Formation of collimated beams behind the woodpile photonic crystal

We experimentally observe formation of narrow laser beams behind the woodpile photonic crystal, when the beam remains well collimated in free propagation behind the crystal. We show that the collimation depends on the input laser beam's focusing conditions, and we interpret theoretically the observed effect by calculating the spatial dispersion of propagation eigenmodes and by numerical simulation of paraxial propagation model.

Figure 1

The woodpile PhC sample. Every second layer of piles of the same orientation is half-period shifted: (a) illustration and (b)–(d) micrographs by an electronic microscope showing the top (c) and side (d) views.

Figure 2

(a) The experiment to observe formation of well-collimated beam. The laser beam (532 nm) is focused (with a ×10 microscope objective), and the woodpile-like PhC is positioned at some distance (on order of millimeters) behind the focal plane. Behind the crystal, in the far-field area, the “shadow” of the PhC sample occurs, at the middle of which a narrow spot is observed, indicating formation of a well-collimated beam. (b), (c) The snapshots of typical observed transmitted field distribution on large (c) and small (d) spatial scales. The shown domain in (d) corresponds to the “shadow” of the PhC.

Figure 3

Top intensity of the collimated beam depending on the distance between the focal plane and the PhC for moderate-contrast (a) and low-contrast (b) sample. Insets show typical shapes of the beam for different positions of the PhC. Thin horizontal lines mark the intensity of the light without the crystal. The shown domains in the inset corresponds to the “shadow” of the PhC.

Figure 4

(a)–(c) Spatial dispersion curves as calculated by diagonalization of expansion (4) in 2D. (d) Spatial dispersion surface in 3D as obtained by factorization of the upper 2D spatial dispersion curve from the case (a). Parameters are $Q_x=0.5$ and $\Delta n_{0}f=0.025$. The difference between the calculated cases is detuning from three-wave resonance (equivalently the distance from the corner of a Brillouin zone): $\Delta Q_z=\left(Q_z-Q_x^2/\phantom{Q_x^{2}2}2\right)=0$ for (a) and (d), $\Delta Q_z=0.03$ for (b), and $\Delta Q_z=-0.03$ for (c). The site of square domain in plot (d) corresponds to coordinate range in (a).

Figure 5

Beam propagation inside and behind the PhC as obtained by numerical integration of the paraxial model (2) in 2D. The one-dimensional (1D) profile of beam in the far-field domain and the 2D profile in the far-field domain (as obtained by factorization) are shown on the right. The site of square 2D domain corresponds to the coordinate range in 1D plots. The parameters for (a)–(c) cases correspond to these in Fig. 4.