A tunable polarization state converter/synthesizer based on an anisotropic resonant metasurface (ARM) is demonstrated. The semiconductor-based metasurface is designed to have a thermo-optically actuated optical mode excited by an incident free-space optical field. A diversity of output polarization states is reversibly generated and controlled by the temperature-dependent phase retardance between the two principal linear polarization states. The effect of metasurface inhomogeneity on the range of achievable polarization states is discussed and quantified, and the potential routes to “perfect” efficiency are suggested. By virtue of having a thickness of a fraction of the operating wavelength, ARMs represent a novel class of tunable polarization states’ generating devices poised to find use in free-space communications and other applications.
Optically anisotropic (birefringent) materials can transform the polarization state of light by introducing phase delays between its principal linearly polarized components. Birefringent materials, such as electrically-controlled liquid crystals, electro-optical, and photoelastic materials, are used in various optical components and devices that enable on-demand changing of the polarization state of light. In general, a phase delay of between the two linear polarizations requires a birefringent material of thickness , where is the wavelength of light and is the refractive index difference between the two principal polarizations, which, in the context of uniaxial materials, are often referred to as ordinary and extraordinary rays. Typically, for most natural birefringent materials. Therefore, for , thereby making most polarization-transforming components (e.g., half- or quarter-wave plates) fairly thick. Thus, efficient polarization conversion using ultrathin ( m) components may seem impossible.
This conclusion was overturned by the recent introduction of ultrathin, judiciously engineered resonant nanostructures: electromagnetic metasurfaces.1–3 By employing one or more resonant modes whose spectral positions and free-space coupling coefficient are carefully engineered,4–6 arbitrary phase shifts can be produced for either one of the polarization components. The flexibility in phase control and the mode of operation (transmission or reflection) enabled various flat-optics devices, such as lenses,7–9 phase plates,10 diffraction gratings,11–13 polarization converters,14–20 and many others.
In most cases, the properties of metasurfaces are predefined by their geometry. Several tuning approaches have been suggested for metasurfaces, including mechanical,21–24 liquid-crystal,25–29 phase-change,30 magnetic,31,32 electric,33,34,15,35–38 and all-optical39,40 mechanisms. Among all, thermo-optics41–45 is arguably most straightforward because the refractive index of any material depends on its temperature , where is the thermo-optic coefficient. Germanium, one of the most promising materials for nanophotonics46–50 due to its infrared transparency and high refractive index, also has one of the largest thermo-optic coefficients: C in the mid-infrared range.51 Although it has been used for tuning optical properties of dielectric microcavities,45 its tunability has not been utilized in polarization conversion devices.
Here, we exploit the thermo-optic properties of germanium (Ge) to design and fabricate a resonant dielectric metasurface that acts as a tunable polarization state (PS) generator. We design an anisotropic Ge-based metasurface that supports a spectrally-sharp high quality factor resonance that can be excited by one of the principal linear polarizations of an incident light, and whose spectral position can be tuned by almost its full width at half-maximum bandwidth by heating the metasurface by C. The phase difference between the resonant (horizontal) and nonresonant (vertical) polarizations induced by the transmission through the metasurface is thermally varied in the range, thereby enabling an output elliptical PS tuned by the metasurface’s temperature. The role of geometrical disorder is analyzed, as a path toward tunable perfect polarization converters of subwavelength thicknesses. The rest of the paper is organized as follows. In Sec. II, we describe the design principles and the optical response of a Ge-based high- anisotropic resonant metasurface (ARM) used for demonstrating polarization state generation in the mid-IR. Experimentally obtained transmission spectra shown in Fig. 1 are analyzed and compared with simulation results. The -factors of the sharply-resonant (high-) and weakly-resonant (low-) resonances, corresponding to the two principal polarization states of the ARM, are extracted by fitting the transmission data to the standard Fano interference expressions. The expected effects of the geometric disorder on the optical properties of the ARM are calculated, including the reduction of the -factor and the depth of the resonant transmission dip for the incident polarization states. Experimental results demonstrating thermo-optically actuated polarization states generation are presented in Sec. III. Section IV considers the effect of geometric disorder on the depolarization of the transmitted light. Conclusions and possible approaches to improving the performance of the ARM-based polarization states generators are presented in Sec. V.
II. GENERAL TRANSMISSION PROPERTIES OF RESONANT ANISOTROPIC METASURFACES
A. Polarization-dependent Fano resonances
The schematic of the ARM used in this work is depicted in Fig. 1(a). A linearly polarized state, with the polarization plane at an angle with respect to the axis of the sample frame, is converted into an elliptic polarization state after transmission through the ARM. The output state, characterized by the tilt angle and ellipticity , defined as the ratio between the long and short axes of the polarization ellipse, is modulated by changing the temperature of the metasurface. Such thermal modulation enables generating a wide range of the possible output polarization states. Note that, while thermal modulation of the transmission modulation has been recently demonstrated for high- metasurface, temperature-tunable polarization state generation has not been experimentally realized.
The metasurface consists of a rectangular subwavelength array of germanium bar resonators deposited on CaF (see Sec. VI for fabrication details). A scanning electron microscope (SEM) image of the fabricated metasurface is shown in Fig. 1(b). Individual resonators have a thickness of m, length m, and width m. The resonators are arranged in an array with lattice constants m and m. The arrangement allows for stronger coupling between the resonators along , where the interresonator gap is 0.5m, than along , where the gap is 0.9m.
Linear transmittance spectra were acquired using Fourier transform infrared spectroscopy (FTIR) with an external transmittance setup shown in Fig. 1(c) and described elsewhere.52,53 The experimental room-temperature transmittances and (where the indices and correspond to the orientation of the incident polarization along the two principal axes) are plotted in Fig. 1(d) as a function of the wavelength . We have verified that no polarization conversion takes place for these incident polarization states, as expected based on the mirror symmetry of the metasurface.54
Fano-type transmission spectra are observed for both incident polarizations owing to the coherent interferences between the background (and broadband) signals transmitted through the sample and the Lorentzian resonances of the metasurface. The experimentally obtained transmission spectra were fitted to the standard functional form of the asymmetrical Fano lineshapes that are given by55
where are the resonance bandwidths, are the central frequency of the resonance, and the Fano parameter characterizes the amount of the spectral asymmetry. The quality factors reveal that a sharp Fano resonance with a occurs at m for the -polarized incident light. A much broader resonance with is found at m for the -polarized transmittance.
Simulated broadband and transmittances are plotted in Fig. 2(a) as a function of . The local electric field maps corresponding to the two resonant modes were calculated from driven simulations using COMSOL and are plotted in Figs. 2(b) and 2(c), revealing the unique character of the two respective modes. Due to strong mode coupling between the adjacent cuboids in the -direction, the -polarized mode exhibits a long-lived magnetic quadrupole mode56,57 with a corresponding high numerically calculated -factor . Conversely, weak mode coupling between neighboring rods in the -direction accommodates a strongly radiating dipole mode for polarization with smaller calculated -factor . These behaviors are consistent with the earlier demonstrations of high- resonances in high-index (polaritonic) antennas.58 An alternative interpretation of the high- resonances involves the excitation of quadrupole dipole moments which are known59,60 in the case of plane wave illumination to have much higher -factors than the electric dipole moments.
Notably, simulations indicate that , , and spectra of phase difference between the two polarizations are nearly insensitive to the angle of incidence for angles up to 15. Figures 2(d) and 2(e) show the simulated room-temperature transmission spectra for -polarized and -polarized light as a function of wavelength for incident angles in the range 0. The lack of sensitivity to the incidence angle implies that our experiment, which uses a collimated light source with a beam divergence angle smaller than , can be accurately modeled by assuming normally incident light.
B. The influence of structural disorder on transmission properties of Fano-resonant metasurfaces
Metasurface imperfections, such as manufacturing defects and structural inhomogeneity, introduce an undesired randomized distribution of the geometric dimensions of meta-atoms which contributes toward the broadening and shallowing of narrow resonant spectral features.61,62 In Fig. 1(b), the metasurface SEM images show filleted corners, tilted sidewalls, and near-20 nm variance in the lengths and widths rectangular meta-atoms. The effects of disorder-induced broadening are explored using COMSOL finite-element method simulations. Transmission coefficients and phases were calculated separately for the incident E-field oriented along the -axis and the incident E-field oriented along the -axis as a function of wavelength and temperature for three metasurfaces: one with perfect homogeneity and two with randomized bar-resonator dimensions following normal distributions with nm and nm variance (calculation details are included in Sec. VI).
Figure 2(f) compares the calculated transmittance spectra of the ideal and disordered samples at room-temperature for incident light polarization oriented along the two symmetry axes of the metasurface. Fluctuation in meta-atom sizes can lead to increased diffuse scattering losses.61 As a result, while the ideal sample exhibits near-zero at m, the disordered samples demonstrate the broadening and shallowing of intensity with increasing disorder. For nm and nm, the calculated minima are 0.22 and 0.37, respectively, nearby to experimental resonance depth, 0.30. The respective -factors of the disordered structures are and . Within the nm range, the lineshape did not vary appreciably.
III. THERMAL TUNING OF METASURFACE RESONANCES AND POLARIZATION STATES OF THE TRANSMITTED LIGHT
To enable thermal tuning of the resonant wavelength , the metasurface was mounted on a metal-ceramic heater, and the transmission spectrum was measured in the temperature range, where C and C. The results of this measurement presented in Fig. 3(a) indicate that the spectral position of the Fano resonance redshifts with temperature at a constant rate of change: nm C. This redshift is a direct result of the thermo-optic effect in Ge45 and is in agreement with numerical simulations. The results presented in Fig. 3(a) demonstrate a modulation depth of approximately for m. Such deep transmission amplitude modulation is accompanied by negligibly small resonance broadening. Within the same temperature range, we found that does not significantly change for any wavelength in the m m spectral range (not shown). This finding is consistent with the low-Q nature of the -polarized resonance. Although the apparatus used in this work is not suitable for localized heating (see Sec. VI), designs exist to enable localized and rapid thermal control over individual elements; using integrated resistive microheaters, for instance, localized heating of silicon microring resonators has been demonstrated with 10–100 s thermal modulation time constants.63
Due to the anisotropy of the metasurface, light that is initially linearly polarized becomes elliptically polarized upon transmission through the ARM. The output polarization is determined by the Jones matrix of the ideal metasurface,64
Here, all matrix elements are functions of the metasurface temperature and the wavelength ; and are the phase retardances of the sample along orthogonal sample axes; and are the transmission coefficients for the - and -components of the incident E-field, respectively. The off-diagonal elements of vanish because of the presence of at least one in-plane mirror symmetry. The output state vector is then defined as , where, in our case, is the incident linearly polarized state with the electric field oriented at an angle with respect to the axis [see Fig. 1(a)].
The polarimetry setup depicted in Fig. 1(c) was utilized to experimentally investigate the polarization state of the transmitted light. A polarizer was used to define the incident state . Light transmitted through the sample was collected by a HgCdTe (MCT) photodetector after passing through an analyzer. The intensity of light at the detector was recorded as a function of analyzer angle for various sample temperatures (see Sec. VI for details). The measured transmitted intensity was fitted to Malus’s law64 for a range of temperatures in order to extract the characteristic parameters , , and of the transmitted light’s polarization ellipse. Note that the ellipticity , defined as the ratio between the polarization ellipse’s long and short axes, can only be determined up to a sign using the rotating-analyzer technique used in this work. Differently stated, the rotating-analyzer polarimetry cannot be used to determine the handedness of the transmitted light. Therefore, a quarter-wave plate was placed before the analyzer to identify the handedness of the output light. The tilt angle and ellipticity of the transmitted light are plotted in Figs. 3(b) and 3(c) as a function of wavelength for five different temperatures. The incident state was chosen to be antidiagonal, i.e., corresponding to . From Figs. 3(b) and 3(c), we observe that as the wavelength is swept across the sharp Fano resonance, the tilt angle and ellipticity of the generated elliptically-polarized state undergo an absolute change of and , respectively. At room temperature, the ellipticity and the tilt angle are maximized for m; for longer wavelengths, the polarization ellipse becomes less circular and the tilt angle saturates at around . Increasing sample temperature redshifts the and lineshapes. The relative retardance phase between the diagonal components of is plotted as a function of wavelength for different temperatures in Fig. 3(d). The magnitude of the thermally-actuated change in the relative retardance phase is maximized for m: from () to ().
To visualize the experimental polarization modulation by the ARM at different wavelengths, input polarizations, and temperatures, we mapped the normalized Stokes parameters of the transmitted light–, , and – onto the surface of Poincaré spheres, as shown in Fig. 4. Note that such representation is possible for fully polarized light: . The assumption of the output beam being close-to-perfectly polarized is justified below in the Discussion section of the paper. Nonzero indicates elliptical polarization, while = 0 corresponds to linear polarization. For example, the antidiagonal state has and , with , , and . The three primary axes of the Poincare sphere correspond to horizontal , diagonal , and left circular polarizations, characterized by the respective nonvanishing Stokes parameters of , , and .
Individual curves represent the transformation of the transmitted polarization state with increasing sample temperature for different incident polarizations. Notably, the polarization states of transmitted light (solid circles in Fig. 4) appear on the side of the Poincaré sphere opposite to the incident polarization (hollow circles in Fig. 4). Figure 4(a) shows the polarization modulation corresponding to m, i.e., away from the high- resonance. In this regime, the metasurface device behaves like a horizontal polarizer: the polarization of the transmitted light tends toward polarization, regardless of the incident polarization or temperature. The minimal spread of individual trajectories in this regime demonstrates the importance of operating near a high-Q resonance for thermal tuning. An intermediate span of polarization states, shown in Fig. 4(b), occurs for m.
Figure 4(c) corresponds to the maximum Poincaré sphere coverage at m. In this regime, the metasurface behaves as a tunable polarization converter with a variable retardance. For example, for the incident polarization, the transmitted light [purple curve in Fig. 4(c)] begins near polarization at room temperature and nearly reaches polarization at 125 C. The full polarization tuning range covers of the upper Poincaré hemisphere surface by varying both temperature and .
IV. THE INFLUENCE OF STRUCTURAL DISORDER ON POLARIZATION PRESERVATION
To investigate the role of disorder on the span of transmitted polarization states, the simulated transmittance and phase components of the transmitted E-field for the ideal and disordered structures in Sec. II were used to construct wavelength and temperature-dependent Jones matrices. From these, the evolution of the transmitted polarization states with increasing temperature for the range C–C is mapped to the Poincaré sphere [Fig. 5(a)]; the experimental data are added for comparison. For brevity, only the case is considered. While the calculated polarization trajectory for the nm disordered system is comparable to the experimental measurements, the nm disordered metasurface demonstrates a decreased span of polarization states, and the ideal metasurface exhibits an increased span of polarization states, covering a path that starts near polarization and diverts toward polarization before returning to polarization. Notably, the ideal metasurface is capable of operating at perfect quarter-wave and half-wave retardances.
Inhomogeneous size distribution can lead to light scattering and an aplanar wavefront of the transmitted wave. This, in turn, may be a source of light depolarization, as the polarization state Jones vector may be a function of the scattered wavevector. We estimated the potential depolarization of the transmitted beam in the experiment by calculating the resulting Stokes vector of a beam diffracted off the nm randomized structure at different collection angles (see Sec. VI for details). The results of the calculation are given in Fig. 5(b). For a collection angle close to zero, only the normally transmitted wave is detected, producing a perfectly polarized state. As more scattered light is allowed into the detection system, the polarization states mix, producing a partially depolarized wave, with the minimum degree of polarization of approximately 0.94 observed when all the forward-scattered light is detected. In the experiment, the collection angle was limited to a value of 8, as indicated by the vertical dashed line. The estimated experimental degree of polarization is, therefore, on the order of , supporting our initial choice of using the Jones formalism to analyze the data.
V. OUTLOOK AND CONCLUSIONS
On a final note, one of the disadvantages of the current design is low efficiency, with the lowest experimental transmittance being around 0.3. While this may not be a serious obstacle for applications where the overall intensity of light is not of concern, it is generally desirable to have polarization elements transmit close to all of the input intensity. While the cavities proposed in this work do not fulfill this condition, there are designs based on the so-called Huygens’ metasurfaces that allow efficient transmittance and phase control simultaneously.16,65 The so-called extreme Huygens’ metasurfaces, based on quasibound states in the continuum,66 provide a potential platform for an efficient thermo-optic polarization conversion. Using, for example, arrays of anisotropic zig-zag meta-atoms,67 along with germanium as the constituent material, one can envision a highly-efficient, tunable waveplate that is subwavelength in thickness and performs on par with conventional, optically thick polarization modulators.
To conclude, we have demonstrated a thermally tunable germanium-based metasurface that acts as a polarization converter. Enabled by high thermo-optic coefficient of germanium and a high- collective mode of the metasurface, the mode position can be adjusted by almost its bandwidth within a 100 C window. The anisotropic nature of the mode affects the polarization state of light transmitted through the metasurface, which can be widely tuned by heating the sample. We have analyzed the effect of disorder and outlined a strategy to create an efficient and widely tunable subwavelength polarization converter device.
A. Sample fabrication
For metasurface fabrication, two layers of PMMA (100 nm of 950 K over 600 nm of 495M) were spun on a crystalline CaF substrate, baked at 170 C for 15 min each, covered by a layer of Espacer 300Z spun at 6000 rpm, e-beam exposed at 1000 C/cm (JEOL 9500FS), and developed in MIBK:IPA 1:3 for 90 s. A 5-s mild descum oxygen-plasma etch (Oxford Plasmalab 80) was performed to remove 10–20 nm of undeveloped resist to ensure good adhesion of Ge to the substrate. By electron-beam evaporation, 300 nm of Ge was deposited through the mask, which was then subject to lift-off in the sonicated acetone for 60 s at room temperature. The resulting samples, after being coated by Espacer 300Z, were characterized with a scanning electron microscope (Zeiss Ultra). Before the measurements, the Espacer layer was removed in DI water.
B. Optical characterization
Spectroscopic measurements were performed using a Bruker Vertex 70V Fourier-transform infrared spectrometer (FTIR). A wire-grid polarizer was used to select the initial polarization of the FTIR spectrometer beam and a ZeSe lens was used to normally focus the beam onto the metasurface. The metasurface was mounted on a Thorlabs HT19R metal ceramic heater, and its temperature was controlled with a TH100PT thermistor through a feedback loop controller. Light from the sample was collected and refocused on a liquid nitrogen-cooled mercury-cadmium-telluride (MCT) detector with a gold elliptical mirror. All spectra were taken at 8 cm spectral resolution in the mid-IR range ( cm) and averaged over 512 scans; the equivalent wavelength range is 2–10 m and the equivalent resolution near 3.2 m is 8.2 nm.
The intensity of light transmitted through the metasurface for -polarized incident light was recorded for temperatures ranging from 25 C to 125 C in steps of 12.5 C.
A standard rotating-analyzer polarimetry setup was utilized to investigate the polarization state of the transmitted light for six incident linear polarizations between 90 and 180. For this technique, the transmitted light from the metasurface was directed into the MCT detector after passing through an analyzer. The sample temperature was increased from C to C in steps of 12.5 C; at each fixed temperature, transmission spectra were collected as analyzer orientation was varied between and 360 in increments. A 30-s wait was implemented after each temperature increment to ensure the thermal stability and homogeneity of the sample prior to optical measurements.
The intensity the of the transmitted light as a function of analyzer angle is given by Malus’s law: . The parameters , , and are determined by fitting the measured intensity as a function of . Upon transmittance through the sample, light becomes elliptically polarized. The ellipticity (defined as the ratio between the ellipse’s long and short axes) and the tilt angle of the polarization ellipse are given by and , respectively. Then, the polarization state of transmitted light can be expressed in terms of its normalized Stokes parameters: , , and .
The sign of , that is, the handedness of the polarization ellipse, cannot be determined from the polarizer-analyzer measurements. In order to resolve this ambiguity, a liquid-crystal polarization modulator (Meadowlark Optics) with a known direction of the fast axis was inserted between the sample and analyzer. The sample was heated to 125 C to access the near-quarter waveplate regime, and the polarization modulator was set to operate as a quarter-wave plate (QWP) with a horizontal fast axis near the wavelength of interest (3.2 m). The Jones matrix = (3.2m, 125 C) of the sample-QWP configuration is then given by
where the positive and negative signs correspond to the sample having a vertical and horizontal fast axis, respectively. Assuming that the fast axis of the sample is aligned with the fast axis of the polarization modulator, then the composite sample-modulator configuration behaves as a half-wave plate; alternatively, if the fast axis of the sample is perpendicular to the fast axis of the modulator, then the polarization of the transmitted light should be unaffected. Transmittance spectra were collected for linearly-polarized incident light oriented at 135 and a series of analyzer angles. The maximum transmittance for 3.2 m was demonstrated at an analyzer angle of 45, confirming a horizontal fast axis of the sample and the left-handedness of transmitted light.
If the Jones vector of the transmitted light is expressed as , then the relative phase difference between its - and -components is given by , where , , and . Since the relative phase of the incident light in our experiment is , the amount of phase retardation by the sample (equivalently, the phase difference between the diagonal components of the sample Jones matrix) is given by . Although the degree of polarization transmitted through the polarizer is high (extinction ratio 1:1000), we do not exclude the depolarization of light by the sample. The degree of polarization was estimated by the deviation between polarizations of simulated diffracted plane waves from a resonator array (details in Sec. VI C). The degree of polarization is given by , where are the normalized simulated Stokes parameters of the collected light. The collection angle of the analyzer was estimated to be 8.3, based on 1 in. analyzer aperture size and 3.5 in. separation between the metasurface and analyzer, corresponding to a degree of polarization of approximately 0.98.
Full wave simulations of optical properties of the metasurface were performed on a commercially available FEM software, COMSOL multiphysics. In the periodic single cell simulation, the Ge-metasurface is modeled as a periodic array of Ge bars with dimensions—the thickness, width and length, as well as the lattice constants—being variable. In the initial design stage, we have taken into account the fact that a high Q resonance for -polarization is needed for the sensitive temperature tuning. One way to achieve high quality factor resonances is by utilizing high-order multipole modes which can store more energy per volume than lower-order multipole modes.59,60,68 The energy dissipation can be further tailored by interresonator coupling in the farfield, known as the constructive or destructive interference.69 In metasurfaces, this effect can also help to reduce radiative loss of a particular resonance.70 Such interference determines the strength of directional radiation by the lattice sum of a metasurface, when using radiating plane wave basis.71 The quality factor needs to be around 300 so that it is not extremely large and can be resolved in experiment but sufficient for a tuning through temperature of resonance frequency by half of its full width. Therefore, the bar array needs to have a narrow gap along the y direction for introducing a small net radiating dipole in the plane of the sample. Given this consideration, random geometric dimensions were used in eigenfrequency simulations to find such a resonance. Finally, we chose the following parameters as shown in the main text: thickness m, length m, width m, lattice constants m, and m.
We note that the refractive index of Ge can change by 1 due to a 100 C temperature increment, while its shape can change only by 0.0672 due to the same temperature increment. Likewise, the CaF substrate can expand in size by only 0.2.73 To estimate the sensitivity of the resonance wavelength to thermal expansion, we simulated the resonance shift due to the shape change of 0.06 in each dimension of the brick and due to the 0.2 change in the unit cell periodicity. We find the combined effects of Ge and substrate thermal expansion redshift resonance by , verifying that the dominant contributor to the resonance shift is an optical effect.
To simulate the optical response of a realistic structure, we performed simulations of bar arrays with randomized dimensions. A 10 by 10 array of Ge bars lying in the plane is implemented in a COMSOL wave optics model. The lateral dimensions of the bars were generated by a random number generator that follows a normal distribution with the mean values being the designed dimension, and a small variance that can be judiciously chosen from 0 to 50 nm to fit the experimental data. Periodic boundary conditions are enforced along both x and y directions, and a normal incident plane wave is launched from the air side by implementing a surface current source. The transmitted signal is collected by averaging the farfield and components of -polarized or -polarized plane waves with a normally emitting . Both amplitudes of the transmitted light and phase difference between the two polarizations can be numerically extracted from the full wave simulation. We performed multiple simulations on various random configurations and found that the spectral feature and phase retardation due to the high-Q resonance is generally comparable.
To estimate the degree of polarization, we performed a plane-wave expansion on the transverse field components of the 10 by 10 resonator array for the resonance wavelength (3.28 m) and an incident [1,1,0] polarization. The Fourier components of the diffracted plane waves are used to obtain the complex amplitudes of the p- and s-polarized fields of each diffraction order. Here, and define the unit vectors of the s- and p-polarizations in the lab frame, where is the diffraction angle, is defined as where and are the lattice vectors of the array and and are integers corresponding to the diffraction order. The normalized Stokes parameters of each diffracted beam are determined in the plane defined by each of orders individual E and H fields according to , , , and , where and are the complex amplitudes of the s and p-polarized fields, respectively. Since the diffracted beams are collimated by a lens in experiment, we sum the normalized Stokes parameters of each diffracted beam to acquire the effective Stokes parameters of light measured by the detector (i.e., with ).56 Beams having a diffraction angle exceeding the experimental collection angle are excluded from the sum. The degree of polarization is then given by .
This work was supported by the Office of Naval Research (ONR) (Grant No. N00014-17-1-2161), by the University of Dayton Research Institute (UDRI) (Grant No. RSC17004), and by the Global Research Outreach program of Samsung Advanced Institute of Technology. This work was performed in part at the Cornell NanoScale Science & Technology Facility (CNF), a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant No. NNCI-1542081).