Mueller matrix polarimetry, valued for its ability to reveal structural information of samples, is gaining prominence in vectorial optics and sample analysis. The 16 elements of a Mueller matrix comprehensively describe the polarization properties of a sample, which are crucially linked to the microstructure of the material. This renders it particularly useful in medical diagnostics1^–23 or material analysis.24^–31 However, due to its mathematical complexity and abstract nature, being able to extract meaningful physical features directly from the Mueller matrix remains a challenge. To this end, methods have been developed to analyze the Mueller matrix, with typical examples including Mueller matrix symmetric decomposition,32 Mueller matrix differential decomposition,33^,34 Mueller matrix transformation,35 and Mueller matrix polar decomposition (MMPD).36 Among these methods, MMPD stands out as a prevalent method, which has been used and validated in characterizing various biomedical and optical materials, particularly through its retarder component.37^–50

The retarder component is conventionally described by three parameters: the linear retardance δ, fast-axis orientation θ for a linear retarder, and the circular retardance ϕ for a circular retarder, with its sign indicating the handedness.16 Previous studies have shown that the linear retarder strongly correlates with fibrous structures in biomedical tissues.8^,51^–56 Specifically, the orientation of the linear axis can reveal disorganized fiber orientations, indicating symptoms of disease, whereas linear retardance measures the density of the fibrosis in various tissues.52^–55^,57^,58 These properties change markedly across different disease stages, making linear retarder parameters strongly correlated with the diagnosis and staging of diseases such as breast cancer, bowel disease, cervical cancer, and liver cancer.52^,53^,59^–65 Meanwhile, circular retardance primarily occurs in chiral molecules such as glucose, skin, and soft tissue membranes,66^–68 through which the magnitude and distribution of circular retardance provide insights into the composition and structure of the material.67^,69^–71

Despite the obvious importance of these parameters, misinterpretations can occur when attempting to decipher the properties of the retarder using this approach. This has been demonstrated in several prior studies72^,73 but is revisited here to emphasize the limitations of conventional parameters. Specifically, due to reciprocity and predefined formats, the conventional MMPD parameter set can accurately represent the configuration only when the sample comprises a circular retarder followed by a linear retarder. In practice, some retarders exhibit a multilayered structure as seen in bulk tissue, and specific information about their hierarchical structure, such as the number of layers and the properties of each layer, is often unknown. In addition, some retarders do not have a distinct layered structure, making layer-based modelling inherently unsuitable. Without prior knowledge of the target, assuming that the retarder element is a fixed bilayer structure may yield unreliable results. Therefore, it is essential to adopt parameters that can reliably characterize retarder properties without prior knowledge of the actual structure.

This study focuses on the appropriate characterization of retarders when no prior knowledge is available. In the theory section, we first highlight the limitations of conventional parameters. Then, we describe and advocate the use of an elliptical retarder parameter set—previously proposed by Lu and Chipman36 but less commonly utilized—to characterize the overall effect of retarders as elliptical retarders, thereby circumventing the ambiguities associated with further decomposition. In the results section, the above-mentioned approach is validated using liquid crystal (LC) samples, which serve as well-defined retarder models due to their stable and well-characterized birefringent properties.

MMPD decomposes a Mueller matrix into a sequential multiplication of a diattenuator matrix (MD), a retarder matrix (MR), and a depolarizer matrix (MΔ), as shown in Eq. (1) [see further details in Sec. 1 in the Supplementary Material]: M=MΔMRMD.(1)

In nature, the real structure of some retarders consists of discrete layers, whereas others may lack a well-defined layered structure and can be modeled as continuously layered media, as shown in the upper part of Fig. 1. Various metrics have been developed to characterize retarder properties. Conventionally, MR is further decomposed into the product of a linear retarder matrix MLR and a circular retarder matrix MCR,74 as shown in the lower-left panel of Fig. 1 and Eq. (2): $$\begin{gathered} MR=MLRMCR \\ =(10000cos22θ+sin22θcosδsin2θcos2θ(1−cosδ)−sin2θsinδ0sin2θcos2θ(1−cosδ)sin22θ+cos22θcosδcos2θsinδ0sin2θsinδ−cos2θsinδcosδ)·(10000cosϕsinϕ00−sinϕcosϕ00001). \end{gathered}$$(2)However, for layered retarders, both the number of layers and the properties of each layer must be known to construct an accurate model. Simply characterizing a layered retarder using a predefined dual-layer model may lead to misinterpretation of its properties. For nonlayered retarders, applying a layer-based model is inherently inappropriate. Therefore, we need a more general and appropriate framework to describe such samples. In this context, we find that the elliptical retarder model offers unique advantages for this purpose, as shown in the lower-middle panel of Fig. 1.

For a retarder with an unknown structure, regardless of how complex the internal structure of the material may be, its overall effect on polarized light can be described as a single elliptical retarder. Subsequently, the elliptical axis orientation φ, degree of ellipticity χ, and elliptical retardance ρ are commonly used to describe the properties of the elliptical retarder.36

The function of an elliptical retarder can be seen as rotating along a specific axis on the Poincaré sphere by a certain angle. The rotation axis, which is by definition the fast axis of the retarder, corresponds to a certain polarization state UR on the Poincaré sphere. According to the polarization ellipse, this polarization state can be determined by two parameters, the elliptical axis orientation φ (in the range of [−π2,π2]) and the degree of ellipticity χ (in the range of [−π4,π4]). The fast axis of the retarder SR and the corresponding polarization state UR, are given in Eq. (3) [see details in Sec. 1 in the Supplementary Material]. The parameters a1, a2, and a3 are the three components of the normalized fast axis vector. Subsequently, φ and χ can be calculated using Eqs. (4) and (5) as follows: SR=(1a1a2a3)=(1cos2φcos2χsin2φcos2χsin2χ),UR=(cos2φcos2χsin2φcos2χsin2χ),(3)φ=0.5atan2(a2,a1),(4)χ=0.5sin−1a3.(5)

After determining the rotation axis, the elliptical retardance ρ describes the angle of rotation around that axis. The physical meaning of the retardance ρ is the phase shift introduced between the fast axis and slow axis when light propagates through the retarder. The value of ρ can be calculated using Eq. (6) from the bottom right 3×3 submatrix of MR, denoted as mR. ρ=cos−1[tr(mR)−12].(6)

The elliptical retarder model described above was originally proposed by Lu and Chipman.36 Building on this foundation, the core of our work is to demonstrate that, for a pure retarder system without prior knowledge, employing the elliptical retarder model not only offers an overall characterization of the system’s properties but also helps avoid potential misinterpretations that may arise from further decomposition.

In this section, we first demonstrate the limitations of the conventional dual-layer model in characterizing both layered and nonlayered structures. For samples with a layered structure, two experiments [see details in Sec. 2 in the Supplementary Material] show that a mismatch between real structure and the conventional dual-layer model can lead to misinterpretation. For samples with a nonlayered structure [see details in Sec. 3 in the Supplementary Material], the layer-based models are inherently unsuitable.

Then, we apply the elliptical retarder parameter set to characterize the samples in all three cases mentioned above. These results demonstrate the effectiveness and suitability of the elliptical retarder parameter set for describing unknown retarders beyond the constraints of conventional modeling assumptions.

The structure of the first layered sample, media A, consists of a linear retarder followed by a circular retarder, forming a combined elliptical retarder (MR=MCRMLR), as shown in Fig. 2(a) [see details in Sec. 2 in the Supplementary Material]. For the fabrication of the linear retarder, we used a nematic LC mixture (E7, Synthon Chemicals Ltd., Wolfen, Germany) capillary-filled into a glass cell with anti-parallel rubbed polyimide alignment layers. For the fabrication of the circular retarder, we used a chiral nematic LC mixture consisting of 83 weight percent (mass fraction) E7 LC and 17% (mass fraction) R811 chiral dopant (R811, Merck KGaA, Darmstadt, Germany).75 The thickness of the glass cell was 5μm for the linear retarder and 9 μm for the circular retarder, both of which were determined from the transmission spectrum of a UV–Vis spectrometer. We treat the LC glass cells as a whole, and the effect of the glass on the designed retarders is considered negligible.

Then, the reference metrics of the linear and circular retarder components, as well as the metrics obtained from the conventional dual-layer model of the combined sample, were calculated over the entire imaging area and presented as mean value ± standard deviation in Table 1.

It can be seen that the linear and circular retardances in the conventional metrics are close to the corresponding reference values. However, the linear axis orientation results from the conventional model show large deviations from the reference values, which may lead to misinterpretation. For example, in certain biomedical imaging, such deviations may result in incorrect interpretation of fiber orientation. Note that in the special case of a two-layer structure with a sequence opposite to that of the conventional dual-layer model, the conventional model can be expected to yield correct values for the linear retardance δ and circular retardance ϕ, but the linear axis orientation will shift from θ to θ−0.5ϕ [see details in Sec. 4 in the Supplementary Material]. This experiment demonstrates that when the sample’s layer sequence differs from the predefined model, conventional parameters may lead to misinterpretations.

The structure of the second sample, media B, consists of two overlapping linear retarders, forming an overall elliptical retarder (MR=MLR2MLR1), as shown in Fig. 2(b). The fabrication process of these linear retarders is the same as in the first experiment. As before, the reference metrics of the two linear retarder components, as well as the metrics obtained from the conventional dual-layer model of the combined sample, were calculated over the entire imaging area and presented as mean value ± standard deviation in Table 2.

By comparing with the reference, it can be seen that the conventional dual-layer model produces a circular retarder with a high circular retardance (ϕ′=−84.755deg±0.348deg) that does not exist in the actual configuration, and a linear retardance (δ′=16.035deg±0.232deg) that is significantly smaller than in the actual configuration (δ1=142.969deg±0.242deg, δ2=142.277deg±0.316deg). As in the previous experiment, this demonstrates that conventional parameters may lead to misinterpretations.

It should be noted that the data show some deviations in these two experiments. We believe that these deviations may be caused by three factors. First, the LC samples we used are not ideal linear or circular retarders. Second, the stacked sample thickness (with glass) is too high to achieve accurate focusing, given that each sample is 1.44 mm thick. Third, there is a slight spatial nonuniformity in the distribution of the LC. In the measurement process, it is hard to consistently image the exact same region every time. Despite these deviations, we are still able to derive convincing conclusions from the experimental results.

The third sample, media C, is an LC droplet, which was printed onto a glass substrate coated with a rubbed polyvinyl alcohol alignment layer using a MicroFab inkjet system [see details in Sec. 3 in the Supplementary Material]. It serves as an example of a nonlayered sample, featuring a spatially varying, locally twisted director field that lacks well-defined macroscopic layering, as shown in Fig. 2(c). The parameters calculated from the ER model, along with those from the dual-layer model, are presented in the right part. However, because the sample lacks a layered configuration of linear or circular retarders, the conventional dual-layer model is inherently unsuitable for its characterization.

To evaluate the elliptical retarder parameter set, we applied it to three samples described above: two layered-structure samples and the LC droplet sample with a nonlayered structure.

First, the layered-structure sample media A—composed of an LR followed by a CR—was re-examined here, as shown in Fig. 3(a). The distribution of the elliptical axis is illustrated in the middle panel of Fig. 3(a). To make the fast axis more visually intuitive, we also sampled points within the region and plotted the ellipses of the fast axes. The result suggests that the fast axis of the sample is close to linear, with a low degree of ellipticity. The distribution of the sample’s elliptical retardance is illustrated in the right panel of Fig. 3(a). The elliptical retardance across the imaging region was ρ=143.830deg±1.205deg.

Second, the layered-structure sample media B—composed of two different LRs—was re-examined here, as shown in Fig. 3(b). The distribution of the elliptical axis is illustrated in the middle panel, and the distribution of the sample’s elliptical retardance is illustrated in the right panel. We can see that this sample exhibits a relatively high degree of ellipticity in its fast axis. In addition, the elliptical retardance across the imaging region was ρ=86.451deg±0.350deg. The resulting optical properties are similar to those of a quarter-wave plate; its fast axis corresponds to left-handed circular polarization.

Third, the non-layered-structure sample media C—an LC droplet—is re-examined here, as shown in Fig. 3(c). The distribution of the elliptical axis is illustrated in the middle panel, and the distribution of the sample’s elliptical retardance is illustrated in the right panel. It can be observed that the fast axis primarily exists in the form of elliptical polarization states. Specifically, the central region is dominated by nearly right-handed elliptical polarization, whereas the arrangements in the peripheral regions are more complex. In addition, the distribution of the elliptical retardance aligns well with the fast axis distribution. We also conducted imaging of a biomedical slice as an additional demonstration [see details in Sec. 5 in the Supplementary Material].

The choice of decomposition method for a retarder depends on prior knowledge of the sample; without such information, no single decomposition method can be considered universally correct. To circumvent this issue, we propose to use a set of elliptical retarder parameters to characterize a retarder with an unknown structure as a whole. This set of parameters was originally proposed by Lu and Chipman in 1996,36 where elliptical retardance and the corresponding fast axis were used to describe an arbitrary retarder. However, as noted in Sec. 1, these parameters have seen limited use in current polarization analysis, where further decomposition of the retarder is more commonly adopted. In this work, we aim to show that this elliptical retarder parameter set is effective and more suitable for characterizing the polarization properties of various samples.

However, further work is needed to fully characterize elliptical retarders. To uniquely characterize a retarder, two additional issues need to be considered. First, due to phase wrapping, a 2π range needs to be assumed, and the retardance values are derived within this range. Second, even within a 2π range (e.g., 0 to 2π), any given retarder has a corresponding retarder with different properties but the same Mueller matrix, and we need to choose one of them. The fast axis of this corresponding retarder is the slow axis of the original retarder, and its retardance is 2π minus the retardance of the original retarder. These issues arising from the limitation of Stokes–Mueller formalism may be addressed by obtaining more prior knowledge of the sample or through the introduction of other optical parameters such as wavelength and absolute phase.76^,77

This work is not confined to the analysis of MR within the MMPD framework; instead, it can be applied to any pure retarder system, with special importance in cases involving system-level modulation of polarized light. Compared with the conventional decomposition model, this approach offers several advantages. First, it is advantageous for characterizing bulk biomedical samples, where the specimen typically contains many layers of unknown structures.8 Second, it is advantageous for characterizing waveplates or SLM arrays used for vector beam modulation, where the overall modulation effect of the array is important.30^,39 Third, recent studies have used this set of metrics to construct physical fields with topological properties, which may have applications in data storage.78 In summary, the elliptical retarder parameter set provides a new perspective for future research, particularly in the characterization of media with complex structures. By offering a comprehensive characterization of a retarder, this work lays the foundation for potential applications in areas such as biomedical analysis and material characterization.

Acknowledgment. The authors would like to thank the support of St John’s College, the University of Oxford, and the Royal Society (Grant No. URF\R1\241734) (C.H.). The authors would also like to thank Prof. Martin Booth and Prof. Daniel Royston at the University of Oxford for their valuable support.

Biographies of the authors are not available.