Directional Emissivity from Two-Dimensional Infrared Waveguide Arrays
(Supplemental Material)
Fig. SI1 RCWA simulations for waveguides with a=b=6m, d = 7m and h= 15m of (a) The parallel
component of the reflectivity;(b) The perpendicular component of the reflectivity;(c)The parallel
component of the transmission;(d) The perpendicular component of the transmission; (e) The parallel
component of the absorption (A=1-R-T); (f) the perpendicular component of the absorption (A=1-R-T).
According to Kirchoff’s Law, absorption and thermal emission are equal in thermal equilibrium.
Conservation of energy dictates that absorption, A, is given by A = 1 − T − R where R is the reflectivity
and T is the transmission. So by measuring R and T, we can calculate A. While we can readily measure
the reflectivity of our structure, transmission is more difficult. Our waveguides were fabricated on
single-side polished silicon wafers, so that transmitted light scatters diffusely, leading to no measureable
signal even though silicon is largely transparent at these wavelengths. For this reason, we evaporated
3000 gold to eliminate transmission of incident light out of the back surface, reflecting this energy back
into the top half-space.
In modeling, however, we have access both reflection and transmission, so we can calculate the
absorption exactly. Fig. SI1 contains the reflection maps (a,b), transmission maps (c,d) and absorption
maps (e,f) for a square waveguide with sides a=b=6 m, period d = 7 m, and depth h = 15 m. The
transmission signal was collected immediately after the bottom of the waveguides. Both the
transmission and absorption maps exhibit characteristics of the four regions identified in the main text.
As expected, for wavelengths longer than the waveguide cutoff, transmission is zero, while for
wavelengths shorter than the cutoff, transmission increases depending on wavelength and angle.
Interestingly, the same four regions identified in the main text for reflection are apparent in both the
transmission and absorption maps. From Fig. SI1 (e,f) it is apparent that both the spectral and angular
features of high absorption correspond very well to the features of low reflection, reduced by the
transmission in these regions.
To further investigate the impact of transmission, we evaporated 1000 Angstroms of gold on the back of
a waveguide sample, setting the transmission identically to zero. While this caused slight changes in the
reflectivity magnitude, it neither created new spectral features nor eliminated any of the relevant
existing features.
As a final experimental verification of the angular dependence of thermal emission, a waveguide with
𝑎 = 6𝜇𝑚 and 𝑑 = 9𝜇𝑚 was placed on a hotplate kept at 50 °C and imaged with a thermal infrared
camera placed on a goniometer so that the viewing angle could be adjusted (see experimental setup in
Fig.SI2(a). In addition to the sample, the hotplate also had a low emissivity gold reference sample and a
high emissivity black reference sample for calibration of the thermal images. Fig.SI2 (b) shows the series
of thermal images over 8 angles of incidence spanning 5 degrees to 75 degrees. The sample appears
cooler with increasing off-normal viewing angle, with a measured emissivity varying from 0.3 to 0.1 over
the range of measurements, corresponding well to the calculated emissivity over the camera bandwidth.
Fig. SI2 Schematic diagram of the IR camera measurement experiment; and (b) IR images taken at 8
different angles of incidence. The camera was calibrated using low and high emissivity reference
samples. The measured 2D IR waveguides demonstrate an average polarization emissivity ranging from
0.3 ( = 5 degrees) to 0.1 (= 75 degrees).
Fig. SI3 Graph of the emissivity vs angle. The graph was generated by integrating the RCWA modeled
absorption over the 8-14 mm band of the IR camera used to generate the images in Fig. 4(b) of the main
text.
Figure SI2 was created with a calibrated Tamarisk 320 IR camera with wavelength range 8-14 m.
Estimates for the sample emissivity as perceived by the camera ranged from 0.3 at a viewing angle of 5
degrees to 0.1 at a viewing angle of 75 degrees. The ability to calculate the absorption versus angle and
wavelength presented in Fig. SI1, allows us to calculate the emissivity over the same band. Fig. SI2
shows a graph of the integrated emissivity over this band, and generates values for emissivity in line
with those from the camera. In addition to the absolute values being close to those of the camera
measurements, the trend of steadily decreasing emissivity from normal incidence to 35 degrees and the
relative plateau from 45 to 75 degrees is also observed in the modeled data.