We investigate the modulational instability of plane waves in quadratic
nonlinear materials with linear and nonlinear quasi-phase-matching gratings.
Exact Floquet calculations, confirmed by numerical simulations, show that the
periodicity can drastically alter the gain spectrum but never completely
removes the instability. The low-frequency part of the gain spectrum is
accurately predicted by an averaged theory and disappears for certain gratings.
The high-frequency part is related to the inherent gain of the homogeneous
non-phase-matched material and is a consistent spectral feature.Comment: 4 pages, 7 figures corrected minor misprint

A. Hasegawa

A. Saher Helmy

A. V. Buryak

C. B. Clausen

C. R. Menyuk

D. E. Pelinovsky

G. I. Stegeman

H. Fang

H. He

J. F. Corney

J. Klinger

L. A. Ostrovskii

L. Bergé

M. M. Fejer

N. J. Smith

N. N. Rosanov

O. Bang

P. D. Miller

P. Di Trapani

R. A. Fuerst

S. N. Vlasov

S. Trillo

T. B. Benjamin

T. H. Coskun

T. J. Alexander

Y. S. Kivshar

English

arXiv

We investigate the modulational instability of plane waves in quadratic nonlinear materials with linear and nonlinear quasi-phase-matching gratings. Exact Floquet calculations, confirmed by numerical simulations, show that the periodicity can drastically alter the gain spectrum but never completely removes the instability. The low-frequency part of the gain spectrum is accurately predicted by an averaged theory and disappears for certain gratings. The high-frequency part is related to the inherent gain of the homogeneous non-phase-matched material and is a consistent spectral feature

Corney, Joel Frederick

Bang, Ole

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Modulational instability  in periodic quadratic  nonlinear materialsCorney, Joel Frederick; Bang, OlePublished in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.87.133901Publication date:2001Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Corney, J. F., & Bang, O. (2001). Modulational instability  in periodic quadratic  nonlinear materials. PhysicalReview Letters, 87(13), 133901. DOI: 10.1103/PhysRevLett.87.133901VOLUME 87, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER 2001Modulational Instability in Periodic Quadratic Nonlinear MaterialsJ. F. Corney and O. BangDepartment of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark(Received 20 March 2001; published 7 September 2001)We investigate the modulational instability of plane waves in quadratic nonlinear materials with linearand nonlinear quasi-phase-matching gratings. Exact Floquet calculations, confirmed by numerical simu-lations, show that the periodicity can drastically alter the gain spectrum but never completely removesthe instability. The low-frequency part of the gain spectrum is accurately predicted by an averaged the-ory and disappears for certain gratings. The high-frequency part is related to the inherent gain of thehomogeneous non-phase-matched material and is a consistent spectral feature.DOI: 10.1103/PhysRevLett.87.133901 PACS numbers: 42.65.Tg, 42.65.Jx, 42.65.KyThe modulational instability (MI) of waveforms is a fun-damental phenomenon in nonlinear media and is closelyassociated with the concept of self-localized waves, or soli-tons. Similar to solitons, MI occurs due to an interplaybetween nonlinear and dispersive or diffractive effects.It involves the exponential growth of weak perturbationsthrough the amplification of sideband frequencies, whichcauses a destructive modulation of the carrier wave. Manydiverse physical systems exhibit MI, e.g., fluids [1], plas-mas [2], nonlinear optics [3], molecular chains [4], andFermi-resonant interface waves and waveguide arrays [5].In nonlinear optics MI may appear as a transverse instabil-ity that breaks up a broad optical beam, thereby acting as aprecursor for the formation of stable bright spatial solitons[6]. Conversely, the stable propagation of dark solitons re-lies on the stability of the constant-intensity backgroundand thus requires the absence of MI [7].Here we study optical media with a purely quadratic(x 2) nonlinearity. These materials are of significanttechnological interest due to their strong and fast cas-caded nonlinearities [8], which can support stable brightsolitons in all dimensions [9]. Moreover, they providethe most general means of studying nonlinear processes,because varying the phase mismatch between the fun-damental and second-harmonic (SH) waves changes thenonlinearity from being distinctly quadratic to effectivelycubic. The generality is further enhanced when theproperties of the medium are modulated with long-periodquasi-phase-matching (QPM) gratings. This periodicitynot only allows one to tune the effective mismatch, but italso induces effective cubic nonlinearities [10,11], whichmay be engineered to different strengths and signs [12].Except in special cases, MI is unavoidable in x2 ma-terials. Optical pulses can be stable if the dispersion hasopposite sign at the fundamental and SH frequencies [13].Transverse inhomogeneities, as in x2 waveguide arrays[5], and spatial incoherence [14] also eliminate MI. How-ever, coherent cw beams propagating in pure homogeneousx2 materials are always unstable [7]. Naturally, a cubic(Kerr) nonlinearity is always present in x2 materials, and,if defocusing and sufficiently strong, it may eliminate MI[15]. Unfortunately, the cubic nonlinearity in conventionalx2 materials is usually focusing, though it may be a fac-tor in the recent observation of apparently stable quadraticoptical vortex solitons [16].We consider transverse MI of coherent cw beams inx2 materials whose linear and nonlinear properties aremodulated and study the instability gain spectra, which cannow be measured in the laboratory [17]. It is known that,in a system with cubic nonlinearity, periodic dispersion[18] (as in dispersion-managed fibers) or nonlinearity [19]can reduce the growth rate of fluctuations. We find that, inquadratic materials, the periodicities also drastically alterthe MI gain spectrum but they do not entirely remove theinstability.We numerically find exact plane-wave solutions andapply Floquet theory to determine their MI gain spectrum.Numerical simulations, with exact plane-wave solutionsseeded with noise as initial conditions, confirm the Floquetgain spectra. In the regime of efficient quasi phase match-ing, we find that the MI gain spectrum contains twodistinct and well-separated features with fundamentallydifferent physical origins. The low-frequency part of thegain is accurately predicted by an averaged theory anddisappears for certain grating modulations. The high-frequency part of the spectrum is related to the inherentgain of the non-phase-matched material (i.e., with no grat-ings) and appears to be unavoidable. However, becausethey are consistently small, the high-frequency peaks canbe ignored under a less stringent definition of experimentalstability. This gives the possibility of stable plane wavesas predicted by the averaged theory.The system under consideration consists of a cw beam(carrier frequency v) and its SH propagating in a lossless1D slab waveguide under conditions for QPM type-Isecond-harmonic generation (SHG). Both the linear andquadratic nonlinear susceptibilities are periodic alongthe Z direction of propagation, as illustrated in Fig. 1,but are constant along the transverse X direction. Themodulation of the refractive index is assumed to be weak:DnjZn¯j ø 1, where njZ  n¯j 1 DnjZ and jrefers to the frequency jv. For many materials and etch-ing techniques, such a linear grating appears inevitablywhen the nonlinear susceptibility is altered [20]. We133901-1 0031-90070187(13)133901(4)$15.00 © 2001 The American Physical Society 133901-1VOLUME 87, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER 2001FIG. 1. Normalized linear and quadratic nonlinear gratings,ajZ and xZ, with domain length L0  pjkgj.consider gratings for forward QPM only, such that thegrating period is much longer than the optical period andwe can neglect Bragg reflections. The evolution of theslowly varying beam envelopes is then given by [21]i≠ZE1 112≠2XE1 1 a1ZE1 1xZE1E2eibZ  0 ,i≠ZE2 114≠2XE2 1 2a2ZE2 1(1)xZE21e2ibZ  0 ,where E1  E1X,Z and E2  E2X,Z are the enve-lope functions of the fundamental and SH, respectively.The coordinates X and Z are in units of the input beamwidth X0 and the diffraction length Ld  k1X20 , respec-tively. The parameter b  DkLd is proportional to themismatch Dk  k2 2 2k1, where kj  jvn¯jc is theaverage wave number. The normalized refractive-indexgrating is ajZ  LdvDnjZc, and the normalizednonlinear grating is xZ  LdvdeffZn¯1c, wheredeff  x 22 is given in SI units.To consider a plane wave with longitudinal wave numberoffset L and transverse wave number V, we transform tothe new variables wx, z  E1X,Z expiVX 2 iLZ,yx, z  E2X,Z exp2iVX 2 2iLZ 1 ib˜Z, x ph X 1 VZ, and z  hZ, where h  jL 1 12V2jfor L ﬁ 2 12V2 and h  1 otherwise. We have alsointroduced a residual effective mismatch b˜  b 2 kg.This gives the equationsi≠zw 112≠2xw 2 rw 1 a01zw 1x 0zwyeikz  0 ,i≠zy 114≠2xy 2 sy 1 2a02zy 1(2)x 0zw2e2ikz  0 ,where r  sgnL 1 12V2, s  2r 2 b˜h, and k kgh. Note that r  0 whenL  2 12V2. We Fourier ex-pand the rescaled gratings, a0jz  ajZh and x 0z xZh:a0j  a0jXngneinkz, x 0  d00 1 d0Xngneinkz, (3)where, for the square waveform that we consider (Fig. 1),gn  2sipn for n odd and gn  0 for n even. The signs  sgnk is included to ensure that the expansions givethe same grating form for both signs of k. The gratingswill drive a periodic variation in z of spatial frequency kin the fields of system (2). We explicitly include this fastvariation by expanding the fields in the Fourier series also:w Xnwnx,zeinkz , y Xnynx, zeinkz. (4)For plane-wave solutions w,y  ws,ys, all the coeffi-cients wn and yn are constants determined by2nk 1 rwn 1 a01Xlgn2lwl 1Xl,pDn1l2p21wl yp  0 ,2nk 1 syn 1 2a02Xlgn2lyl 1Xl,pDn2l2p11wlwp  0 ,where Dn  d0gn 1 d00dn, dn being Kronecker’s delta.We find the solution numerically using a standard relax-ation technique based on Newton’s method.We determine the linear stability of the plane-wave solu-tions by using exact Floquet theory [6]. For perturbationsof transverse frequency offset n:wx, z  wsz 1 dw1ze2inx 1 dw2 zeinx,yx, z  ysz 1 dy1ze2inx 1 dy2 zeinx,(5)the linear growth is governed by the linearized equation≠zP  MzP, where Pz and the matrix Mz areP 0BBB@dw1dw2dy1dy21CCCA, M  i0BBB@a b c 02b 2a 0 2c2c 0 d 00 22c 0 2d1CCCA ,(6)and where we have defined the components a  2n22 2r 1 a01z, b  yszxzeikz, c  wszxze2ikz, andd  2n24 2 s 1 2a02z.Because of the periodicity in a0j , x 0, ws, and ys, thematrix M is also periodic with period zp  2pjkj. Todetermine the stability, we need only find the eigenvaluesli of the mapping dT of the solution over one period:Pz 1 zp  dTPz. We calculate dT numerically byintegrating the linearized equation [6] with a fourth-orderRunge-Kutta method. If all the li lie on the unit circle forevery n, then the solutions ws and ys are stable; otherwise133901-2 133901-2VOLUME 87, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER 2001there is a gain whose profile is g  maxRelnlizp.We note that the actual growth rate at short distances maydiffer from g, depending on the overlap of the initial per-turbation with the unstable eigenvector [22]. However,for larger distances (z . 1g), as we consider here, thegrowth rate will approach the limiting maximum value g.As a first example, we calculate the Floquet gainspectrum for a conventional domain-reversal grating inLiNbO3 (a01  a02  d00  0) for r  21, k  100, andexact phase matching (s  22). As Fig. 2(a) shows, thespectral features fall into two well-separated regions: twolow-frequency bands around n  2.0 and n  3.4, and anarrow high-frequency band at n  14.2. For comparisonFig. 2(b) plots the gain for a GaAsAlAs superstructurewith a nonlinear grating etched though quantum-welldisordering [23] (d00d0  4.6) and a linear grating chosento be a01 2 a02k  219.2. High-frequency gainbands are again present, but the low-frequency bands areabsent. Further results show that such narrow, high-nbands are a general feature of the gain spectrum forQPM gratings. They are related to the inherent MI in thenon-phase-matched, gratingless x 2 material: Each gainpeak that appears in the gratingless spectrum usually alsoappears in the Floquet spectrum, typically with the sameheight and spectral location [Fig. 2(d)]. An exception isthe gain from the symmetric grating case (d00  0), asin LiNbO3, where the original “homogeneous” peak issuppressed. Often, as in Fig. 2(b), this peak is split intotwo or more closely spaced components by the grating.The low-frequency features in the gain spectrum areaccurately predicted by approximate averaged-field equa-tions for the dc components w¯, y¯  w0,y0. FromFIG. 2. Floquet gain spectra for (a) the LiNbO3 and (b) theGaAsAlAs gratings, for s  22. The diamonds give theaveraged-field results in (c) and the equivalent non-phase-matched homogeneous results in (d).Eqs. (2) these equations are [10,11]i≠zw¯ 112≠2xw¯ 2 rw¯ 1 rw¯y¯ 1gjy¯j2 2 jw¯j2w¯  0 ,i≠z y¯ 114≠2xy¯ 2 sy¯ 1 rw¯2 1(7)2gjw¯j2y¯  0 ,where r  2isd0 1 2d00a01 2 a02kp and g  d020 1d021 2 8p2k. The averaged equations are valid forsmall residual mismatch (b˜ ø b) and small grating pe-riod (k ¿ 1). They show that the linear grating maydecrease the strength of the effective quadratic nonlinear-ity, possibly even to zero [11,20]. More importantly, thenonlinear grating induces cubic nonlinearities. These canbe defocusing (g . 0) [11] and may thus stabilize planewaves [15]. We find that Eqs. (7) indeed have modula-tionally stable plane-wave solutions in a large regime forr  21, as shown in Fig. 3(a), where g˜  gjrj2 givesthe relative strength of the cubic nonlinearity. The fig-ure also shows that, since the mismatch s cannot be large(otherwise the averaged equations are invalid), the effec-tive quadratic nonlinearity jrj must be reduced (i.e., g˜increased) to get a stable solution.The case shown in Fig. 2(a) corresponds to having g˜ 44.5, s  22, which lies outside the region of stabilitypredicted by the averaged equations. The gain profile,shown in Fig. 2(c), matches exactly the low-frequency partof the Floquet spectrum. The case in Fig. 2(b) correspondsto r  0, in which case the averaged equations reduce totwo coupled nonlinear Schrödinger equations. The plane-wave solution w0 p2rg, y0  0 is stable for g . 0(r  21), in agreement with the absence of low-frequencybands in the exact Floquet spectrum.Figure 3(b) shows the maximum gain calculated fromFloquet theory for a given modulation depth. Where theaveraged equations predict MI, the Floquet gain is large, inagreement with the averaged results. However where theaveraged equations predict stability, Floquet theory findsa residual gain, due to the high-frequency gain bands.However, because the residual gain is often small andnarrow, it may be neglected under a reasonable definition0 2 4 6 8 100510γσStableUnstable~(a)0.511.551005~(b)ln γσgainFIG. 3. (a) Region of MI predicted by the averaged-field the-ory with r  21 and r ﬁ 0. (b) Maximum gain predicted byFloquet theory for 5 # k # 30, with d00d0  53 and a01 2a02k  0.158. The region of zero gain predicted by averaged-field theory is shaded.133901-3 133901-3VOLUME 87, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER 2001050−2020250(a)zx|w|2  d m2050−2020100(b)zx|w|2  dm2  FIG. 4. Propagation of dark solitons with (a) a modulated non-linearity (d00d0  4.6, k  112) and (b) no modulation (d0 0). In both cases b  100 and Ld  200 mm. dm  d0 1 d00.of experimental stability, which allows for a small growthover typical length scales in experiments. Neglecting thehigh-frequency gain bands causes the results of Floquetand average-field calculations to coincide and wholeregions in parameter space of “experimentally stable”plane waves to appear. These regions are even larger thanshown in Fig. 3 and also appear for r  11 and r  0.We demonstrate this experimental stability by simulatingthe propagation of dark solitons with Eq. (1), shownin Fig. 4(a). The parameters correspond to a nonlineargrating induced in GaAsAlAs through quantum-welldisordering [23], as in Fig. 2(b) but with no linear modu-lation. Stable propagation of the dark soliton can be seenfor over 60 diffraction lengths. Now because of the smallgain in highly non-phase-matched materials, such stablepropagation can also be achieved without the nonlineargrating [Fig. 4(b)]. But the SHG efficiency is then verylow, as evidenced by the relative SH intensity (not shown)being a factor of 10 less than with the grating.The Floquet spectral results are confirmed by numeri-cal simulations of Eqs. (1). We launch plane wavesseeded with Gaussian white noise to excite all spa-tial frequencies. The gain spectrum is then calculatedfrom the Fourier-transformed fundamental w˜ as g˜n lnjw˜z2,nj 2 lnjw˜z1,njz2 2 z1 [24]. As shownin Fig. 5, the simulations agree with the Floquet curves,in both the low-n [Fig. 5(b)] and the high-n [Fig. 5(c)]regions.FIG. 5. (a) Gain profile calculated from propagative simula-tions, for the same parameters as in Fig. 2(a). Comparisonswith the Floquet theory (solid curve) are given in (b) and (c).In summary, we have used an exact Floquet techniqueto find the MI gain spectrum in x2 materials withgeneral QPM gratings. Because of the periodicity, novelhigh- and low-frequency gain bands appear in thespectrum. A simple approximate averaged-field theorycorrectly predicts the low-frequency gain bands. The high-frequency peaks that are consistently present in the spec-trum are typically small and may be neglected under arealistic definition of experimental stability. The predic-tions of the averaged and the exact Floquet analyses thenagree, and stable plane waves and dark solitons becomepossible, even under conditions close to phase matching.Because it is now possible to directly probe the gainspectrum in experiments [17], these interesting spectralprofiles may be verified in the laboratory.This work is supported by the Danish Technical Re-search Council under Grant No. 26-00-0355. We thankY. S. Kivshar for useful discussions.[1] T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417(1967).[2] A. Hasegawa, Plasma Instabilities and Nonlinear Effects(Springer-Verlag, Heidelberg, 1975).[3] L. A. Ostrovskii, Sov. Phys. JETP 24, 797 (1967).[4] Y. S. Kivshar and M. Peyrard, Phys. Rev. Lett. 46, 3198(1992).[5] P. D. Miller and O. Bang, Phys. Rev. E 57, 6038 (1998).[6] R. A. Fuerst et al., Phys. Rev. Lett. 78, 2756 (1997).[7] S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).[8] For a review, see G. I. Stegeman, D. J. Hagan, andL. Torner, Opt. Quantum Electron. 28, 1691 (1996).[9] L. Bergé et al., Phys. Rev. A 52, 28 (1995); D. E.Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev.Lett. 75, 591 (1995).[10] C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett.78, 4749 (1997).[11] J. F. Corney and O. Bang, Phys. Rev. E (to be published).[12] O. Bang et al., Opt. Lett. 24, 1413 (1999).[13] A. V. Buryak and Y. S. Kivshar, Phys. Rev. A 51, 41 (1995).[14] T. H. Coskun et al., Phys. Rev. Lett. 84, 2374 (2000);J. Klinger, H. Martin, and Z. Chen, Opt. Lett. 26, 271(2001).[15] T. J. Alexander, A. V. Buryak, and Y. S. Kivshar, Opt. Lett.23, 670 (1998).[16] P. Di Trapani et al., Phys. Rev. Lett. 84, 3843 (2000).[17] H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman,Phys. Rev. Lett. 86, 4528 (2001); Opt. Lett. 25, 1786(2000).[18] N. J. Smith and N. J. Doran, Opt. Lett. 21, 570 (1996).[19] S. N. Vlasov, Sov. J. Quantum Electron. 6, 245 (1976).[20] M. M. Fejer et al., IEEE J. Quantum Electron. 28, 2631(1992).[21] C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B11, 2434 (1994); O. Bang, ibid. 14, 51 (1997).[22] N. N. Rosanov, Prog. Opt. 35, 1 (1996).[23] A. Saher Helmy et al., Opt. Lett. 25, 1370 (2000).[24] H. He et al., Phys. Rev. E 59, 6064 (1999).133901-4 133901-4

Modulational instability in periodic quadratic nonlinear materials

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