Single-shot on-chip spectral sensors based on photonic crystal slabs

Miniaturized spectrometers have significant potential for portable applications such as consumer electronics, health care, and manufacturing. These applications demand low cost and high spectral resolution, and are best enabled by single-shot free-space-coupled spectrometers that also have sufficient spatial resolution. Here, we demonstrate an on-chip spectrometer that can satisfy all of these requirements. Our device uses arrays of photodetectors, each of which has a unique responsivity with rich spectral features. These responsivities are created by complex optical interference in photonic-crystal slabs positioned immediately on top of the photodetector pixels. The spectrometer is completely complementary metal–oxide–semiconductor (CMOS) compatible and can be mass produced at low cost.


Supplementary Note 1: Spectrum Reconstruction
The unknown spectrum I(λ) is measured by recording its transmitted intensity through each PC structure by a CMOS sensor, with λ denoting the wavelength. Mathematically, for the ith PC, the measured signal S i can be written as Here, is reduced Planck constant, c is the speed of light in vacuum, and T i (λ) the transmission spectrum of the ith structure, which could be predetermined in the calibration process. The quantum efficiency of the sensor Q(λ) is calculated as where P in (λ) is the power of the incident light at wavelength λ, S CMOS (λ) is the signal recorded by the CMOS sensor without PC slabs. Combining Eqs. (1) and (2), we get We then name η(λ) = S CMOS (λ)/P in (λ), which is measured by taking the ratio of signal received by the CMOS sensor and a power meter illuminated by the same monochromatic light for each wavelength.
For recovery purposes, the unknown spectrum I(λ) is discretized into a N -dimension vector I n = I(λ n ), n = 1, 2, · · · , N . Then Eq. (1) is turned into M linear equations with N unknowns We can find the input spectra I that minimizes the l 2 norm ||S − T ηI|| 2 2 subject to I n > 0.
In practice, S is subject to experimental noise, then a smooth signal I that approximates S can be reconstructed as a solution to a regularized objective function min I ||S − T ηI|| 2 2 + k||DI|| 2 2 subject to I n > 0, where k > 0 is the weight. DI is the second-order derivative of the signal I [1, 2]. Minimizing ||DI|| 2 2 forces I to be smooth.

Supplementary Note 2: Structure Design
The proposed PC-spectrometer integrates an array of PC structures with a variety of periodicities and shapes. The parameters for each PC structure are chosen so that the spectral responsivity is substantially different between the various PC slabs. In order to achieve this goal, we simulated the spectral responsivities for hundreds of PC structures, and selected ones that less correlated.
The simulation was performed in S 4 [3] by solving Maxwells equations using rigorous coupled-wave analysis (RCWA).
To determine the number of structures needed for the recovery of an incident spectrum, we generated several spectra and simulated the measurement process with different numbers of independent PC slabs. In each test, the spectrum is digitalized with 1-nm resolution in a 200-nm  Figure 2 in the main text. For each angle, the spectral response of each PC structure is calibrated before measuring the unknown spectrum. Regardless of the incident angle, the spectrum can be obtained accurately, which indicates that the device sustains good performance for varying incident angles when calibration and signal measurement are performed at the same angle.
We also analyzed the cases when the unknown signal is measured at angles that are different from the calibration angle. Supplementary Figure 5 displays the test results for the measurements of two different spectra. For both cases, the incident beams are tilted by 1 to 4 degrees with respect to the calibration angle. The device maintains good performance within a 1-degree shift.
When the incident angle deviates more and more from the calibration angle, the recovered signals degrade while the main features persist.