1. Introduction

The field of metasurfaces explores spatially distributed nanoresonators to completely control the optical wavefront [1–7]. One surface can perform multiple wavefront manipulations, promising multifunctional, lightweight and dynamic planar optics. It has the potential to replace bulky optical components such as lenses, mirrors, beam-splitting polarizers and others [4–7]. Their applications are versatile and can be used in, e.g. imaging optics [8–10], polarization optics [11–14], holography [15], vector beam synthesis [16], nonlinear optics [17] and more, with fast expanding applications.

In this work, we investigate metasurfaces that control the polarization states of light, either in terms of a complete polarization state generator (PSG) or polarization state analyzer (PSA) [18]. Such surfaces have already been proposed by Gori [19] and further revised by Bomzon et al. [20]. The full potential of these works was implemented using spatially distributed dielectric resonators for transmissive components [11–14] or plasmonic resonators for reflective components [3,21]. Although the polarization properties are typically addressed, the designs are often reported at specific wavelengths (815 nm [21] and 915 nm [12]), and spectroscopic information appears limited. The need for calibration of imperfections has already been reported when operating polarimeters as Stokes vector analyzers [14]. It is necessary to develop an accurate method for optical characterization, as mass production by deep ultraviolet lithography or nanoimprint lithography of metasurface lenses is becoming a reality [22–24].

However, many of the recently reported metasurface designs operate in a diffractive mode, which makes proper characterization of the diffractive orders necessary. Spectroscopic Ellipsometry (SE) is a well-established tool for accurate monitoring and control of the growth of thin film systems ranging from electronics, optoelectronics, and photovoltaics to chemistry and biophysics [18,25–27]. It is thus expected that ellipsometry will play a key role in the commercialization of thin film-based metasurface technology.

In this work, we focus on transmission metasurfaces that split the orthogonal polarization states into the first diffraction orders ($+1$ and $-1$), as illustrated in Figs. 1(a-c). Such structures are known as beam splitting polarizers [11–13]. The design studied in this work is similar to the layout proposed by Arbabi et al. [11,12] using a-Si pillars. However, our design consists of only diffracted plane waves, which allows for a precise determination of the Mueller matrices (MMs) by exploring already well-established, accurate polarimetric instrumentation. Although this type of structure has been proposed in several recent publications, it seems important to investigate the efficiency and accuracy of these nano-structured optical devices.

 

Fig. 1. a-c) Design of the three metasurfaces where a) MS1 works as a horizontal/vertical polarizer, b) MS2 works as a +45$^\circ$/−45$^\circ$ polarizer and c) MS3 works as a left and right circular light polarizer in the $-1$ ($\mathbf {k}_{\text {-1}}$) and $+1$ ($\mathbf {k}_{\text {+1}}$) diffracted orders, respectively. The $\mathbf {k}_{\text {0}}$ vector indicates the 0-order (specular) transmission, while the schematic insets gives the trace of the electric field vector in the ($\mathbf {\hat {s}}$, $\mathbf {\hat {p}}$), coordinates of the detector. The top Figures a)-c) show the layout of the pillars in one super-period with the designed pillar dimensions $D_{\text {x}}$ and $D_{\text {y}}$. d) Phase and transmission maps from sweeps over pillar dimensions $D_{\text {x}}$ and $D_{\text {y}}$ for p- and s-polarized light. The crosses indicate the dimensions used in one super-period for MS1. e) Transmission values for the pillars in one super-period, for MS1 found by taking the dimensions $D_{\text {x}}$ and $D_{\text {y}}$ from the average found from SEM images. f) Phase shifts for MS1 similar to e).

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Two off-specular (or diffracted order) spectroscopic Mueller matrix ellipsometry (MME) configurations are employed in order to characterize the metasurfaces. The first configuration uses a standard plane wave detection scheme (collimated incident light) and direct readout of the MMs at each diffraction angle. The second configuration, which is used throughout the main paper, involves a pair of focusing probes, mounted on both the illumination and collection sides. This geometry allows for the complete measurement of many diffracted plane waves collected at one selected diffraction angle with corresponding MMs recovered in the spectral analysis. The results obtained using the first method are outlined in detail in the Supplement 1 (SI).

The experimentally measured MMs are addressed by reporting the degree of polarization, where the measured response of the metasurfaces is expected to be affected by both the spatial coherence of the incoming light source and the detector system [28]. We further measure the efficiency in the diffracted orders, the transmission in the zero order and the symmetry properties of the MMs, in addition to performing a comprehensive polar decomposition.

The accuracy of the fabricated structures is investigated in terms of the polarization ellipsis parameters calculated from the MM for a metasurface operating as a PSG or PSA device. Furthermore, it is demonstrated that the finite element method (FEM), used in the design and the metrology, reveals distinct fabrication-induced changes to the measured MMs.

This paper aims to demonstrate that the spectroscopic MME approach can aid in both the design of metasurfaces and the evaluation of the fabrication process. It is therefore expected to provide essential feedback to simulations and manufacturing parameters. It is thus possible that spectroscopic MME will become a critical step for both successful commercial production and towards a widespread adoption of metasurface technology.

2. Experimental

2.1 Design, layout and fabrication of a-Si metasurfaces

Three samples per operation wavelength were designed and fabricated, with the notation of Pors et al. [21]; MS1 for beam splitting of horizontal and vertical polarization (Fig. 1(a)), MS2 for +45$^\circ$ and −45$^\circ$ polarization (Fig. 1(b)), and MS3 for splitting of left and right circular polarized light (Fig. 1(c)). The reported design in this work is intended to operate at 915 nm, supported by additional designs and measurements at 800 nm (reported in SI), 1000 nm and 1100 nm wavelengths. The design process for similar metasurface gratings is described in detail elsewhere [11–13], but is briefly reviewed here.

The phase shift ($\delta _{\text {ss}}$) and transmission ($T_{\text {ss}}$) of each pillar was calculated from a square array with given pillar-diameters $D_{\text {x}}$ and $D_{\text {y}}$, using FEM (COMSOL with the wave optics module) with TE-polarized (s-polarized) light at normal incidence. The calculations used a constant pillar height of $h$ = 715 nm and period $d$ = 650 nm. The corresponding $\delta _{\text {pp}}$ and $T_{\text {pp}}$ follows readily by a 90$^\circ$ rotation. The result of sweeping over all pillar dimensions are shown in Fig. 1(d).

MS1 and MS2 are based on a transmission phase grating, imposing a linear phase shift from 0 to 2$\pi$ in each supercell structure. MS2 is identical in dimensions to MS1, but with a 45$^\circ$ rotation induced on each pillar, see the layout of the top Figs. 1(a) and (b). This supercell is repeated and makes up a phase-only grating if the transmission is assured to be 1. Using a-Si:H resonators, in the form of conical pillars as meta-atoms (in the limit of no interaction between the resonators), a discrete phase shift can be imposed on the wavefront of the incoming light, where the dimension of each pillar ($D_x$ and $D_y$) determines the phase shift and transmission. MS3, on the other hand, is a Pancharatnam–Berry phase (or geometric phase) metasurface [12,13,19,20] and consists of one chosen $D_x$ and $D_y$ with a relative phase shift of $\pi$ radians between the s- and p-components, where each nanoresonator is operating as a half wave retarder. In one super-period, each pillar is rotated by $\pi /12$ radians with respect to the previous one, ending with a full rotation of $\pi$ radians, as schematically shown in the layout of the top Fig. 1(c).

Figures 1(e) and (f) show the phase profiles and transmission in one super-period of MS1 ($\Lambda$ = 12d, d = 500 nm) at $\lambda _0 = 915~nm$, where we have used the dimensions $D_x$ and $D_y$ experimentally obtained by averaging over pillar dimension found from scanning electron microscopy (SEM) images (see Supplement 1 for more details). These corresponding diameters are also indicated in the simulated phase and transmission maps in Fig. 1(d) as crosses with corresponding pillar numbers. Note that shortening the distance between the pillars to $d$ = 500 nm in the super-period of the MS resulted in a better efficiency.

The ensemble of the three designs make up a complete set of Stokes vectors or analysis states for a complete Stokes polarimeter, operating by splitting the orthogonal polarization states into the $-1$ and $+1$ diffracted orders [12]. These metasurface samples were fabricated on a 1 mm thick double-side polished SiO$_{\text {2}}$ wafer, where the a-Si:H layer was deposited using plasma enhanced chemical vapour deposition (PECVD). The patterns were produced by electron beam lithography (EBL), while the etching of the pillars were conducted using inductively coupled plasma-reactive-ion-etch (ICP-RIE). The fabrication details are outlined in detail in section S1.2. The pillar dimensions $D_{\text {x}}$, $D_{\text {y}}$ and height $h$ are schematically defined and tabulated for a super-period in the Supplemental, along with the complete SEM images of witness samples.

2.2 Diffractive order Mueller matrix ellipsometry

A commercial spectroscopic MME (double-rotating compensator ellipsometer Woollam RC2-XI operating in the range 210 nm to 1700 nm) with an azimuthal rotation stage was used to measure the MMs of the samples facing away from the incident beam, see Fig. 2(a). The instrument uses a 150W Xe source that is focused through a pin hole, which in this work can be considered as the extended incoherent source, which is collimated by a source lens ($f_s\approx$ 2 cm), as illustrated in Fig. 2(a). The detector uses a combination of a silicon and indium gallium arsenide spectrograph having a resolution of 1 nm below 1000 nm and 2.5 nm above. The initial collimated beam has a waist of approximately 3 mm. MME is an extension and generalization of standard SE and enables one to fully characterize the polarimetric response of a sample, including its depolarizing properties [18,29]. The measured MM is represented by

 

Fig. 2. a) Experimental ellipsometric setup. Light from a Xe source is focused through a pinhole with diameter 100 $\mu$m operating as the incoherent source area, followed by a lens (SL) at $f_{\text {s}}$=20 mm (focal length of SL). The source is followed by a polarizer (P) and a rotating compensator (RC). For the focus probe measurements, an additional focusing lens (FL) is inserted while excluded in the plane wave method. The sample is placed at $f_{\text {f}}$=80 mm from FL (if inserted). There, the light is scattered into $\theta _{\text {s}}$ for the $-1$ order and $-\theta _{\text {s}}$ for the $+1$ order. The light detected in the $-1$ order goes through the collection lens (CL) if the focusing probes are applied, followed by the rotating compensator (RC) and polarizer/analyzer (P) before entering the detector. Again, if CL is included, the distance is $f_{\text {f}}$ to the metasurfaces. The specularly transmitted signal is indicated with $\mathbf {\hat {k}}_{\text {0}}$. b) Closer look at the area around the MS in the plane wave configuration. The incoming light is diffracted into the $-1$, 0 or $+1$ order. The detector arm is placed at an angle of $\theta _{\text {s}}$ and will measure additional k-vectors corresponding to $\Delta \theta _{\text {s}}$, given by the entrance pupil (EnP) of the detector. c) Focus probe measurement depicting the incoming light being focused by the FL. The diffracted signal enters the CL, where it continues into the detector. This configuration picks up a larger $\Delta \theta _{\text {s}}$ span. d) The scattering plane of the metasurface sample not aligned with the x-z plane of the ellipsometer, where $\phi$ is the azimuthal rotation of the sample. e) The scattering plane of the MS aligned with the x-z plane of the ellipsometer.

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The off-axis movement of the detector arm in the plane of incidence is currently not automated. Therefore, each measured spectrum was recorded manually at an angle of $\theta _{\text {s}}$ and was limited to steps of 0.1$^\circ$. For the focus probe measurements, the spectroscopic MM at the scatter angle with maximum intensity was selected, while plane wave measurements with collimated light used all scatter angles, see below. Assuming the incidence plane aligns with the grating and at normal angle of incidence, the relation between the wavelength at maximum signal and the diffracted angle is given by [34]

The incoming light is limited to normal incidence throughout this work ($\theta _{\text {s}}~=~0^\circ$), and the scattered light is only measured in the plane of incidence. Figure 2 shows that $\mathbf {\hat {s}}~=~\mathbf {\hat {k}}_{\text {i}} \times \mathbf {\hat {p}}$ with $\mathbf {\hat {s}}~=~\mathbf {\hat {y}}$ and $\mathbf {\hat {p}}~=~\mathbf {\hat {x}}$, where $\mathbf {\hat {k}}_{\text {i}}$ is parallel to the wave vector of the incoming light. Further, $\mathbf {\hat {x}}, \mathbf {\hat {y}}$ and $\mathbf {\hat {z}}$ are the unit vectors of the metasurface sample and are presumed aligned with the laboratory axis after the geometric alignment. For the scattered light, $\mathbf {\hat {s}}~=~\mathbf {\hat {k}}_{\text {m}} \times \mathbf {\hat {p}}$, where $\mathbf {\hat {k}}_{\text {ss}}$ is congruent with the scattered wave vector and m is either $-1$ or $1$, as illustrated in Fig. 2.

The detector arm movement in our system is also limited solely to positive scattering angles (typical for the standard ellipsometric configuration). Therefore, the $-1$ and $+1$ scattered orders were measured at an azimuthal angle of $\phi ~=~0^\circ$ and $\phi ~=~180^\circ$, respectively. This is shown in Figs. 2(d) and 2(e). Rotating the sample as such is equivalent to moving the detector arm to a negative scattering angle.

2.2.1 Plane wave measurement configuration

The typical collimated beam is 3 mm in diameter. The plane wave measurement beam size was reduced to 1 mm diameter using a pin-hole centered in the beam (not illustrated), as the samples were 0.42 mm$^2$. Although the beam covers an area much larger than the structures, diffracted orders can only arise from the nanopillars themselves. With an ideal detector, all signals measured at $\theta ~\not =~0^\circ$ are diffracted from the metasurface grating. Since the setup utilizes a collimated beam, the spatial coherence length is estimated to be $200\lambda _{\text {0}}$ [31], where $\lambda _{\text {0}}$ is the analyzed wavelength. The detected beam passes through a pinhole before being analyzed by the spectrograph. The finite size of the latter pinhole allows for angles $\pm \Delta \theta _s$ to enter the detector, as illustrated in Fig. 2(b). Results using this method are presented in the SI. Note that the spectroscopic MME method is less sensitive to broadband illumination and thus the temporal coherence properties of incident light due to the spectral analysis on the detector side.

2.2.2 Focus probe measurements

To ensure adequate intensity in the $-1$, $0$ and $+1$ orders, measurements inside the patterned areas were performed using focusing and collection lenses operating at a numerical aperture of approximately 0.017 ($\approx 1^\circ$), as is illustrated in Figs. 2(a) and (c). This allows for illumination over a small area and a high quality signal from the metasurfaces, making it suitable for measuring prototype structures. The collection lens also allows for a larger k-vector span to be collected, leading to a larger $\Delta \theta _{\text {s}}$. Therefore, only one measurement angle, chosen where the highest intensity occurred was necessary to recover the full spectrum of all diffracted wavelength. A pinhole of 100 $\mu$m in diameter is found on the source side and is regarded as the incoherent source area used in the estimation of the spatial coherence [31], see schematic drawing on Fig. 2(a). It reduces drastically the coherence length to $\lambda _0/0.017\approx 30\lambda _0$, but this still covers 7 and 10 super-periods for the shortest and longest wavelengths measured, respectively.

2.2.3 Geometric alignment

Each sample was aligned manually for the scattering measurements. As it is necessary for the sample to be normal to the beam, they were first aligned in a reflection configuration. As the detector in standard ellipsometry only moves in the x-z plane, the incidence plane and the scattering plane must overlap. Figure 2(d) illustrates poor alignment of the sample ($\phi \neq 0^\circ, 180^\circ$), in which the full signal cannot be measured. Figure 2(e) shows the correct aligned system, where the rotation of the sample is $0^\circ$ and $180^\circ$ for the $-1$ and $+1$ order, respectively.

3. Results and discussion

Figure 3 shows the Mueller matrices $\mathbf {M}_{\textrm {MS1}}^{\pm 1} (\lambda )$, $\mathbf {M}_{\textrm {MS2}}^{\pm 1}(\lambda )$ and $\mathbf {M}_{\textrm {MS3}}^{\pm 1}(\lambda )$, where the azimuthal orientation of the sample $\phi ~=~0^\circ$ gives the $-1$ order, and $\phi ~=~180^\circ$ gives the $+1$ order. It is observed that above 760 nm (to approximately 1000 nm) the MSs generate reasonably pure orthogonal linear states (MS1 and MS2) and circular states (MS3). The vertical red and blue dashed lines indicate the maximum transmission for the $-1$ and $+1$ orders, respectively, while the black vertical dashed line indicates the expected peak transmission at the design wavelength, as shown below. The observed maximum diffraction efficiency was found rather at $\theta _s~=~8.0^\circ$, while the designed profile (given $\lambda$ = 915 nm, super-period $\Lambda$=12d, d = 500 nm and Eq. (2)) corresponds to a peak efficiency at $\theta _s~=~8.7^\circ$. Additionally, interesting symmetries in the spectral response of the MMs are observed and will be discussed below. It was discovered that the recorded MMs had slightly non-physical elements (barely larger than 1). The non-physical part was here removed by Cloude filtering [37]. Some simple results can easily be extracted directly from the measured MMs. The top row of Fig. 4 shows the spectrally dependent depolarization index $P_{\Delta }$, given by [38]

 

Fig. 3. a) Normalized Mueller matrices $\mathbf {M}_{\textrm {MS1}}^{\pm 1}$, b) $\mathbf {M}_{\textrm {MS2}}^{\pm 1}$ and c) $\mathbf {M}_{\textrm {MS3}}^{\pm 1}$, for the $-1$ ($\theta _{\text {s}}~=~8.0^\circ$, $\phi ~=~0^\circ$) and $+1$ ($\theta _{\text {s}} = -8.0^\circ$, $\phi ~=~180^\circ$) diffracted orders. The MMs are normalized against $M_{\text {11}}$, as shown in Eq. (1). The black dotted line represents the design wavelength of 915 nm. The red and blue dashed lines represent the wavelengths of the highest transmission intensity of the $-1$ and $+1$ scattered orders, respectively.

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Fig. 4. The top row shows the degree of depolarization $P_{\textrm {GB}}$ using Eq. (3) for a) MS1, b) MS2 and c) MS3, respectively. The middle row shows the selected MM elements d) ${\mid }m_{\text {12}}{\mid }$ and ${\mid }m_{\text {21}}{\mid }$ for MS1, e)${\mid }m_{\text {13}}{\mid }$ and ${\mid }m_{\text {31}}{\mid }$ for MS2 and f) ${\mid }m_{\text {14}}{\mid }$ and ${\mid }m_{\text {41}}{\mid }$ for MS3. Red and blue represents the $-1$ and $+1$ orders, respectively. Full and dashed lines represent upper and lower block diagonal elements, respectively. The bottom row shows the $M_{\text {11}}^{\pm 1}$ (also denoted transmission) for g) MS1, h) MS2 and i) MS3. The black dotted line represents the design wavelength of 915 nm. The red and blue dashed lines represent the wavelength of the highest transmission intensity of the $-1$ and $+1$ scattered orders, respectively.

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The measurement system supplies an incident beam with a reasonable spatial coherence, while the detection system ensures temporal coherence. Within these specifications, it is experimentally confirmed that polarization beam splitting MSs can be regarded as coherent devices not depolarizing light. The only contributions to depolarization are spatially manufactured inhomogeneities within the beam, diffracted light from outside the coherence area and the finite resolution of the spectrograph possibly averaging over fast variations in the spectroscopic Mueller matrix. In addition, a reduced spatial coherence would also be correlated to a reduction in the diffraction efficiencies.

The first row of Fig. 4 shows the depolarization index, defined from Eq. (3). The second row shows the absolute value of the spectrally dependent primary MM elements of the two nearly orthogonal diffracted states. This means $m_{\text {12}}$ and $m_{\text {21}}$ for MS1, $m_{\text {13}}$ and $m_{\text {31}}$ for MS2 and $m_{\text {14}}$ and $m_{\text {41}}$ for MS3, for both $\pm 1$ diffracted orders. The MM elements relate to the measured transmission into the diffracted orders, i.e. $M_{\text {11}}^{\pm 1}$, which are shown in the bottom rows of Fig. 4. It is clearly observed that peak efficiencies are located between 830 nm and 840 nm, not at the designed wavelength of 915 nm. Furthermore, several dips in the transmission are noted and identified as Rayleigh lines (at the onset or closing of a diffracted order) [39], where energy becomes redistributed, hence reducing the transmission.

For MS1, the $+1$ order transmission is slightly red-shifted in comparison to the $-1$ order. The peak wavelength position for the $-1$ and $+1$ diffracted orders occurred at 828 nm and 848 nm, with peak values of 0.3. The depolarization index is found to be above 0.99 between the wavelengths 760 nm and 952 nm. The important Mueller elements $|m_{12}|$, $|m_{21}|>0.95$ for the $-1$ order in the spectral range 815 nm to 846 nm. These elements for the $+1$ order are above 0.95 in the range 800 nm to 867 nm and again from 887 nm to 905 nm. These are thus the spectral ranges in which the MS1 perform best.

For MS2, the $+1$ and $-1$ order transmissions are nearly identical, where the peak values are 0.32 and 0.30 at 830 nm, respectively. The depolarization index is found to be above 0.99 between 764 nm and 914 nm for the $-1$ order, and between 764 nm and 954 nm for the $+1$ order. The Mueller elements $|m_{13}|$ and $|m_{31}|$ are closest to their ideal values adjacent to the transmission peak, and are all above a value of 0.95 between 808 nm to 841 nm. There is thus a 25 nm overlapping wavelength range of MS1 and MS2 where both has a satisfactory response.

For MS3, the $-1$ and $+1$ orders have nearly identical transmission, with both peak values being 0.37 at 838 nm. The depolarization index is found to be above 0.995 in the spectral range 760 nm to 986 nm, with the exception of a dip (780 nm to 824 nm), reducing it to 0.985. The $|m_{14}|\geq$0.95 for the entire plotted spectral range of 750 nm to 1000 nm, while $|m_{41}|$ experiences two large dips, resulting in the sample only being useful as a PSG in the range 820 nm to 850 nm. Moreover, it is clear that MS3 is most affected by the Rayleigh phenomenon, as demonstrated by the large dips in transmission. Indeed, since the depolarization index, the MM elements and the transmission are strongly affected when encountering a Rayleigh line, the phenomenon must be carefully introduced into the design process after establishing manufacturing-induced issues.

The peak diffraction-efficiencies ($\eta ^{\pm 1}$) into a given polarization state are estimated to 60 ${\%}$ for MS1 compared to 80 ${\%}$ of the design, while 74 ${\%}$ for MS3 compared to 84 ${\%}$ of the design (90 ${\%}$ if neglecting reflection losses at the first air-glass interface).

Finally, the ensemble of the three fabricated MSs has a relatively narrow operational range of 20 nm located near 830 nm, where the polarimetric properties are satisfactory. However, this range has clearly room for improvement both in terms of design and control of manufacturing.

The following MMs correspond to the filtered values at the highest intensity wavelengths from Fig. 3 for the $-1$ and $+1$ scattered orders. These matrices are later used for analyzing the polar decomposition.

3.1 Symmetry, polarization ellipsis angles and polar decomposition

The MMs for the polarization beam splitting metasurfaces appear to overall follow the symmetry that is obtained by setting $t_{\textrm {ps}}~=~t_{\textrm {sp}}$ in the Jones matrix [31,40], resulting in

3.1.1 Polarization ellipse of generator and analyzer states

The polarization ellipsis angles, ellipticity ($\epsilon$) and azimuthal tilt-angle ($\psi$) of the ellipsis, as illustrated in Fig. 5(g), for a generative and an analyzing system are given as [29]

 

Fig. 5. Polarization ellipsis angles and transmission for the generating (subscript G) and analyzing (subscript A) states for the $-1$ diffracted order. The top row shows the azimuthal-tilt angles $\psi$ for a) $\mathbf {M}_{\textrm {MS1}}^{-1}$ and b) $\mathbf {M}_{\textrm {MS2}}^{-1}$. It has been excluded for $\mathbf {M}_{\textrm {MS3}}^{-1}$, as the circular polarization state is independent of rotation. The middle row shows the ellipticity angles $\epsilon$ for c) $\mathbf {M}_{\textrm {MS1}}^{-1}$, d) $\mathbf {M}_{\textrm {MS2}}^{-1}$ and e) $\mathbf {M}_{\textrm {MS3}}^{-1}$. The bottom row shows the transmission (i.e. $M_{\text {11}}^{-1}$) for c) MS1, f) MS2 and i) MS3. The left, dashed line represents the wavelength of the highest transmission of the $-1$ order, while the right, dashed line represents the design wavelength of 915 nm.

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For MS1, the ideal ellipsis angles are both 0$^\circ$ and the experimental results in Fig. 5(a) and (b) show that both the analyzing and generative angles remain within $\pm 1.5^\circ$, with $\psi _A$ increasing linearly with wavelength. The analyzing state appears to be closest to its ideal value when the transmission is at its maximum, while the generative state is closer at the design wavelength.

The polarization ellipsis angles for MS2, as shown in Fig. 5(d) and (e), deviate from the ideal angles inconsistently for different wavelengths. Both angles for the generative and analyzing states highly deviate at the Rayleigh line near 850 nm. The generating state remains close to the ideal $\psi$ value at the highest intensity, while the analyzing state is closer to the ideal $\epsilon$ at the same wavelength. The analyzing state also maps a polarized state close to the optimal value at the designed wavelength. However, at this wavelength, the transmission is too low for efficient usage.

MS3 experiences large differences between the generative and analyzing states. The $\epsilon _{\text {A}}$ remains close to 45$^\circ$ for the entire spectrum and is at the expected value at the peak in transmission. The $\epsilon _{\text {G}}$ experiences two minima when encountering Rayleigh lines at 805 nm and 915 nm. Still, the sample reaches 45$^\circ$ at the designed wavelength of 915 nm. However, the transmission in this region is extremely low, leaving the sample too inefficient to use.

The ellipticity angles reveal that the analysis states have higher purity than the generator states and also provide insight into which polarization states are achieved at a given wavelength. The corresponding ellipsis angles for the Mueller matrices $\mathbf {M}_{\textrm {MS1}}^{+1}$, $\mathbf {M}_{\textrm {MS2}}^{+1}$ and $\mathbf {M}_{\textrm {MS3}}^{+1}$ are shown in Section S2 in Supplement 1 (SI). The ellipticity angles for the $+1$ order are close to the expected values. However, the angles appear to be of higher purity at longer wavelengths, in the range of low efficiency. Similar results were found using the plane wave method, showing how both methods supply a complete representation of all polarization-altering properties of the beam-splitting metasurfaces over a large wavelength range.

3.1.2 Polar decomposition of experimental MMs of metasurfaces

The forward and reverse decomposition splits a MM into the following two configurations

The forward decomposed matrix of $\textbf {M}_\mathrm {MS1}^{-1}$ is

In the case of $\textbf {M}_\mathrm {MS2}^{-1}$, the forward decomposed measured Mueller matrix is given as

Finally, in the case of $\textbf {M}_\mathrm {MS3}^{-1}$, the reverse decomposition gives a more reasonable depolarization matrix for the system than the forward decomposition and is given as

3.2 Simulation of MMs: conical pillar anomalies

The most striking deviations prevailed by the MME measurements are the shift of the peak efficiency and operation of the metasurfaces to lower wavelengths. The peak transmission into the $\pm 1$ diffracted orders is downshifted from a design wavelength of 915 nm, indicated by the vertical, dashed lines in Fig. 6, to roughly 840 nm. This was accompanied by a smaller degradation in the MMs away from ideal components. In addition, similar samples designed at 1000 nm and 1100 nm were produced and showed the same trend of blue shift in the transmission peak. On the other hand, the blueshift in the maximum efficiency for the 800 nm (only shifted to 780 nm) sample was significantly less than for the other samples. It is suspected that the presence of the absorption edge of a-Si:H is in this case filtering the signal. The measurements of the 800 nm design are presented in S4 using the plane wave configuration.

 

Fig. 6. Experimental and simulated a) $\mathbf {M}_{\textrm {MS1}}^{-1}$, b) $M_{\text {11}}^{-1}$, c) $\mathbf {M}_{\textrm {MS1}}^{+1}$ and d) $M_{\text {11}}^{+1}$. The red lines represent the corrected, conical-shaped pillars. The blue line represents straight pillars with, $D_{\text {x}}$ and $D_{\text {y}}$ corresponding to SEM images of finished manufactured samples. The black lines represent the measured experimental results. The dashed vertical lines indicate the initial design wavelength.

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If no computational design errors arise due to e.g. convergence issues, then the manufacturing steps are responsible for the measured deviations. In fact, this is the biggest advantage of the MM spectroscopic method. Not only does it allow for both monitoring and control of the manufacturing process, but it also verifies the initial design, in addition to producing feedback into the calculations. By investigating toppled pillars (see Supplement 1), it was discovered they had a reversed conical shape, meaning the top diameters were larger than the bottom of the pillars.

Figures 6(a and c) show the experimental and simulated Mueller matrices $\textbf {M}_{\mathrm {MS1}}^{\pm 1}$. The corresponding experimental and simulated transmission $M_{\text {11}}^{\pm 1}$ are shown in Figs. 6(b) for the $-1$ order and d) for the $+1$ order. The experimental data is compared to simulations with straight pillars and simulations with conically shaped pillars. The experimentally determined SEM dimensions were used in the straight pillars, while the initial design dimensions were used on top for the conically shaped pillars. For MS1, the $D_{\text {x}}$ and $D_{\text {y}}$ parameters at the bottom of the pillars are reduced by 45 nm and 50 nm compared to the top, respectively. For MS3, both $D_{\text {x}}$ and $D_{\text {y}}$ are reduced by 50 nm at the bottom of the pillars compared to the top of the pillars. The dimensions are specified in the SI.

It is observed from Fig. 6 that the peak transmission (proportional to the diffraction efficiency), initially designed at 915 nm for straight pillars, is effectively downshifted to 840 nm for the conical-shaped pillars in addition to reducing the transmission, which matches well with the experimental data. Figure 7 shows the corresponding results for the experimental and simulated Mueller matrices $\textbf {M}_{\mathrm {MS3}}^{\pm 1}$. Further, the results for MS3 show the same trend as established for MS1. Irrefutably, a reasonable model that allows for MM simulation in response to conically shaped pillars has been established.

 

Fig. 7. Experimental and simulated a) $\mathbf {M}_{\textrm {MS3}}^{-1}$, b) $M_{\text {11}}^{-1}$, c) $\mathbf {M}_{\textrm {MS3}}^{+1}$ and d) $M_{\text {11}}^{+1}$. The red lines represent the corrected, conical-shaped pillars. The blue line represents straight pillars with $D_{\text {x}}$ and $D_{\text {y}}$ corresponding to SEM images of finished manufactured samples. The black lines represent the measured experimental results. The dashed vertical lines indicate the initial design wavelength.

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The properties of a sawtooth phase profile should cause the specular transmission to sharply dip at the wavelength corresponding to the maximum diffraction efficiency. Figure 8 shows the experimentally measured and simulated 0-order transmission ($M_{\text {11}}^0$) corresponding to MS1 and MS3. The dip in the transmission has a good correspondence between the experimental and simulated data after the conical shape has been introduced, while the wavelength of the transmission dip of the straight pillars experience a major offset from the measured data. The additional Rayleigh lines are also clearly observed from the transmission spectra. Inspection of the diffracted orders from simulations of the conical pillars (for MS1 with p-polarized incidence) give 63 ${\%}$ transmission efficiency into the −1 order, with 9 ${\%}$ loss into other forward diffracted orders (including the forward specular one), while more than 22 ${\%}$ is lost into all reflected orders. The reflectance appears as the dominant loss channel. Both forward and backward diffraction losses increase strongly upon moving away from the wavelength of the peak efficiency of the design-order.

 

Fig. 8. Unpolarized, transmitted light $M_{\text {11}}^0$ for experimental and simulated data for a) MS1 and b) MS3. The red lines represent the corrected, conically shaped pillars taken from SEM images of tipped-over pillars. The blue line represents straight pillars, with $D_{\text {x}}$ and $D_{\text {y}}$ corresponding to SEM images of finished manufactured samples. The black lines represent the measured experimental results.

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Inspection of the diffracted orders from simulations of the conical pillars (for MS1 with p-polarized incidence) give 63 ${\%}$ transmission efficiency into the − order, with 9 ${\%}$ loss into other forward diffracted orders (including the forward specular one), while more than 22 ${\%}$ is lost into all reflected orders. The reflectance appears as the dominant loss channel. Both forward and backward diffraction losses increase strongly upon moving away from the wavelength of the peak efficiency of the design-order.

The MME method used in this work is true scattering ellipsometry, while it is more common to focus on only the specular order [30–32,43–45]. Indeed, the measurements performed here of both the 0 (specular) and $\pm$1 diffracted orders give a better understanding of the metasurfaces. A full fitting of the data enables an accurate determination of the pillar profiles and is believed to be useful for many other structures. Even though the FEM was used here, it is a computationally heavy process. Other numerical techniques, such as the rigorous coupled wave analysis (RCWA) are more efficient when the parameters describing the structures are close to the real values [44]. With sufficient computational power, one can then numerically inverse the measured MMs to reveal manufacturing errors and produce direct feedback regarding the manufacturing steps.

There are some curious deviations in the off-diagonal elements (experimental data shows off-diagonal elements with $|m_{\text {ij}}|$ < 0.04) that are not explained by the model. The origin of this remains unknown, but it is noted that the effect is wavelength-dependent. They are not believed to be a simple result of alignment artifacts, leading to an uncomplicated, rotated MM, as discussed above. Furthermore, the measurement of the diffracted order imposes quite a strict alignment in the incidence plane. However, the measurement of the diffracted order, instead of the specular one, allows for separation between the purely transmitted and the diffracted signals from the nanostructures.

4. Conclusions

A commercial MME system has been modified into a high-precision, diffracted order configuration, in order to allow for accurate characterization of the diffraction polarimetric performance of metasurface beam splitters. Three metasurfaces (MS1, MS2 and MS3) were designed and nanomanufactured using resonators made of elliptically shaped a-Si:H pillars on glass, and the ensemble of the three MSs constitutes a complete Stokes polarization state generator or analyzer. Using low numerical aperture focus probes on the source and receiver sides resulted in remarkably accurate measurements of MMs and diffraction efficiencies. This allowed a large number of diffracted angles to be obtained in a single measurement. This measurement was also demonstrated without the additional usage of focusing optics, here denoted a plane wave configuration, which was demonstrated as an accurate alternative and presented in Supplement 1 (SI). The emphasis was placed on the analysis of the experimental MMs and the identification of their promising potential for addressing design and fabrication errors in metasurface systems.

The experimental spectroscopic MMs of the $\pm 1$ diffracted orders reveal metasurface components with an excellent polarimetric purity, i.e, the MSs are generating or analyzing nearly orthogonal polarization states of the $\pm 1$ diffracted orders at the wavelength of peak efficiency, but this peak wavelength was found to vary in the manufactured samples from 828 nm to 838 nm. The ensemble of the MSs was found to have reasonable polarimetric pure properties ($|m_{12}^{MS1}|$, $|m_{13}^{MS2}|$, $|m_{14}^{MS3}|$>0.95) across an overlapping window of 20 nm. Both an increased overlapping spectral range and overlapping peak efficiencies are envisaged through strict control of both the design and the fabrication process. A lowered polarimetric purity (e.g. accepting only $|m_{12}^{MS1}|$, $|m_{13}^{MS2}|$, $|m_{14}^{MS3}|$>0.9), can be obtained across a much larger spectral region (760 nm to 1000 nm), but with low efficiency at the edges of the spectral range.

The MMs were further analyzed in detail by extracting the polarization ellipsis parameters, indicating that the samples operate better as a PSA than a PSG and gives in all measured cases a few degrees of deviation from pure or ideal states. The polar decomposition of the experimental MMs revealed highly ideal diattenuator matrices with diattenuation values close to 1 (0.99, 0.97 and 0.98 for MS1-3), a mainly diagonal depolarizer matrix (with depolarization index near unity), and an average retarder matrix at the peak efficiency wavelength. It was noted that the reverse decomposition appeared favorable for the Pancharatnman-Berry geometry (MS3).

It was observed that the purity of the states is in certain cases strongly affected by the Rayleigh phenomenon, as exemplified here for the generative states of MS3, while the analyzing state remains close to its ideal values over a large spectral range. The Rayleigh lines must therefore be carefully included in the design and characterization for optimal performance both in terms of efficiency and polarization purity.

The recorded MMs were found to have a near-unity degree of polarization, even outside their operational wavelength. This confirms experimentally that metasurface beamsplitting devices operate correctly, given the spatial coherence of the source and the resolution of the spectral analysis in the current experiment, and thus confirming that the MSs can be valid candidates to replace many bulk optical components.

The recorded MMs generally display the symmetry obtained by setting the complex transmission coefficients as $t_{\text {sp}} = t_{\text {ps}}$ in the Jones matrix, resulting in the $m_{\text {14}} = -m_{\text {41}}$ asymmetry, which is particularly striking for MS3. This results in the device generating right circular polarized light, but analyzing left circular polarized light.

It was experimentally established that all the MS samples experienced a blue shift and a lowered intensity for the scattered orders compared to the design. Simulations revealed this to be caused by manufacturing errors, leading to conical pillars. Surprisingly, the sawtooth profile and transmission at the new operational wavelengths still show promising results and polarization abilities. This is the strongest asset behind MME in metasurface technology, as it supplies feedback between experimentally realized metasurfaces and simulations, allowing both design issues and manufacturing errors to be identified.

In summary, these measurements expose MME as an excellent choice in the characterization of metasurfaces and could be essential for future mass production of metasurface-based optics, especially in combination with computational tools.