Optimal ultra-miniature polarimeters in silicon photonic integrated circuits

Measurement of the state of polarization of light is essential in a vast number of applications, such as quantum and classical communications, remote sensing, astronomy, and biomedical diagnostics. Nanophotonic structures and integrated photonic circuits can, in many circumstances, replace conventional discrete optical components for miniature polarimeters and chip-scale polarimetry systems, and thus signiﬁcantly improve robustness while minimizing footprint and cost. We propose and experimentally demonstrate two silicon photonic (SiP) four-photodetector (PD) division-of-amplitude polarimeters (4PD-DOAPs) using a CMOS-compatible photonic fabrication process. The ﬁrst design targets minimizing the number of optical components. The second design makes use of a slightly more complex circuit design to achieve an optimal frame for measurements; this measurement frame minimizes and equalizes estimation variances in the presence of the additive white Gaussian noise and the signal dependent shot noise. Further theoretical exami-nation reveals that within the optimal measurement frames for Stokes polarimeters, the DOAP with four PDs has the minimal equally-weighted variance compared to those with a greater number of PDs.


I. INTRODUCTION
State of polarization (SoP) is one of the essential properties of light. It conveys unique information on optical sources and on light-matter interactions 1-4 . Measurement of SoP thus finds crucial application in communications, quantum information, astronomy, and biomedical and chemical sensing 5 . Polarimeter performance has been improved over and again during its long development history 6 . While theory on optimization of polarimeter parameters is well established, the implementation of polarimeters in bulky, discrete optical components has hindered their broad application.
Recently, the rapid development of nanoscience and nanotechnology has led to a significant progress towards ultra-miniaturization of polarimeters. Some miniature polarimeters have been demonstrated, such as a metasurface polarimeter [7][8][9] , a circular polarization imager using chiral structure 10 , all-fiber polarimeter 11 , and a sub-wavelength polarimeter exploiting the spin-orbit interaction of light 12 . The photonic integrated circuit (PIC) technology has immense advantages for realizing miniature, solid-state polarimeters as it is capable of integrating a vast number of optical components on a single chip [13][14][15][16] .
A complete chip-scale polarimetry system can be achieved with all the components, including photo-detectors (PDs) and electric signal processing circuits, integrated on micro-chips, leading to substantially improved robustness with reduced cost and footprint. Various materials can be adopted in PICs, such as indium phosphide 17 , silicon 13,18 , silicon nitride 19 , and germanium 20 , covering from visible to long-wave-infrared regions, while sharing the same principles and waveguide architectures. We recently demonstrated a chip-scale full-Stokes polarimeter in a silicon PIC, consisting of a surface polarization splitter (SPS) and an on-chip optical interferometer circuit, producing the complete analysis matrix of an optimally conditioned polarimeter 21 . Silicon photonic Stokes a) Electronic mail: wei.shi@gel.ulaval.ca receivers 22,23 were also demonstrated using integrated waveguide components such as an edge coupler, a polarization splitter, an optical hybrid, and Ge PDs .
However, compared to conventional solutions, optimization of the PIC-based polarimeter parameters has not been extensively explored, which will be addressed in this paper. We use integrated optical components to realize an optimal PICbased polarimeter with the CMOS-compatible silicon photonics technology. Our design achieves classical optimal measurement frames for SoP reconstruction.
Here we focus on division-of-amplitude polarimeters (DOAPs) that split the light beam into several paths for fast, simultaneous measurement 24 . In our previous work, we proposed and demonstrated an optimally conditioned silicon photonic DOAP 21 requiring six PDs. Nevertheless, the full reconstruction of the Stokes vector in principle requires only four measurements of optical intensity. Therefore, the signal processing cost was increased as we generated more measurements than the minimal four for a DOAP. In this paper, we propose two chip-scale silicon photonic four-photodetector DOAPs (SiP-4PD-DOAPs). The first design minimizes the number of optical components for an ultra-compact design. The second exploits an asymmetrical power splitter (APS) 25 to produce an optimal reference frame with minimized and equalized estimation variances. Both devices are designed for a standard 220-nm-thick silicon-on-insulator (SOI) wafer and are optimal in the presence of both additive Gaussian noise and signal-dependent (Poisson) shot noise.
The rest of this paper is organized as follows. In Section II, we review the fundamentals of Stokes polarimeters and define conventions that will be used in the following sections. In Section III and IV, we present the design and experimental results of the two proposed SiP-4PD-DOAPs. While photodiodes were not integrated in the device under test, our results clearly establish the viability of an ultra-compact solution. In Section V, we discuss various structures for noise minimization in the scope of optimal measurement frames of Stokes polarimeters. Section VI is the conclusion.

II. STOKES POLARIMETER PRINCIPLES
The SoP is typically characterized by a 4 × 1 Stokes vector. Therefore, complete reconstruction of the SoP requires a minimum of four distinct measurements, which can be realized by projecting the Stokes vector onto four or more analysis states determined by the Mueller matrix (analysis matrix) of the polarimeter. In classical free-space optical systems, this operation can be achieved via rotating polarizers or via retarders in combination of a fixed polarizer. In a PIC, this can be realized through waveguide interferometers without mechanical moving parts. Figure 1 shows PIC counterparts of some free-space optical components commonly used in Stokes polarimeters. A SPS 21 can decompose the two orthogonal E-field components (E x and E y ), each coupling (ideally with equal power) into two single-mode waveguides that guide in opposite directions. As shown in Fig. 1a, the SPS functions as combined polarization beam splitter (PBS) and half-wave plate (HWP) in a conventional free-space optical system. An on-chip beam combiner (i.e., 3-dB Y-branch 26 ) coherently combines E x and E y (orthogonal in free space, but coupled to the same mode in two waveguides), as shown in Fig. 1b, outputting √ 2 2 (E x + E y ), which is equivalent to a 45 • linear polarizer. A retarder can be simply replaced by two optical waveguides with various lengths that introduce an phase difference between E x and E y (as shown in Fig. 1c).
A polarimeter transforms the Stokes vector into a series of intensities that can be detected by PDs. The analysis matrix W defines the transformation where S = (S 0 , S 1 , S 2 , S 3 ) T is the input Stokes vector. I = (I 1 , I 2 , ..., I N ) T is an N-dimensional vector representing the measured intensities, not to be confused with the identity matrix. The noise contribution of the PDs is n. The estimated Stokes vectorŜ is given bŷ where W † denotes the generalized inverse of W, also known as the synthesis matrix. Here, we only consider the case of N = 4, so that W † = W −1 . The error of the estimated Stokes vector can be obtained by Equation (3) shows estimation error is influenced by noise level and the synthesis matrix. For noise n with covariance matrix Γ, taking expectations, we have In the presence of additive white Gaussian noise (AWGN), and when the noise at each PD is zero-mean and identically distributed with variance σ 2 n , we have The condition number 27,28 κ = W · W −1 is a figure of merit often used to evaluate polarimeter performance, where * is the matrix norm (taken as the L 2 norm throughout this work). The detection signal-to-noise ratio (SNR) is maximized when the condition number is minimized.
In the presence of shot noise (i.e., Poisson noise), assuming independent noise in each PD, the noise covariance matrix is diagonal with i th entry proportional to the i th detected signal power. For A i j denoting the i, j element of matrix A, this means Therefore, the variance of the Stokes estimate is SoP dependent for Poisson noise. As the signal power varies across PDs, the Poison noise is not identically distributed (unlike the AWGN). For best performance, the polarimeter would equalize the noise variances. Following 29 , we define matrix Q by ) T , and P as the degree of polarization. As the Poison variance depends on S, each component ofŜ will have some maximum variance, γ max i , and minimum variance, γ min i , associated with that component. The mean excursion between these extrema, ∆γ, is given by Ideally the polarimeter would equalize the noise for zero excursion, i.e., where maximum noise variance γ max i equal to minimum noise variance γ min i on each Stokes vector component. We therefore seek an optimal structure 29 minimizing the condition number κ and the variance difference ∆γ. The schematic of the proposed SiP-4PD-DOAP is shown in Fig. 2. Firstly, the incoming light is split into four waveguides by SPS. The optical waves propagating in the four paths (i.e., 2 E x , and √ 2 2 E y ) carry the full information of SoP of the incoming light. Two of the optical paths are split into four paths by two 50:50 Y-branches, and then they separately pass through unique θ i -phase-retard waveguides.
The optical waves which pass through θ 1 -phase-retard and θ 3 -phase-retard waveguides are by construction coherent with each other and they combine to yield intensity I 2 . The optical waves passing through θ 2 -phase-retard and θ 4 -phase-retard waveguides are also coherent, yielding I 3 . This section of the polarimeter is called a crossing coherent analyzer.
The remaining paths do not pass through any components. A 3-dB optical attenuator is added before the PDs to distribute unpolarized light equally among the four outputs, intensities I 1 through I 4 .
The analysis matrix W 1 of the proposed SiP-4PD-DOAP is therefore We plot the condition number for this polarimeter as a function with (θ 1 − θ 3 ) and (θ 4 − θ 2 ) in Fig. 3. The minimum condition number can be obtained when where m is any integer. We next improve the device by minimizing ∆γ, or equivalently, minimizing ∆ γ = ∑ 3 n=1 u i . The variation of ∆ γ with θ up for Eq. (10) is depicted in Fig. 4. For the m = 0 case, we observe minimum ∆ γ at (θ 1 − θ 3 ) = 0.1825π or 0.3175π. For our design, we selected (θ 1 − θ 3 ) = 0.1825π; see arrow in

B. Experiment and results
The device was fabricated using a commercial CMOScompatible SOI process with electron-beam lithography at Applied Nanotools Inc. The thicknesses of the silicon and oxide layers are 220 nm and 2 µm, respectively. The scanning electron microscope (SEM) image of the fabricated devices is presented in Fig. 5a. The size of the strip waveguides are 500 nm × 220 nm. The SPS is formed using a 30 × 30 array of cylindrical holes fully etched through silicon with a period Λ of 695 nm and a hole diameter D of 440 nm (as shown in the inset of Fig. 5a). The geometry of this 2-D array is based on the Huygens-Fresnel principle. When the period of the cylindrical holes matches the Bragg condition for a certain wavelength, the light with normal incidence can be coupled into the waveguide 30 .
We define the numerical efficiency by (P x1 + P x2 + P y1 + P y2 ) /P 0 where P 0 is the incident optical power, and P x1 , P x2 , P y1 , P y2 are the optical power coupled into the four paths, respectively. The SPS numerical efficiency is given in Fig. 5b. Its 3-dB bandwidth is 35 nm, and the center wavelength is 1550 nm. More details about the design of SPS are shown in our previous paper 21 .
The experiment setup is shown in Fig. 6. A linearly polarized light beam is generated using a tunable laser. The Through randomly rotating the HWP and QWP, we can generate a series of SoPs that spread widely over the surface of Poincaré sphere, as shown in Fig. 7a and b. The fabricated device was used to measure these SoPs. The measured results and the corresponding input SoPs are depicted in Fig. 7c. An excellent agreement is observed between the measured and input SoPs. Because our device is unpackaged, the experimental set-up vibrations would cause near 0.8-dB of intensity measurement relative errors, which would bring near 0.114 of root-mean-square (RMS) error of the SoP measurements 21 . Therefore, the RMS error of the Stokes vector reconstruction is very high and achieves 0.147 in this demonstration. The RMS error can be significantly reduced after packaging or us-ing integrated PDs 31,32 on the chip.
To study the property of our device responds to other wavelengths, we fix the orientations of HWP and QWP at 20 • and 60 • with respect to the x-axis, respectively, and tune the wavelength from 1540 nm to 1565 nm. The input SoPs as a function with wavelength are shown in the dashed line of Fig. 8. The dots with error bar in the Fig. 8 are the measured results of our device. We can observe that the measured results also agree well with the corresponding input SoPs at other wavelengths.

A. Design
The condition number of the above device (shown in Fig.5) is 1.65 √ 3, which is higher than the theoretical minimum value for a full-Stokes polarimeter 27 . The noise variances of each Stokes element estimate are sensitive to the incoming SoP in the presence of signal-dependent Poisson shot noise 29 . To obtain a minimal and equalized noise variance on each Stokes channel, another structure is proposed and presented in Fig. 9. It includes two crossing coherent analyzers and two APS. The APS are located between the SPS and crossing coherent analyzer. The schematic of APS is presented in Fig. 10a. For APS, we denote the values of the weaker and the relatively stronger output power ratio by PR and (1 − PR), respectively. The length (L) and width (2w) of the splitting region of APS are equal to 2.32 µm and 1.4 µm, respectively. Controlling the asymmetry of APS 25 , we can control the output power ratio PR. Besides, a 2-ports SPS is designed to replace the 4-ports SPS. To increase the efficiency of the 2ports SPS, two distributed Bragg reflection (DBR) gratings 33 are added at two idle ports of SPS. DBR gratings can reflect the lights back to the desired waveguides.
The SEM image of the improved device with a footprint of 350 × 460 µm 2 is presented in Fig. 11a. The enlarged SEM images of APS and SPS are shown in Fig. 11 b and c, respectively. In Fig. 11 c, we can observe that the DBR consists of 8 alternating layers of silicon and silicon oxide. The width of silicon layer, and the lattice period are 160 nm, and 360 nm, respectively. The improved device were used to measure a series of SoPs, and the corresponding results are depicted in Fig. 11 d. The measured results agree well with the input SoPs. Its RMS error is near 0.081 which is 44 % lower than that of the unimproved device under the same 0.8-dB of intensity measurement relative errors caused by the vibrations of the experimental set-up. Next, we will discuss the architecture of polarimeter with the measurement frame. The condition number as a function with θ 1 − θ 3 and θ 4 − θ 2 , when τ = 2 − √ 3. The structure with the parameter of (π/4 3π/4) was chosen to be fabricated.

V. DISCUSSIONS: NOISE MINIMIZATION
The polarimeter can be regarded as a projector that projects the input Stokes vector onto an intensity vector of measurement 36 . For simplicity, we normalize the analysis matrix W so that W 2 i1 = W 2 i2 + W 2 i3 + W 2 i4 = 1, where i means the i th row of the matrix 37 . Therefore, the endpoints of the reduced vectors w i = (W i2 ,W i3 ,W i4 ) are located on the surface of the Poincaré sphere. The measurement frame (i.e. the set of vectors {w i }) can be described by a polyhedron whose vertexes are defined by the endpoints of the reduced vectors w i . It has been demonstrated that the Platonic polyhedron can achieve the minimum condition number 38 . Our first proposed SiP-4PD-DOAP (Fig. 2) whose measurement frame is an irregular tetrahedron (as shown in Fig.12a) does not have the minimum condition number. Figure 12b and c show the mea-surement frames of the polarimeter desgined by Savenkov 35 and our proposed second SiP-4PD-DOAP with APSs (including W b and W b ), respectively. Both of them provide a regular tetrahedron and the minimum condition number. The regular tetrahedron is a spherical 2-design 39 with N=4, which has been proven not able to realize noise variance equalization except for two particular orientations (i.e. the tetrahedrons are shown in Fig. 12c) 36 in the presence of Poisson noise. However, this limitation can be broken via the regular octahedron, which is the simplest spherical 3-design. The regular octahedron presented in Fig. 12d is one example: when rotated to another orientation, it remains such a property 36 .
All the polyhedrons shown in Figs. 12c and d can realize a minimal and equalized Poison noise variance, but suffer from different additive Gaussian noise. Here we examine the impact of the detection number N on the total variance of the four Stokes channels (termed as equally weighted variance, EWV). Consider the cases of Platonic polyhedrons. The optical power received by each PD is proportional to S 0 /N (i.e. the DOAP, and the division of time polarimeter, DOTP which used in the scenario of "photon-starved"). In these cases, the analysis matrix W has the following properties: 36 and where W T is the transpose of W. For AWGN, EWV add is given by 40 where σ 2 n is the variance of the additive noise, and Tr ( * ) means the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of * . Based on Eq. 17 and 18, we can obtain that Based on Eqs. 16, 17 and 20, we can obtain that From Eq. 19, we can know that in the presence of the additive noise, EWV add increases with N. Therefore, the regular tetrahedrons in the two specific orientations are the best architectures ( Fig. 12c and d). On the other hand, Eq. 20 indicates that the EWV Poi is independent of the numbers of PDs. Overall, 4PD-DOAP not only has a relatively low cost in signal processing, but also is less influenced by noise in the reconstruction of SoP. Notice that the conclusion may be opposite for a DOTP, where no power splitting is required and SoP is detected at a relatively low speed 34 . In this case, people usually take more measurements to suppress noise.

VI. CONCLUSION
In conclusion, we have demonstrated, for the first time, a chip-scale, solid-state full-Stokes polarimeter with an optimal frame in presence of both Gaussian and Poisson noises. Two ultra-compact full-Stokes polarimeters with a minimum number of power detection have been proposed and experimentally demonstrated using a CMOS-compatible fabrication process. Their designs were optimized taken into consideration both the condition number and estimation variance. A polarimeter architecture for an optimal 4PD-DOAP analysis matrix (W b or W b ) with the minimum condition number (κ = √ 3) and Poisson shot noise equalization (∆γ = 0) has been achieved in a PIC for the first time. Excellent agreement has been shown between the measured results using our devices and a bench-top commercial instrument. We show that, within the optimal frames of Stokes polarimeters, increasing the number of detection beyond four through power division causes a higher additive Gaussian noise while the Poisson shot noise is not affected. Therefore, 4PD-DOAP offers a theoretically optimal DOAP design.
Integrated polarimeters are still in the experimental demonstration phase. Comparing performance is difficult due to the wide variation in reported measurement set-ups. For example, our polarimeters are not packaged, and therefore error is Copyright (c) 2019 AIP. Personal use is permitted. For any other purposes, permission must be obtained from the American Institute of Physics by emailing rights@aip.org. National Science and Engineering Council of Canada (NSERC) Funding: STPGP 494358-16 induced both from the measurement set-up and the polarimeter itself. The theoretical optimality of our proposed architecture could be established experimentally vis-á-vis others methods only if the same measurement conditions could be put in place. A summary of recently reported integrated full-Stokes polarimeters is provided in Supplementary Material.
The proposed structures can also be extended to other materials platforms, such as silicon nitride, and germanium for the visible and mid-infrared ranges 20,41 . Furthermore, these compact polarimters can be readily integrated with other silicon photonics devices such as spectrometers 42 so that an comprehensive optical vector analysis can be achieved on a single chip for even broader applications.

VII. FUNDING
Natural Sciences and Engineering Research Council of Canada (STPGP 494358 -16).

VIII. SUPPLEMENTARY MATERIAL
See supplementary material for the previously reported integrated full-Stokes polarimeters.