Abstract
Localized energy states such as two-color gap solitons are theoretically and numerically predicted in a periodic structure in the presence of a frequency doubling nonlinearity. These parametric solitons exhibit appealing features as compared to the Kerr case. Novel effects such as merging and all-optical buffering are envisaged.
©1998 Optical Society of America
1.. Introduction
The interplay between nonlinearity and periodicity allows for the propagation of envelope waves, gap solitons (GSs) after Chen and Mills [1], at carrier frequency within a stopband (gap) such that any linear wave turns out to be exponentially decaying. It is well-known that this rather universal phenomenon finds its optical application in distributed feedback structures where the Bragg effect is responsible for coupling counterpropagating waves whose nonlinear phase shift is affected by self- and cross-induced terms of the Kerr type (for a review see Ref. [2]). The recent experiments performed succesfully in fiber Bragg gratings [3,4] have brought a fresh perspective to the investigation of such new fascinating manifestation of nonlinearity.
In the Kerr case, the simplified notion of the shift of the Bragg condition caused by an intensity-dependent refractive index supports somehow the intuition that the propagation becomes allowed thanks to the nonlinearity. In general, however, this picture is oversimplified, and gap solitary envelopes must be directly searched as solutions of complex nonlinear wave models. Among these, it has been recently shown that GSs of genuine parametric nature can propagate in quadratic media, even far from the cascading limit (i.e., self action first investigated for transverse gratings [5]). These GSs are two-color envelopes bound together through second-harmonic generation (SHG, the generalization to the nondegenerate case can be obviously carried out), either in doubly-resonant (i.e., twin gap) nonharmonic grating [6–12] or in singly-resonant gratings with a negligible resonance at second-harmonic [13,14]. This paper is aimed at discussing some interesting aspects of these parametric GSs. As it will be clear parametric GSs, besides being important per se, constitute also a promising laboratory for investigating the properties of the quadratic solitons recently observed in homogeneous media [15,16].
Usually GSs are studied by means of two different approaches employing either the basis of linear Bloch eigenfunction, or the coupled-mode equations with first-order derivative terms [2]. In Section 2, we show in what sense the two approaches can be considered equivalent in quadratic media. In Section 3 we obtain a reduced coupled-mode model which is widely studied in the theory of parametric solitons in homogeneous media, and discuss the excitation of some class of solutions. Finally Section 4 contains a dicussion of GSs in singly resonant gratings.
2.. Derivation of the governing equations
In this section, following the results obtained for Kerr media [17], we outline the derivation of the coupled-mode equations obtained by means of the Bloch function approach in a periodic medium with quadratic nonlinearity (for more details on the derivation we also refer the reader to Ref. [17]). This method is based on the exact solutions of the linear problem and hence it is better suited than the usual coupled-mode approach to deal with situations where the grating and the nonlinearity are not necessarily a weak perturbation (i.e., shallow and weakly nonlinear gratings) of the propagation. Although Bragg gratings in quadratic materials have already been developed several years ago (see Ref. [18] and references therein), new structures exploiting quasi-phase matching to improve SHG efficiency [19], poled fibers [20], or gratings written through the photorefractive effect [21], appear to be of great interest to assess the experimental feasibility of quadratic gap solitons. The Bloch function approach offers, for instance, the advantage to be easily adaptable to study quasi-phase matched in which also the nonlinear coefficient vary periodically.
In a nondispersive periodic nonlinear medium, the Maxwell’s equations for plane waves read as
with the periodic index n(z) = n(z + d) of period d. After introducing the fields
where n _{0} is a reference index, Eqs. (1) are rewritten in terms of A = [A ^{+}, A ^{-}]^{T} as
where
n̲(z) ≡ n(z)I̲ and B = [B ^{+}, B ^{-}]^{T} acccounts for the nonlinear terms, i.e., explicitly
We assume that only quadratic parametric interactions are effective, as described by the second-order polarization
Few assumptions are implicit in the preceeding relations, the most important being the plane wave approximation and the scalar approach. The former is widely accepted and can be relaxed by introducing an effective area or width in waveguides. The latter is made for simplicity and could be relaxed to account for vectorial or type II SHG.
Furthermore we will neglect the coupling with evanescent or leaky modes.
The starting point in the following derivation is the solution of the linearized (i.e., B = 0) Eqs. (3), A = Ψ _{μ}(z)exp(-iω_{μ}t), written in terms of the orthonormal set Ψ _{μ} (Ψ _{μ} have one to one correspondence to the canonical Bloch eigenfunctions ϕμ of the Sturm-Liouville problem ∂_{zz} ϕ_{μ} + n(z)^{2}(${\omega}_{\mu}^{2}$/c ^{2})ϕ_{μ} = 0 [17]), which obey the eigenvalue equation
The eigenvalues ω_{μ} cannot be inside certain intervals which are named photonic band gaps, whose existence is a direct consequence of the periodic index n(z). In the following we will distinguish two sets of eigensolutions, Ψ _{1μ} and Ψ _{2μ}, for the fundamental frequency (FF) and second harmonic (SH), respectively. The following othornormality condition is satisfied (L = N d, with N integer, is associated to box normalization)
Then we expand A as
with ω _{o1} and ω _{o2} two center gap frequencies such that ω _{o2} = 2ω _{o1} + δω and δω/ω _{o1,o2} ≪ 1 (the actual carrier frequencies at FF ω and SH 2ω are close to ω _{o1} and ω _{o2}, respectively). Then the fields A _{j} , with j = 1, 2 are expanded as
where we adopted the Einstein convention over the summatory. In Eq. (10), Ψ _{jp} is the generic eigenvector of the linearized Maxwell equations for the frequency jω, Ψ _{ju} (Ψ _{jl}) is the eigenvector associated to the eigenvalue ω_{uj} (ω_{lj} ) which correspond to the upper (lower) edge of the gap around ω_{oj} (see Fig. 1). Note that Eq. (10) is general enough to allow for the investigation of the propagation effects in the whole gap.
To obtain a self-consistent system from Eqs. (3), we neglect the generation of higher-order harmonics, hence posing B = B _{1} exp(-iω _{o1} t) + B _{2} exp(-iω _{o2} t) + c.c. for the nonlinear term in Eqs. (3). Then, to find the equations that rule the dynamics of the coefficients ${f}_{j}^{h}$ , assumed to be slowly-varying with z and t in the presence of the nonlinearity, we apply a multiple scale expansion (MSE) with smallness parameter η ≪ 1, and slow variables t_{n} = η^{n}t and z_{n} = η^{n}z with n = 0,1,2… (the fastest scale is associated with the periodic linear response n(z) = n(z _{0})). At second-order (with exp(-iω_{oj}t) → exp(-iω_{oj}t) Eqs. (3) become
where V̲ = diag(c, -c). With the hypotesis that the effects of two Bloch functions bordering the two bandgaps (those denoted by the subscripts u and l in Eq. (10)) are predominant (i.e., ${f}_{j}^{p}$ ≪ ${f}_{j}^{l}$ ,${f}_{j}^{u}$ with p ≠ u, l) and using the orthonormality condition between the Ψ _{jh} , one ends up with the following equations (in terms of the original variables)
$$i\frac{\partial {f}_{j}^{l}}{\partial t}-\frac{{\mathbf{\Delta}}_{j}}{2}{f}_{j}^{l}-{V}_{j}\frac{\partial {f}_{j}^{u}}{\partial z}+{\vartheta}_{j}^{l}=0,$$
with Δ_{j} = ω_{uj} - ω_{lj} the gap widths, V_{j} = iN ^{-1}〈${\mathbf{\Psi}}_{\mathit{\text{jl}}}^{\u2020}$·V̲·Ψ _{ju} 〉 plays the role of group velocity, the nonlinear terms ${{\mathrm{\vartheta}}_{1}}^{r}$ ≡ ω _{o1} ${{\alpha}^{r}}_{\mathit{\text{pq}}}$ (${{f}_{1}}^{p}$)*${{f}_{2}}^{q}$ exp(+iδωt) and ${{\mathrm{\vartheta}}_{2}}^{r}$ ≡ ω _{o1} ${{\alpha}_{\mathit{\text{pq}}}}^{r}$ ${{f}_{1}}^{p}$ ${{f}_{1}}^{q}$ exp(+iδωt) (hereafter all implicit summation are between indeces varying in set {u, l} or equivalently in the set {1, 2}) and the coefficients
$${{\alpha}_{pq}}^{r}\equiv \sqrt{{n}_{0}}\chi {\int}_{0}^{d}{\varphi}_{p,1}\left(z\right){\varphi}_{q,1}\left(z\right){\varphi}_{r,2}\left(z\right)dz.$$
To retrace the usual coupled mode equations for χ^{(2)} Bragg gratings [6,7,8], we make the following change of variables
which can be inverted and interpreted as the application of a vectorial homography Θ to the spinor field E_{j} ≡ [${E}_{j}^{+}$ ${E}_{j}^{-}$]^{T} = [${{E}_{j}}^{1}$ ${{E}_{j}}^{2}$]^{T}, resulting into the spinor field
or equivalently ${{f}_{j}}^{p}$ = ${{\mathrm{\Theta}}^{p}}_{q}$ ${{E}_{j}}^{q}$ , p,q = 1, 2. In terms of these new variables Eqs. (12) are written as
$$i\left(\frac{\partial {E}_{j}^{-}}{\partial t}-{V}_{j}\frac{\partial {E}_{j}^{-}}{\partial z}\right)+{V}_{j}{\mathbf{\Gamma}}_{j}{E}_{j}^{+}+{\tau}_{j}^{-}=0,$$
with V_{j} Γ_{j} ≡ Δ_{j}/2 (Γ_{j} are the usual coupling strengths of the gratings) and
i.e. ${{\tau}_{j}}^{p}$ = ½(${{\mathrm{\Theta}}^{p}}_{m}$ )^{†} ${{\vartheta}_{j}}^{m}$ = (Θ^{-1}${{)}^{p}}_{m}$${{\vartheta}_{j}}^{m}$ . Then it is necessary to express the nonlinear terms of Eqs. (15) in terms of the fields E and this results into the following relations
$$\phantom{\rule{3em}{0ex}}{{H}^{j}}_{hk}\equiv {\left({{\mathbf{\Theta}}^{j}}_{m}\right)}^{\mathbf{\u2020}}{\left({{\mathbf{\Theta}}^{p}}_{h}\right)}^{*}{{\mathbf{\Theta}}^{q}}_{k}{{\alpha}^{m}}_{pq};\phantom{\rule{.2em}{0ex}}{H}_{hk}j\equiv {\left({{\mathbf{\Theta}}^{j}}_{m}\right)}^{\mathbf{\u2020}}{{\mathbf{\Theta}}^{p}}_{h}{{\mathbf{\Theta}}^{q}}_{k}{{\alpha}_{pq}}^{m}.$$
Equations (16) show that the nonlinear terms are in general constituted by all the products of the form E ^{*} E or EE, for the fundamental and the harmonic field, respectively. After some algebra we can write the following explicit expressions for the coefficients H
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+i\left\{{-\left(-1\right)}^{h+k}{{\alpha}_{21}}^{1}-{{\alpha}_{22}}^{2}+{\left(-1\right)}^{j}\left[{\left(-1\right)}^{h}{{\alpha}_{11}}^{2}-{\left(-1\right)}^{k}{{\alpha}_{12}}^{2}\right]\right\},$$
$${{H}_{hk}}^{j}=\left\{{\left(-1\right)}^{j}\left[{\left(-1\right)}^{h+k}{{\alpha}_{11}}^{1}-{{\alpha}_{22}}^{1}\right]+{\left(-1\right)}^{h}{{\alpha}_{21}}^{2}+{\left(-1\right)}^{h}{{\alpha}_{12}}^{2}\right\}+$$
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+i\left\{{-\left(-1\right)}^{h+k}{{\alpha}_{11}}^{2}+{{\alpha}_{22}}^{2}+{\left(-1\right)}^{j}\left[{\left(-1\right)}^{k}{{\alpha}_{21}}^{1}+{\left(-1\right)}^{h}{{\alpha}_{12}}^{1}\right]\right\}.$$
Let us consider a simplification of such formalism. Tipically one considers shallow gratings, i.e., a profile of the refractive index in a doubly resonant structure [6] arising from the first two terms in the Fourier expansion of n(z)
with n _{1,2}/n_{B} ≪ 1. In order to calculate the form of equations, we need to know the Bloch functions in proximity of the band edge of the gaps, in this context they can be written as
$${\varphi}_{lj}\left(z\right)={N}_{j}\mathrm{sin}\left(\frac{j\pi z}{d}\right),$$
where N_{j} are real-valued normalization coefficients. To evaluate the particular form of the system (15) we need to calculate the integrals in Eqs. (13), through the expressions in Eqs. (18). We obtain
$${\alpha}_{ll}^{l}={{\alpha}_{ul}}^{u}={{\alpha}_{lu}}^{u}=-{{\alpha}_{uu}}^{l}=\frac{d}{4}\sqrt{{n}_{0}}{\chi}_{NL}{N}_{1}^{2}{N}_{2}\equiv \alpha .$$
By substituting these relations in Eqs. (17), we find the only nonvanishing coefficients to be
and hence Eqs. (15) reduce to the system
$$i\left(\frac{\partial {E}_{1}^{-}}{\partial t}-{V}_{1}\frac{\partial {E}_{1}^{-}}{\partial z}\right)+{V}_{1}{\mathbf{\Gamma}}_{1}{E}_{1}^{+}-2i\alpha {\omega}_{o1}{{\left({E}_{1}^{-}\right)}^{*}E}_{2}^{-}{e}^{-i\delta \omega t}=0,$$
$$i\left(\frac{\partial {E}_{2}^{+}}{\partial t}+{V}_{2}\frac{\partial {E}_{2}^{+}}{\partial z}\right)+{V}_{2}{\mathbf{\Gamma}}_{2}{E}_{2}^{-}+2i\alpha {\omega}_{o1}{\left({E}_{1}^{+}\right)}^{2}{e}^{i\delta \omega t}=0,$$
$$i\left(\frac{\partial {E}_{2}^{-}}{\partial t}-{V}_{2}\frac{\partial {E}_{2}^{-}}{\partial z}\right)+{V}_{2}{\mathbf{\Gamma}}_{2}{E}_{2}^{+}+2i\alpha {\omega}_{o1}{\left({E}_{1}^{-}\right)}^{2}{e}^{i\delta \omega t}=0.$$
Equations (20) are equivalent to the equations obtained with the standard coupled mode theory. Therefore they demonstrate the equivalence of the two approaches in the case of quadratic gap solitons.
3.. Gap solitons resembling parametric walking solitons in homogeneous media
The purpose of this section is to show how to derive a wide class of GS solutions. By introducing the scaled variables τ = tV _{1}Γ_{1}, ξ = zΓ_{1}, and fields ${E}_{1}^{\pm}$ = [(V _{1}Γ_{1})/(2√2ω _{O1} α)]${w}_{1}^{\pm}$, ${E}_{2}^{\pm}$ = [(iV _{1}Γ_{1})/(2ω _{o1} α)]${w}_{2}^{\pm}$, Eqs. (20) are conveniently rewritten in dimensionless units, as
$$i\left(\pm {w}_{2,\xi}^{\pm}+\frac{{w}_{2,\tau}^{\pm}}{{v}_{2}}\right)+{\kappa}_{2}{w}_{2}^{\mp}+\frac{{\left({w}_{1}^{\pm}\right)}^{2}}{2}{e}^{i\partial \omega \tau}=0,$$
where ∂ω = δωV _{1}Γ_{1} and v_{1} = κ _{1} = 1. The carriers of the envelopes ${w}_{1,2}^{\pm}$ are the Bragg frequencies ω _{o1,o2}. However, it is convenient to introduce explicitly the normalized detunings δ _{1,2} from the Bragg conditions, by means of the transformation ${u}_{m}^{\pm}$(ξ, τ) = ${w}_{m}^{\pm}$(ξ,τ) exp(iδ_{m}v_{m}τ), with m = 1, 2. The new variables ${u}_{m}^{\pm}$ obey the system
$$i\left(\pm {u}_{2,\xi}^{\pm}+\frac{{u}_{2,\tau}^{\pm}}{{v}_{2}}\right)+{\delta}_{2}{u}_{2}^{\pm}+{\kappa}_{2}{u}_{2}^{\mp}+\frac{{\left({u}_{1}^{\pm}\right)}^{2}}{2}=0,$$
where ∂ω ≡ v _{2} Δ _{2} - 2v_{1} δ _{1}. Equations (22) will be routinely used for obtaining the numerical results discussed later.
Returning to the ${w}_{m}^{\pm}$ variables, by introducing a spinor field ϕ_{m} ≡ [${w}_{m}^{+}$ ${w}_{\mathrm{m}}^{-}$] >^{T} and the linear spinor operator L_{m} = L_{m} (ξ,τ) ≡ iσ ^{3} ∂_{ξ} + $\frac{i}{{v}_{m}}$σ^{0} ∂ _{τ}, which makes use of the Pauli matrices
Eqs. (21) may be recast into the compact form
where the nonlinear terms read as
We apply a MSE with the indexed slow variables ξ_{n} = η^{n}ξ and τ_{n} = η^{n}τ (η ≪ 1), by expanding L_{m} as L_{m} = ∑_{n}=0,1,…η^{n} ${L}_{m}^{\left(n\right)}$ where ${L}_{m}^{\left(n\right)}$ = L_{m} (Θ_{n} , τ_{n} ) are “slow” operators. By expanding also the solution in powers of η and retaining first orders which correspond to “small solutions” of Eqs. (21)), we set
$$={\eta}^{2}{\varphi}_{m}^{\left(2\right)}({\xi}_{j},{\tau}_{j};j\ge 0)+{\eta}^{3}{\varphi}_{m}^{\left(3\right)}({\xi}_{j},{\tau}_{j};j\ge 0)=$$
$$=\left[{\eta}^{2}{a}_{m}({\xi}_{j},{\tau}_{j};j\ge 1){f}_{m}^{(\pm )}+{\eta}^{3}{b}_{m}({\xi}_{j},{\tau}_{j};j\ge 1){f}_{m}^{(\mp )}\right]\phantom{\rule{.2em}{0ex}}\mathrm{exp}\phantom{\rule{.2em}{0ex}}\left(i{Q}_{m}{\xi}_{0}-i{\mathbf{\Omega}}_{m}^{(\pm )}{\tau}_{0}\right),$$
where ${\mathrm{\Omega}}_{m}^{(\pm )}$ = ±v_{m} $\sqrt{{{\kappa}_{m}}^{2}+{{Q}_{m}}^{2}}$ are the eigenvalues associated with the linear (i.e., N _{1,2} = 0) solution of Eqs. (24), which yield the dispersion relation. In Eq. (26) one must choose the upper or lower signs of the normal modes f_{m} , depending whether the relative carrier frequency mω is in the proximity of the upper or lower edge of the relative gap, respectively. The normal modes fm ^{(±)} (Q_{m} ) are given by the corresponding eigenvectors
$${f}_{m}^{(-)}\left({Q}_{m}\right)=\frac{1}{\sqrt{2\sqrt{{\kappa}_{m}^{2}+{Q}_{m}^{2}}}}\left[\frac{\underset{{\kappa}_{m}}{\sqrt{\sqrt{{\kappa}_{m}^{2}+{Q}_{m}^{2}}-{Q}_{m}}}}{\sqrt{\sqrt{{\kappa}_{m}^{2}+{Q}_{m}^{2}}-{Q}_{m}}}\right].$$
By developing the MSE at different orders, we obtain equations in terms of the unknowns a_{m} , b_{m} in Eq. (26). The first order yields the link between the b_{m} and the a_{m}
and at next order, the following set of partial differential equations for a_{m} is obtained,
$$i\frac{\partial {a}_{2}}{\partial {\tau}_{2}}+i{\mathbf{\Omega}}_{2}^{\prime}\frac{\partial {a}_{2}}{\partial {\xi}_{2}}+\frac{{\mathbf{\Omega}}_{2}^{\u2033}}{2}\frac{{\partial}^{2}{a}_{2}}{\partial {\xi}_{1}^{2}}+{v}_{2}\chi {a}_{1}^{2}{e}^{-i\left(\overline{\mathbf{\Delta}Q}{\xi}_{1}-\overline{\mathrm{\Delta \Omega}}{\tau}_{1}\right)}=0,$$
$$i\frac{\partial {a}_{1}}{\partial {\tau}_{1}}+i{\mathbf{\Omega}}_{2}^{\prime}\frac{\partial {a}_{1}}{\partial {\xi}_{1}}=0;i\frac{\partial {a}_{2}}{\partial {\tau}_{1}}+i{\mathbf{\Omega}}_{2}^{\prime}\frac{\partial {a}_{2}}{\partial {\xi}_{1}}=0,$$
where ηΔQ̅ ≡ Q _{2} - 2Q _{1} = ΔQ, ηΔΩ̅ = ${\mathrm{\Omega}}_{2}^{\pm}$ - 2${\mathrm{\Omega}}_{1}^{\pm}$ + ∂ω = ΔΩ, and χ, Ω_{m}, Ω"_{m} are nonlinear coefficients, group-velocities, and dispersions, respectively, that is
$${\Omega}_{m}^{\prime}=\frac{d{\Omega}_{m}^{\pm}}{d{Q}_{m}}=\pm \frac{{v}_{m}{Q}_{m}}{\sqrt{{\kappa}_{m}^{2}+{Q}_{m}^{2}}}\phantom{\rule{.2em}{0ex}};{\Omega}_{m}^{\u2033}=\frac{{d}^{2}{\Omega}_{m}^{\pm}}{d{Q}_{m}^{2}}$$
The result of the MSE is better understood by writing explicitly the following approximate solution of Eqs. (22) that permits a direct comparison with the numerics
where ρ_{m} = -sgn [Ω_{'' m}(Q_{m} )], and 𝜜_{1,2} is the solution of the system
$$i\frac{\partial {\bm{\Alpha}}_{2}}{\partial \tau}+i{\Omega}_{2}^{\prime}\frac{\partial {\bm{\Alpha}}_{2}}{\partial \xi}+\frac{{\Omega}_{2}^{\u2033}}{2}\frac{{\partial}^{2}{\bm{\Alpha}}_{2}}{\partial {\xi}^{2}}+{v}_{2}\chi {\bm{\Alpha}}_{1}^{2}{e}^{-i\left(\stackrel{}{\mathbf{\Delta}Q}\xi -\stackrel{}{\mathrm{\Delta \Omega}}\tau \right)}=0.$$
In Eqs. (30), the upper (lower) sign holds for ω_{m} being close to the upper (lower) edge of the relative gap, where also ρ_{m} = -1 (ρ_{m} = 1).
This result, the validity of which arise from Eqs. (26), shows that the two frequency must be in proximity of one of their respective band edges for the nonlinearity to be effective, and the solution must evolve “slowly” in time and space, being “small”. This implies that, in this framework, the allowed solitary solution can only travel at small velocities and carry a limited amount of energy; furthermore, “in-gap” solutions are limited to frequencies not too close to the Bragg resonances, that means, in terms of the detunings δ_{m} , |δ_{m} | ≃ κ_{m} .
Each frequency may be in proximity of the upper branch (UB) or lower branch (LB) of the dispersion curve associated to the linear periodicity (see Fig. 1), i.e., four cases are possible:UB-UB (ω UB and 2ω UB), LB-UB(ω LB and 2ω UB), UB-LB, and LB-LB. Equation (31) can be recast in a form which uses a reduced number of parameters by introducing the variables s = (ξ - Ω_{1} τ) and σ = τ|Ω"_{1}| and the rescaled fields
$${\bm{\Alpha}}_{2}(\xi ,\tau )=\frac{\mid {\Omega}_{1}^{\u2033}\mid}{{v}_{1}}{u}_{2}[\left(\xi -{\Omega}_{1}^{\prime}\tau \right),\tau \mid {\Omega}_{1}^{\u2033}\mid ]\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left\{-i\left[\mathbf{\Delta}Q\xi -\left({\Omega}_{2}^{\u2033}\mathbf{\Delta}{Q}^{2}/2-\mathbf{\Delta}Q{\Omega}_{2}^{\prime}\right)\tau \right]\right\}.$$
We obtain the reduced equation
$$i\frac{\partial {u}_{2}}{\partial \sigma}-\frac{{\rho}_{2}}{2}\gamma \frac{{\partial}^{2}{u}_{2}}{\partial {s}^{2}}-i{\delta}_{W}\frac{\partial {u}_{2}}{\partial s}+\chi \frac{{u}_{1}^{2}}{2}{e}^{i\beta \tau}=0,$$
where γ = |Ω"_{2}| / |Ω"_{1}| is the dispersion ratio, δ_{w} ≡ (Ω_{1} - Ω_{2} + Ω"_{2}ΔQ)/|Ω"_{1}| is the walk-off coefficient, and β = (Ω"_{2} ΔQ ^{2}/2 - ΔQΩ'_{2} + ΔΩ) / |Ω"_{1}| is the equivalent mismatch.
Equations (32) are formally equivalent to the equations which govern the propagation of temporal envelopes in uniform media in the presence of a nonvanishing walk-off, except for the fact that the role of time and space is now interchanged. The solitary solutions of Eqs. (32) are known in the literature [24,25]: they are numerically found as a two-parameter family of bound states. A nonzero walk-off δ_{w} causes the envelope solutions of Eqs. (32) to be complex with a phase curvature which allows the two waves (four, counterpropagating ones, in the case of GSs) at different carrier frequency to travel locked together at a certain velocity in spite of the group velocity difference. While we refer the reader to Refs. [24,25] for a detailed discussion of these solutions which is beyond the scope of this paper, we will discuss below a particular and important case.
3.1 . Gap solitons with strong group-velocity dispersion
Let us consider the case in which the two frequencies experience the maximum available grating dispersion and they are group-velocity matched, that is they are inside their gaps and close to the band edges with Q _{1,2} = 0 (see Fig. 1). In this case Eqs. (32) become reminiscent of those governing SHG in uniform media without walk-off [6]
$$i\frac{\partial {u}_{2}}{\partial \tau}-\frac{{\rho}_{2}}{2}\frac{{v}_{2}}{{\kappa}_{2}}\frac{{\partial}^{2}{u}_{2}}{\partial {\xi}^{2}}+\frac{{v}_{2}}{\sqrt{2}}\frac{1+{\rho}_{2}}{2}\frac{{u}_{1}^{2}}{2}{e}^{-i\beta \tau}=0,$$
with the equivalent mismatch β = 2v_{1} κ _{1} (δ_{1}/κ _{1 } + ρ_{1}) - v _{2} κ _{2}(δ_{2}/κ _{2} + ρ _{2}). Having assumed positive coupling coefficients κ _{1,2}, whenever ρ _{2} = - 1 the nonlinearity is inefficient and no energy localization takes place within the structure, i.e. doubly-resonant localization requires a SH close to its LB to allow a nonzero overlap between the Bloch eigensolutions of the linear periodic wave equation. Looking for solutions ${u}_{m}^{\pm}$ = ${u}_{m}^{\pm}$(ξ, τ) of Eqs. (33) travelling with a velocity V in the cases (ρ _{2} = -1) UB-LB and LB-LB, Eqs. (30) yield [11]
where ζ = √|p _{1}|(ξ-V_{τ} ), A _{1} = |p _{1}|/√2κ _{2}, A _{2} = -ρ _{1} p _{1}/√2, with p_{m} = m ^{2} V ^{2} + 2${\kappa}_{m}^{2}$ (1 + ρ_{m} δ_{m} /κ_{m} ). Here (U _{1},U _{2}) gives the soliton profile as the separatrix solution of the system s_{m} Üm= -∂P/∂U_{m} where the equivalent potetial 2P (U _{1},U _{2}) = -${U}_{1}^{2}$ -${\alpha U}_{2}^{2}$ + ${U}_{1}^{2}$ U _{2}, s_{m} = ρ _{3}-_{m} sign(p_{m} ), which depends ton the single reduced parameter α = ρ _{1} P _{2}/P _{1} (see Ref. [22] for SHG in homogeneous media).
A detailed discussion of the possible GS waveforms is contained in Ref. [6]. Here we point out that in the LB-LB case (ρ _{1} = ρ _{2} = 1) when both fields are inside the gap (δm ∼ -κ_{m} ), U _{1,2} are bright-bright real envelopes (note, however, that from Eqs. (34), GS have a nonreal correction), which are stable at least in the framework of Eqs. (33). They can travel with a (small) velocity V subject to the constrain κ _{2}/v _{2} = 2. In particular, for α = 1, the amplitude of envelopes U _{1,2} have a sech^{2} profile.
It is important to note that is the two frequency are in proximity of the band edges (1 + ρ_{m} δ_{m} /κ_{m} ≃ 0) the parameter p _{1,2} are small numbers in the case of low velocity and so are the amplitudes and the inverse widths of the components of the solution. This ensures the validity of the MSE as previously discussed. Note also that whenever 1 + ρ_{m} δ_{m} /κ_{m} ≃ 0, the equivalent mismatch β is small, and as a consequence one cannot apply the cascading or Kerr-equivalent limit of Eqs. (33).
3.2 . Excitation of slowly moving or still GSs in doubly resonant structures
Focusing onto bright-bright LB-LB solitons with V≠0 (and κ _{2}/v _{2} = 2, without loss of generality) the excitation of such localized distributions inside a reflecting Bragg structure is a key point to address. To this extent, we integrated numerically Eqs. (22) with an FF pulse incident on a semi-infinite nonlinear grating from a medium with the same average index. Launching a Gaussian pulse ${u}_{1}^{+}$ = ${G}_{1}^{+}$ exp[-(τ - τ _{0})^{2}/W ^{2}] in ξ = 0 with ${u}_{1}^{-}$=${u}_{2}^{\pm}$ = 0 and = δ_{1} = δ _{2} = -0.9, as shown in Fig. 2, the SH generated at the interface and inside the grating is partially reflected and partially locked to the transmitted FF, forming a two-color parametric gap soliton, with shape well described by Eqs. (34) and propagation speed V ≃ 0.3, (i.e., 30 % of the natural FF group-velocity). The excitation of a nondispersive wave is a signature of soliton existence for gratings of comparable coupling strengths at FF and SH in the LB-LB case. By changing the sign of the de-tunings and operating UB-UB (similar results are obtained by operating UB-LB and LB-UB), i.e. δ _{1} = δ _{2} = 0.9, no significant light is trapped into the grating, most of the incident power being reflected.
It appeals to intuitition, however, that a slowly-moving simulton cannot evolve into a stationary state, because momentum conservation has to be satisfied. With such constraint in mind, the inelastic collision of two counterpropagating but otherwise similar gap simultons is expected to conserve total momentum and give rise through merging to a still localized state in the grating. Such a possibility, readily inspired by the non-integrable nature of the governing system of Eqs. (33) and by numerical results obtained with reference to in-phase quadratic spatial solitons interacting with opposite transverse velocities [26], was explored by launching identical GSs from opposite ends of the Bragg structure with ${u}_{1}^{\pm}$ (∓L/2, τ) = ${G}_{1}^{\pm}$ exp[-(τ - τ _{0})^{2}/W ^{2}] in a grating extending from ξ = - L/2 to ξ = L/2. It is known, in fact, that nearly in-phase quadratic solitons of Eqs. (33) can merge if their (transverse or temporal) velocities are small [26].
Figure 3 shows the result of a typical numerical simulation carried out for ${G}_{1}^{+}$ = ${G}_{1}^{-}$ = 1 and W = 10. After generating the required SH the input pulses travel as two symbiotic LB-LB simultons which collide in the middle of the structure and coalesce into a self-trapped two-color stationary gap-soliton. We observe a weak breathing typical of the excitation of an internal mode of the soliton. We remark that the final bound state carries about 70 % of the energy associated with the incoming gap-simultons, despite radiation of energy occurring at both FF and SH. In physical units we can estimate the duration of the pulse by t_{FWHM} = 100 ps, and the input peak intensity as 100 MW/cm^{2} (when ${G}_{1}^{\pm}$ = 1) assuming d_{eff} = 12 pm/V (KNbO_{3}), Γ_{1} ∼ Γ_{2} = 0.5 mm^{-1} and V _{1} = c/2.
3.3 . Interrogation process and all optical buffers
Once a still soliton has been excited in a grating structure, the problem that raises is how to measure it. Of course any measurement process needs an interaction, at least in a classical context, and the only way to interact with some energy trapped inside the structure is to send other energy into it. Inside the gap, the only way to reach the still soliton is to use nonlinear waves (i.e., another soliton). Though one could consider the interaction with out gap frequencies which can freely propagate in the system, this is a more difficult task since the condition for an efficient interaction between different frequencies should be satisfied. We consider the interaction between a moving soliton and a still one. The physical insight suggests that the merging process will be repeated, if the new moving soliton is in phase with the old one. What happens is reported in Fig. 4. The still simulton merges with the moving soliton and they form a new moving soliton which travel very slowly since its “mass” is greater than the parent solitons (think about momentum conservation). This process could be interpreted as a reading process in an all optical buffer in which the bit is represented by a still solitary wave. By measuring the time of flight of a soliton in the structure one can infer on the presence of localizations of energy in the system. This is a manifestation of the inherent bistability of nonlinear photonics crystal, indeed the response will depend on “the history” of the structure.
4.. Parametric gap solitons in a singly resonant Bragg grating
There exist interesting physical situations in which only the FF feels a Bragg action due to a periodic linear response, whereas the propagation of the second harmonic is not affected by the grating. This type of structures are obviously refered to as “singly resonant Bragg grating”. Besides the trivial case of a purely harmonic grating there are different situations which fall into this case. Think, for instance, about the polarization dependence of the Bragg gratings usually fabricated in waveguides [23]. If the mode at FF is TE-polarized and the SH mode is TM-polarized and an etching grating is being used, the Bragg strength at SH can be order of magnitudes less than that at FF. In these cases the interaction can be modeled by Eqs. (22) with κ _{2} = 0.
It can be easily seen that, when this condition is satisfied, the MSE previously outlined is no longer valid (terms like 1/κ2 diverges), hence calling for a a different approach. First of all, let us point out that that our intuition might suggest that no bound two-color solitaries should exist in this case, since the linearized equations admits no exponential decaying solution for the SH fields. However, a deeper analysis reveals that the decay can be of nonlinear origin (the SH decays in regions where the FF is strong), as also confirmed by analytical solutions of the bright type [10].
The simplest way to deal with this case is to consider the cascading or Kerr-equivalent limit of Eqs. (22), by expanding the SH fields in power of the inverse mismatch 1/δ _{2}, the leading-order term being given by ${u}_{2}^{\pm}$ = - (${u}_{1}^{\pm}$)^{2}/(2δ _{2}). By using this result in the equations for the FF fields and introducing the Bragg centered amplitudes ${w}_{1}^{\pm}$ = exp(-iv _{1} δ _{1} τ) ${u}_{1}^{\pm}$/√|δ_{2}|, we obtain the following cubic-equivalent model [13,14]
where σ = - sgn(δ _{2}) is the sign of the nonlinearity. Though Eqs. (35) possess also different solutions on a nonvanishing background, we limit ourselves to the case of bright-bright solutions which exist within the region ${\delta}_{1}^{2}$+V ^{2} < 1. This condition defines a “dynamical gap” for solitons moving with velocity V in terms of the detuning δ_{1} from Bragg frequency at FF. The explicit two-parameter solutions of Eqs. (35) reported in Ref. [14] generalize those obtained by Aceves and Wabnitz for focusing nonlinearities (see Ref. 2).
Important enough, applying MSE to Eqs. (35), one obtains the following NLS equation which describes the dynamics of the system when the FF frequency is in proximity of the band edge (i.e., |δ _{1}| ≃ 1)
where the fields read as
From Eqs. (36), it is clear that the NLS-type GSs do exist if the constraint δ _{1} δ _{2} < 0 is fulfilled. However exact solutions of Eqs. (35) do exist even for parameters such that δ _{1} δ _{2} > 0, clearly showing that MSE cannot capture the whole existence domain. To understand why this happens, we observe that near the band-edge (|δ _{1}| → 1), when δ _{1} δ _{2} < 0 the solution of Eqs. (35) tends to zero amplitude (this is typically referred to as the “low amplitude limit”), while on the other hand for δ _{1} δ _{2} > 0 the solutions tends to a finite non zero amplitude. Hence, in the latter case, the requirement of“small solutions” is not satisfied and the corresponding solitary wave cannot be described by a MSE approach. Nevertheless the constraint δ _{1} δ _{2} < 0 appears to be strongly related to the stability of GSs. Indeed the stability analysis of Eqs. (35) has shown that the “low amplitude solutions” appear to be stable (as expected from integrability of the NLS model) against the build-up of both translational and oscillatory instabilities, whereas this is not the case for the “high amplitude solutions” [14]. This fact turns out to be of extreme importance for the understanding of numerical experiments on the excitation of travelling parametric GSs in the singly resonant regime.
With these results in mind we can proceed to establish the existence of singly resonance moving GSs in the regime of small mismatches. The most direct way to handle this problem is to solve numerically the model looking for travelling wave solutions. However some heuristic considerations are in order. In the limit of large mismatches, for any given detuning |δ _{1}| < 1 of the FF, GSs with any absolute velocity lower than the upper bound value V_{max} = √1 - ${\delta}_{1}^{2}$ are admitted. By reducing δ _{2} in principle this maximal velocity can change. Since exponentially decaying linear waves for the FF must be allowed for GSs to exist (SH linear waves do not experience any exponential decay due to the Bragg effect thereby decaying when the FF is still strong), this velocity cannot increase but could evetually decrease. This turns out to be in agreement with the numerical solutions of Eqs. (22). Qualitatively, we find that the maximal velocity V_{max} below which GS solutions of Eqs. (22) do exist, is reduced when |δ _{2}| is decreased, or in other words GSs no longer fill the entire dynamical gap. For large values of mismatches the existence region approach the Kerr-like or cascading limit, while lower values of δ _{2} results into a considerable reduction of critical velocities. When δ _{2} approaches the phase-matching condition (δ _{2} = 0) no bright solution does exist. The results of this analysis tell us that the phase-mismatch is a crucial parameter that rules the dynamics of these GSs, by means of which one can control the velocity of an excited GS.
Let us consider the process of excitation of moving solitons by external FF pulses ${u}_{1}^{+}$ = √psech [(ξ - τ)/W]. We show results that were obtained for W = 5, which are representative of a realistic pulse with FWHM t_{W} = 1.76 × W/(Γ_{1} V _{1}) ≈ 100ps (for typical values of Γ_{1} = 0.5 mm^{-1}, Γ_{2} = 0, and V _{1} = c/2). The dimensionless input peak intensity p = 10 corresponds to 500 MW/cm^{2} for KNbO_{3} (d_{eff} = 12 pm/V). In our numerics we have not been able to observe GSs propagation in the case δ _{1} δ _{2} > 0, whereas in the case δ _{1} δ _{2} < 0 the excited solitons have always velocity comparable with the critical velocity calculated numerically. In this respect the dynamics of the formation process is not that different from parametric GSs with a double resonance. However, some important differences are related to the linear waves present in the system. The known mechanism for the excitation of still solitons in the case of a doubly resonant structure is the inelastic collision of two solitons. This type of behavior cannot be straightforwardly extended to the χ ^{(3)} case, and consequently to the cascading regime of χ ^{(2)} GSs, as it relies on peculiar features of the collision properties. So far we have not observed any merging with Kerr or χ ^{(2)} Kerr-equivalent nonlinearities.
In general, a moving soliton carries a certain amount of total linear momentum, that is conserved (at least in media which can be considered as infinite) during propagation provided the GS interacts neither with other solitons nor with linear waves. During collision the momentum changes and moving GSs can generate a still one. As discussed previously, in the doubly resonant case it is the interaction between solitary waves that allows for the soliton fusion into a still one. In the case of single resonance it is the interaction process with linear waves that plays an important role. The long range propagation in Fig. 5 shows that the formation of a GS with extremely low velocity (vanishing for a sufficiently long time) occurs with sufficiently low mismatches. In this case the deceleration process is likely to be caused by FF radiation which is Bragg reflected toward the GS envelope, thereby carrying momentum contribution of opposite sign. Since the process is entirely spontaneous we propose the term “lazy” GS for this kind of nonlinear waves.
5.. Conclusions
We have presented a theoretical and numerical analysis of ligth propagation in periodic structures in materials with a nonlinear response of the χ ^{(2)} type. Recently, as also witnessed by a large body of literature, solitary waves in quadratic media on one hand, and χ ^{(3)} Bragg solitons on the other hand, have been blooming areas of research. These two fields are enriched by several experiments which confirm the theoretical analisys and open up new areas of investigation. The system that we have considered here can be viewed as a link between these two topical areas. Indeed the dynamics of nonlinear waves exhbits the main features of both quadratic solitons (e.g., two-color trapping) and gap solitons (e.g., nonlinear self-trasparency or tunneling, and low velocity propagation). However, the combination of the quadratic nonlinearity and the bandgap structure offers the possibility to observe new peculiar features, such as, for instance, the mechanisms of generation of nonlinear stationary states discussed in the text. The Bragg effect is responsible for the existence of these states, whereas the dynamics resembles the dynamics of quadratic solitons in homogeneous media.
This work can be extended in several directions to include other effects such as losses, quasi-phase matching structures, higher dimensional structures leading to the propagation of light bullets, etc.. At present, however, the crucial issue is to assess the ideal experimental geometry for the observations of the phenomena described in one spatial dimension. Quadratic gap solitons appear promising in view of the numbers which should require long pulses (of the order of 100 ps, as those used in Kerr Bragg solitons experiments [3,4]), and lower powers due to the large χ ^{(2)} nonlinearity. Furthermore, the technologies for materials which yield efficient second-harmonic generation is mature. On one hand, one possible environment could be very well-known nonlinear crystals such as LiNbO_{3}, with channel waveguides, and gratings written either by reactive ion etching [18], or by phorefractives effects [21]. The necessary phase-matching could be achieved by means of temperature-tuning and/or QPM in the former structure and through QPM in the latter one. On the other hand, also fibers looks promising due to the well-established grating fabrication tecniques (i.e., holographic gratings written through photosensitivity [3,4]) and their potentially long length. Indeed, they have permitted the first observation of cubic gap solitons in spite of their low χ ^{(3)} nonlin-earity. Quadratic nonlinearity as large as 1 pm/V, can be also achieved by means of poling techniques [20], and hence appears very appealing also in view of experiments on quadratic gap solitons.
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