Chiral objects cannot be superimposed onto their own mirror images and behave differently under left circularly polarized (LCP) and right circularly polarized (RCP) light incidences [1]. Although many natural materials exhibit chiral behaviors, they are typically very weak. Therefore, to drastically enhance chiral responses, various chiral metasurfaces have been studied, including plasmonic and dielectric metasurfaces [2–4]. Circularly polarized (CP) emission can also be achieved by integrating emissive materials into chiral metasurfaces [5-7]. Such CP light sources in integrated device platforms are important for many potential applications, including displays, optical communication, and biomedical diagnosis [88–12]. However, the exact characterization of the polarization state in CP emission may include some errors because of the unwanted polarization distortion caused by optical components in the optical setup. Therefore, it is highly desirable to determine a degree of polarization distortion and conduct compensation if necessary.

Among others, a beam splitter can induce the distortion of the polarization state of light [13-17]. Beam splitters divide incident light into two different directions (reflected and transmitted beams). They are often indispensable to save the limited space in the optical setup and share the beam path of the source and signal light beams. In our measurement setup (Fig. 1), photoluminescence (PL) from the chiral metasurface sample passes through a beam splitter. Therefore, the polarization state of CP emission can be distorted, and the measured Stokes parameters of CP emission may include some errors. For that reason, it is important to determine the degree of polarization distortion caused by the beam splitter.

Figure 1.Schematic of the measurement setup. The sample emission passes through the beam splitter, and the polarization distortion of the sample emission can occur. BFP, back focal plane; LP, linear polarizer; BS, beam splitter.

Here, we directly measure the Mueller matrix of the beam splitter used in our Fourier-plane measurement setup (Fig. 1) and characterize how the Stokes parameters are modified after passing through the beam splitter. The Stokes vector (consisting of the four Stokes parameters) provides a complete description of the polarization state of light [18]. When the input beam with the Stokes vector S_input passes an optical component (e.g. beam splitter), the Stokes vector S_output of the output beam can be slightly modified: S_output = M_BS S_input, where the Mueller matrix (4 × 4 matrix) MBS of the beam splitter connects the Stokes vectors of the input and output beams. Therefore, by directly measuring the Mueller matrix of the beam splitter, we can retrieve the original polarization state of the CP emission from the metasurface sample. In particular, we consider one of our recent experiments on a perovskite metasurface [19] and directly evaluate how a degree of circular polarization (DCP) of the sample emission is affected by the beam splitter. We find that the polarization distortion of S3 /S0 caused by the beam splitter was very small (~1%) in our experiments. Using the measured Mueller matrix of the beam splitter, we specifically identify what contributes to polarization distortion in CP emission.

In this paper, we first explain the Stokes parameters in detail and describe how to measure them in experiments (Sections II and III). Then, we discuss how to determine the Mueller matrix of the beam splitter and present the measurement results (Section IV). In Section V, we present the experimental data from our metasurface sample (Stokes parameters of CP emission) and characterize how Stokes parameters are affected by the beam splitter.

Two linearly polarized optical waves with orthogonal polarizations can be described as [20]

(1)E→xz,t=i^E0xcoskz−ωt

(2)E→yz,t=j^E0ycoskz−ωt+ε

where ε is a relative phase difference between the two orthogonal waves. At time t, E_y and E_x can be written as follows:

(3)Ey/E0y=coskz−ωtcosε−sin(kz−ωt)sinε

and

(4)Ex/E0x=cos(kz−ωt)

(5)sin(kz−ωt)=[1−(ExE0x)2]1/2

From these equations, it follows:

(6)(EyE0y−ExE0xcosε)2=[1−ExE0x2]sin2ε

(7)EyE0y2+ExE0x2−2 Ex E0x Ey E0y cosε=sin2ε

These expressions can be rewritten in the general elliptical form with the orientation angle (Ψ) and ellipticity angle (χ) (Fig. 2):

Figure 2.Schematic of polarization ellipse. Orientation angle (Ψ) and ellipticity angle (χ), respectively.

(8)tan2Ψ=2E0xE0ycosεE0x2−E 0y2, 0≤Ψ≤π

(9)sin2χ=2E0xE0ysinεE20x+E20y,  −π4≤χ≤π4

At a particular ε, we can obtain linear and circular polarizations. When ε = ±nπ with an integer n, linear polarization occurs. If n is an even (odd) integer, the total electric field can be expressed as Eq. (10) [Eq. (11)].

(10)E→=i^E0x+j^E0ycoskz−ωt

(11)E→=i^E0x−j^E0ycoskz−ωt

Please note that if ε > 0, E^→y lags E^→x in time. Meanwhile, when E_0x = E_0y = E₀ and ε = −π/2 + 2mπ (ε = π/2 + 2mπ), where m is integer, the circular polarization appears and can be described as follows, respectively:

(12)E→=E0i^coskz−wt+j^sinkz−wt

(13)E→=E0i^coskz−wt−j^sinkz−wt

Depending on the order of E_x and E_y phases, it can be called RCP light or LCP light. According to our convention, the electric field vector is rotating clockwise in time for RCP light when we are looking at an incoming wave [20].

The Stokes parameters are convenient and effective to represent the polarization state of light. From the above equations, the Stokes parameters are defined as [21]:

(14)S0=E0x2+E0y2

(15)S1=E0x2−E0y2

(16)S2=2E0xE0ycosε

(17)S3=2E0xE0ysinε

where S0 represents the total radiant illumination, S1 the horizontal or vertical linear polarization state illumination, S2 the ±45° direction linear polarization state illumination, and S3 the RCP or LCP state illumination. Organizing them in vector form, we obtain the Stokes vector as follows:

(18)S→=S0S1S2S3

By dividing the Stokes parameters by the total intensity S0, the normalized Stokes vector is obtained, which is often used in experiments.

It is not easy to directly measure individual electric fields (E_x, E_y) and the phase difference ε in an experiment because a photodetector or charge-coupled device (CCD) usually detects the light intensity. Therefore, it is more desirable to reformulate the Stokes parameters in terms of light intensities that can be directly measured using typical optical components (such as a linear polarizer and quarter-wave plate).

The initial Stokes vector (S→i) of incident light can be modified after passing through an optical element, and the modified Stokes vector (S→f) can be obtained by multiplying a 4 × 4 matrix (Mueller matrix) to the Stokes vector. Table 1 shows the Mueller matrices for the linear polarizer and quarter-wave plate [20].

Table 1 Mueller matrices of linear polarizer and quarter-wave plate

Name	Mueller Matrix
Linear Polarizer 0°	121100110000000000
Linear Polarizer 90°	121−100−110000000000
Linear Polarizer +45°	121010000010100000
Linear Polarizer −45°	1210−100000−10100000
Quarter-wave Plate, Fast Axis Vertical	10000100000−10010
Quarter-wave Plate, Fast Axis Horizontal	10000100000100−10

The detected intensity from the CCD corresponds to the total intensity of light or the first row of the Stokes vector (i.e. S0 of S→f). Using the linear polarizer and quarter-wave plate (or multiply the corresponding Mueller matrix to S→i), we can move S₁, S₂, S₃ of S→i to the first row of S→f, so that it can be detected by the CCD. Figure 3 shows the overall procedures (a total of six measurements). By combining the measured data, we can determine S→i in terms of light intensities. In the following, we describe how to determine S→i in measurements [22-25].

Figure 3.Schematic of linear polarization (LP) and quarter-wave plate (QWP) positions and each state for measuring six Stokes parameters. (a), (b), (c) are related to the S1, S2, S3, respectively.

First, to obtain the value of S_1,i, the linear polarizer is set to 0° and 90° [Fig. 3(a)]. This is expressed through the following Mueller matrix. Note that the operation [1 0 0 0] corresponds to the CCD detection (i.e. reading the first row of the Stokes vector).

(19a)I0=1 0 0 0S→f=1 0 0 012 1 100 1 10000000000 S 0,i S 1,i S 2,i S 3,i=12 S0,i+ S1,i

(19b)I90=1 0 0 0S→f=1 0 0 012 1 −100 −1 10000000000 S 0,i S 1,i S 2,i S 3,i=12 S0,i− S1,i

(19c)I0−I90=S1,i

In a similar way, S2,i can be obtained with the linear polarizer oriented at 45° and 135° [Fig. 3(b)]:

(20a)I45=1 0 0 0S→f=1 0 0 012 10 100000 10 100000 S 0,i S 1,i S 2,i S 3,i=12 S0,i+ S2,i

(20b)I135=1 0 0 0S→f=1 0 0 012 10 −100000 −10 100000 S 0,i S 1,i S 2,i S 3,i=12 S0,i− S2,i

(20c)I45−I135=S2,i

S3,i is related to DCP. To measure it, a quarter-wave plate is used together with a linear polarizer [Fig. 3(c)].

(21a)Ircp=1 0 0 0S→f=1 0 0 012 10 −100000 −10 100000 10000 100000 −100 10 S 0,i S 1,i S 2,i S 3,i=12 S0,i+ Sa,i

Ilcp=1 0 0 0S→f=1 0 0 012 10 −100000 −10 100000 10000 100000 100 −10 S 0,i S 1,i S 2,i S 3,i=12 S0,i− Sa,i

(21c)Ircp−Ilcp=Sa,i

The sum of the two intensities for the orthogonal polarizations becomes S0,i:

(22)S0,i=I0+I90=I45+I90=Ircp+Ilcp

Then, the Stokes vector S→i can be expressed in terms of light intensities:

(23)S→i= S0,i S1,i S2,i S3,i= S0,iI0−I90I45−I135Ircp−Ilcp

Normalized Stokes parameters are given by:

(24)S1,iS0,i=I0−I90I0+I90S2,iS0,i=I45−I135I45+I135S3,iS0,i=Ircp−IlcpIrcp+Ilcp

In this way, the polarization state of light can be determined with a total of six measurements. The obtained Stokes parameters are directly related to the orientation angle (Ψ), ellipticity angle (χ), the degree of polarization (DOP), the degree of linear polarization (DLP), and the DCP as follows [20, 26]:

(25)Ψ=12tan−1S2S1

(26)χ=12tan-1S3S12+S22

(27)DOP=S12+S22+S32S0

(28)DLP=S12+S22S0

(29)DCP=S3S0

When CP light passes through an optical element (such as a beam splitter), a small distortion in the polarization state of light (or Stokes parameters) can happen, and this can cause unwanted errors in polarization measurements. Therefore, it is important to find the degree of polarization distortion and compensate for it if necessary. To do this, we first need to measure the Mueller matrix of the optical component. Among others, it is known that a beam splitter can cause errors in CP light measurements. In this section, we describe the procedure to measure the Mueller matrix of the beam splitter and present the measured Mueller matrix in our lab. Then, in the next section, we analyze the degree of polarization distortion caused by the beam splitter in our CP emission experiments.

Note that we only consider the transmission mode of the beam splitter (Fig. 4). In our lab setup (Fig. 1), since the sample emission (or PL) passes through the beam splitter and is detected by the CCD, we only consider transmission mode (but not reflection mode). In our measurements, Stokes parameters indicate the normalized values. In our experiment, we only consider the normalized Stokes parameters to simplify the discussion.

Figure 4.Schematic of measurement setup in lab for Stokes parameter of (a) after passing the beam splitter (Soutput) and (b) reference (Sinput).

Figure 4 shows the procedure of the Mueller matrix measurements. Specifically, we measure the Stokes vectors with or without the beam splitter [Figs. 4(a) and 4(b)]. The measured Stokes vectors correspond to the output Stokes vector (S_output) and input Stokes vector (S_input), respectively, and are described by the following relation:

(30)Soutput= S0 S1 S2 S3 output=MBSSinput= M11 M12 M13 M14 M21 M22 M23 M24 M31 M32 M33 M34 M41 M42 M43 M44 S0 S1 S2 S3 input

where M_BS is a Mueller matrix of the beam splitter. The Mueller matrix of the ideal beam splitter (M_{BS_ideal}) is a 4 × 4 identity matrix:

(31)MBSideal=1000010000100001

However, a real beam splitter has small deviation from this ideal case, as we show below.

To accurately measure M_BS, we consider S_{input[0°,45°,90°,rcp]} and S_{output[0°,45°,90°,rcp]} consisting of four input Stokes vectors (S_input) and four output Stokes vectors (S_output), respectively [27]:

(32)Soutput,0°Soutput,45°Soutput,90°Soutput,rcp=MBSSinput,0°Sinput,45°Sinput,90°Sinput,rcporSoutput0°,45°,90°,rcp=MBSSinput0°,45°,90°,rcp

Here, S_{input[0°,45°,90°,rcp]} and S_{output[0°,45°,90°,rcp]} are 4 × 4 matrices and consist of four input or output normalized Stokes vectors measured by linear (at 0°, 45°, and 90°) and circular (rcp) polarization measurement settings.

For measurement, we used a 50:50 non-polarized beam splitter (BS013; Thorlabs, NJ, USA). Table 2 shows the measured Mueller matrix M_BS in normal incidence.

Table 2 M_BS measured at 530 nm, 550 nm, and 600 nm

Wavelength (λ) (nm)	MBS
530	1000−0.002611.00243−0.04682−0.078150.00591−0.001860.990640.028260.006720.00696−0.018800.99114
550	1000−0.022861.00060−0.01013−0.02889−0.002080.001020.999490.031200.003390.03074−0.043280.98938
600	10000.000361.00027 0.00503 −0.01623−0.003370.001541.002570.064200.000800.00945−0.051550.99693

We find that the diagonal terms in the Mueller matrix are still major components and remain close to unity, while small nonzero off-diagonal terms exist. Because of the off-diagonal terms, small errors can occur in Stokes parameter measurements. For example, S_3,o is affected by S_0,i, S_1,i, S_2,i of the sample emission as well as S_3,i:

(33)S3,o=M41So,i+M42S1,i+M43S2,i+M44S3,i

where S_m,i is the input Stokes parameters from the sample and Sm,o is the output Stokes parameter measured after passing through the beam splitter. m (= 1, 2, 3) indicates the Stokes parameter S₁, S₂, and S₃, respectively.

From Table 2, we find that the off-diagonal terms (M₄₁, M₄₂ and M₄₃) of the Mueller matrix are orders of magnitude smaller than the diagonal term (M₄₄). Therefore, we can expect that polarization distortion in circular polarization measurements would be relatively small. We also notice that M43 is larger than the other off-diagonal terms. Therefore, S₂ of the sample emission (S_2,i) contributes more errors to the S₃ measurement, as also indicated in other studies [28].

Four measurements are enough to determine four Stokes parameters (S₀, S₁, S₂, and S₃). However, it is known that more measurements can further improve the accuracy of Stokes parameter measurements. For example, see [29-33] for details.

Using the measured M_BS, we can deduce the original Stokes parameters of emitted light from the chiral metasurface. From Eq. (30), we obtain the original Stokes parameters (S_input) using the inverse matrix:

(34)Sinput= MBS −1Soutput

In this way, we can correct the polarization distortion caused by the beam splitter and acquire the original Stokes parameters before passing through the beam splitter.

Now, using the measured Stokes parameters (S_output) of PL in our chiral metasurface measurements, we retrieve the Stokes parameters (S_input) of the original sample emission. Using the Fourier-plane measurement, we image the back focal plane in a microscope objective instead of the sample surface [34]. Then, we can obtain the angle-resolved PL (or reflection) spectrum and directly measure the photonic bands in periodic optical structures.

In particular, we consider one of our recent experiments on a perovskite metasurface [19] and directly evaluate the degree of polarization distortion caused by the beam splitter. CP emission with a large DCP was recently demonstrated from perovskite metasurfaces [35, 36]. Figures 5(a) and 5(b) are schematics of our chiral metasurface made of an emissive perovskite material. Arrays of paired rectangular bars were patterned on a glass substrate using electron beam lithography and reactive ion etching. The bar tilt angle (θ) was varied to control the chiral response. Grayscale lithography was employed to control the etching depths in the substrate and create out-of-plane symmetry breaking (D), which induced strong intrinsic chiral responses in the normal direction. Subsequently, a perovskite layer was spin-coated on the patterned glass substrate. A 2D organic–inorganic hybrid perovskite [(C₆H₅C₂H₄NH₃)₂PbI₄, PEPI] was used as a light-emitting layer. Then, the whole sample was covered by a ~200-nm-thick poly(methyl methacrylate) (PMMA) layer to protect the perovskite layer. The perovskite material has a higher index than the surrounding medium (i.e., the refractive index of SiO2 substrate is 1.5), and thus the perovskite material, which filled the patterned substrate, formed a dielectric metasurface.

Figure 5.Schematic of sample. (a) 3D view of the sample. (b) yz (left) and xy (right) view with parameters. The two rods have different heights ha = 135 nm, hb = 160 nm and widths a = 110 nm, b = 95 nm. Both have the same length, L = 150 nm with period P = 320 nm and are tilted to the θ. (c) and (d) are a measured photoluminescence (PL) emission dispersion map, when θ is 0° and 16°, respectively.

Figure 5(c) and (d) show the measured Stokes parameters of PL (S_1,o /S_0,o, S_2,o /S_0,o and S_3,o /S_0,o) in the Fourier plane. The measured PL spectrum shows the mode dispersion of the metasurface. As the bar tilt angle increases, the chiral eigenstate appears near λ = 553 nm at k_x /k₀ = 0 (normal direction). The measured stokes parameters show this feature clearly. At zero bar tilt angle, the emission is linearly polarized; The quadratic and linear bands have orthogonal linear polarizations (0° and 90°), and thus they appear as red and blue colors. In this case, S_3,o /S_0,o remains very small (close to zero). However, at θ = 10°, S_3,o /S_0,o increases drastically near λ = 553 nm due to the chiral eigenstate, while S_1,o /S_0,o of the quadratic band is reduced (especially at k_x /k₀ = 0).

Figures 6(a) and 6(b) compare the Stokes parameter from PL measurements (S_3,o /S_0,o, red line) with the compensated (i.e., original sample emission) Stokes parameter (S_3,i /S_0,i, blue dotted line) at θ = 0° and 10°, respectively. Figure 6 corresponds to the normal direction (k_x /k₀ = 0, the white dotted line) in Fig. 5. In both cases, we notice that the measured and compensated results are very similar, which indicates that the polarization distortion was very small in our CP emission measurements.

Figure 6.Polarization compensation in measurement (kx /k0 = 0). (a), (b) are the measured (S3,o /S0,o, red line) and compensated (S3,i /S0,i, blue dotted line) Stokes parameter at θ = 0° and 10°, respectively. (c), (d) deviation ∆(S3 /S0).

We note that S₂ in our chiral metasurface is very small. However, in general, S₂ can be more substantial depending on the design of chiral structures. In that case, we expect more errors can occur for S₃ measurements. Therefore, it indicates that circular polarization measurements need careful attention to avoid such errors caused by optical components in the setup.

To quantify the degree of polarization distortion induced by the beam splitter, we also evaluated the deviation from the original Stokes parameter S₃ /S₀ as follows:

(35)ΔS3/S0=S3,o/S0,o−S3,i/S0,i

Larger deviations mean more polarization distortions happened. The deviations are shown in Figs. 6(c) and 6(d) for θ = 0° and 10°, respectively. We find that the deviations of S₃ /S₀ in our measurements remain as small as 0.01 (= 1%).

As a reference, the deviations for S₁ /S₀ and S₂ /S₀ are also presented in Fig. 7. We find that the deviations are small at θ = 0° but the magnitude of the deviations increases up to ~0.03 at θ = 10°. It is because the emission from our metasurface becomes chiral at θ = 10° (i.e., S3 becomes significant) and the measurements of S1 and S2 are also affected by S3 due to polarization distortion [in a similar way to Eq. (33)].

Figure 7.Polarization compensation in measurement (kx /k0 = 0). (Upper row) Measured, compensated Stokes parameter and (lower row) their deviation. (a) and (b) are related to the S1 while (c) and (d) are related to the S2. (a) and (c) are for θ = 0° while (b) and (d) are for 10°.

In the current work, we investigated how the Stokes parameter measurements are affected by the beam splitter. Specifically, we first detailed the general procedures for the Stokes parameter and Mueller matrix measurements. Then, we presented the Mueller matrix of the beam splitter used in our lab. Using the measured Mueller matrix, we retrieved the original polarization state of the CP emission from our metasurface sample and found that the polarization distortion of S₃ /S₀ was as small as ~1% in our measurements. We also analyzed what contributes to polarization distortion in CP emission. Our work may provide useful guidelines on how to determine a degree of polarization distortion in general Stokes parameter measurements and how to retrieve the original polarization state of the sample emission.

The authors declare no conflicts of interest.

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

National Research Foundation (NRF) of Korea (NRF-2022R1F1A1074532)