Wide-angle coupling into rod-type photonic crystals with ultralow reflectance

We describe the surprising phenomenon of near-perfect coupling from free space into uniform two-dimensional rod-type photonic crystals over a wide range of incident angles. This behavior is shown to be a generic feature of many rod-type photonic crystal structures that is related to strong forward scattering resonances of the individual cylinders. We explain these results using both semianalytic analysis and two-dimensional numerical calculations and identify the conditions under which efficient, wide-angle coupling can occur. The results may lead to more efficient designs for in-band photonic crystal devices such as superprisms and self-collimation based photonic circuits.

Figure 1

Geometry used to calculate the transmission of a plane wave incident on a semi-infinite PC at an angle ${\varphi }_0$.

Figure 2

(a) Transmittance ${\epsilon }_T$ vs dimensionless frequency $d∕\lambda$ and incident angle ${\varphi }_0$ for the 2D PC described in the text. The dashed white line indicates the maximum frequency for which the single diffracted grating order condition is satisfied, as discussed in Sec. 5. (b) Transmittance vs ${\varphi }_0$ at $d∕\lambda =0.34$, indicated by the dashed black line in (a).

Figure 3

Maximum acceptance angle ${\varphi }_{\mathit{max}}$ plotted as a function of normalized cylinder radius for a square lattice of rods of index ${\nu }_c=3.0$ in air. The three curves correspond to minimum transmittance levels of 90%, 95%, and 99% for an incident plane wave of TM polarization.

Figure 4

Single cylinder scattering geometry: plane wave incident from above is scattered in directions $\theta$, where $\theta =-\pi ∕2$ corresponds to forward scattering.

Figure 5

(a) Normal incidence transmittance for the PC lattice considered in Fig. 2. (b), (c) Single cylinder scattering cross section and asymmetry parameter, respectively, plotted as a function of $r_c∕\lambda$ for the same frequency range as (a). Both parameters are calculated for multipole truncation orders $N_m=0,1,2,3$, where the $N_m=3$ curve (solid) is converged to within graphical accuracy. (d) Multipole scattering coefficients $\mid b_n\mid$ plotted for $n=0–3$. Shaded regions in (a)–(d) indicate the band gaps of the bulk PC. (e)–(i) Scattering patterns given by ${\mid T\left(\theta \right)\mid }^2∕k$ for $r_c∕\lambda =0.034$, 0.085, 0.116, 0.163, and 0.204 for light incident from above.

Figure 6

Normal incidence grating (dashed curve) and PC (solid curve) transmittance spectra for the PC considered in Sec. 2.

Figure 7

Comparison of single cylinder scattering parameters $g$ (solid) and ${\tilde{g}}_c$ (dashed) for a cylinder of index ${\nu }_c=3.0$ and radius $r_c=0.34d$, and ${\tilde{g}}_G$ (dotted) for a grating of period $d$ formed from identical cylinders. The horizontal axis is expressed in units of the normalized grating frequency, $d∕\lambda$.

Figure 8

Plot of the numerator of the backscattering amplitude, $T_G(\pi ∕2-{\varphi }_0)$, defined in Appendix (dotted), the reflectance of a single grating (dashed) and the semi-infinite PC (solid) as a function of normalized frequency $d∕\lambda$ for ${\varphi }_0=0°–70°$ in 10° increments. Calculations are to dipole order accuracy.