Bloch Cavity Solitons in Nonlinear Resonators with Intracavity Photonic Crystals

We predict a novel type of cavity solitons, Bloch cavity solitons, existing in nonlinear resonators with the refractive index modulated in both longitudinal and transverse directions and for both focusing (at normal diffraction) and defocusing (at anomalous diffraction) nonlinearities. We develop a modified mean-field theory and analyze the properties of these novel cavity solitons demonstrating, in particular, their substantial narrowing in the zero-diffraction regime.

Figure 1

(a) A nonlinear resonator with an intracavity two-dimensional photonic crystal. The dispersion relation of waves, here the isofrequency cureve, $k_z=f(k_x)$ for a homogeneous resonator (b) and for a 2D intracavity PPS (c) (the longitudinal mode structure of the cavity is depicted by stripes). The dashed lines in (c) indicate the dispersion curves in the limit of a vanishing modulation.

Figure 2

Intensity profiles of the forward wave at the output of the resonator (upper plots), and the spatial distributions of the total (forward plus backward) intracavity field (bottom plots), calculated from the model (1) for (a) homogeneous cavity; (b) a weakly modulated cavity ($v=1$) and a focusing nonlinearity ($\gamma =1$); and (c) a strongly modulated cavity ($v=5$) and a defocusing nonlinearity ($\gamma =-1$). Other parameters are: $l=1.5l_0$, $q_x=5.45\sqrt{2k_0/l_0}$, $q_z=16.76l_0^{-1}$, $\rho =0.98$, $\varphi =0.52\pi$, and $H=1$, where $l_0$ is a normalization constant.

Figure 3

(a)–(c) Dispersion for waves in resonance, and (d)–(f) intensity profiles of the corresponding CSs for different values of the modulation parameter ($M$). The profiles are plotted for a CS in a homogeneous cavity (a),(d) and for Bloch CSs in a cavity with PPS ($M=600$) for a focusing nonlinearity $\delta =1$, ${\Delta }^{\prime }=-2$ (b, e); and a cavity with PPS ($M=800$) for a defocusing nonlinearity $\delta =-1$, ${\Delta }^{\prime }=2$ (c),(f); Thick lines correspond to the central spatial harmonic ($|a_0{|}^2$) and the shaded profiles result from the superposition of all three modes. Holding beam amplitude $E=1.37$. Other parameters are: $D=100$, $P=1257$, $\Omega =-2000$.

Figure 4

The width of Bloch CSs vs the modulation parameter $M$ for different effective detunings ${\Delta }^{\prime }$. The vertical gray stripe marks the zero-diffraction regime. The left side of the zero-diffraction regime corresponds to normal diffraction and focusing nonlinearity: $\delta =1$, $E=1.37$ for ${\Delta }^{\prime }=-2$ and $E=2.3$ for ${\Delta }^{\prime }=-4$. The right side is the case of anomalous (inverted) diffraction and defocusing nonlinearity: $\delta =-1$, $E=1.37$ for ${\Delta }^{\prime }=2$, and $E=2.3$ for ${\Delta }^{\prime }=4$.