Kinematic light scattering by 3D photonic crystals

In 1912, Max von Laue discovered that illumination of a natural crystal with polychromatic x-rays results in a pattern of discrete diffraction spots. He was awarded a Nobel prize for his achievement, which proved that x-rays are electromagnetic waves and crystals are composed of periodic arrays of atoms. Laue scattering is now commonly used to determine the symmetry and structure of natural crystals. Extending the approach towards the visible regime has thus far failed because of a lack of natural crystals with appropriate lattice parameters. Although artificial, 3D photonic crystals (PCs) may be used, PC scattering of visible light suffers from both high contrast and the tensor character of the refractive index, leading to multiple scattering of the incident light (in contrast to x-ray diffraction by small crystals).

We have demonstrated Laue scattering by PCs using visible light with an internal refractive-index contrast of less than 1%. The resulting Laue pattern exhibits color separation and displays the crystal's symmetry. Scattering curves of individual Laue spots measured under monochromatic light can be described adequately by kinematic-scattering theory. These results support the potential applicability of such crystals for spectral filtering.

The samples we used were manufactured using holographic lithography. To create a complete 3D PC, we used four laser beams to generate an interference pattern inside a commercially available photopolymer.¹ At the positions of maximum intensity in the resulting interference pattern, the material becomes polymerized, which is accompanied by a change in refraction index. Because of the spatially resolved light-induced polymerization, the refractive index of the pure polymer (n = 1.5) is modulated by approximately Δn = 10⁻³. The refraction-index profile is nearly sinusoidal in all three spatial directions and the volume covered can be extended to a lateral size of several millimeters and a depth of a few microns. The specific beam configuration allows creation of a lattice of trigonal symmetry with surface orientation parallel to the trigonal axis.

For our diffraction measurements we used a fiber laser (SuperK Compact, Koheras) that emitted a continuous spectrum of white light between 500 and 1700nm with an angular divergence of less than 0.045^°. The probe light was positively polarized. The sample was mounted onto a diffractometer (Stadi MP, STOE & Cie.) equipped with a four-axial goniometer. Both the sample and detector axes could be moved in steps of 0.001^° (see Figure 1 for the instrument's geometry and definitions of the axes involved). In addition, the goniometer was equipped with a sample-translation stage in the x direction, enabling sample movement by ±2cm.

Figure 1. Experimental setup and definition of the angles involved.

We recorded the diffracted intensities of a number of Laue spots (see Figure 2) using a detector setup consisting of a photodiode and an electronic readout system. The detector diode was mounted onto the goniometer's 2θ rotation axis. The sample and detector axes could both be moved independently. Any spatial variation of the Laue spot caused by a change in the incident angle at the sample circle, ω, could be compensated for by detector motion around the 2θ axis.

Figure 2. Laue diffraction for white laser light, showing the PC's trigonal symmetry. Because of saturation, the rainbow-like color separation is only visible for low-intensity Laue spots.

We aligned the sample by rotation around the φ and ψ axes such that the plane spanned by the incident beam and the diffracting vector of the Laue spot of interest was perpendicular to the ω axis. Because of the small number of stacked layers along the surface normal, the colored spots are smeared out, as seen for the −1^st-order spots. On the other hand, the +1^st-order Laue spots are too intense and saturate the CCD so that their spectral distribution cannot be resolved. Figure 3 shows Rocking curves of selected Laue spots recorded with monochromatic laser beams. The curves clearly reproduce the shape of a Laue function.

Figure 3. Rocking scans using monochromatic laser light through selected Laue spots. Each maximum is accompanied by a fringe structure, as predicted by kinematic theory. (ω in degrees.)

The results of our experiment exemplify the validity of the kinematic-scattering approach for PCs with a small refractive-index contrast. This opens up possibilities to use such crystals as dispersive optical elements, which forms part of our future research directions. Such elements would enable parallel processing of color-coded optical information transported using white light, without the need to introduce additional filters.

We are grateful for collaborations with S. Orlic and C. Müller at the Department of Optical Technologies of the Technical University of Berlin, particularly for supplying us with the photopolymer samples, and to Marcel Roth for preliminary experiments. We acknowledge funding from the German ministry of research and education through the NAMIROS (Nano- und Mikrostrukturierung von Polymeren und Nanokompositen als Raumgitter für die optische Sensorik) project.