Slope matched highly birefringent hybrid dispersion compensating fiber over telecommunication bands with low confinement loss

This article reveals a best possible design for hybrid dispersion compensating fiber with high birefringence established on modified broadband compensating structure through S, C and L telecommunication bands. The simulation outcome exhibits relatively higher birefringence of 3.76 × 10−2 at wavelength of 1550 nm. The suggested fiber also has dispersion compensation characteristics in an inclusive series of wavelengths which covers 1400–1625 nm. The reported design can achieve dispersion quantity of − 606 ps/(nm·km) at 1550 nm effective wavelength. The reported fiber design matches the relative dispersion slope 0.003694 nm−1 similar to single-mode fiber at 1550 nm operating wavelength. This fiber demonstrates negatively flattened effective dispersion of − 2.703 ± 0.734 ps/(nm·km) within 180 nm flat band ranging from 1460 to 1640 nm wavelength. It is also convenient to optical high bit rate communication systems. The low confinement loss is found 3.756 × 10−10 dB/m at the operating wavelength. This design also achieves highly nonlinear coefficient of 50.34 W−1 km−1. In some cases, it can also be used in sensing applications.


Introduction
Optical fiber waveguides are very suitable for fast data transmission in communication systems. The photonic crystal fibers are the type of optical fiber waveguide which consists of different periodic air hole arrangement in the silica base region [1]. Microstructure PCF designs are broadly used for various optical applications like sensing, dispersion compensation, ultrafast data transmission and nonlinearity. Dispersion property of optical fiber is a major problem in wavelength division multiplexing (WDM) and high bit rate transmission systems as it occurs pulse broadening and bandwidth limitation. In high-speed WDM systems, compensation is necessary to the transmission characteristics like the dispersion and its slope of singlemode fiber (SMF) [2]. Conventional fibers show high losses in data transmission systems too. Dispersion compensating fiber (DCF) with high negative dispersion is a good candidate to reduce the losses and costs which occurs in optical waveguides [3][4][5][6][7]. It is evident that for typical dispersion compensating fibers negative dispersion values varies from -100 to -300 ps/(nmÁkm) at 1550 nm [8]. PCFs or holey fibers can be engineered to change the dispersion values in different wavelength parameters. The optical property like dispersion can be varied by changing the dimension of air holes and their corresponding positions, [9,10] that is vital for dispersion compensation in a PCF design.
Birefringence is a property of maintaining optical polarization in fiber which is suitable for the optical sensing applications [11]. High birefringence can be achieved on a PCF structure when the asymmetrical shape is implemented in fiber core by engaging deliberate artificial defects at the center core or the air holes in elliptical rather than circular shapes are made [12]. Achieving high birefringence the circular air holes were used rather than elliptical ones just to overcome the fabrication challenge. Elliptical air holes are difficult to implement in a microstructure or nanostructure designs.
In recent times, there are some proposal of PCF designs by researchers to encounter various optical characteristics like high birefringence and dispersion compensation. Such as, an octagonal-shaped photonic crystal fiber recommended in [13] has encountered negative dispersion of -588 ps/(nmÁkm) and the birefringence was reported 1.81 9 10 -2 at 1550 nm. This design possesses less than 10 dB/m confinement loss. Kaijage et al. [14] proposed a modified octagonal PCF structure attaining the dispersion coefficient and birefringence -239.5 ps/(nmÁkm) and 1.67 9 10 -2 , respectively, at 1550 nm operating wavelength. This modified PCF contains elliptical air holes near the center core which has complexity in fabrication but has low confinement loss in the order of 10 -5 . In the past, a golden spiral design was proposed which accomplished 1.60 9 10 -2 as its birefringence and -400 ps/(nmÁkm) as its quantity of negative dispersion both observed at the same 1550 nm effective wavelength [15]. But confinement loss is not measured in this spiral structure because in previous spiral structures show relatively high confinement losses. In 2016, a defected core PCF that consisted of four elliptical air holes near the center core had been proposed which achieved birefringence of 3.373 9 10 -2 with negative dispersion of -837.8 ps/(nmÁkm) over E, S, C and L communication bands in the same 1550 nm wavelength [16]. As it used elliptical air holes, some complexities were faced in fabrication of the fiber. Moreover, in some current works, 2.75 9 10 -2 birefringence and -331 ps/(nmÁkm) amount of negative dispersion at 1550 nm operating wavelength through S ? C ? L bands have been offered [17] using a defected core circular PCF. A PCF design offered by Habib et al. [18] demonstrated negative dispersion varied from -134 to -385 ps/(nmÁkm) through E, S, C, L and U wideband and showed birefringence 2.13 9 10 -2 at 1550 nm with relatively high confinement loss of 4.54 dB/m. In the same year a defected core decagonal PCF was also prescribed showing negative dispersion quantity of -390 ps/(nmÁkm), has a very negligible birefringence in the order of 10 -5 and confinement loss is about 3 9 10 -6 dB/m at 1550 nm [19]. In the year 2016, Ali et al. proposed a hybrid PCF in [20] which reported birefringence of 2.78 9 10 -2 having a negative dispersion of -345 ps/(nmÁkm) where an asymmetric core was created on a circular-based cladding. This structure maintains confinement loss is about less than 10 -2 dB/m at effective wavelength of 1.55 lm.
In this article, we offer a hybrid dispersion compensating PCF (HyDCF) that is very satisfactory for compensating the dispersion of single-mode fiber over the S, C and L communication bands. The reported fiber also acquired birefringence of 3.76 9 10 -2 along with -606 ps/ (nmÁkm) negative dispersion at 1550 nm operating wavelength over the S, C and L communication bands. The residual dispersion slope was matched to 0.003694 nm -1 with respect to single-mode fiber. The nonlinear property of this proposed HyDCF was also observed, and its value is 50.34 W -1 km -1 at 1550 nm. As the proposed design shows the single-mode property from 1350 to 1700 nm wavelength and so on, it may be very beneficial for high bit stream rate fast transmission over the long distance of an optical fiber communication channel. As it exhibits the dual property, the suggested fiber is a best nominee for both sensing and telecommunication applications effectively. This fiber also demonstrates confinement loss in the order of 10 -10 which is comparatively lower than some hybrid dispersion compensating PCF structures.

Materials and methodology
The designed microstructure of HyDCF was based on a pure fused silica material. The circular air hole cladding consists of hexagonal inner structure with octagonal outer structure. Circular air holes structured in a periodic order has been used in the fiber cladding. Figure 1 displays the transverse cross-sectional view of the proposed fiber design. The artificial defection in core was made with six air hole rings in the cladding comprised of two hexagonal rings in inner cladding layer and four octagonal rings in outer cladding layer to construct the hybrid PCF. The inner core was made asymmetrical by reducing the diameter of two air holes among the six air holes in the first ring.
Four octagonal circular air hole rings were used in the outer cladding to minimize the confinement loss of the fiber, and the defected inner core was introduced to achieve high birefringence. The diameter of smaller air holes was d c = 0.24 lm. First and second ring air hole diameter of the hexagonal structure was d 1 = d 2 = 0.74 lm. The air holes from the third to sixth of the octagonal structure were d = 0.52 lm. The two smaller air holes in the first ring were adjusted to attain high birefringence with negative dispersion solidity as this has an impact over these optical characteristics. Air-hole to air-hole distance of two consecutive rings is recognized as the pitch (K) of the structure.
In this structure, pitch was taken 0.8 lm as optimum which made this design proper for practical fabrication process as it is closer to 1 lm. There is a significant development of fabrication process in recent times, specifically drilling, sol-gel casting, tapering and lithography [21,22] without any major difficulty it is possible to draw our HyDCF structure. In recent times, sol-gel technique is used to reduce the fabrication challenges of asymmetrical dimensions and positions of air hole construct on a silica base in PCF designs [23]. Wiederhecker et al. [24] fruitfully implement standard stack-and-draw technique and have a high air-filling fraction cladding and a tube-like core at nanoscale as 110 nm and 200 nm. The complex proposed PCF structures like Arif et al. [25], and Ahmed et al. [26] also suggested sol-gel technique is a better technique to fabricate this type of structures. In this structure, hybrid cladding was presented and there were two small air holes closer to the core region which was problematic to fabricate in stack and draw method. But Ruan et al. [27] successfully fabricated as small as 20 nm air hole in fiber core by using extrusion technique. So, both sol-gel and extrusion technique are a better way to fabricate the proposed PCF structure.

Simulation techniques and equations
Finite element method (FEM) has been applied to exemplify the performance parameters of the HyDCF. A commercial full vectorial software has been implemented to simulate and calculate the modal characteristics of the proposed fiber. Cross sections of PCF designs with a static number of air holes are separated into consistent subspaces where Maxwell's equations are solved by FEM method for the neighboring subspaces. The subspaces are triangular in shape, and it provides a good approximation in PCF structures. Following vectorial equation is obtained from Maxwell's curl equations of electromagnetics [28].
Here, electric field vector and free-space wave number is denoted by Ẽ and k 0 , respectively, operating wavelength Free-space wave number can be computed by this equation k 0 ¼ 2p=k. In this computation process, Sellmeier equation was used to calculate the effective refractive index of the fiber in different wavelengths which leads to compute the dispersion, confinement loss and normalize frequency. The equation is as follows where n and k represent the refractive index and the wavelength, respectively; B 1 , B 2 , B 3 , C 1 , C 2 , and C 3 are the coefficients of fiber background material silica.
Circular perfectly matched layer (PML) boundary is used in the FEM method to simulate the wavelength dependence PCF optical properties. FEM solves the Maxwell equations directly in best approximation to acquire the value of the effective refractive index, n eff . Once, is computed for each wavelength, the dispersion D, birefringence B, effective area A eff and confinement loss L c can be calculated through the following equations.
L c ¼ 8:686 Â k 0 Im n eff ½ ð4Þ where Re n eff ½ and Im n eff ½ are the real and imaginary parts of the effective refractive index n eff , respectively; k is the space wavelength, velocity of light in vacuum is c and k 0 is the free-space number.
The effective area A eff can be calculated as follows [29]: where A eff is measured in lm 2 and Ẽ is electric field in the medium. Therefore, nonlinear phenomena in PCFs can be studied by effective area too.
As single-mode fiber shows positive dispersion and dispersion slope, the requirements for a DCF should be large negative dispersion and a dispersion slope over a broad series of wavelengths. The total residual dispersion of the fiber is computed by the following equation [30]: where D SMF and D DCF are denoted as the dispersion coefficients of the SMFs and DCFs, respectively. If it is required to compensate the total dispersion, the length of DCFs (L DCF ) should be like that it makes the residual dispersion zero. Also the matching of dispersion slope as same as single-mode fiber is needed. The residual dispersion slope can be calculated by the following equation [31]: where S SMF and S DCF are the dispersion coefficients of the single-mode fiber and dispersion compensating fiber, respectively. Deliberately investigating the mode property of the proposed HyDCF, it was seen that single-mode of the fiber was varied according to effective V parameter inside the telecommunication band. Normalize frequency V eff factor for the HyDCF can be computed by using the ensuing equation [32]: Equation (9) is executed to authenticate the single-mode behavior of the proposed structure where represents the wave number in the free space and K is pitch; n core and n cl denote the effective refractive index of core and cladding, correspondingly computed by Sellmeier equation and mode analysis by finite element method. The cutoff frequency of the effective V parameter is 2.405 for singlemode fibers [33].
The numerical calculations like chromatic dispersion, birefringence, effective area, confinement loss, total residual dispersion, residual dispersion slope and V parameter are accomplished by Matlab software with the assistance of equations related to each term stated above.     Fig. 2a. The global diameter of the smaller air holes of the first ring of the hexagonal cladding structure is d c /K = 0.3 and other air holes of the first and second rings are d 1 /K = d 2 /K = 0.92. The octagonal ring structure from the third to the sixth air hole rings has the diameter parameter of d/K = 0.65. To reduce the complexity of fabrication, the adjacent air holes are separated to a nominal distance. The plot of the wavelength versus dispersion shows the proposed HyDCF has negative dispersion coefficient about -193.9 and -606 ps/(nmÁkm) along the x and y polarization, respectively, at 1550 nm. The y polarization dispersion is considered as the optimum value of dispersion in this work. Due to high negative dispersion, the proposed HyDCF could be an appropriate candidate for dispersion compensating in high bit rate transmission system. The property of the proposed fiber is affected by the diameter pitch as a function of wavelength for this reason the pitch value has been varied to observe the deviations in dispersion and birefringence. The effect by varying global diameter of pitch 0.8, 0.9, 1 and 1.1 lm while the other parameters are kept constant is revealed by Fig. 2b. When the value of pitch is varied 0.8, 0.9, 1 and 1.1 lm, their corresponding y polarization dispersion value becomes -606, -277, -116.7 and -27.52 ps/(nmÁkm), respectively.

Simulation results and discussions
Birefringence characteristics of the proposed HyDCF are shown in Fig. 3. This particular property of the proposed HyDCF design will provide sensing applications [34]. The asymmetrical design of the proposed fiber core design reveals high birefringence, which is necessary for polarization maintaining applications. However, existing conventional PM fibers show a modal birefringence of about 5 9 10 -4 [35]. Moreover, our proposed PCF demonstrates birefringence of about 3.76 9 10 -2 in optimum design parameters, which could be a proper contender in sensing applications. In Fig. 3, the effect of pitch on birefringence is also measured. As the value of pitch was taken 0.8, 0.9, 1 and 1.1 lm, the birefringence at 1550 nm in this value becomes 0.03755, 0.03257, 0.02767 and 1. 31 1.34 1.37 1.4 1.43 1.46 1.49 1.52 1.55 1.58 1.61 1.64 1.67 1.7 1.73 1.73 1 31 1.34 1.37 1.4 1.43 1.46 1.49 1.52 1.55 1.58 1.61 1.64 1.67 1.7 1. The reported microstructure HyDCF reveals singlemode operation over the band of interest starting from 1350 to 1800 nm in Fig. 4. The proposed HyDCF shows that at 1550 nm operating wavelength the value of V parameter is 2.282 (i.e., V eff \ 2.405) which fulfills the criteria to be a single-mode fiber. It is also evident that the fiber demonstrates endlessly single-mode operation through 1350-1800 nm wavelengths covering E, S, C, L and U telecommunication bands.
From the analysis of Fig. 5a, it is evident that the relative dispersion slope (RDS) value of our proposed HyDCFs is 0.003694 nm -1 which matches to the single-mode fibers' RDS slope. To attain better dispersion compensation, it is necessary to match the RDS of single-mode fiber. Effective dispersion curve of the proposed HyDCF shown in Fig. 5b is negative dispersion flattened, and the value of flattened dispersion coefficient is of -2.703 ± 0.734 ps/ (nm km) within 180 nm flat band ranging from 1460 to 1640 nm wavelength. Compensation ratio (CR) of dispersion of fiber is also analyzed in Fig. 5c to prove the accomplishment of negative flattened dispersion at the range of 1460 to 1640 nm wavelength. After compensating of dispersion through 1-km-long suggested HyDCF, it is found to be equal to the total dispersion in 50-km-long SMF, and the residual dispersion versus wavelength plot is shown in Fig. 5d. The value of residual dispersion is negatively flatten at the range of 1460 to 1640 nm wavelength band. Analyzing the results, it is evident that this HyDCF design could be an appropriate nominee for high bit rate transmission systems over the band of interests.
It is perceived in Fig. 6a that the optimal value of effective area of the HyDCF is 2.327 lm 2 at 1550 nm wavelength. The nonlinearity plot of the reported HyDCF for optimum pitch of the design parameters is observed in Fig. 6b. The measure of nonlinearity at optimum pitch value is 50.34 W -1 km -1 at 1550 nm operating wavelength. According to the results demonstrated, the suggested design of HyDCF contains decent effective area and nonlinearity for the use of nonlinear applications.
The observation of optimal and varied confinement loss with respect to wavelength for varying pitch from optimum value of the reported HyDCF is displayed on Fig. 7. From the plot, 3.756 9 10 -10 dB/m is observed as optimum value of confinement loss at 1550 nm wavelength. Varying the pitch value 0.8, 0.9, 1 and 1.1 lm, the confinement loss changes 3.756 9 10 -10 dB/m, 3.176 9 10 -10 dB/m, 2.369 9 10 -10 dB/m and 2.258 9 10 -10 dB/m, respectively. Confinement loss is a main drawback of a photonic crystal fiber [36]. From the curve, it is evident that this modified hybrid fiber design exposes very low confinement loss too. The design shows the confinement loss in the range of 10 -10 range which is lower in other periodic structures referenced in the table.
The spreading of fundamental optical field for x and y polarization modes at the effective wavelength of    Fig. 8a and b. Conferring to simulation results, it is evident that modes in x and y polarization are firmly bounded at the center core area due to high availability of index in the center than the cladding.
The matching of residual dispersion slope of the reported HyDCF has shown dispersion compensation over S, C and L bands of telecommunication is validated in the residual dispersion curve. The commercially available standard optical fibers face the dispersion problem while transmitting signal into long distance. The dispersion compensating fiber allows to compensate the dispersion, but there are some residue of dispersion while transmitting signal to long distance. This residual dispersion can be compensated by accomplishing single-mode operation to certain distance, and here, it is computed successfully through 50 km distance. Also, commercially available standard fibers are not polarization maintaining fibers. This HyDCF microstructure photonic crystal fiber contains the high birefringence property which is suitable for sensing application, and high nonlinearity property is the primary attribute of supercontinuum generation. The real-world usage of the proposed HyDCF will be significant in the field of communication applications and optical sensing.
Lastly, an appraisal is prepared concerning birefringence and residual dispersion characteristics between the offered HyDCF and other fibers formerly studied for dispersion compensation and sensing applications. Table 1 displays the assessment of these optical characteristics.

Conclusions
The proposed artificially defected core hybrid photonic crystal fiber design concurrently demonstrates high birefringence and nonlinearity for use as an optical sensor. Dispersion coefficient in high negative value is achieved in this design for attaining dispersion compensating applications. The stated HyDCF achieves 3.76 9 10 -2 as high birefringence and highly nonlinear quantity of 50.34 W -1 km -1 at the effective wavelength of 1550 nm which recommends it as a good nominee for optical sensor. The simulation results also illustrate the negatively flattened residual dispersion of -2.703 ± 0.734 ps/(nmÁkm) within the S, C and L telecommunication bands. The design possesses low confinement loss which is very much important in photonic crystal fiber waveguide designs. The  Ref. [20] -345 2.78 9 10 -2 \ 10 -2 Proposed HyDCF -606 3.76 9 10 -2 3.756 9 10 -10 J Opt (June 2020) 49 (2): 187-195 193 optical properties are very much suitable for endlessly single-mode operation over the range of 1350-1800 nm wavelength. The confinement loss is obtained 3.756 9 10 -10 dB/m which is comparatively low than the other hybrid and complex structures of photonic crystal fiber. The excellent guiding properties lead this fiber design to be capable of other optical sensing utilization and wideband dispersion compensation in high bit stream rate in wavelength division multiplexing transmission systems.