Abstract
First-order supersymmetry (SUSY) adapted from quantum physics to optics manipulates the transverse refractive index of guided-wave structures using a nodeless ground state to obtain intended modal content. Second-order SUSY can be implemented using excited states as a seed function, even with the presence of nodes. We apply second-order SUSY to the coupled-mode equations by recasting them as the Dirac equation. This enables the engineering of non-uniform surface corrugation of waveguide gratings and coupling potential, which encapsulates the Bragg interaction between counterpropagating modes. We show that the added bound states appear as transmission resonances inside the bandgap of the finite grating. The probability density of each state provides the longitudinal modal energy distribution in the waveguide grating. The smooth modal energy distribution of the states obtained by SUSY can mitigate longitudinal spatial hole burning in high power laser operation. We demonstrate that degenerate second-order SUSY allows the insertion of two states, which can coalesce into Friedrich-Wintgen type bound states in the continuum (BIC) for one-dimensional grating. We show that the eigenfunctions of BIC states are doubly degenerate with opposite parity, and the corresponding transmission resonances have phase changes of 2π across these states. One-dimensional BIC states can find application as robust high-speed all-optical temporal integrators by lifting restrictions on the length of various sections in the phase-shifted grating.
Highlights
First-order supersymmetry (SUSY) adapted from quantum physics to optics manipulates the transverse refractive index of guided-wave structures using a nodeless ground state to obtain intended modal content
Supersymmetric (SUSY) transformations, which originated in quantum physics and were recently adapted to photonics, offer a robust, physics-based approach to design photonic structures[11]
We demonstrate that the second-order SUSY (2-SUSY) offers an efficient approach to design waveguides with engineered surface corrugation to obtain desired functionalities
Summary
First-order supersymmetry (SUSY) adapted from quantum physics to optics manipulates the transverse refractive index of guided-wave structures using a nodeless ground state to obtain intended modal content. Used 1-SUSY transforms the potential V0 of the Hamiltonian H0 1⁄4 À∂2z þ V0 in its nonsingular partner potential V1 whose spectra can differ at most in the ground state energy level. This type of transformation is known as unbroken SUSY and is defined by first-order intertwining operator A 1⁄4 ∂z þ WðzÞ, where W(z) is the superpotential. Each SUSY operation transforms the refractive index distribution (potential) and the propagating modes (eigenfunctions) using a nodeless fundamental mode (ground state). If the two energies are unequal and real ε1 ≠ ε2 2 R, the
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