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

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 non-Hermitian 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 non-Hermitian scattering that have recently been put forward.

ARTICLE

Received 13 Oct 2014 | Accepted 21 Apr 2015 | Published 8 Jul 2015

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 waves 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 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 non-Hermitian scattering that have recently been put forward.

DOI: 10.1038/ncomms8257

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

K.G. Makris1,2, Z.H. Musslimani3, D.N. Christodoulides4 & S. Rotter1

1 Institute for Theoretical Physics, Vienna University of Technology, Vienna A-1040, Austria. 2 Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544 USA. 3 Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510, USA. 4 College of OpticsCREOL, University of Central Florida, Orlando, Florida 32816, USA. Correspondence and requests for materials should be addressed to K.G.M. (email: mailto:[email protected]

Web End [email protected] ; mailto:[email protected]

Web End [email protected] ).

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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

waves 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 reects an acoustic pressure wave. All of these cases lead to scattering, diffraction and wave interference, resulting in the highly complex variation of a waves spatial prole 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 device1 or the research in adaptive optics2, and in wavefront shaping through complex media3 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 non-Hermitian potential regions4,5, 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 experiment610, are the unidirectional invisibility of a gainloss potential11, devices that can simultaneously act as laser and as a perfect absorber1214 and resonant structures with unusual features like non-reciprocal light transmission10 or loss-induced lasing1517. In particular, systems with a so-called parity-time (PT ) symmetry18, where gain and loss are carefully balanced,

have recently attracted enormous interest in the context of non-Hermitian photonics1924.

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 Schrdinger equation (NLSE). In the linear regime, such constant-intensity waves resemble Bessel beams of free space25. They carry innite energy, but retain many of their exciting properties when being truncated by a nite-size input aperture. In the nonlinear regime, this class of waves turns out to be of fundamental importance, as they provide the rst instance to investigate the best known symmetry breaking instability, that is, the so-called modulational instability (MI)2631, in inhomogeneous potentials. Using these solutions for studying the phenomenon of MI, we nd that in the self-focusing 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.

ResultsOne-dimensional constant-intensity waves. Our starting point is the well-known NLSE. This scalar wave equation encompasses many aspects of optical wave propagation as well as the physics of matter waves. Specically, we will consider the NLSE with a general, non-Hermitian potential V(x) and a Kerr nonlinearity,

i @c

@z

@2c

@x2 V x

c g c

dx 2

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

PT -symmetric, since V(x) V*( x). We emphasize, however,

that our analysis is also valid for conned, 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

R

1 1 Im V x

dx 0 in the case of

localized or periodic potentials.). 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,

c x; z

A eigA

2z i

R

Wxdx 3 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 constant-intensity

wave given by equation 3 is one of the radiation eigenmodes (not conned) of the potential with propagation constant equal to zero. (non-zero propagation constants are obtained by adding a constant term to the potential Vx in equation 2). Another

interesting point to observe is that the above solutions exist only for non-Hermitian potentials, since for W(x)-0 we also have V(x)-0. Therefore, these families of counterintuitive solutions are a direct consequence of the non-Hermitian nature of the involved potential V(x) and as such exist only for these complex structures with gain and loss. The fact that such constant-intensity waves are a direct generalization of the fundamental concept of free-space plane waves to complex environments can be easily understood by setting W(x) c1 const. with c1 2 R: In

this case, the potential V(x) c12 corresponds to a bulk dielectric

medium for which the constant-intensity waves reduce to the plane waves of homogeneous space, c eic1x. It can also be

shown that the potential W(x) determines the power ow in the transverse plane that physically forces the light to ow from the gain to the loss regions. In particular, the transverse normalized Poynting vector dened as S (i/2)(cqc*/qx c* qc/qx) takes

on the following very simple form: S A2W(x).

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

Hn x

e Bx2. The results for

vanishing non-linearity (g 0) and n 1, B 0.5 are shown in

Fig. 1. Note that the corresponding localized optical potential V(x) is not PT -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 c(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 constant-intensity solutions with the correct phase, where diffraction is found to be strongly suppressed. Similar to the diffraction-free beams25, we nd 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).

Two-dimensional constant-intensity waves. Similar constant-intensity solutions can also be derived in two spatial dimensions

j j2c 0 1

The scalar, complex valued function c(x,z) describes the electric eld 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) (ref. 32), which are determined by the following relation,

V x

W2 x

i

dW x

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a b

1

5

0

|[afii9852]|2

Potential

0

1

50 0 20

40

z

2

0

2

x

50

x

c d

|[afii9852]|2

1

0.5

0

50

|[afii9852]|2

1

0.5

0

40

z

0

20

20

40

z

x

50

x

50 0 50

Figure 1 | Constant-intensity waves in a linear waveguide coupler. (a) Real part (green line) and imaginary part (black line) of the complex potential V(x) satisfying equation 2 (blue lled regions depict loss, whereas the red one depicts gain). (b) Evolution of a constant amplitude without the correct phase at the input at z 0. (c,d) Spatial diffraction of the truncated constant-intensity solution satisfying the correct phase relation of equation 3. Two different input

truncations are shown for comparison. The lines in the x z planes of (b,c,d) around x 0 depict the real refractive index of the potential as shown in a.

Note the different vertical axis scale in b.

x,y. The family of these complex potentials V(x,y) and the corresponding constant-intensity solutions c(x,y,z) of the two-dimensional NLSE i @c@z @2c@x2 @2c@y2 V x;y

c g c

j j2c 0

are:

V x; y

i ~r

4 ~r

0 5

c x;y;z

function is y(x,z) gA2z

R

W(x) dx. This expression describes the stationary constant-intensity wave under the perturbation of the eigenfunctions Fl(x) and Gl(x) with eoo1. The imaginary part of l measures the instability growth rate of the perturbation and determines whether a constant-intensity solution is stable (l 2 R) or unstable (lAC). To leading order in e, we

obtain a linear eigenvalue problem for the two-component

perturbation eigenmodes ul(x) [Fl(x) Gl(x)]T, the eigenvalues

of which are l. This eigenvalue problem and the operator matrix

M

AeigA

2z i

R C d~x 6 where ~x Wx Wy with Wx,Wy 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 one-dimensional case, these solutions are valid in both the linear and the nonlinear domain. For the particular case of irrotational ow Wx cosx

siny, Wy 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 constant-intensity 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 dened as i=2c

~rc c

~rc,

$ are dened in the Methods section. So far the presented MI-analysis is general and can be applied to any real W(x) (periodic or not). To be more specic, we now apply this analysis to study the MI of constant-intensity waves in

PT -symmetric optical lattices19,20, assuming that W(x) is a

periodic potential with period a. In particular, we consider the example of a PT -symmetric photonic lattice where

W x

V02 V1cos x

(the resulting optical potential and the

corresponding constant-intensity solution are given in the Methods section). For all the subsequent results we will always assume (without loss of generality) that V0 4 and V1 0.2. It is

important to note here that for these parameters our PT -lattice

V(x) is in the so-called unbroken PT -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

illustrating that the wave ux follows stream line patterns from the gain to the loss regions. Once the nite beam starts to diffract, this balanced ow is disturbed and the waves are concentrated in the gain regions.

Modulation instability of constant-intensity waves. Quite remarkably, the above diffraction-free and uniform intensity

waves are also solutions of the NLSE for both the self-focusing 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 c x;z

A eFl x

eilz eG l x

e il z

eiy

x;z

, where the phase

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a b

5

2

0

z

y

y

10

8

6

2

0

y

50

50

50

50

50

50

0

0

0

50

50

50

x

5

50

5

0

5

y

0

x

x

4

c

0

0

x

50

y

0

50 50

50 50

50

0 0

2

x

y

2

0

2

x

Figure 2 | Constant-intensity waves in a two-dimensional linear optical lattice. (a) Imaginary part of the complex potential V(x,y) discussed in the text. Red and blue regions correspond to gain and loss, respectively. (b) Iso-contour of the beam intensity launched onto the potential in a through a circular aperture of radius B40l0, where l0 is the free space wavelength. Also shown are three transverse intensity plots (from bottom to top) at z 0, z 5,

z 10. (c) Transverse power ow pattern (indicated by arrows) of the beam at z 5.

0 0 0.2 0.4

we can expand the perturbation eigenvectors ul(x) in a Fourier series and construct numerically the bandstructure of the stability problem (different from the physical band-structure of the optical lattice). Based on the above, the FloquetBloch theorem implies that the eigenfunctions ul(x) can be written in the form ul(x) /(x,k)eikx, where /(x,k) /(x a,k) with k being the

Bloch momentum of the stability problem (see Methods). The corresponding results are illustrated in Fig. 3a,b for a self-focusing nonlinearity (g 1) and for different values of the amplitude A.

More specically, we show the instability growth rate |Im{l(k)}| as a function of the Bloch wavenumber k of the perturbation eigenvector in the rst half Brillouin zone. We see that the constant-intensity 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 constant-intensity solutions are linearly stable and their instability dependence forms bands reminiscent of the bands appearing in conventional MI results for bulk or periodic potentials26,29,30, but quite different and profoundly more complex.

To understand the physical consequences of such instabilities and how they lead to lament formation, we have performed independent numerical simulations for the dynamics of the constant-intensity solutions against specic perturbations with results being shown in Fig. 4. More specically, we examine the intensity evolution of a constant-intensity solution when being perturbed by a specic FloquetBloch stability mode. In other words, at the input of the waveguide structure at z 0,

we have c x;z 0

A eFl x

0.25

1

0.2

0.8

|Im()|

|Im()|

0.15

0.6

0.1

0.4

k

k

|Im()| |Im()|

0.05

0.2

0 0 0.2 0.4

0 0 0.2 0.4

k

k

0.06

0.08

0.06

0.04

0.04

0.02

0.02

0 0 0.2 0.4

Figure 3 | Modulation instability diagrams for self-focusing and defocusing nonlinearity. Growth rate of the instability |Im{l(k)}| as a function of the Bloch momentum (half of rst Brillouin zone), for self-focusing nonlinearity and amplitudes (a) A 0.5, (b) A 1 and for

defocusing nonlinearity (c) A 1 and (d) A 2. Different colours in

a and b denote different instability bands.

y x

V0x2 V1 sin x

and we want to know whether the linear

stability analysis captures the exponential growth of the imposed perturbations correctly. For the considered PT -symmetric lattice

with self-focusing nonlinearity, we examine the nonlinear

eG l x

eiy

x;0

, with phase

dynamics of the constant-intensity solution and the result is presented in Fig. 4a. For a perturbation eigenmode with Bloch momentum k 0 and A 1, e 0.01, we can see from Fig. 3b

that Im{l(0)}B1. Therefore, we can estimate the growth for a propagation distance of z 5 to be around |1 0.01 e1 5|2B6.1,

which agrees very well with the dynamical simulation of

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a b

4.05

4

6

[afii9852]2

4

2

3.95

20 20

40

5

0

0 10

20

x z

[afii9852]2

5 2

4 6

8

x z

Figure 4 | Perturbed constant-intensity waves propagating in nonlinear media. Numerical results for the intensity evolution of a constant-intensity wave for (a) a self-focusing nonlinearity (g 1) with parameters k 0, A 1, e 0.01, and (b) for a defocusing nonlinearity (g 1) with parameters k 0.22,

A 2, e 0.001 The peak values are indicated on the vertical axes and match very well with the results of our perturbation analysis.

Fig. 4a. Similarly, for the defocusing nonlinearity (Fig. 4b), and for parameters k 0.22 and A 2, e 0.001, we estimate

the growth for a propagation distance z 35 to be around

|2 0.001 e0.046 35|2B4.02, which matches very well with the

numerical propagation result of Fig. 4b.

DiscussionSymmetry breaking instabilities belong to the most fundamental concepts of nonlinear sciences. They lead to many rich phenomena such as pattern formation, self-focusing and lamentation 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 bres as well as its spatial counterpart in nonlinear Kerr, quadratic, biased photorefractive media with both coherent and partially coherent beams and in discrete waveguide arrays2631. Up to now, most of the mathematical modelling of MI processes has been focused on wave propagation in homogeneous nonlinear media, where an exact constant-intensity 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 constant-intensity 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 FloquetBloch 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 difculties by introducing the above family of constant-intensity 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 non-Hermitian waveguide structures. They also remain valid for any sign of Kerr nonlinearity and thus allow us 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 optics611,14,1922,24. We nd that in the self-

focusing regime, the waves are always unstable, whereas in the defocusing regime the instability appears for specic values of Bloch momenta. In both regimes (self-focusing, defocusing), the constant-intensity solutions break up into laments following a complex nonlinear evolution pattern.

We expect that our predictions can be veried by combining recent advances in shaping complex wave fronts3 with new techniques to fabricate non-Hermitian scattering structures with gain and loss710. As the precise combination of gain and loss in the same device is challenging, we suggest using passive structures with only loss in the rst place. For such suitably designed passive systems6, 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 simplications an experimental demonstration of our proposal should certainly be within reach of current technology.

Methods

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

c(x,z) f(x)exp(imz), where f(x) is the complex eld prole and m the corre

sponding propagation constant, to be found. By substitution of this last relation into equation 1, we get the following nonlinear equation mf fxx V(x)f g|f|2

f 0. We assume a solution of the form f(x) r(x)exp[iY(x)], with r(x),Y(x) real

functions of position x. Since V(x) VR(x) iVI(x), the last nonlinear equation can

be separated in real and imaginary parts. As a result we get the following two

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coupled equations for the real and the imaginary part of the complex potential, respectively:

rxx VR x

Y2x m

r

gr3 0 7 Yxxr 2Yxrx rVI 0 8 where Yx dY/dx and rx dr/dx. By choosing VR(x) Yx2, and by solving

equation 8 to get VI x Yxx 2Yx rr , we can reduce the above system of

coupled nonlinear ordinary differential equations to only one, namely rxx m r

gr3 0. If we assume now a constant amplitude solution, namely r(x) A const.,

we have the following general solution for any real-valued phase function Y(x): c(x,z) A exp[iY(x) igA2z], where VR x

Y2x, and VI(x) Yxx. By setting

W(x) Yx(x), we can write the optical potential, for which the constant-intensity

solution exists, as V(x) W2(x) idW/dx, and the constant-intensity solution

itself reads as follows c x; z

Aexp igA2z

i R

W x

ei qnkx 16

X 1n 1Wneiqnx 17 where q 2p/a 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 coefcients un,un and the

band eigenvalue l(k) that depends on the Bloch momentum k:

On k un

X 1m 1Un;m k un-m gA2un lun

gA2un On k un

X 1m 1Un;m k un-mlun

W x

dx . 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. 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: c x; z

A eFl x

eilz

eG l x e il z eiy x;z, where y(x,z) gA2z

R

W(x)dx and eoo1. Here, Fl(x) and Gl(x) are the perturbation eigenfunctions and the imaginary part ofl measures the instability growth rate of the perturbation. By dening the perturbation two-component eigenmode ul(x) [Fl(x) Gl(x)]T, we obtain the

following linear eigenvalue problem (to leading order in e):

M

$ ^

L

18

where Un,m(k) 2[q(n m) k]Wm and On(k) gA2 (qn k)2. The family of

constant-intensity wave solutions of the NLSE is modulationally unstable if there exists a wave number k for which Im{l(k)}a0, while they are stable if l(k) is real.

For our case, the periodic function W is given by W x

V2 V1cos qx

for

which equation 18 becomes:

mn k

un an 1 k

un 1 an1 k

un1 gA2un lun

gA2un nn k un an 1 k un 1 an1 k un1 lun

19

wl x

l wl x

9

where the operator matrix M

$ ^

L is

dened by the following expression

M

$ ^

L

^

L gA2 gA2

^

L

10

^ L0 i

^

^ L1 11 ^

L0 gA2 d2=dx2 12

^

L1 2W x

d=dx 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 specic

example of a well-known non-Hermitian potential, that is, that of a

PT -symmetric optical lattice19,20. More specically for the particular case of

W x

V2 V1 cos x

, we get the corresponding optical potential and constant-

intensity wave:

V x

V20

where an(k) V1(qn k), mn(k) On(k) V0(qn k) and

nn(k) On(k) V0(qn k).

Direct eigenvalue method. An alternative way (instead of the plane wave expansion method that was used above) of solving the innite dimensional eigenvalue problem of equation 9 is to directly apply the FloquetBloch theorem on the eigenfunctions ul(x), employ the Born-von-Karman boundary conditions (periodic boundary conditions at the end points of the nite lattice) and construct numerically the bandstructure of the instability growth for every value of the Bloch momentum. In particular, the eigenfunctions can be written asul(x) [u(x)eikx u(x)eikx]T, where u(x) u(x a), u(x) u(x a). Substituting

this form of the perturbation eigenfunctions into equation 9, we get the following eigenvalue problem:

^

L 2ikd=dx k2 2kW x

and the related linear operators are dened by the relationships:

L

!

u

u

gA2

gA2

^

L 2ikd=dx k2 2kW x

20

4 V21 cos2x V0V1 cosx

iV1 sinx 14

c x; z

l k

u

u

Aexp igA2z i

V0x

2 iV1sinx

15

where the Bloch momentum takes values in the rst Brillouin zone kA[ p/a,p/a] and the operators

It is obvious that this potential is PT -symmetric as it satises the symmetry

relation V(x) V*( x). In order for the constant-intensity solution to be periodic

in x with the same period as the lattice, the term V0 must be quantized, namely V0 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

c x; z

x; z

eizV

^ L are dened by equations 1113.

By applying the nite difference method, we restrict our analysis to one unit cell xA[ a/2,a/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 l(k) valid near the Bragg points based on perturbation theory. These points are given by (for self-focusing nonlinearity):

ln k

=4. 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 a) that leads to equations 14 and 15. As we are interested in the MIs of the constant-intensity wave solution of the NLSE under self-focusing and defocusing nonlinearities, we want the PT -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 V0 4 and V1 0.2, which lead to an unbroken

spectrum with real eigenvalues. As W(x) is periodic, we can expand the perturbation eigenvectors ul(x) in Fourier series and construct numericallythe band-structure of the stability problem. So at this point, we have to distinguish

between the physical band-structure of the problem and the perturbation band-structure of the stability problem of equation 9. Based on the above,the FloquetBloch theorem implies that the eigenfunctions ul(x) can be written in the form ul(x) /(x,k)eikx, where /(x,k) /(x a,k) with k being the Bloch

momentum of the stability problem. Applying the plane wave expansion method, the wavefunctions /(x,k) and the potential W(x) can be expanded in Fourier series as:

wl x

X 1n 1un k

un k

q

21

2 k nq

k nq

2 kc

ln k

k nq

q

22

2 k nq

2 kc

l0 k

q

k2 k2 kc

23

where kc 2A2 and n 1,2,3,.... The above analytical formulas lead to an excellent

match with the numerical approaches in the shallow lattice limit (V0, V1oo1). Complex lament formation. To understand better the complex lament formation of a constant-intensity solution in a PT -symmetric lattice for both signs

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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 lament formation were based on the perturbation eigenmode proles. In particular, we have at z 0, the following

initial eld prole in terms of Bloch eigenfunctions ux; ux : c x; z 0

A euk x

eikx eu k x

e ikx

eiy

x

, for specic values of Bloch momentum k and the constant-intensity wave amplitude A.

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Acknowledgements

K.G.M. is supported by the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/20072013) under REA grant agreement number PIOF-GA-2011303228 (project NOLACOME). Z.H.M. was supported in part by NSF Grant No. DMS-0908599. The work of D.N.C. was partially supported by NSF Grant No. ECCS-1128520 and the Air Force Ofce of Scientic Research Grants Nos. FA9550-12-1-0148 and FA9550-14-1-0037. S.R. acknowledges nancial support by the Austrian Science Fund (FWF) through Project SFB NextLite (F49-P10) and Project GePartWave (I1142).

Author contributions

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.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Makris, K. G. et al. Constant-intensity waves and their modulation instability in non-Hermitian potentials. Nat. Commun. 6:7257doi: 10.1038/ncomms8257 (2015).

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