Asymmetric cubic nonlinearities induced by long period
gratings: Signature and importance

Ole Bang and Joel F. Corney

Department of Informatics and Mathematical Modelling
Technical University of Denmark, building 321
DK-2800 Kongens Lyngby, Denmark.

Since the observation of nonlinear phase-shifts larger than pi, quadratic nonlinear or x(2) materials have been of significant interest in photonics. With the maturing of the quasi-phase-matching (QPM) technique, in particular by electric-field poling of ferro-electrics, such as LiNbO3, the number of applications of x(2) materials has increased even more. It is therefore more important than ever to have precise models of QPM samples.

In addition to providing effective phase-matching, QPM gratings generate asymmetric cubic nonlinearities (ACN) in the equations for the average field [1,2]. This cubic nonlinearity is focusing or defocusing according to the sign of the phase mismatch [2], and its strength can be increased (e.g., dominating the Kerr nonlinearity) by modulating the grating.

In CW-operation ACN induce an intensity dependent phase mismatch, which implies a nonzero so-called separatrix intensity, the crossing of which abruptly changes the phase-shift of the fundamental over a period by pi, with obvious use in switching applications [3]. We have derived a formula for this QPM-induced separatrix intensity, which corrects earlier estimates by a factor of 5.3, and we have found the crystal lengths necessary for an optimal flat phase-versus-intensity response on either side of the separatrix [3]. Clearly, when operating close to or on both sides of this separatrix, ACN become important - the simple averaged model with merely an effective mismatch, and thus no separatrix, is inadequate.

The most startling example appears when the competition between a linear and a nonlinear QPM grating eliminates the effective quadratic nonlinearity. With no nonlinearity solitons should not exist but, as shown in the figure, both bright and dark solitons do exist and they are stable under propagation [2]. This paradox is elegantly explained by including ACN in the model, which then correctly supports simple bright and dark nonlinear Schrödinger solitons. In describing modulational instability (MI), ACN become important if the nonlinear part of the QPM grating has a dc value and/or if the QPM grating has both a nonlinear and a linear part. Examples are quantum-well disordering in semiconductors and alternating linear and nonlinear domains in polymers. We have shown that ACN are necessary to correctly describe the MI gain spectrum in such samples, and in particular to predict the novel QPM-induced regimes in which long-wave instabilities disappear and plane waves become modulationally stable over hundreds of diffraction lengths [4].

All these effects are confirmed numerically and thus abundant theoretical evidence supports the presence of ACN. Furthermore, ACN are a general effect of non-phase-matched interaction between any number of waves and as such appear also in homogeneous x(2) materials (no QPM grating) in the cascading limit [4]. In fact, in this case the asymmetric signature of ACN may be measured as the difference between the properties in upconversion (SHG) and downconversion, since there is no effective quadratic nonlinearity. Such an experiment was just reported [5] and thus ACN have now been confirmed experimentally.

This research is supported by the Danish Technical Research Council through Talent Grant No. 26-00-0355.


  1. C. Balslev Clausen, O. Bang, Y.S. Kivshar, "Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media", Phys. Rev. Lett. 78, 4749 (1997).
  2. J.F. Corney, O. Bang, "Solitons in quadratic nonlinear photonic crystals", Phys. Rev. E 64, 047601 (2001).
  3. O. Bang, T.W. Graversen, J.F. Corney, "Accurate switching intensities and optimal length scales in quasi-phase-matched materials", Opt. Lett. 26, 1007 (2001).
  4. J.F. Corney, O. Bang, "Modulational instability in periodic quadratic nonlinear materials", Phys. Rev. Lett. 87, 133901 (2001).
  5. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing", Phys. Rev. Lett. (to appear).