Diffractive order Mueller matrix ellipsometry for the design and manufacture of polarization beam splitting metasurfaces

Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 703 Diffractive order Mueller matrix ellipsometry for the design and manufacture of polarization beam splitting metasurfaces VICTORIA M. BJELLAND,1,2 NATHAN HALE,1 NIKLAS SCHWARZ,1DANIEL VALA,3JENS HØVIK,1 AND MORTEN KILDEMO1,* 1Department of Physics, Norwegian University of Science and Technology (NTNU), 9491 Trondheim, Norway 2European Organization for Nuclear Research (CERN) - 1211 Geneva 23, Switzerland 3IT4Innovations, National Supercomputing Center, and Faculty of Materials Science and Technology, VSB – Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic *[email protected] Abstract: Optical metasurface technology promises an important potential for replacing bulky traditional optical components, in addition to enabling new compact and lightweight metasurfacebased devices. Since even subtle imperfections in metasurface design or manufacture strongly affect their performance, there is an urgent need to develop proper and accurate protocols for their characterization, allowing for efficient control of the fabrication. We present non-destructive spectroscopic Mueller matrix ellipsometry in an uncommon off-specular configuration as a powerful tool for the characterization of orthogonal polarization beam-splitters based on a-Si:H nanopillars. Through Mueller matrix analysis, the spectroscopic polarimetric performance of the ± 1 diffraction orders is experimentally demonstrated. This reveals a wavelength shift in the maximum efficiency caused by fabrication-induced conical pillars while still maintaining a polarimetric response close to ideal non-depolarizing Mueller matrices. We highlight the advantage of the spectroscopic Mueller matrix approach, which not only allows for monitoring and control of the fabrication process itself, but also verifies the initial design and produces feedback into the computational design. Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. 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 #501709 https://doi.org/10.1364/OE.501709 Journal © 2024 Received 31 Jul 2023; revised 22 Oct 2023; accepted 25 Oct 2023; published 21 Dec 2023 Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 704                                                                                            Fig. 1. a-c) Design of the three metasurfaces where a) MS1 works as a horizontal/vertical polarizer, b) MS2 works as a + 45 ◦ / − 45 ◦ polarizer and c) MS3 works as a left and right circular light polarizer in the − 1 ( k-1 ) and + 1 ( k+1 ) diffracted orders, respectively. The k0 vector indicates the 0-order (specular) transmission, while the schematic insets gives the trace of the electric field vector in the ( ˆ s , ˆ 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 x and D y . d) Phase and transmission maps from sweeps over pillar dimensions D x and D y for pand 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 x and D y from the average found from SEM images. f) Phase shifts for MS1 similar to e). 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. Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 705 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. 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 ◦ and − 45 ◦ 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 ( δss ) and transmission (T ss ) of each pillar was calculated from a square array with given pillar-diameters D x and D 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 δpp and T pp follows readily by a 90◦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 π in each supercell structure. MS2 is identical in dimensions to MS1, but with a 45 ◦ 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 Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 706 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 π radians between the sand p-components, where each nanoresonator is operating as a half wave retarder. In one super-period, each pillar is rotated by π/ 12 radians with respect to the previous one, ending with a full rotation of π 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 ( Λ= 12d, d = 500 nm) at λ0= 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 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 x ,D y and height hare 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≈ 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 M=M11 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 1m12 m13 m14 m21 m22 m23 m24 m31 m32 m33 m34 m41 m42 m43 m44 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ =M11 ⎡⎢⎢⎢⎢⎣ 1Dt P m3×3⎤⎥⎥⎥⎥⎦, (1) where P and D are the polarizance and diattenuation vectors, respectively [29]. The corresponding Mueller-Jones matrix used in the calculations contains intensity-like products of the complex amplitude transmission coefficients for each element and is given elsewhere [30–32]. In brief, given a Jones matrix for the metasurface J , the corresponding Mueller-Jones matrix is given by M=A(J⊗J∗)A−1 [33], where ⊗ is the Kronecker product and the square matrix A is given in the literature [33]. The element M 11 =1 2(︁|tpp|2+|tsp|2+|tps|2+|tss |2)︁ , with t pp ,t sp ,t ps and t ss the amplitude transmission coefficients of the whole system, gives the total transmission Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 707 ^ k -1 ^ k-1 ^ k+1 ^ k+1 ^ k0 ^ k0 EnP ^ k+1 ^ k-1  s - s s,y p,x ^^ ^ ^^ k ,z in Source P P RC RC FL SL MS Inserted Detector Inserted fsf f f f ^ k0 CL ^ CL ^ ^ ^ z x y  s  s p ^ s ^ p ^ s ^  s  s Incident Light Focused Incident Light F ^ ^ ^ z x y 0o, 180o Unaligned 0o, 180o Aligned  s  s a) b) c) d) e) Fig. 2. a) Experimental ellipsometric setup. Light from a Xe source is focused through a pinhole with diameter 100 µ m operating as the incoherent source area, followed by a lens (SL) at f 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 f= 80 mm from FL (if inserted). There, the light is scattered into θs for the − 1 order and −θs 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 f to the metasurfaces. The specularly transmitted signal is indicated with ˆ k0 . 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 θs and will measure additional k-vectors corresponding to ∆θs , 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 ∆θs span. d) The scattering plane of the metasurface sample not aligned with the x-z plane of the ellipsometer, where ϕ is the azimuthal rotation of the sample. e) The scattering plane of the MS aligned with the x-z plane of the ellipsometer. when normalized against the lamp intensity. M 11 is used below to give a good estimate of the transmission into diffracted orders and the diffraction efficiency of the metasurface samples. 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 θs and was limited to steps of 0.1 ◦ . 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] sin(θs)=sin(θi)+mλ Λ, (2) Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 708 where θi is the incidence angle and θs is the scatter-angle for orders m, λ is the wavelength detected by the spectrograph on the detector side and Λ is the super-period of the sawtooth phase grating. For an ideal sawtooth phase grating, both the generalized version of Snell’s law [35,36] and simple diffraction theory [35], shows that all light is in this case diffracted into a single order, i.e. m = + 1 or m =− 1. The diffraction efficiency of the orthogonal polarization states into the first diffraction orders can then be defined by η±1= 2M ±1 11 , where M ±1 11 is the measured total transmission into the orders m=±1. The incoming light is limited to normal incidence throughout this work ( θs= 0 ◦ ), and the scattered light is only measured in the plane of incidence. Figure 2shows that ˆ s=ˆ ki׈ p with ˆ s=ˆ y and ˆ p=ˆ x , where ˆ ki is parallel to the wave vector of the incoming light. Further, ˆ x , ˆ y and ˆ 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, ˆ s=ˆ km׈ p, where ˆ kss 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 ϕ= 0 ◦ and ϕ= 180 ◦ , 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 θ≠ 0 ◦ are diffracted from the metasurface grating. Since the setup utilizes a collimated beam, the spatial coherence length is estimated to be 200 λ0 [31], where λ0 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 ±∆θs 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 ( ≈ 1 ◦ ), 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 ∆θs . 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 µ 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 λ0/ 0.017 ≈ 30 λ0 , 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 ( ϕ≠ 0 ◦ , 180 ◦ ), Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 709 in which the full signal cannot be measured. Figure 2(e) shows the correct aligned system, where the rotation of the sample is 0◦and 180◦for the −1 and +1 order, respectively. 3. Results and discussion Figure 3shows the Mueller matrices M±1 MS1(λ) , M±1 MS2(λ) and M±1 MS3(λ) , where the azimuthal orientation of the sample ϕ= 0 ◦ gives the − 1 order, and ϕ= 180 ◦ 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 θs= 8.0 ◦ , while the designed profile (given λ= 915 nm, super-period Λ= 12d, d = 500 nm and Eq. (2)) corresponds to a peak efficiency at θs= 8.7 ◦ . 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. 4shows the spectrally dependent depolarization index P∆, given by [38] P∆=√︄∑︁4 i,j=1m2 ij −1 3, (3) where P ∆= 1 corresponds to a non-depolarizing system and P ∆= 0 to a fully depolarizing system. The depolarization index observed in Figs. 4(a-c) appears to be near unity over a large spectral range. It starts to drop when the signal is low, which is either outside the efficiency range of the grating or at the onset of diffracted orders (near Rayleigh lines [39]). Hence, the samples generally convey an excellent degree of uniformity. 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. 4shows 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 12 and m 21 for MS1, m 13 and m 31 for MS2 and m 14 and m 41 for MS3, for both ± 1 diffracted orders. The MM elements relate to the measured transmission into the diffracted orders, i.e. M ±1 11 , 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. Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 710 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| ≥ 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.    m 41 m 4 l    m 31 m 3 k    m 21 m 2 j    m 42    m 32    m 22    m 12 m 1 i    m 43    m 33    m 23    m 13    m 44    m 34    m 24    m 14 M   M      m 41 m 4 l    m 31 m 3 k    m 21 m 2 j    m 42    m 32    m 22    m 12 m 1 i    m 43    m 33    m 23    m 13    m 44    m 34    m 24    m 14 M   M         m 41 m 4 l    m 31 m 3 k    m 21 m 2 j       m 42    m 32    m 22    m 12 m 1 i       m 43    m 33    m 23    m 13       m 44    m 34    m 24    m 14 M   M         Fig. 3. a) Normalized Mueller matrices M±1 MS1 , b) M±1 MS2 and c) M±1 MS3 , for the − 1 ( θs= 8.0 ◦ , ϕ= 0 ◦ ) and + 1 ( θs=− 8.0 ◦ , ϕ= 180 ◦ ) diffracted orders. The MMs are normalized against M 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. Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 711 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 ( η±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. 3for the − 1 and + 1 scattered orders. These matrices are later used for analyzing the polar decomposition. M   M                                                                                       Fig. 4. The top row shows the degree of depolarization P GB using Eq. (3) for a) MS1, b) MS2 and c) MS3, respectively. The middle row shows the selected MM elements d) | m 12| and | m 21| for MS1, e) | m 13| and | m 31| for MS2 and f) | m 14| and | m 41| 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 ±1 11 (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. Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 718             M 11  M 0   M 11  M 0   M 11  M 0        M 11  M 0   M 11  M 0   M 11  M 0    Fig. 8. Unpolarized, transmitted light M 0 11 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 x and D y corresponding to SEM images of finished manufactured samples. The black lines represent the measured experimental results. 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 ± 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 ± 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 MS1 12 | , | m MS2 13 | , | m MS3 14 |> 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 MS1 12 | , | m MS2 13 | , Research Article Vol. 32, No. 1 / 1 Jan 2024 / Optics Express 719 | m MS3 14 |> 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 sp = t ps in the Jones matrix, resulting in the m 14 =− m 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. 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