discussion
The COMSOL software which is commercially available and works on the basis of finite element method, is employed for mathematical computations and exploring the performances of the fiber. Conventional trial and error method has been adopted to optimize different structural parameters of the proposed fiber focusing on Hi-Bi, low CL, and high PER. A perfectly-matched layer (PML) of cylindrical shape is placed at the outside domain of the fiber to absorb the radiated energy and represent the modal characteristics precisely [13]. Firstly, a convergence test at around 1.515 µm is conducted on the proposed highly birefringent AR-HCF based on the confinement loss performance to justify the numerical accuracy by selecting the PML layer thickness along with the mesh size parameter properly as depicted in Fig. 2. Figure 2(a) represents the CL characteristics with the variation of PML boundary thickness and it is observed that a steady loss performance is obtained from 1 µm thickness and beyond, which leads us to choose a PML thickness of 4 µm for further analysis of this work. A user defined maximum mesh size value of λ/6m for the region of silica and silicon material and λ/4m for the portions filled with air is considered in this work, where m represents the mesh size parameter. Variation of CL performance with varying the value of m is illustrated in Fig. 2(b) and after $m = 0.9$, CL almost stabilizes, hence $m = 1$ is chosen for the rest of study.
Fig. 2. Test of convergence: analysis of the performance of confinement loss of the proposed AR-HCF at a wavelength of around 1.515 µm for the variation of (a) the thickness of the PML boundary and (b) mesh size parameter (m) where λ/6m and λ/4m has been employed as the maximum mesh size element, respectively, for silica/silicon and air region.
Phase Birefringence (B), which is one of the most significant criteria of a fiber having PM characteristics, can be calculated as following [25,27,36]
where nx and ny denote the effective refractive index of a fiber towards x and y axis respectively.Relative effective index (Δn) is defined as [27]
where neff denotes the real part of effective RI of x or y polarized FMs.In this form of fiber, CL is a vital loss to calculate which emerges because of the leaky tendency of the light directed in the fiber core, and calculation can be executed as following [14]
3.1 Optimization of the wall thicknesses of silica tubes (t1 and t3)
We start our analysis through the optimization procedure of x axis cladding tube thickness, t3, along with y axis cladding tubes and elliptical nested tube thickness, t1 at around 1.515 µm wavelength with a core diameter of 26 µm and a uniform gap separation of g between outer cladding tubes. These wall thicknesses play a very crucial role to achieve high birefringence, polarization maintaining, low losses, and single polarization characteristics. PER is defined as the ratio of loss between x polarization fundamental mode and y polarization fundamental mode, which characterizes the single polarization operation of a birefringent fiber. Over a sufficiently long propagation distance, a birefringent fiber with a PER value of > 100 is able to remove any unwanted cross coupled polarization [12,24]. A fiber is claimed to be a highly birefringent one if the phase birefringence of that fiber is in the range of ∼ 1 × 10−4 with a confinement loss of < 1 dB/m [12].
The effect of changing the wall thickness, t3 on birefringence, loss and PER is demonstrated in Figs. 3(a) and (b). In Fig. 3(a), the proposed fiber offers a high birefringence of > 1.6 × 10−4 over a wide range of 591 nm < t3 < 610 nm. It is observed from Fig. 3(b) that the loss of the FM of y polarization is < 0.009 dB/m over the above-mentioned thickness range of t3 with a minimum loss of 0.002 dB/m. The loss of the x polarized FM is < 1.2 dB/m in this thickness range which is larger than y polarized FM due to the coupling of light between x polarized core and cladding modes. This is because the cladding tube along x axis is thicker than y axis cladding tubes and an additional silicon layer is placed alongside x axis tubes. In this entire range of t3, the PER value is ∼ > 100 which ensures single polarization operation. By sacrificing the loss of y polarized FM, birefringence can be increased further with the increase of t3. Hence, the optimized t3 is chosen as 605 nm by considering high birefringence, low loss and high PER for further investigation of this work.
Fig. 3. (a) Relative effective index (Δn) and birefringence, and (b) confinement loss of x and y polarized FMs along with PER for the variation of x axis cladding tube thickness of t3, (c) relative effective index (Δn) and birefringence, and (d) confinement loss of x and y polarized FMs along with PER for the variation of y axis cladding tube and elliptical nested tube thickness of t1. All the simulations are performed at around 1.515 µm with a core diameter of 26 µm considering a uniform gap separation, g between the cladding tubes.
In Figs. 3(c) and (d), t1 is varied from 350 nm to 380 nm whereas t3 is set to 605 nm and the effect of this variation on birefringence, loss, and PER is observed. Elliptical nested tubes have been mainly used in this structure to reduce the number of resonator layers in the cladding elements and provide low loss guidance than circular nested elements without hampering the desired birefringence level [12,20]. The achieved birefringence is > 2.57 × 10−4 over a wide range of 350 nm < t1 < 380 nm as shown in Fig. 3(c), and faces a very little impact due to change in t1 as mentioned earlier. From 350 nm to 380 nm of t1, y polarized FM loss is sustained as low as < 0.007 dB/m with a minimum loss of 0.005 dB/m and a loss of < 0.78 dB/m for x polarized FM, and maintaining a PER value of ∼ > 100 within this t1 range. Therefore, a wall thickness t1 = 372 nm is selected for our proposed model by considering the above-mentioned performance parameters.
3.2 Optimization of wall thicknesses of silicon tubes, t2 and gap separation, g
In this portion, the effect of changing the wall thickness of silicon layer, t2, and separation between cladding tubes, g, on birefringence, x and y polarized FMs loss, and PER is discussed and shown in Figs. 4(a)-(d) while keeping the other dimensions as Dc = 26 µm, t3 = 605 nm, and t1 = 372 nm.
Fig. 4. (a) Relative effective index (Δn) and birefringence, and (b) confinement loss of x and y polarized FMs along with PER for the variation of silicon wall thickness of t2, (c) relative effective index (Δn) and birefringence, and (d) confinement loss of x and y polarized FMs along with PER for the variation of gap separation, g. All the simulations are performed at around 1.515 µm with Dc = 26 µm, t3 = 605 nm, and t1 = 372 nm.
In our proposed fiber, the silicon layer is added only with the cladding tubes along x axis to induce high birefringence, hence, the thickness of this layer is a crucial parameter for the performance of the fiber. It is varied from 290 nm to 302 nm and an increasing trend of birefringence is observed with the increase of wall thickness, hence, obtained birefringence is > 1.3 × 10−4 in this entire range as depicted in Fig. 4(a). This is due to the fact that light interaction with silica and silicon cladding tubes increases for x polarization with the increasing thickness of high refractive index silicon layer. The FM loss for y polarization is < 0.01 dB/m in this range of t2 with a minimal loss of 0.001 dB/m, and for x polarization FM loss is < 2 dB/m which yields to PER value ∼ > 100 in this entire range as demonstrated in Fig. 4(b). Here, optimized silicon wall thickness is chosen as 299 nm by making a trade-off among birefringence, loss, and PER.
High birefringence, low loss, and single polarization guidance depend strongly on the gap between cladding tubes, g for the proposed fiber which is observed in Figs. 4(c) and (d). The reason for this is the hybrid characteristics of silica-silicon cladding components in which light is well confined for the FM of one polarization state while for the other polarization state light spreads to the cladding area when g is varied. It is evident from Fig. 4(d) that the proposed fiber has low loss FM guidance for y polarization whereas high loss for x polarization within a broad range of g. The y polarized FM loss maintains a value of < 0.04 dB/m for a wide range of 0.25 µm < g < 3 µm with a minimum FM loss of 0.005 dB/m, whereas for x polarized FM the loss is maintained below 1.05 dB/m. Again, from the PER graph of Fig. 4(d), the proposed fiber exhibits a comparatively higher PER value for smaller g, as expected because small gap separation results in more interaction of light with both the silicon-silica tubes which provides high loss for x polarization and low loss guidance for y polarization, but this effect is lessened as g starts to increase. High birefringence of > 2.33 × 10−4 is maintained for a very wide range of 0.25 µm < g < 3 µm, and from Fig. 4(c), it is also observed that g has less impact on birefringence because the relative effective index for x and y polarized cases remain almost constant with the wide variation of g. Hence, considering the concerned parameters of our proposed structure, a value of g = 1 µm is taken as the optimized value.
3.3 Optimization of Z and ellipticity, α
In this portion, the effect of changing the distance between outer cladding tubes and elliptical nested tubes (Z), and ellipticity of the nested tubes (α) on birefringence, x and y polarized FM loss, and PER is discussed and shown in Figs. 5(a)-(d) while keeping the other dimensions as Dc = 26 µm, t3 = 605 nm, t1 = 372 nm, t2 = 299 nm, and g = 1 µm. From Figs. 5(a) and (b), it is observed that the birefringence is > 1 × 10−4 and FM loss of y polarization is < 0.02 dB/m with the lowest loss of 0.004 dB/m for a very wide range of 3 µm < Z < 9 µm. However, x polarized FM has a loss of < 1.44 dB/m in this range of Z which is higher than y polarized FM having an increasing loss tendency for both x and y polarized FMs with an increasing value of Z. Reason behind this is the antiresonance effect of the nested tube is lessened with the increasing distance between outer cladding tubes and nested tubes, and this effect is more severe for x polarized FM because of the presence of thicker cladding tubes as well as the silicon layers alongside x axis. So, the obtained PER value is ∼ > 100 in the range of 5.75 µm < Z < 8.5 µm. PER curve faces some random fluctuations because of fluctuating CL curves which occurs due to Fano-resonance [37]. Birefringence also shows similar increasing behavior with increase in distance between cladding and nested tubes because the light interaction between core and cladding for x polarization is enhanced with increasing value of Z, hence, the relative effective mode index is decreased accordingly, whereas it remains almost unchanged for y polarization. Here, the optimized value of Z is taken as 6.50 µm for further investigation of this work.
Fig. 5. (a) Relative effective index (Δn) and birefringence, and (b) confinement loss of x and y polarized FMs along with PER for the variation of distance between cladding tube and nested tube, Z, (c) relative effective index (Δn) and birefringence, and (d) confinement loss of x and y polarized FMs along with PER for the variation of ellipticity, α. All the simulations are performed at around 1.515 µm with Dc = 26 µm, t3 = 605 nm, t1 = 372 nm, t2 = 299 nm, and g = 1 µm.
Furthermore, ellipticity, α is varied from 0.55 to 1: the birefringence is maintained as > 2.53 × 10−4, and the y and x polarized FMs sustain loss as < 0.1 dB/m and < 3.28 dB/m, respectively, along with a minimum loss of 0.004 dB/m for y polarized FM in this wide range of α as demonstrated in Figs. 5(c) and (d). PER is maintained as ∼ > 100 for a broad range of 0.73 < α < 1. Around α = 0.85, the loss is lowest which yields to highest PER, because the elliptical nested tubes provide better antiresonance at this point, and loss gradually increases as the ellipticity approaches to 1 i.e., to a circular shape of the nested tube which is well discussed in the literature [20]. Some random fluctuations are observed in PER curve as discussed earlier. Birefringence faces very little impact due to ellipticity variation as the elliptical tubes are mainly inserted to reduce the loss of the fiber by keeping both polarized effective indices unchanged, therefore, ellipticity of 0.85 is chosen as the optimized value to fulfill our goals.
3.4 Effect of changing core diameter, Dc
The effect of changing the core diameter, Dc on birefringence, loss, and PER is discussed in this section while the other parameters (Dc, t1, t2, t3, g, Z, and α) are in the optimized level. Core diameter is varied from 14 µm to 30 µm, and from Fig. 6(a) it is evident that the birefringence increases as core diameter decreases. The reason behind this kind of characteristics is the increased interaction of the core guided light with its surrounding cladding structure when core size decreases [25]. This is also reflected in the loss curve of Fig. 6(b), where a steep increase behavior in loss with decreasing core size is observed. Hence, PER value is also lower for smaller core sizes and increases accordingly. After 26 µm core diameter, light interaction of y-polarized FM with its surrounding cladding increases, thus y-polarized CL also increases which results into decreasing PER. From Figs. 6(a) and (b), it is observed that, when core diameter is set to 26 µm, the proposed AR-HCF provides best performance in terms of the desired performance parameters. Hence, a core diameter of 26 µm is chosen for this work.
Fig. 6. (a) Relative effective index (Δn) and birefringence, and (b) confinement loss of x and y polarized FMs along with PER for the variation of core diameter, Dc. All the simulations are performed at around 1.515 µm with t3 = 605 nm, t1 = 372 nm, t2 = 299 nm, g = 1 µm, Z = 6.5 µm, and α = 0.85.
3.5 Effect of wavelength variation
In this section, the wavelength dependence of various performance parameters like birefringence, confinement loss, bandwidth, and PER are calculated under optimized conditions and discussed based on the graphical representation of Figs. 7(a)-(d). In Fig. 7(a), the vital property of our proposed fiber, birefringence, is illustrated with the variation of wavelength for the geometrical optimized parameters. The highest birefringence of 4.7 × 10−4 is obtained at 1.51 µm wavelength for our proposed AR-HCF while keeping the birefringence level of > 1 × 10−4 for a wide bandwidth of 100 nm ranging from 1.48 µm to 1.58 µm. Again, in the wavelength region beyond 1.58 µm to 1.70 µm, the birefringence level is in the range of ∼ 0.8 × 10−4, which is very close to 1 × 10−4 and can also be considered as a high birefringence region [27,29]. Hence, for our proposed AR-HCF, a broad wavelength region of 220 nm, ranging from 1.48 µm to 1.70 µm is expected to exhibit high birefringence characteristics. The obtained birefringence level of our suggested AR-HCF offers an upliftment than all the recently proposed structures on AR-HCFs platform within this wavelength region of interest [12,22–25,27–29]. The usage of different wall thicknesses in the cladding area along with high refractive index layers (silicon) alongside x axis introduces asymmetry in the structure and leads into different amount of light interaction for x and y polarized cases, therefore, enhances the level of birefringence of our suggested structure.
Fig. 7. (a) Birefringence, (b) confinement loss of x and y polarized FMs, and (c) PER with the variation of wavelength, (d) Mode field profile of x polarized FM (blue border solid line) and y polarized FM (red border dotted line) at around 1.515 µm wavelength. The AR-HCF has a diameter of core, Dc = 26 µm, x axis cladding tube thickness of t3 = 605 nm, y axis cladding tube as well as elliptical nested tube thickness of t1 = 372 nm, silicon layer thickness of t2 = 299 nm, gap separation of g = 1 µm, Z = 6.5 µm, and ellipticity, α = 0.85.
Confinement loss of both polarized (x and y) FMs of our suggested Hi-Bi fiber is demonstrated in Fig. 7(b), and from the graph, it is evident that the CL is lower in our desired optimized region of wavelength and it increases gradually as the wavelength shifts from optimum point. At the highest birefringent wavelength (1.51 µm) for our proposed fiber, a very low CL of 0.007 dB/m is obtained for y polarized FM and 2.09 dB/m for the x polarization FM. The lowest CL over the wavelength of interest for our proposed fiber is 0.001 dB/m at 1.53 µm having a high birefringence of 1.88 × 10−4. The x polarized FM is lossy comparing to the y polarized FM because of the addition of silicon layers alongside x axis which is responsible for producing Hi-Bi. A lower loss performance in our proposed structure is achieved, compared to the recent related structures [12,22–25,27–29], because of the proper selection of elliptical nesting elements in the cladding area. The wavelength region over which the CL of y polarized FM sustains < 1 dB/m is 210 nm (ranging from 1.49 µm to 1.7 µm) and the CL around 1.45 µm is higher for both x and y polarized FMs due to the occurrence of resonance around that wavelength [13].
The highest PER value of 300 is obtained for our proposed structure at 1.51 µm while covering a single polarization BW of 25 nm (from 1.5 µm to 1.525 µm). The highly birefringent wavelength region in which PER and loss are < 100 and < 1 dB/m, the fiber offers highly birefringent polarization maintaining characteristics (Fig. 7(c)). The mode field profiles for x and y polarized cases can be seen in Fig. 7(d). Overall, our proposed fiber exhibits both single polarization and polarization maintaining characteristics simultaneously over a wide range of wavelength.
3.6 Bending loss
The bending loss is an important aspect of an AR-HCF, which occurs due to the coupling of light between the hollow air core and surrounding cladding components as a consequence of bending. The analogous refractive index model is adopted for the computation of bend loss in the x direction regarding from the air core as [14]
where ${n_{eqv}}$ denotes the equivalent RI due to bend, ${n_m}$ is the RI of the material before applying stress, ${R_{bend}}$. defines the radius of curvature, and x defines the bending direction. At around 1.515 µm of wavelength, Fig. 8(a) displays the bending loss characteristics of y polarized FM for varying bend radius. With the increasing bend radius, bending loss gradually decreases as expected, and at a smaller bend radius of 6 cm, a lower bend loss of 0.009 dB/m is obtained. A very low bend loss of < 0.01 dB/m is sustained for the proposed AR-HCF from a very short bending radius of 4 cm and above. Again in Fig. 8(b), variation of bend loss as a function of wavelength for a fixed bend radius of 6 cm is illustrated, and from 1.50 µm to 1.69 µm, bending loss is maintained as < 1 dB/m with the lowest loss of 0.002 dB/m at 1.53 µm wavelength.Fig. 8. (a) Calculated bending loss (y polarized) with the variation of bend radius for our proposed AR-HCF at the wavelength of around 1.515 µm. Bending is along x direction and the field profiles are inset for bend radius of 1 cm (red border) and 6 cm (black border), and (b) bend loss spectra at a short bend radius of 6 cm. The optimized geometrical parameters of the proposed AR-HCF have been used for this simulation.
3.7 Single mode performance analysis
Higher order mode extinction ratio (HOMER) identifies the standard single mode operation of our proposed AR-HCF and it is calculated through the ratio of the lowest loss among the higher order modes (HOMs) to the loss of FM [38]. For effective single mode operation, HOMER value of > 10 is desired [38]. In Fig. 9(a), the CL spectra of the y polarized FM along with three HOMs is shown, and it is observed that the loss of HOMs is higher than the FM loss due to the more leakage of light from core to the cladding area which is evident from the mode field profiles of HOMs and FM. However, HOMER is calculated regarding the HE11 (FM) and TM01 mode as it yields lower CL than other HOMs. From HOMER curve of Fig. 9(b), the highest HOMER value of 63 is obtained at 1.51 µm of wavelength maintaining a value of > 10 for a wide bandwidth of 130 nm (from 1.50 µm to 1.63 µm) which ensures effective single mode operation of our proposed fiber in the highly birefringent region. Hence, our proposed AR-HCF can perform as an effective single mode fiber for a wide range of wavelengths.
Fig. 9. (a) CL spectra for the first four core modes (one FM and three HOMs) along with the mode field profile in the top panel where the color of frame correlates with the color of line, and (b) HOMER as a function wavelength of our proposed fiber. In this study, the HE11 mode and TM01 mode are considered for HOMER calculation using optimized geometrical parameters of the proposed AR-HCF.
3. Asymmetrical analysis
In this section, the birefringence and loss characteristics of the proposed AR-HCF is analyzed when asymmetry is created on the structure and compared with the symmetrical one. Asymmetry on our structure is created in two ways: removing one nested tube from x axis cladding as depicted in Fig. 10(a) and removing one silicon layer from x axis cladding area as depicted in Fig. 10(b) while keeping rest of the structure unchanged for both cases. The asymmetric structure of Figs. 10(a) and (b) is named as A1 and A2, respectively. The birefringence and loss characteristics of structure A1 is shown in Figs. 10(c) and (d): birefringence of A1 is lower than the symmetric one within wavelength range of 1.48 µm to 1.53 µm, then it increases compared to the symmetric one and again decreases by keeping almost similar values obtained from the symmetric structure. This is because of the lower relative effective index of x polarized FM of A1 than the symmetric structure and vice-versa, whereas the relative effective index of y polarized FM for both A1 and symmetric structure is almost similar (not shown here graphically). Loss of both x and y polarized FMs for structure A1 is higher than the symmetric structure because of removing the nested elliptical tubes from x axis cladding element. Again, in Fig. 10(e), the birefringence of structure A2 is compared with the symmetric one, and as expected the birefringence is lower than symmetric structure due to removal of one high index silicon layer from the cladding area of x axis. However, from the loss spectra of Fig. 10(f), it is evident that the loss of both x and y polarized FM of structure A2 is lower than the symmetric one as a consequence of removing the silicon layer which is responsible for making the structure lossy.
Fig. 10. (a) Asymmetric structure A1 after removal of one nested tube alongside x axis, (b) asymmetric structure A2 after removal of one silicon layer alongside x axis, (c) birefringence and (d) loss spectra for symmetrical and asymmetrical structure (A1), (e) birefringence, and (f) loss spectra for symmetrical structure and asymmetrical structure (A2).
4. Fabrication tolerance
From fabrication viewpoint, the evaluation of the structural parameters variation is an important factor so that the proposed fiber can hold down the availed results post fabrication. The proposed AR-HCF contains junction points between circular cladding tubes, silicon layers, and the inner nested elliptical tubes for the cladding components alongside x axis. Again, for y axis cladding components the junction point is created between the inner elliptical nested tubes and outer circular cladding tubes. These junction points may be influenced during the fabrication process; hence the displacement effect of the junction points has been evaluated by following the literature [39] and considering an identical shift of θ (in degree) ranging from 0° to 120°. In the top panel of Fig. 11, the structural orientation for different rotational angles is shown. In Figs. 11(a) and (b), birefringence and CL spectra of the proposed AR-HCF as a function of θ are characterized at around 1.515 µm wavelength, and it is clear that the birefringence of the proposed AR-HCF is very well tolerated with the variation of rotation angle. Birefringence is maintained as > 2 × 10−4 for 0° < θ < 80°, and > 1.1 × 10−4 in the range of 80° < θ < 120°. On the contrary, loss increases with the increase in θ as depicted in Fig. 11(b), because the junction points come closer to the core as θ increases, hence, create extra loss due to the resonance at nodes, which in turn reduces the birefringence as both x and y polarized FMs interaction with cladding area goes higher. However, the CL sustains as < 0.09 dB/m in a wide range of 0° < θ < 95°, and maintains below 1 dB/m till 110°. Thus, the displacement effect of junction is well tolerated by our proposed AR-HCF.
Fig. 11. (a) Birefringence and (b) CL with the variation of rotation angle, θ, at around 1.515 µm of wavelength; (c) birefringence and (d) CL for ideal fiber, fiber with penetration amount k = t1/10, t1/4 and t1/2 as a function of wavelength. Structure for different rotation angle, θ and penetration amount, k, are attached in the top panel.
Again, the penetration of the elliptical nested tube to the outer circle may be influenced during fabrication as shown in the top panel of Fig. 11 (blue border square shape where k denotes the amount of penetration): birefringence and CL spectra as a function of wavelength at different penetration amount, k is exhibited in Figs. 11(c) and (d). Birefringence is not affected due to the increase in k as t1/10, t1/4, and t1/2, as shown in Fig. 11(c), because the increased penetration of nested tube increases the junction width which is far from core and mainly responsible for loss performance. Thus, the highest loss is faced when k = t1/2, as antiresonance element is decreased when k increases. Despite of this, loss for k = t1/2, sustains below 0.01 dB/m for a wide range of 180 nm (from 1.50 µm to 1.68 µm) and below ∼ 1 dB/m for 200 nm (1.49 µm to 1.69 µm) as almost the ideal one. Loss for k = t1/10 and k = t1/4, is slightly higher than the ideal one and little bit lower than k = t1/2 for a wide variation of wavelength, as expected. Hence the proposed AR-HCF could tolerate the penetration effect of the nested tubes during fabrication.
For practical implementations of the proposed fiber, different techniques like 3D printing, stack-and-draw, extrusion, and others are presently available [20], and are used to fabricate different types of AR-HCF geometries. For example, Cregan et al. [15] utilized a strategy based on stack and draw method to make PBG-HCF. In the fabrication of ice-cream cone-shaped HCF [40], nested structure [41], conjoined tube ARF [37], two split cladding fiber [42], etc., the stack and draw procedure has been adopted. For complicated structures fabrication, other techniques (e.g., extrusion [43], 3D printing [44], etc.) are also being used. Kosolapov and co-authors [45], constructed a revolver HCF and mentioned that the nested tubes are changed to elliptical shape when the tubes are bent towards the inner capillary. They also proposed that the ellipticity inhibits to preform torsion during the drawing process, thus the possibility is made to obtain an elliptical nested inner tube within the tubes of outer periphery with nominal chirality [45]. Moreover, the elliptical nested tubes aid to duly orient them in between the outer tubes [45]. As our fiber incorporates nested tubes of elliptical shape, so there would be a few complications while fabricating the fiber which could be resolved with the aid of recent advanced fabrication technologies as discussed above. Also, our fiber contains bi-thickness cladding elements along with a silicon layer, and this type of AR-HCFs have already been constructed using existing fabrication technologies [25,29,32]. The post processing steps can be helpful for tuning the results; hence, it can be hoped that the proposed AR-HCF could potentially sustain the simulated outcomes after the process of fabrication. To fabricate our fiber, few tubes are required but it is impossible to buy few tubes instead of a bundle of tubes. Therefore, price of individual tube will be very low, hence, cost of the proposed fiber will not be high. Due to the above complexity, it is hard to estimate the actual cost and reusability of our proposed fiber.
Here, the performance of our proposed AR-HCF is compared with recent related structures in terms of birefringence, CL, PER, bandwidth of birefringence and PER, bending loss, and effectively single mode operation (HOMER) as depicted in Table 1. It is evident from this comparison table that the birefringence of our proposed fiber is nearly one order higher and CL is one to two orders lower than the recently proposed related structures [12,23,25,27–29]. A very few literatures discussed bending loss and single mode characteristics whereas our fiber exhibits very good performance in these aspects as well. From the above analysis, we can conclude that our proposed fiber outperforms the other structures with respect to the almost all aspects of comparison.
Table 1. Relative Study on Proposed and Existing Fibers
5. Conclusion
We have demonstrated an AR-HCF which is capable of performing better in terms of different performance parameters (high birefringence over wide bandwidth, low confinement and bend loss, single mode, and single polarization). Birefringence of our proposed fiber is improved significantly, nearly by one order, comparing to the formerly proposed fiber as discussed in the literature with its highest value of 4.7 × 10−4 at 1.51 µm and sustains > 1 × 10−4 for a broad bandwidth of 100 nm. Also, from 1.58 µm to 1.70 µm, the birefringence is kept as ∼ 8 × 10−5 which is very close to the birefringence level of SCFs (∼ 10−4 [12]). Hence, our proposed fiber exhibits high birefringence characteristics for a very wide bandwidth of 220 nm. Besides that, loss is reduced by one order (CL is 0.007 dB/m and 0.001 dB/m at 1.51 µm and 1.53 µm respectively) and covers a broad bandwidth of 210 nm with a loss level of < 1 dB/m. The PER of our fiber is 300 at 1.51 µm, and can maintain > 100 for a bandwidth of 25 nm which assures single polarization operation. The fiber also achieves a HOMER value of 63 at 1.51 µm and maintains an effective single mode behaviour in the entire highly birefringent region. Moreover, very good bend robust characteristics is observed with a low bend loss of 0.009 dB/m and able to maintain < 1 dB/m for a wide bandwidth of 190 nm for a small bend radius (6 cm). Therefore, the current discussion has led to the conclusion that the presented AR-HCF having the above excellent properties might be useful for future polarization controlled optical devices.
Funding
Department of EEE, Research & Extension (RUET) (DRE/7/RUET/528(39)/PRO/2021-22/17).
Acknowledgments
Authors gratefully recognize the contribution of the Department of EEE, Research & Extension (DRE/7/RUET/528(39)/PRO/2021-22/17) of RUET, Bangladesh and the Department of EE, Penn state, USA.
Disclosures
The authors declare no conflict 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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