Quantum-mechanical analogy and supersymmetry of electromagnetic wave modes in planar waveguides

Abstract

We study an analogue of the equations describing TE and TM modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schrödinger equation. The effective potential energies involved in the Schrödinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permeability is constant, it means that the refractive index is constant, we find that the TE and TM modes are described by the supersymmtric quantum mechanics.

Supersymmetric Transformations in Optical Fibers

“…Specifically, the true-mode (de)multiplexer is designed using optical potentials with partial degeneracy induced by the violation of the paraxial approximation. Remarkably, this concept has not been previously observed in the optical SUSY literature [23][24][25][26][27][28][29]. In this vein, these results might be extrapolated to SUSY planar waveguides, which may open different possibilities for signal-processing applications in integrated photonics [54,55].…”

“…The unbroken SUSY procedure and its intriguing features and applications can also be exploited by using planar waveguides [24][25][26][27][28]. In this context, a planar SUSY mode converter has been experimentally demonstrated in Ref.…”

“…Despite these interesting features of supersymmetric models, there has been no experimental evidence of SUSY in nature so far [13]. Nevertheless, the ideas of SUSY have been profitably extended to diverse areas of physics and engineering [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] with applications to, e.g., quantum cascade lasers [20], graphene field theory [21], or matter-wave circuits [22].…”

“…One-dimensional (1D) SUSY has attracted particular attention in the fields of nonrelativistic quantum mechanics and photonics [14,[23][24][25][26][27][28][29]. The superalgebra underneath 1D SUSY transformations (also known as Darboux or DarbouxCrum transformations) allows us to relate two different real quantum or optical potentials with identical spectra and scattering properties in the Helmholtz regime [14,24].…”

“…Thanks to these interesting features, real and complex quantum potentials have been found to be analytically solvable [14][15][16][17][18][19], and a class of supersymmetric 1D optical devices has been recently proposed with intriguing and unexpected properties for mode-multiplexing, modeconversion, and cloaking applications [24][25][26][27][28][29].…”