Ray Tracing Analysis of Light Radiation from Sharp Bends in Graded-Index Multimode Fibers (original) (raw)
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We obtain an asymptotic expression for calculation of radiation fields of the waveguide modes of a multimode fiber with a step-like profile of the refractive index at high normalized frequencies. The distribution of intensity of the output radiation is calculated numerically. Statistical characteristics of the calculated speckles are found and compared with the experiment. Rigorous calculation of the intensity distribution of radiation from a multimode optical fiber in the far diffraction region with allowance for the edge effects is a complicated problem [1, 2]. If the number of propagating waveguide modes is large, then application of the classical numerical methods requires long computation time. Therefore, it is difficult or even impossible to implement such methods on a personal computer. For optical fibers (OFs) with step-like profiles of the refractive index, the problem under consideration can be solved analytically using a number of approximations, which reduces the duration of calculations by several orders of magnitude. Correspondingly, new possibilities for studying the parameters of the output radiation by means of numerical simulation appear. Actually, the posed problem consists of two parts, namely, determination of the intensity of the electromagnetic field at the output end of the fiber and calculation of the field in the far diffraction region. The expressions for field intensity of separate waveguide modes are well known. For example, the longitudinal component E z of the electric field can be written as [3, 4] E z (r, ϕ, z) = A ls J l (u ls r/r 0) cos(lϕ + ϕ 0) exp(iβ ls z)/J l (u ls), where l and s are the azimuthal and radial indices of the mode, respectively, A ls is the normalization factor, J l is a Bessel function of the first kind of order l, r 0 is the radius of the OF core, r, ϕ and z are cylindrical coordinates, and u ls and β ls are, respectively, the eigenvalue of the characteristic equation and the axial propagation constant for the mode with indices l and s. For the classical numerical transformation, it is necessary to calculate beforehand the field at the OF output end by summing up the fields of all waveguide modes and, then, to determine the field in the far diffraction region. In the case of analytical solution, it is convenient to calculate at first the field of each waveguide mode in the far zone and, then, to perform summation with allowance for the phase and polarization. Due to the cylindrical symmetry of the initial distribution, it is expedient to use cylindrical coordinates when performing integration in the plane of the OF output end. The concept of calculating the directional pattern and the field intensities by the stationary-phase method is well known [5]. We neglect both the penetration of the fields of the waveguide modes into the fiber cladding and a change in the field as it escapes into free space and assume that cos θ ≈ 1, where θ is the angle of radiation escape. Representing the expression for a plane electromagnetic wave as a sum of Bessel functions [6] and following [5-7], we can reduce the considered problem of radiation escape to the well-known problem of radiation injection [6]. In