Experimental and theoretical analysis of the intensity of beams diffracted by three-dimensional photonic crystals

An analysis of the diffracted beams emerging from three-dimensional photonic crystals is herein presented. The wave vectors of nonspecular beams are calculated for a triangular two-dimensional lattice and the change in their directions as a function of the wavelength is confirmed experimentally for the case of face-centered-cubic colloidal crystals illuminated under normal incidence. A fluctuating behavior of beam intensity as a function of the wavelength of the incident light is predicted for perfectly ordered lattices. As it is the case for specularly reflected and ballistically transmitted beams, this modulation arises from multipole resonances of the sphere ensemble that are smoothed out via the diffuse light scattering produced by imperfections in the crystalline structure. When optical extinction is introduced in order to model the effect of imperfections, it is possible to accurately reproduce experimental observations.

Figure 1

(Color online) Scheme of the experimental setup used to measure the efficiency of nonspecular reflected beams. Transmitted beams are also indicated.

Figure 2

(Color online) (a)–(b) Diffraction patterns of reflected beams projected on the screen shown in Fig. 1 ($xy$ plane) for: (a) $\lambda =484\text{nm}$ and (b) $\lambda =565\text{nm}$. (c) Diffraction pattern of transmitted beams projected on a screen parallel to the $xy$ plane for $\lambda =539\text{nm}$. (d)–(f) Diffraction patterns of reflected and transmitted beams projected on a screen parallel to the $yz$ plane for: (d) $\lambda =622\text{nm}$, (e) $\lambda =593\text{nm}$, and (f) $\lambda =512\text{nm}$.

Figure 3

(Color online) Calculated efficiencies of nonspecular reflected beams for (a) a perfect self-standing crystal 10 layers wide $\left({\epsilon }_i=0\right)$ and (b) the same lattice after introducing extinction $\left({\epsilon }_i=0.04\right)$.

Figure 4

(Color online) Measured (solid lines) and calculated (dashed lines, ${\epsilon }_s=2.45+0.03i$) reflection efficiencies of diffracted channels (1,0) (black lines) and $\left(1,-1\right)$ (red lines).