We present rapid Mueller matrix polarimetry that can extract twelve Muller matrix elements from a single intensity image in real time and with high spatial resolution. This is achieved by parallelizing the respective polarization state generation and polarization state detection processes, which in existing polarimeters is performed sequentially. Parallelization of the polarization state generation process is accomplished through the use of vector beams, for which this work represents a new application domain. Polarization state detection is parallelized by uniquely combining a microscope/array detector setup with a specialized algorithm that simultaneously utilizes information from multiple spatial regions of the array detector. Simulated results applying this technique to two anisotropic samples including metamaterial yield material parameters that are consistent with those reported in the literature.
© 2009 Optical Society of America
In recent years the demand for improved optical characterization techniques for material processing has increased due to, among other things, the continued miniaturization of components in the semiconductor industry, and the rapid development of a variety of nanostructured materials for photonic applications [1–7]. Precise characterization of such materials is necessary for process control, performance optimization, and device integration [7, 8]. One approach to characterizing material optical properties is through the determination of its Mueller matrix, a mathematical description of a material’s linear optical properties including anisotropy and optical activity [7, 9]. To achieve this, Mueller matrix polarimetry (MMP) has been successfully invoked by way of a variety of experimental techniques [10–21].
Determination of the elements of the Mueller matrix in MMP is typically done by analyzing the polarization of the light reflected from a sample as a function of the polarization of the incident light . This involves two processes: polarization state generation (PSG), whereby the polarization of the incident light is varied systematically, and polarization state detection (PSD), in which the polarization of the reflected light is determined. Over the years a variety of optical components, such as rotating retardation plates [10, 17, 22], Pockels cells [23, 24], photoelastic modulators [13, 25, 26], and liquid crystal variable retarders [15, 27] have been used for PSG and PSD. Some common configurations in which these components have been arranged include rotating polarizer/rotating analyzer (RP/RA), rotating polarizer/rotating compensator fixed analyzer (RP/RCFA), phase modulator/phase modulator (PM/PM), dual phase modulator/dual phase modulator (DPM/DPM) and the like. However, irrespective of the optical elements used, or the particular configuration employed, all existing MMPs carry out the respective processes of PSG and PSD sequentially. For example, in the RP/RCFA configuration, the input polarization state is sequentially changed by way of a rotating polarizer, and likewise the reflected polarization state is analyzed sequentially via a rotating compensator (waveplate) followed by a fixed linear polarizer (analyzer) . Similarly, in the DPM/DPM arrangement, the incident polarization is changed in time by changing the retardation of two variable retardation elements, and the reflected polarization is inferred by varying in time the retardation of two additional variable retardation devices [13, 15–17]. This sequential operation in existing MMP approaches ultimately limits the speed at which the sample of interest can be characterized, thus making it difficult for real-time polarimetric characterization of dynamic processes such as the thin film growth [14, 28]. One way to overcome this limitation is to parallelize the respective PSG and PSD processes.
In order to parallelize the PSG process two requirements seem evident. First, it will be necessary to simultaneously deliver the required polarization diversity onto the sample—at least four linearly independent polarizations for complete Mueller matrix determination [17, 19]. Second, it should be possible to uniquely relate each incident polarization to a corresponding reflected polarization. A practical solution that satisfies both constraints is to use vector beams. Unlike scalar beams that have the same polarization throughout their cross-section, vector beams exhibit spatially nonuniform polarization [29, 30]. Their potential applications in surface plasmon excitation , microscopy [29, 32], laser machining , linear acceleration of electron beams , and addressing/switching mechanisms in magnetic cores memories  have over the years motivated the development of several ways to generate them [29, 30, 34–36]. Now it is possible to generate vector beams with arbitrary polarization distribution with high accuracy [30, 35] which makes their use in quantitative polarization studies such as polarimetry a practical proposition.
In Mueller matrix formalism, the polarization of light is represented by a four element Stokes vector . To parallelize the PSD, the individual Stokes vector elements of the reflected light needs to be determined simultaneously. A straightforward approach to doing this might be to divide the reflected light into four beams. These beams could then be passed through separate optical setups designed to measure a different element of the Stokes vector. Though this approach based on amplitude division is conceptually straightforward, it would require multiples of optical elements potentially increasing unwanted error contributions from each element. Another approach might be to design an optical setup that modifies the polarization of the reflected light as a function of the position on the beam. Given a fixed analyzer and an array detector with such a setup, the intensity recorded at each point on the array detector would be a projection of the polarization along different polarization components. Thus, these intensity values, along with a priori knowledge of the optical setup, could be used to completely determine the Stokes vector.
In this paper, we propose rapid Mueller matrix polarimetry (RAMMP) that achieves parallelization of both the PSG and PSD processes in real time and with high spatial resolution. We achieve parallelization of PSG by using vector beams, whereas to parallelize PSD a specially designed optical setup consisting of a microscope objective, an array detector, and an algorithm that combines information from different parts of the array detector is used. Our proposed scheme permits the extraction of twelve elements of the Mueller matrix from a single intensity image from the array detector, thereby reducing the experimental measurement time to the acquisition time of the array detector. To our knowledge the use of vector beams in polarimetry to improve polarization diversity on the sample has not previously been reported, thus this paper adds a new application domain. Moreover, use of a microscope objective provides diffraction-limited (spatial) sensitivity, which is very useful for the characterization of integrated circuits with ever decreasing feature size [28, 38], and of nanostructures and nanomaterials that continue to become technologically and scientifically important . In Section 2, we present the Mueller matrix description of the RAMMP along with a procedure to extract the Muller matrix elements from a single intensity image. Finally, in Section 3, we present the results of numerical studies on two anisotropic samples including a metamaterial, and discuss the properties of the RAMMP setup.
Figure 1 shows the proposed RAMMP optical setup. An input scalar polarized laser beam is converted into a collimated vector beam by the vector beam generator (VBG). The vector beam is then reflected by a non-polarizing beam splitter (BS) onto a low numerical aperture (NA) microscope objective (OBJ). The focused beam first passes through the waveplate (WP) before being reflected by the sample, and subsequently re-collimated by OBJ after passing again through WP. Finally, the beam is analyzed by the fixed linear polarizer (LP) and imaged onto a CCD camera.
Typically, to calculate the electric field distribution on the sample one uses vector diffraction theory [39–41]. However, we are more interested in analyzing the polarization of the output beam as a function of the polarization of the input beam, and since both beams are collimated they can be treated as ensembles of rays . Moreover, since we are interested only in the linear polarization properties of the sample, each ray in the output beam can be traced to a unique ray in the input beam. Therefore, we use a modified ray optics model in which each ray is associated not only with direction but also with intensity and polarization . A Mueller matrix description of the setup is presented below.
To analyze the schematic shown in Fig. 1, we begin with the optical field on the back focal plane of OBJ. Let us consider a general incident ray located at radial distance r and azimuth angle Φ in the XOY coordinate system as shown in Fig. 2 . Let its Stokes vector be39–41]. Although the focusing action of OBJ on a ray can be represented in any coordinate system, it takes particularly simple form in a coordinate system which has its X axis lying on the plane of incidence of the ray. Because of this, for a ray at (r, Φ) we define a new coordinate system X1OY1 obtained by rotating the XOY coordinate system by an angle Φ counterclockwise. The Stokes vector in the new coordinate system X1OY1 is related to that in the old coordinate system XOY through a standard Muller matrix of the form Fig. 2). Due to this one-to-one transformation, the refraction operation of the lens can simply be written as,40,41]. Here θ is the cone angle of the ray and is defined as , where f is the focal length of OBJ . The refracted ray then passes through WP with its fast axis aligned along OX thereby making an angle of -Φ with OX1 (refer to Fig. 2). The Mueller matrix of a waveplate with its fast axis at 0° and retardance δ is given by 9], is given by MR(-Φ)MWP(0, δ)MR(Φ). Generally, the retardance of a waveplate changes with the angle of incidence , and as such the setup has been designed to work for retardance values several degrees (to within 10°) around that of a quarter waveplate. In general, the Muller matrix of any optical system is a function of both the angle of incidence and azimuth angle. The Mueller matrix for the sample at azimuth angle Φ =0 and cone angle of θ is, in general, given by,Fig. 2 (compare Es1 with Es2 and Ep1 with Ep2). This change in polarization is represented by a Mueller matrix of the formFig. 2) which is related to X1OY1 through a clockwise rotation by π. For the reflected ray, the fast axis of WP makes an angle (π-Φ), and its effect on the ray therefore is given by MR(π-Φ)MWP(0, δ)MR(-π+Φ). The collimation effect of the lens is modeled by Mueller matrix40,41]. The Stokes parameters obtained after all these transformations are in the X2OY2 coordinate system. To bring them back to our initial fixed coordinate system XOY, a counterclockwise rotation of the axes by (π-Φ) is required, i.e., MR(-π+Φ). The collimated ray then passes through LP with its transmission axis along OX and it transforms the Stokes vector according to 9]. This equation constitutes the forward model of the system. To determine the Muller matrix for a particular angle of incidence θ' i.e., MS(θ=θ', Φ=0), intensities on the array detector corresponding to that angle of incidence, i.e., intensities along a radial distance r=f*tan(θ') on the detector are arranged in the polarimetric data reduction equation  as9]Eq. (11).
3. Results and discussion
To test the validity of RAMMP some straightforward numerical studies were carried out on two anisotropic samples, including a metamaterial. As shown in Fig. 3 numerical analysis involves two steps: generation of synthetic data which comprise the expected intensities at the array detector for a given sample, and the retrieval of the Mueller matrix elements from these intensities. To generate the synthetic data, [as shown in Fig. 3(a)], values for the permeability (μ), permittivity (ε), and rotation (ρ, ρ ') tensors  of the samples were obtained from the literature. These values were then used to calculate the Jones matrix for reflection using the Berreman formalism, which is widely used to analyze the reflection and transmission of polarized light from stratified planar structures [43–46]. Furthermore, this formalism is particularly useful for analyzing arbitrary anisotropic samples irrespective of their orientation which is not possible using the Fresnel approach . Since the Jones matrix for reflection for a given sample is a function of angle of incidence, and since in RAMMP the light is incident on the sample over a wide angular range due to the use of a microscope objective, an array of Jones matrices [JM(θ, Φ=0)] for reflection were calculated. These Jones matrices were then converted to Mueller matrices [MS(θ, Φ=0)] using the standard Jones-to-Mueller transformation . Synthetic data [P(θ, Φ)] was finally obtained by using the forward model of the system [Eq. (11)] in conjunction with the calculated Mueller matrices and field distribution of vector beams. To retrieve the Mueller matrix elements for angle of incidence θ', [ as shown in Fig. 3(b)] synthetic data corresponding to that angel of incidence were taken as a function of azimuth angle Φ, and matrices W and P [defined in Eq. (12)] were constructed. The Mueller vector was estimated using Eq. (13), and was constructed by rearranging its elements.
Schematics of the samples numerically studied are shown in Fig. 4 . Figure 4(a) represents a thin film of a transparent uniaxial crystal deposited on top of a crystalline silicon substrate . It is representative of materials like quartz and calcite which are widely used in optical components. The second sample depicted in Fig. 4(b) is a stratified metal-dielectric metamateial fabricated using Ag and MgF2 . This material is magnetically active, i.e., the relative permittivity is not equal to 1, and is magnetically and electrically anisotropic. Reference  shows that it shows negative (optical) refraction at a photon energy of 3.7 eV. This metamaterial structure has found its use, among other things, in the emerging hyperlens research [48,49], which provide spatial resolution beyond the diffraction limit.
RAMMP uses vector beams to deliver polarization diversity on the sample. In our numerical studies we used a vector beam with polarization state distribution as shown in Fig. 5 . Such a vector beam can be obtained from simply modifying the setups used to generate a radial vector beam to impart a relative phase shift of π/2 and π to the third and fourth quadrants of the constituent HG01 mode, respectively. Such a beam can easily be implemented by using spatial light modulators .
Figures 6 and 7 show a comparison between the Mueller matrix elements used to generate the synthetic data and the ones retrieved by solving the inverse problem as described earlier. The retrieved values are in exact correspondence with the original. Moreover, for each sample, only one intensity image is used to retrieve the Mueller matrix elements shown. This is made possible because of the parallelization of the PSG and PSD. Furthermore, RAMMP is flexible with type of vector beam that can be used, as long as the input beam delivers four linearly independent polarization states on the sample simultaneously. However, to characterize isotropic samples less exotic beams can be used
As can be seen in Figs. 6 and 7, RAMMP retrieves only twelve of the sixteen Mueller matrix elements, corresponding to the first three rows of the Mueller matrix. However, this is not a severe limitation. The elements from these three rows are sufficient to find all four complex elements of the Jones matrix of the sample within an absolute phase term . Since a non-diagonal Jones matrix can accurately describe the polarization property of any type of non-depolarizing sample  including the emerging material structures that are magnetically active, this scheme is widely applicable. It is interesting to note that with a slight modification RAMMP can allow determination of all the Mueller matrix elements. To do so, an additional intensity image needs to be taken by changing the orientation of the polarizer. Then, assuming that the polarizer is rotated by an angle ψ, MLP in Eq. (10) will be replaced by MR(ψ)* MLP* MR(-ψ).
For experimental realization of RAMMP, in the presence of noise, one has to be careful to deliver a comparable amount of different polarizations onto the sample. To improve the spatial resolution further high NA focusing can be used; however, since high NA focusing can enhance certain polarization components at the expense of others, one might have to use polarization manipulation techniques such as the one presented in . Moreover, high NA focusing will require modification of the system description to account for the effect of index matching fluids that are used in such systems.
The Mueller matrix of a general anisotropic sample is a function of both the azimuth angle and the angle of incidence. However, for a fixed angle of incidence one can relate the Mueller matrices at different azimuth angles by using Mueller matrix transformations for optical elements. This fact allowed us to take advantage of information overlap between intensities recorded for various azimuth angles for a fixed angle of incidence. However, the information overlap between intensities recorded for different angles of incidence is still unexploited. It is likely that combining this information will improve the robustness of the approach. This, however, is not possible within the framework of the Mueller matrix as no general relations exist to relate the Mueller matrices of different angles of incidence. An alternative approach might be to work directly with the optical matrix  that is composed of permittivity tensor, permeability tensor and rotation tensors.
In this paper, we presented a rapid Mueller matrix polarimetry that can extract twelve Mueller matrix elements from a single intensity image. It achieves this by parallelizing the operation of the polarization state generator and polarization state detector, which is in stark contrast to the existing Mueller matrix polarimeters in which the respective polarization state generation and detection processes are done sequentially. Parallelization of the polarization state generation was achieved by using vector beams, for which this paper provides a new application domain. Polarization state detection was parallelized by using an array detector, a specially designed optical setup, and the realization that the Mueller matrices of optical elements with the same angle of incidence but different azimuth angles are related by standard Mueller matrix transformations. Numerical studies carried out on two anisotropic samples, one of which was a metamaterial, verified the approach. Future work includes experimental realization of RAMMP, and derivation of its description in the framework of the optical matrix  to make it a more information rich and robust characterization tool.
This work was supported by the University of Illinois at Urbana-Champaign (UIUC) research start-up funds. We thank Dr. Brynmor Davis and members of the PROBE Lab for the useful discussions.
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