Silicon Channeled Spectropolarimeter for On-Chip Single-Detector Stokes Spectroscopy

and low-cost spectropolarimeters are desirable in many ﬁ elds for probing the wavelength-dependent polarization state of light. Conventional spectropolarimetry systems use bulky, free-space optical components or at least four spectrometers. Herein, a chip-scale spectropolarimeter capable of reconstructing input Stokes spectrum with a single spectrometer is demonstrated. the a a on a The is using a complementary semiconductor (CMOS)-compatible a resolution of 0.1 nm across a wavelength range of 56 nm near 1550 This work provides a path toward on-chip, low-cost Stokes spectroscopy. simulations of the RMSE varying with f , we set SNR at 25dB according to our experimental results. Figure 8 shows the RMSE as a function of SNR when using the Fourier transform method and the compressive sensing method to reconstruct the Stokes spectra when f is equal to 0.003 mm. We can observe that increasing the SNR, the difference of the RMSE between the compressive sensing method and the Fourier transform method ﬁ rst increases, then decreases, and eventually approaches the same level at a high SNR.


Introduction
The Stokes spectrum describes the polarization state of light as a function of wavelength. It provides rich information on optical sources and light-matter interactions in various physical and chemical processes, such as the behavior of polarized exciton emissions, [1] the magnetic field of sunspots, [2,3] biosignatures, [4] the polarization mode dispersion of fiber, [5] and the absolute configuration of chiral molecules. [6,7] Spectropolarimetry is therefore applied in many fields such as biology and pharmacology, where it is used to distinguish the chirality of molecules, a critical feature of proteins, enzymes, DNA, RNA, amino acids, [7] and the majority of drugs on the market. [8] The reconstruction of a Stokes spectrum typically necessitates at least four spectral measurements. Low cost of spectral measurement can be realized either by splitting light into multiple spatial paths or in a time sequence. [9,10] Both approaches require an optical system with a mechanical or modulation mechanism that is usually bulky and expensive. Recently, a new concept, the channeled spectropolarimetry (CSP), has garnered considerable attention. [11][12][13][14][15] This technique allows for the reconstruction of the four spectrally resolved Stokes parameters using one measured spectrum which is called as channeled spectrum. This unique feature of a single spectral measurement significantly simplifies the optical system. Nevertheless, unlike the general type spectropolarimeter which has come into the chip level, [16] the on-chip CSP devices have never been reported because its architecture is difficult to be realized by integrated optics.
With the capability of monolithically integrating a wide variety of photonic functions, silicon photonics technology provides a massive advantage for enabling ultracompact photonic systems on a chip. [17][18][19][20][21][22][23][24] In this work, we will propose and demonstrate a monolithically integrated silicon photonic CSP (SiPh-CSP) for the first time. This device uses only one spectrometer with a single photodetector to reconstruct four spectrally resolved Stokes parameters, leading to substantially reduced power consumption, footprint, and cost. To produce a channeled spectrum, the conventional channeled spectrum generators typically apply two birefringent retarders and a polarizer with optical axes that need to be carefully aligned at specifically rotated angles. [11] In contrast, the configuration of our proposed SiPh-CSP uses only integrated waveguide components, eliminating the need to align birefringent axes. Using the SiPh-CSP, we examine two methods for channeled spectrum analysis and Stokes spectrum reconstruction, namely, the Fourier transform method [11,25] and the compressive sensing method. [14,[26][27][28] S 0 ðσÞ ¼ jE x ðσÞj 2 þ jE y ðσÞj 2 S 1 ðσÞ ¼ jE x ðσÞj 2 À jE y ðσÞj 2 S 2 ðσÞ ¼ E x ðσÞ ⋅ E Ã y ðσÞ þ E Ã x ðσÞ ⋅ E y ðσÞ S 3 ðσÞ ¼ i½E x ðσÞ ⋅ E Ã y ðσÞ À E Ã x ðσÞ ⋅ E y ðσÞ (1) where i indicates an imaginary unit, and E Ã x ðσÞ and E Ã y ðσÞ represent the complex conjugate of E x ðσÞ and E y ðσÞ, respectively. Equation (1) shows that the Stokes spectrum includes four independent spectra. In the following, we will detail how does the proposed device encode the four spectra on a single channeled spectrum. Figure 1 shows the schematic of our proposed SiPh-CSP. It consists of a surface polarization splitter, a channeled spectrum generator circuit, and an on-chip Vernier dual-microring spectrometer (VDMS). The surface polarization splitter splits two orthogonal linearly polarized states in free space, E x ðσÞ and E y ðσÞ, and couples them, respectively, into two different waveguides with fundamental mode. Then, the waveguide leads the light to the channeled spectrum generator to produce a channeled spectrum. The channeled spectrum generator plays an essential role in determining the performance of the channeled spectropolarimeter, and these will be detailed later in this section. The VDMS consists of two tunable microring resonators (MRs). By varying the electrical powers that are linked to the microheaters of MRs, the resonance wavelength of the MRs can be tuned using the thermo-optic effect and the channeled spectrum then can be measured. Using the Vernier effect, the two cascaded MRs with slightly different radii can dramatically increase the free-spectral range (FSR) of the dual-microring system. In our design, all the optical waveguides are single mode and operate in the quasi-transverse electric (TE) mode.
As shown in Figure 1, the channeled spectrum generator comprises two pairs of differential delay lines that are connected by a 2 Â 2 multimode interference (MMI) coupler. The differential length of the first and second pairs is L 1 and L 2 , respectively. At the output of the channeled spectrum generator, the second pair of differential delay lines is combined by a waveguide Y-junction, yielding the channeled spectrum IðσÞ where S 23 ðσÞ ¼ S 2 ðσÞ þ iS 3 ðσÞ and arg½S 23 ðσÞ represents the argument of S 23 ðσÞ; σ 0 is the center wavenumber; ϕ 1 ðσÞ ¼ 2πL 1 ⋅ ½ðn eff ,0 À n g,0 Þ ⋅ σ 0 þ oðσ À σ 0 Þ, and ϕ 2 ðσÞ ¼ 2πL 2 ⋅ ½ðn eff ,0 À n g,0 Þ ⋅ σ 0 þ oðσ À σ 0 Þ; oðσ À σ 0 Þ represents the Peano's form of the remainder and is neglected in our calculation for the first-order approximation; n g,0 and n eff ,0 are the group and effective index of the waveguide at σ 0 , respectively; κðσÞ and τðσÞ indicate the cross-coupling and straightthrough coefficients of the MMI coupler, respectively, with τðσÞ 2 þ κðσÞ 2 ¼ 1, and τðσÞ ¼ κðσÞ ¼ ffiffi 2 p 2 for a broadband 3 dB MMI coupler. The derivation of Equation (2) is provided in Supplementary Note 1, Supporting Information.
From Equation (2), we can see that the Stokes parameters are modulated onto different optical path difference (OPD) carriers in a cosinusoidal manner. Specifically, S 0 ðσÞ, the total power of the incoming light, is carried by the zero OPD; S 1 ðσÞ is modulated onto n g,0 ⋅ L 2 which is determined by the second pair of differential delay lines; and S 2 ðσÞ and S 3 ðσÞ are modulated onto n g,0 ⋅ jL 1 AE L 2 j which is determined by the combination of the two pairs of differential delay lines. The choices for L 1 and L 2 and their ratios are important for avoiding the overlap of these carriers. To demonstrate the proposed device, we have selected L 2 ¼ 3L 1 for our design and in the following discussion. In this case, the OPD carriers are spaced well apart from each other and set at 0, 2n g,0 L 1 , 3n g,0 L 1 , and 4n g,0 L 1 .
Using the channeled spectrum, either the Fourier transform [11] or compressive sensing [14,[26][27][28] methods can be applied to reconstruct the input Stokes spectrum. These two methods have their respective advantages and disadvantages. The principles behind these methods are discussed later.

Principle of the Fourier Transform Method
We will first introduce the principle of the Fourier transform method, and then discuss the principle of the compressive sensing method. Assuming that L 2 ¼ 3L 1 , the Fourier transform of IðσÞ (Equation (2)) can be expressed as www.advancedsciencenews.com www.adpr-journal.com where ℱ represents the operator of the Fourier transform, l denotes the OPD, and C 0 ðlÞ ¼ ℱ½S 0 ðσÞ C 1 ðlÞ ¼ ℱf2κðσÞ ⋅ τðσÞ ⋅ S 1 ðσÞe i½ π 2 þϕ 2 ðσÞ g C * 2 ðlÞ ¼ ℱfκ 2 ðσÞ ⋅ S * 23 ðσÞe Ài½ϕ 1 ðσÞÀϕ 2 ðσÞ g C 3 ðlÞ ¼ ℱfτ 2 ðσÞ ⋅ S 23 ðσÞe i½ϕ 1 ðσÞþϕ 2 ðσÞ g (4) and CðlÞ mathematically includes seven OPD channels centered at 0, AE2n g,0 L 1 , AE3n g,0 L 1 , and AE4n g,0 L 1 . A digital filter that selects one of the channels can then be applied in the OPD domain. Next, an inverse Fourier transform can be applied to reconstruct the Stokes parameter that is carried by the selected channel. While S 0 ðσÞ and S 1 ðσÞ can be directly reconstructed by C 0 ðlÞ and C 1 ðlÞ, respectively, S 2 ðσÞ and S 3 ðσÞ can be obtained by S Ã 23 ðσÞ and S 23 ðσÞ via C 2 ðlÞ and C 3 ðlÞ. All the design parameters, such as 2κðσÞ ⋅ τðσÞe i½ π 2 þϕ 2 ðσÞ , κ 2 ðσÞ ⋅ e i½ϕ 2 ðσÞÀϕ 1 ðσÞ , and τ 2 ðσÞ ⋅ e i½ϕ 1 ðσÞþϕ 2 ðσÞ , can be calibrated using a known Stokes spectrum such as (3) indicates that the space between two adjacent channels is equal to n g,0 L 1 . A larger channel space between channels allows ones to use a wider filter bandwidth when selecting an OPD channel, avoiding significant crosstalk from other channels. In this case, the proposed device can capture fine features that rapidly change along with the wavelength in a Stokes spectrum. But on the contrary, the spectral resolution of the spectrometer, which determines the OPD boundaries in the Fourier transformed channeled spectrum CðlÞ, does not allow the L 1 is set at a super-large value. For a given spectral resolution Δσ of a measurement, 1 2Δσ is the maximum resolvable OPD according to the Fourier transform theory. Therefore, the spacing between the channels (i.e., n g,0 L 1 ) is constrained so that all channels can be resolved within the resolution-determined range. A proper choice of L 1 should be To illustrate this trade-off, Figure 2 shows the simulated results using varied values for L 1 (with L 2 ¼ 3L 1 ) and the spectral resolution Δσ is fixed at 83.24 m À1 (0.2 nm, according to the performance of the proposed spectrometer). A Stokes spectrum, shown in Figure 2a, is used as the input. Figure 2b-d shows the channeled spectra generated by the channeled spectrum generator with L 1 ¼ 0.11 mm, L 1 ¼ 0.3 mm, and L 1 ¼ 0.5 mm, respectively. Figure 2e-g shows the Fourier transformed channeled spectra with L 1 ¼ 0.11 mm, L 1 ¼ 0.3 mm, and L 1 ¼ 0.5 mm, respectively. We can see that in the case of L 1 ¼ 0.11 mm, the channels are very close to each other, leading to strong crosstalk. However, if the differential delay is too large, such as the case of L 1 ¼ 0.5 mm (shown in Figure 2g), information carried by the larger OPD channels (in this case, S 23 and part of S 1 ) would be lost. A good balance can be achieved at L 1 ¼ 0.3 mm ( Figure 2c) according to Equation (5). A design that uses L 1 ¼ 0.3 mm was selected and fabricated for our experiment.
The Fourier transform method described earlier is very intuitive. However, this method is constrained by the spectral resolution and involves a filtering process that causes loss of information. [14,29] As a result, the Fourier transform method is more suited for a Stokes spectrum in which the polarization state changes gradually with the wavelength. [14,29]

Principle of the Compressive Sensing Method
Using the proposed device, we also can reconstruct the Stokes spectrum by the compressive sensing method. The principle of the compressive sensing method is described as follows. Following Equation (2), the channeled spectrum can be simplified as where a 0 ðσÞ, a 1 ðσÞ, a 2 ðσÞ, and a 3 ðσÞ are linear operators that are determined by the structure of the device and independent of the input Stokes spectrum. These linear operators can be calculated by measuring four known Stokes spectra. Based on the linear relationship defined in Equation (6), we can convert the reconstruction of a Stokes spectrum to a convex optimization problem [14,30] using 1) the measured channeled spectrum as the optimization target, 2) the estimated Stokes spectrum as the variable to calculate a virtual channeled spectrum in each iteration, and 3) the difference between the virtual and measured channeled spectrum as the cost function. By minimizing the cost function, the estimated Stokes spectrum converges to an optimum as the eventual measured result of the CSP. Details about the compressive sensing method are provided in Supplementary Note 2, Supporting Information. Note that Equation (5) also works for the choice of L 1 using the compressive sensing method (more details can be found in Figure S3 of Supplementary Note 2, Supporting Information). Note that the device cannot measure the four Stokes parameters of light with a single frequency because it is impossible to reconstruct four independent parameters only using one single sample point. But it is possible to reconstruct the four Stokes parameters of light with more than four separated frequencies. Figure 3a shows a pseudocolor micrograph of the fabricated chip. The surface polarization splitter is formed using a 30 Â 30 array of cylindrical holes that are etched through the silicon with a period of 695 nm and a diameter of 440 nm. Two distributed Bragg reflection gratings are added in the two idle ports of the surface polarization splitter to improve the optical coupling efficiency. [31] The period and filling factor of the distributed Bragg reflection gratings are 360 nm and 0.44, respectively. More details about the design of the surface polarization splitter are provided in our previous work. [17,32] A scanning electron microscope (SEM) image of the fabricated surface polarization splitter is shown in Figure 3b. The channeled spectrum generator consists of two pairs of differential delay lines, connected by a 3 dB MMI coupler with a footprint of 4.1 μm Â 157.5 μm. Details about the design and performance of the MMI coupler can be found in Supplementary Note 3, Supporting Information. Two VDMSs can be observed in the optical micrograph ( Figure 3a): one is used for VDMS characterization, with the optical path indicated by the long red arrow; the other one is used to measure the channeled spectrum from the channeled spectrum generator, with the optical path indicated by the long green arrow. They share the same first-stage resonator (MR1) but have separate second-stage resonators (MR2). The diameters for MR1 and MR2 are 57.1 and 60.1 μm, respectively. Both MR1 and MR2 have a coupler-length of 3 μm and a coupler-gap of 300 nm. The SEM images of the MR1 and MR2 before cladding are shown in Figure 3c,d, respectively. Figure 3e shows a micrograph of the device after being packaged with fiber arrays and electrical connections. In the following section, we will first demonstrate the proposed VDMS before characterizing our SiPh-CSP. We used an off-chip photodetector to read the optical intensity in our experiment. However, a Ge-on-silicon photodetector, [33,34] which has been well developed and is widely available in silicon photonics foundry processes, can be readily integrated onto the same chip.

Performance of the VDMS
The FSRs of the fabricated MR1 and MR2 were measured to be 3.08 and 2.94 nm, respectively. Due to the Vernier effect, as shown in Figure 4a, the cascaded dual MRs achieved a substantially extended FSR of greater than 56 nm. The inset of Figure 4a shows the filter shape of the cascaded dual MRs with a 3 dB bandwidth of 0.1233 cm À1 , indicating that the best achievable resolution of the fabricated VDMS is 0.1233 cm À1 . In our experiment, a sweeping  www.advancedsciencenews.com www.adpr-journal.com step of 0.4162 cm À1 (0.1 nm) was selected to accommodate the impact of thermal fluctuation during the spectrum measurement. By controlling the heating powers that were applied to MR1 and MR2, we were able to tune the resonance wavenumber of the cascaded dual-MR system. The experimental calibration of the heating powers for each resonance wavenumber is provided in Supplementary Note 4.1 and 4.2, Supporting Information. The input spectrum could be obtained by sweeping the resonance wavenumbers with a 0.4162 cm À1 step and simultaneously reading the optical intensity of the dual-MR drop port. The experimental setup of the spectrum measurement is provided in Supplementary Note 4.4, Supporting Information. Figure 4b compares the measured results that were obtained using a commercial optical spectrum analyzer (black line) and the fabricated VDMS (red line). The small discrepancy is due to fluctuations in the resonance position caused by fluctuations in the temperature, [16] which can be improved by adding an integrated temperature sensor to calibrate the temperature fluctuation.

Performance of the SiPh-CSP
We experimentally demonstrate the proposed SiPh-CSP with 19 input Stokes spectra. Here, we only show the reconstructed results of three input Stokes spectra whose components were evenly distributed from À1 to 1 (see Figure 5 and 6). The other results are provided in the Supplementary Note 4.6, Supporting Information.
The reconstructed results of the three input Stokes spectra using the Fourier transform are shown in Figure 5. Details about the calibrations, the generation of input Stokes spectra, and the experimental setup are presented in Supplementary Note 4.3 and 4.5, Supporting Information. Figure 5a,d,g shows measured channeled spectrum, Fourier inversion of the channeled spectrum, and reconstructed Stokes spectrum, respectively, for the first input Stokes spectrum. Figure 5b,e,h shows measured channeled spectrum, Fourier inversion of channeled spectrum, and reconstructed Stokes spectrum, respectively, for the second input Stokes spectrum. Figure 5c,f,i shows measured channeled spectrum, Fourier inversion of channeled spectrum, and reconstructed Stokes spectrum, respectively, for the third input Stokes spectrum. Figure 5g-i shows the final reconstructed results by using the Fourier transform method. The root mean squared error (RMSE) of the reconstructed results of using the Fourier transform method is 0.0664, corresponding to an orientation angle RMSE of 3.2 and an ellipticity angle RMSE of 1.85 . Here, we define the RMSE of the reconstructed results as where n and m mean the total numbers of the spectral points and the measured Stokes spectra, respectively. As shown in Figure 5 and Supplementary Note 4.6, Supporting Information, a total of 19 Stokes spectral samples (i.e., m ¼ 19) were used to characterize our device. S j ðσ q Þ re,u and S j ðσ q Þ in,u represent the jth reconstructed and input Stokes parameter, respectively, at the wavenumber σ q for the u th measured Stokes spectrum sample. Improving the performance of the proposed spectrometer in the future, the reconstructed accuracy of the Stokes spectrum can be significantly improving. However, the reconstructed Stokes spectra showed a slight sinusoidal variation along the input Stokes spectra due to the intrinsic limitations of the Fourier transform method, as discussed earlier. This variation can be improved by simultaneously using longer differential delays (i.e., L 1 and L 2 ) and a better spectral resolution.
Using the same measured channeled spectra (Figure 5a-c), we also applied the compressive sensing method to reconstruct the input Stokes spectra. The compressive sensing method formulates the reconstruction of a Stokes spectrum as a multivariate optimization problem following Equation (6). In our optimization model, the cost function is defined as the deviation of the estimated channeled spectrum from the measured one and the outcome variables are wavelength-dependent Stokes parameters. [14] As shown in Figure 6a, the estimated Stokes spectrum slowly closes to the input Stokes spectrum after each iteration. Details about the compressive sensing method are provided in Supplementary Note 2, Supporting Information. The reconstructed results are shown in Figure 6. The RMSE of the reconstructed results using the compressive sensing method is 0.0367, corresponding to an orientation angle RMSE of 2.12 and an ellipticity angle RMSE of 1.18 . In theory, the accuracy of the Stokes spectral reconstruction is independent of the input Stokes spectrum. However, the reconstructed result shown in    www.advancedsciencenews.com www.adpr-journal.com Figure 5c is worse than other cases, which may be due to the fabrication imperfection. As discussed in Supplementary Note 4.5, Supporting Information, the fabrication imperfection can change the weighting applied to each Stokes parameter (i.e., a 0 ðσÞ, a 1 ðσÞ, a 2 ðσÞ, and a 3 ðσÞ in Equation (6)), which may degrade the accuracy and cause performance dependency on the input polarization state. This problem can be mitigated by improving the fabrication process.
Comparing the results in Figure 5 and 6, we can see that the Stokes spectrum can be more accurately reconstructed using the compressive sensing method, which, however, comes at a computational cost. In our experiment, the data processing time (Intel Core i9-9900K) per sample of using the Fourier transform method and the compressive sensing method are 0.0004 and 5 s, respectively. In theory, the time complexity of the Fourier transform method and the compressive sensing method are OðN log 2 NÞ and OðN 3 Þ, respectively, where N is the total number of frequency points. [27] The time consumption of the compressive sensing method will significantly increase with the operation wavelength range or the wavelength resolution. Moreover, the performance of the two methods may vary under different measurement conditions, depending on noise level, operating bandwidth, and spectral resolution, as discussed in Experimental Section. [35] However, we have demonstrated that both of the Fourier transform and the compressive sensing methods can be used by our device to reconstruct the Stokes spectrum.

Reconstructing Stokes Spectra in the Presence of Strong Wavelength Dependence
In the aforementioned experiment, we only examined the Stokes spectra in which the polarization states are constant or vary slowly with the wavenumber. In this section, we performed some simulations to study the properties of two methods when they are used to reconstruct the Stokes spectrum in which the polarization state quickly varies with the wavenumber. The Stokes spectra are expressed as where j ¼ 1, 2, and 3; f determines the dependence of the input Stokes vector on wavenumber; b j0 ðσÞ, b j1 ðσÞ, b j2 ðσÞ, b j3 ðσÞ, and φ j1 , φ j2 , φ j3 are the random values. Details about the simulations including how to randomly generate such Stokes spectra are provided in Experimental Section. We introduced f to describe the dependence of the input Stokes vector on wavenumber. For each f, 10 000 of randomly generated Stokes spectra were used to characterize the properties of two methods. According to Equation (7), the simulated RMSE of the reconstructed results as a function of f is shown in Figure 7b. Figure 7b shows that the reconstructed RMSE for both Fourier transform and compressive sensing methods increases with frequency f. When f reached 0.11 mm which corresponds to the channel bandwidth limitation of the Stokes spectrum, the Fourier transform method starts to lose some information of the input Stokes spectrum, while the compressive sensing method remains at a low RMSE level. The reason for the Fourier transform method's sensitivity to the frequency f is that this method requires the high-frequency detail to be cut off to sort the channels. Simultaneously, the channel overlap increased with frequency f, as shown in the images embedded in Figure 7b. Figure 7c,d shows examples of Stokes spectrum reconstruction with an f of 0.465 mm using the Fourier transform and the compressive sensing methods, respectively. We can clearly observe from these figures that the Fourier transform method with a RMSE of 0.327 loses high-frequency detail. Therefore, the compressive sensing method is a better choice for our device when f is large. The downside of this method is its computationally intensive operation. As discussed previously, the compressive sensing method requires more time to process data than the Fourier transform method. Therefore, the compressive sensing method can be used in some fields which do not have stringent requirements for signal processing speeds, such as the search for Earth-and super Earth-like planets, [36] the study of the chiral molecule configurations, [6,7] and RNA sensors. [37] However, some applications do require fast signal processing for real-time monitoring, such as probing the activity of diguanylate cyclases and c-di-GMP phosphodiesterases, [38] for which the Fourier transform method may be more appropriate. Moreover, to measure the Stokes spectra with very fine features such as the atomic absorption lines, we can increase the differential delays (i.e., L 1 and L 2 ) of the channeled spectrum generator and the spectral resolution of the proposed spectrometer.

Conclusion
In conclusion, we have demonstrated an ultracompact, solid-state spectropolarimeter and have shown that on-chip reconstruction of a full-Stokes spectrum can be realized with a single spectral measurement. All the optical components, including surface polarization splitter, channeled spectrum generator, and spectrometer, were monolithically integrated on a silicon chip with a footprint of 0.3 mm Â 1.5 mm. Our device achieves a resolution of 0.1 nm over a bandwidth of 56 nm. Because only one spectrometer is required, the power consumption decreased by a factor of four. Fabricating a similar architecture using the other materials could extend the operating wavelength range into the visible and midinfrared regions. [39][40][41] As the device was implemented using a technology compatible to the CMOS process, it can be readily integrated with microelectronics and leverage the quickly evolving silicon photonics ecosystem. [42] The advantages of ultracompactness, low power consumption, and CMOS-compatibility indicate great potential for new applications such as low-cost bioanalysis of racemic drugs and wearable and implantable sensors.

Experimental Section
Fabrication: The device was fabricated using a commercial CMOScompatible silicon on insulator (SOI) process with electron-beam lithography at Applied Nanotools Inc. The SOI wafer includes a 2 μm buried oxide layer, a 220 nm silicon layer, and a 2 μm oxide cladding layer. The devices were subsequently packaged at our lab. The electrical connections were achieved using the Westbond 7400A Wire Bonder.

Experiment:
The heating powers that were applied to the MRs were calibrated using a tunable laser source (Agilent 81600B) with an optical power of around 3 dBm. The heating powers of the heaters were driven using a Keithley 2612B Souremeter. To decrease the amount of time required for heating power calibration, we proposed a two-step method that first searched the heating power of single MR and then swept the heating powers of two MRs in a small power range. Details are provided in Supplementary Note 4.1, Supporting Information. The light from a high-power wide-band Erbium ASE source (INO) was used to characterize the proposed spectrometer. A commercial optical spectrum analyzer (OSA, Yokogawa AQ6370D) was used to measure the spectrum. The experimental setup used to calibrate the heating powers that were applied to the MRs of the SiPh-CSP is provided in Figure S7 of Supplementary Note 4.3, Supporting Information.
The components that were used to control the input Stokes spectrum include: a polarizer (LPNIR100-MP2, Thorlabs), an HWP (WPH10M-1550, Thorlabs), and a QWP (WPQ10M-1550, Thorlabs). Two stepper motor rotators (K10CR1/M, Thorlabs) were used to control the angles of the HWP and QWP, respectively. The experimental setup for measuring the Stokes spectrum using the proposed SiPh-CSP and the calibrated a 0 ðσÞ, a 1 ðσÞ, a 2 ðσÞ, and a 3 ðσÞ of the fabricated device are presented in Supplementary Note 4.5, Supporting Information. The optimization algorithm is based on CVX which is a Matlab-based modeling system for convex optimization and was provided by CVX Research, Inc.
Simulation: For the simulated results shown in Figure 7, the target Stokes spectra are randomly generated by emulating the operation of two birefringent retarders, as shown in Figure 7a. The thickness of the first retarder is two times that of the second one. Assuming a linearly polarized state passing through the two retarders, the output Stokes vector is given by where M k (k ¼ 1, 2) is the Mueller matrices of the first (or second) retarder and can be expressed as www.advancedsciencenews.com www.adpr-journal.com where θ k is the angle of the fast axis of the kth retarder, and δ k is the phase difference between the fast and slow axes, given by where D k represents the thickness of the kth retarder with D 1 ¼ 2D 2 and Δn eff is the effective index difference between the fast and slow axes of the retards. By defining f ¼ ðD 1 þ D 2 Þ ⋅ Δn eff and following Equation (9)-(11), the Stokes spectrum produced by the two retarders can be expressed as where j ¼ 1, 2, 3 and b j0 , b j1 , b j2 , and b j3 are functions of θ 1 and θ 2 . Using Equation (12), we can generate a series of Stokes spectra with an arbitrary polarization state using the three free variables f, θ 1 , and θ 2 . In our simulations (Figure 7b), for each f, 1000 Stokes spectra were generated using 1000 random pairs of ½θ 1 , θ 2 . To make sure that all S j ðσÞ included the cosine term with f, we set θ 1 6 ¼ 0 ∘ or90 ∘ and θ 1 6 ¼ θ 2 . Each target Stokes spectrum was simulated 3 times. There are 3000 samples for each value of f.
Here, the signal-to-noise ratio (SNR) is defined by [14,35] SNR ¼ 10log 10 " P n q¼1 I 2 ðσ q Þ À P n q¼1 G 2 ðσ q Þ P n q¼1 G 2 ðσ q Þ # where Iðσ q Þ and Gðσ q Þ are the optical intensity and white additive Gaussian noise, respectively, at the wavenumber σ q . For the simulations of the RMSE varying with f, we set SNR at 25 dB according to our experimental results. Figure 8 shows the RMSE as a function of SNR when using the Fourier transform method and the compressive sensing method to reconstruct the Stokes spectra when f is equal to 0.003 mm. We can observe that increasing the SNR, the difference of the RMSE between the compressive sensing method and the Fourier transform method first increases, then decreases, and eventually approaches the same level at a high SNR.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author. www.advancedsciencenews.com www.adpr-journal.com