Design and Theoretical Analysis of Highly Negative Dispersion-Compensating Photonic Crystal Fibers with Multiple Zero-Dispersion Wavelengths

Tis paper presents a highly negative dispersion-compensating photonic crystal fber (DC-PCF) with multiple zero dispersion wavelengths (ZDWs) within the telecommunication bands. Te multiple ZDWs of the PCF may lead to high spectral densities than those of other PCFs with few ZDWs. Te full-vectorial fnite element method with a perfectly matched layer (PML) is used to investigate the optical properties of the PCFs. Te numerical analysis shows that the proposed PCF, i.e., PCF (b), exhibits multiple ZDWS and also achieves a high negative chromatic dispersion of − 15089.0ps/nm · km at 1.55 μ m wavelength, with the multiple ZDWs occurring within the range from 0.8 to 2.0 μ m range. Other optical properties such as the confnement loss of 0.059 dB/km, the birefringence of 4 . 11 × 10 − 1 , the nonlinearity of 18.92W − 1 km − 1 , and a normalized frequency of 2.633 was also achieved at 1.55 μ m wavelength. Tese characteristics make the PCF suitable for high-speed, long-distance optical communication systems, optical sensing, soliton pulse transmission, and polarization-maintaining applications


Introduction
Te discovery of a photonic crystal fber (PCF) has presented numerous opportunities for optical communication systems than conventional fbers [1][2][3]. PCFs operate by guiding light based on the properties of photons. Tese guiding mechanisms of light in PCF can either be index-guiding photonics [4], photonic bandgap guiding [5], or a hybrid of the two guiding principles [6]. Te index-guiding PCFs are a subclass of PCFs with a central solid core [7] surrounded by microstructured cladding. Te core has a higher refractive index than that of the cladding. On the other hand, the photonic bandgap PCFs confne light based on the bandgap efects [5]. Each type of PCF has its unique strength; however, a hybrid of the two results in a PCF with fexible features [8]. Also, based on the fexible structure design, material use, and wide range of applications, PCF has recently received much more attention than conventional optical fbers [9][10]. Achieving the desired property in a PCF is often obtained by tailoring the geometrical properties of air holes in the cladding [11][12][13] and the optical properties of the dopant. Tese fexibilities give PCFs due advantage over the conventional fbers in applications such as optical communication [14], refractive index and temperature measurement [15], liquid chemical sensing [16,17], gas sensing [18], biosensing [19][20][21], nonlinear optic devices, high-speed terahertz propagation of the optical signal to remote distances [22], coherent optical tomography [23], optical transmissions (lasers) [24], submarine communications [25], chromatic dispersion management [26,27], reduction in confnement and bending losses [28,29], high birefringence, high nonlinearity [30,31], as well as single or multimode [27,32,33] operations of optical signals. Despite all these benefts that PCF ofers, it is faced with challenges such as chromatic dispersion, confnement, and bending losses when propagating signals over long distances. To deal with these limitations, one can design a fber with zero dispersion within the C-band in the telecommunication spectrum. One could also design a fber with a high negative dispersion coefcient to counteract the efects of any dispersion that may arise in the optical link.
Te authors in [30] reported a hexagonal-structure PCF with four rings using circular air holes around the core to achieve a negative chromatic dispersion of −47.72 ps/ nm·km, a birefringence of 2.02 × 10 − 2 , and a nonlinear coefcient of 40.68 W − 1 km − 1 at 1.55 µm wavelength. Te results achieved a high nonlinear coefcient. However, it recorded a low negative chromatic dispersion. Similarly, the authors in reference [34] proposed a PCF of dual-concentric design. Tey reported a −78010 ps/nm·km dispersion at a wavelength of 1.55 µm. However, the dual-core concentric nature of the PCF will pose fabrication challenges. Besides, their design did not report nonlinearity, an equally important property of PCFs for optical communication.
A signal's power spectral density (PSD) is the distribution of optical energy in a communication system. High PSD is ideal for reducing fber losses [35]. One way to achieve that is by designing PCF with many zero-crossing dispersion wavelengths (ZDW). Te authors of reference [26] reported the design of a PCF with three ZDWs using circular air holes, while the authors of reference [27,31,36] proposed and designed PCFs with two zero-crossing dispersion wavelengths. Specifcally, Amoah et al. [26] designed a PCF with three ZDWs and a dispersion of −220.39 ps/ nm·km at 1.55 μm using nondefective air holes in the core. Teir design also achieved an ultrafat chromatic dispersion variation of ±0.9 ps/nm·km. However, they achieved a low negative dispersion within the wavelength of 1.53-1.8 μm.
Similarly, the authors of reference [37] proposed a similar PCF designed as in the study by authors of reference [21] for generating a supercontinuum coherent optical spectrum. Teir results achieved three ZDWs within 1200 nm to 2200 nm wavelength. Even though their investigations used a few air holes, no specifc dispersion coefcient was reported at 1.55 µm wavelength.
In reference [38], Yang et al. proposed a PCF to analyze the efects of self-steepening on soliton tunneling of PCFs. Te results achieved three ZDWs at 771 nm, 924 nm, and 1014 nm wavelengths. However, designing PFC with ZDW within the entire telecommunication band is essential. Besides, their results did not report a specifc dispersion coefcient at 1.55 µm.
Te reviewed literature shows that most of the proposed PCF designs could not achieve more than three ZDWs and high negative dispersion at 1.55 µm. Tese two characteristics are ideal for high-speed long-haul optical communication systems.
In light of all the abovementioned assertions, we propose a novel dispersion-compensating photonic crystal fber (DC-PCF) to achieve high negative chromatic dispersion at 1.55 μm and we observed multiple ZDWs over the O-E-S-C-L-U communication bands than reported in the literature.

Design of the Proposed DC-PCF
In this investigation, silica-based index-guiding PCFs with a hexagonal arrangement of circular air holes in the claddings are designed. An inner diameter, D 1 � 9.8 μm and an outer diameter, D 2 � 11.8 μm, is used throughout the three designs. Each cladding consists of an equilateral triangle of sides equivalent to the pitch (Λ). Detailed designs of each of the structures are illustrated below.
(i) In Figure 1(a), uniform air holes of diameter (d 0 ) � 1.5 μm were used in this design without inserting tiny air holes in the cladding while the pitch(Λ) was kept constant at 2.2 μm (ii) Similarly, in Figure 1 Maxwell's equation [30] shown in equation (1) with the FV-FEM in PML is used to simulate the PCFs.
where E is the electric feld vector, n is the refractive index of silica determined by the Sellmeier equation, International Journal of Optics wave number in a vacuum, and λ is the wavelength. Te refractive index of silica material for our proposed PCF was determined using the Sellmeier equation [30] as shown by the following equation: where λ is the wavelength in (μm), n is the refractive index of pure silica material, and B 1,2,3 and C 1,2,3 are the Sellmeier constants as shown in Table 1.

Optical Properties of PCFs.
Chromatic dispersion is a major inhibiting factor for the transmission of optical signals. It causes the spread of signals in the fber link, resulting in a high bit-error rate (BER). Te dispersion may result from the waveguides, D w (λ), or the optical property of the material, D m (λ). Chromatic dispersion, D (λ), is the summation of the waveguide and material dispersion. Each of the components of dispersion is computed using the following equations [39]: where λ is the wavelength and speed of light in free space and n silica is the refractive index of silica material.
where c is the speed of light in free space, Re (nef) is the real value of the efective refractive index, and λ is the wavelength. Te net dispersion is calculated as follows: Te refractive index of the optical material is afected by the polarization of light. Tis efect results in the diference in refractive indexes of the orthogonally x-y polarized modes. Such an absolute diference is known as birefringence. A high birefringence value is necessary for designing PCF with sensing applications [40]. Birefringence is determined numerically by using the following equation [41]: where n x and n y are the refractive indices of the material's two fundamental polarization modes. When light propagates through a transparent medium, some of its optical power may be lost due to absorption, bending, or leaking from the core due to cladding or due to an infnite number of air holes in the cladding. Te associated optical energy thus will be leaked from the core to the cladding. Tis phenomenon constitutes confnement loss. Confnement loss is calculated using the imaginary part of the efective mode index as shown by the following equation [33]: where λ is the wavelength and Im (nef) is the imaginary part of the efective refractive index. Te efective mode area is the area in optical fber over which the electric feld (E) is optimally distributed. Te efective mode area is an essential parameter in analyzing fber nonlinearity. A large efective mode area is essential in systems with high-bit data transmission. However, for nonlinear systems small efective mode is required. Te efective mode area of the fber is computed using the following equation [42]: Te nonlinear coefcient [33] is computed with the following formula: where n 2 is the coefcient of the refractive index of silica material, 2π/λ � k0 is the wave number in free space, λ is the wavelength, and A ef is the efective mode area. For this research study, n 2 is taken as 2.76 ×10 − 20 m 2 W − 1 [30].

Results and Discussion
Te results in Figure 2 depict the confnement of the optical energy in the core of the three PCF structures when simulated at 1.55 um wavelength. It can be justifed that the optical energy is confned in the core in all three cases. Te intensity of the optical energy can be deduced from the legend of the 2D plot λ � 1.55 µm. However, the results of X-Y polarized modes shown in Figure 4 depict that the PCF with geometry (b) and the proposed PCF gives clear polarized modes among the three PCF structures.

Efects of Variation of the Diameter of Air Holes on the Efective Index of PCFs.
Here, we analyse the efects of variation of the diameter of air holes on the efective index of the diferent designs, since the efective index plays a sensitive role in the analysis of all other characteristics of PCFs. We varied the diameter of the bigger air holes from 1.0 to 1.5 μm while keeping the pitch size constant at 2.2 μm. We simulated the designs at operating wavelengths of 1.3 and 1.55 μm. It is observed from Figure 5 that the efective refractive index is high when the diameter of the corresponding air holes is relatively smaller in all three cases. Te modal indexes decrease monotonically as the diameter of the air holes increases across the wavelength from 1.3 to 1.55 μm. However, we investigated other characteristics while keeping the diameter of the air holes constant at 1.5 μm for all three designs. Figure 6 shows a graph of the efective refractive index as a function of wavelength for the three PCF structures. We deduced a gradual decline in the efective refractive index from 1.2 to 2.0 μm as in the other three structures, which justifes the relation n eff � k/λ. It is also worth noting that in PCF (b) and PCF (c), the efective refractive indexes decrease steeply as the wavelength increases. In contrast, PCF (a) results show the least decrement.

Efects of Variation of the Diameter Air Holes on
Dispersion. Figure 7 shows chromatic dispersion as a function of operating wavelength for geometric variations    International Journal of Optics of the number and diameter of bigger air holes (d 0 ) for the three PCF structures. As the diameter of air holes, d 0 varied from 1.0 to 1.5 μm, the dispersion coefcient decreased gradually at a 1.5 μm wavelength. At a geometry specifed below the graph, PCF (b) produced high negative dispersion than the other two PCFs. Since we aim to design a high negative dispersion-compensating (DC-PCF) with multiple zero-crossing dispersion wavelengths, we included smaller air holes for further analysis.

Efects of Inclusion of Tiny Air Holes on Chromatic
Dispersion. Te results plotted in Figure 8  Te results in Figure 9 show the confnement loss against the wavelength for PCF (a) and PCF (b) at a constant air hole size of d 0 � 1.5 µm and a pitch Λ � 2.2 µm. Te results show no signifcant change in the confnement losses for the two PCF structures. Furthermore, tiny air holes are included in PCF (b) and are compared to the confnement loss to nonlinearity at diferent geometrical properties. It is observed that confnement loss decreases as the corresponding wavelength decreases while the nonlinear coefcient increases monotonically, as seen in Figure 10. Te implication is that neither inclusion of tiny air holes nor the variation of diameter of such tiny air holes has any signifcant efects on confnement loss. Figure 11 shows the nonlinearity coefcient as a function of operating wavelength for all the diferent PCF structures. It is observed that the nonlinearity decreases as the x-polarized y-polarized (b) x-polarized y-polarized     International Journal of Optics wavelength increases for all three PCFs at constant air holes of d 0 � 1.5 μm and Λ � 2.2 µm. PCF, with the inclusion of 10 tiny air holes, shows signifcantly higher values of nonlinear coefcient. Te PCF without tiny air holes shows minimum nonlinearity compared to those with 6 and 10 tiny air holes. Figure 12 shows the birefringence against the wavelength for all PCFs. As can be observed from the fgure, birefringence is higher and increases sharply for the PCF with 10 tiny air holes. It, however, remains fairly constant in a zigzag manner in PCF (a) and PCF (c). At λ � 1.55 µm, PCF(b) yielded a birefringence of 0.411, while PCF (a) and PCF (c) yielded a birefringence of 0.069 and 0.045, respectively. Cleary, PCF (a) and PCF (c) could not produce a high birefringent and hence will not be suitable for sensor applications.     International Journal of Optics Te efective mode area, A eff , is essential in analyzing the nonlinearity and confnement losses in PCF. Here, we plot a graph of A eff to variation in operating wavelength for all three PCF designs. A highly efective area is essential in reducing the confnement loss. However, it is inversely related to nonlinearity. In Figure 13, the efective area of the PCF (a) and PCF (b) increases monotonically with an increase in the wavelength while that of PCF (c) remains constant from 1.2 to 2.0 µm wavelengths. Te graph depicts that PCF without the inclusion of tiny air holes yielded a relatively high efective mode area than the others at d 0 � 1.5 μm and Λ � 2.2.
Te normalized frequency, also known as the Vnumber, as seen in Figure 14, decreases with an increase in the operating wavelength in the PCF with 10 tiny air holes in all three cases of variation in the geometric dimension of the PCF. It is worth noting that the V-number decreases linearly in PCF (b). Te V-number is the character used to determine the number of modes of a PCF. When the V-number is less than 3.1415 throughout the entire length of the fber, then such fber is endlessly a single mode; otherwise, it is a multimode PCF. It observed that at d 0 � 1.5, d 1 � 0.75, and Λ � 2.2 μm, the proposed PCF has a V-number of less than 3.1415 from 1.2 to 2.0 μm thus, exhibiting single-mode characteristics. Te V-numbers at 1.3 μm and 1.55 μm are 2.912 and 2.633, respectively. However, when d 0 � 1.3, d 1 � 0.70, and Λ � 2.2 μm, the PCF behaves as a multimode at lower wavelengths and moves towards a single mode as wavelength increases. Terefore, the proposed PCF can support both single-mode and multimode operations. Table 2 shows our proposed PCF's point of zerocrossing dispersion wavelengths. Te literature has established that multiple ZDWs give PCFs a high power spectral density (H-PSD), making our proposed PCF a good candidate for chromatic dispersion compensation, applicable for high-speed optical communication systems. Table 3 shows the performance comparison between the PCFs reviewed and our proposed PCF at 1.55 μm. Our proposed PCF achieved a relatively high negative chromatic dispersion and multiple ZDWs within a wide range of wavelengths.

. Conclusion
A novel silica-based dispersion-compensating photonic crystal fber (DC-PCF) with multiple zero dispersion wavelengths (ZDWs) within the telecommunication bands was investigated and proposed. Te multiple ZDWs of the PCF may lead to high spectral densities than those of other PCFs with few ZDWs. Using the full-vectorial fnite element method with an anisotropic perfectly matched layer (PML), the numerical analysis shows that the proposed PCF, i.e., PCF (b), achieved a high negative chromatic dispersion (−15089.0 ps/nm·km) at λ � 1.55 μm which is higher than the results in [43][44][45]. Besides, the multiple zero-crossing dispersion wavelengths occurring within the wavelength range from 0.8 to 2.0 μm give the proposed PCF a high power spectral density (HPSD) making it suitable for dispersion compensation. Other optical properties such as the confnement loss of 0.059 dB/km, the birefringence of 4.11 ×10 − 1 , the nonlinearity of 18.92 W − 1 km − 1 , and a normalized frequency of 2.633 was also achieved at 1.55 μm wavelength. Tese characteristics make the PCF suitable for high-speed, long-distance optical communication systems, optical sensing, soliton pulse transmission, and polarization-maintaining applications.

Data Availability
Te supplementary data used to support the fndings of the study can be obtained from the corresponding author via e-mail upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.