Constant-intensity waves and their modulation instability in non-Hermitian potentials

In all of the diverse areas of science where waves play an important role, one of the most fundamental solutions of the corresponding wave equation is a stationary wave with constant intensity. The most familiar example is that of a plane wave propagating in free space. In the presence of any Hermitian potential, a wave's constant intensity is, however, immediately destroyed due to scattering. Here we show that this fundamental restriction is conveniently lifted when working with non-Hermitian potentials. In particular, we present a whole new class of waves that have constant intensity in the presence of linear as well as of nonlinear inhomogeneous media with gain and loss. These solutions allow us to study, for the first time, the fundamental phenomenon of modulation instability in an inhomogeneous environment. Our results pose a new challenge for the experiments on non-Hermitian scattering that have recently been put forward.

Our intuition tells us that stationary waves, which have a constant intensity throughout an extended region of space, can only exist when no obstacles hamper the wave's free propagation. Such an obstacle could be an electrostatic potential for an electronic matter wave, the non--uniform distribution of a dielectric medium for an electromagnetic wave or a wall that reflects an acoustic pressure wave. All of these cases lead to scattering, diffraction and wave interference, resulting in the highly complex variation of a wave's spatial profile that continues to fascinate us in all its different manifestations. Suppressing or merely controlling these effects, which are at the heart of wave physics, is a challenging task, as the quest for a cloaking device [1] or the research in adaptive optics [2], and in wavefront shaping through complex media [3] make us very much aware. New strategies in this direction are thus in high demand and would fall on a fertile ground in many of the different disciplines of science and technology in which wave propagation is a key element.
A new avenue to explore various wave phenomena has recently been opened up when it was realized that waves give rise to very unconventional features when being subject to a suitably chosen spatial distribution of both gain and loss. Such non--Hermitian potential regions [4,5], which serve as sources and sinks for waves, respectively, can give rise to novel wave effects that are impossible to realize with conventional, Hermitian potentials. Examples of this kind, that were meanwhile also realized in the experiment [6--10], are the unidirectional invisibility of a gain--loss potential [11], devices that can simultaneously act as laser and as a perfect absorber [12--14] and resonant structures with unusual features like non--reciprocal light transmission [10] or loss--induced lasing [15--17]. In particular, systems with a so-called parity--time (PT) symmetry [18], where gain and loss are carefully balanced, have recently attracted enormous interest in the context of non--Hermitian photonics [19--23].
Inspired by these recent advances, we show here that for a general class of potentials with gain and loss, it is possible to construct constant--intensity wave solutions. Quite surprisingly, these are solutions to both the paraxial equation of diffraction and the nonlinear Schrödinger equation. In the linear regime, such constant--intensity waves resemble Bessel beams of free space [24]. They carry infinite energy, but retain many of their exciting properties when being truncated by a finite--size input aperture. In the nonlinear regime, this new class of waves turns out to be of fundamental importance, as they provide the first instance to investigate the best known symmetry breaking instability, i.e., the so--called modulational instability (MI) [25--30], in inhomogeneous potentials. Using these new solutions for studying the phenomenon of MI, we find that in the defocusing case, unstable finite size and periodic modes appear causing the wave to disintegrate and to generate a train of complex solitons. In the self--focusing regime, the uniform intensity solution is modulationally unstable for all wavenumbers.

Results
Our starting point is the well known nonlinear Schrödinger equation (NLSE). This scalar wave equation encompasses many aspects of optical wave propagation as well as the physics of matter waves. Specifically, we will consider the NLSE with a general, non--Hermitian potential V x ( ) and a Kerr nonlinearity, The scalar, complex valued function ψ x,z ( ) describes the electric field envelope along a scaled propagation distance z or the wave function of a matter wave as it evolves in time. The nonlinearity can either be self--focusing or defocusing, depending on the sign of g . For this general setting, we now investigate a whole family of recently introduced potentials V x ( ) [31], which are determined by the following relation, where W x ( ) is a given real function. In the special case where W x ( ) is even, the We emphasize, however, that our analysis is also valid for confined, periodic or disordered potentials W x ( ) , which do not necessarily lead to a PT--symmetric form of V x ( ) (but for which gain and loss are always balanced since Im V x ( ) For the entire non--Hermitian family of potentials that are determined by equation (2) (see Methods) we can prove, that the following analytical and stationary constant--intensity wave is a solution to the NLSE in equation (1), notably with a constant and real amplitude A. We emphasize here the remarkable fact that this family of solutions exists in the linear regime ( g = 0 ) as well as for   In order to illustrate the properties of such constant--intensity solutions, we consider the following one--dimensional potentials (not counting the direction of propagation z ) generated by Hermite polynomials choosing W x . The results for n = 1, B = 0.5 are shown in Fig. 1. Note that the corresponding localized 1a) and physically describes a waveguide coupler with optical gain in the middle and lossy arms in the evanescent region around it. If the initial beam is not designed to have the correct phase (as given by equation (3)) but is instead ψ x,0 ( ) = A then the light diffracts fast to the gain region, as can be seen in Fig. 1b. In Figs. 1c,d, we show the results for the constant--intensity solutions with the correct phase, where diffraction is found to be strongly suppressed. Similar to the diffraction--free beams [24], we find that the wider the width of the truncation aperture is at the input facet, z = 0 , the larger is the propagation distance after which the beam starts to diffract (compare Fig. 1c with Fig. 1d). Similar constant--intensity solutions can also be derived in two spatial dimensions x, y . The family of these complex potentials V x, y ( ) and the corresponding constant--intensity solutions ψ x, y,z This expression describes the stationary constant--intensity wave under the perturbation of the eigenfunctions F λ x ( ) and G λ x ( ) with ε << 1 . The imaginary part of λ measures the instability growth rate of the perturbation and determines whether a constant--intensity solution is stable ( λ ∈  ) or unstable ( λ ∈ C ). To leading order in ε we obtain a linear eigenvalue problem for To be more specific, we now apply this analysis to study the MI of constant--intensity waves in PT--symmetric optical lattices [19,20], assuming that W x ( ) is a periodic potential with period α . In particular, we   Fig. 4a. Similarly, for the defocusing nonlinearity (Fig. 4b), and for parameters k = 0.22 and A = 2 , ε = 0.001, we estimate the growth for a propagation distance z = 35 to be around 2 + 0.001⋅ e 0.08⋅35 2~4 .06 , which matches very well with the numerical propagation result of Fig. 4b.

Discussion
Symmetry breaking instabilities belong to the most fundamental concepts of nonlinear sciences. They lead to many rich phenomena such as pattern formation, self--focusing and filamentation just to name a few. The best known symmetry breaking instability is the modulational instability. In its simplest form, it accounts for the break up of a uniform intensity state due to the exponential growth of random perturbations under the combined effect of dispersion/diffraction and nonlinearity.
Most of the early work on MI has been related to classical hydrodynamics, plasma physics and nonlinear optics. Soon thereafter, it was realized that the idea of MI is in fact universal and could exist in other physical systems. For example, spatial optics is one particular area that provides a fertile ground where MI can be theoretically using a broad beam as an initial condition whose nonlinear evolution is monitored.
However, none of these alternatives amount to true MI.
We overcome such difficulties by introducing the new family of constant-intensity waves, which exists in a general class of complex optical potentials. These novel type of waves have constant intensity over all space despite the presence of non--Hermitian waveguide structures. They also remain valid for any sign of Kerr nonlinearity and thus allow us, for the first time, to perform a modulational stability analysis for non--homogeneous potentials. The most appropriate context to study the MI of such solutions is that of PT--symmetric optics [6--11,14,19--22]. We find that in the self--focusing regime, the waves are always unstable, while in the defocusing regime the instability appears for specific values of Bloch momenta. In both regimes (self--focusing, defocusing), the constant--intensity solutions break up into filaments following a complex nonlinear evolution pattern.
We expect that our predictions can be verified by combining recent advances in shaping complex wave fronts [3] with new techniques to fabricate non--Hermitian scattering structures with gain and loss [7--10]. Since the precise combination of gain and loss in the same device is challenging, we suggest using passive structures with only loss in the first place. For such suitably designed passive systems [6] solutions exist that feature a pure exponential decay in the presence of an inhomogeneous index distribution. This exponential tail should be observable in the transmission intensity as measured at the output facet of the system. Another possible direction is that of considering evanescently coupled waveguide systems. Using coupled mode theory one can analytically show, that our constant--intensity waves exist also in such discrete systems with distributed gain and loss all over the waveguide channels. In this case the constant--intensity waves are not radiation modes but rather supermodes of the coupled system. With these simplifications an experimental demonstration of our proposal should certainly be within reach of current technology.

Constant--Intensity solutions of the non--Hermitian NLSE
We prove here analytically that stationary constant--intensity solutions of the NLSE exist for a wide class of non--Hermitian optical potentials (which are not necessarily PT--symmetric). We are looking for solutions of the NLSE of the form is the complex field profile and µ the corresponding propagation constant, to be found. By substitution of this last relation into equation (1) we get the following nonlinear equation

the last nonlinear equation can be separated in real and
imaginary parts. As a result we get the following two coupled equations for the real and the imaginary part of the complex potential, respectively: where Θ x ≡ dΘ dx and ρ x ≡ dρ dx . By choosing V R x ( ) = Θ x 2 , and by solving equation (8) to , we can reduce the above system of coupled nonlinear ordinary differential equations to only one, namely ρ xx − µ ρ + gρ 3 = 0 . If we assume now a constant amplitude solution, namely ρ x ( ) = Α = const. , we have the following general solution for any real--valued phase function Θ x We can easily see that in the special case where W x ( ) is even, the actual optical potential V x ( ) is PT--symmetric.

Modulation instability analysis in optical potentials
In order to study the modulation instability of the uniform intensity states for any given W , we consider small perturbation of the solutions of the NLSE of the form: Here, F λ x ( ) and G λ x ( ) are the perturbation eigenfunctions and the imaginary part of λ measures the instability growth rate of the perturbation. By defining the T , we obtain the following linear eigenvalue problem (to leading order in ε ): where the operator matrix and the related linear operators are defined by the relationships: (13) So far the above discussion is general and applies to any (periodic or not) potential W x ( ) that is real.

Properties of the PT--symmetric optical lattice
We choose a specific example of a well--known non--Hermitian potential, i.e., that of a PT--symmetric optical lattice [19,20]. More specifically for the particular we get the corresponding optical potential and constant-- It is obvious that this potential is PT--symmetric since it satisfies the symmetry In order for the constant--intensity solution to be periodic in x with the same period as the lattice, the constant term V 0 must be quantized, namely Even though this is the case, this term is important because it also appears in the real part of V x ( ) . It determines if the PT--lattice is in the broken or in the unbroken phase, regarding its eigenspectrum. For the considered parameters, the lattice is below the exceptional point and its eigenvalue spectrum is real.

Plane Wave Expansion Method
Even though our methodology is general, we apply it to study the modulation instability of constant--intensity waves in PT--symmetric optical lattices. In particular we consider the periodic W x ( ) (with period α ) that leads to equations (14) and (15).
where q = 2π α is the dual lattice spacing. Substitution of equation (16) and equation (17) into the eigenvalue problem of equation (10), leads us to the following nonlocal system of coupled linear eigenvalue equations for the perturbation modes u n ,υ n and the band eigenvalue λ k ( ) that depends on the Bloch momentum k : where U n,m k  (18) becomes: where a n k

Direct Eigenvalue Method
An alternative way (instead of the plane wave expansion method that was used above) of solving the infinite dimensional eigenvalue problem of equation (10) where the Bloch momentum takes values in the first Brillouin zone k ∈ −2π α ,2π α , in order to calculate the growth rate of the random perturbations for every value of the Bloch momentum. We have checked explicitly that both approaches, i.e., the plane wave expansion method based on equation (19) and the direct eigenvalue analysis based on equation (20), give the same results.
Analytical results in the shallow lattice limit In the limit of a shallow optical lattice (the refractive index difference between the periodic modulation and the background refractive index value is very small), one can gain substantial insight into the structure of the unstable band eigenvalues by deriving an approximate analytical expression for λ k ( ) valid near the Bragg points based on degenerate perturbation theory. These points are given by: where k c = 2A 2 and n = 1,2,3,... . The above analytical formulas lead to an excellent match with the numerical approaches in the shallow lattice limit ( V 1 << 1 ).

Complex filament formation
In order to understand better the complex filament formation of a constant--intensity solution in a PT--symmetric lattice for both signs of nonlinearity, we performed nonlinear wave propagation simulations based on a spectral fast Fourier approach of the integrating factors method for NLSE. The initial conditions that were used to examine the filament formation were based on the perturbation eigenmode profiles. In particular, we have at z = 0 , the following initial field profile in terms of Bloch eigenfunctions u x ( ) ,υ x ( ) :