Abstract
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 nonHermitian potentials. In particular, we present a whole 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 the fundamental phenomenon of modulation instability in an inhomogeneous environment. Our results pose a new challenge for the experiments on nonHermitian scattering that have recently been put forward.
Introduction
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 nonuniform 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. 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 nonHermitian 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, which were meanwhile also realized in the experiment^{6,7,8,9,10}, are the unidirectional invisibility of a gain–loss potential^{11}, devices that can simultaneously act as laser and as a perfect absorber^{12,13,14} and resonant structures with unusual features like nonreciprocal light transmission^{10} or lossinduced lasing^{15,16,17}. In particular, systems with a socalled paritytime () symmetry^{18}, where gain and loss are carefully balanced, have recently attracted enormous interest in the context of nonHermitian photonics^{19,20,21,22,23,24}.
Inspired by these recent advances, we show here that for a general class of potentials with gain and loss, it is possible to construct constantintensity wave solutions. Quite surprisingly, these are solutions to both the paraxial equation of diffraction and the nonlinear Schrödinger equation (NLSE). In the linear regime, such constantintensity waves resemble Bessel beams of free space^{25}. They carry infinite energy, but retain many of their exciting properties when being truncated by a finitesize input aperture. In the nonlinear regime, this class of waves turns out to be of fundamental importance, as they provide the first instance to investigate the best known symmetry breaking instability, that is, the socalled modulational instability (MI)^{26,27,28,29,30,31}, in inhomogeneous potentials. Using these solutions for studying the phenomenon of MI, we find that in the selffocusing case, unstable periodic modes appear causing the wave to disintegrate and to generate a train of complex solitons. In the defocusing regime, the uniform intensity solution is modulationally unstable for some wavenumbers.
Results
Onedimensional constantintensity waves
Our starting point is the wellknown 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, nonHermitian 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 selffocusing or defocusing, depending on the sign of g. For this general setting, we now investigate a whole family of recently introduced potentials V(x) (ref. 32), 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 actual optical potential V(x) turns out to be symmetric, since V(x)=V*(−x). We emphasize, however, that our analysis is also valid for confined, periodic or disordered potentials W(x), which do not necessarily lead to a symmetric form of V(x) (but for which gain and loss are always balanced since in the case of localized or periodic potentials.). For the entire nonHermitian family of potentials that are determined by equation 2 (see Methods), we can prove that the following analytical and stationary constantintensity 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 arbitrary strength of nonlinearity (g=±1). Under linear conditions (g=0), the constantintensity wave given by equation 3 is one of the radiation eigenmodes (not confined) of the potential with propagation constant equal to zero. (nonzero propagation constants are obtained by adding a constant term to the potential in equation 2). Another interesting point to observe is that the above solutions exist only for nonHermitian potentials, since for W(x)→0 we also have V(x)→0. Therefore, these families of counterintuitive solutions are a direct consequence of the nonHermitian nature of the involved potential V(x) and as such exist only for these complex structures with gain and loss. The fact that such constantintensity waves are a direct generalization of the fundamental concept of freespace plane waves to complex environments can be easily understood by setting W(x)=c_{1}=const. with In this case, the potential V(x)=c_{1}^{2} corresponds to a bulk dielectric medium for which the constantintensity waves reduce to the plane waves of homogeneous space, ψ=. It can also be shown that the potential W(x) determines the power flow in the transverse plane that physically forces the light to flow from the gain to the loss regions. In particular, the transverse normalized Poynting vector defined as S=(i/2)(ψ∂ψ*/∂x−ψ* ∂ψ/∂x) takes on the following very simple form: S=A^{2}W(x).
To illustrate the properties of such constantintensity solutions, we consider the following onedimensional potentials (not counting the direction of propagation z) generated by Hermite polynomials choosing . The results for vanishing nonlinearity (g=0) and n=1, B=0.5 are shown in Fig. 1. Note that the corresponding localized optical potential V(x) is not symmetric (Fig. 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 Fig. 1c,d, we show the results for the constantintensity solutions with the correct phase, where diffraction is found to be strongly suppressed. Similar to the diffractionfree beams^{25}, 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).
Twodimensional constantintensity waves
Similar constantintensity solutions can also be derived in two spatial dimensions x,y. The family of these complex potentials V(x,y) and the corresponding constantintensity solutions ψ(x,y,z) of the twodimensional NLSE are:
where with W_{x},W_{y} being real functions of x,y and C being any smooth open curve connecting an arbitrary point (a,b) to any different point (x,y). As in the onedimensional case, these solutions are valid in both the linear and the nonlinear domain. For the particular case of irrotational flow W_{x}=cosx siny, W_{y}=cosy sinx, the resulting periodic potential V(x,y) is that of an optical lattice with alternating gain and loss waveguides. The imaginary part of such a lattice is shown in Fig. 2a. In Fig. 2b, we display the diffraction of a constantintensity beam with the correct phase (as in equation 6) launched onto such a linear lattice (g=0) through a circular aperture. As we can see, the beam maintains its constant intensity over a remarkably long distance. In Fig. 2c, we present the corresponding transverse Poynting vector defined as , illustrating that the wave flux follows stream line patterns from the gain to the loss regions. Once the finite beam starts to diffract, this balanced flow is disturbed and the waves are concentrated in the gain regions.
Modulation instability of constantintensity waves
Quite remarkably, the above diffractionfree and uniform intensity waves are also solutions of the NLSE for both the selffocusing and the defocusing case. This allows us to study the modulation instability of such solutions under small perturbations. In particular, we are interested in understanding the linear stability of the solutions of equation 1 of the form , where the phase function is θ(x,z)=gA^{2}z+∬W(x) dx. This expression describes the stationary constantintensity wave under the perturbation of the eigenfunctions F_{λ}(x) and G_{λ}(x) with ɛ<<1. The imaginary part of λ measures the instability growth rate of the perturbation and determines whether a constantintensity solution is stable () or unstable (λ∈). To leading order in ɛ, we obtain a linear eigenvalue problem for the twocomponent perturbation eigenmodes ϕ_{λ}(x)≡[F_{λ}(x) G_{λ}(x)]^{T}, the eigenvalues of which are λ. This eigenvalue problem and the operator matrix are defined in the Methods section. So far the presented MIanalysis is general and can be applied to any real W(x) (periodic or not). To be more specific, we now apply this analysis to study the MI of constantintensity waves in symmetric optical lattices^{19,20}, assuming that W(x) is a periodic potential with period α. In particular, we consider the example of a symmetric photonic lattice where (the resulting optical potential and the corresponding constantintensity solution are given in the Methods section). For all the subsequent results we will always assume (without loss of generality) that V_{0}=4 and V_{1}=0.2. It is important to note here that for these parameters our lattice V(x) is in the socalled ‘unbroken symmetric phase’ with only real propagation constants (see Methods). In the broken phase, some of these eigenvalues are complex and the instabilities due to nonlinearity are physically expected. As W(x) is periodic we can expand the perturbation eigenvectors ϕ_{λ}(x) in a Fourier series and construct numerically the bandstructure of the stability problem (different from the physical bandstructure of the optical lattice). Based on the above, the Floquet–Bloch theorem implies that the eigenfunctions ϕ_{λ}(x) can be written in the form ϕ_{λ}(x)=φ(x,k)e^{ikx}, where φ(x,k)=φ(x+α,k) with k being the Bloch momentum of the stability problem (see Methods). The corresponding results are illustrated in Fig. 3a,b for a selffocusing nonlinearity (g=1) and for different values of the amplitude A. More specifically, we show the instability growth rate Im{λ(k)} as a function of the Bloch wavenumber k of the perturbation eigenvector in the first half Brillouin zone. We see that the constantintensity waves are linearly unstable for any value of the Bloch momentum of the imposed perturbation and that instability band gaps form due to the periodic nature of the imposed perturbations. The different bands are illustrated in Fig. 3a,b with different colours.
The situation is different for the defocusing case (g=−1) where the results are presented in Fig. 3c,d. For some values of k, the constantintensity solutions are linearly stable and their instability dependence forms bands reminiscent of the bands appearing in conventional MI results for bulk or periodic potentials^{26,29,30}, but quite different and profoundly more complex.
To understand the physical consequences of such instabilities and how they lead to filament formation, we have performed independent numerical simulations for the dynamics of the constantintensity solutions against specific perturbations with results being shown in Fig. 4. More specifically, we examine the intensity evolution of a constantintensity solution when being perturbed by a specific Floquet–Bloch stability mode. In other words, at the input of the waveguide structure at z=0, we have , with phase and we want to know whether the linear stability analysis captures the exponential growth of the imposed perturbations correctly. For the considered symmetric lattice with selffocusing nonlinearity, we examine the nonlinear dynamics of the constantintensity solution and the result is presented in Fig. 4a. For a perturbation eigenmode with Bloch momentum k=0 and A=1, ɛ=0.01, we can see from Fig. 3b that Im{λ(0)}∼1. Therefore, we can estimate the growth for a propagation distance of z=5 to be around 1+0.01·e^{1·5}^{2}∼6.1, which agrees very well with the dynamical simulation of Fig. 4a. Similarly, for the defocusing nonlinearity (Fig. 4b), and for parameters k=0.22 and A=2, ɛ=0.001, we estimate the growth for a propagation distance z=35 to be around 2+0.001·e^{0.046·35}^{2}∼4.02, 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, selffocusing and filamentation just to name a few. The best known symmetry breaking instability is the MI. 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 modelled (mainly within the framework of the NLSE) and experimentally realized. Indeed, temporal MI has been observed in optical fibres as well as its spatial counterpart in nonlinear Kerr, quadratic, biased photorefractive media with both coherent and partially coherent beams and in discrete waveguide arrays^{26,27,28,29,30,31}. Up to now, most of the mathematical modelling of MI processes has been focused on wave propagation in homogeneous nonlinear media, where an exact constantintensity solution for the underlying governing (NLSE type) equations can be obtained. In this context, inhomogeneities are considered problematic as they provide severe conceptual limitations that hinder one from constructing a constantintensity solution, a necessary condition to carry out the MI analysis. Several directions have been proposed to bypass this limitation. They can be organized into three distinct categories: (i) the tight binding approach, in which case the NLSE equation in the presence of an external periodic potential is replaced by its discrete counterpart that, in turn, admits an exact plane wave solution (a discrete Floquet–Bloch mode), (ii) MI of nonlinear Bloch modes and (iii) direct numerical simulations 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 above family of constantintensity waves, which exists in a general class of complex optical potentials. These type of waves have constant intensity over all space despite the presence of nonHermitian waveguide structures. They also remain valid for any sign of Kerr nonlinearity and thus allow us to perform a modulational stability analysis for nonhomogeneous potentials. The most appropriate context to study the MI of such solutions is that of symmetric optics^{6,7,8,9,10,11,14,19,20,21,22,24}. We find that in the selffocusing regime, the waves are always unstable, whereas in the defocusing regime the instability appears for specific values of Bloch momenta. In both regimes (selffocusing, defocusing), the constantintensity 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 nonHermitian scattering structures with gain and loss^{7,8,9,10}. As 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 constantintensity waves exist also in such discrete systems with distributed gain and loss all over the waveguide channels. In this case, the constantintensity 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.
Methods
Constantintensity solutions of the nonHermitian NLSE
We prove here analytically that stationary constantintensity solutions of the NLSE exist for a wide class of nonHermitian optical potentials (which are not necessarily symmetric). We are looking for solutions of the NLSE of the form ψ(x,z)=f(x)exp(iμz), where f(x) 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 −μf+f_{xx}+V(x)f+gf^{2} f=0. We assume a solution of the form f(x)=ρ(x)exp[iΘ(x)], with ρ(x),Θ(x) real functions of position x. Since V(x)=V_{R}(x)+iV_{I}(x), 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 get , 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 realvalued phase function Θ(x): ψ(x,z)=A exp[iΘ(x)+igA^{2}z], where , and V_{I}(x)=−Θ_{xx}. By setting W(x)≡Θ_{x}(x), we can write the optical potential, for which the constantintensity solution exists, as V(x)=W^{2}(x)−idW/dx, and the constantintensity solution itself reads as follows . We can easily see that in the special case where W(x) is even, the actual optical potential V(x) is symmetric.
Modulation instability analysis in optical potentials
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: , where θ(x,z)=gA^{2}z+∬W(x)dx and ɛ<<1. 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 perturbation twocomponent eigenmode ϕ_{λ}(x)≡[F_{λ}(x) G_{λ}(x)]^{T}, we obtain the following linear eigenvalue problem (to leading order in ɛ):
where the operator matrix is defined by the following expression
and the related linear operators are defined by the relationships:
So far the above discussion is general and applies to any (periodic or not) potential W(x) that is real.
Properties of the symmetric optical lattice
We choose a specific example of a wellknown nonHermitian potential, that is, that of a symmetric optical lattice^{19,20}. More specifically for the particular case of , we get the corresponding optical potential and constantintensity wave:
It is obvious that this potential is symmetric as it satisfies the symmetry relation V(x)=V*(−x). In order for the constantintensity solution to be periodic in x with the same period as the lattice, the term V_{0} must be quantized, namely V_{0}=0,±2,±4,.... This constant term that appears in the potential W(x) results in another constant term in the actual potential V(x) and can be removed (with respect to the NLSE) with a gauge transformation of the type . 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 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 constantintensity waves in symmetric optical lattices. In particular, we consider the periodic W(x) (with period α) that leads to equations 14 and 15. As we are interested in the MIs of the constantintensity wave solution of the NLSE under selffocusing and defocusing nonlinearities, we want the lattice V(x) to be in the unbroken phase. In the broken phase some eigenvalues are complex and the instabilities are physically expected. That is the reason why we choose (without loss of generality) the parameters V_{0}=4 and V_{1}=0.2, which lead to an ‘unbroken’ spectrum with real eigenvalues. As W(x) is periodic, we can expand the perturbation eigenvectors ϕ_{λ}(x) in Fourier series and construct numerically the bandstructure of the stability problem. So at this point, we have to distinguish between the physical bandstructure of the problem and the perturbation bandstructure of the stability problem of equation 9. Based on the above, the Floquet–Bloch theorem implies that the eigenfunctions ϕ_{λ}(x) can be written in the form ϕ_{λ}(x)=φ(x,k)e^{ikx}, where φ(x,k)=φ(x+α,k) with k being the Bloch momentum of the stability problem. Applying the plane wave expansion method, the wavefunctions φ(x,k) and the potential W(x) can be expanded in Fourier series as:
where q=2π/α is the dual lattice spacing. Substitution of equations 16 and 17 into the eigenvalue problem of equation 9, leads us to the following nonlocal system of coupled linear eigenvalue equations for the perturbation coefficients u_{n},υ_{n} and the band eigenvalue λ(k) that depends on the Bloch momentum k:
where U_{n,m}(k)=2[q(n−m)+k]W_{m} and Ω_{n}(k)=gA^{2}−(qn+k)^{2}. The family of constantintensity wave solutions of the NLSE is modulationally unstable if there exists a wave number k for which Im{λ(k)}≠0, while they are stable if λ(k) is real. For our case, the periodic function W is given by for which equation 18 becomes:
where a_{n}(k)=V_{1}(qn+k), μ_{n}(k)=Ω_{n}(k)−V_{0}(qn+k) and ν_{n}(k)=Ω_{n}(k)+V_{0}(qn+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 9 is to directly apply the Floquet–Bloch theorem on the eigenfunctions ϕ_{λ}(x), employ the BornvonKarman boundary conditions (periodic boundary conditions at the end points of the finite lattice) and construct numerically the bandstructure of the instability growth for every value of the Bloch momentum. In particular, the eigenfunctions can be written as ϕ_{λ}(x)=[u(x)e^{ikx} υ(x)e^{ikx}]^{T}, where u(x)=u(x+α), υ(x)=υ(x+α). Substituting this form of the perturbation eigenfunctions into equation 9, we get the following eigenvalue problem:
where the Bloch momentum takes values in the first Brillouin zone k∈[−π/α,π/α] and the operators are defined by equations 11, 12, 13. By applying the finite difference method, we restrict our analysis to one unit cell x∈[−α/2,α/2], 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, that is, 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 perturbation theory. These points are given by (for selffocusing nonlinearity):
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_{0}, V_{1}<<1).
Complex filament formation
To understand better the complex filament formation of a constantintensity solution in a 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 , for specific values of Bloch momentum k and the constantintensity wave amplitude A.
Additional information
How to cite this article: Makris, K. G. et al. Constantintensity waves and their modulation instability in nonHermitian potentials. Nat. Commun. 6:7257 doi: 10.1038/ncomms8257 (2015).
References
Leonhardt, U. Optical conforming mapping. Science 312, 1777–1780 (2006).
Tyson, R. K. Principles of Adaptive Optics 3rd edn CRC Press (2001).
Mosk, A.P., Lagendijk, A., Lerosey, G. & Fink, M. Controlling waves in space and time for imaging and focusing in complex media. Nat. Phys. 6, 283–292 (2012).
Moiseyev, N. NonHermitian Quantum Mechanics Cambridge Univ. (2011).
Bender, C. M. Making sense of nonHermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007).
Guo, A. et al. Observation of symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009).
Rüter, C.E. et al. Observation of paritytime symmetry in optics. Nat. Phys. 6, 192–195 (2010).
Regensburger, A. et al. Paritytime synthetic photonic lattices. Nature 488, 167–171 (2012).
Feng, L. et al. Experimental demonstration of a unidirectional reflectionless paritytime metamaterial at optical frequencies. Nat. Mat. 12, 108–113 (2013).
Peng, B. et al. Paritytimesymmetric whispering gallery microcavities. Nat. Phys. 10, 394–398 (2014).
Lin, Z. et al. Unidirectional invisibility induced by symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011).
Chong, Y. D., Ge, L., Cao, H. & Stone, A. D. Coherent perfect absorbers: timereversed lasers. Phys.Rev. Lett. 105, 053901 (2011).
Wan, W. et al. Timereversed lasing and interferometric control of absorption. Science 331, 889–892 (2011).
Sun, Y., Tan, W., Li, H.Q., Li, J. & Chen, H. Experimental demonstration of a coherent perfect absorber with phase transition. Phys. Rev. Lett. 112, 143903 (2014).
Liertzer, M. et al. Pumpinduced exceptional points in lasers. Phys. Rev. Lett. 108, 173901 (2012).
Brandstetter, M. et al. Reversing the pumpdependence of a laser at an exceptional point. Nat. Commun 5, 4034 (2014).
Peng, B. et al. Lossinduced suppression and revival of lasing. Science 346, 328–332 (2014).
Bender, C. M., Brody, D. C. & Jones, H. F. Complex extension of Quantum Mechanics. Phys. Rev. Lett. 89, 270401 (2002).
Makris, K. G., ElGanainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics of symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008).
Musslimani, Z. H., Makris, K. G., ElGanainy, R. & Christodoulides, D. N. Optical solitons in periodic potentials. Phys. Rev. Lett. 100, 030402 (2008).
Chong, Y. D., Ge, L. & Stone, A. D. symmetry breaking and laser absorber in optical systems. Phys. Rev.Lett. 106, 093902 (2011).
Kottos, T. Optical physics: broken symmetry makes light work. Nat. Phys. 6, 166–167 (2010).
Makris, K. G., Ge, L. & Türeci, H. E. Anomalous transient amplification in nonnormal photonic media. Phys. Rev. X 4, 041044 (2014).
Ambichl, P., Makris, K. G., Ge, L., Chong, Y., Stone, A. D. & Rotter, S. Breaking of PTSymmetry in Bounded and Unbounded Scattering Systems. Phys. Rev. X. 3, 041030 (2013).
Durnin, J., Miceli, J. J. Jr. & Eberly, J. H. Diffractionfree beams. Phys. Rev. Lett. 58, 1499 (1987).
Tai, K., Hasegawa, A. & Tomita, A. Observation of modulational instability in optical fibers. Phys. Rev. Lett. 56, 135 (1986).
Malendevich, R., Jankovic, L., Stegeman, G. & Aitchison, J. S. Spatial modulation instability in a Kerr slab waveguide. Opt. Lett. 26, 1879 (2001).
Kip, D., Soljacic, M., Segev, M., Eugenieva, E. & Christodoulides, D.N. Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000).
Meier, J. et al. Experimental observation of discrete modulational instability. Phys. Rev. Lett. 92, 163902 (2004).
Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M. Nonlinearly induced transition in photonic systems. Phys. Rev. Lett. 111, 263901 (2013).
Zakharov, V. E. & Gelash, A. A. Nonlinear stage of modulation instability. Phys. Rev. Lett. 111, 054101 (2013).
Wadati, M. Construction of paritytime symmetric potential through the soliton. theory. J. Phys. Soc. Jpn. 77, 074005 (2008).
Acknowledgements
K.G.M. is supported by the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007–2013) under REA grant agreement number PIOFGA2011–303228 (project NOLACOME). Z.H.M. was supported in part by NSF Grant No. DMS0908599. The work of D.N.C. was partially supported by NSF Grant No. ECCS1128520 and the Air Force Office of Scientific Research Grants Nos. FA95501210148 and FA95501410037. S.R. acknowledges financial support by the Austrian Science Fund (FWF) through Project SFB NextLite (F49P10) and Project GePartWave (I1142).
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All authors have contributed to the development and/or implementation of the concept, discussed and analysed the results. K.G.M. and Z.M. carried out the analytical and numerical calculations. S.R. and D.C. provided theoretical and conceptual support. K.G.M., Z.M. and S.R. wrote the manuscript with input from all authors.
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Makris, K., Musslimani, Z., Christodoulides, D. et al. Constantintensity waves and their modulation instability in nonHermitian potentials. Nat Commun 6, 7257 (2015). https://doi.org/10.1038/ncomms8257
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DOI: https://doi.org/10.1038/ncomms8257
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