Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

Abstract

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian: Special Cases

“…The "solvable" cases object of our next study are those preserving time reversal invariance and parity inversion, with the previously discussed senses. This implies that at the same time our determinations of −D 2 verify two conditions: (i) m 2 = 0, for the time reversal invariance and (ii) any or several of ( 17)- (19).…”

“…As a result, many articles have been published. For the benefit of the reader, we have selected to list in the bibliography just a few [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. This kind of supersymmetry intends the construction of a sequence of Hamiltonians, derived from an original Hamiltonian, with a closely related energy spectrum, which could be obtained from the spectrum of this original Hamiltonian.…”

New Families of Complex Hyperbolic-Secant Refractive-Index Profiles through the Factorization Method

“…Consider a Hamiltonian of the form in (8), where the optical potential V (r) depends on a discrete parameter ℓ > 1, such that V (r) = V (r; ℓ) ≡ V ℓ (r) and consequently H = − d 2 dr 2 + V ℓ (r) ≡ H ℓ . Also consider the following [13] factorization condition…”

“…Considering the existing formal analogy between quantum mechanics and classical electrodynamics (see for instance Ref. [7][8][9]), different techniques for the generation of quantum potentials, such as the factorization and supersymmetric methods [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], are being performed in order to generate exactly-solvable refractive-index profiles in the optical context. In particular, complex-valued refractive-index profiles can be also constructed [8,9,20].…”

“…[7][8][9]), different techniques for the generation of quantum potentials, such as the factorization and supersymmetric methods [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], are being performed in order to generate exactly-solvable refractive-index profiles in the optical context. In particular, complex-valued refractive-index profiles can be also constructed [8,9,20]. As the corresponding differential operators are non-Hermitian, neither the orthogonality nor the completeness of its eigenvectors are a priori ensured.…”