Photorefractive surface nonlinearly chirped waveguide arrays

We report an alternate type of nonlinear waveguides, photorefractive surface nonlinearly chirped waveguide arrays, which can be directly induced by photorefractive surface waves in virtue of diffusion and drift nonlinearities. The amplitude of such nonlinearly chirped waveguide arrays has an apodized envelope owing to the diffusion nonlinearity. The refractive-index change of the apodized tails converges to a nonzero value which can be handily adjusted by an external electric field. Moreover, the chirp parameters such as amplitude, sign (positive or negative), and initial position can be conveniently adjusted by an external electric field, background illumination, incident beam, etc. Then the guided-wave properties of this type of waveguide arrays are analyzed by using the transfer matrix method. Owing to the flexible tail and the nonlinear chirp, the dispersion curves of the index-guided modes can be tailored by an external electric field and the dispersion curves of ordinary and extraordinary Bragg guided modes couple, intertwine, and anticross with each other. Meanwhile, there is a clear “competition” in the coupling hybrid mode near anticrossing.

Figure 1

(a1) and (b1) The PR SWs modes with $E_0=-{10}^4\mathrm{and}{10}^4\mathrm{V}/\mathrm{m}$, respectively. (a2) and (b2) The distributions of refractive-index change corresponding to (a1) and (b1), respectively. (a3) and (b3) The distributions of the local spatial frequency $K\left(x\right)$ of PR SWs corresponding to (a1) and (b1) and the distributions of the channel width ${\mathrm{\Lambda }}_i$ corresponding to (a2) and (b2), respectively.

Figure 2

(a) Effective restoring force $F$ as a function of normalized amplitude $A^{\prime }\left(x\right)$: solid and dotted curves correspond to $E_0=15000\mathrm{V}/\mathrm{m}$ ($a>g$) and 5000 V/m ($a<g$), respectively; (b) threshold electric field $E_{\mathrm{th}}$ (red oblique line) and critical electric field $E_c$ (green solid line) as a function of propagation constant $\beta /n{k}_0$, respectively, the green dotted line correspond to $\beta =2.730118\times {10}^7{\mathrm{m}}^{-1}$; (c) and (d) the distribution of channel width ${\mathrm{\Lambda }}_i$ to channel order $i$ for the PR SNCWGAs induced by the PR SWs with $E_0=-2000$, 0, 2000, 3574, 6000, 8000, and 10 000 V/m from bottom to top for lower $I^{\prime }\left(x\right)$ and higher $I^{\prime }\left(x\right)\sim {10}^5$, respectively.

Figure 3

(a) and (b) Dispersion relation of the PR SNCWGAs corresponding to PR SWs with $E_0=-{10}^4$ and ${10}^4\mathrm{V}/\mathrm{m}$, respectively, the horizontal dotted line corresponds to $N_{\mathrm{eff}}=-n^2{r}_{\mathrm{eff}}{E}_0/2$; (c) the fine structure of the green box in the (b); (d) the scatterplot of the anticrossing between the dispersion curves of $\mathrm{T}{\mathrm{E}}_{21}$ and $\mathrm{T}{\mathrm{E}}_{02}$, corresponding to the green box in (c); and $N_{\mathrm{eff}}=\beta /n{k}_0-1$ is the normalized effective refractive index.

Figure 4

The modes for $\omega /{\omega }_0=2.66$. (a)–(f) Correspond to intersections of dispersion curves and the right edge of picture box in Figs. 3 and 3 from top to bottom except the high order ($m\ge 2$) index-guided modes, respectively. Index-guided modes: (a) $\mathrm{T}{\mathrm{E}}_{00}$ with $N_{\mathrm{eff}}=7.611\times {10}^{-5}$; (c) $\mathrm{T}{\mathrm{E}}_{10}$ with $N_{\mathrm{eff}}=1.088\times {10}^{-5}$. First Bragg-guided modes: (b) $\mathrm{T}{\mathrm{E}}_{01}$ with $N_{\mathrm{eff}}=5.228\times {10}^{-5}$; (d) $\mathrm{T}{\mathrm{E}}_{11}$ with $N_{\mathrm{eff}}=-1.571\times {10}^{-5}$; (f) $\mathrm{T}{\mathrm{E}}_{21}$ with $N_{\mathrm{eff}}=-2.840\times {10}^{-5}$. Second Bragg-guided modes: (e) $\mathrm{T}{\mathrm{E}}_{02}$ with $N_{\mathrm{eff}}=-2.338\times {10}^{-5}$.

Figure 5

The mode evolution of each dispersion branch associated with the anticrossing between $\mathrm{T}{\mathrm{E}}_{21}$ and $\mathrm{T}{\mathrm{E}}_{02}$. Left-hand side: The modes evolution from $\mathrm{T}{\mathrm{E}}_{21}$ to $\mathrm{T}{\mathrm{E}}_{02}$ corresponding to the upper and lower branch in Fig. 3. Right-hand side: The modes evolution from $\mathrm{T}{\mathrm{E}}_{02}$ to $\mathrm{T}{\mathrm{E}}_{21}$ corresponding to the lower branch in Fig. 3. The green dotted lines show the boundary of IGM region and BGM region for $\mathrm{T}{\mathrm{E}}_{21}$.