Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 2, 2024



Article Number  35  
Number of page(s)  11  
DOI  https://doi.org/10.1051/jeos/2024036  
Published online  11 October 2024 
Research Article
FEM analysis of a highly birefringent modified slotted core circular PCF for endlessly single mode operation across E to L telecom bands
^{1}
Department of Electrical and Electronic Engineering, Rajshahi University of Engineering and Technology, Rajshahi 6204, Bangladesh
^{2}
Department of Electrical and Electronic Engineering, World University of Bangladesh, Dhaka 1230, Bangladesh
^{3}
Department of Computer Science and Engineering, World University of Bangladesh, Dhaka 1230, Bangladesh
^{4}
Department of Physics, COMSATS University Islamabad, Islamabad, Pakistan
^{*} Corresponding author: amit.rueten@gmail.com
Received:
30
May
2024
Accepted:
28
August
2024
This paper describes an exceptionally high birefringent modified slotted core circular photonic crystal fiber (MSCCPCF). At the 1.55 μm telecommunication wavelength, the proposed fiber structure aims to achieve exceptional birefringence performance through the thoughtful placement of air holes and the incorporation of slots. The optical properties of the proposed MSCCPCF are rigorously simulated using the finite element method (FEM). The FEM simulations show high birefringence of up to 8.795 × 10^{−2} at 1.55 μm. The suggested fiber exhibits single mode behavior in the E to L communication bands (V_{eff} < 2.405). Numerous geometric factors and their effects on other optical properties, such as birefringence, beat length (17.62 μm) and dispersion coefficient (−310.8 ps/(nm · km)) have been meticulously studied. The proposed fiber’s viability and potential uses are evaluated by analyzing modal features like nonlinearity (21.76 W^{−1} km^{−1}), confinement loss (5.615 × 10^{−11} dB/cm), and dispersion. The proposed fiber structure has potential for use in polarizationmaintaining devices, sensors, and other photonic applications requiring high birefringence and tailored optical properties.
Key words: Birefringent / Circular photonic crystal fiber / Finite element method / Polarizationmaintaining fiber / Single mode / Slotted core
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Photonic crystal fibers (PCFs) have gained popularity in recent years due to their capacity to alter light in ways that regular optical fibers cannot [1]. PCFs have distinct qualities such as large mode area, strong nonlinearity, and customized dispersion characteristics due to the periodic placement of air holes or voids within a solid core [2]. Different forms of PCF or microstructure optical fibers have been proposed in various literature and research. Some of them have also been invented and used to reallife situations. The most common periodic air hole arrangements in cladding designs are hexagonal [3], octagonal [4], decagonal [5], square [6], circular [7], spiral [8], honeycomb [9], and other hybrid patterns [10–12]. These holey fibers can effectively manage dispersion and reduce pulse widening in communication systems [13]. Highly nonlinear and large mode PCFs are effective in sensing and biosensing applications [14]. The necessity for great precision and efficiency in optical technology has resulted in the development of PCFs with high birefringence. Birefringence is an important feature of optical fibers that allows for independent propagation channels for orthogonal polarization modes, which is critical for applications in polarizationsensitive devices, telecommunications, and sensors [15]. Recent research has focused on producing PCFs with extremely high birefringence. Halder (2016) developed a broadband dispersion compensating PCF with ultrahigh birefringence of 3.373 × 10^{−2}, strong negative dispersion coefficient, and large nonlinear coefficient, ideal for highspeed transmission systems. The design includes four elliptical air holes in the core region there still complexity in design procedure remains. Fabrication method is not included in the design procedure. Realizing elliptical air holes in a small pitch value is very difficult. The optical properties seems to be dependent on the ellipticity of the elliptical air holes the effect of this is not addressed and analyzed for that PCF design [16]. Islam (2017) developed a hexagonal PCF with strong birefringence (3.34 × 10^{−2} at 1550 nm), low confinement loss, and a large negative dispersion coefficient, making it suitable for sensing applications. The design includes asymmetric air holes in the core region. The ellipticity of the elliptical air holes is narrow possessing a difficulty in fabrication and the results will also effect in real world scenario [17]. Amin (2016) proposed a pure silica index guiding PCF with 3.474 × 10^{−2} birefringence at 1.55 μm, ideal for coherent optical communication and sensing applications. The complexity of the design makes it difficult to fabricate by stack and draw method. The last air hole ring and the Perfectly Matched Layer (PML) is kept at a distance in this design so the scattered light reflections and extra mode formation is not suitable for long haul operation [18]. Liu et al. (2018) designed PCFs with an elliptical tellurite core, resulting in unprecedented gains in birefringence (7.57 × 10^{−2}), nonlinearity (188.39 W^{−1} km^{−1}), and negligible confinement loss (10^{−9} dB/m) at 1.55 μm wavelength. Tellurite glass has been infiltrated in an elliptical air hole to achieve high birefringence and nonlinearity but absorption loss in the telecommunication band is not calculated. Absorption loss and temperature coefficient should have to be taken into account to realize the proposed fiber in reality otherwise the outcomes may vary [19]. Chaudhary (2019) constructed a hybrid PCF with a birefringence of 12.046 × 10^{−3} at 1550 nm. It has two zero dispersion wavelengths and a strong nonlinear coefficient, making it ideal for use in optical systems. This structure is complex and aperiodic in nature and loss profile is not measured for this design which is very crucial in telecommunication data transfer [20]. Halder (2020) developed a hybrid dispersion compensating fiber with high birefringence of 3.76 × 10^{−2} at 1550 nm wavelength, low confinement loss, and a relative dispersion slope similar to singlemode fiber. This efficiently addresses dispersion across S, C, and L telecommunication bands. The design of this hybrid model includes very small air holes around the center core which seems very difficult to fabricate. Higher order mode extinction ratio should be analyzed to verify single mode operation [21]. Liang et al. (2020) demonstrated a unique PCF with a sandwich structure, displaying ultrahigh birefringence of 3.85 × 10^{−2} and considerable negative dispersion at 1550 nm wavelength. This suggests possible applications in longdistance optical communication and polarization preserving fibers. The sandwich structure is squeezed so while implementing stack and draw method it is difficult to materialize and confinement loss needs to be addressed in those structures [22]. Benlacheheb et al. (2021) developed a polymer PCF with a triangular shape and circular air pores, resulting in a high birefringence of 4.9 × 10^{−2} at 1550 nm. Polymer PCF is more suitable in terahertz band than visible light band due to their high absorption loss in the communication bands. Other loss profiles should be taken into account while analyzing these polymer materials in telecommunication bands [23]. Wang (2021) introduced a new tellurite glass PCF design with a nearelliptic core and six small air holes. The fiber achieves high birefringence (5.05 × 10^{−2}) and nonlinear coefficient (up to 1896 W^{−1} km^{−1}) at 1.55 μm wavelength, with zero dispersion points around the same wavelength. The design is complex to realize and fabricate also loss profile should have to be measured [24]. Du et al. (2022) designed a tellurite glassbased PCF with a substantial birefringence of 3.79 × 10^{−2} and a nonlinear coefficient of 1672.36 W^{−1} km^{−1} at 1.55 μm wavelength, with potential applications in many domains. The absorption coefficient and temperature coefficient should have to be considered while fabricating this PCF structure. Fused silica is widely used background material for PCF in telecommunication bands. Using tellurite glass in telecom bands is not clear in this literature [25]. Priyadharshini et al. (2023) introduced a crosscore octagonal shaped fiber to achieve birefringence of 1.5 × 10^{−3} and beat length of 1.04 × 10^{−3} meters. Confinement loss is not measured for this crosscore octagonal shaped PCF also there is a need of chromatic dispersion analysis for long haul optical fiber communication [26]. Halder et al. (2023) presented a defected core hybrid (hexagonal–decagonal) cladding PCF structure which showed a birefringence of 2.372 × 10^{−2} and large negative dispersion of −3534 ps/(nm · km). The authors use single pitch to design the whole hybrid PCF cladding structure which leads difficulty to adjust the core and cladding air hole rings if the air filled fraction is changed sometimes hexagonal and decagonal air holes may overlap. So adjustability is a limitation in this proposed hybrid PCF structure [27]. Liu et al. (2023) proposed a simple central trielliptic core inside a hexagonal microstructure holey fiber to achieve a high birefringence of 3.56 × 10^{−2}. The effective area values should have to be included to verify the nonlinear coefficients of this fiber. The design possesses higher complexity to achieve higher birefringence in order of 10^{−2} which can be simply achieved with a less complex structure [28]. Recently, Agbemabiese and Akowuah (2024) reported four different types of hexagonal holey fiber structures with different combinations of air holes to attain birefringence of 3.5 × 10^{−3} and nonlinearity of 15.64 W^{−1} km^{−1}. Several PCF structures were analyzed in their study and the structures are aperiodic in nature to achieve lower birefringence and nonlinearity than our previous studies and current study showed. The air hole diameters should show some periodicity while designing such PCF structures. In an air hole ring there are some holes bigger and some are smaller but not in a periodic manner even the authors did not clear the design methodology to some extent why they are motivated to such designs and what are the advantages of it. The clarity of the design methodology should be on focus in these PCF designs [29]. These studies demonstrate the promise of PCFs with extraordinarily high birefringence in a wide range of applications. But some shortfalls related to confinement loss, propagation loss, absorption loss, material selection, design complexity and fabrication methodology has been found in these studies. In response to the stringent requirements of these applications, this work presents a unique design for a PCF with ultrahigh birefringence: a modified slotted core circular photonic crystal fiber (MSCCPCF). The strategic incorporation of slots within the core region, as well as changes to the crystal fiber’s inner rings, makes this fiber design unique. This configuration has been meticulously designed to significantly increase the fiber’s intrinsic birefringence. Using the Finite Element Method (FEM) for simulation, the design achieved a birefringence level of up to 8.795 × 10^{−2} at 1.55 μm, which is relevant for telecom. The design also helps to retain lower confinement loss (5.615 × 10^{−11} dB/cm at 1.55 μm). It also demonstrates continuous single mode operation by keeping the normalized frequency V_{eff} below 2.405 across the telecommunication window, i.e. the E to L telecom bands. The design goal of this study is to develop a highly birefringent PCF that maintains singlemode operation across the E to L telecom bands while minimizing losses and nonlinear effects. This paper addresses these challenges by introducing a modified slotted core circular PCF structure that strategically optimizes the placement of air holes and slots, achieving exceptionally high birefringence, low confinement loss, and continuous singlemode operation.
2 Design procedure of proposed MSCCPCF
The design procedure begins with the determination of important geometric parameters that influence the construction of the PCF. The parameters include the pitch (Λ) and relative air hole diameters (d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, d_{6}, and d_{7}) that are normalized to the pitch. Geometrically modified circular air hole rings have been constructed in seven distinct circular layers. From the center point to each and every layer of air hole arrangements are spaced with a distance equal to pitch (Λ) value. A circular arrangement is formed with different diameter of air holes in each layer. Each layer there is a different type of angular spacing is introduced to form such arrangement. The rectangular slots are structured in the core region horizontally and placed at a distance equal to pitch value. The seven air hole layers are constructed in this manner to confine the light within the core region and lessen the propagation loss during transmission. The structure also exhibits negative dispersion coefficient due to this layer arrangement. The rectangular slots have been intruded to upscale the birefringence value. Figure 1 depicts the cross sectional view of MSCCPCF, with a comprehensive outline of the core and cladding. In the proposed MSCCPCF, the pitch Λ is set to 0.9 μm and the relative diameters are determined as follows: d_{1}/Λ = d_{2}/Λ = d_{4}/Λ = 0.45, d_{3}/Λ = d_{6}/Λ = 0.7, d_{5}/Λ = 0.6, and d_{7}/Λ = 0.85. Two rectangular slots are placed into the core portion of the MSCCPCF to increase birefringence and adjust the fiber’s optical properties. The dimensions of these slots are carefully designed to maximize the influence on birefringence and other optical characteristics. In this design, the slot dimensions are as follows: The rectangular slot’s width (a) is half of the pitch Λ/2, while its height (b) is Λ/2√3. The foundation material of this holey fiber is fused silica, and the aircores are geometrically drilled. To address reflection and backscattering issues, a circular PML with a thickness of 10% of the cladding radius was applied along the structure’s perimeter. The PML layer effectively absorbs undesirable reflections. The PML thickness around the proposed designs was modified to 1 μm with a scaling factor of 1.
Figure 1 Cross sectional view of proposed MSCCPCF. Where, Λ = 0.9 μm, d_{1}/Λ = d_{2}/Λ = d_{4}/Λ = 0.45, d_{3}/Λ = d_{6}/Λ = 0.7, d_{5}/Λ = 0.6 and d_{7}/Λ = 0.85. Rectangular slot dimensions: a = Λ/2 and b = Λ/2√3. 
Figure 2 demonstrates the angle arrangement of seven air hole rings. The distance of the two rectangular slots is kept equal to the pitch (Λ) of the proposed MSCCPCF. This modified circular arrangement is made with different angular orientation. The internal angle between two adjacent air holes from 4th to 7th outer cladding rings is set to 7.5°. The adjacent air hole angle is set to 15° for 2nd to 3rd cladding rings and 30° angle distance is kept within the first air hole ring.
Figure 2 Quarter transverse crosssectional view of air hole arrangement technique. 
3 Computational methodology
The fullvector FEM is a powerful numerical method for analyzing the physical properties of a PCF; anisotropic PML boundary conditions are chosen independently for each direction. The fundamental equation for the FEM can be expressed using Maxwell’s equations as [30]:$$\nabla \times \left(\frac{\nabla \times \stackrel{\u20d7}{H}}{{\epsilon}_{r}}\right)=\frac{{\omega}^{2}{\mu}_{r}\stackrel{\u20d7}{H}}{{c}^{2}},$$(1)where $\stackrel{\u20d7}{H}$ represents the intensity of the magnetic field. ε_{r} and μ_{r} represent the relative dielectric permittivity and magnetic permeability, respectively. ω represents the angular frequency of the light wave. In a vacuum, light has a velocity of c. The derivation yields the propagation constant β and the plural form of the effective index n_{eff}. The properties of the PCF can be calculated using the equations below.
The refractive index ${n}_{{\mathrm{SiO}}_{2}}\left(\lambda \right)$ of the substrate material, silica, was considered wavelengthdependent and computed using the Sellmeier equation, which takes the form [31]:$${n}_{{\mathrm{SiO}}_{2}}\left(\lambda \right)=\sqrt{1+\frac{0.6961{\lambda}^{2}}{{\lambda}^{2}0.004679}+}\frac{0.4079{\lambda}^{2}}{{\lambda}^{2}0.01351}+\frac{0.8974{\lambda}^{2}}{{\lambda}^{2}97.9340}.$$(2)
With the propagation constant β determined, the effective refractive index n_{eff} was subsequently computed using the following formula [32].$${n}_{\mathrm{eff}}=\frac{\beta (\lambda ,n(\lambda \left)\right)}{{k}_{0}}.$$(3)
The propagation constant of a wave is denoted as β, while k_{0} denotes the wave number in free space, where the wavelength λ is included.
The second order dispersion (β_{2}) and confinement loss (α_{CL}) can be calculated using the following formulas [33, 34]:$${\beta}_{2}\left(\lambda \right)=\frac{\lambda}{c}\left(\frac{{\partial}^{2}\mathrm{Re}\left[{n}_{\mathrm{eff}}\right]}{\partial {\lambda}^{2}}\right)\mathrm{ps}/\mathrm{nm}\xb7\mathrm{km},$$(4) $${\alpha}_{\mathrm{CL}}=8.686\times \frac{2\pi}{\lambda}\times \mathrm{Im}\left[{n}_{\mathrm{eff}}\right]\times 1{0}^{2}\mathrm{dB}/\mathrm{cm},$$(5)where, c is the velocity of light measured in meter per second. The real and imaginary part of the effective refractive index is denoted by Re[n_{eff}] and Im[n_{eff}], respectively.
The formula for calculating the modal birefringence of an optical fiber is as follows [35]:$$B=\left{n}_{\mathrm{eff}}^{x}{n}_{\mathrm{eff}}^{y}\right,$$(6)where ${n}_{\mathrm{eff}}^{x}$ and ${n}_{\mathrm{eff}}^{y}$ is the effective refractive index for x and ypolarization modes which also refers to the slow and fast axis of the refractive index in the polarization maintaining fibers.
The effective mode area A_{eff} is the region which is occupied by the fundamental mode. The formula of effective area is as follows [36]:$${A}_{\mathrm{eff}}=\frac{{\left[\iint \left\stackrel{\u20d7}{E}\right(x,y){}^{2}\mathrm{d}x\mathrm{d}y\right]}^{2}}{\iint \left\stackrel{\u20d7}{E}\right(x,y){}^{4}\mathrm{d}x\mathrm{d}y}\mathrm{\mu}{\mathrm{m}}^{2}.$$(7)
Here, A_{eff} is measured in square micrometers (μm^{2}) and is dependent on the strength of the electric field $\stackrel{\u20d7}{E}(x,y)$ in the medium.
The effective mode area is directly related to the nonlinear coefficient, which can be computed in the manner given below [37]:$$\gamma =\frac{2\pi {n}_{2}}{\lambda {A}_{\mathrm{eff}}}\times 1{0}^{3}\mathrm{}{\mathrm{W}}^{1}\mathrm{k}{\mathrm{m}}^{1},$$(8)where λ is the operating wavelength, c is the light’s velocity, and n_{2} = 31 × 10^{−21} m^{2}/W (for fused silica) is the nonlinear refractive index coefficient.
The numerical aperture (N_{A}) can be expressed mathematically in the following way [38]:$${N}_{\mathrm{A}}=\frac{1}{\sqrt{1+\frac{\pi {A}_{\mathrm{eff}}}{{\lambda}^{2}}}}.$$(9)
The following formula is used to find the V_{eff} parameter for the proposed PCF [39]:$${V}_{\mathrm{eff}}=\frac{2\pi R{N}_{\mathrm{A}}}{\lambda}.$$(10)Here N_{A} is the numerical aperture and R is the radius of the core of the PCF.
The beat length in a PolarizationMaintaining Fiber (PMF) is an important parameter because it represents the distance that the polarization state of light propagating through the fiber takes to complete one full cycle. In other words, it is the distance over which the polarization of light in the fiber repeats itself. The formula for the beat length (L_{b}) in a birefringent optical fiber is given by [40]:$${L}_{b}=\frac{\lambda}{{n}_{\mathrm{eff}}^{x}{n}_{\mathrm{eff}}^{y}}=\frac{\lambda}{\Delta {n}_{\mathrm{eff}}},$$(11)where, L_{b} is the beat length, λ is the wavelength of light in the fiber, and Δn_{eff} is the difference in effective refractive indices between the two principal polarization modes of the fiber.
4 FEM outcome of optical properties
In this section, the numerical results of optical properties obtained through finite element method analysis are demonstrated and discussed. The results were obtained by constructing the structure in the COMSOL multiphysics simulation software. The optical properties such as dispersion, birefringence, nonlinear coefficient, effective area, numerical aperture, Vnumber, confinement loss, and beat length have been calculated using the EL communication band. During PCF fabrication, global diameters may vary by ±1% [41]. We analyzed the impact of varying parameters by ±1% on dispersion, birefringence, and confinement loss, as discussed in the section below. The parameter which is related to air hole diameters and slot dimensions is a function of pitch. In the design procedure it is evident that each of the air filled fractions (d/Λ) and slot dimensions are related to pitch value. Pitch change will change the air hole diameters and rectangular slot dimension. In fabrication there may be slight variation can be seen in the fiber parameters during preform formation. So this effect is analyzed by varying pitch value to ±1%. The results for wavelength were plotted using MATLAB software.
4.1 Effective refractive index
The effective refractive index of a PCF is a crucial parameter that determines its guiding properties. The effective refractive index was analyzed for both x and ypolarization fundamental modes by mode analysis. In Figure 3 wavelength dependent effective refractive index (n_{eff}) plot is demonstrated. The n_{eff} value ranges from 1.3200888801638233 to 1.2939963177717213 (for xpolarization) and 1.2546651162545484 to 1.1829793668811053 (for ypolarization) in the E to L communication bands. These values will be used in finding other optical properties like dispersion and confinement loss.
Figure 3 Wavelength vs. effective refractive index curve. 
4.2 Dispersion characteristics
In Figure 4, the wavelengthversusdispersion plot illustrates the polarizationdependent dispersion characteristics of the proposed MSCCPCF. At 1550 nm wavelength, the xpolarization exhibits a dispersion coefficient of −310.8 ps/(nm · km), while the ypolarization shows −77.96 ps/(nm · km), indicating pronounced birefringence and offering insights into the fiber’s polarization sensitive applications.
Figure 4 Wavelengthdependent dispersion characteristics of the proposed MSCCPCF. 
At 1550 nm operating wavelength, the dispersion of −295.2 ps/(nm · km) resulting from a +1% pitch variation and −326.3 ps/(nm · km) from a −1% pitch change for xpolarization underscores the significant impact of even slight modifications in the MSCCPCF fiber’s pitch parameter, as depicted in Figure 5a.
Figure 5 Dispersion changes in response to a ± 1% variation in the pitch parameter of the MSCCPCF for (a) xpolarization and (b) ypolarization. 
Similarly, for ypolarization after ±1% variation of pitch value the dispersion coefficient varies from −74.06 to −81.86 at 1.55 μm operating wavelength is shown in Figure 5b.
Figure 6 presents the wavelengthdependent dispersion curve with a ±1% variation in the air hole diameters (d_{1} to d_{7}). In Figure 6a, the dispersion coefficient for the xpolarization ranges from −270.4 to −351.2 ps/(nm · km) at a wavelength of 1.55 μm. In Figure 6b, the dispersion coefficient for the ypolarization varies from −67.82 to −88.09 ps/(nm · km) at the same wavelength of 1.55 μm. This sensitivity emphasizes the necessity for precise control over structural parameters to tailor dispersion characteristics for specific applications.
Figure 6 Dispersion changes in response to a ± 1% variation in the d_{1} to d_{7} diameter of the MSCCPCF for (a) xpolarization and (b) ypolarization. 
4.3 Birefringence
Figure 7 illustrates the wavelengthdependent birefringence plot for the optimized parameters of the MSCCPCF, showcasing its crucial role in polarizationmaintaining fiber. At 1550 nm wavelength, the birefringence value peaks at 8.795 × 10^{−2}, signifying its remarkable magnitude. Additionally, the figure demonstrates a notable trend of increasing birefringence as the wavelength ascends within the E to L communication bands.
Figure 7 Wavelengthdependent birefringence plot of the optimized MSCCPCF parameters. 
Figure 8a illustrates how the birefringence of the proposed MSCCPCF changes with a ±1% variation in the pitch value. This slight pitch adjustment results in birefringence values ranging from 8.663 × 10^{−2} (for +1% of pitch) to 8.927 × 10^{−2} (for −1% of pitch) at 1550 nm wavelength.
Figure 8 Variation of birefringence with (a) ±1% pitch adjustment and (b) ±1% variation of d_{1} to d_{7} in the proposed MSCCPCF. 
Figure 9 Depiction of the wavelengthdependent relationship between effective area and nonlinear coefficient in the proposed MSCCPCF structure. 
Figure 8b illustrates the impact of a ±1% variation in d_{1} to d_{7} on birefringence. At a wavelength of 1550 nm, the birefringence shifts from 8.681 × 10^{−2} to 8.909 × 10^{−2} as the air hole diameters vary by +1% and −1%.
The birefringence analysis underscores the pivotal role of pitch parameter control in modulating the birefringence of the MSCCPCF, revealing its potential for tailored polarizationmaintaining applications.
4.4 Effective area and nonlinear coefficient
Figure 9 represents a single plot that shows the relationship between wavelength, effective area, and nonlinear coefficient. At 1550 nm wavelength, the effective area and nonlinear coefficient show an inverse correlation with values of 21.76 W^{−1} km^{−1} and 5.085 μm^{2}, respectively.
Figure 10 Plot demonstrating the wavelengthdependent numerical aperture of the proposed MSCCPCF. 
The nonlinear coefficient and effective area values demonstrate that the proposed MSCCPCF structure has the potential to facilitate efficient nonlinear optical processes. This combination shows a balance between strong nonlinear effects and a relatively large effective area, which is useful for nonlinear frequency conversion and supercontinuum generation.
4.5 Numerical aperture
At a wavelength of 1550 nm, Figure 10 illustrates a numerical aperture of 0.3616 for the proposed MSCCPCF under optimal parameters. It highlights the fiber’s ability to capture and transmit light efficiently within its core, indicating favorable conditions for optical signal propagation and coupling efficiency in photonic applications.
Figure 11 Plot depicting the relationship between wavelength and confinement loss in the proposed MSCCPCF. 
4.6 Confinement loss
Figure 11 displays the relationship between wavelength and confinement loss, showing a value of 5.615 × 10^{−11} dB/cm at 1.55 μm wavelength.
Figure 12 Plot illustrating the relationship between wavelength and confinement loss, demonstrating the impact of ±1% variation in (a) pitch value and (b) air hole diameters from d_{1} to d_{7} on the proposed MSCCPCF. 
Figure 12a depicts the plot of wavelength versus confinement loss with a ±1% variation of pitch value. For a −1% pitch change, the confinement loss increases to 6.458 × 10^{−11} dB/cm, whereas for a +1% pitch change, it decreases to 4.773 × 10^{−11} dB/cm at 1550 nm wavelength. Figure 12b displays the effect of a ±1% variation in the air hole diameters (d_{1} to d_{7}) on confinement loss. At a wavelength of 1550 nm, the confinement loss ranges from 3.931 × 10^{−11} to 7.3 × 10^{−11} dB/cm when all the air hole diameters are adjusted by +1% and −1%, respectively.
Figure 13 Plot illustrating the wavelengthdependent variation of the Vnumber for the proposed MSCCPCF. 
The finding emphasizes the fiber’s ability to effectively confine and transmit light within its core, demonstrating its suitability for highperformance optical communication systems and other photonic applications.
4.7 Vnumber
From the wavelengthdependent Vnumber plot in Figure 13, the Vnumber is determined to be 1.319 at 1550 nm wavelength for optimal parameters of the proposed fiber.
Figure 14 Wavelength dependent beat length plot of proposed MSCCPCF for optimum parameters. 
The Vnumber value is less than 2.405 across the E to L communication bands. It demonstrates fibers’ single mode behavior, despite the fact that this communication band causes it to be endlessly single mode.
4.8 Beat length
Figure 14 illustrates the relationship between wavelength and beat length. At an operating wavelength of 1550 nm, the beat length of this fiber is measured to be 17.62 μm. It represents the periodicity of polarization mode coupling within the proposed fiber structure.
Figure 15 Electromagnetic field distribution in the proposed MSCCPCF for (a) xpolarization, ${\mathrm{LP}}_{01}^{x}$ and (b) ypolarization, ${\mathrm{LP}}_{01}^{y}$ mode. 
This finding is significant because it sheds light on the fiber’s polarizationmaintaining properties and its suitability for applications requiring precise control over polarization states, such as fiber optic sensing and telecommunications systems.
4.9 Electromagnetic field distribution
The compactness of the fundamental mode LP_{01} is also investigated. In Figure 15 the electromagnetic field distribution is shown for ${\mathrm{LP}}_{01}^{x}$ and ${\mathrm{LP}}_{01}^{y}$ fundamental modes.
4.10 Comparison with other PCF structures
In Table 1 the comparison has been shown among some recent literatures and researches with the proposed PCF structure.
Comparative assessment of optical characteristics at λ = 1.55 μm for proposed and prior PCFs.
Table 1 reveals that the proposed MSCCPCF produced higher birefringence in a single silica background material. The effective area is larger than in previous PCFs, allowing for larger mode area applications. Finally, because the proposed PCF has a shorter beat length than previous PCFs, it is a good candidate for maintaining polarization in an optical fiber.
5 Fabrication challenge and technique
Amouzad Mahdiraji et al. (2014) both discuss the stackanddraw technique, presenting improved methods for preform preparation and fiber fabrication [42]. Yajima et al. (2013) introduces a slurry casting method for silica PCF preform production, which, when combined with an OH^{−} reduction process, results in a lowloss PCF [43]. Kim et al. (2003) describes the fabrication process of a PCF, including stacking, jacketing, collapsing, and drawing, and presents the measured optical properties of the fiber [44]. These studies collectively highlight the diverse approaches to fabricating PCFs, each with its advantages and potential applications. Circularhole structures are typically crafted using drilling or solgel techniques, whereas stack and draw methods are preferred for circular patterns [45, 46]. Asymmetrical holes such as elliptical or rectangular ones are best created through extrusion or 3D printing [47]. The suggested model, incorporating both circular and rectangular air holes, can be manufactured by employing drilling, solgel, stack and draw processes for circular holes, and extrusion, along with 3D printing, for the rectangular air holes. The solgel fabrication method can be used to fabricate the whole design except rectangular slots and later rectangular slots can be made by drilling method in the preform. So the proposed MSCCPCF can be fabricated incorporating both solgel and drilling fabrication methods.
6 Conclusion
The MSCCPCF design has excellent optical properties, including high birefringence at telecommunications wavelengths around 1.55 μm. FEM simulations show significant birefringence of up to 8.795 × 10^{−2} at 1.55 μm, with singlemode behavior across E to L communication bands (V_{eff} < 2.405). This study emphasizes the versatility and potential applications of the proposed fiber structure by conducting a systematic investigation into the impact of various geometric factors on birefringence and other optical properties, including previous results and analyses. The MSCCPCF is suitable for polarizationmaintaining devices, sensors, and other photonic applications that require tailored optical properties. Modal features like nonlinearity (21.76 W^{−1} km^{−1}), confinement loss (5.615 × 10^{−11} dB/cm), and dispersion characteristics have been thoroughly examined, highlighting its feasibility. These findings help to advance our understanding and practical application of highperformance PCFs in a variety of optical systems.
Funding
There is no internal or external funding given for this research.
Conflicts of interest
Authors of this manuscript declare no conflicts of interest.
Data availability statement
The data that support the findings of this article are not publicly available. They can be requested from the author using the email address [amit.rueten@gmail.com].
Author contribution statement
Amit Halder wrote the article with contributions from Yeasin Arafat, who helped in the manuscript writing. Amit Halder, Muhammad Ahsan and Imtiage Ahmed carried out the experiments and testing using COMSOL multiphysics software. Amit Halder developed the simulation software, using conceptualization by Md. Shamim Anower, and implemented the data analysis algorithm. The conceptualization, funding acquisition, and project administration were done by Yeasin Arafat and Md. Riyad Tanshen. Imtiage Ahmed, Muhammad Ahsan, Zubairia Siddiquee, and Md. Riyad Tanshen contributed to data collection, resources, and software validation.
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All Tables
Comparative assessment of optical characteristics at λ = 1.55 μm for proposed and prior PCFs.
All Figures
Figure 1 Cross sectional view of proposed MSCCPCF. Where, Λ = 0.9 μm, d_{1}/Λ = d_{2}/Λ = d_{4}/Λ = 0.45, d_{3}/Λ = d_{6}/Λ = 0.7, d_{5}/Λ = 0.6 and d_{7}/Λ = 0.85. Rectangular slot dimensions: a = Λ/2 and b = Λ/2√3. 

In the text 
Figure 2 Quarter transverse crosssectional view of air hole arrangement technique. 

In the text 
Figure 3 Wavelength vs. effective refractive index curve. 

In the text 
Figure 4 Wavelengthdependent dispersion characteristics of the proposed MSCCPCF. 

In the text 
Figure 5 Dispersion changes in response to a ± 1% variation in the pitch parameter of the MSCCPCF for (a) xpolarization and (b) ypolarization. 

In the text 
Figure 6 Dispersion changes in response to a ± 1% variation in the d_{1} to d_{7} diameter of the MSCCPCF for (a) xpolarization and (b) ypolarization. 

In the text 
Figure 7 Wavelengthdependent birefringence plot of the optimized MSCCPCF parameters. 

In the text 
Figure 8 Variation of birefringence with (a) ±1% pitch adjustment and (b) ±1% variation of d_{1} to d_{7} in the proposed MSCCPCF. 

In the text 
Figure 9 Depiction of the wavelengthdependent relationship between effective area and nonlinear coefficient in the proposed MSCCPCF structure. 

In the text 
Figure 10 Plot demonstrating the wavelengthdependent numerical aperture of the proposed MSCCPCF. 

In the text 
Figure 11 Plot depicting the relationship between wavelength and confinement loss in the proposed MSCCPCF. 

In the text 
Figure 12 Plot illustrating the relationship between wavelength and confinement loss, demonstrating the impact of ±1% variation in (a) pitch value and (b) air hole diameters from d_{1} to d_{7} on the proposed MSCCPCF. 

In the text 
Figure 13 Plot illustrating the wavelengthdependent variation of the Vnumber for the proposed MSCCPCF. 

In the text 
Figure 14 Wavelength dependent beat length plot of proposed MSCCPCF for optimum parameters. 

In the text 
Figure 15 Electromagnetic field distribution in the proposed MSCCPCF for (a) xpolarization, ${\mathrm{LP}}_{01}^{x}$ and (b) ypolarization, ${\mathrm{LP}}_{01}^{y}$ mode. 

In the text 
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