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Numerical Analysis of Photonic Crystal Fiber of Ultra-High Birefringence And www.nature.com/scientificreports OPEN Numerical analysis of photonic crystal fber of ultra‑high birefringence and high nonlinearity Patrick Atsu Agbemabiese* & Emmanuel Kof Akowuah A numerical analysis of a hexagonal PCF structure with four circular air hole rings around the core has been presented in this paper. By utilizing a full vectorial fnite element method with perfectly matched layers, propagation properties such as birefringence, chromatic dispersion and confnement losses are numericaly evaluated for the proposed PCF structure. Specifcally, birefringence of 2.018 × ­10–2, nonlinear coefcients of 40.682 ­W−1 ­k m−1, negative chromatic dispersion of − 47.72 ps/km.nm at 1.55 µm and − 21 to − 105 ps/km.nm at the telecommunication band of C‑U have been reported. Photonic crystal fbers (PCFs) have outperformed its conventional step index fber counterpart. Tis is due to the fexibility in the design of the structure, the material used and application. PCFs referred to as microstructures or holey fbers have received growing interest in recent years. Tey are classifed into two categories according to its light guiding mechanism; photonic band gap guiding and refractive index guiding which is based on total internal refection. Both band gap and index guiding PCFs have claddings which consist of air holes, although they difer in the structure of the core1,2. In Solid core PCfs, the refractive index contrast is achieved by the arrangement of air holes in the cladding3. PCFs have a range of degrees of freedom and design fexibility to control optical properties such as endless single mode4, high birefringence5, chromatic dispersion management6, large mode area7, high nonlinearity 8 and low confnement loss 9. Tese properties open the door for a lot of applications like; nonlinear optics10, sensing1, high power technology11 and telecommunications12,13. Te optical properties can be achieved through optimisation of air hole shape, size and position14–17. Birefringence is generally categorized into two; these are geometrical birefringence and stress birefringence18,19. Optical fbers are doped in order to increase the refractive index, change the melting properties and induce birefringence1. High Birefringence is required to keep two linear orthogonal polarisation states over long dis- tance. High birefringent PCFs are highly sensitive to temperature, a characteristic which is important in sensing applications20. Dispersion is a measure of the temporal broadening of the optical pulse and it is one of the main sources of penalty for efcient transmission of optical signals. A PCF of high Birefringence and negative dispersion com- pensation is important in optical amplifcation as it maintains linear polarisation and useful for double Raman gains and sensing21. Furthermore, research has shown that achieving high birefringence, high nonlinearity and negative dispersion can be useful for polarization control in fber optic sensors, telecommunication applica- tions and broadband dispersion compensation in high-bit-rate transmission systems 22. A lot of work has been reported recently to achieve high birefringence, high nonlinearity and dispersion management through the use of elliptical holes around the core 23,24, using diferent shapes of air holes other than circular in or around the core25–27, defective core21, doping the core with gas or liquid and hybrid cladding28,29, double cladding22, dual core photonic bandgap30, triangular lattice 31 and other shapes of air hole rings apart from the widely used hexagonal lattice32–35. Rhombic and elliptical hole PCFs have been presented26 to achieve birefringence (B) of 8 × ­10–3 and Confnement loss(CL) of 5 × ­10–4 dB/m at 1.55 µm. However chromatic dispersion, which plays a key role in opti- cal communication was not reported on by 26. Furthermore, fabrication of such structures are relatively difcult considering the complex designs rhombic desings present26. Tere are also reports of PCF structures combining circular and elliptical holes to achieve 36 B of 10 –3 and a CL of 10 −7 dB/m. Despite the impressive performance of the structures presented by36, the chromatic dispersion properties are not presented by the authors36. Similar to the strutctures presented by26, these structures are bulky, complex and ccostly to fabricate36. Tere is a growing interest in hybrid PCF structures which employ two or more lattice structures. One such example is presented by28, which consists of hexagonal and decagonal structures with fve air hole rings. Te structure reported CL of 8.13 × ­10−3 dB/m and B of 2.6 × ­10–2 with a very high negative compensation at 1.55 µm, which is similar to 37. In the PCF structure by 5,38 a very high B of 10 –3 at a low CL have was reported but elliptical Department of Computer Eng, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. *email: [email protected] Scientifc Reports | (2020) 10:21182 | https://doi.org/10.1038/s41598-020-77114-x 1 Vol.:(0123456789) www.nature.com/scientificreports/ Parameters Constants B1 0.69675 B2 0.408218 B3 0.890815 C1 4.67914826e-3 C2 1.35120631e-2 C3 97.9340025 Table 1. Values of sellmeier coefcients for background silica material. air holes incorporated in the core of the structures make fabrication difcult. A design with seven rectangular air holes located in the core has been presented in 27 with very low loss. However the fabrication of rectangular holes would be difcult. Paul et al.23, designed a PCF of square lattice with fve air hole rings made up of both circular and elliptical air holes. A very high value of dispersion compensation (DC) and a high B of 4.74 × ­10–3 has been achieved however, owing to the number of elliptical holes present in the design, the fabrication of the structure would be difcult. Another high B in the order of 10–2 and very large negative dispersion is presented in24,39,40 but, the structure contains elliptical holes around the core which increases fabrication difculty. A dual concentrated core with fve layer air hole ring has been presented with high DC and low CL of 10 −4 dB/m but no birefringence results were shown. In21, seven rings of square lattice structure of circular air holes was proposed. A high B of the order of 10–2 was obtained with negative chromatic dispersion. Furthermore, Arif et al. 5 demonstrated a nonlinearity of 39.330 W−1 km−1 but with a low B of 2.83 × ­10–3. In22, high B of 3.12 × ­10–2, nonlinearity of 24.89 W−1 km−1 and high negative dispersion has been proposed however the nonlinearity is low and the structure is complex and fabrication would be a challenge. It has also been demonstrated in41 that nonlinearity of 26.67 W−1 km−1 can be achieved at a pump wavelength of 1.3 µm. Bored core PCF reported in 42, achieved a nonlinearity of 118.4 W−1 km-−1 and a negative dispersion value of − 2221 ps/km.nm but the birefringence (B) was not discussed. It is clear from the aforementioned research works that it is very difcult to simultaneously achieve excellent birefringence, chromatic dispersion and nonlinear coefcient for a given PCF structure. Tis paper presents a proposed PCF structure with ultra-high B of 2.018 × ­10–2 and nonlinear coefcients of 40.72w−1 km−1 at 1.55 µm. Chromatic dispersion of − 47.7 ps/km.nm at 1.55 µm has been demonstrated. Our proposed PCF structure with high B and high nonlinearity within optical communication wavelengths is very desirable in high bit rate data communication, polarization maintaining and sensing applications. Design methodology Refractive index of pure silica is dependent on the wavelength, and for this structure, SellMeier equation43 used: B 2 B 2 B 2 2 1 1 2 3 n = + 2 + 2 + 2 (1) − c1 − c2 − c3 where n is the refractive index of the silica and λ is the wavelength in µm. B 1,2,3 and C1,2,3 are SellMeier coef- fcients as shown in Table 1. In this paper in order to determine the optical properties of the PCF the cross section is segregated into subspaces where Maxwell equation are computed by determining the adjacent subspaces. Te vectorial Eq. (2) is determined using anisotropic PML14,44: −1 2 2 ∇× S ∇×E − k0n [S]E = 0 (2) where ko = 2π/λ is the wave number in vacuum, λ is the wavelength, E is the electric feld vector, n is the refrac- tive index of the domain, [s] is the PML matrix, [s]−1 is an inverse matrix of [s]. Dispersion is a key factor that reduces the information that the fber cable can carry. Te presence of disper- sion results in the spreading of the pulse creating intersymbol interference. Dispersion can categorised into two ,that is intermodal and intramodal dispersion. Intermodal occurs in multimode fbers while intermodal occurs in single mode fbers. Dispersion is made up of the material dispersion and waveguide dispersion Te chromatic dispersion is calculated with the formula1; 2 ∂ Re(neff ) D =− (3) c ∂2 where c, is the speed of light in free space and Re is the real value of the efective refractive index. Birefringence(B) is the diference in the refractive indices of the two polarisation modes. It is calculated as 45; B =|nx − ny| (4) where nx and ny are the efective refractive indices for x and y polarization modes respectively. Te confnement loss is calculated from the imaginary part of the efective refractive index. Scientifc Reports | (2020) 10:21182 | https://doi.org/10.1038/s41598-020-77114-x 2 Vol:.(1234567890) www.nature.com/scientificreports/ In PCFs the light propating through the core is due to the fnite rings of air holes in the bulk silica which extends to infnity.
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