## Abstract

We introduce simple model equations describing the dynamics of light in thin photonic crystal films with Kerr nonlinearity. We report modulational instabilities and bright and dark localized structures of light that exist in this system in the proximity of Fano resonances.

© 2005 Optical Society of America

Nondispersive localized structures (LSs) of light in coherently pumped dissipative systems have been actively studied over the last several decades, see, e.g., [1, 2, 3] for reviews. These structures have been successfully observed experimentally in semiconductor microcavities [4], single-mirror feedback loops with nonlinear elements [5] and other systems. The recent wave of activity in the area of photonic crystals (PCs) has also led to several theoretical studies of LSs and pattern formation in cavities with periodically structured nonlinear elements [6, 7, 8].

Optical bistability is one of the precursors for the existence of LSs [1, 2]. It has been recently demonstrated that an optical beam incident from the top on the thin film with the periodic modulation of the refractive index, i.e. photonic crystal (PC) film, can resonantly couple to long-lived states in the film [9, 10, 11]. These are sometimes referred to as Fano resonances [11, 12]. If the PC film is nonlinear, the propagation constant of the low-leakage modes can be off-resonant at low intensity, while being resonant at high intensity. This results in bistable transmission through and reflection from the nonlinear PC film [11]. In this paper we investigate LSs in nonlinear PC films resonantly excited from the top.

Resonant modes cause a strong build up of energy inside PC films, with quality factors that can be of order hundreds of thousands. Here we study nonlinear properties of this system under the condition, when the counter-propagating resonant modes are coupled coupled together by a specially designed Bragg grating, see Fig. 1. Using this new grating we can excite Bloch modes close the edge of the Brilluoin zone, where the group velocities are close to zero. This means that we can pump the film from the top by a broad beam of light and observe the slow moving or even stationary bright or dark spots, which are the LSs of light. In general only one grating can be used but, as it will be mentioned below, two gratings provide better quality factors.

Let *κ*_{f}
and *κ*_{b}
be the grating vectors chosen to ensure the existence of the Fano resonances and Bragg coupling of the counter-propagating waves, respectively. The condition for Fano resonance between the pump wave and the film modes is given by *k*
_{0} sin(*θ*)=*lκ*_{f}
+*β*, where *k*
_{0} is the wavevector of the pump wave, *θ* is the angle of incidence, *l* is an integer and *β* is the wavevector of the Fourier component dominating the positive part of the spectrum of the exact mode guided in the film. As it has been already mentioned above there are two counter-propagating resonant modes. For one of those the amplitude of the first harmonic with *β*>0 is greater than the amplitude of the first harmonic with *β*<0, and for the other one the situation is reversed. This means that the energy of the two modes flows along the film in opposite directions. The *κ*_{b}
grating provides the Bragg resonance between these waves. Condition for this resonance is *mκ*_{b}
=2*β*, where *m* is another integer. Obviously the Bragg and Fano conditions should be satisfied for the same frequency ω_{0}. In order to further reduce losses, one can eliminate coupling of the guided modes with all modes of the free space, apart from the pump wave, by designing a film obeying *β*+*mκ*_{b}
>*k*
_{0}. Under this condition it is possible to develop a coupled mode approach to describe the suggested system.

As this paper is only a proof of principle, we restrict ourselves to a simple one dimensional model. A two dimensional generalization is possible along the lines outlined in Ref. [13], where two-dimensional gap solitons have been considered. Starting from Maxwell’s equations and using the slowly-varying amplitude approximation we have been able to derive the following system for the dimensionless amplitudes *A*± of the two waves counter-propagating along the PC film pumped from the top:

$$\left(i{\partial}_{t}-i{\partial}_{x}+i\Gamma +{\mid {A}_{-}\mid}^{2}+2{\mid {A}_{+}\mid}^{2}\right){A}_{-}+{A}_{+}=0.$$

This system is analogous to the coupled-mode equations describing conventional gap solitons [14], but it includes the external pump and leakage losses. We consider a 4*µ*m thick film made from a highly nonlinear soft glass with refractive index ≃3.1 and *n*
_{2}≃5·10^{-16}
*m*
^{2}/*W*, [15]. The pump wavelength is 1.55*µ*m, the periods of the two gratings are 597nm and 751nm and the two characteristic coupling lengths are 8.2*µ*m and 6.2 *µ*m. In our analysis the space coordinate *x* is measured in units of the second grating coupling length 6.2*µ*m. The time *t* is measured in units of the time 64fs required for the wave envelopes to travel one coupling length and Γ~10^{-3}. The pump parameter *I*=*µB* is the product of the pump field amplitude *B* and the coupling coefficient between the free space waves and the guided waves, *µ*. The fields *A* ± and *B* are normalized so that self phase modulation shifts the phase of the guided mode by one after the propagation distance equals the coupling length. For the chosen parameters the coupling coefficient *µ*=0.0244. The wave amplitudes are normalized so that |*A*±|^{2} and |*B*|^{2} are measured in units of 5.76 · 10^{12}
*W*/*m*
^{2}. *q* is the normalized detuning of the x-projection of the wavevector of the pump wave from *β*, *δ* is the normalized detuning of the pump frequency from the resonant frequencyω_{0}.

First we consider how our system behaves in the linear regime. We seek solutions of the linearized Eqs. (1) in the form *A*
_{±}=*a*
_{±}
*e*
^{iqx-iδt}. The dependences of the energy densities |*A*
_{+}|^{2}+|*A*
_{-}|^{2} vs *δ* are shown in Fig. 2(a,b) for *q*≠0 and in Fig. 3(a,b) for *q*=0. In each of these cases we find two sharp resonances. Each of the peaks corresponds to the resonant excitation of the guided mode of the film with grating. Neglecting the losses and the pump we recover the well known dispersion law $\delta =\pm \sqrt{1+{q}^{2}}$ with the forbidden gap for *δ*∈(-1,1) centered around *q*=*δ*=0. The upper and low branches of this dispersion profile correspond to the right and left resonance peaks, respectively. At *q*=0, see Fig. 3, the eigenmode consists of the two counter-propagating waves with the nearly equal amplitudes. Therefore it does not matter to which of the two waves the external pump is coupled to. That is why the reflection coefficients are practically the same for *δ*=1 and *δ*=-1. As we deviate |*q*| from 0 the non-propagating Bloch mode gradually transforms into a travelling wave with one of the amplitudes |*a*
_{±}| tending to zero. It makes a noticeable difference to the reflection coefficients around *δ*=±1, see Figs. 2(c,d).

The reflection coefficient *R* of the thin film can be expressed as *R*=|*R*
_{0}+*ρA*
_{+}/*B*|, where *R*
_{0} is the reflection coefficient from the homogeneous film and *ρ* is the coupling coefficient between the guided mode and the free space mode for the upper half-space. This reflection coefficient characterizes the power reflected into the main reflection maximum. For the chosen film parameters *R*
_{0}≈-0.58-*i*0.4 and *ρ*=-0.0448-0.027*i*. Note, that there is also the second reflection maximum corresponding to the reflection of the backward component of the Bloch mode. Higher order reflection maxima do not exist for our choice of parameters. The dependence of *R* on *δ* for the two different values of *q* is shown in Fig. 2(c,d) and Fig. 3(c,d), and it has an asymmetric shape, which distinguishes Fano resonances in PC films from Fabry-Perot ones [9, 10, 12, 11]. The asymmetry is less obvious, but still present, in the energy density plots.

Now reintroducing nonlinearity we find that the dependence of the energy density (see the full lines in Fig. 2(a,b) and Fig. 3(a,b)) and the reflection coefficients (see the full lines in Fig. 2(c,d), Fig. 3(c,d)) on the frequency detuning becomes multivalued, i.e., our system is bistable in the vicinity of the Fano resonances. Fig. 4 shows the bistable dependence of the energy density on the pump intensity. This behavior is similar to the one reported in [12] for the Fano resonances found in the channel waveguide resonantly coupled to a nonlinear resonant cavity. To study the stability of the possible homogeneous solutions we make the substitution *A*
_{±}=(*a*
_{±}+*ε*
_{±}
*e*
^{iQx+λt})*e*
^{iqx-iδt}, where *Q* is the wavevector of the perturbation and λ(*Q*) is its growth rate. The middle branch solution has been found unstable already for the spatially homogeneous perturbations with *Q*=0. The stability of the upper and low branches of the bistability curve depends, however, on whether we study the left (*δ*=-1) or the right (*δ*=1) resonance. The former corresponds to lower branch of the dispersion characteristic, $\delta =-\sqrt{1+{q}^{2}}$, and the latter to the upper one. ${\mathit{\partial}}_{k}^{2}$*δ*<0 for the lower branch and >0 for the upper, which corresponds to negative and positive diffraction, respectively. Since the nonlinearity is focusing, we expect and indeed find that the modulational instabilities peak at *Q*≠0 in the spectral proximity of the *δ*=1 resonance. On the other hand, the low branch solution is stable if ${\mathit{\partial}}_{k}^{2}$*δ*<0.

By analogy with other nonlinear optical systems with driving and damping [1, 2] bright localized structures of light superimposed on a stable background exist when the upper branch of the bistability curve is modulationally unstable and the lower one is stable. When the upper branch is stable, as in the spectral proximity of *δ*=-1, then dark LSs are expected. Both types of LSs have been found by direct numerical simulation of Eqs. (1). The transverse profiles of the LSs are shown in Fig. 5 and location of the branch of the bright LS relative to the branch of the homogeneous solution is shown in Fig. 4. Bright solitons move in a spatially nonuniform pump. If the amplitude of the pump is a constant then the velocity of the soliton obeys the law *v*=*q*/(1+*q*
^{2})^{1/2} with very good precision. This relation can be obtained within the framework of the slow varying amplitude approach written for the slow amplitude of the *Bloch* wave, although very precise simulations reveal that there is a small deviation from this law and solitons with *q*=0 are not at rest but moving with very small velocity. This motion is due to the asymmetry of the pump. LSs come to rest only for some critical value of the *q*-parameter, when spatial inhomogeneity of the pump exactly compensates for the imbalance of the pump going into the counter-propagating waves. Another way to arrest the motion is to pump the thin film with the two beams, such that the pump terms in Eqs. (1) are symmetric. More details on control and stability of LSs in nonlinear PC films will be published elsewhere.

This work is supported by the EPSRC grant GR/R74918/01.

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