Spatially Shaping Waves to Penetrate Deep inside a Forbidden Gap

It is well known that waves with frequencies within the forbidden gap inside a crystal are transported only over a limited distance—the Bragg length—before being reflected by Bragg interference. Here, we demonstrate how to send waves much deeper into crystals in an exemplary study of light in two-dimensional silicon photonic crystals. By spatially shaping the wave fronts, the internal energy density—probed via the laterally scattered intensity—is enhanced at a tunable distance away from the front surface. The intensity is up to $100\times$ enhanced compared to random wave fronts, and extends as far as $8\times$ the Bragg length, which agrees with an extended mesoscopic model. We thus report a novel control knob for mesoscopic wave transport that pertains to any kind of waves.

Figure 1

Schematic of wave transport in a 2D photonic crystal consisting of pores with unavoidable disorder. At frequencies in the gap, light is quickly Bragg attenuated; by shaping the wave fronts, light is transported much deeper into the crystal.

Figure 2

(a) Left: scanning electron microscope image of a 2D photonic crystal. The lower half shows the 2D array of pores in the $yz$ plane, and the upper half shows the $xy$ cleavage plane with $6\mu \mathrm{m}$ deep pores (scale bar: $5\mu \mathrm{m}$). Right: $yz$ plane with the centered rectangular crystal with lattice parameters $a=695\mathrm{nm}$ and $c=a/\sqrt{2}$ with a pore diameter $2R=290\mathrm{nm}$ (or $R/a=0.21$). Light is incident in the $z$ direction, corresponding to the $\mathrm{\Gamma }K$ high-symmetry direction. (b) Reflectivity measured on the photonic crystal (open circles) for TE-polarized light, compared to unstructured silicon wafer (filled squares). The prominent photonic crystal reflectivity peaks match well with the stop gaps in the calculated band structure. The dashed line highlights the frequency near the gap center where many steering experiments were performed.

Figure 3

Linearly polarized light from a tunable source is directed via a spatial light modulator (SLM) onto the sample through microscope objective $M1$. The incident polarization is tuned using the half-wave plate ($\lambda /2$). Light reflected by the crystal is imaged onto an infrared camera using the same objective and lens $L1$ ($f=500\mathrm{mm}$). A long working distance objective ($M2$) images the lateral scattered light onto an infrared camera using lens $L2$ ($f=500\mathrm{mm}$). Broadband linear polarizers are used to analyze the reflected and the lateral scattered light. The inset shows the image taken on the camera in reflection. The bright spot in the center (highlighted in red) is the focused coherent light with a uniform phase pattern on the SLM.

Figure 4

Camera images of the lateral scattered light $(\nu /c=6250{\mathrm{cm}}^{-1})$ with (a) an uncontrolled and (b) the controlled wave front. The red arrow points to the target location for steering the light, which exhibits as a bright spot. (c) The integrated intensity in a 5 pixel high ($yz$ plane) strip around the target location shows a clear increase in the intensity at the target depth of $10\mu \mathrm{m}$. The dashed gray line indicates the detector dark counts. The gray band is the intensity expected at $z=10\mu \mathrm{m}$ upon wave front shaping as predicted by our model.

Figure 5

Measured wave front shaping enhancement $E_W$ versus penetration depth $z$ into the crystal at a frequency at the center of the stop gap $\nu /c=6250{\mathrm{cm}}^{-1}$. The calculated enhancement using a model that accounts for the Bragg interference and the random multiple scattering agrees well with the measurements.