Advances in Materials Science and Engineering

Volume 2018, Article ID 5806947, 11 pages

https://doi.org/10.1155/2018/5806947

## Analysis of Spatially Doped Fused Silica Fiber Optic by Means of a Hamiltonian Formulation of the Helmholtz Equation

Escuela de Física, Universidad Nacional de Colombia, Medellín, Colombia

Correspondence should be addressed to Raúl E. Jiménez-Mejía; oc.ude.lanu@enemijer

Received 16 January 2018; Accepted 10 April 2018; Published 14 May 2018

Academic Editor: Andrey E. Miroshnichenko

Copyright © 2018 Raúl E. Jiménez-Mejía et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper discusses an alternative method for calculating modal parameters in optical fibers such as propagation constants, transverse distributions, and anisotropy, due to linear and nonlinear phenomena acting as perturbations caused by doped silica regions. This method is based on a Hamiltonian formulation of the Helmholtz equation and the stationary perturbation theory, which allows a full-vectorial description of the electric field components when linear anisotropic inhomogeneities and Kerr nonlinearity are included. Linear and nonlinear parameters can be found for each propagating mode, and its accuracy has been successfully tested when compared to numerical calculations from the vector finite element method, and the results are published in the literature. This method facilitates the calculation of the spatial-distributed perturbation effects on individual electric field components for each propagating mode.

#### 1. Introduction

The characteristics of propagation of light in optical waveguides are governed by Maxwell’s equations. In the case of a monochromatic wave with time dependence of , these equations can be written aswhere and are the electric and magnetic field vectors and represents the electric displacement vector given by , being the polarization vector that includes linear and nonlinear responses given by: [1]. In uniform waveguides, with dependence attached to appropriated boundary conditions, solutions for radiated and guided modes in (1) can be found from an eigenproblem written in terms of the wavenumber . These eigenfunctions are characterized by a specific set of parameters to describe the propagation characteristics as an unique entity such as spatial distribution for each field component, an effective refractive index, and the optical power distribution for each of the propagating modes [1–3]. Analytical solutions for the transverse electric field distributions found for some basic dielectric profiles , such as rectangular slabs and cylindrical waveguides, can be used as basis solutions for finding analytical approaches of relatively more complex transverse dielectric distributions by using the perturbation theory [4–6]; this method allows spanning the perturbed scenario through a linear combination of the unperturbed solutions. In order to implement the perturbation theory to optical waveguides in the same fashion as in the Hamiltonian eigenproblem in quantum mechanics [4], a Hamiltonian formulation of Maxwell’s equations can be proposed [5–8]. In this formulation, the propagating parameters of a waveguide with perturbed transverse dielectric profile are calculated from the unperturbed waveguide , assuming that the perturbed dielectric profile can be considered as a small change from the unperturbed one as . Such perturbation is included into the equations by means of a pair of very elaborated operators constructed to obtain an eigenproblem representation of (1) with a Hermitian Maxwell Hamiltonian [7, 8]. The first operator acts as a longitudinal projector and the second one includes the transverse characteristics of the waveguide [7]. By using this procedure, Maxwell’s equations can be written as a generalized eigenproblem, expressed in Dirac’s notation, can be handled by means of the standard perturbation theory, and can be written aswith an orthogonality condition between two eigenkets given bywhere and are nondegenerate propagation constants, and , , , and are their corresponding transverse mode distributions. The result in (3) coincides with the well-known orthogonality condition between two propagating modes in the coupled mode theory (CMT) [2, 9] and constitutes the basis function form for the perturbation expansions [7].

Although the formulation in (2) allows the application of the stationary perturbation theory, one of the difficulties in its application is the assessment of the perturbation effects upon the field distribution individually, because the orthogonality condition cannot be applied in the usual sense between electric or magnetic field independently. This is mainly because the definition of the eigenkets in (2b) leads to the perturbation expansion coefficients to be calculated based in a condition of power-mode independence between two propagating modes, as stated in (3), instead of the effects upon the field individually. Evaluation of the perturbation effects over the electric and magnetic field in a separated way constitutes an advantage in the analysis of dielectric waveguides, particularly for the electric field which can be modified not only in its propagation characteristics but also in its transverse distributions, when immersed in dielectric profile perturbations. In the attempt to deal with (1a) and (1b) as independent eigenproblems, only (1b) satisfies the hermiticity condition for the resultant operator [8]. However, perturbative terms associated to linear and nonlinear contributions of the polarization vector cannot be included into the formulation [8]. On the other hand, (1a) permits the inclusion of the polarization vector effects, but resultant operator does not satisfy hermiticity [7, 8].

In this paper, an alternative formulation for calculating the propagating parameters for a perturbed waveguide, as well as the distortions on the transverse electric field distribution is discussed. In the proposed formulation, the effects of perturbative terms upon the modal parameters can be calculated by using the electric field uniquely; thus, expansion coefficients found for the linear combination are relating the expected contribution of the perturbation over each modal electric field distribution. This fact offers a main advantage of the current formulation over conventional methods based on the CMT theory because instead of considering the whole transversal field in expansion coefficients, current formulation simplifies their calculation and facilitates the physical interpretation of the external perturbation over each electric field component that propagates throughout the fiber. As a consequence, the effects of the perturbation terms upon the waveguide modal parameters are more intuitive to be understood and calculated. This formulation can be obtained by operating over (1a) to yield the Helmholtz equation, and after an analogous Hamiltonian formulation of the Helmholtz equation (HFHE), the resultant Hamiltonian operator is reduced to be the Laplacian, and the rest of the terms can be included in the perturbation operator in a very natural manner. By means of this mathematical treatment, field discontinuities and full-vectorial perturbations, concerned to the polarization vector, can be directly included into the perturbative term. Therefore, it is possible to calculate full-vectorial perturbed waveguides scenarios from simple solutions to the Helmholtz equation for each electric field component in the unperturbed situation. A set of numerical experiments for fiber optics waveguides are discussed to show the accuracy of the HFHE when different types of perturbative terms such as inhomogeneities, anisotropies, and nonlinearities are included. Results obtained with HFHE shows an excellent agreement with FEM simulations, and the results are reported in literature.

#### 2. Hamiltonian Formulation for the Full-Vectorial Helmholtz Equation

The polarization vector is responsible to include linear and nonlinear response of dielectrics. In the case of isotropic materials, the linear term of the polarization vector can be expressed through a scalar relation with the electric field, that is, , where is the first-order electrical susceptibility of the material. However, when anisotropies are present, the polarization vector and the electric field vector must be related through a tensorial relationship given by [1, 10]. On the other hand, nonlinear response of the material implies to take into account a set of relatively more complex relations with the electric field [1]. In the majority of dielectrics, the nonlinear polarization vector can be written in terms of powers of the electric field as , where corresponds to the high-order susceptibility tensors. Apart from their natural resonances, this relation can be given by a sum of these terms: [1]. In the following, we show how to deal with polarization vector effects in the resultant eigenvalue problem when calculating modal parameters in optical waveguides.

Consider a *z*-directed propagating wave with propagation constant *β* and applying the vectorial identity , the eigenproblem in (1a) can be written aswhere is the wavenumber, the speed of light in vacuum; is the propagation constant through the waveguide with as the mode effective refractive index and is the linear, homogeneous, and isotropic susceptibility. The additional term includes any inhomogeneity associated to the spatial distribution of the permittivity and polarization vector effects. Linear anisotropies are included through the term , and the nonlinearities are taken into account according to the electric field dependence [1]. This mathematical artifice permits to deal with the Helmholtz equation in the same way to the Hamiltonian eigenvalue problem in quantum mechanics [4], where the unperturbed scenario corresponds to the assumption of isotropic, linear, and lossless material, which reduces (4) to the well-known Helmholtz equation for each electric field component:where is the refractive index of the linear and isotropic medium and acts as the eigenvalue. It is worth noting that the full-vectorial characteristics of the electromagnetic field can still be considered in (5) as long as each electric field component is included in the formulation, any anisotropic effect that involves different electric field components can be addressed by means of the perturbative terms in the polarization vector as presented in (4b). Using Dirac’s notation, (5) can be written aswhere are the normalized kets that represent the modal spatial distribution for each electric field component projected on the coordinate system. These normalized kets can be calculated from the solution to (5b) after imposing the corresponding boundary conditions associated to the waveguide characteristics, which leads to a set of functions that are able to propagate throughout the waveguide. For a step-index fiber optic, the set of solutions of each electric field component, , , and , describes the spatial distribution associated to the propagating mode, which are typically divided in four different families depending on the electric and magnetic field configuration, namely: , , , and modes [2, 3]. Once the solutions are found, normalized kets can be calculated bywhere is commonly called the modal area.

As it is stated in the perturbation theory, eigenfunctions are able to form a basis that allows to span the space of solutions, if completeness and orthogonality are satisfied [4], which are the main difficulties in applying the perturbation theory to (6). In the proposed formulation, completeness requirement is relaxed by means of a reduced basis analysis where both conditions can be achieved. It must be highlighted that even in the more elaborated formulation in (2), it is not possible to guarantee the existence of a complete orthogonal basis for the Maxwell’s equations [6, 7]. Therefore, in some scenarios, a reduced set of elements taken from the set of solutions could be enough to describe the perturbed scenario. In practice, few-mode fibers which allow applications with superior features over the standard single-mode fiber (SMF) in optical communications and sensors, are highly suitable for this type of analysis in as much as only some propagating modes can be excited [11].

On the other hand, orthogonality condition must be tested between the set of solutions to (6). In the case of EH and HE families, it is easy to demonstrate orthogonality since their angular dependence leads to a null inner product independently of the radial functions [3]. However, when the orthogonality condition is tested between solutions of the same family, it will depend on Bessel’s functions associated to the radial coordinate [3]:where are complex constants; is the fiber core radius; , , , and are different roots of the transcendental equation for the EH or HE hybrid mode [3]. A simple inspection over the integral in (8) shows that the inner product between two general modes vanishes only when and are the roots of the Bessel function with , which is not the case for the step-index fibers because and are determined by the boundary conditions at the core-cladding interface and in general, they do not coincide with a root of the Bessel function [12]. This fact shows that hermiticity of the Laplacian operator in step-index fiber waveguides underlies on the boundary conditions, which supports the discussion presented in [5] about the nonhermiticity of the operator that results from (2a) after multiplying from the left by . Since the Laplacian operator in step-index fibers is non-Hermitian, it is not possible to use the typical procedures for finding their corresponding correction terms to the eigenvalue and eigenstate [4]. Indeed, it is required more sophisticated expressions in the formulation of perturbative terms in order to deal with non-Hermitian operators and propose an approached solution by using the whole set of nonorthogonal eigenstates [13]. Another possibility is to find a set of orthonormal functions through a systematic orthogonalization process such as Gramm–Schmidt [4], which allows to span each eigenstate as a linear combination of a new orthonormal set of functions that could be used as an orthogonal basis. This procedure works fine when degeneracy is present. However, in the case of nondegeneracy, this new-basis elements will not constitute eigenstates of the zeroth order Hamiltonian [13, 14].

A more practical approach is to select some elements from the set of solutions in order to form a convenient orthogonal reduced basis that guarantees the diagonal matrix representation of the Laplacian operator [4, 8]. This assumption can make sense physically in fiber optics since propagating modes constitute a finite basis and under a small external perturbation, these modes will interact among each other instead of changing the mathematical nature of the solutions. The procedure for selecting solutions consists in testing the hermiticity of the operator between the functions of the set of solutions. By solving (6), a finite number of functions with different eigenvalues is found, . We can write two different eigenvalue equations: , and . Assuming that each ket has associated a corresponding bra , Hermitian condition of the Laplacian operator can be tested through

Since the eigenvalues obtained in (6) are real valued, conjugation makes no effects in the test. When there is not degeneracy between propagating modes, , (9) is satisfied only by these spatial distributions that are strictly orthogonal each other. Whether the inner product between two nondegenerate solutions is different from zero, that is, , hermiticity test of the Laplacian operator upon these solutions will fail, and they are not suitable functions to be used in the reduced basis. An important case to take into account appears when , (9) is satisfied but eigenvalues could be degenerated, , which implies to use a different formulation of perturbation theory [4]. Although this is not found in step-index fibers because eigenvalues are different each other, this could happen under the weakly-guiding approximation [3], where degeneracy can be overcome grouping degenerated modes into a new set of propagating modes, known as linearly polarized (LP) modes, which can also be analyzed by using the same formulation. Certainly, the accuracy of the result will depend on the number of modes that are included into the basis as well as the perturbation nature. It will be shown that perturbations involving spatial inhomogeneities, as well as some anisotropies and nonlinearities can be analyzed through this simplification leading to accurate results in the propagation parameters, as long as the optical fiber propagates few modes only. However, other type of perturbations that impose strong changes over the number of propagating modes must be addressed by different approaches.

Once the basis of eigenkets, is constructed by orthonormal and nondegenerated eigenfunctions, the first-order stationary perturbation theory permits to calculate the correction terms for the propagating parameters through the expressions [4]where and are the propagation constant and mode field distribution after perturbation, respectively. From (10a), we can see thatwhere . Here, it was assumed that .

It is worth noting that perturbation operator can include any inhomogeneity, linear anisotropy, and nonlinear effects, depending on which terms are included in (4b).

#### 3. Numerical Experiments

This section presents a set of numerical experiments to show the accuracy of the HFHE method described above, in the calculation of modal propagation parameters under the presence of external perturbations. Inhomogeneity and anisotropies of refractive index profiles can be designed to enhance the intermodal interaction, which can exploit the induced distortion on the mode field profile in the design, for instance, of spatial-multiplexing processes and all-optical switching. By means of the present formulation, the external perturbations can be conveniently engineered, such as each component of the electric field in the propagating mode is affected selectively. These perturbations can be achieved by introducing selectively external materials to the fused silica matrix by different doping techniques [15]. Optical poling has been also one of the possible techniques to induce anisotropies and spatially modulated inhomogeneities to oxide glasses [16]. The application of the current formulation is very suitable in few-mode fibers where the amount of supported propagating modes constitutes an orthogonal basis for each of the electric field components. However, its application on single-mode fibers with arbitrary polarization is also possible since the electric field components can be conveniently selected to constitute an orthogonal basis. Different types of analysis were performed in order to illustrate how the perturbations can be included into the analysis, and the accuracies are also discussed.

##### 3.1. Linear Inhomogeneities

This section considers a step-index few-mode fiber with the following parameters: , , and radius ; the wavelength was set to be in order to allow only one propagating mode for each family. In this case, the set of solutions for the unperturbed Helmholtz equation is composed by orthogonal functions with propagating modes . Perturbation consisted in the inclusion of spatial inhomogeneities for the linear refractive index in both the core and the cladding region, such as the perturbation strength, , was imposed from the unperturbed case , to refractive index changes about . This term can be included in (4b), where only the first two terms are considered. The presence of inhomogeneities in the spatial distribution of the permittivity makes to appear a polarization charge density at the interfaces between the inhomogeneous regions. It is worth noting that the first term of the operator in (4b) represents the polarization charge that can be found by . In the case under study, the azimuthal change of susceptibility will impose a surface charge density at each azimuthal interface . Using cylindrical coordinates, polarization charge density at each azimuthal interface can be written bywhere subscript defines both regions at the interface in which the azimuthal component of the electric field is directed from 1 to 2. Subscript relates the mode function for the corresponding propagating mode. As it is stated in (12), the magnitude of the perturbation depends on the relative change of permittivity due to the inhomogeneity. For the refractive index contrasts under consideration, induced polarization charge at the interfaces can be neglected; thus, the perturbation operator can be given by a simple expression , which is a spatial-dependent perturbation.

In order to compare the accuracy of the proposed approach in the prediction of the effects due to the inhomogeneities , the results obtained with the HFHE formulation were compared with those obtained from the vector FEM approach. Comparisons were performed for both the effective refractive index and the distortion in the electric field distribution that undergoes each mode. It is worth noting that the simulations with the FEM are performed assuming that fiber does not suffer any perturbation, that is, inhomogeneities are included as initial conditions of the problem, so it is not rigorously speaking an induced distortion by a perturbation, but an initial distribution of refractive index. This is an important advantage of the HFHE method because it can describe the transition from an initial spatial distribution of the guided mode into a distorted one due to the presence of an external perturbation, which can be used as an strategy for modal division multiplexing [17].

Figure 1 shows the spatial dependence of the perturbative terms. Once the perturbation is included, both the effective refractive index and the change in the field distribution for each propagating mode are calculated as a function of the perturbation magnitude. Figure 2 presents the dependence of the effective refractive index for each propagating mode as a function of the perturbation strength. From this figure, we can see that our results are in good agreement with the vector FEM solutions. We found a maximum absolute error about between them at the strongest perturbation. Sources of the mismatch can come from neglecting the induced polarization charges due to the imposed inhomogeneities in the polarization vector. An additional comparison was performed on the calculation of the field profiles of the guided modes due to the presence of the inhomogeneity. In [7] correction terms were applied only to propagation constants. However, by using HFHE formulation and through the perturbation method described in Section 2, the spatial-distribution correction can be also well estimated after perturbation as presented in Figure 3.