Band-specific phase engineering for curving and focusing light in waveguide arrays

Band-specific design of curved light caustics and focusing in optical waveguide arrays is introduced. Going beyond the discrete, tight-binding model, which we examined recently, we show how the exact band structure and the associated diffraction relations of a periodic waveguide lattice can be exploited to phase-engineer caustics with predetermined convex trajectories or to achieve optimum aberration-free focal spots. We numerically demonstrate the formation of convex caustics involving the excitation of Floquet-Bloch modes within the first or the second band and even multiband caustics created by the simultaneous excitation of more than one band. Interference of caustics in abruptly autofocusing or collision scenarios are also examined. The experimental implementation of these ideas should be straightforward since the required input conditions involve phase-only modulation of otherwise simple optical wavefronts. By direct extension to more complex periodic lattices, possibilities open up for band-specific curving and focusing of light inside two-dimensional or even three-dimensional photonic crystals.

Figure 1

(a) Band structure and (b) GV ($v_g=-d\beta /dK$) versus BM for a WGA described by the potential $V\left(x\right)=5\cos \left(2\pi x\right)$.

Figure 2

Caustics in a WGA under excitation of the first band. The design parameters are (a) $\delta =1.8$, ${\xi }_0=-20$, $v_{g0}=1.5$ and (b) $\delta =2.2$, ${\xi }_0=-20$, $v_{g0}=1.2$. The equations of the caustics are $f\left(z\right)=0.055z^{1.8}$ and $f\left(z\right)=0.009z^{2.2}$, respectively, and are superposed as red curves. In both cases, the envelope of the input beam is $A\left(\xi \right)={\left(e^{-0.1\xi }+e^{5\xi }\right)}^{-1}$.

Figure 3

Caustics in a WGA under excitation of the second band. The design parameters are (a) $\delta =1.8$, $v_{g0}=5$, ${\xi }_0=-20$, $\alpha =0$ and (b) $\delta =1.8$, $v_{g0}=5$, ${\xi }_0=-20$, $\alpha =2$. The corresponding equations are $f\left(z\right)=0.4790z^{1.8}$ and $f\left(z\right)=2z+0.1910z^{1.8}$, respectively, and are superposed as red curves. The envelope of the input beam and the lattice potential are the same as Fig. 2.

Figure 4

Caustics in a WGA under excitation of the second band. The input beam amplitude $A\left(\xi \right)$ is a continuous FB mode envelope in (a) and $A\left(x\right)={\left(e^{-0.1x}+e^{5x}\right)}^{-1}$ in (b). The input phase is the same in both cases and results from Eq. (9), where GV is given by Eq. (10) with the parameters $\delta =1.8$, ${\xi }_0=-30$, and $v_{g0}=5$. The bottom blue curves are the intensity of the input beam, while the superposed red curves are the caustic $x=0.346z^{1.8}$.

Figure 5

Focusing in a WGA using (a) the parabolic phase $\phi \left(\xi \right)=-{\xi }^2/70.4$ imposed on the envelope $A\left(\xi \right)=\exp \left[-{\left(\xi /76\right)}^{20}\right]$ and (b) the phase resulting from Eq. (9) with $v_g\left(\xi \right)=-\xi /40$, imposed on the envelope $A\left(\xi \right)=\exp -{\left(\xi /60\right)}^{20}]$. The dashed red curves in (a) are the twin fold type caustics that create the cusp.

Figure 6

Focusing in a WGA using the phase resulting from Eq. (9) with (a) $v_g\left(\xi \right)=-\xi /6$ and (b) $v_g\left(\xi \right)=-(\xi -0.5)/6$, imposed on the envelope $A\left(\xi \right)=\exp \left[-{\left(\xi /30\right)}^{20}\right]$. The input BM assumes values $K⩾0$ in (a), hence exciting the first and second bands, and $K⩾\pi$ in (b) hence exciting only the second band.

Figure 7

(a) Caustic in a WGA under the excitation of two bands. The parameters for the phase engineering are $\delta =2.5$, $v_{g0}=1.5$, and ${\xi }_0=-12$. (b) BM (solid, blue line, left ordinate) and phase (dashed, red line, right ordinate) of the input beam. The amplitude of the input beam is $A\left(\xi \right)=\exp \left[-{\left(\frac{x+45}{50}\right)}^{30}\right]$.

Figure 8

(a) Interference of two symmetric, opposite accelerating caustics. The parameters of the caustics are the same as Fig. 2a. (b) Collision of caustics from the first and second bands excited simultaneously by the same input wavefront. The input BM varies linearly with slope $0.688\pi /20$ and a $0.5\pi$ jump to the second band at $\xi =-20$. The amplitude of the input beam in (b) is $A\left(\xi \right)=\exp \left[-{\left(\frac{x+21}{22}\right)}^{20}\right]$.