2320 OPTICS LETTERS / Vol. 31, No. 15 / August 1, 2006

Spatial supercontinuum generation in nonlinearphotonic lattices

Ofer Manela, Guy Bartal, and Mordechai SegevDepartment of Physics and Solid State Institute, Technion, Haifa 32000, Israel

Hrvoje BuljanDepartment of Physics, University of Zagreb, PP 331, 10000 Zagreb, Croatia

Received January 17, 2006; revised March 13, 2006; accepted May 2, 2006;posted May 11, 2006 (Doc. ID 67268); published July 10, 2006

We show that two Bloch modes launched into a nonlinear photonic lattice evolve into a comb or a supercon-tinuum of spatial frequencies, exhibiting a sensitive dependence on the difference between the quasi-momenta of the two initially excited modes. This phenomenon results from four-wave mixing combined withexchanges of momentum between the optical field and the lattice. © 2006 Optical Society of America

OCIS codes: 190.4420, 190.4380.

Wave propagation in nonlinear periodic structuresexhibits many interesting phenomena.1 The periodic-ity of the refractive index alters diffraction, while thenonlinearity may lead to various effects such as theformation of lattice solitons2–4 and modulationinstability.2,5 Such systems have been a centraltheme in science for more than half a century, start-ing from the famous work of Fermi, Pasta, and Ulam6

(FPU) and continuing nowadays as an extensive fieldof research.7 In their work, FPU excited one mode ofa discrete periodic system and studied the way en-ergy spreads among the modes of the linear latticeunder the influence of nonlinearity. Instead of the an-ticipated equipartition of energy among the differentlattice modes, the system evolved in an almost peri-odic fashion, exhibiting near recurrence to the initialstate.

Here we study how energy spreads among the lin-ear [Floquet–Bloch (FB)] modes of a continuous peri-odic system, under the action of nonlinearity. We ad-dress the following question: starting with theexcitation of two FB modes (of quasi-momenta k1 andk2), to which other FB modes will the energy flow andhow? We show that, through four-wave mixing(FWM), energy spreads to more and more FB modesthat are evenly spaced in reciprocal space. After suf-ficiently large propagation distances, the energy dis-tribution among the FB modes attains a comb or acontinuum structure, displaying a sensitive depen-dence on the momentum difference between the twoinitial modes. The dynamics described here also ap-plies to other nonlinear periodic systems, such asnonlinear photonic crystals and Bose–Einstein con-densates in periodic lattices.

The propagation of a coherent wave in a generallinear system is best understood by considering theevolution of the linear modes of the system. Each oneof these modes evolves in a trivial way, merely accu-mulating phase at a constant rate as it propagates,with no energy exchange between modes. The linearmodes of a periodic system are the FB modes.8 Ac-

cording to the FB theorem these modes can be

0146-9592/06/152320-3/$15.00 ©

written as Ank�x ,z�=exp�i�nkz��nk�x�=exp�i�nkz��exp�ikx�unk�x�, where unk�x+D�=unk�x� are periodicfunctions with the same period D as the periodicity ofthe system; k is the quasi-momentum (QM) of themode, which may be chosen to be in the first Brillouinzone (BZ): −G /2�k�G /2 (G�2� /D is the width ofthe BZ); n is the band number; and �nk is the propa-gation constant of this mode. If nonlinearity is intro-duced into the system, the linear modes becomecoupled. Let us consider a 1D waveguide array that isperiodic in the transverse dimension x and invariantalong the z axis, displaying a Kerr nonlinearity. Theevolution of a coherent wavepacket in this systemis described by the dimensionless nonlinearSchrödinger equation

i�A

�z+

�2A

�x2 + �n�x�A ± �A2�A = 0, �1�

where A is the slowly varying amplitude of the fieldand �n�x� is a linear refractive index with periodicityD. Expanding A as a superposition of the FB modesA�x ,z�=�n�k�an�k��z��n�k��x�exp�i�n�k�z�, substitutingA into Eq. (1), multiplying by the complex conjugateof �nk, and integrating over x yields coupled-modeequations for the amplitudes ank�z�:

idank�z�

dz± �

n1k1�

n2k2�

n3k3an1k1�z�an2k2�z�an3k3

* �z�

�Cn1k1n2k2n3k3nk

�exp�i��n1k1 + �n2k2 − �n3k3 − �nk�z� = 0, �2�

using the orthogonality of the FB modes. The tensorCn1k1n2k2n3k3nk is defined by the overlap integral,��n1k1�x��n2k2�x��n3k3

* �x��nk* �x�dx, integrated over the

whole lattice. Thus, quartets of modes are coupled,and for each quartet the extent of energy transfer de-pends on the overlap integral between the four modesof the quartet. In the case of FB modes, owing to their

symmetry properties, these overlap integrals vanish

2006 Optical Society of America

August 1, 2006 / Vol. 31, No. 15 / OPTICS LETTERS 2321

for most combinations of modes; so energy is trans-ferred only between groups of modes that have spe-cial relations between their QM. Utilizing the prop-erties of the FB modes, the tensor C can be written as

Cn1k1n2k2n3k3nk = C0one cell

un1k1�x�un2k2�x�un3k3* �x�

�unk* �x�exp�i�k1 + k2 − k3 − k�x�dx, �3�

where the coefficient C0 is given by

C0 = exp i�k�N − 1�D

2 � sin��kND/2�

sin��kD/2�

= exp i�k�N − 1�D

2 �N �n=−�

�

sincND

2��k − nG��

� � 1 N is odd

�− 1�n N is even, �4�

where �k�k1+k2−k3−k4 and N is the number of lat-tice sites. That is, the coefficient C0 is a sum of sincfunctions centered around the origin in k space andaround every reciprocal lattice vector. The first zeroof each sinc occurs exactly at 1/N of the width of theBZ, and in the limit of an infinite lattice, the sincfunction becomes a delta function. For a finite latticewith periodic boundary conditions, the sinc functionis nullified exactly at the allowed k values (apartfrom k=0). Hence, quartets of modes are coupled ifand only if �k�0, ±mG (m is an integer), which is aQM conservation rule up to an integer multiple of areciprocal lattice vector. Surely this result is antici-pated: the QM’s value is fixed just up to a reciprocallattice vector, and the nonlinear term in Eq. (1) yieldsFWM.9–12 In addition, Eq. (2) shows that energy istransferred efficiently when the phase-matching con-dition �n1k1+�n2k2−�n3k3−�n4k4 0 is satisfied. Al-though the results are presented in one transversedirection, they can be readily generalized to higherdimensions, and also for other nonlinearities (satu-rable, quadratic, etc.).

Up to this point our discussion is general and doesnot specify the initial conditions. From this point onwe consider an initial condition where two FB modesare excited at z=0, and we examine how a nonlinearinteraction leads to redistribution of energy into theother (initially nonpopulated) FB modes. Implement-ing the QM conservation rule for this case, one cansee that two new FB modes with QM k=2k1−k2 andk=2k2−k1 will be generated. When the propagationdistance is large enough, the nonlinear interactioncauses a cascaded excitation of modes, where everynewly generated FB mode interacts with the FBmode that excited it, and their interaction generatesmodes with QM k1−n�k12 and k2+n�k12 (n=1,2. . .,�k12�k2−k1). If the resulting QM k is outside thefirst BZ (i.e., k�G /2 or k−G /2), the FB mode towhich energy is transferred is folded back into the BZto k−G (or to k+G). The coupling is mainly to theband of the originally excited FB modes because of a

higher value of the integral of Eq. (3) (describing the

fourfold overlap among the wave functions of the fourmodes) and better phase matching. Since the value ofthe QM is meaningful up to an integer multiple of thereciprocal lattice vector, the QM space has the topol-ogy of a torus (which in one dimension is a circle). Inone dimension, this implies that if the difference be-tween the QM of the two initial FB modes �k12 iscommensurable with the width of the BZ (i.e., �k=G with a rational =m /n, m and n being coprimeintegers), then the nonlinear interaction gives rise toa comb of FB modes, consisting of only n modes ineach band [as an example, see Fig. 1(a), for which =1/10]. On the other hand, if �k12 is incommensu-rable with G, then the nonlinear interaction excitesan infinite dense set of modes13 [Fig. 1(d)]. Clearly,the dynamics described above is sensitive to initialconditions because the commensurable initial condi-tions are densely embedded between the incommen-surable ones. For a finite lattice with N sites, insteadof commensurability the relevant criterion is thevalue of the greatest common divisor (GCD) of�k12/ �2� /DN� and N. The number of excited statesin this case is N /GCD��k12/ �2� /DN�,N�.

To quantify the difference between the commensu-rate and the incommensurate cases, we calculate theparticipation number PN=1/�kpk

2, where pk

= �A�k��2 /��A�k��2dk and A�k� is the Fourier transformof A. This function measures the number of apprecia-bly excited modes and is approximately equal to theexponent of the Shannon entropy.14 In both cases, PNevolves initially in the same fashion, starting withthe value 2 and increasing as more and more modesbecome populated (Fig. 2). After some energy flows atleast one cycle around the BZ (which in our exampleoccurs at z 16), the evolution of PN differs dramati-cally between the two cases: in the commensuratecase it stays within the range 5–10 (bounded by thedenominator of ��k12/G=1/10), whereas in the in-commensurate case it increases rapidly.

To contrast this situation with propagation in a ho-mogeneous medium, we launch two plane waves of

Fig. 1. Spatial power spectrum of the light, evolving withpropagation. Excitation of two FB modes with QM differ-ence (a), (b) commensurate or (d), (e) incommensurate withthe width of the BZ. (a), (d) One-band FB power spectrum;(b), (e) Fourier power spectrum; (c), (f) Evolution of theFourier power spectrum of the field when two plane wavesare launched into a nonlinear homogeneous medium. In allpanels the horizontal axis is the propagation �z� axis and

the vertical axis is the (quasi-) momentum �kx� axis.

2322 OPTICS LETTERS / Vol. 31, No. 15 / August 1, 2006

the same amplitudes and k’s of Figs. 1(a) and 1(d)into a homogeneous medium with the same nonlin-earity and compare the evolution in Fourier space be-tween the nonlinear propagation in the lattice [Figs.1(b) and 1(e)] and in the homogeneous medium [Figs.1(c) and 1(f)]. Since every FB mode is a comb in Fou-rier space, the comb (continuum) evolving in FBspace is a comb (continuum) in Fourier space [Figs.1(a) and 1(b)] �or Figs. 1(d) and 1(e)�. In contrast tothe lattice case, for propagation in homogeneous me-dia, the nonlinear interaction in both cases alwayspopulates a discrete set of plane waves, i.e., a comb inFourier space. That is, in all our simulations, the twoplane waves interacting via FWM in homogeneousmedia never evolve into a continuum and never dis-play any sensitive dependence on momentum differ-ence. We note that for both self-focusing and self-defocusing nonlinearities, the nonlinear dynamicsand supercontinuum generation (SCG) are generi-cally the same. In addition, we note that the intro-duction of a “prismatic” term (i.e., linear in x) into thelinear refractive index does not change the physicalbehavior described above, since it causes only a col-lective transport of the energy in the reciprocal spacewith a k-independent rate. Hence, fulfillment of thecommensurability condition is unaffected by thisterm. This means that SCG also appears in settingswhere (linear) Bloch oscillations occur,15,16 with theresults being similar to those without the prismaticterm.

The maximum propagation distance for studies onSCG generated by the mechanism described above islimited by nonlinear interaction with noise, givingrise to the appearance of modulation instability. Thisphenomenon is inevitable numerically as well as ex-perimentally, because the noise is never zero: thenonlinearity amplifies the noise and gives rise to theformation of a periodic pattern (determined by the in-terplay between lattice dispersion and nonlinearity2),thereby affecting the energy redistribution amongthe FB modes. Here these noise-driven effects mani-fest themselves by excitation of a continuum ofmodes for both the commensurate and incommensu-rate cases, diminishing the difference between thetwo generic cases. We also reemphasize that our

Fig. 2. (Color online) Participation number as a functionof propagation for incommensurate (solid curve) and com-mensurate (dashed curve) QM difference.

studies here are performed in a continuous periodicmedium, rather than discrete as in the original FPUproblem. Naturally, one may ask how the energyflows among lattice modes for different types of exci-tation (single-mode excitation, etc.). In this sense, theproblem that we addressed is the continuous versionof the FPU problem. Altogether, understanding SCG,modulation instability, and other nonlinear phenom-ena in continuous nonlinear periodic systems shouldhave important implications for the long-standingFPU problem of energy equipartition in nonlinear pe-riodic systems.

In summary, we have shown that two optical Blochmodes interacting nonlinearly in a photonic latticeredistribute their energy through FWM and foldingin reciprocal space, populating a comb of an increas-ing number of evenly spaced FB modes (if the mo-mentum difference between the initial modes is com-mensurate with the lattice momentum) or asupercontinuum of FB modes (if the initial momen-tum difference is incommensurate with the latticemomentum). This behavior is independent of the signof the nonlinearity and does not occur in homoge-neous nonlinear media. Finally, we suggest imple-menting these ideas with Bose–Einstein condensatesin optical lattices.

References

1. D. N. Christodoulides, F. Lederer, and Y. Silberberg,Nature 424, 817 (2003).

2. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13,794 (1988).

3. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R.Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383(1998).

4. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N.Christodoulides, Nature 422, 147 (2003).

5. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y.Silberberg, R. Morandotti, H. Yang, G. Salamo, M.Sorel, and J. S. Aitchison, Phys. Rev. Lett. 92, 163902(2004).

6. E. Fermi, J. Pasta, and S. Ulam, Los Alamos ReportLA-1940 (Los Alamos Scientific Laboratory, 1955).

7. D. K. Campbell, P. Roseanu, and G. M. Zaslavski,Chaos 15, 015101 (2005).

8. N. W. Ashcroft and N. D. Mermin, Solid State Physics(Saunders, 1976).

9. C. M. de Sterke and J. E. Sipe, Phys. Rev. A 39, 5163(1989).

10. L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y.Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S.L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).

11. M. Trippenbach, Y. B. Band, and P. S. Julienne, Phys.Rev. A 62, 023608 (2000).

12. K. M. Hilligsøe and K. Mølmer, Phys. Rev. A 71,041602(R) (2005).

13. V. I. Arnold, Mathematical Methods of ClassicalMechanics, 2nd ed. (Springer-Verlag, 1989), Chap. 3.

14. J. Pipek and I. Varga, Phys. Rev. A 46, 3148 (1992).15. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F.

Lederer, Phys. Rev. Lett. 83, 4752 (1999).16. R. Morandotti, U. Peschel, J. S. Aitchison, H. S.

Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756(1999).