Abstract
A plane multilayered waveguide structure is considered. The layers are located between two half-spaces with constant permittivities. The permittivity in each layer can be a constant or nonlinear (depends arbitrarily on modulus of the electric field intensity). We consider propagation of polarized electromagnetic waves in such a structure. The physical problem is reduced to (nonlinear) boundary eigenvalue problem in a multiply-connected domain. We suggest a numerical approach to calculate propagation constants (eigenvalues) for (nonlinear) layered waveguide structures based on numerical solution of a Cauchy problem in each layer. By means of transmission conditions on the layer boundaries we can define initial data for each Cauchy problem. When all Cauchy problems are solved we construct a function that depends on the spectral parameter. The zeros of this function which can be effectively calculated are the sought-for propagation constants (eigenvalues).
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Notes
- 1.
Let us describe why in this nonlinear problem it is impossible to consider complex values of γ. As \(\mathbf{E} = \left (0,\mathrm{ E}_{y}(x){e}^{i\gamma z}, 0\right ) = {e}^{i\gamma z}(0,\mathrm{ E}_{y}(x), 0)\), then \(\vert \mathbf{E}{\vert }^{2} = \vert {e}^{i\gamma z}{\vert }^{2} \cdot \vert \mathrm{E}_{y}{\vert }^{2}\). As it is known \(\vert {e}^{i\gamma z}\vert = 1\) if \(\mathop{\mathrm{Im}}\nolimits \gamma = 0\). Let \(\gamma =\gamma ^{\prime} + i\gamma ^{\prime\prime}\) and \(\mathop{\mathrm{Im}}\nolimits \gamma \neq 0\). Then \(\vert {e}^{i\gamma z}\vert = \vert {e}^{i\gamma ^{\prime}z}\vert \cdot \vert {e}^{-\gamma ^{\prime\prime}z}\vert = {e}^{-\gamma ^{\prime\prime}z}\), that is Eq. (3) contains \(z\). This means that the function Y (x) depends on z. This contradicts to the choice of E y (x). In the linear problem it is possible to consider complex γ.
- 2.
Definition 1 is a nonclassical analog of the known definition of the characteristic number of a linear operator function depending nonlinearly on the spectral parameter [5]. This definition, on the one hand, is an extension of the classic definition of an eigenvalue to the case of a nonlinear operator function. On the other hand, it corresponds to the physical nature of the problem.
It is well known that electromagnetic waves in a layer propagate on dedicated frequencies, and there are finite numbers of such frequencies. Fixed values of the spectral parameter γ correspond to these dedicated frequencies. This situation takes place both for linear and nonlinear cases. This is the reason why it is necessary (in theory and applications) to determine eigenvalues γ and eigenfunctions.
- 3.
The nonlinear problem under consideration depends essentially on the initial condition Y (h 0 − 0). Similar problem when the permittivity inside each layer is constant does not depend on an initial condition. This means that in the linear problem the “bundle” of waves with different amplitudes corresponds to each eigenvalue γ. In the nonlinear problem eigenvalues depend on amplitudes.
- 4.
It is clear that the value of the function F depends on the solutions of considered Cauchy problem only.
- 5.
Let us describe why in this nonlinear problem it is impossible to consider complex values of γ. As \(\mathbf{E} = \left (\mathrm{E}_{x}(x){e}^{i\gamma z}, 0,\mathrm{ E}_{z}(x){e}^{i\gamma z}\right ) = {e}^{i\gamma z}\left (\mathrm{E}_{x}(x), 0,\mathrm{ E}_{z}(x)\right )\), then \(\vert \mathbf{E}{\vert }^{2} ={ \left \vert {e}^{i\gamma z}\right \vert }^{2} \cdot \left (\vert \mathrm{E}_{x}{\vert }^{2} + \vert \mathrm{E}_{z}{\vert }^{2}\right )\). As it is known \(\vert {e}^{i\gamma z}\vert = 1\) if \(\mathop{\mathrm{Im}}\nolimits \gamma = 0\). Let \(\gamma = \gamma ^{\prime} + i\gamma ^{\prime\prime}\) and \(\mathop{\mathrm{Im}}\nolimits \gamma \neq 0\). Then \(\vert {e}^{i\gamma z}\vert = \vert {e}^{i\gamma ^{\prime}z}\vert \cdot \vert {e}^{-\gamma ^{\prime\prime}z}\vert = {e}^{-\gamma ^{\prime\prime}z}\), that is system (19) contains \(z\). This means that the functions X(x), Z(x) depend on z. This contradicts to the choice of \(\mathrm{E}_{x}(x)\) and \(\mathrm{E}_{z}(x)\). In the linear problem it is possible to consider complex γ.
- 6.
As the functions f i n and g i are arbitrary it is impossible to integrate system (25). However, there are conditions when the first integral of system (25) can be found. For example, the following condition \(\frac{\partial f_{i}} {\partial \left ({\left \vert E_{z}\right \vert }^{2}\right )} = \frac{\partial g_{i}} {\partial \left ({\left \vert E_{x}\right \vert }^{2}\right )}\), pointed out in [8], leads to the fact that the equation
$$\displaystyle{\frac{dX} {dZ} = \frac{\left ({\gamma }^{2}\left (\varepsilon _{i}^{z} + g_{i}\right ) + 2\left (\varepsilon _{i}^{x} {-\gamma }^{2} + f_{i}\right ){X}^{2}f_{iv}^{{\prime}}\right )Z} {\left (2{X}^{2}f^{\prime}_{iu} +\varepsilon _{ i}^{x} + f_{i}\right )\left( {\gamma }^{2} -\varepsilon _{i}^{x} - f_{i}\right )X} }$$can be transformed into a total differential equation. Using this condition in [11] allows to find DE.
- 7.
It is clear that the value of the function F depends on the solutions of considered Cauchy problem only.
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Acknowledgements
I should like to thank Prof. Yu. G. Smirnov for his bright ideas and valuable advice. The work is supported by Russian Federation President Grant (MK-2074.211.1). I also thank to the referee for his remarks.
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Valovik, D.V. (2013). Electromagnetic Wave Propagation in Nonlinear Layered Waveguide Structures: Computational Approach to Determine Propagation Constants. In: Beilina, L., Shestopalov, Y. (eds) Inverse Problems and Large-Scale Computations. Springer Proceedings in Mathematics & Statistics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-00660-4_6
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