Abstract
Propagation of the temporal soliton in Kerr-type photonic crystal waveguide is investigated theoretically and numerically. An expression describing the evolution of the envelope of the soliton based on the full-wave modal analysis, taking into account all space-harmonics, is rigorously obtained. The nonlinear coefficient is derived, for the first time, based on a modification of the refractive indices for each space-harmonic due to the Kerr-type nonlinearity. For illustrating the general formulation and results, we performed extensive computational electromagnetics simulations for the propagation of gap solitons in an experimentally feasible photonic crystal waveguides, endorsing the correctness and usefulness of the proposed formalism.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
After John Scott Russell’s experimental discovery and the subsequent mathematical formulations by Rayleigh and Boussinesq, the generality of solitonic behaviour has been recognized in many branches of physics, electronics and biology [1–3]. Several theoretical and numerical techniques have been developed so far to analyze the formation and the propagation of solitons (kink-solitons, spatial and spatio-temporal solitons, gap solitons, etc.) in different kind of media [4]. In general, the main requirement for the propagation of solitons is the one- or quasi-one dimensionality of the medium except for the spatial soliton [5] for which a beam propagation direction plays the role of time. In photonics a quasi-one dimensionality can be achieved either by sharp refractive index change between the guiding layer and the surrounding medium [6], or by the photonic crystal (PhC) along (PhC fiber [7]) or perpendicular (PhC waveguide [8]) to the direction of the soliton propagation.
The transmission of localized energy in a structure with a periodically varying refractive index, such as 1D Bragg grating [9] or 2D PhC [10], occurs due to the modulation of corresponding Bloch modes. In weakly nonlinear systems, the problem could be solved by multiple-scale perturbation method and the soliton propagation is described by the nonlinear Schrödinger equation for a slowly-varying envelope of the soliton [11,12]. Planar PhC waveguides, which are designed by removing one or a few rows from the original PhCs, are considered as very promising candidates for various realizations of optical devices on a chip. Moreover, PhC waveguides guarantee a very good confinement of the modes in the slow light regime [13].
The main tool to deal with the nonlinearities is variational Ansatz, from which a soliton shape is obtained [14,15]. Here, for the first time, a theoretical framework for the propagation of temporal solitons in PhC waveguides based on a multi-harmonic treatment of the nonlinear setup is presented. The method is based on a full-wave modal analysis using the Floquet-Bloch theorem. A rigorous expression for the nonlinear coefficient (due to Kerr-type nonlinearity) based on a modification of the refractive indices of each space-harmonic is derived. For the sake of confirmation and illustration we performed numerical simulations for the propagation of gap-solitons. The gap-solitons are formed if the power of the launched signal exceeds a certain threshold value [16–18]. Their formation time is relatively short and the registered gap soliton inside the guiding structure can be used for comparison with the analytical solution of the nonlinear Schrödinger equation. A very good agreement between the numerical and theoretical results is demonstrated.
2. Formulation of the problem
Without loss of generality we analyze three coupled symmetric PhC waveguides composed of a hexagonal lattice made of circular air-holes that are periodically distributed along the ${x}$-axis with a common period ${h}$, as shown in Fig. 1. The structure as such is already well studied [19]. The guiding regions ${(a)}$, ${(b)}$ and ${(c)}$ having the same width ${w}$ are separated by the barrier layers of two PhCs with a barrier thickness according to the number of layers $N_B$. The radius of the air-holes is ${r}$ and $\epsilon _s=\ n_s^{2}$ is the relative dielectric permittivity of the background material. The number of layers ${N}$ of the upper and lower PhCs is taken large enough in order to minimize the leakage of the power along the transverse ${y}$ - axis. The modulated ${x}$ component of the electric field $E_x(x,y,t)$ propagating in the nonlinear coupled PhC waveguides can be written in the following form:
The linear system of Eqs. (6)–(11) can be re-written in the following compact form:
Once the non-linear dispersion relation $\omega (k_{x0},|\Psi |^{2})$ for weak nonlinearities is defined from (13) and (14), there exists a well-established procedure to reduce the problem to a nonlinear Schrödinger equation using the multiple-scale treatment of the wave equations [11]. Following the analysis described in [24], we find the following expression:
3. Numerical results and discussions
We analyze a 2D model system of three coupled PhC waveguides composed of experimentally feasible planar hexagonal lattice of air holes formed in a dielectric nonlinear background medium with a linear refractive index $n_s = 2.95$ (crystalline silicon) in conjunction with a Kerr-type nonlinearity [19]. This 2D model has proven to be a very good approximation of the original 3D structure. The thickness of the upper and lower PhCs is taken $N=5$ and the radius of the air-holes is $r = 0.32h$, where $h$ is a period of the PhCs. The barrier layers are composed of 1-layered structures, $N_B=1$, and the length of the PhC is $30h$. The dispersion diagram of the structure is shown in Fig. 1. Here, we are interested only in the symmetric mode described by the blue line, since this mode is responsible for the formation of the gap soliton. The dispersion diagram of this mode can be well approximated by the parabola taking into account the terms up to the square of the angular frequency and the group velocity dispersion amounts to $\frac {\partial ^{2}\omega }{\partial k_{x0}^{2}} \approx 1.9 \frac {hc}{2\pi }$. A full-wave computational electromagnetics analysis is conducted based on the finite-difference time-domain (FDTD) method [25] together with uniaxial (perfectly matched layer) PML at the operating normalized frequency $\frac {h\omega }{2\pi c} = 0.232$ (Fig. 1).
A continuous wave (CW) signal with $(H_z,E_x,E_y)$ is launched through the middle waveguide of the coupled PhC waveguides [guiding region ${(b)}$]. The injected peak power of the CW signal is chosen as $\chi ^{(3)}E_0^{2} = 0.1410$, which means that for silicon with the nonlinear refractive index $n_2={3}\cdot {10^{-18}}[m^{2}\cdot W^{-1}], E_0^{2} = {2.7}\cdot {10^{18}} [V^{2}\cdot m^{-2}]$ (we excite the waves very close to the left edge inside the PhCs). A nonlinear medium due to the Kerr effect results in a dispersion shift of the symmetric mode (blue line in Fig. 1) to the lower frequencies yielding the formation of the gap soliton. The shift of the angular frequency due to the Kerr-type nonlinearity can be calculated theoretically based on (17) and it amounts to $\frac {h\Delta \omega }{2\pi c}=0.0016$. Figure 3(a) depicts the magnetic field distribution of the gap soliton propagation. From the numerical simulations it follows that the maximum $F$ and the width $\Lambda$ of the gap soliton exhibiting a stable profile propagating inside the waveguide, amounts to $F=300 A/m$ and $\Lambda =4.10 h$, respectively.
Figure 3(b) illustrates the longitudinal dependence of the magnetic field $H_z$ versus the dimensionless parameter $x/h$ at $y=0$ [cf. Figure 3(a)], where [a.u] stands for arbitrary units. Numerical results from the FDTD analysis are shown by the blue line, whereas the theoretical result is indicated by a dashed red line. The latter we find by substituting (16) and (17) together with (6)–(11) into (1) and (2) from which the magnetic field $H_z$ is easily retrieved. Calculating the nonlinear coefficient $\gamma$ from (14), the width of the bandgap soliton $\Lambda$ is directly obtained from (17). Based on our theoretical analysis, the width of the bandgap soliton amounts to $\Lambda =3.95 h$, which is in a very good agreement with the numerical result ($\Lambda =4.10 h$). It is worth emphasizing that only the scattering amplitudes of the space harmonics $m = - 1$ and $m = 0$ gives rise to the formation of the electromagnetic field (1), while the scattering amplitudes of other space-harmonics are very small and can be neglected. All scattering amplitudes are calculated by solving for the eigenvalue problem in the linear regime defined as ${\mathbf {\Omega }} (\omega , k_{x0}, n_s) \cdot {\mathbf {A}}^{T} = 0$.
Figure 3(c) depicts the the transversal dependence of the magnetic field $H_z$ versus $y/h$ at $x=0$ [cf. Figure 3(a)]. The numerical results of the FDTD calculations are depicted as blue line, whereas the theoretical results are indicated by the dashed red line. The latter is obtained by substituting (6)–(11) into (1) from which the magnetic field $H_z$ is inferred. A very good agreement is observed between the results obtained from our theory and those that follow from the corresponding full-wave computational electromagnetics simulations.
4. Conclusion
A rigorous theoretical formulation describing the evolution of the envelope of temporal solitons propagating in Kerr-type nonlinear PhC waveguides has been proposed. The formalism is based on the full-wave modal analysis and is very general in nature and can be applied to various configurations of planar PhCs including the most challenging ones, such as plasmonic crystals with intrinsic losses. We investigated an experimentally feasible hexagonal lattice formed by air-holes in crystalline silicon. The expression for the nonlinear coefficient has been rigorously derived while accounting for all the space-harmonics and the interactions between them. The width of the soliton and the shift of the angular frequency due to Kerr nonlinearities have been analytically calculated. Extensive computational electromagnetics simulations based on the FDTD were performed demonstrating the correctness of the proposed formalism.
Appendix A
In case of a multilayered periodic structure such as a PhC (Fig. 1), the scattered space harmonics are conceptualized as new incident waves on the neighbouring PhC arrays and thus scattered into another set of space harmonics, which then impinges back on the original array. The scattering process from each layer is described by reflection and transmission matrices, which relate a set of the incident space harmonics to a set of reflected and transmitted ones. Reflection ${\mathbf {r}_{j}}$ and transmission ${\mathbf{f}_{j}}$ matrices for ${j}$-th layer are derived as follows [22]:
Appendix B
A schematic view of a guiding region ${(a)}$ having a refractive index $\tilde {n}_s=n_s + \delta n$, whereas the refractive index of other regions is equal to $\ n_s$ is depicted in Fig. 2. In this case we need a slight modification only of (6) and (7) by implementing the diagonal Fresnel matrices. Other expressions are the same as in (8)–(11). Expressions in (6) and (7) are re-written in the following form:
Acknowledgments
V. Jandieri would like to express his sincere gratitude to Professor Kiyotoshi Yasumoto from Kyushu University, Japan for fruitful discussions about the theory of photonic crystals.
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