All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures

We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide side-coupled to a Kerr-nonlinear photonic crystal resonator. First, we extend the standard coupled-mode theory analysis to photonic crystal structures and obtain explicit analytical expressions for the bistability thresholds and transmission coefficients which provide the basis for a detailed understanding of the possibilities associated with these structures. Next, we discuss limitations of standard coupled-mode theory and present an alternative analytical approach based on the effective discrete equations derived using a Green’s function method. We find that the discrete nature of the photonic crystal waveguides allows a geometry-driven enhancement of nonlinear effects by shifting the resonator location relative to the waveguide, thus providing an additional control of resonant waveguide transmission and Fano resonances. We further demonstrate that this enhancement may result in the lowering of the bistability threshold and switching power of nonlinear devices by several orders of magnitude. Finally, we show that employing such enhancements is of paramount importance for the design of all-optical devices based on slow-light photonic crystal waveguides.

Figure 1

(Color online) Three types of the geometries of a straight photonic-crystal waveguide side coupled to a nonlinear optical resonator, $A_{\alpha }$. Standard coupled-mode theory is based on the geometry (a) which does not account for discreteness-induced effects in the photonic-crystal waveguides. For instance, light transmission and bistability are qualitatively different for (b) on-site and (c) inter-site locations of the resonator along waveguide and this cannot be distinguished within the conceptual framework of structure of type (a).

Figure 2

(Color online) Dependencies of the transmission coefficient, $T$, the outgoing light intensity, ${\mathcal{J}}_{\mathrm{out}}$, and the resonator’s mode intensity, ${\mathcal{J}}_{\alpha }$, on the incoming light intensity, ${\mathcal{J}}_{\mathrm{in}}$, for ${\sigma }^2\left(\omega \right)=1$ and negative product $\left[\sigma \left(\omega \right){\mathcal{J}}_{\alpha }\right]$.

Figure 3

(Color online) Dependencies of the transmission coefficient, $T$, the outgoing light intensity, ${\mathcal{J}}_{\mathrm{out}}$, and the resonator’s mode intensity, ${\mathcal{J}}_{\alpha }$, on the incoming light intensity, ${\mathcal{J}}_{\mathrm{in}}$, for several different values of ${\sigma }^2\left(\omega \right)$ and positive product $\left[\sigma \left(\omega \right){\mathcal{J}}_{\alpha }\right]$.

Figure 4

(Color online) Dependencies of the threshold incoming light intensity and the corresponding transmission coefficients on ${\sigma }^2\left(\omega \right)$ for the four critical points (1)–(4) indicated by circles in Fig. 3. Here, we assume that $\left[\sigma \left(\omega \right){\mathcal{J}}_{\alpha }\right]>0$.

Figure 5

(Color online) Linear transmission through a photonic-crystal waveguide that is created by removing every second rod in a row $(\vec{s}=2{\vec{a}}_1)$ side coupled to a one-site resonator created by removing a single rod. The underlying 2D photonic crystal is described in Appendix . We compare exact numerical results (solid line) with the analytical results based on Eq. (29) (dotted-dashed line) and Eq. (B1) (dashed line).

Figure 6

(Color online) Liner transmission through a photonic-crystal waveguide created by removing every second rod in a row $(\vec{s}=2{\vec{a}}_1)$ side coupled to an inter-site resonator created by removing a single rod. The underlying 2D photonic crystal described in Appendix . We compare exact numerical results (solid line) with the analytical results based on Eq. (39) (dashed line).

Figure 7

(Color online) Linear transmission spectrum for a photonic-crystal waveguide created by removing every second rod in a row $(\vec{s}=2{\vec{a}}_1)$ side coupled to a single on-site (a) or inter-site (b) polymer-rod resonator (marked by the open circle in the insets). The underlying 2D photonic crystal is described in Appendix and results for three different values of the resonator dielectric constant ${\epsilon }_{\alpha }$ are shown.

Figure 8

(Color online) (a) Linear transmission spectrum and (b) nonlinear bistable transmission for three different side-coupled waveguide-resonator photonic-crystal structures whose topology is shown in (c). The rods consist of Si or GaAs (full black circles) or polymer (open red circles). Example $A$ represents a close to optimal structure with inter-site location of the $\alpha$ resonator whose resonance frequency lies close to the edge $k=\pm \pi ∕s$ of the passing band; example $B$ represents a suboptimal structure with an inter-site location of the $\alpha$ resonator whose resonance frequency lies near the center of the passing band; example $C$ represents a suboptimal but commonly used structure with an on-site location of the $\alpha$ resonator whose resonance frequency lies near the center of the passing band. Closed circles in (a) indicate frequencies with $T\left(\omega \right)=80\%$ that are used for achieving high-contrast bistability in (b). Red circles in (c) indicate positions of the nonlinear polymer rods with ${\epsilon }_{\alpha }=2.56$. Other parameters of the 2D photonic crystal are described in Appendix .

Figure 9

(Color online) Frequency dependence of the detuning coefficient $D_n\left(\omega \right)$ for the 2D photonic crystal described in Appendix , for two types of resonators: Removing a single rod (${\epsilon }_n=1.0$; solid line) leads to a localized mode at ${\omega }_n=0.392$, while replacing a single rod by a geometrically identical rod made of polymer (${\epsilon }_n=2.56$, dashed line) leads to a localized mode at ${\omega }_n=0.374$.

Figure 10

Frequency dependence of the nonlinear feedback parameter ${\kappa }_n\left(\omega \right)$ for the 2D photonic crystal described in Appendix . The nonlinear resonator is created by replacing a single rod with a polymer rod of the same radius which supports at ${\epsilon }_n=2.56$ a localized mode with frequency ${\omega }_n=0.374$.

Figure 11

(Color online) Frequency dependence of the coupling coefficients $V_{n,m}\left(\omega \right)$ for the 2D photonic crystal described in Appendix with a on-site side-coupled waveguide-resonator system shown in Fig. 1b. The notations are the same as those in Eq. (24) and we assume that the waveguide is created by removing every second rod in a row (located at ${\vec{R}}_n=2{\vec{a}}_{1}n$) whereas the on-site resonator is created by replacing a single rod at ${\vec{R}}_{\alpha }=-2{\vec{a}}_2$ with a polymer rod $({\epsilon }_n=2.56)$ of the same radius. The dispersion relation for such a waveguide is displayed in Fig. 12.

Figure 12

(Color online) Dispersion relation for a photonic-crystal waveguide created by removing every second rod in a row $(\vec{s}=2{\vec{a}}_1)$ in the 2D photonic crystal described in Appendix . Numerical exact results (solid line) are calculated with the supercell plane-wave method [47] and the approximate results are obtained from Eq. (A5) with $L=1$ (dotted line) and $L=2$ (dashed line), using the coupling coefficients from Fig. 11.