Dr. Nixon

Nonlinear Waves, Optics, Asymptotic Analysis

Parity-Time Symmetry in Optics

Pyramid Diffraction in 2D PT-symmetry lattice.

Pyramid Diffraction in 2D PT-symmetry lattice.

Continuous spectrum for a PT-symmetric lattice. Here µ is the wave number (eigenvalue) and k is a parameter related to the period of the eigenfunction.  Left: Below "phase transition" with gaps in the continuous spectrum.  Right: At "phase transition" with degeneracies where the gaps closed.

Continuous spectrum for a PT-symmetric lattice. Here µ is the wave number (eigenvalue) and k is a parameter related to the period of the eigenfunction. 

Left: Below "phase transition" with gaps in the continuous spectrum. 

Right: At "phase transition" with degeneracies where the gaps closed.

        Linear Schrodinger operators with complex but parity-time (PT) symmetric potentials have the unintuitive property that their spectra can be completely real. Interest in PT-symmetry originated from a push in Quantum Mechanics to weaken the assumption of hermitian potentials (a non-physical requirement), where a real
spectrum is required to guarantee real energy. However, the same governing equations apply equally to optical lattices, where PT-symmetric potentials can be realized by employing symmetric index guiding and an antisymmetric gain/loss profile. In this setting, the real spectrum manifests as the stable propagation of light waves even in the presence of gain and loss. Furthermore, in materials with a nonlinear index of refraction, families of localized solutions called solitons can exist; a very unusual property for a dissipative system. Unlike Quantum Mechanics, optical PT potentials have been experimentally realized for both spatial and temporal optics. As the strength of the gain-loss profiles is increased the system reaches a point of spontaneous symmetry breaking known as the "phase transition" point. At this phase transition point the band gaps, gaps in the continuous spectrum of the linear operator, all close and a series of degeneracies develop at the points where the Bloch bands intersect. 

My work in this field has involved detailed stability analyses for solitons, finding reduced equations for the envelope of light packets propagating through these lattices, and designing new classes lattices that with real spectra or soliton families.

Stability analysis for solitons in a PT-symmetric lattice. Left: Power Diagram - P is the power of the soliton, and µ is the wave number. Stable solitons (blue line), unstable solitons (dotted red), continuous spectrum (shaded light blue). Top Right: Soliton profiles for the wave numbers indicated in the power diagram. Bottom Right: Spectral stability of the solitons shown above.

Stability analysis for solitons in a PT-symmetric lattice.

Left: Power Diagram - P is the power of the soliton, and µ is the wave number. Stable solitons (blue line), unstable solitons (dotted red), continuous spectrum (shaded light blue).

Top Right: Soliton profiles for the wave numbers indicated in the power diagram.

Bottom Right: Spectral stability of the solitons shown above.

ASYMPTOTICS BEYOND ALL ORDERS (AND BEYOND 1 DIMENSION)

Soliton solution in an optical lattice (solid blue), approximation of the envelopes (dotted purple) Left: Localized soliton. Right: Soliton where exponentially small terms in the asymptotic expanstion result in a growing tail.

Soliton solution in an optical lattice (solid blue), approximation of the envelopes (dotted purple) Left: Localized soliton. Right: Soliton where exponentially small terms in the asymptotic expanstion result in a growing tail.

        The study of asymptotitcs ”beyond-all-orders” of ε is over a century old, dating back to the works of George Stokes. Early development progressed slowly along disparate tracks and only in the last 20 years has the proliferation of more unified efforts been seen. Here the terms of interest are exponentially small in some limiting parameter ε ≪ 1, i.e., on the order of exp(−c/ε) for c > 0 a constant. Since no power series in ε exists for such terms (all their derivatives are zero at ε = 0) the effects of such terms elude traditional asymptotic methods. While it seems that such minuscule contributions should be ignorable, there has been ample evidence to indicate their physical relevance. In 1928, Oppenheimer calculated the exponentially small decay rate for hydrogen atoms under perturbation, part of what lead to the description of quantum tunneling. In 1985, Kruskal and Segur used exponential asymptotics to resolve a long standing question on the existence of dendritic fingers in crystal formation. These are some of the early successes; many more examples maybe be found in the review article. Theoretical work in exponential asymptotics for wave systems has previously been limited to one-dimensional problems. Beyond the importance of this research for expanding our knowledge of how gap solitons bifurcate and behave near the band edge, it also represents a fundamental advancement in the field of exponential asymptotics.  

LineSolitons.png

 I've work on developing a theory for exponential asymptotics in two-dimensional periodic media where solutions are composed of a soliton envelope over a periodic carrier wave. Previously, the use of exponential asymptotics has been limited to one-dimensional problems owing to its inherently difficult nature. The theory has been successful for the study of line solitons where the carrier waves are fully two-dimensional and the envelope is one-dimensional. I've also worked on extending the understanding on exponential asymptotics in 1D, tackling coupled mode problems and several new issues that arise in this case.