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In this research, the influence of the parameters design, such as the refractive index of the core, the cladding, and the radius of the core on propagation constant (β) of single-mode optical fiber in optical communication region (1.2-1.6) m have been studied and investigated. Material, waveguide, and profile dispersions are analyzed and investigated. Three models of optical fibers with different relative refractive indices () (0.004, 0.007, 0.01) at a wavelength equal 1.55 m, and three models of core radius (3,4,5) m are taken in the count. Numerical simulations and modeling are arranged depending on weakly guiding approximation for solving homogeneous wave equation derived from Maxwell’s equations. Our modeling has been solved by the aid of MATLAB software. Material and profile dispersion have no significant change for various relative refractive index, while waveguide dispersion is affected by the change of relative refractive index. the waveguide dispersion increased by increasing core diameter and the profile dispersion decreased as the core diameter increased. There is no effect on martial dispersion by increasing the core diameter. Keyword: Single-mode optical fiber, Propagation constant, Material Waveguide, and profile dispersion. لا لماعم ريثأت ن لا صئاصخ ىلع بلقلا رطقو يبسنلا راسك فايل ةيئوضلا طمنلا ةيداحأ غابدلا زاتمم امس 1 * ، ديعس مناغ فانم 2 1،2 ءايزيفلا مسق ، ةفرصلا مولعلل ةيبرتلا ةيلك ، لصوملا ةعماج قا رعلا ،لصوملا ، ةصلاخلا : لماعم لثم ميمصتلا تاملعم ريثأت ةسا رد إ ىلع يئوضلا فيلل بلقلا رطق فصنو فلاغلاو بلقلا راسكن ث تبا لأا ( راشتن β ) ةيئوضلا تلااصتلاا لاجم يف طمنلا ةيداحأ ةيئوضلا فايللأل (1.2-1.6) m مت يتلا إ لك قيقدتو ليلحت مت .ثحبلا اذه يف اهئاصقتس ،يبناجلاو يجوملاو يداملا حزقتلا نم جذامن ةثلاثل ا نم لأ ( يبسنلا راسكنلاا لماعم ريغت بسح ةيئوضلا فايل  ) (0.004, 0.007, 0.01) ( يجوملا لوط دنع 1.55 m فاصنأب طامنأ ةثلاثلو ) أ يئوضلا فيلل بلقلا راطق m ( 3,4,5 ذخا يتلا ) مت .ةسا ردلا يف ت تلاداعم نم ةدمتسملا ةسناجتملا تاجوملا تلاداعم لحل فيعضلا هيجوتلا بيرقت ىلع دامتعلااب ةجذمنلاو ةيددعلا ةاكاحملا بيترت Journal of Education and Science (ISSN 1812-125X), Vol: 29, No: 4, 2020 (124-139) 125 ةغلب هدادعا مت جمانرب مادختساب ةسا ردلا هذه يف ةجذمنلا تمتو .ليوسكام MATLAB . انظحلا ثيح أ لا يبناجلاو يداملا حزقتلا ن ريغتي ا ريثك ظوحلم لكشب كلذكو .يئوضلا فيلل يبسنلا راسكنلاا لماعم ريغتب رثأتي يجوملا حزقتلا امنيب ،يبسنلا راسكنلاا لماعم تا ريغتل لا حزقتلا رثأتي لا نيح يف .يئوضلا فيللا رطق ةدايزب يبناجلا حزقتلا لقي امنيب يئوضلا فيلل بلقلا رطق ةدايزب يبناجلا حزقتلا دادزي ام يد .يئوضلا فيلل بلقلا رطق ةدايزب :ةلادلا تاملكلا فايللاا ةيئوضلا .يبناجلاو يجوملاو يداملا حزقتلا ،راشتنلاا تباث ،طمنلا ةيداحأ Introduction Step Index The step-index fiber is represented by cylindrical waveguide dielectric core that is surrounded by the cladding. This fiber has a core refractive index higher than the cladding refractive index and radius (a). The core and cladding refracting indices are uniform as shown in equation (1). n(r) = { n1 when r < a (core) n2 when r ≥ a (cladding) (1) The step-index fiber has two types on the basic model: ● Single-mode step-index. ● Multimode step-index Single-mode step-index Single-mode step-index fiber has a very small central core of diameter which is between (2-10)μm , being very small, this diameter leads to one path for light rays through the cable while there is more than one path for light ray in multimode as shown in figure 1. The difference of refractive index between the layers in fiber is called the relative refraction index (∆); it is very small and can be written as in equation (2) [1]: Figure (1) The step-index fiber for (a) multimode step-index fiber; (b) single-mode step-index fiber [2, 3]. Journal of Education and Science (ISSN 1812-125X), Vol: 29, No: 4, 2020 (124-139) 126 ∆= n1 2 − n2 2 2n2 2 (2) In weakly guiding fiber ( n1 ≅ n2 ), the relative refractive index value: ∆= n1 − n2 n1 (3) For the step-index, the single-mode fiber must be satisfied with the condition ( 0 < V ≤ 2.405 ). Where V is Normalized frequency, it is the very beneficial parameter in optical fiber, which briefs all the important characteristics of the fiber in a single number. the normalized frequency used to calculate the number of possible modes and can be used to calculate the cut off wavelength. The relation normalized frequency is given by equation (4). V = 2πa λ √n1 2 − n2 2 (4) Where: a is the core radius, λ is the wavelength, n1 the refractive index of core and n2 the refractive index of the cladding. The ratio between the angular momentum and the phase velocity is the propagation constant (β) of the guided modes; it lies in the values of (n2k < β < n1k), where k = 2π λ . To find the Normalized propagation constant (b) [4] b = (neff 2 − n2 ) 1 2 ⁄ (n1 2 − n2 2) 1 2 ⁄ [0,1] (5) The change of the fundamental linearly polarized mode (LP01) propagating along the fiber is defined by the propagation constant β. It is suitable to define the effective refractive index for single-mode fiber as a ratio propagation constant of the fundamental mode to that of the wavenumber as shown in equation (6) [3]. neff = β k (6) Where: neff the range between n1 and n2 values. Dispersion In the optical fiber communication system, there are many problems such as dispersion in which the light pulse is spread out when it propagates through the channel of the fiber transmission. The greatest effect of dispersion occurs in the case of digital systems in the form of broadening in the width of transmitter pulses over the fiber; the broadening increased as the traveling distance in the fiber is increased. The negative phenomenon leads to inter symbol interference between traveled pulses and it leads to increase the errors. By increasing the length of the fiber, we get an increase in Dispersion; it is measured by the unit of time per unit of length, such as ns/km, ps/km, or time/km [5]. Journal of Education and Science (ISSN 1812-125X), Vol: 29, No: 4, 2020 (124-139) 127 In optical fiber communication, the total dispersion parameter is given by: D = − λ c dneff dλ2 (7) where c is light velocity and neff is the effective refractive index. There are many types of dispersion: 1Material dispersion 2Waveguide dispersion 3Profile dispersion Material dispersion The light emitted from a source with different wavelengths each wavelength propagates with different group velocity in fiber; this occurs due to the change in the refractive index through the fiber material. This effect is called material dispersion [6]. The pulse which is composed of different wavelengths also has the same effect as it propagates through the fiber. This broadening in the pulse depends upon the variation of the refractive index in the fiber along the transmission line [2]. In the dispersive medium, the phase velocity is different from group velocity depending on the amount of the dispersion in this media, when there is no dispersion. Vphase = ω k (8) νg = δω δβ (9) Where ω the angular frequency. The phase and group velocities are the same Vg = Vph. The amount of dispersion in the medium depends on the variation of refractive indices through the fiber which tends to spread out or delay the light wave depends on its wavelength; this is called the group velocity delay [4]. The group index shows the way of light behavior when the light pulses are considered the group index that become important because the light pulse broadening in time of the input signals of a different wavelength. N ≡ n − λ dn dλ (10) The plane wave propagated in the core dielectric medium of fiber which refractive index n(λ), which is infinitely extended in the direction of propagation , these wave can be used in the evalulation of the dispersion material in the fiber [7]. Dm = 1 c [ dN1 dλ A(V) + dN2 dλ 〈1 − A(V)〉] (11) Journal of Education and Science (ISSN 1812-125X), Vol: 29, No: 4, 2020 (124-139) 128 Where A(V) ≡ 1 2 [ d(bV) dV + b] (12) Dm is material dispersion, and A(V) is the fraction of LP01 mode power that is carried in the core. The total material dispersion in the fiber, which can be considered as sum material dispersion of both core and glad regin. Tg = 1 c {N1A(V) + N2[1 − A(V)] + N2Δ[A(V) − b]} (13) Figure (2), explains the relationship between the material dispersion and wavelength for pure silica. It is observed that the material dispersion tends to zero region around 1.3 μm. Waveguide Dispersion: The waveguide dispersion, in single-mode fiber, takes place as the light passes across the core and the cladding because the refractive index in the core is higher than the cladding [8]. The light moved through the core to cladding is more slowly, the difference between the refractive index of the core and the cladding is very small <<1. The different refractive index for the core and the cladding in the single-mode caused the spreading of the light at different speeds and led to the propagation delay; thus the waveguide dispersion. The chromatic dispersion defined by the combination of the material and waveguide dispersion. [9] Figure (2). The function between wavelength and material dispersion for silica [2]. Journal of Education and Science (ISSN 1812-125X), Vol: 29, No: 4, 2020 (124-139) 129 The material and waveguide dispersion is zero at wavelength λ = 1.3 μm, therefore, the best wavelength in terms of dispersion is λ = 1.3 μm where is called the zero-dispersion, but the best wavelength in terms of attenuation is λ = 1.55μm. The optical fibers' design with the zero-dispersion at a wavelength λ = 1.55μm which is called the dispersion-shifted fibers.

The influence of relative refractive index and core diameter on properties of single-mode optical fiber.