Non-Bragg-gap solitons in one-dimensional Kerr-metamaterial Fibonacci heterostructures

A detailed study of non-Bragg-gap solitons in one-dimensional Kerr-metamaterial quasiperiodic Fibonacci heterostructures is performed. The transmission coefficient is numerically obtained by combining the transfer-matrix formalism in the metamaterial layers with a numerical solution of the nonlinear differential equation in the Kerr slabs, and by considering the loss effects in the metamaterial slabs. A switching from states of no transparency in the linear regime to high-transparency states in the nonlinear regime is observed for both zero-order and plasmon-polariton gaps. The spatial localization of the non-Bragg-gap solitons is also examined, and the symmetry properties of the soliton waves are briefly discussed.

Figure 1

Pictorial view of some sequences $S_m$. The sequences ${\mathrm{\Omega }}_m$ resulting from suppressing the two last elements of $S_m$ (see the end of each sequence) are depicted as bold letters. Arrows indicate the approximate position of the inversion center of each ${\mathrm{\Omega }}_m$. A schematic representation of the building blocks A and B, as well as the Fibonacci heterostructure corresponding to the sequence $S_6$, is also displayed.

Figure 2

Photonic band structure, for normal incidence, of periodic heterostructures with unit cells satisfying the Fibonacci building rule with $a=b=10$ mm. Results were obtained, in the vicinity of the zero-order ($\langle n\rangle =0$) gap, by using ${\epsilon }_{\text{A}}={\epsilon }_{\text{A}}^0=2$ [i.e., $\alpha =0$ in Eq. (1)] and ${\mu }_{\text{A}}=1$ in layers A, and $\gamma =0$ in Eqs. (2) and (3) for slabs B. Solid, dashed, and dotted lines correspond to $S_{12}, S_{10}$, and $S_8$ Fibonacci heterostructures in the unit cell, respectively.

Figure 3

Transmission coefficient as a function of the wave frequency, for the same set of parameters used in Fig. 2. Dashed and solid lines correspond to the Fibonacci heterostructures $S_{10}$ and $S_{12}$, respectively, with $a=b=10$ mm. The vertical dashed line is located at $\nu =3.0557$ GHz in the vicinity of the lower edges of the zero-order ($\langle n\rangle =0$) gaps of the $S_{10}$ and $S_{12}$ Fibonacci heterostructures. At this frequency value the transmission coefficients of the $S_{10}$ and $S_{12}$ heterostructures coincide ($T=0.1458$).

Figure 4

Transmission coefficient for normal incidence, as a function of the defocusing nonlinearity power, corresponding to the Fibonacci heterostructures (a) $S_{10}$ and (b) $S_{12}$, with $a=b=10$ mm. The linear parameters are the same as the ones used in Fig. 3. Results were obtained for $\nu =3.0557$ GHz, in the vicinity of the bottom of the zero-order ($\langle n\rangle =0$) gap (cf. the vertical dashed lines in Fig. 3). Calculations were performed for phenomenological loss and absorption parameters $\gamma =0, \gamma ={10}^{-3}$ GHz, and $\gamma ={10}^{-2}$ GHz. In each panel, for each value of $\gamma$, dots are at the local maxima of the transmission coefficient. In the inset of (b) we have enlarged the interval of the defocusing nonlinearity power in order to display the three first local maxima of the transmission coefficient for $\gamma ={10}^{-2}$ GHz.

Figure 5

Zero-order ($\langle n\rangle =0$) gap soliton for normal incidence corresponding to the first, second, and third local maxima [(a), (b), and (c), respectively] of the transmission as a function of the defocusing nonlinearity power [cf. Fig. 4] in the $S_{10}$ Fibonacci heterostructure with $a=b=10$ mm at $\nu =3.0557$ GHz. Solid, dashed, and dotted lines correspond to phenomenological loss and absorption parameters $\gamma =0, \gamma ={10}^{-3}$ GHz, and $\gamma ={10}^{-2}$ GHz, respectively. Results for $\gamma ={10}^{-2}$ GHz are not depicted in (b) because of the absence of the two-soliton peak in Fig. 4 for this level of absorption.

Figure 6

Zero-order ($\langle n\rangle =0$) gap soliton for normal incidence corresponding to the first, second, and third local maxima [(a), (b), and (c), respectively] of the transmission as a function of the defocusing nonlinearity power [cf. Fig. 4] in the $S_{12}$ Fibonacci heterostructure with $a=b=10$ mm at $\nu =3.0557$ GHz. Solid and dashed lines correspond to phenomenological loss and absorption parameters $\gamma =0$ and $\gamma ={10}^{-3}$ GHz, respectively. Results for $\gamma ={10}^{-2}$ GHz are not displayed because of the absence of the one-soliton, two-soliton, and three-soliton peaks in Fig. 4 for this level of absorption.

Figure 7

Photonic band structure for oblique incidence ($\theta =\pi /24$) and TE configuration in the linear regime [$\alpha =0$ in Eq. (1)] of periodic heterostructures with unit cells satisfying the Fibonacci building rule. Results were obtained for frequency values around the PP gap, with parameters ${\epsilon }_{\text{A}}={\epsilon }_{\text{A}}^0=2$ and ${\mu }_{\text{A}}=1$ in layers A, $\gamma =0$ in Eqs. (2) and (3) for slabs B, and $a=b=10$ mm. Solid, dashed, and dotted lines correspond to $S_6, S_4$, and $S_2$ Fibonacci heterostructures in the unit cell, respectively. (b) displays a zoom of the numerical results depicted in (a) around the PP gap, in order to capture the fine details of the band structure.

Figure 8

Transmission coefficient for the TE configuration with incidence angle $\theta =\pi /24$, as a function of the wave frequency, for $\alpha =0$ in Eq. (1) and $\gamma =0$ in Eqs. (2) and (3). The calculated result corresponds to the Fibonacci heterostructure $S_{12}$, with $a=b=10$ mm. The vertical dashed line is located at $\nu =5.0490$ GHz in the vicinity of the lower edge of the TE longitudinal magnetic PP gap. At this frequency value one has $T=0.01$.

Figure 9

Transmission coefficient of the TE modes with incidence angle $\theta =\pi /24$ as a function of the defocusing nonlinearity power in the Fibonacci heterostructure $S_{12}$, with $a=b=10$ mm. Results were obtained at $\nu =5.0490$ GHz in the vicinity of the lower edge of the TE PP gap (cf. the vertical dashed line in Fig. 8), and for three different values of the phenomenological loss and absorption parameters $\gamma$ of layers B. In all panels, for each value of $\gamma$, dots are located at the local maxima of the transmission coefficient.

Figure 10

TE plasmon-polariton gap solitons for oblique incidence ($\theta =\pi /24$) in the Fibonacci heterostructure $S_{12}$, with $a=b=10$ mm. Solid, dashed, and dotted lines correspond to phenomenological loss and absorption parameters $\gamma =0, \gamma ={10}^{-5}$ GHz, and $\gamma ={10}^{-4}$ GHz, respectively. Results corresponding to the different values of the damping constant $\gamma$ were obtained for the first five local maxima of the transmission coefficient as labeled in Fig. 9.