Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 39 This work is licensed under a Creative Commons Attribution 4.0 International License. Birefringence for Elliptical-Core Fibers with Low Ellipticities Miami A. Mohammed dr.miamiph@uomustansiriyah.edu.iq Department of Physics, College of Science, Mustansiriyah University, Bahgdad, Iraq. Abstract Model birefringence was measured for elliptical-core fibers with low ellipticities, note the birefringence depends strongly on the frequency, especially when fiber is being operated near the higher mode cutoff where ν ≤ 2.4 for circular fiber of the single-mode type that correspond to the birefringence maximum. When ν> 2.4 this also correspond to the birefringence maximum that can be introduced in an elliptical core fiber while still operating in the single-mode regime near the higher mode cutoff. Also the birefringence is proportional to the fiber core ellipticity when core ellipticity is much less than unity, but this birefringence deviates from the linear for the large core ellipticities. Keywords: Modal Birefringence, Frequency, elliptical-Core Fibers, Single-Mode Fiber, Mode Cutoff. 1. Introduction The possibility of using polarization-sensitive components in optical systems based on single–mode fiber waveguides accounts for the great interest that elliptic fibers are now attracting .Ellipticity of the core cross section is on of the main imperfections in real optical fibers, which frequency lifts the degeneracy between the two main orthogonally polarized modes of an ideal circular fiber end give rise to birefringence .In the case when specific polarization characteristics of optical waveguides are achieved by geometric effects it would be of interest not only to study other axially asymmetric fibers, but also to investigate more fully the phenomenon of mode birefringence of elliptic fibers[1]. We shall obtain an approximate formula for the mode birefringence of a low-ellipticity fiber. The formula is derived using approximate vector solutions of the Maxwell equations for two dominant orthogonally polarized modes. It is shown that there are considerable changes in the mode birefringence due to changes in the parameters of the single–mode modes. It is shown that the ranges of the approximate of a single-mode elliptic fiber in which the mode birefringence Doi: 10.30526/35.1.2798 Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Article history: Received 30, December 2021, Accepted 9, January, 2022, Published in January 2022. https://creativecommons.org/licenses/by/4.0/ mailto:dr.miamiph@uomustansiriyah.edu.iq Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 40 is maximal and the intermodal dispersion minimal overlap near the limit of the single–mode regime [2, 3]. An experimental study was conducted in which the geometry of an optical fiber with a hollow core with a high ellipticity results in a high geometric refraction in the optical fiber at a wavelength of 1550 µm [4].There was a theoretical study that included numerical analysis using the finite element method. The high birefringence of photonic crystal fibers with an elliptical air hole resulted in better than using a circular hole to obtain high birefringence in photonic crystal fibers, and it was later compared with ordinary optical fibers. It has an elliptical air hole and a circular hole, where the highest birefringence was obtained for the photonic crystal fiber [5]. A theoretical study was used to fabricate an elliptical gap optical fiber with a compressed lattice that results in high binary refraction with a wide frequency of terahertz [6], the directional losses of this fiber are effectively reduced by the absorption of the material, as long as there is a partial dominance of the modular power distribution in the air gaps inside the dielectric material [7]. A new study was proposed to measure sea water temperature sensor based on the Sagnac Loop of high birefringent elliptical fibers. After optimizing the test parameters at a wavelength, it achieves the highest sensitivity and the highest detection range with low cost, small size and ease of manufacture of this type of fiber [8]. A numerical analysis was used by applying the finite element method FEM to design two models of optical crystal fibers, each containing five rings of air holes of circular and oval shapes. As a result, we will obtain high binary refraction and negative scattering at wavelength 1.55 μm and nonlinearity with Small loss of optical crystal fibers It was found that the elliptical model gives more small confinement loss than the circular model at wavelength [9]. The fibers have been designed with a fiber with a highly elliptical core surrounded by a trench designed to obtain the optimum effective indicators. In addition to the manufacture of asymmetric structures to increase the birefringence in the fiber through thermal stress that occurs during the manufacturing process, this results in a high birefringence in the fiber for all guided spatial modes, which can be used in the multiplex- input transmission system.Also in the same year, two types of fibers with semi-circular holes were designed in the Sagnac ring, where the design is designed to coincide with torsion, malleability and temperature. This new combination of fibers results in a high sensitivity to torsion-induced binary breakage at a certain temperature. The presence of such physical parameters as torsion, axial pressure and temperature allows resonance depressions in the interference pattern with different rates of wavelength shift [10, 11]. A high-birefringence fiber was designed due to the stress-induced polarization -maintaining fibers (PMF) fibers, which are designed from an elliptical core, in addition to four elliptical side holes distributed around it, and two symmetrical parts in a circular. What was compared to the typical birefringence of conventional polarization -maintaining fibers (PMF) as Panda fibers,this type of PMF proposed design improves the ability to maintain polarization, as it is considered a promising candidate that can be used in a wide range in high quality optical fiber sensors and also in optical communication systems [12]. In this paper, the effect of a change in the ellipticity of the core on the value of the birefringence modal was studied. We note that the ellipticity of the core of the fiber changes through the fast and slow propagation axes and results in retardation that depends on the degree of ellipticity, and also the phase difference strongly depends on the frequency. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 41 The paper is organized as follows. In section 2, the theoretical part about birefringence in the optical fiber and the design of the elliptical core fibers is presented. Then, the practical section is presented in Section 3, which includes the optical system for conducting the workflow, followed by the most important findings and discussions in Section 4, and then finally, the most important conclusions that we reached through the workflow are presented in Section 5. 2. Theory Part 2.1 Birefringence insingle –mode fiber The birefringence normally-exhibited by single-mode optical fibers is the result of a difference δβ between the propagation constant of the two orthogonally polarized state of the fundamental mode.Each mode travels at different velocity, known as birefringence. The fiber model birefringence is defined as [13, 14].   2 )( YXB   (1) Where βX, βYis the mode propagation constants, the difference in phase velocities cause the fiber to exhibit linear retardation, the value of which depends on the fiber length. This leads to a polarization state which is, in general,elliptical but which varies periodically along the fiber with characteristiclength L where  2 L (2) Typical single- mode fibersare found to have a period of a few centimeters. 2.2 EllipticalCore Fibers Since the stress layer is separated from the core by more than five times the core radius, the influence of stress layer on the geometrical anisotropy can be ignored. Figure 1shows the geometry of the elliptical core fiber. The relative difference in refraction indexes between core and cladding ∆ and the ellipticity of the core ϵ are defined as [15, 16]. n nn 2 2 1 2 2 2 1   (3) a ba   (4) where 𝑛1 is the refractive index of the core, 𝑛2 is the refractive index of the cladding, and a, bare the semi major and semi minor axes of the core, respectively .The two dominant modes of the ellipticalcoresfiber are the 𝐻𝐸11 𝑋 (𝐻𝐸11 𝑜𝑑𝑑) mode, in which the direction of the Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 42 electric field vector lies along the major( x-coordinate ) axis Figure 2 (a), and the 𝐻𝐸11 𝑌 ( 𝐻𝐸11 𝑒𝑣𝑒𝑛) mode , in which the electric field vectoralong the minor(y-coordinate) axis Figure 2(b). In calculating propagation constant, delay time, waveguide dispersion, a numerical method was used based on the point-matching principle. The point-matching method is a useful technique to analyze dispersion characteristics for homogeneous optical fibers with deformed core boundaries[18] .The hybrid electromagnetic fields were expanded in terms of linear combination of circular harmonics (Bessel’s functions) and the boundary conditions are imposed on the field at a finite number points on the boundary. The propagation constant is given as the eigenvalue of the determinant equation whose elements are obtained by using Bessel’s functions. For a fiber having slightly elliptical core(∆ <<1, ϵ<< 1), the normalized birefringence B is expressed approximately in a form [14, 18,19]: )2( 2 3 2   e a B  (5) a X Y b n₁ n₂ Core Cladding Figure 1 .Cross section of elliptical core. (a ) HE11 X Mode (b) HE11 Y Mode Figure 2.Field pattern for HE11 X (left), and HE11 Y (right) modes [17]. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 43 )2( 3 )2( 4 22 2 3 2     e a (6) The normalized phase difference 𝛿𝛽 is expressed as a function of the normalized frequencyν for elliptical core fibers with various ellipticties. )2( 3)2( 4 222 3 2      a e (7) Where  2 1 ak n (k=2𝜋/ 𝜆is a wavelength of light in vacuum). 3. Experimental Part The experimental apparatus setup used for the measurement is shown in Figure 3. Linearly polarized light from a He-Ne laser operating at (632.8) nm is focused on a Fresnel rhombus adjusted to transform the state of polarization from linear to circular. The circularly polarized light is focused on polarizer to select the polarization plane of the emergent linearly polarized beam. The light with selected plane of polarization is coupled to the fiber through a microscope objective. The emerging light of the optical fiber end is focused on analyzer through a microscope objective .The light beam then reaches the detection system,which constants of a PIN-SI photo detector connected to a low-noise pre-amplifier. The electric signal from the detection system fed a lock-in amplifier whose reference is the signal of the optical chopper placed in front of laser [18, 19]. A fiber was used having the characteristics shown in the Table 1, the deviation from circularity was determined by etching the fiber end in hydrofluoric acid and examining it under a high- power optical microscope to ensure from deformed that obtain in the fiber end. Also, we take various value of core ellipticitywith small ellipticity of fiber core (0.19, 0.227, 0.38, 0.45, 0.77) and index differentbetween core and cladding is (∆=2.3× 10−3) [20]. Table 1 .Properties of used optical fiber [21] Property Fiber GSB2 Core composition Ge𝑂2/𝑆𝑖𝑂2 Core diameter 5 µm Cladding composition B2𝑂3/𝑆𝑖𝑂2 Cladding diameter 16.5 µm Substrate tube Silica Relative index different 2.3× 10−3 Overall fiber diameter 210 µm V-Value at 632.8 nm 2.4 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 44 The experimental procedure for the determination of the linear birefringence consists to obtain the polarization state of the emerging light beam for a specific polarization plane of the coupled light beam. The polarization plane is obtained turning the polarizer (45°) with respect to the principle axes of the fiber , so that the two electric field components will be equally excited a long these axes .The principle axes of the fiber are the polarization planes a long of which ,the linear polarization state of propagation beam is preserved. Therefore these axes can be determined adjusting the polarizer and verifying the polarization state of emergent beam through the analyzer [20, 22]. 4. Results and Discussion The phase difference between the two polarization states of the fundamental mode on an elliptical fiber in the limit of small ellipticity and weak guidance. However, calculation the phase difference δβ as a function of normalized frequency ν for small values of ∆, a and ϵ from equation (7) as shown in Table 2. Also Figure 4show the phase difference as a function of normalized frequency for various valuesof ellipticity core(0.185,0.227,0.37,0.45,0.77) and index different between core and cladding (∆=2.3× 10−3), this birefringence depends strongly on the frequency or ν value. These graphs demonstrate that when the parameters of elliptic fiber vary only within the limits of the single –mode fiber regime, the maximum δβ near 2< ν < 2.4 at which the single -mode fiber is being operated near the high mode cutoff are considered (ν≈2.4 for step –index fiber). Figure 4 also shows dependence of the absolute value of the relative difference ∆ between different the two propagation constants are very and sensitive, when the different between the refractive indices is few. For value ν slight greater than 2.4 this is typical of elliptic fibers with small ellipticity of the core, while still near the limit of the single-mode regime, the difference between the group velocity is zero and consequently, the dispersion of a signal transmitted by a single –mode fiber with the parameters for this range is minimal .Moreover, Microscope Object Laser He-Ne Lock –in Amplifier Desk Top Computer Photo- detection System Analyzer Microscope Object Single Fiber Micropositioner Micropositioner Polarizer Chopper Fresnel Rhombus Plotter Figure 3 .Experimental apparatus used to perform the measurements [20]. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 45 in this range modal birefringence is close to its maximum value. By increasing ν value give lower figures than all other graphs this mean reach to implies that the birefringence produced by a given core ellipticity is here found, by a considerable margin in some cases. In addition that, The phase difference δβ as a function of The core ellipticity in fiber having ν=2.4 at λ=0.632.8 nm show in Figure 5, the birefringence for fiber with small- core ellipticity is proportional to ellipticity when ϵ is much than less unity (ϵ<<1) for operation at the high mode cutoff normalized frequency ν=2.4 for near-circular step –index fibers. But for more elliptical cores(ϵ >>1) the birefringence is not proportional to ellipticities, then that the birefringence is no longer very sensitive to the core ellipticity and,in the range still near the high mode cutoff. Table 2 .Show phase difference δβ as a function of normalized frequency for various and a=2.55 µm × 10−3of core ellipticity, ∆=2.3 values ν δβ ϵ2 = 0.185 δβ ϵ2= 0.227 δβ ϵ2= 0.37 δβ ϵ2= 0.45 δβ ϵ2= 0.77 0 0 0 0 0 0 0.2 0.011 0.014 0.022 0.034 0.04 0.4 0.031 0.052 0.063 0.085 0.1305 0.6 0.051 0.071 0.104 0.138 0.209 0.8 0.067 0.101 0.138 0.188 0.267 1 0.08 0.125 0.164 0.233 0.328 1.2 0.089 0.145 0.184 0.274 0.365 1.4 0.099 0.156 0.194 0.299 0.389 1.6 0.1 0.168 0.202 0.308 0.405 1.8 0.1016 0.177 0.206 0.314 0.413 2 0.1012 0.178 0.2079 0.315 0.415 2.2 0.099 0.176 0.207 0.313 0.414 2.4 0.098 0.168 0.2043 0.306 0.408 2.6 0.095 0.157 0.2007 0.302 0.401 2.8 0.093 0.142 0.1962 0.29 0.392 3 0.091 0.134 0.1912 0.284 0.382 3.2 0.088 0.125 0.186 0.271 0.372 3.4 0.085 0.117 0.18 0.262 0.361 3.6 0.082 0.111 0.175 0.253 0.3501 3.8 0.08 0.107 0.169 0.237 0.339 4 0.077 0.102 0.164 0.228 0.328 4.2 0.075 0.098 0.158 0.215 0.316 4.4 0.072 0.092 0.153 0.205 0.306 4.6 0.069 0.087 0.148 0.188 0.296 4.8 0.067 0.083 0.143 0.172 0.286 5 0.065 0.081 0.138 0.165 0.276 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 46 5.2 0.063 0.075 0.133 0.154 0.267 5.4 0.061 0.072 0.129 0.148 0.258 5.6 0.058 0.069 0.125 0.139 0.25 5.8 0.056 0.065 0.12 0.134 0.241 6 0.055 0.062 0.116 0.13 0.233 6.2 0.053 0.059 0.112 0.124 0.226 6.4 0.05 0.057 0.109 0.122 0.219 6.6 0.048 0.054 0.105 0.119 0.211 6.8 0.047 0.053 0.102 0.114 0.205 7 0.045 0.052 0.099 0.109 0.198 7.2 0.044 0.051 0.096 0.102 0.192 7.4 0.042 0.05 0.093 0.101 0.186 7.6 0.041 0.048 0.09 0.099 0.181 7.8 0.039 0.046 0.087 0.098 0.175 8 0.038 0.044 0.081 0.093 0.17 Figure 4 .Phase difference δβ as a function of normalized frequency for various values of core ellipticity (0.185, 0.227, 0.37 0.45, 0.77) and ∆=2.3× 10−3. ϵ= 0.77 ϵ= 0.45 ϵ= 0.37 ϵ= 0. 227 ϵ= 0. 185 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 47 5. Conclusion An ellipticity in the fiber core establishes preferred fast and slow axes of propagation and produces a retardance which depends on the degree of ellipticity and also the phase difference depends strongly on the frequency. The maximum δβ near ν value 2< ν < 2.4 is consistent with the usual operating condition for conventional single –mode fibers. When ν > 2.4 this is also the maximum birefringence that can be introduced in an elliptical core fiber while still operating in the single mode regime. Increase ν value, the mode birefringence decrease until reach to implies near the limits of the single –mode fiber regime, then the parameters of elliptic fiber is very sensitive because of the small absolute difference between the two propagation constants. When core ellipticity is much than less unity, the phase difference is proportional to core ellipticity and for elliptical cores more than unity phase difference is no longer very sensitive to the core ellipticity is not proportional to core ellipticity and deviation from linearity for the large core ellipticity. References 1. Kalosha, V .P.; Khapalyuk, A. P.; Mode Birefringence in a Single-Mode Elliptical Optical Fiber, Journal of Quantum Electronics, 1983, 13, 1, 109–112. 2. Marcuse, D.; Theory of Dielectric Optical Waveguide, Academic Press, New York, 1991. 3. Elion, G. R.;Elion, H. A.; Fiber Optical in Communication Systems, Marcel, Dekker Inc., New York, 1978. 4. Jung, Y.; Han, S. R.; Kim, S.; Paek, U. C.; Oh K.; Versatile control of geometric birefringence in elliptical hollow optical fiber, Opt. Lett., 2006, 31, 18,2681-2683. Figure 5.Phase difference δβ as a function of core.ellipticity in fiber having ν=2.4 at λ=0.632.8 nm. 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 48 5. Yuh-Sien, S.; Yuan-Fong, C.; Han-Hsuan, Y.; Lin-Fang, S.; Tzong-Jer, Y.; Din Ping T.;High birefringence photonic crystal fiber with a complexunit cell of asymmetric elliptical air hole cladding, Appl. Opti., 2007, 46, 22, 5276-5281. 6. Chen, H.; Chen, D.; Hong, Z.; Squeezed lattice elliptical-hole terahertz fiber with high birefringence, Appl. Opt., 2009, 48, 20, 3943-3947. 7. Kim, S. E.; Kim, B. H.; Lee, C. G.; Lee, S.; Oh K.; KeeC.;Elliptical defected core photonic crystal fiber with high birefringence and negative flattened dispersion,OPTICS EXPRESS, 2012,20, 2, 1385-1391. 8. Wang, X. ; Yang, H. J. ; Wang, S.-S.; Liao, Y: P.;WangJ.; Seawater Temperature Measurement Based on a High-Birefringence Elliptic FiberSagnac Loop, IEEE Photonics Technol. Lett, 2015, 27, 16. 9. Saha, R.; Hossain, M. M.; Rahaman, M. E.; Mondal, H. S.;Design and analysis of high birefringence and nonlinearity with small confinement loss photonic crystal fiber,Frontiers of Optoelect.,2019, 12, 163-175. 10. Corsi, A.; Chang, J. H.; Wang, R.; Wang, L.; Rusch, L. A.; LaRochelle S.;Highly elliptical core fiber with stress-induced birefringence for mode multiplexing, Opt. Appl., 2020, 45, 10, 2822-2825. 11. Htein, L.; Gunawardena, D. S.; Liu, Z.; Hwa, Y. T.; Twosemicircular-hole fiber in a Sagnac loop for simultaneous discrimination of torsion, strainand temperature, OPTICS EXPRESS,2020, 28, 23,33841-33853. 12. Li M.; Li H.; Design of ultrahigh birefringent stress-induced polarization- maintaining fiber with hole-assistance, Optical Fiber Technology, 2021, 67, 102707. 13. Shah Alam, M.; Anwar ,S. R. M.; Modal Properties of Elliptical Core Optical Fibers Considering Stress-Optic Effects, J. Electrical and Computer Engineer., 2010, 5, 4, 257-261. 14. Adams, M. J.; Payne, D.N.; Ragdaale C. M.; Birefringence in Optical Fibers with Elliptical Cross-Section, Electron. Lett.,1979, 15, 10, 298-299. 15. Okamoto, K.; Hosaka, T.; Sasakl, Y; Linearly Single Polarization Fibers with Polarization Zero Mode Dispersion, Journal of Quantum Electronics,1982, 18, 4, 496-503. https://ieeexplore.ieee.org/author/37085383488 https://ieeexplore.ieee.org/author/37085374047 https://ieeexplore.ieee.org/author/37085390283 https://ieeexplore.ieee.org/author/37085375717 https://link.springer.com/article/10.1007/s12200-018-0837-6#auth-Rekha-Saha https://link.springer.com/article/10.1007/s12200-018-0837-6#auth-Md__Mahbub-Hossain https://www.springer.com/journal/12200/ https://www.sciencedirect.com/science/article/abs/pii/S1068520021002571#! https://www.sciencedirect.com/science/article/abs/pii/S1068520021002571#! Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (1)2022 49 16. Okashi ,T.; Single- Polarization Single Mode Fibers, Journal of Quantum Electronics, 1981, 17, 6, 879-884. 17. Love, J. D.; Sammut, R.A.; Snyde A. W.; Birefringence in Elliptically deformed Optical Fibers”, Electron. Lett.,1979, 15, 20, 615-616. 18. Summat, R. A.; Birefringence in Slightly Elliptically Optical Fibers, Electron. Lett.,1980, 16, 19, 615-616. 19. Kumar, A.; Varshney, R. K.; Thyagarajan, K.; Birefringence Calculation in Elliptical-Core Optical Fibers, Electron. Lett.,1984, 20, 3, 112-113. 20. Vieria A. J. C.; Siliva M. T. C.; Low Birefringence Single Optical Fiber Polarimetiric Properties, J. Electrical and Computer Engineer, 1988, 3, 10, 57-62. 21. Adam M. J.; Payne D.N.; Fabrication of Single –Mode Fiber Exhibiting Extremely Low Polarization Birefringence, Electron. Lett.,1979, 15, 11, 309-311. 22. Hwang I. K.; Lee Y. H.; High birefringence in elliptical hollow optical fiber, Appl. Opt., 2004, 12, 3, 1916–1923. https://ieeexplore.ieee.org/author/37085375023 https://ieeexplore.ieee.org/author/37085375023