Planar photonic crystal gradient index lens, simulated with a finite difference time domain method

We demonstrate the performance of a planar gradient index (GRIN) lens in a two-dimensional photonic crystal. The gradient index is achieved by spatially varying the air-hole radius in the photonic crystal, resulting in a smooth variation of the effective refractive index. We simulate the GRIN lens using a two-dimensional finite difference time domain method to show that it performs as conventional GRIN lenses do. Some of the differences between conventional GRIN lenses and the photonic crystal GRIN lens are discussed.

Figure 1

(Color online) Band structure for a hexagonal lattice with hole-sizes $0.3a$ (solid lines) and $0.32a$ (dashed lines), showing the second band [green(bottom)], third band [red(middle)], and fourth band [blue(top)]. The normalized frequency where the device is operated (0.37) is shown as a horizontal black line.

Figure 2

(Color online) FDTD simulation result (E-field) of a Gaussian beam with ${\omega }_0=2.26\mu \mathrm{m}$ and ${\lambda }_0=\pi ∕2\mu \mathrm{m}$ propagating through a 2D photonic crystal GRIN lens. The dashed line represents an interface between the homogeneous background medium on the left-hand side and the photonic crystal on the right-hand side.

Figure 3

(Color online) The $\mathbf{k}$-space representation of the E-field in the GRIN lens shown in Fig. 2. The hexagonal grid of dashed lines represent the primitive cell boundaries. The primitive cell in the center is the first Brillouin zone. The inset shows a blown-up version of the peaks in one of the primitive cells, with the dominant peak labeled by $\mathbf{A}$.

Figure 4

(Color online) Demodulated reconstruction of the forward propagating beam in the primitive cell labeled by $\mathbf{A}$ in Fig. 3. The dashed line represents an interface between the homogeneous background medium on the left-hand side and the photonic crystal on the right-hand side.