Supersymmetry-generated complex optical potentials with real spectra

Abstract

We show that the formalism of supersymmetry (SUSY), when applied to parity-time (PT) symmetric optical potentials, can give rise to novel refractive index landscapes with altogether non-trivial properties. In particular, we find that the presence of gain and loss allows for arbitrarily removing bound states from the spectrum of a structure. This is in stark contrast to the Hermitian case, where the SUSY formalism can only address the fundamental mode of a potential. Subsequently we investigate isospectral families of complex potentials that exhibit entirely real spectra, despite the fact that their shapes violate PT-symmetry. Finally, the role of SUSY transformations in the regime of spontaneously broken PT symmetry is investigated.Comment: 6 pages, 4 figure

Supersymmetry-generated one-way-invisiblePT-symmetric optical crystals

“…Moreover, the unidirectional * bikash.midya@gmail.com invisibility is ambiguous for a crystal with length L > 2π 3 /b 2 a 3 [22]. Thus, at the PT -symmetry breaking point the sinusoidal crystal appears to be one-way invisible solely for a shallow grating which indeed is realized by recent experiments on a PT -synthetic photonic lattice [24,25].On the other hand, nonrelativistic supersymmetry (SUSY) transformations are shown [26][27][28][29][30][31][32] to be useful in the framework of optics to synthesize new optical structures. In particular, SUSY has provided a method to generate an optical medium with defects that can not be detected by an outside observer [30], to obtain transparent interface separating two isospectral but different crystals [29], and to create a family of isospectral potentials to optimize quantum cascade lasers [31].…”

An analysis of the zero energy states in graphene

“…It will be seen that after diagonalization the components obey Schrödinger like equations with complex potentials which have real spectrum but are not PT symmetric [15][16][17][18]. Also the examples considered here would illustrate that complex potentials possessing real spectrum and without PT symmetry [19][20][21] are not merely of theoretical interest but they can be related to Hermitian systems. Furthermore it will be seen that one may reproduce the results of a previous study [12,14] by suitably choosing the potential parameters.…”

Supersymmetric mode converters

“…1c). Along these same lines, additional degrees of freedom 16 could be used in addressing other design goals. SUSY phase-matching can be employed to facilitate high-fidelity mode conversion over a broad spectral range, for example, throughout the telecommunication-relevant C-band (see Supplementary Fig.…”

“…This asymmetric approach allows for the direct removal of any eigenvalue l k by factorizing H ð1Þ ¼ H À l k ¼ QR to obtain H ð2Þ ¼ RQ. Note that in continuous one-dimensional settings, complex-valued potentials are required to address states other than the fundamental mode 16,17 . In contrast, the QR formalism allows one to accomplish this task without resorting to nonHermitian configurations involving the interplay between gain and loss.…”

“…In its optical manifestation, SUSY can potentially establish close relationships between seemingly different dielectric structures 14 . For example, two refractive index profiles interrelated via supersymmetric transformations share identical scattering characteristics 15 and therefore can become virtually indistinguishable to an external observer, even in the presence of losses 16,17 . In the context of guided wave optics, supersymmetric partner waveguides are characterized by perfect global phase-matching conditions: With the exception of the fundamental mode, each guided mode of the original multimode waveguide has a counterpart in the partner arrangement with exactly the same propagation constant, or effective index (see Fig.…”