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 real-life 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 polarization-sensitive 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 ultra-high birefringence of 3.373 × 10⁻², strong negative dispersion coefficient, and large nonlinear coefficient, ideal for high-speed 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 single-mode 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 long-distance 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 near-elliptic 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 glass-based 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 cross-core 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 cross-core 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 ultra-high 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 single-mode 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 single-mode operation.

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 (d₁, d₂, d₃, d₄, d₅, d₆, and d₇) 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: d₁/Λ = d₂/Λ = d₄/Λ = 0.45, d₃/Λ = d₆/Λ = 0.7, d₅/Λ = 0.6, and d₇/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 air-cores 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 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.

The full-vector 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]:

∇×∇×H⃗εr=ω2μrH⃗c2,(1)where H⃗ 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 nSiO2(λ) of the substrate material, silica, was considered wavelength-dependent and computed using the Sellmeier equation, which takes the form [31]:

nSiO2λ=1+0.6961λ2λ2-0.004679+0.4079λ2λ2-0.01351+0.8974λ2λ2-97.9340.(2)

With the propagation constant β determined, the effective refractive index n_eff was subsequently computed using the following formula [32].

neff=β(λ,n(λ))k0.(3)

The propagation constant of a wave is denoted as β, while k₀ denotes the wave number in free space, where the wavelength λ is included.

The second order dispersion (β₂) and confinement loss (α_CL) can be calculated using the following formulas [33, 34]:

β2λ=-λc∂2Reneff∂λ2ps/nm·km,(4)

αCL=8.686×2πλ×Im[neff]×10-2dB/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=neffx-neffy,(6)where neffx and neffy is the effective refractive index for x- and y-polarization 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]:

Aeff=∬|E⃗(x,y)|2dxdy2∬|E⃗(x,y)|4dxdyμm2.(7)

Here, A_eff is measured in square micrometers (μm²) and is dependent on the strength of the electric field E⃗(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]:

γ=2πn2λAeff×103 W-1km-1,(8)where λ is the operating wavelength, c is the light’s velocity, and n₂ = 31 × 10⁻²¹ m²/W (for fused silica) is the nonlinear refractive index coefficient.

The numerical aperture (N_A) can be expressed mathematically in the following way [38]:

NA=11+πAeffλ2.(9)

The following formula is used to find the V_eff parameter for the proposed PCF [39]:

Veff=2πRNAλ.(10)Here N_A is the numerical aperture and R is the radius of the core of the PCF.

The beat length in a Polarization-Maintaining 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]:

Lb=λ|neffx-neffy|=λΔneff,(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.

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, V-number, confinement loss, and beat length have been calculated using the E-L 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.

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 y-polarization 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 x-polarization) and 1.2546651162545484 to 1.1829793668811053 (for y-polarization) in the E to L communication bands. These values will be used in finding other optical properties like dispersion and confinement loss.

In Figure 4, the wavelength-versus-dispersion plot illustrates the polarization-dependent dispersion characteristics of the proposed MSCCPCF. At 1550 nm wavelength, the x-polarization exhibits a dispersion coefficient of −310.8 ps/(nm · km), while the y-polarization shows −77.96 ps/(nm · km), indicating pronounced birefringence and offering insights into the fiber’s polarization sensitive applications.

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 x-polarization underscores the significant impact of even slight modifications in the MSCCPCF fiber’s pitch parameter, as depicted in Figure 5a.

Similarly, for y-polarization 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 wavelength-dependent dispersion curve with a ±1% variation in the air hole diameters (d₁ to d₇). In Figure 6a, the dispersion coefficient for the x-polarization 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 y-polarization 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 7 illustrates the wavelength-dependent birefringence plot for the optimized parameters of the MSCCPCF, showcasing its crucial role in polarization-maintaining 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 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 8b illustrates the impact of a ±1% variation in d₁ to d₇ 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 polarization-maintaining applications.

Figure 10 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 μm², respectively.

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.

At a wavelength of 1550 nm, Figure 11 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 12 displays the relationship between wavelength and confinement loss, showing a value of 5.615 × 10⁻¹¹ dB/cm at 1.55 μm wavelength.

Figure 13a 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 13b displays the effect of a ±1% variation in the air hole diameters (d₁ to d₇) 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.

The finding emphasizes the fiber’s ability to effectively confine and transmit light within its core, demonstrating its suitability for high-performance optical communication systems and other photonic applications.

From the wavelength-dependent V-number plot in Figure 14, the V-number is determined to be 1.319 at 1550 nm wavelength for optimal parameters of the proposed fiber.

The V-number 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.

Figure 15 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.

This finding is significant because it sheds light on the fiber’s polarization-maintaining properties and its suitability for applications requiring precise control over polarization states, such as fiber optic sensing and telecommunications systems.

The compactness of the fundamental mode LP₀₁ is also investigated. In Figure 15 the electromagnetic field distribution is shown for LP01x and LP01y fundamental modes.

In Table 1 the comparison has been shown among some recent literatures and researches with the proposed PCF structure.

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.

Amouzad Mahdiraji et al. (2014) both discuss the stack-and-draw 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 low-loss 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. Circular-hole structures are typically crafted using drilling or sol-gel 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, sol-gel, stack and draw processes for circular holes, and extrusion, along with 3D printing, for the rectangular air holes. The sol-gel 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 sol-gel and drilling fabrication methods.

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 single-mode 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 polarization-maintaining 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 high-performance PCFs in a variety of optical systems.