We present new numerical methods for the solution of inverse spectral problem to determine the dielectric constants of core and cladding in optical fibers. These methods use measurements of propagation constants. Our algorithms are based on approximate solution of a nonlinear nonselfadjoint eigenvalue problem for a system of weakly singular integral equations. We study three inverse problems and prove that they are well posed. Our numerical results indicate good accuracy of new algorithms.
In this paper, we investigate the problem of determination of dielectric constants of core and cladding in optical fibers using measurements of propagation constant. Such kinds of problems arise in determination of electromagnetic parameters in nanocomposite materials in nanoengineering. These parameters cannot be measured experimentally because of nanocomposite type of materials [
There are many nondestructive material characterization techniques for obtaining permittivities of dielectric materials (see a short review of recent results in [
Inverse problems for determining of dielectric permittivity were studied in many works; see, for example, [
The new approximately globally convergent method for reconstruction of permittivity function was developed in [
The method investigated in this work significantly differs from the above described techniques. Compared with [
We note that two mathematical models of optical waveguides were investigated in detail by methods of integral equations in [
An outline of this paper is a follows. In Section
Let us consider an optical fiber as a regular cylindrical dielectric waveguide in a free space. The cross-section of the waveguide’s core is a bounded domain
The cross-section of a cylindrical dielectric waveguide in a free space.
Eigenvalue problems of optical waveguide theory [
In forward eigenvalue problems, the permittivity is known and it is necessary to calculate longitudinal wavenumbers
In inverse eigenvalue problems, it is necessary to reconstruct the unknown permittivity
The domain
Denote by
Denote by
Let
The next theorem follows from results of [
For any
For waveguides of circular cross-section, the analogous results about the localization of the surface modes spectrum and about the continuous dependence between the transverse wavenumbers
The plot of the dispersion curves computed by the spline-collocation method for surface eigenmodes in a weakly guiding dielectric waveguide of the circular cross-section. The exact solutions are plotted by solid lines. The solution for the fundamental mode is plotted by the lower red solid line. Numerical solutions of the residual inverse iteration method are marked by red crosses. Initial approximations for SVDs are marked by blue circles.
The following statements hold.
(1) For any
(2) For any
(3)
For a given
To compute eigenmodes, we use the representation of eigenfunctions of problem (
Suppose that the boundary
Let
It follows from results of [
Let
(1) If a function
(2) Each eigenfunction of problem (
Let us formulate the nonlinear spectral problem for transverse wavenumbers as follows. Suppose that the boundary
A spline-collocation method was proposed in [
Any iterative numerical method for computations of the nonlinear eigenvalues
In this case, we can investigate spectral properties of the matrix
In the next step, for each
On the base of solution of nonlinear eigenvalue problem (
Clearly, if for given
The red solid line presents the plot of the computed function
In this subsection, we present three algorithms for approximate solution of three inverse spectral problems. The algorithms are based on previous numerical solution of nonlinear eigenvalue problem (
The first inverse problem is formulated as follows. Suppose that the boundary
The solution of this inverse spectral problem is calculated in the following way. First, we compute the number
Suppose that the permittivity of the core is given and that the propagation constant
The solution of this problem is calculated in the following way. First, we compute the number
The full variant of our problem is the reconstruction of both permittivities of the core and of the cladding. The solution for the fundamental mode of nonlinear eigenvalue problem (
The blue and the green solid lines are plots of function
The intersection of these lines marked by the red circle is the unique exact solution of our problem. Therefore, we calculate the permittivities
Let us describe numerical results obtained for a nonlinear eigenvalue problem (
We started our numerical calculations with computations of initial approximations for nonlinear eigenvalues
Note here that, applying previously calculated initial approximations for nonlinear eigenvalues
Our next numerical experiment shows reconstruction of core’s permittivity using the algorithm in Section
The red solid line is the plot of function
In our computations, by analogy with [
Some numerical results of reconstruction of
Absolutely analogous results were obtained for reconstruction of cladding’s permittivity using the algorithm in Section
The red solid line is the plot of function
Finally, we show simultaneous reconstruction of both permittivities of core and of cladding using the algorithm in Section
In this work, we have formulated three inverse spectral problems and proved that they are well posed. It is important to note that, in our algorithms, any information on specific values of eigenfunctions is not required. For solution of these inverse problems, it is enough to know that the fundamental mode is excited and then to measure its propagation constant for one or for two frequencies. This approach satisfies the practice of physical experiments because usually the fundamental mode is excited in waveguides in real applications [
For the approximate solution of the inverse problems, we propose to solve the nonlinear spectral problem for transverse wavenumbers in order to compute the dispersion curve for the fundamental mode. These calculations are done accurately by the spline-collocation method. Finally, we have demonstrated unique and stable reconstruction of the permittivities using new inverse algorithms.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Michael Havrilla and George Hanson for helpful discussions concerning the topics covered in this work. This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities; the work was supported also by RFBR and by Government of Republic Tatarstan, Grant 12-01-97012-r_povolzh’e_a. The research of Larisa Beilina was supported by the Swedish Research Council, the Swedish Foundation for Strategic Research (SSF) through the Gothenburg Mathematical Modelling Centre (GMMC), and by the Swedish Institute.