Supplementary Information Compact spectrometer based on a disordered photonic chip Brandon Redding, Seng Fatt Liew, Raktim Sarma, Hui Cao* Department of Applied Physics, Yale University, New Haven, CT 06520, USA *E-mail: hui.cao@yale.edu 1. Spectrum reconstruction algorithm The intensity distribution on the detector array is I = T S, where T is the transmission matrix and S is the input spectrum. After calibrating T, the input spectrum can be reconstructed simply by multiplying the measured I by the inverse of the transmission matrix: S = T -1 I. However, the spectra reconstructed in this manner are susceptible to experimental noise, as shown in Fig. S1(a). To improve the accuracy of the reconstructed spectrum, we used a nonlinear optimization algorithm based on simulated annealing to find the spectrum S which minimized I TS 2. We used the spectrum obtained from the inversion procedure as the initial guess for S and used the simulated annealing algorithm to minimize I TS 2. The final reconstructed spectrum is shown in Figure S1(b). Figure S1 Spectrum reconstruction. The spectrum for a probe signal at λ=1512 nm reconstructed using the matrix inversion procedure (a) and the simulated annealing algorithm (b). In the presence of experimental noise, the simulated annealing algorithm improves the quality of the reconstructed spectrum. 2. Measuring arbitrary spectra with the random spectrometer In addition to accurately identifying the wavelength of narrowband input, the random spectrometer can also measure spectra consisting of multiple wavelengths or even a continuous broad band. Using the random spectrometer shown in Fig. 1, we synthesized various probe spectra as the weighted sum of speckle patterns recorded at individual wavelengths and applied the same reconstruction algorithm as described in Section 1. In Fig. 2(e), we present a reconstructed spectrum of 3 narrow lines with different intensity. The random spectrometer was able to accurately identify the wavelength and relative amplitude of the three lines. Figure 2(f) showed the reconstructed spectrum of a broad continuous spectrum. In this case, since the NATURE PHOTONICS www.nature.com/naturephotonics 1
spectral feature was broad, we sampled the transmission matrix at the wavelength interval of 1 nm, yielding a transmission matrix with 25 spectral channels and 25 spatial channels (detectors). This result demonstrated that the random spectrometer was able to accurately reconstruct the broadband spectra. 3. Spectral range of the random spectrometer Multiple scattering in a random structure occurs over an extremely broad range of frequencies. This enables the same random spectrometer to operate at different spectral regions simply by using the transmission matrix calibrated for the desired spectral region. The spectral resolution will change for different spectral regions since the transport mean free path of the scattering structure is wavelength dependent. However, the first Mie resonance of the scatterer used in the random spectrometer (75 nm radius air hole in a silicon membrane) is at the wavelength of 440 nm, much shorter than the probe wavelength (~1500 nm). Figure S2 plots the calculated scattering cross section of the air cylinder in the wavelength range of 1250 nm to 1750 nm. The scattering cross section changes by ~50% over the wavelength range of 500 nm. Since the mean free path scales with the scattering cross section, it varies gradually in the wavelength range of 1250 nm to 1750 nm. Therefore, the random spectrometer can operate over a very large spectral range with only a modest change in resolution. Figure S2 Broadband scattering. Calculated scattering cross section of a 75 nm radius air cylinder embedded in a silicon membrane. The scattering cross section decreases gradually from λ=1250 nm to 1750 nm, allowing the random spectrometer to provide similar performance and resolution over a large spectral range. 4. Thermal stability of the random spectrometer One advantage of fabricating the random structure on-chip is that the speckle pattern is very stable, since the scatterers (air holes etched into the silicon membrane) do not move. However, temperature variation will introduce a change in the refractive index of the Si layer and alter the speckle pattern. To estimate the change in temperature that can be tolerated for spectrum reconstruction, we performed numerical simulation of the 20 μm radius random spectrometer using the FDFD method. Specifically, we simulated the speckle pattern generated for a fixed input wavelength (λ = 1512.5 nm) at different temperatures by changing the refractive index of the Si according to the thermo-optic coefficient, dn/dt 1.8 10-4 K -1 [S1]. Figure S3(a) plots the intensity distribution over the detectors as a function of temperature. The speckle pattern gradually decorrelates as the temperature changes. We then reconstructed the input spectra from the speckle patterns using the transmission matrix obtained at one specific temperature. As shown in Fig. S3(b), up to ±4 o K change of temperature, the input wavelength can be recovered 2 NATURE PHOTONICS www.nature.com/naturephotonics
SUPPLEMENTARY INFORMATION to the accuracy of the spectral resolution (~ 0.5 nm). To achieve better thermal stability, on-chip temperature stabilization techniques, which have been developed for microcavity-based devices, may be used. Alternatively the spectrometer can be calibrated at multiple temperatures, allowing the appropriate transmission matrix to be selected in software as the temperature changes. Figure S3 Modeling thermal stability of the 20 m radius random spectrometer. (a) Calculated intensity of light reaching 20 output waveguides as a function of temperature change for an input signal at λ=1512.5 nm. As the temperature varies, the speckle pattern gradually decorrelates. (b) Reconstructed spectra for an input light of wavelength λ = 1512.5 nm at different temperatures. When the change of temperature exceeds 4 o K, the recovered wavelength deviates from the input wavelength by 0.5 nm, which is approximately the spectral resolution. 5. Numerical simulation of optical transmission In the random spectrometer, we surrounded the disordered photonic structure with the photonic crystal boundary to reduce the insertion loss. The triangular lattice of air holes that form the boundary were designed to have a 2D photonic bandgap (PBG) for TE polarized light in the wavelength range of 1478 nm 1560 nm. The photonic band structure was calculated with the plane wave expansion method [S2]. To compare the optical transmission with and without the photonic crystal boundary, we performed 2D numerical simulations using the finite-difference frequency-domain method. As shown in Fig. S4, the random spectrometer has the semicircular shape. The input waveguide injects light from the center of the semicircle, and the output waveguides are distributed along the circumference. The density and size of air holes are identical to those of the fabricated spectrometer. The effective index of refraction of the silicon layer is 2.86, it is obtained by matching the PBG of a triangle lattice of air holes (with the same density and size of air holes as in the random array) in the 220 nm silicon layer on top of silica to that of the approximate 2D structure. In Fig. S4(a), the boundary in between the waveguides are open and light can leak out. The boundary at the flat base of the semicircular area containing the scattering medium is also open. In Fig. S4(b), the boundary reflects light back into the random medium. For simplicity, the photonic crystal lattice is replaced by a perfect magnetic conductor at the boundary, which also reflects the TE polarized light from all directions. We summed the flux of light reaching the detectors indicated in green in Fig. S4(a,b) and divided by the input flux to obtain the transmission. For the open random structure of radius 25 μm, the transmission is merely 21% in the wavelength range of 1500 nm to 1525 nm. With the reflecting boundary, the transmission NATURE PHOTONICS www.nature.com/naturephotonics 3
increases to 60%. In this 2D simulation, the out-of-plane scattering was not taken into account. Nonetheless, the result illustrates that the introduction of reflecting borders can significantly improve the transmission through a scattering medium. Fig. S4 Numerical simulation of optical transmission through a random spectrometer. (a) Schematic of a semicircular random medium with open boundary. The semicircle has a radius of 25 μm. The random medium consists of a random array of air holes with a radius of 75 nm. PML: Perfectly matched layer, PMC: Perfect magnetic conductor. (b) Schematic of the same random medium with reflecting boundary. The photonic crystal structure that formed the boundary in the fabricated spectrometer was replaced with PMC to reduce the computational load. In both (a) and (b), the optical signal enters the random medium through an input waveguide at the center of the semicircle marked by the black arrow. The output waveguides, distributed along the circumference, delivers the transmitted light to the detectors marked by the green line. (c) Calculated amplitude of the H z field for the TE polarized light at λ = 1500 nm diffusing through the scattering medium shown in (a). Without the reflecting boundary, most of the input light leaves the scattering medium through the base of the semicircle before reaching the output waveguides at the circumference. (d) Calculated amplitude of the H z field for the TE polarized light at λ = 1500 nm diffusing through the random medium shown in (b). The reflecting boundary efficiently channels the input light to the output waveguides. The scale bars indicate 10 μm. 6. Estimation of out-of-plane scattering loss Since 3D FDFD simulations of the on-chip spectrometers are computationally heavy, we estimated the relative out-of-plane leakage of the completely random structure, the photonic amorphous structure, and the golden-angle spiral lattice from 2D simulations using the method in Ref. [S3]. For the TE polarized light, we first calculated the in-plane electric field components (E x and E y ) inside the three scattering structures. Then we conducted a 2D spatial Fourier transform of E x and E y and compared the fraction of wavevectors present within the light cone (these wavevectors can leak out of the plane). Since the field distribution obtained with the 2D simulation which neglects out-of-plane scattering differs from the actual field distribution in the presence of out-of-plane scattering, we do not expect to obtain quantitatively accurate 4 NATURE PHOTONICS www.nature.com/naturephotonics
SUPPLEMENTARY INFORMATION estimations of the out-of-plane scattering loss, but we can make qualitative comparisons of the three scattering media. The spectrometers in the simulation had the geometry shown in Fig. S4(b) with a radius of 20 μm. Although the simulation included the output waveguides, we extracted the fields (E x and E y ) across the scattering structures within the semicircle to estimate the out-of-plane loss. We then computed ( F(E x ) 2 + F(E y ) 2 ), where F represents the 2D Fourier transform. There are two light cones, one for air above the silicon layer ( k > ω/c, ω is the angular frequency and c is the speed of light in air), the other for silica underneath ( k > n s ω/c, n s = 1.5 is the refractive index of silica). Since the light cone for the silica is larger than that of the air, we computed the fraction of the Fourier components within the silica light cone. We found that for the random, amorphous, and spiral structures, 0.82%, 0.59%, and 0.49% of the electric field intensities were within the light cone, respectively. Hence, the random structure is more likely to scatter light into the light cone, leading to larger out-of-plane leakage than the amorphous and spiral structures. 7. Characterization of amorphous and spiral spectrometers In addition to reducing the out-of-plane scattering loss, the spectrometers based on the photonic amorphous structure and the golden-angle spiral lattice had similar performance to the random spectrometer. We performed the same calibration procedure for transmission matrices of the amorphous and spiral structures, and characterized their operation as a spectrometer. Specifically, the transmission matrices for the 20 m radius spectrometers were recorded for TE polarized light in the wavelength range of 1500 nm to 1525 nm in steps of 0.25 nm. We tested their ability to reconstruct a series of narrow lines. As shown in Fig. S5(a,c), both amorphous and spiral spectrometers accurately recovered the spectral lines over 25 nm bandwidth. We also tested their spectral resolution by measuring two closely spaced lines. Figure S5(b,d) exhibited a wavelength resolution of 0.75 nm around = 1512 nm for both spectrometers, which is comparable to the random spectrometer. Figure S5 Characterization of amorphous and spiral based spectrometers. (a,c) Reconstructed spectra for a series of narrow spectral probes across the 25 nm bandwidth using the spectrometers based on the photonic amorphous structure (a) and the golden-angle spiral lattice (c). The black dotted lines mark the center wavelength of each probe line. (b,c) Reconstructed spectrum (blue line) of two narrow spectral lines separated by 0.75 nm using the amorphous (b) and spiral (d) spectrometers. The red dotted lines mark the center wavelengths of the probe lines. NATURE PHOTONICS www.nature.com/naturephotonics 5
References [S1] Komma, J., Schwarz, C., Hofmann, G., Heinert, D., Nawrodt, R., Thermo-optic coefficient of silicon at 1550 nm and cryogenic temepratures, Applied Physics Letters, 101, 041905 (2012). [S2] Johnson, S. G. & Joannopoulos, J. D. Block-iterative frequency-domain methods for Maxwell s equations in a planewave basis. Optics Express 8, 173-190 (2001). [S3] Boriskina, S., Gopinath, A., Dal Negro, L. Optical gap formation and localization properties of optical modes in deterministic aperiodic photonic structures. Optics Express 16, 18813-18826 (2008). 6 NATURE PHOTONICS www.nature.com/naturephotonics