Observation of Light Transmission Through Randomly Rough Glass Surfaces Up to and Beyond the Critical Angle of the Glass-Air Interface

Mark Harrison, Martin Griswold, David Saltzberg (UCLA)

November 13, 2018

Abstract

When light propagates through a medium and encounters a boundary to a medium with a lower index of refraction, it will totally internally reflect if the angle of incidence is larger than the critical angle as defined by Snell’s Law. We have added to our roughness teststand the ability to probe incident angles up to and beyond the critical angle. In this series of experiments, we show that if the interface between the two media is a random rough surface with surface features on the order of the wavelength of the incident light, transmission of light will be observed at angles of incidence well beyond the critical angle.For light not beyond the critical angle, the peak does not obey Snell’s law and

is not bent as far away from the normal. Light that would have passed through specularly is spread into a larger solid angle, with less intensity at any given angle. These two properties are relevant to the ANITA experiment because we measure the transmission of radiation through a rough surface.

Setup

This experiment was largely identical with the one described in a previous note[1] with one major difference. In order to have light incident upon the interior surface of the glass at angles larger than the critical angle, we affixed a half-cylinder lens so that incident laser light would not refract when entering the glass (Fig. 1).


In this setup, the laser was a He-Ne type with a wavelength of 633nm and 580uW power output. The polarizer was oriented so that the laser light is incident upon the glass surface with s-polarization (out of the page in the above diagram). We rotated the lens-glass element about the center of curvature of the half-cylinder lens to that the laser beam was always normal to the surface of the lens. The photometer was moved along a line perpendicular to the incident laser in order to measure the scattering of light after passing through the optical elements. Not shown are extra masks and apertures used to block extraneous reflections from the various elements.

Glass has an index of refraction of 1.5. Accordingly, the critical angle is 41.8º.

Diffuser Surface Features

Using a VEECO Dimension 3100 Atomic Force Microscope, we characterized the surface features of the diffusers. An explanation of the meaning of the grit numbers appears in the previous note1. Briefly, the inverse of the grit number is the diameter in inches of the largest particle of grit used to grind the diffuser.

Each scan produced a 256×256 array of heights. Table 1 below gives the RMS heights and correlation lengths of the two diffusers over various scan areas. The RMS height is simply the RMS of the distribution of every height in the scan. The RMS correlation length was found by calculating the two-dimensional autocorrelation function of the height data and finding the average half-width at half-maximum (HWHM). Assuming a roughly Gaussian shape for the curve, we divided the HWHM byto get what we call the RMS Transverse Correlation Length. Images of the diffuser surfaces appear in figures 2-4.

Grit / Scan size (μm) / RMS Height (μm) / RMS Transverse
Correlation Length (μm)
400 / 50×50 / 1.0 / 2.5
75×75 / 1.2 / 3.1
100×100 / 1.4 / 5.1
1000 / 10×10 / 0.32 – 0.34 / 0.9 – 1.3
20×20 / 0.41 – 0.57 / 1.2 – 4.3
30×30 / 0.50 – 0.54 / 1.3 – 3.1
50×50 / 0.50 – 0.60 / 4.1 – 7.7
1500 / 10×10 / 0.23 – 0.26 / 0.4 – 0.5
20×20 / 0.28 – 0.42 / 0.7 – 1.1
30×30 / 0.36 – 0.45 / 0.9 – 1.2
50×50 / 0.42 – 0.47 / 3.8 – 4.1

Table 1: Feature sizes of ground glass diffusers

We performed scans at two different locations for the 1000- and 1500-grit diffusers to get an idea of the homogeneity of the surface roughness. Only one set of scans was performed on the 400-grit diffuser because of time constraints. Within a factor of 2, the feature sizes are relatively homogenous across the surface of the diffuser.

Since the ratio of the Cherenkov radiation wavelength to the laser’s wavelength is about 106 (~0.5m to ~0.6μm), these diffusers can be said to model Antarctic features with sizes given by Table 1 in meters. For example, the 1500-grit diffuser models an Antarctic terrain with an RMS height of 0.23 – 0.47m and RMS transverse correlation length of 0.4 – 4.1m.[2]

Results

Figures 5-7 show comparative charts of the light transmissions at 0º, 40º, and 70º angle of incidence (finer data could be taken in the future if deemed helpful). The flat glass refracted light according to Snell’s Law at angles of incidence smaller than the critical angle and showed no transmission of light at angles larger than the critical angle. The diffusers, however, skewed the distribution of light towards the normal of the diffuser surface and continued to transmit light even when the laser light was incident at angles much larger than the critical angle.

At angles of incidence that are not totally internally reflected, the 1500-grit diffuser has the highest peak transmission and the 400-grit diffuser lets through the lowest. At angles that are totally internally reflected in flat glass, these observations are reversed.

Conclusion

Our experiments demonstrate that a rough interface will transmit light incident upon it at angles of incidence normally totally internally reflected by a flat surface. Further studies and other simulations incorporating this data are required to determine the significance of these findings to the ANITA project. Although we did not observe any differences between a coherent and incoherent light source, perhaps some difference could be observed between this continuous wave source versus a pulsed source. For ANITA we would like to quantify the compensating effects of larger aperture due to viewing beyond TIR vs. the diffusion of the specular power, with loss of intensity, into a larger solid angle. Figures 5-7 show that for our diffuser the latter effect is dramatic, but this diffuser is also a significant overestimate of the expected roughness. The lower index of refraction of snow (~1.3), where present, might also reduce these effects.

Figure 2: Surface of a 400-grit diffuser

Figure 3: Surface of a 1000-grit diffuser

Figure 4: Surface of a 1500-grit diffuser

Figure 5: Comparison of scattering at 0º incidence

Figure 6: Comparison of scattering at 40º incidence

Figure 7: Comparison at 70º incidence (well beyond the critical angle where flat glass does not transmit light)

[1]Daub, Everson, et al., “Optical Modeling of the Effect of Surface Roughness on Detection of Signals from Within the Ice with ANITA.” March 10, 2006.

[2]This can compared to continental data taken from a variety of traverses (mostly near the coasts) and backscatter properties of radar altimetry (80% of the surface) from satellites. Near the coasts more roughness is observed due to katabatic winds and snow accumulation. Large scale (>30cm) features occur most often near the coasts but with smaller frequency than our diffuser. Smaller features (~3cm) appear most everywhere but are smaller than the corresponding features even on the 1500 grit diffuser. So our diffuser is about an order of magnitude rougher than much of the ice we will be viewing. We plan to post a literature summary of traverse and radar-altimeter data in the elog soon.