The development of new artificial structures and materials is today one of the major research challenges in optics. In most studies so far, the design of such structures has been based on the judicious manipulation of their refractive index properties. Recently, the prospect of simultaneously using gain and loss was suggested as a new way of achieving optical behaviour that is at present unattainable with standard arrangements. What facilitated these quests is the recently developed notion of 'parity-time symmetry' in optical systems, which allows a controlled interplay between gain and loss. Here we report the experimental observation of light transport in large-scale temporal lattices that are parity-time symmetric. In addition, we demonstrate that periodic structures respecting this symmetry can act as unidirectional invisible media when operated near their exceptional points. Our experimental results represent a step in the application of concepts from parity-time symmetry to a new generation of multifunctional optical devices and networks.

Parity–time synthetic photonic lattices

The concept of parity-time symmetric systems is rooted in non-Hermitian quantum mechanics where complex potentials obeying this symmetry could exhibit real spectra. The concept has applications in many fields of physics, e.g., in optics, metamaterials, acoustics, Bose-Einstein condensation, electronic circuitry, etc. The inclusion of nonlinearity has led to a number of new phenomena for which no counterparts exist in traditional dissipative systems. Several examples of nonlinear parity-time symmetric systems in different physical disciplines are presented and their implications discussed.

Nonlinear waves in PT -symmetric systems

Solitons, nonlinear self-trapped wavepackets, have been extensively studied in many and diverse branches of physics such as optics, plasmas, condensed matter physics, fluid mechanics, particle physics and even astrophysics. Interestingly, over the past two decades, the field of solitons and related nonlinear phenomena has been substantially advanced and enriched by research and discoveries in nonlinear optics. While optical solitons have been vigorously investigated in both spatial and temporal domains, it is now fair to say that much soliton research has been mainly driven by the work on optical spatial solitons. This is partly due to the fact that although temporal solitons as realized in fiber optic systems are fundamentally one-dimensional entities, the high dimensionality associated with their spatial counterparts has opened up altogether new scientific possibilities in soliton research. Another reason is related to the response time of the nonlinearity. Unlike temporal optical solitons, spatial solitons have been realized by employing a variety of noninstantaneous nonlinearities, ranging from the nonlinearities in photorefractive materials and liquid crystals to the nonlinearities mediated by the thermal effect, thermophoresis and the gradient force in colloidal suspensions. Such a diversity of nonlinear effects has given rise to numerous soliton phenomena that could otherwise not be envisioned, because for decades scientists were of the mindset that solitons must strictly be the exact solutions of the cubic nonlinear Schrodinger equation as established for ideal Kerr nonlinear media. As such, the discoveries of optical spatial solitons in different systems and associated new phenomena have stimulated broad interest in soliton research. In particular, the study of incoherent solitons and discrete spatial solitons in optical periodic media not only led to advances in our understanding of fundamental processes in nonlinear optics and photonics, but also had a very important impact on a variety of other disciplines in nonlinear science. In this paper, we provide a brief overview of optical spatial solitons. This review will cover a variety of issues pertaining to self-trapped waves supported by different types of nonlinearities, as well as various families of spatial solitons such as optical lattice solitons and surface solitons. Recent developments in the area of optical spatial solitons, such as 3D light bullets, subwavelength solitons, self-trapping in soft condensed matter and spatial solitons in systems with parity-time symmetry will also be discussed briefly.

http://www.physics.sfsu.edu/~laser/pdf/rop347172p20.pdf

Optical spatial solitons: historical overview and recent advances

We review quantum mechanical and optical pseudo- Hermitian systems with an emphasis on PT-symmetric systems important for optics and electrodynamics. One of the most

https://iopscience.iop.org/article/10.3367/UFNe.0184.201411b.1177/pdf

PT-symmetry in optics

Families of analytical solutions are found for symmetric and antisymmetric solitons in a dual-core system with Kerr nonlinearity and parity-time (PT)-balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT-symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the “supersymmetric” case, with equal coefficients of gain, loss, and intercore coupling. These solitons feature a subexponential instability, which may be suppressed by periodic switching (“management”).

Stability of solitons in parity-time-symmetric couplers

Dynamics of a chain of interacting parity-time-invariant nonlinear dimers is investigated. A dimer is built as a pair of coupled elements with equal gain and loss. A relation between stationary soliton solutions of the model and solitons of the discrete nonlinear Schrodinger (DNLS) equation is demonstrated. Approximate solutions for solitons whose width is large in comparison to the lattice spacing are derived, using a continuum counterpart of the discrete equations. These solitons are mobile, featuring nearly elastic collisions. Stationary solutions for narrow solitons, which are immobile due to the pinning by the effective Peierls-Nabarro potential, are constructed numerically, starting from the anticontinuum limit. The solitons with the amplitude exceeding a certain critical value suffer an instability leading to blowup, which is a specific feature of the nonlinear parity-time-symmetric chain, making it dynamically different from DNLS lattices. A qualitative explanation of this feature is proposed. The instability threshold drops with the increase of the gain-loss coefficient, but it does not depend on the lattice coupling constant, nor on the soliton's velocity.

/pdf/solitons-in-a-chain-of-parity-time-invariant-dimers-3mck885m85.pdf

Solitons in a chain of parity-time-invariant dimers.

We demonstrate that light propagation in waveguide arrays that include PT-symmetric structures can exhibit strongly nonlocal sensitivity to topology of the array at fixed other parameters. We consider an array composed of lossless waveguides, that includes a pair of PT-symmetric waveguides with balanced gain and loss, and reveal that PT-symmetry breaking thresholds are different for planar and circular array configurations. These results demonstrate that PT-symmetric structures can offer new regimes for optical beam shaping compared to conservative structures.

/pdf/nonlocality-in-pt-symmetric-waveguide-arrays-with-gain-and-3rvxuyl7pg.pdf

Nonlocality in PT-symmetric waveguide arrays with gain and loss.

We study the scattering of linear and nonlinear waves in a long waveguide array with a parity-time ($\mathrm{PT}$)-symmetric defect created by two waveguides with balanced gain and loss. We present exact solutions for the scattering of linear waves on such a defect, and then demonstrate numerically that the linear theory can describe, with a good accuracy, the soliton scattering in the case of weak nonlinearity. We reveal that the reflected and transmitted linear and nonlinear waves can be amplified substantially after interaction with the $\mathrm{PT}$-symmetric defect thus allowing an active control of the wave scattering in the array.

/pdf/scattering-of-linear-and-nonlinear-waves-in-a-waveguide-58tfckah29.pdf

Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect

We study wave propagation in linear arrays composed of pairs of conjugate waveguides with balanced gain and loss, that is, arrays of $\mathcal{PT}$-symmetric couplers, where the linear spectrum is known to feature high-frequency and low-frequency branches. We introduce a domain wall by switching the gain and loss in half of the array and analyze the scattering of linear waves on this defect. The analysis reveals two major effects: amplification of both reflected and transmitted waves and excitation of the reflected and transmitted low-frequency and high-frequency waves by the incident high-frequency and low-frequency waves, respectively.

Wave scattering on a domain wall in a chain of PT-symmetric couplers

Photonic structures composed of coupled waveguides with loss and gain regions offer new possibilities for shaping optical beams and pulses compared to conservative structures [1, 2]. Such structures can be designed as optical analogues of complex parity-time (or PT) symmetric potentials, which can have a real spectrum corresponding to the conservation of power for optical eigenmodes, for the magnitude of gain/loss below a certain threshold. When gain/loss is increased above the threshold, PT symmetry breaking occurs, corresponding to amplification of certain eigenmodes. The PT symmetry breaking transition was recently demonstrated experimentally for directional couplers composed of two waveguides [3, 4]. PT-symmetric potentials appear in many physical contexts, and one feature actively investigated in the context of quantum theories is the property of nonlocality, where PT-defect dynamics can be sensitive to potential profile at distant locations, and it was questioned the observability of such behavior in real physical systems [5, 6]. In this work, we reveal that effective nonlocality of PT-symmetric structures with gain and loss elements can lead to pronounced differences for optical beam dynamics in arrays of coupled waveguides with the same characteristics but different topology.

/pdf/nonlocal-effects-in-waveguide-arrays-with-pt-symmetric-2mfvt6fnv5.pdf

Sergey V. Suchkov

Papers

Solitons in a chain of parity-time-invariant dimers.

Nonlocality in PT-symmetric waveguide arrays with gain and loss.

Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect

Wave scattering on a domain wall in a chain of PT-symmetric couplers

Nonlocal effects in waveguide arrays with PT-symmetric defects