1. Introduction

The appropriate patterning of a transparent dielectric material offers the opportunity to control the properties of light in this medium to a large extent. These properties may go beyond those of any homogenous medium. Even in the simplest case of light propagation in an one-dimensional array of coupled waveguides (photonic lattice) striking anomalies can be observed. It was shown both theoretically and experimentally [1, 2] that diffraction can be controlled in size and sign by the input conditions. Consequently, the leading diffraction term can be suppressed for a certain beam inclination, namely if the central transverse spatial frequency of the beam is located at the centre of the linear band (inflection point of the dispersion relation). Exploiting this property several schemes for efficient switching and routing of low-power signals have been demonstrated in waveguide arrays with cubic and quadratic nonlinearities [3, 4]. Moreover, it has been shown that a novel type of localized solutions exists in this so-called subdiffractive regime in photonic lattices. First, such midband solitons were found in lasers with saturable absorbers [5] and later in an array of passive Kerr cavities (discrete midband cavity solitons - DMCS) [6]. They move with a constant velocity and their central transverse wavevector corresponds to the site of maximum linear group velocity. Because these midband solitons experience a reduced diffraction their size is likewise reduced. The subdiffractive regime can be accessed by a particular tilt of the holding beam. On the other hand discrete cavity solitons (DCS) existing in arrays of coupled cavities have been primarily studied for normal incidence of the holding beam both in the cubic [7] and the quadratic case [8, 9]. Very recently for an arbitrarily inclined holding beam a rich variety of resting and moving DCS was identified [10].

In the limit of strong coupling some DCSs converge to their continuous counterparts (cavity solitons - CS), which are well studied localized solutions in wide-aperture nonlinear cavities (see for reviews [11, 12]). They can be switched on and off at arbitrary positions in the transverse plane of the resonator. Usually they form zero-parameter families and their features are entirely fixed by the underlying optical system [12]. The direction and velocity of CS motion is fixed either by the tilt of the holding beam or by any transverse gradients of the system parameters [13]. However, unlike CSs in continuous systems moving and resting DCSs can coexist [9, 10].

In this paper, we shall demonstrate the existence of equivalent, but counterpropagating DCSs in arrays of quadratically nonlinear cavities driven by a second harmonic (SH) holding beam (Fig. 1). Like for DMCS in the Kerr case [6], the mean transverse spatial frequency of the FH is situated at the inflection point where diffraction is suppressed. The balance between fast fundamental harmonic (FH) and slow SH waves determines the velocity of the DMCS and influences drastically their existence domain. We shall consider the collision of DMCSs.

 

Fig. 1. (Color online) An array of coupled cavities endowed with quadratic nonlinearity and driven by the holding beam at SH.

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2. Discrete midband cavity solitons

We assume that the cavities are resonant for both the FH and SH waves and that a mean-field approach can be applied. The appropriately scaled evolution equations for the slowly varying envelopes of the transmitted FH and SH fields read as [8, 9]

where Δ_1,2 are the detunings of both fields from the cavity resonance. The time T, the coupling constants C _1,2 and the detunings are scaled by the FH photon lifetime and δ is the ratio of the FH/SH photon lifetimes. The doubly-resonant degenerated optical parametric oscillator (DOPO), considered here, requires the pump at the SH (amplitude E ₀) instead at FH for the SHG case [8, 9]. We also allow for an inclination of the holding beam by introducing a phase shift q between the field incident on adjacent cavities. In what follows, we assume that the nearest cavities are pumped out-of-phase (q=π). Because of the much weaker field confinement of FH guided modes in comparison to SH, the diffraction properties are exclusively determined by the FH field (C ₂=0).

Equation (1) admits stationary discrete plane wave (PW) solutions in the form u_n=b ₁exp(iq ₁ n), ν_n=b ₂exp(i q ₂ n). Note that the phase gradient q ₁ induces an additional detuning. Therefore, it is convenient to introduce an effective detuning as Δ′₁=Δ₁+2C ₁(cosq ₁-1). Then the PW amplitude does not depend on q _1,2 provided that the effective detuning Δ′₁ remains constant. The analysis resembles that in the continuous DOPO [14–20]. One of the PW solutions is trivial with the zero FH amplitude (b ₁=0). The nontrivial branch bifurcates at E _B=[(Δ′² ₁+1)(Δ² ₂+δ²)]^1/2 either supercritically, if Δ′₁Δ₂-δ>0, or subcritically, if Δ′₁Δ₂-δ>0, from the trivial branch of PW solutions. Taking into account the inclination of the holding beam (we assume q=π) the nontrivial PW solution has the form:

where the moduli are |b ₁|²=Δ′₁Δ₂-δ±[E ² ₀-(δΔ′₁+Δ²)²]^1/2 and|b ₂|²=Δ′² ₁+1 (see [19]). The FH component of these nontrivial PW solutions is degenerate, with phase difference π. As a result the holding beam with q=π generates two PW solutions with equal amplitudes but with opposite signs of inclination [solutions u ⁺ _n and u ^- _n in Eq. (2)].

Like in the Kerr case [6] we expect that there are robust bright DMCSs which propagate across the array with constant amplitude and width provided that the PW response curve is bistable. Like the nontrivial PW solution [Eq. (2)] the DMCS centre represents a local nonlinear cavity resonance, whereas the trivial PW provides the background. Therefore the nontrivial PW solution [Eq. (2)] should contribute considerably to the spatial spectrum of the DMCS. However the group velocities of the spatial harmonics (q ₁=π/2 and -π/2) have opposite directions. Which direction the DMCS would prefer in the DOPO? To identify this we performed numerical simulations of the discrete model (1). The narrow initial FH pulse without inclination excites counterpropagating nonlinear waves which under certain conditions move with constant amplitude and width and are therefore DMCSs [Fig. 2(a)]. There are two DMCSs corresponding to the PW solution of Eq. (2) with ‘+/-’ signs. The exciting beam has to be sufficiently narrow to ensure that its spatial spectrum covers both spatial components (π/2 and -π/2). For example, the excitation of DMCSs fails if the exciting beam is wider than two lattice periods [Fig. 2(b)]. Note that counterpropagation of DMCS is due to the quadratic nonlinearity, in contrast to the Kerr case where the propagation direction of DMCS is unambiguously fixed by the system parameters [6].

 

Fig. 2. (Color online) Excitation of the counterpropagating MDCSs by means of a narrow additional pulse with the width equal to one (a) or two (b) cavities. (c) Amplitude profile of the MDCS. The diamonds depict the results of discrete model (1) whereas the full line corresponds to the results of quasi-continuous approximation (3). Parameters: E ₀=7.25, Δ′₁=-2, Δ₂=-3, δ=2, C ₁=0.7.

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Only one of the two counterpropagating DMCSs can be excited provided that an appropriate inclination of the exciting FH beam is chosen. Fig. 2(c) shows the amplitude profile of DMCS which moves to the right. The mirror image of this DMCS is also a solution of Eq. (1) moving to the left. Like in the case of cubic nonlinearity [6] the DMCS profile exhibits an asymmetric shape and it sheds radiation to diffracting waves, which leads to the formation of characteristic oscillating tails in the wake of the moving localized structure.

3. Existence domain of DMCSs

To study the existence domain of the DMCSs we use a quasi-continuous model similar to that derived for the cubic case [6, 10]. We introduce a new transverse coordinate x and the continuous functions u(x), ν(x), which are by definition the envelopes of the discrete solution of Eq. (1) with the central spatial harmonics q ₁ and q ₂: u(x=n)exp(i q ₁ n)≡u_n, ν(x=n)exp(i q ₂ n)≡ν_n. Then we expand the functions u _n±1 up to the third order in the Taylor series. The inclination of the holding beam determines the central spatial harmonic of SH waves as q ₂=q, whereas the nonlinear parametric effects are responsible for the generation of the FH waves with inclination q ₁=q ₂. Then we obtain the equations for the envelopes:

where the diffraction coefficients read as D ⁽¹⁾=2C ₁sin(q/2), D ⁽²⁾=C ₁cos(q/2), D ⁽³⁾=(C ₁/3)sin(q/2). The odd diffraction terms disappear for normal incidence of the holding beam (q=0 and D ⁽¹⁾=0, D ⁽³⁾=0) and the model (3) becomes similar to the ordinary continuous one which describes the doubly resonant planar cavity endowed with the quadratic nonlinearity. For example, modulational instability, pattern formation [14–16] and CSs [17–20] were studied in the continuous doubly resonance DOPO.

 

Fig. 3. (Color online) Maximal amplitude of the bright MDCS depending on the coupling constant (a) and on the holding beam amplitude (b). The solid parts of these branches correspond to stable and the dashed ones to unstable solutions. The parameters are as in Fig. 2.

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Third-order diffraction must be accounted for if the holding beam inclination increases. It reaches the maximum D ⁽³⁾=C ₁/3 provided that adjacent waveguides are pumped out-of-phase (q=π) where the FH second-order diffraction term is equal to zero. It corresponds to the case of diffractionless beam propagation in an array of evanescently coupled linear waveguides [1, 2]. The FH group velocity attains the maximum D ⁽¹⁾=2C ₁.

Equation (3) admits localized stationary solutions in a reference frame moving with a constant velocity, yet undetermined, to be calculated self-consistently together with the field profile. The continuous profile, calculated from the model (3), coincides almost exactly with the profile of the discrete model (1) [full curve in Fig. 2(c)]. Third-order diffraction is responsible for the spreading which is compensated by the nonlinear effects. Therefore the asymmetric shape of DMCS is not surprising.

Using the quasi-continuous model (3) we investigate the existence domains of DMCSs depending on the key control parameters like the coupling strength and the amplitude of the holding beam. The nonlinear coupling between waves of FH and SH forms the moving DMCS. However, FH and SH waves have different transverse group velocities. The balance between fast FH and slow SH waves (because of C ₁≫C ₂) determines the velocity of the DMCS and restricts their existence domain. Therefore, unlike the solutions found for cubic nonlinearity [6], stable DMCSs exist only for a small interval of the coupling constant C ₁ [Fig. 3(a)].

In conventional planar cavities CSs belong to branches which bifurcate from critical points of the PW solution [11]. We identified two qualitatively different types of branches of DMCSs depending on the coupling constant. For the coupling constant increasing beyond a critical value the DMCS branch bifurcates subcritically from the limiting point of bistability of the PW [curve C ₁=0.7 and inset in the Fig. 3(b)]. This is the usual behaviour for continuous CSs [11]. However if the coupling constant is smaller than a critical value the DMCS branch does not connect with the PW solution at all [curve C ₁=0.5 in the Fig. 3(b)]. The DMCSs branch is multistable, in other words, there is a discrete set of stable solutions which exists for the same system parameters. In general the bistability of the PW solution is a necessary condition for the existence of DMCSs. Note that models (1) and (3) allow for the existence of bright DMCSs not only for the negative cavity detunings but for positive ones as well. For fixed absolute values of the detunings these solitons are identical.

4. Collision of DMCSs

Because of the dissipative nature of the system the continuous DOPO shows reach interaction dynamics of CSs. Numerical studies revealed regimes in which CS pairs fuse, repel, or form bound states [20]. However in the continuous cavity case CSs have similar (close to zero) drift velocities and start to interact only in the close vicinity of each other. In contrast the existence of identical, but counterpropagating localized solutions in the discrete system allows to realize a novel soliton-based all-optical switching scheme. As it was mentioned both DMCSs can be excited separately by means of appropriately tilted excitation pulses [Fig. 4(a)]. A typical interaction scenario of DMCSs is presented in the Fig. 4(b). Both DMCSs disappear usually after collision. Indeed, the FH components are out-of-phase; therefore, they are annihilated due to destructive interference.

 

Fig. 4. (Color online) (a) Moving DMCS. (b) Interaction dynamics of the counterpropagating MDCSs (b). The parameters are as in Fig. 2.

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5. Conclusions

In conclusion we predicted the existence of discrete midband cavity solitons in an array of coupled quadratically nonlinear cavities driven by a SH holding beam. We have shown that the quadratic nonlinearity allows for the formation of identical but counterpropagating soliton solutions. This is in contrast to the Kerr case. Moreover the balance between fast FH and slow SH waves determines the velocity of the DMCS and restricts their existence domain. Using the effective equations for the continuous envelopes we investigated the existence domain and calculated the profile of the DMCSs. The collision dynamics of the counterpropagating DMCS was studied as well.