A comprehensive study of large negative dispersion and highly nonlinear perforated core PCF: theoretical insight

A photonic crystal fiber (PCF) containing circularly organized square-shaped air holes in the cladding region is investigated. The fiber core is perforated with four circular air-filled holes to instate high nonlinearity and large negative dispersion. The numerical analysis is done with a finite element method based COMSOL Multiphysics tool to investigate different optical properties of the propounded PCF. The simulation outcome verifies a high nonlinear coefficient value of 85 W−1 Km−1 at telecommunication window 1.55 μm which is, the highest ever achieved value on comparing with the other existing literature without using any nonlinear materials or liquids to the best of the authors’ knowledge. In parallel, the large negative value of dispersion −597 ps nm−1 km−1 is achieved for S/Λ equals 0.70 at the same communication window. However, the highest achieved nonlinearity and negative dispersion are 300 W−1 Km−1 and −1689 ps/nm/km. Moreover, birefringence, numerical aperture, and propagation loss are also measured as 2.40 × 10−3, 0.59, and 4.12 × 10−11 dB m−1 respectively along with an extremely high core power fraction of 99.98%. Hence, the propounded PCF is suitable for residual dispersion compensation, supercontinuum generation, and high bitrate transmission.


Introduction
Nowadays PCF is a hot research topic as it finds applications in numerous fields such as biomedical imaging, supercontinuum generation, and sensing enactment [1][2][3][4]. PCF realization obviated the structural and background material limitations [5]. Based on PCF's operation mechanism, it can be classified into two categories that are index guiding (IG) and photonic bandgap (PBG). With the aid of the periodic cladding arrangement, light can be confined in the core region of IG-PCF via modified total internal reflection (TIR). While in PBG, light can be confined in the core of lower RI compared to cladding via the bandgap effect. PCF exhibits various optical properties contrary to conventional optical fiber such as high birefringence, low confinement loss, high nonlinearity, low waveguide dispersion, etc [6,7]. PCF consists of many tiny periodic holes in its cladding region which runs throughout the fiber length. PCF shows an interesting attribute that optical properties can be amended by changing its structural parameters without any change in its background material. PCF can be applicable for dispersion compensating fiber in the field of high-speed communication as the desired dispersion properties can be obtained with the variation in structure design [8]. Reeves et al reported in their research article that a very low dispersion variation of +0.24 ps/nm/km with a low fiber loss of 0.08 dB m −1 could be obtained by proper designing of PCF [9]. Reeves in another piece of literature found that low loss and flattened dispersion could be observed with several air hole rings in the cladding of fiber [10]. In the same way, Saitoh et al demonstrated that dispersion can also be controlled by the core air hole defects in a PCF [11]. Marcos et al in their article obtained a very high average negative dispersion for microstructure PCF [12]. In 2015, a research article admitted that PCF with square-shaped air holes in the core and cladding region can achieve large negative dispersion of −780 ps nm −1 km −1 at 1.0 μm while in the L band he obtained dispersion of −360 ps nm −1 km −1 [13]. But in this manuscript, the authors did not show any findings related to the nonlinear coefficient. In 2019, Das et al showed that liquid infiltration in the core hole of PCF displayed large negative dispersion of −1600 ps nm −1 km −1 along with the shift in the zero-dispersion wavelength [14]. Later, in the same year, G D Krishna showed that desirable dispersion and confinement loss could be achieved with air hole defects in the core of PCF [15]. While in 2020, Anurag et al achieved dispersion of −810 ps/nm/km for PCF with square air holes in the core as well as in the cladding [16]. Birefringence and nonlinearity are other significant optical properties of PCF that can be amended by modifying its core and cladding air holes. Bo et al in their research article proposed a hybrid PCF and achieved birefringence of 1.56×10 −3 at the wavelength of 1.20 μm [17]. Whereas, Yu et al showed birefringence and nonlinearity of 0.92×10 −3 and 20 W −1 Km −1 at 1.33 μm, respectively for the elliptical rhombus air-core PCF [18]. Arif et al proposed simple hexagonal lattice PCF and obtained high birefringence and nonlinearity equal to 1.83×10 −3 and 20.04 W −1 Km −1 [19]. Hexagonal lattice PCF with elliptical core holes has been realized for liquid sensing and achieved birefringence of 1.95×10 −3 at 1.20 μm [20]. High birefringence of 1.45×10 −3 with a low loss figure has been reported for H-PCF by Cheng et al in their research article [21]. While Ahmed et al recommended an octagonal PCF with a square core for liquid sensing with high nonlinearity of 3.26 W −1 Km −1 and low confinement loss at a 1.33 μm operating wavelength [22]. Hao et al in their research showed birefringence of 2×10 −3 with negative dispersion of −20 ps/nm/km for the PCF with an elliptical lattice pattern of circular air holes [23]. X Bai in his literature coined that with the aid of square holes in the cladding region of the PCF leads to very low confinement loss and relatively flat dispersion that is useful in mode division multiplexing (MDM) [24]. However, he obtained a very minimal amount of nonlinear coefficient value of less than 2.58 W −1 Km −1 .
In this manuscript, five circular rings of square air holes are proposed with circular airhole core defects. Four airhole core defects are placed at the vertices of a square as shown in figure 1. The entire structure is conceptualized based on FEM. Simulation results show that large negative dispersion of −1689 ps/nm/km is achieved for the propounded structure. Moreover, high birefringence of 2.40×10 −3 and high nonlinearity of 300 W −1 Km −1 with very low confinement loss of 10 −11 are resulted in the recommended structure. Apart from this, some other essential optical parameters like core power fraction (CPF), group birefringence, numerical aperture (NA), effective V-parameter and beat length are also investigated for the proposed structure. Fabrication feasibility is considered while choosing the dimensions and shape of propounded PCF and hole packing density (HPD) has been deliberately chosen well within practical fabrication limit.

Design of structure
The proposed PCF cross-section is shown in figure 1 with an enlarged view of its core. The cladding of the recommended PCF consists of a periodic arrangement of square air holes. Five circular rings of square air holes are preferred in the cladding for concentrating the mode light within the core of PCF. Such an annular ring PCF shows the behavior of step-index fiber [25]. In the current design, we prefer square air holes instead of circular air holes because square air holes provide larger HPD (i.e.S/Λ) as compared to the circular ones [16,24]. The radius of square airhole rings is given below where n R denotes the number of rings (1 to 5) from the core of the proposed PCF and Λ shows the used pitch in this manuscript. Pitch is generally termed for either distance between two adjacent rings of air holes or the radius of the first air hole ring. In this manuscript, we have taken the pitch (Λ) and side of the square airhole (S) equal to 1.2 μm and 0.85 μm, respectively. Hence, the HPD ratio (S/Λ) for the propounded PCF is 0.70. Although many current fabrication technologies are available larger than the HPD ratio (i.e. more than 0.90) is cumbersome to be realized practically via fabrication. Therefore, we have proposed a simple fashioned structure and all parameters are selected well within the fabrication realization limit. The background should be transparent and displays a very low loss profile for the entire communication range. SiO 2 fulfills the above-stated requirements. Hence, SiO 2 is selected as background material whose refractive index can be calculated as given below [14]. A layer of 1 μm for absorbing the unwanted scattering losses is used at the outer boundary of the proposed structure. Such a layer is commonly known as a perfectly matched layer (PML). This structure has very low complexity with an HPD ratio of value 0.70 and fabrication complexity will be low with such a small HPD ratio. Many advanced techniques are available in this modern era for PCF fabrication. 3D printing and extrusion techniques are cutting edge technology for PCF fabrication [26,27]. Recently in 2011, Atakaramians et al in their literature showed that the extrusion method is suitable for square hole fabrication [27]. In this method, the fiber is drawn from bulk glass and with the help of a mandrel element any required shape of air holes, especially square ones can be easily fashioned. Fused preforms is another substitute technique for fiber fabrication [28]. Preforms technique provides minimum distortion error while designing the fiber and it is the best suitable technique for low as well as high HPD ratios. In this method, fiber can be drawn by fusing two or more dissimilar glasses. Etching is applied subsequent to this and after this process PCF of required material is only left. By applying performs nearly any shape of air hole can be delineated.
Stack & draw and Sol-gel methods are other prime PCF fabrication techniques [13,29]. With the use of a square/ circular shaped mandrel element, we can fabricate the square/circular air holes with fine precision in the PCF.

Numerical analysis
We have probed the entire structure with PML by using fully vectored FEM. To analyze the structure, the whole cross-section is dissected into triangular subsections with the PML layer. The Mesh analysis is applied to these triangular subsections of the propounded structure with the perfectly matched electric field and magnetic field boundary conditions. PCF cross-section is subdivided into 10345 domain elements, 1821 boundary elements and 73166 degrees of freedom. Following is the solution of Maxwell equations for PCF subsections with the help of FEM that is as follows [16].
Here, E denotes electric field, The phase or modal birefringence can be calculated with equation (5) [16].
Group birefringence G (λ) is closely associated with phase birefringence B(λ). An important property of fiber in telecommunication is polarization mode dispersion (PMD=G(λ)/c) which is directly related to G(λ) and it should be minimum to minimize PMD. Group birefringence can be estimated using the following relation [30].
The dispersion profile for the recommended structure can be evaluated by the following relation [16].
Where c denotes the speed of light, λ shows the used wavelength and n ef displays the effective refractive index of the recommended structure.
Nonlinear properties of the fiber are associated with the nonlinear coefficient of PCF and many properties such as supercontinuum generation, four-wave mixing and solitons generation depend on high nonlinear coefficient values. It can be calculated as follows where I(r) = | | E tr 2 denotes transverse electric field (E tr ) intensity distribution in the proposed structure. The loss incurred by deviation of mode light intensity from core to the cladding of PCF is known as confinement loss which can be evaluated with equation (10)  where Img(n ef ) displays the imaginary part of obtained effective refractive index (ERI).
High NA ensures the high power carrying capacity of the fiber. The modal area of the fiber is related to the numerical aperture (NA) as given below [16]. V-parameter of PCF can be calculated by using equation (13). A fiber behaves as a single-mode fiber for fiber displays multimode nature [32].
Here, r denotes the radius of the fiber core, n co represents ERI of core and n clad depicts ERI of PCF cladding. Beat length of the fiber is nearly associated with birefringence and can be formulated as follows [32].

Results and discussions
This section thoroughly describes the change in optical specifications such as birefringence, group birefringence, nonlinearity, confinement loss, dispersion, numerical aperture, and effective V-parameter for the changes in core airhole diameter (d) and HPD ratio (S/Λ). Structural parameters can vary with an anomaly in the designing parameter during the fabrication process. Hence, fabrication tolerance is taken up to  2%.
Figures 2(a)-(b) describes the electric field placement in the core of the proposed structure for x-and ypolarization, respectively. The fundamental core mode is located well in the core holes which defines that mode light concentrates in the small effective core area. Such mode field confinement is required for the low loss fiber.
The change in phase birefringence with variation in operating wavelength is revealed in figures 3(a)-(b). Birefringence value increases with an increase in the HPD ratio. As we increase the HPD ratio, the difference between x-and y-polarized core electric fields increases. Therefore, the highest birefringence of 2.40×10 −3 is observed for S/Λ = 0.70. However, maximum birefringence of 2.40×10 −3 is also obtained for core airhole diameter (d) equals 0.30 μm as shown in figure 3(b). Therefore, the propounded structure can be used as a polarization-maintaining fiber and polarization splitter for these high birefringence values.
Group modal birefringence variation with wavelength is shown in figures 4(a)-(b). Group birefringence is almost increasing with a wavelength which is the contrast behavior as compared to phase birefringence. The negative slope of birefringence versus wavelength contributes to the increasing behavior of group birefringence. The nature of the curve is negative over 0.8-1.2 μm and increases to a maximum value of 6.93×10 −3 for S/Λ = 0.70 as shown in figure 4(a). However, the minimum G(λ) is obtained for S/Λ = 0.63. The same behavior of group birefringence is observed for the core airhole diameter variation as shown in figure 4(b) and it is noticeable that for solid core, the achieved G(Λ) is of 10 −7 order. Hence, the proposed fiber can be used in high bit data rate communication.
The dispersion profile of a structure over a communication window should be as small as possible to negate the positive material dispersion. The material dispersion of fiber is always positive. While waveguide dispersion of fiber can be amended with the structural change in fiber waveguide. Here, figures 5(a)-(b) displays the dispersion profile variation of the recommended structure for the operating wavelength. Improving the HPD ratio of the proposed structure also improves negative dispersion as shown in figure 5(a). The largest negative dispersion value achieved with the configured structure is −597 ps/nm/km for S/Λ equals 0.70 at 1.55 μm. While almost negative dispersion is obtained for the core diameter variation as shown in figure 5(b). However, the maximum negative dispersion of −1689 ps/nm/km is obtained for the solid core at 1.95 μm wavelength. Thus, such a large negative dispersion endorsed that the propounded structure could be applicable as a residual dispersion compensator.  The change in nonlinearity for divergent values of operating wavelength is shown in figures 6(a)-(b) for the change in HPD ratio and core airhole diameter, respectively. From the figures, it is obvious that numeric nonlinearity decreases with operating wavelength for both types of structural variations. The highest nonlinearity of 300 W −1 Km −1 is obtained for both HPD ratio (S/Λ) and core airhole diameter (d) values of 0.70 and 0.24 μm at 0.75 μm. It is the highest achieved value of nonlinearity for any proposed structure without introducing an ellipse in PCF core or without doping of any foreign high nonlinear RI material to the best of the author's knowledge. With such huge nonlinearity, the proposed PCF can be used for supercontinuum generation as well as for solitons generation.
The effective area or modal area of fundamental core mode describes how well a core mode intensity is confined with a small core area. Figures 7(a)-(b) shows the A eff variation with wavelength for change in HPD ratio and core airhole diameter, respectively. It is seen from the figures that A eff increases with wavelength. From figure 7(a), it is seen that A eff decreases with the HPD ratio. In the same way, increasing the core airhole diameter also increases the modal area as shown in figure 7(b). The lowest obtained value is 0.734 μm 2 for both S/Λ and d equal to 0.70 and 0.24 μm at 0.75 μm operating wavelength, respectively.
The fundamental mode of light diffuses into the cladding area as it propagates along the fiber length. Such deviation of light from the core region induces loss which is generally termed as confinement loss (CL). CL increases with an increase in wavelength. CL decreases with the HPD ratio as shown in figure 8(a). While CL increases with the core airhole diameter as shown in figure 8(b). The minimum obtained values of CL are 2.01×10 −8 dB m −1 and 4.12×10 −11 dB m −1 , respectively for S/Λ = 0.70 and solid core at 0.75 μm. Henceforth, the propounded PCF could be a potential candidate for low loss waveguide application with such an ultra-low CL.
The change in NA with operating wavelength is shown in figures 9(a)-(b). From figure 9(a), it is seen that increasing the HPD ratio increases the NA of the structure. The maximum NA of 0.48 is obtained for S/ Λ = 0.70. While from figure 9(b), it is noted that increasing the hole diameter decreases the NA of the structure. Therefore, the highest NA of 0.59 is achieved for the solid core variant of the proposed PCF. Such a high value of NA of the propounded structure makes it suitable for biomedical imaging applications.   Table 1 shows an inclusive summary of optical parameter variations against variations in the structural parameter of the proposed PCF. Here, we have observed that maximum birefringence and nonlinear coefficient are obtained for the perforated core of propounded PCF at S/Λ = 0.70 and d = 0.24 μm. Whereas, maximum NA, dispersion, confinement loss and minimum G(λ) are observed for the solid core variant of the propounded   structure. Therefore, for high birefringence and nonlinear coefficient, we have to choose a perforated core while in the case of dispersion, confinement loss and NA, we should go for a solid core variant. Figure 10 shows the variation in core power fraction (CPF) with wavelength. The power decreases with an increase in operating wavelength. The x-pol mode power decreases faster as compared to the y-pol mode power. Whereas from figure 10, it is to be mentioned that cladding and material power fraction is increasing with wavelength because   the core mode power diffuses into the cladding area and background material. The highest core mode power of 99.98% is obtained at 0.75 μm which ensures the high power carrying capacity of the proposed structure. Figure 11 shows the effective V-parameter variation with wavelength. From figures 11(a)-(b), it is to be noticed that V eff decreases with wavelength for both HPD ratio (S/Λ) and diameter (d) of core air holes. The   proposed PCF behaves as a single-mode fiber for the whole operating wavelength range as seen in figure 11. Hence, it can be easily said that the submitted PCF shows endlessly single-mode behaviors. Figure 12 depicts the variation in beat length values against change in operating wavelength values and it can be calculated by using equation (14). The minimum observed value of 542 μm is achieved at a 1.15 μm operating wavelength. To use the propounded structure as polarization-maintaining fiber, we have to select a lower beat length value. Such a considerable low beat length value allows the recommended structure to be used as polarization-maintaining fiber.
From table 2, it is concluded that the proposed work displays better numeric values of listed optical parameters as compared to existing research articles. Thus, the propounded PCF is far better than the previously existing literature. Here, it is worth to be mentioned that group birefringence i.e. G(λ) is included for the first time in any simple PCF structure consisting of air holes in its core and cladding regions.

Conclusion
An unadorned optimized perforated core PCF is designed using FEM in this manuscript. The recommended structure shows a very high nonlinear coefficient of 300 W −1 Km −1 with a high core power fraction equal 99.98%. Large negative dispersion of −1689 ps/nm/km is obtained for the solid core variant of the propounded PCF. The structure has a very high phase birefringence equals 2.35×10 −3 with a high numerical aperture equals 0.59. The propounded structure behaves as an endlessly single-mode fiber with a very low confinement loss of 4.12×10 −11 dB m −1 and a beat length of 542 μm. We have to choose a solid core for large negative dispersion while a perforated core must be chosen for high birefringence and nonlinearity. With such desirable characteristics of optimized PCF ensures a variety of applications such as supercontinuum generation, dispersion compensation, solitons generation, polarization maintenance and high bit data rate transmission.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.