Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects

We consider a Fabry-Perot resonator (FPR) comprised of two off-channel nonlinear defects coupled to the photonic waveguide. For the linear case FPR can support bound states in the form of standing waves between the defects if a distance between them is quantized. For the nonlinear case the bound states appear for arbitrary distance between the defects if electromagnetic intensity is quantized. For transmission through the FPR we reveal additional resonances which are the result of coupling of incident wave with the bound states because of nonlinearity of the defects. The resonances are spaced at the eigenfrequencies of bound states with a width proportional to the input amplitude. The theory of the FPR based on the simple Wang and Fan model [Phys. Rev. E 68, 066616 (2003)] is complemented by the tight-binding one. The results for the transmission and bound states in these models agree with computations in real two-dimensional photonic crystal waveguide coupled with two off-channel defects fabricated from a Kerr-type material.

Figure 1

(a) Photonic crystal consists of a square lattice of dielectric rods. A single row of rods is extracted to form one-dimensional directed waveguide. Two nonlinear defect rods marked by filled circles are inserted into the photonic crystal. (b) The nonlinear model of FPR. (c) The tight-binding version of the photonic crystal system in (a), respectively.

Figure 2

Transmission spectra of the single off-channel nonlinear defect given by Eq. (9) for $\lambda =0.2$ and $\Gamma =1$. The dash line shows the linear case. The nonlinearity shifts the eigenfrequency of the defect at which the transmission equals zero.

Figure 3

(Color online) (a) Numerical solution of Eq. (15) for the parameters $\Gamma =1$, ${\theta }_0=0.7\pi$, ${\theta }_1=\pi$, $\lambda =0.2$, and $E_{in}=0.1$. (b) Blow up of the solution for $X$ (red line) and $Y$ (blue line) around the second BSC point $X_{c2}=Y_{c2}=6.5$ and ${\omega }_{c2}=1.3$. (c) Change in the intensity at the second defect under increasing of the input amplitude two times. Respectively, solid line shows $Y$ at $E_{in}=0.1$ and dashed line shows $Y$ at $E_{in}=0.2$.

Figure 4

(Color online) (a) Transmission spectra as dependent on frequency in the FPR given by Eq. (11) for the nonlinear case $\lambda =0.2$. The parameters of the FPR are given in Fig. 2. The dash line shows the transmission that slightly differs from the transmission in the linear FPR and solid line does a picket-fence resonances induced by BSCs (BSC induced resonances). Blows up of the BSC resonances for the BSC frequency equaled to (b) 0.3, (c) 1.3, and (d) 2.3, respectively. In Fig. 4b blue lines show unstable branches of the solution while red lines do the stable branches.

Figure 5

(Color online) Transmission spectra and intensities at the off-channel defects as dependent on the frequency in FPR with growth of the incident amplitude [(a) and (c)] $E_{in}=0.12$ and [(b) and (d)] $E_{in}=0.2$. Other parameters of the FPR are given in Fig. 3.

Figure 6

(Color online) The frequency behavior of complex eigenvalues $\left|z_{1,2}-\omega \right|$ given by formula (23). $z={\omega }_0+2.3$ defines the frequency of the BSC in the nonlinear FPR.

Figure 7

(Color online) (a) Transmission spectra of the linear FPR for $\theta =\pi +0.1\pi$ (solid line) and $\theta =\pi -0.1\pi$ (dashed line). [(b)–(e)] The transmission given by Eq. (11) in the nonlinear FPR for the parameters $\Gamma =1$, ${\theta }_0=\pi$, and ${\theta }_1=\pi$. The incident power (b) $E_{in}=0.002$, (c) $E_{in}=0.01$, (d) $E_{in}=1$, and (e) $E_{in}=3$. Red open circles indicate a stable branch of the transmission while blue points do the unstable branch.

Figure 8

(Color online) The solution of Eq. (15) for the nonlinear FPR model with parameters $\Gamma =1$, ${\theta }_0=\pi$, ${\theta }_1=\pi$, and $E_{in}=0.1$. $X$ is shown by red points and $Y$ is shown by blue open circles.

Figure 9

(Color online) Input-output characteristics of the nonlinear FPR for the parameters given in Fig. 8. The case $\omega -{\omega }_0=0.03$ is shown by red open circles (stable solution) and by blue points (unstable solution). The case $\omega -{\omega }_0=0.015$ is shown by pink crosses (the stable solution) and by blue small stars (the unstable one).

Figure 10

Transmission spectra as dependent on frequency in the tight-binding model of FPR shown in Fig. 1c for the case of four $\left(N=4\right)$ sites between nonlinear mirrors. The parameters of the FPR presented in Hamiltonian (28) are $u=0.5$, ${\omega }_0=-1.5$, $\lambda =0.008$, and ${\varphi }_{in}=0.1$.

Figure 11

The dispersion relation of the propagating guided TM mode (Ref. 46) in the PhC described in the text.

Figure 12

The frequency of isolated bound mode for the single isolated defect cylinder as dependent on its dielectric constant ${ϵ}_d$ (Refs. 23, 46).

Figure 13

(Color online) (a) Whole self-consistent solution for the EM field at the nonlinear defects in the PhC structure shown in Fig. 1a. (b) Blow up of the solution near the BSC point shown by blue point. The amplitude at the first nonlinear defect rod $\left|E_1\right|$ is shown by red and the amplitude at the second defect rod $\left|E_2\right|$ is shown by blue. The parameters of the PhC are the following $a=0.5\mu \text{m}$, $r_d=0.18a$, $ϵ=11.56$, ${ϵ}_0=5$, ${\omega }_0=0.333$, and $n_2=2\times {10}^{-12}{\text{cm}}^2/\text{W}$ the input power per length $100\text{mW}/a$.

Figure 14

Transmission spectra in the PhC structure for the same parameters given in previous figure. The BSC frequency ${\tilde{\omega }}_c=0.3279$ is marked by open circle.

Figure 15

(Color online) The absolute values of EM field for (a) the resonant case for $\tilde{\omega }={\tilde{\omega }}_0=0.333$ marked in Fig. 14 by star, (b) for the BSC frequency $\tilde{\omega }={\tilde{\omega }}_c=03279$ marked by filled circle in Fig. 14, and (c) for the same frequency ${\tilde{\omega }}_c=03279$ however for the different branch marked by cross in Fig. 14. The wave function shown in (c) tends to the BSC for $E_{in}\to 0$. The incident power per length is chosen $100\text{mW}/a$.

Figure 16

(Color online) Phase structure of (a) the transport solution shown in Figs. 15b, 15b the solution shown in Fig. 15c which is close to the BSC.

Figure 17

(Color online) [(a) and (b)] The three-dimensional plot of the absolute value of the transport solution for the resonant case $\tilde{\omega }=0.333$ as in Fig. 15a however for the two values of the incident power (a) $25\text{mW}/a$ and (b) $100\text{mW}/a$. The cases (c) and (d) show the absolute value of the solution closed to the BSC shown in Fig. 15c for two amounts of the incident power (c) $25\text{mW}/a$ and (d) $100\text{mW}/a$.

Figure 18

Transmission spectra in the PhC structure with the same parameters as in Fig. 14 but for different the nonlinear defects with the parameters ${ϵ}_0=1.1$, ${\omega }_0=0.3865$, and $n_2=3.5\times {10}^{-10}{\text{cm}}^2/\text{W}$. The incident power per length is $100\text{mW}/a$. The first BSC frequency ${\tilde{\omega }}_c=0.3279$ is marked by star and the second one is marked by open circle ${\tilde{\omega }}_c=0.3751$.

Figure 19

(Color online) Phase structures of the states close to the (a) first even BSCs with the frequency ${\tilde{\omega }}_{c1}=0.3279$ and (b) the second odd one with the frequency ${\tilde{\omega }}_{c2}=0.3751$. The states are marked in Fig. 18 by star and open circle, respectively.