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Abstract
Measuring the states of optical polarization is crucial in many scientific and technological disciplines, and more recently towards the development of chip-scale or nanoscale polarimetry. Here we present a new design of on-chip Stokes polarimetric scheme based on polarization-dependent silicon photonic circuits. The structural elements including polarization rotator and splitter, directional coupler, and phase shifter are assembled to produce polarization-dependent silicon photonic circuits. The orthogonally linear, diagonal, and circular polarization components of the incident light, corresponding to the three Stokes parameters (S1, S2, and S3), can be simultaneously measured based on the Stokes-determined silicon photonic circuit output arrays so as to realize the full measurement of the incident polarization states. This on-chip polarimetry proposed here may enrich the family of micro-nano polarimetric devices, and pave the way to polarization-based integrated optoelectronics, nanophotonics, and metrology.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The fundamental dimensionalities of monochromatic light contain several typical degree of freedoms (DoFs) as electromagnetic waves, such as intensity, frequency, and state of polarization (SoP). Apart from the well-known intensity and frequency DoFs, the polarization DoF also holds a role of paramount importance in countless areas of science and technology [1]. Polarization controlling and metrology becomes indispensable in the fundamentals and applications of light, such as chiral light-matter interaction in nanophotonics [2–6], nanofabrication and characterization [7–9], remote sensing [10], chirality measurement via circular Dichroism [11], astronomy [12], optical communications [13,14], imaging [15,16] and even the nascent quantum information technology [17,18]. It is easy to measure and assess the intensity and frequency DoFs by using suitable detectors and spectrometers, respectively. However, the SoP of light, characterizing the vectorial nature of electromagnetic radiation, is an inherently tricky problem to probe experimentally, because of the completely lost phase information between orthogonal polarization states by the conventional power-sensitive detectors [19].
The measurements of SoPs usually require several measurements for the same signal, and then splitting it into several beam paths in space or in time [10,19,20]. It consequently gives rise to so complex, bulky and expensive polarimetric devices with conventional discrete optical components that the polarization measurement is largely limited in many practical scenarios. Therefore, miniaturization of the polarimeters in nanoscience and nanotechnology is highly desirable for obtaining a more simple, compact, and low-cost way to measure the SoPs. Over the last decade, the possible exploration to miniaturize the traditional free-space optical components have been demonstrated [21–23], but still with the decimeter scale footprint. In order to further realize the miniaturization of SoP measurement, several kinds of polarimetric schemes have been proposed and designed with the development of recent nanoscience and nanotechnology, such as utilization of plasmonic nanostructures or dielectric metasurfaces with polarization-dependent optical responses or diffraction [24–33]. These plasmonic thin films and nanostructures being capable of reducing the overall footprint of photonic devices can be attributed to their unique features of surface plasmon polaritons and localized surface plasmons to overcome the diffraction limit of light.
Here we propose a new on-chip polarimetric design based on the distributed silicon photonic circuits (SPCs) that are completely determined by the Stokes vectors of the input polarization light. Such SPC-based polarimetric design has the advantage of integrating the photodetectors (PDs) into the same chip, which may largely increase the compactness of the on-chip polarimeters [34,35]. Although the SPC-based on-chip polarimeters were deigned and demonstrated through the upright surface polarization splitter and on-chip optical interferometers [35–37], here we attempt to adopt the alternative end-coupler that simultaneously receives all the input polarization components, instead of using the two-dimensional gratings as upright polarization splitter. In this design, the coupled polarization modes can be separated through polarization rotating and directional coupling, and the phase information between orthogonal polarization components can be sensed by 3 dB directional couplers (DCs) with a π/2 phase shifter. This design may benefit the complementary metal oxide semiconductor (CMOS)-compatibility of the micro-nano scale polarimetric devices, and inspire wide applications to polarization-based integrated optoelectronics, nanophotonics and metrology [38–40].
2. Principle and structure
The schematic of the proposed on-chip polarimeter based on the Stokes-determined SPCs is shown in Fig. 1(a), where the waveguide structures, materials, and design parameters are presented in detail. In the analysis and simulation here, we assume that the fundamental TE0 and TM0 modes are well excited with the same coupling efficiency by the x- and y-polarization components of the incident light, respectively. Absolutely, in this situation, the phase difference between the excited TE0 and TM0 is determined by that between the incident x- and y-polarization components for various diagonal and circular SoPs, and vice versa. Therefore, the measurement of the input SoPs is transferred to the analysis of the phase difference dependent eight power outputs. For the incident x-polarization component, it can directly transmit along the bus waveguide in the form of TE0 mode, as shown in Fig. 1(b), and then be split into two paths by the Y-branch splitter. However, for the y-polarization component input, a polarization rotation (PR) with a bi-level taper is effectively used for converting the TM0 mode to TE1 mode here [41]. This PR design can facilitate the standard active silicon photonics platforms with symmetric SiO2 cladding that is used when simulating this waveguide structure below. The y-polarization component is equally split into two paths through TE1 mode directional transmission by the directional coupler (DC) with up and down symmetry, as shown in Fig. 1(c). These four paths based on the TE0 mode transmissions are further split into eight paths by the subsequent four Y-branch splitters. Two identical 3 dB DCs connected with two input ends and two output ends are used for sensing the phase difference between the separated x- and y-components. In order to sense the phase differences within two orthogonally diagonal SoPs (45° and 135°) and two orthogonally circularly polarized SoPs (left-handed and right-handed) as well, a π/2 phase shifter (PS) consisting of two trapezoidal tapers on a bus waveguide is added in the front of one of the input ends, as shown in Fig. 1(a). It should be stressed that the input position of the incident polarized light should be suitably selected so that these two 3 dB DCs can precisely sense the diagonal SoPs and circularly polarized SoPs, respectively. Because the phase differences between these polarization components are also determined by the propagation distance differences due to the modal dispersion. In practical application, this position-dependence issue of incident light can be easily addressed by the thermal-optic tuning technique in a suitable position on the waveguide.
Fig. 1. (a) Schematic, materials, design parameters of the on-chip polarimeter based on Stokes-determined SPCs. Simulated polarization mode evolution and transmission for the incident (b) x-polarization and (c) y-polarization components. Mode patterns in the marked positions 1 to 4 for these two SoPs are exhibited in (b1)-(b4) and (c1)-(c4), respectively. Note that the monitored regions in (b) and (c) are the front part of (a) denoted with the black dotted box.
The Stokes parameters of the input SoPs can be retrieved by three kinds of directionalities based on the eight power output arrays, as shown in Fig. 1(a), as follows,
3. Simulation results
We first simulate the designed polarimetric device through three-dimensional finite-difference time domain (FDTD) simulations when the input SoPs in the wavelength of 1552.5 nm are typically x- and y-linear polarization (LP), orthogonally diagonal LP (45° and 135°) and orthogonally left-handed and right-handed circularly polarization (LCP and RCP), respectively. The simulation results of polarization-dependent SPC distribution are shown in Fig. 2. Note that the SiO2 upper cladding is set when simulating this designed polarimeter in Fig. 1(a), and the front PR structure is outside of the monitored regions. From Figs. 2(a) and 2(b), the DC with up and down symmetry shows good performance of directionally splitting the TE1 mode from the front bus waveguide into two side waveguides, while the TE0 mode remains transmission without coupling into the side waveguides. This complete polarization splitting is the basis of realization of analyzing SoP with the on-chip polarimetric waveguide here. From Figs. 2(c) to 2(f), the outputs of 3 dB DCs are directly connected with the phase difference between the x- and y-polarization components of the input polarized light, which enables the proposed on-chip polarimetry to distinguish the orthogonally 45° and 135° LP, as well as the orthogonally LCP and RCP. All the simulation results show the power outputs of the eight SPC arrays are completely determined by the corresponding incident SoPs. This waveguide design is also suitable to analyze other SoPs, because any other SoPs can be decomposed into these specific SoP bases.
Fig. 2. Simulation results of the polarization-dependent SPCs of the on-chip polarimeter when the input SoPs are (a) x-polarization, (b) y-polarization, (c) 45°-LP, (d) 135°-LP, (e) LCP, and (f) RCP, respectively.
We further simulate other SoP inputs with gradient variation, and calculate the Stokes vectors (S1, S2, S3) given by Eqs. (1) to (4) based on the eight polarization-dependent SPC output arrays. The calculated Stokes vectors of about twenty SoPs are listed in Fig. 3, where the retrieved Stokes values are indicated by the stars, and the set SoPs of the incident light are shown in the top. We also exhibit the SoP retrievement and comparison with the set states in the form of Poincaré sphere, as shown in Fig. 4. The simulation results are well consistent with the set SoPs for the proposed and designed on-chip polarimetry in the wavelength of 1552.5 nm. It should be noted that the whole on-chip polarimeter designed here has the size of about 160 μm*17μm, more compact and smaller than the design with upright surface polarization splitter, even though the end-coupling structure and other auxiliaries such as PDs and thermal-optic tuning electrodes are considered in the practical applications.
Fig. 3. Retrieved Stokes parameters (S1, S2 and S3) based on the polarization-dependent SPCs in the wavelength of 1552.5 nm. Note that the input SoPs are shown in the top of the figure.
Fig. 4. Input and retrieved Stokes parameters exhibited on Poincaré sphere in the wavelength of 1552.5 nm.
4. Wavelength-dependence exhibition and solutions
In this part, we focus on the exhibition and discussion about the wavelength-dependence (or bandwidth) of the proposed on-chip polarimeter. It is well-known that the fundamental waveguide devices, such as power couplers and polarization rotators, have high wavelength-dependence due to waveguide dispersion. Since the proposed polarimeter here consists of the traditional DCs and PS, as shown in Fig. 1(a), it could be highly limited by the incident wavelengths. We simulate and exhibit the wavelength-dependent SoP distortion for three different SoP inputs. The input and retrieved polarization ellipses in five wavelengths are listed in Tab. 1. In addition, we use the ellipse parameters ($\theta$, $e$) in terms of major angle $\theta = \textrm{1/2} \cdot {\tan ^{ - 1}}({S_2}/{S_1})$ and ellipticity $e = \tan [1/2 \cdot {\sin ^{ - 1}}({S_3})]$ to evaluate the analysis errors between the input and retrieved SoPs, as shown in Tab. 1. The major angle $\theta$ indicates the orientation of major axis, and the plus and minus ellipticities describe the left- and right-handed polarization orientation, respectively.
Table 1. Three kinds of input and retrieved polarization ellipses as well as the corresponding ellipse parameters of the SoPs in five different wavelengths.
Furthermore, in order to show the wavelength-dependent SoP errors of the retrieved Stokes parameters with more details, we calculate and present the input and retrieved Stokes parameters in Fig. 5 for the three kinds of input SoP states exhibited in Tab. 1. These results show that the designed on-chip polarimeter here has the best accuracy near the 1552.5 nm, which is consistent with the simulation presentations in the above part. From the calculation errors of Stokes parameters S2 and S3 relative to the set values for the three cases in Fig. 5, the 3 dB DCs give the smallest wavelength-dependent errors when sensing the Stokes parameters S2 and S3 that approach to ${\pm} \textrm{1}$. This implies that the 3 dB DCs have the lowest wavelength-sensitivity when the phase differences between two input arms of them are close to π/2. We also exhibit the retrieved Stokes vectors on Poincaré sphere in the wavelengths of 1550 nm and 1555 nm, respectively, as shown in Fig. 6. It shows that the retrieved Stokes vectors in these two wavelengths undergo bigger errors compared with the results in the wavelength of 1552.5 nm that presented in Fig. 4.
Fig. 5. Input and retrieved Stokes parameters versus varied wavelengths. (a1)-(a3) correspond to the first row, (b1)-(b3) to the second row, and (c1)-(c3) to the third row, respectively, listed in Tab. 1.
Fig. 6. Retrieved Stokes parameters exhibited on Poincaré sphere in the wavelengths of (a) 1550 nm and (b) 1555 nm, respectively.
The high wavelength-sensitivity of our proposed and designed on-chip polarimeter gives rise to a narrow work bandwidth about several nanometers around 1552.5 nm for feasible SoP measurement in practice. This narrow-band issue is caused by the conventional element devices with limited bandwidth, such as the PR, DCs and PS with employing the traditional straight waveguide. Actually, in recent years, all these element devices have been increasingly focused on to optimize and broaden their work bandwidth. For example, the bandwidth of the PR and polarization beam splitter (PBS) can be largely broadened by using cascaded bent waveguides or hetero-anisotropic metamaterials [42,43]. The bandwidth of DC can be broadened through asymmetric waveguides as well [44]. As for the PS, the design scheme by using subwavelength grating waveguides has been proposed to achieve the ultra-broadband nanophotonic phase shift [45]. In our recent work, the fork-type inversely tapered nanowire waveguides have also been proposed to compensate for the out-of-step phase mode coupling in a large wavelength range between TE0 and TM0 (TE1) modes excited from the x- and y-polarization components in free space, respectively, which has been found to enable broadband chiral silicon photonic circuits [46]. Despite the initial version of our proposed on-chip polarimeter with narrow bandwidth, we believe that the bandwidth-broadened design and optimization schemes mentioned above can be organically assembled to improve the bandwidth of our on-chip polarimeter here.
5. Conclusion
In this paper, we proposed a new design of on-chip Stokes polarimeter based on the distributed silicon photonic circuits determined by the SoPs of incident light. We assembled the on-chip PR, DCs, and PS structural elements to produce the polarization-dependent SPC output arrays. The three Stokes parameters (S1, S2, and S3) of the incident SoPs could be simultaneously determined according to the power outputs of SPC arrays. We prresented the detailed design structures and parameters, and exhibited credible simulation results. Furthermore, we analyzed and discussed the wavelength-dependence (bandwidth) of our designed on-chip polarimeter, and provided some feasible solutions of improving the narrow bandwidth for the initial design. This on-chip polarimetric scheme here benefits good compatibility with the well-developed CMOS technology, and may open wide application potentials in polarization-based integrated optoelectronics, nanophotonics and metrology.
Funding
National Key Research and Development Program of China (2019YFB1803902); National Natural Science Foundation of China (11774116, 61905081); China Postdoctoral Science Foundation (2019M662596); Special fund of Chinese Postdoctoral Science Foundation (2020T130221).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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