Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection

doi:10.1364/OE.17.021396

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Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection

Santosh Tripathi and Kimani C. Toussaint »View Author Affiliations

Optics Express, Vol. 17, Issue 24, pp. 21396-21407 (2009)
http://dx.doi.org/10.1364/OE.17.021396

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▸ Article Outline
– Abstract
1. Introduction
2. Theory
3. Results and discussion
4. Conclusion
– References and links

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Abstract

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.

1. Introduction

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

S. Chandola, K. Hinrichs, M. Gensch, N. Esser, S. Wippermann, W. G. Schmidt, F. Bechstedt, K. Fleischer, and J. F. McGilp, “Structure of Si(111)-in Nanowires determined from the Midinfrared Optical Response,” Phys. Rev. Lett. 102(22), 226805 ( 2009). [CrossRef] [PubMed]

–7

M. Losurdo, M. Bergmair, G. Bruno, D. Cattelan, C. Cobet, A. de Martino, K. Fleischer, Z. Dohcevic-Mitrovic, N. Esser, M. Galliet, R. Gajic, D. Hemzal, K. Hingerl, J. Humlicek, R. Ossikovski, Z. V. Popovic, and O. Saxl, “Spectroscopic ellipsometry and polarimetry for materials and systems analysis at the nanometer scale: state-of-the-art, potential, and perspectives,” J. Nanopart. Res. , 1–34 ( 2009).

]. Precise characterization of such materials is necessary for process control, performance optimization, and device integration [7

, 8

W. Osten, ed., Optical inspection of microsystems (CRC Press, 2006).

]. 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

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

]. To achieve this, Mueller matrix polarimetry (MMP) has been successfully invoked by way of a variety of experimental techniques [10

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 ( 1992). [CrossRef] [PubMed]

–21

A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and A. Gioti, “Mueller matrix spectroscopic ellipsometry: formulation and application,” Thin Solid Films 455–456, 43–49 ( 2004). [CrossRef]

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 [17

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]

]. 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

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 ( 1992). [CrossRef] [PubMed]

, 17

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]

, 22

R. W. Collins and J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. 16(8), 1997–2006 ( 1999). [CrossRef]

], Pockels cells [23

F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36(22), 5388–5395 ( 1997). [CrossRef] [PubMed]

, 24

F. H. Delplancke, “Investigation of rough surfaces and transparent birefringent samples with Mueller-matrix scatterometry,” Appl. Opt. 36(30), 7621–7628 ( 1997). [CrossRef] [PubMed]

], photoelastic modulators [13

G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36(31), 8190–8198 ( 1997). [CrossRef] [PubMed]

, 25

G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36(31), 8184–8189 ( 1997). [CrossRef] [PubMed]

, 26

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, and J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314(1-2), 47–52 ( 1998). [CrossRef]

], and liquid crystal variable retarders [15

A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 ( 2003). [CrossRef] [PubMed]

, 27

J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24(1), 64–66 ( 1999). [CrossRef] [PubMed]

] 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) [17

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]

]. 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

G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36(31), 8190–8198 ( 1997). [CrossRef] [PubMed]

, 15

A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 ( 2003). [CrossRef] [PubMed]

–17

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]

]. 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

C. Chen, M. W. Horn, S. Pursel, C. Ross, and R. W. Collins, “The ultimate in real-time ellipsometry: Multichannel Mueller matrix spectroscopy,” Appl. Surf. Sci. 253(1), 38–46 ( 2006). [CrossRef]

, 28

D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 ( 2004). [CrossRef]

]. 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

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]

, 19

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455–456, 112–119 ( 2004). [CrossRef]

]. 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

K. C. Toussaint Jr, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 ( 2005). [CrossRef] [PubMed]

, 30

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]

]. Their potential applications in surface plasmon excitation [31

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 ( 2006). [CrossRef] [PubMed]

], microscopy [29

, 32

A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901–153904 ( 2006). [CrossRef] [PubMed]

], laser machining [33

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 ( 1999). [CrossRef]

], linear acceleration of electron beams [34

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 ( 1990). [CrossRef] [PubMed]

], and addressing/switching mechanisms in magnetic cores memories [35

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 ( 2009). [CrossRef]

] have over the years motivated the development of several ways to generate them [29

, 30

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]

, 34

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 ( 1990). [CrossRef] [PubMed]

–36

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 ( 2005). [CrossRef]

]. Now it is possible to generate vector beams with arbitrary polarization distribution with high accuracy [30

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]

, 35

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 ( 2009). [CrossRef]

] 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 [37

C. Brosseau, “Interaction of radiation with linear media,” in Fundamentals of polarized light: a statistical optics approach (John Wiley and Sons, Inc., New York, 1998), Chap. 4.

]. 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

D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 ( 2004). [CrossRef]

, 38

E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat Nano 2(1), 25–32 ( 2007). [CrossRef]

], and of nanostructures and nanomaterials that continue to become technologically and scientifically important [7

]. 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.

2. Theory

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.

Fig. 1 Schematic of the proposed experimental RAMMP setup. See text for details.

Typically, to calculate the electric field distribution on the sample one uses vector diffraction theory [39

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 ( 1959). [CrossRef]

–41

E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 ( 1959). [CrossRef]

]. 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 [40

L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.

]. 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 [40

L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.

]. 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 be

S (r, Φ) = {[S_{0} (r, Φ), S_{1} (r, Φ), S_{2} (r, Φ), S_{3} (r, Φ)]}^{T},

(1)

where S₀, S₁, S₂, S₃ are the four elements of the Stokes vector, and superscript T represents the transpose of the row vector. Assigning polarization to a ray is for conceptual convenience, and is in accordance with the theory presented in Refs [39

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 ( 1959). [CrossRef]

–41

E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 ( 1959). [CrossRef]

]. 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 [9

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

]

M_{R} (- Φ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos (2 Φ) & \sin (2 Φ) & 0 \\ 0 & - \sin (2 Φ) & \cos (2 Φ) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] .

(2)

In the rotated coordinate system upon refraction E_x1 and E_y1 transform to E_p1 and E_s1 , respectively (refer to Fig. 2). Due to this one-to-one transformation, the refraction operation of the lens can simply be written as,

M_{R F} = \frac{1}{\sqrt{\cos (θ)}} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

(3)

where

1 / \sqrt{\cos (θ)}

follows from the intensity law of geometrical optics [40

L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.

,41

E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 ( 1959). [CrossRef]

]. Here θ is the cone angle of the ray and is defined as

\tan^{- 1} (r / f)

, where f is the focal length of OBJ [40

L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.

]. 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

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

]

M_{W P} (0, δ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos (δ) & \sin (δ) \\ 0 & 0 & - \sin (δ) & \cos (δ) \end{matrix}] .

(4)

Thus, the effect of the waveplate on the ray, following standard Muller matrix transformation rules [9

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

], is given by M_R (-Φ)M_WP (0, δ)M_R (Φ). Generally, the retardance of a waveplate changes with the angle of incidence [42

W.-Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176(1-3), 9–15 ( 2000). [CrossRef]

], 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,

M_{S} (θ, Φ = 0,) = [\begin{matrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} & M_{23} & M_{24} \\ M_{31} & M_{32} & M_{33} & M_{34} \\ M_{41} & M_{42} & M_{43} & M_{44} \end{matrix}],

(5)

where M₁₁, …, M₄₄ are Muller matrix elements. For a ray coming from (r, Φ), the sample appears to have been rotated by –Φ, and thus the effect of the sample on the ray can be calculated by M_R (-Φ)M_S (θ, Φ=0)M_R (Φ). Reflection changes the direction of propagation of the ray and the polarization direction as shown in Fig. 2 (compare E_s1 with E_s2 and E_p1 with E_p2 ). This change in polarization is represented by a Mueller matrix of the form

M_{R L} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}],

(6)

This transformation, however, gives the Stokes vector in a new coordinate system, X₂OY₂ (refer to Fig. 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 M_R (π-Φ)M_WP (0, δ)M_R (-π+Φ). The collimation effect of the lens is modeled by Mueller matrix

M_{C L} = \sqrt{\cos (θ)} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

(7)

where

\sqrt{\cos (θ)}

again follows from the intensity law [40

L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.

,41

E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 ( 1959). [CrossRef]

]. 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., M_R (-π+Φ). The collimated ray then passes through LP with its transmission axis along OX and it transforms the Stokes vector according to [9

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

]

M_{LP} = \frac{1}{2} [\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}],

(8)

Finally, the intensity is recorded at the CCD. This operation is represented by

A = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}],

(9)

The combination of the aforementioned operations together give the final expression for the intensity recorded at the CCD for a ray incident on OBJ at (r, Φ) or equivalently at (θ, Φ) to be

\begin{array}{l} P (r, Φ) = A * M_{L P} * M_{R} (- π + Φ) * M_{C L} * M_{R} (π - Φ) * M_{W P} (0, δ) * M_{R} (- π + Φ) \\ * M_{R L} * M_{R} (- Φ) * M_{S} (θ, Φ = 0) * M_{R} (Φ) \\ * M_{R} (- Φ) * M_{W P} (0, δ) * M_{R} (Φ) * M_{R F} * M_{R} (- Φ) * S (r, Φ) . \end{array}

(10)

One observes that the output intensity is a function of the input Stokes vector, the azimuth angle, the radial distance (or the cone angle) and the elements of the sample Mueller matrix. By segregating the elements of the sample Mueller matrix, one can write the expression in the vector dot product form as

P (r, Φ) = \vec{W} • \vec{M} = [\begin{matrix} \begin{matrix} W_{11} \\ W_{12} \\ W_{13} \\ W_{14} \\ W_{21} \\ ⋮ \\ W {}_{44} \end{matrix} \end{matrix}] • [\begin{matrix} \begin{matrix} M_{11} \\ M_{12} \\ M_{13} \\ M_{14} \\ M_{21} \\ ⋮ \\ M {}_{44} \end{matrix} \end{matrix}],

(11)

where

\vec{W}

and

\vec{M}

are known as the polarimetric measurement vector and the Mueller vector respectively [9

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

]. This equation constitutes the forward model of the system. To determine the Muller matrix for a particular angle of incidence θ' i.e., M_S (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 [9

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

] as

P = W \vec{M} = [\begin{matrix} P_{1} \\ P_{2} \\ ⋮ \\ P_{N} \end{matrix}] = [\begin{matrix} \begin{matrix} W_{1, 11} & W {}_{1, 12} & \dots & W_{1, 44} \\ W_{2, 11} & W_{2, 12} & \dots & W_{2, 44} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ W_{N, 11} & W_{N, 12} & \dots & W_{N, 44} \end{matrix} \end{matrix}] [\begin{matrix} \begin{matrix} M_{11} \\ M_{12} \\ ⋮ \\ M_{44} \end{matrix} \end{matrix}],

(12)

where N is the number of rays considered. Given P and W, one can estimate

\vec{M}

through [9

R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.

]

{\hat{\vec{M}}}_{E s t} = {(W^{T} * W)}^{- 1} W^{T} P = W {}_{P}{^{- 1}} P,

(13)

where

W_{P}^{- 1}

is the pseudoinverse of W and

\overset{⌢}{\vec{M}}

is the estimated Mueller vector. Estimated sample Mueller matrix

{\overset{⌢}{M}}_{S}

can be constructed from

{\overset{⌢}{\vec{M}}}_{}

by using the relation implied in Eq. (11).

Fig. 2 Schematic of the coordinate systems used in the derivation of the Mueller matrix description. The change in the polarization state of a general ray as it passes through the system is also shown.

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 [43

R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.

] 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

R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.

–46

M. Becchi and P. Galatola, “Berreman-matrix formulation of light propagation in stratified anisotropic chiral media,” Eur. Phys. J. B 8(3), 399–404 ( 1999). [CrossRef]

]. Furthermore, this formalism is particularly useful for analyzing arbitrary anisotropic samples irrespective of their orientation which is not possible using the Fresnel approach [45

H. Fujiwara, “Ellipsometry of anisotropic materials,” in Spectroscopic ellipsometry: principles and application (John Wiley and Sons Ltd, West Sussex, 2007), Chap. 6.

]. 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 [J_M (θ, Φ=0)] for reflection were calculated. These Jones matrices were then converted to Mueller matrices [M_S (θ, Φ=0)] using the standard Jones-to-Mueller transformation [37

C. Brosseau, “Interaction of radiation with linear media,” in Fundamentals of polarized light: a statistical optics approach (John Wiley and Sons, Inc., New York, 1998), Chap. 4.

]. 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', [

{\overset{⌢}{M}}_{S} (θ = θ', Φ = 0)

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

{\overset{⌢}{M}}_{S}

was constructed by rearranging its elements.

Fig. 3 Block diagram representation of the steps taken for numerical analysis. Here (a) represents the approach for generating the synthetic data, and (b) the inverse model for retrieving the Mueller matrix elements of the sample. Refer text for details.

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 [45

H. Fujiwara, “Ellipsometry of anisotropic materials,” in Spectroscopic ellipsometry: principles and application (John Wiley and Sons Ltd, West Sussex, 2007), Chap. 6.

]. 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 [47

M. Iwanaga, “Effective optical constants in stratified metal-dielectric metameterial,” Opt. Lett. 32(10), 1314–1316 ( 2007). [CrossRef] [PubMed]

]. This material is magnetically active, i.e., the relative permittivity is not equal to 1, and is magnetically and electrically anisotropic. Reference [47

M. Iwanaga, “Effective optical constants in stratified metal-dielectric metameterial,” Opt. Lett. 32(10), 1314–1316 ( 2007). [CrossRef] [PubMed]

] 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

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 ( 2007). [CrossRef] [PubMed]

,49

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 ( 2006). [CrossRef]

], which provide spatial resolution beyond the diffraction limit.

Fig. 4 Schematic of samples numerically studied. (a) A thin anisotropic film deposited on top of a crystalline silicon sample. (b) Stratified metal-dielectric metamaterial.

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 [30

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]

Fig. 5 Polarization state distribution along azimuth of the vector beam used in the calculations. .

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

Fig. 6 Mueller matrix elements for the sample in Fig. 4(a) as calculated from the (a) Berreman formalism, and as recovered using (b) RAMMP.

Fig. 7 Mueller matrix elements for the sample in Fig. 4(b) as calculated from the (a) Berreman formalism, and as recovered using (b) RAMMP.

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 [17

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]

]. Since a non-diagonal Jones matrix can accurately describe the polarization property of any type of non-depolarizing sample [43

R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.

] 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, M_LP in Eq. (10) will be replaced by M_R (ψ)* M_LP* M_R (-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 [32

A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901–153904 ( 2006). [CrossRef] [PubMed]

]. 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 [43

R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.

] that is composed of permittivity tensor, permeability tensor and rotation tensors.

4. Conclusion

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 [43

R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.

] to make it a more information rich and robust characterization tool.

Acknowledgements

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.

References and links

1.	S. Chandola, K. Hinrichs, M. Gensch, N. Esser, S. Wippermann, W. G. Schmidt, F. Bechstedt, K. Fleischer, and J. F. McGilp, “Structure of Si(111)-in Nanowires determined from the Midinfrared Optical Response,” Phys. Rev. Lett. 102(22), 226805 ( 2009). [CrossRef] [PubMed]
2.	J. A. Fagan, J. R. Simpson, B. J. Landi, L. J. Richter, I. Mandelbaum, V. Bajpai, D. L. Ho, R. Raffaelle, A. R. H. Walker, B. J. Bauer, and E. K. Hobbie, “Dielectric response of aligned semiconducting single-wall nanotubes,” Phys. Rev. Lett. 98(14), 147402 ( 2007). [CrossRef] [PubMed]
3.	M. Gilliot, A. E. Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76(4), 045424 ( 2007). [CrossRef]
4.	Z. M. Huang, J. Q. Xue, Y. Hou, J. H. Chu, and D. H. Zhang, “Optical magnetic response from parallel plate metamaterials,” Phys. Rev. B 74(19), 193105 ( 2006). [CrossRef]
5.	W. Wu, E. Kim, E. Ponizovskaya, Y. Liu, Z. Yu, N. Fang, Y. R. Shen, A. M. Bratkovsky, W. Tong, C. Sun, X. Zhang, S. Y. Wang, and R. S. Williams, “Optical metamaterials at near and mid-IR range fabricated by nanoimprint lithography,” Appl. Phys., A Mater. Sci. Process. 87(2), 143–150 ( 2007). [CrossRef]
6.	M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, G. Malandrino, I. L. FragalÌ€, L. Armelao, D. Barreca, and E. Tondello, “Structural and optical properties of nanocrystalline Er2O 3 thin films deposited by a versatile low-pressure MOCVD approach,” J. Electrochem. Soc. 155(2), G44 ( 2008). [CrossRef]
7.	M. Losurdo, M. Bergmair, G. Bruno, D. Cattelan, C. Cobet, A. de Martino, K. Fleischer, Z. Dohcevic-Mitrovic, N. Esser, M. Galliet, R. Gajic, D. Hemzal, K. Hingerl, J. Humlicek, R. Ossikovski, Z. V. Popovic, and O. Saxl, “Spectroscopic ellipsometry and polarimetry for materials and systems analysis at the nanometer scale: state-of-the-art, potential, and perspectives,” J. Nanopart. Res. , 1–34 ( 2009).
8.	W. Osten, ed., Optical inspection of microsystems (CRC Press, 2006).
9.	R. A. Chipman, “Polarimetry,” in Handbook of Optics , M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.
10.	D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 ( 1992). [CrossRef] [PubMed]
11.	R. A. Synowicki, J. N. Hilfiker, and P. K. Whitman, “Mueller matrix ellipsometry study of uniaxial deuterated potassium dihydrogen phosphate (DKDP),” Thin Solid Films 455–456, 624–627 ( 2004). [CrossRef]
12.	J. N. Hilfiker, B. Johs, C. M. Herzinger, J. F. Elman, E. Montbach, D. Bryant, and P. J. Bos, “Generalized spectroscopic ellipsometry and Mueller-matrix study of twisted nematic and super twisted nematic liquid crystals,” Thin Solid Films 455–456, 596–600 ( 2004). [CrossRef]
13.	G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36(31), 8190–8198 ( 1997). [CrossRef] [PubMed]
14.	C. Chen, M. W. Horn, S. Pursel, C. Ross, and R. W. Collins, “The ultimate in real-time ellipsometry: Multichannel Mueller matrix spectroscopy,” Appl. Surf. Sci. 253(1), 38–46 ( 2006). [CrossRef]
15.	A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 ( 2003). [CrossRef] [PubMed]
16.	S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205(4), 743–747 ( 2008). [CrossRef]
17.	P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 ( 1980). [CrossRef]
18.	C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films 455–456, 14–23 ( 2004). [CrossRef]
19.	A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455–456, 112–119 ( 2004). [CrossRef]
20.	G. E. Jellison Jr., “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314(1-2), 33–39 ( 1998). [CrossRef]
21.	A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and A. Gioti, “Mueller matrix spectroscopic ellipsometry: formulation and application,” Thin Solid Films 455–456, 43–49 ( 2004). [CrossRef]
22.	R. W. Collins and J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. 16(8), 1997–2006 ( 1999). [CrossRef]
23.	F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36(22), 5388–5395 ( 1997). [CrossRef] [PubMed]
24.	F. H. Delplancke, “Investigation of rough surfaces and transparent birefringent samples with Mueller-matrix scatterometry,” Appl. Opt. 36(30), 7621–7628 ( 1997). [CrossRef] [PubMed]
25.	G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36(31), 8184–8189 ( 1997). [CrossRef] [PubMed]
26.	E. Compain, B. Drevillon, J. Huc, J. Y. Parey, and J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314(1-2), 47–52 ( 1998). [CrossRef]
27.	J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24(1), 64–66 ( 1999). [CrossRef] [PubMed]
28.	D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 ( 2004). [CrossRef]
29.	K. C. Toussaint Jr, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 ( 2005). [CrossRef] [PubMed]
30.	C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]
31.	Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 ( 2006). [CrossRef] [PubMed]
32.	A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901–153904 ( 2006). [CrossRef] [PubMed]
33.	V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 ( 1999). [CrossRef]
34.	S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 ( 1990). [CrossRef] [PubMed]
35.	Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 ( 2009). [CrossRef]
36.	S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 ( 2005). [CrossRef]
37.	C. Brosseau, “Interaction of radiation with linear media,” in Fundamentals of polarized light: a statistical optics approach (John Wiley and Sons, Inc., New York, 1998), Chap. 4.
38.	E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat Nano 2(1), 25–32 ( 2007). [CrossRef]
39.	B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 ( 1959). [CrossRef]
40.	L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.
41.	E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 ( 1959). [CrossRef]
42.	W.-Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176(1-3), 9–15 ( 2000). [CrossRef]
43.	R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.
44.	D. W. Berreman, “Optics in stratified and anisotropic media - 4X4 matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 ( 1972). [CrossRef]
45.	H. Fujiwara, “Ellipsometry of anisotropic materials,” in Spectroscopic ellipsometry: principles and application (John Wiley and Sons Ltd, West Sussex, 2007), Chap. 6.
46.	M. Becchi and P. Galatola, “Berreman-matrix formulation of light propagation in stratified anisotropic chiral media,” Eur. Phys. J. B 8(3), 399–404 ( 1999). [CrossRef]
47.	M. Iwanaga, “Effective optical constants in stratified metal-dielectric metameterial,” Opt. Lett. 32(10), 1314–1316 ( 2007). [CrossRef] [PubMed]
48.	Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 ( 2007). [CrossRef] [PubMed]
49.	A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 ( 2006). [CrossRef]

OCIS Codes
(120.5410) Instrumentation, measurement, and metrology : Polarimetry
(180.0180) Microscopy : Microscopy
(310.3840) Thin films : Materials and process characterization
(160.3918) Materials : Metamaterials
(160.4236) Materials : Nanomaterials

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: September 11, 2009
Revised Manuscript: November 1, 2009
Manuscript Accepted: November 2, 2009
Published: November 9, 2009

Citation
Santosh Tripathi and Kimani C. Toussaint, "Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection," Opt. Express 17, 21396-21407 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21396

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References

S. Chandola, K. Hinrichs, M. Gensch, N. Esser, S. Wippermann, W. G. Schmidt, F. Bechstedt, K. Fleischer, and J. F. McGilp, “Structure of Si(111)-in Nanowires determined from the Midinfrared Optical Response,” Phys. Rev. Lett. 102(22), 226805 (2009). [CrossRef] [PubMed]
J. A. Fagan, J. R. Simpson, B. J. Landi, L. J. Richter, I. Mandelbaum, V. Bajpai, D. L. Ho, R. Raffaelle, A. R. H. Walker, B. J. Bauer, and E. K. Hobbie, “Dielectric response of aligned semiconducting single-wall nanotubes,” Phys. Rev. Lett. 98(14), 147402 (2007). [CrossRef] [PubMed]
M. Gilliot, A. E. Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76(4), 045424 (2007). [CrossRef]
Z. M. Huang, J. Q. Xue, Y. Hou, J. H. Chu, and D. H. Zhang, “Optical magnetic response from parallel plate metamaterials,” Phys. Rev. B 74(19), 193105 (2006). [CrossRef]
W. Wu, E. Kim, E. Ponizovskaya, Y. Liu, Z. Yu, N. Fang, Y. R. Shen, A. M. Bratkovsky, W. Tong, C. Sun, X. Zhang, S. Y. Wang, and R. S. Williams, “Optical metamaterials at near and mid-IR range fabricated by nanoimprint lithography,” Appl. Phys., A Mater. Sci. Process. 87(2), 143–150 (2007). [CrossRef]
M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, G. Malandrino, I. L. FragalÌ€, L. Armelao, D. Barreca, and E. Tondello, “Structural and optical properties of nanocrystalline Er2O 3 thin films deposited by a versatile low-pressure MOCVD approach,” J. Electrochem. Soc. 155(2), G44 (2008). [CrossRef]
M. Losurdo, M. Bergmair, G. Bruno, D. Cattelan, C. Cobet, A. de Martino, K. Fleischer, Z. Dohcevic-Mitrovic, N. Esser, M. Galliet, R. Gajic, D. Hemzal, K. Hingerl, J. Humlicek, R. Ossikovski, Z. V. Popovic, and O. Saxl, “Spectroscopic ellipsometry and polarimetry for materials and systems analysis at the nanometer scale: state-of-the-art, potential, and perspectives,” J. Nanopart. Res. , 1–34 (2009).
W. Osten, ed., Optical inspection of microsystems (CRC Press, 2006).
R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.
D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992). [CrossRef] [PubMed]
R. A. Synowicki, J. N. Hilfiker, and P. K. Whitman, “Mueller matrix ellipsometry study of uniaxial deuterated potassium dihydrogen phosphate (DKDP),” Thin Solid Films 455–456, 624–627 (2004). [CrossRef]
J. N. Hilfiker, B. Johs, C. M. Herzinger, J. F. Elman, E. Montbach, D. Bryant, and P. J. Bos, “Generalized spectroscopic ellipsometry and Mueller-matrix study of twisted nematic and super twisted nematic liquid crystals,” Thin Solid Films 455–456, 596–600 (2004). [CrossRef]
G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36(31), 8190–8198 (1997). [CrossRef] [PubMed]
C. Chen, M. W. Horn, S. Pursel, C. Ross, and R. W. Collins, “The ultimate in real-time ellipsometry: Multichannel Mueller matrix spectroscopy,” Appl. Surf. Sci. 253(1), 38–46 (2006). [CrossRef]
A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). [CrossRef] [PubMed]
S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205(4), 743–747 (2008). [CrossRef]
P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980). [CrossRef]
C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films 455–456, 14–23 (2004). [CrossRef]
A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455–456, 112–119 (2004). [CrossRef]
G. E. Jellison., “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314(1-2), 33–39 (1998). [CrossRef]
A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and A. Gioti, “Mueller matrix spectroscopic ellipsometry: formulation and application,” Thin Solid Films 455–456, 43–49 (2004). [CrossRef]
R. W. Collins and J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. 16(8), 1997–2006 (1999). [CrossRef]
F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36(22), 5388–5395 (1997). [CrossRef] [PubMed]
F. H. Delplancke, “Investigation of rough surfaces and transparent birefringent samples with Mueller-matrix scatterometry,” Appl. Opt. 36(30), 7621–7628 (1997). [CrossRef] [PubMed]
G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36(31), 8184–8189 (1997). [CrossRef] [PubMed]
E. Compain, B. Drevillon, J. Huc, J. Y. Parey, and J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314(1-2), 47–52 (1998). [CrossRef]
J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24(1), 64–66 (1999). [CrossRef] [PubMed]
D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 (2004). [CrossRef]
K. C. Toussaint, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 (2005). [CrossRef] [PubMed]
C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007). [CrossRef]
Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 (2006). [CrossRef] [PubMed]
A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901–153904 (2006). [CrossRef] [PubMed]
V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999). [CrossRef]
S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]
Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]
S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005). [CrossRef]
C. Brosseau, “Interaction of radiation with linear media,” in Fundamentals of polarized light: a statistical optics approach (John Wiley and Sons, Inc., New York, 1998), Chap. 4.
E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat Nano 2(1), 25–32 (2007). [CrossRef]
B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959). [CrossRef]
L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.
E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 (1959). [CrossRef]
W.-Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176(1-3), 9–15 (2000). [CrossRef]
R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.
D. W. Berreman, “Optics in stratified and anisotropic media - 4X4 matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972). [CrossRef]
H. Fujiwara, “Ellipsometry of anisotropic materials,” in Spectroscopic ellipsometry: principles and application (John Wiley and Sons Ltd, West Sussex, 2007), Chap. 6.
M. Becchi and P. Galatola, “Berreman-matrix formulation of light propagation in stratified anisotropic chiral media,” Eur. Phys. J. B 8(3), 399–404 (1999). [CrossRef]
M. Iwanaga, “Effective optical constants in stratified metal-dielectric metameterial,” Opt. Lett. 32(10), 1314–1316 (2007). [CrossRef] [PubMed]
Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]
A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]

Abouraddy, A. F.
An, I.
Armelao, L.
Artal, P.
Aspnes, D. E.
Bajpai, V.
Barreca, D.
Bauer, B. J.
Becchi, M.
Bechstedt, F.
Ben Hatit, S.
Bergmair, M.
Bernet, S.
Berreman, D. W.
Bos, P. J.
Bouree, J. E.
Bratkovsky, A. M.
Bruno, G.
Bryant, D.
Bueno, J. M.
Capezzuto, P.
Cattelan, D.
Chandola, S.
Chen, C.
Chu, J. H.
Cobet, C.
Collins, R. W.
Compain, E.
de Martino, A.
Delplancke, F.
Delplancke, F. H.
Dohcevic-Mitrovic, Z.
Dorn, R.
Drevillon, B.
Drévillon, B.
Elman, J. F.
Engheta, N.
Esser, N.
Fagan, J. A.
Fang, N.
Ferreira, G. M.
Fleischer, K.
Foldyna, M.
Ford, D. H.
FragalÌ€, I. L.
Furhapter, S.
Gajic, R.
Galatola, P.
Galliet, M.
Garcia-Caurel, E.
Gensch, M.
Giangregorio, M. M.
Gilliot, M.
Gioti, A.
Goldstein, D. H.
Grob, J. J.
Hauge, P. S.
Hemzal, D.
Herzinger, C. M.
Hilfiker, J. N.
Hingerl, K.
Hinrichs, K.
Ho, D. L.
Hobbie, E. K.
Horn, M. W.
Hou, Y.
Huang, Z. M.
Huc, J.
Humlicek, J.
Iwanaga, M.
Jellison, G. E.
Jesacher, A.
Johann, L.
Johs, B.
Jureller, J. E.
Kim, E.
Kim, Y. K.
Kimura, W. D.
Koh, J.
Landi, B. J.
Laskarakis, A.
Laude, B.
Lee, H.
Leuchs, G.
Liu, Y.
Liu, Z.
Logothetidis, S.
Losurdo, M.
Malandrino, G.
Mandelbaum, I.
Maurer, C.
McGilp, J. F.
Modine, F. A.
Montbach, E.
Muller, D.
Naciri, A. E.
Nesterov, A. V.
Niziev, V. G.
Ossikovski, R.
Parey, J. Y.
Park, S.
Pavlopoulou, E.
Podraza, N. J.
Ponizovskaya, E.
Popovic, Z. V.
Pursel, S.
Quabis, S.
Raffaelle, R.
Richards, B.
Richter, L. J.
Ritsch-Marte, M.
Ross, C.
Salandrino, A.
Saxl, O.
Scherer, N. F.
Schmidt, W. G.
Shen, Y. R.
Simpson, J. R.
Stoquert, J. P.
Sun, C.
Synowicki, R. A.
Tidwell, S. C.
Tondello, E.
Tong, W.
Toussaint, K. C.
Vogel, E.
Walker, A. R. H.
Wang, S. Y.
Whitman, P. K.
Williams, R. S.
Wippermann, S.
Wolf, E.
Wu, W.
Xiong, Y.
Xue, J. Q.
Yu, Z.
Zapien, J. A.
Zhan, Q.
Zhang, D. H.
Zhang, W.-Q.
Zhang, X.

Adv. Opt. Photon.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

Appl. Opt.

Appl. Phys. B

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005). [CrossRef]

Appl. Phys., A Mater. Sci. Process.

W. Wu, E. Kim, E. Ponizovskaya, Y. Liu, Z. Yu, N. Fang, Y. R. Shen, A. M. Bratkovsky, W. Tong, C. Sun, X. Zhang, S. Y. Wang, and R. S. Williams, “Optical metamaterials at near and mid-IR range fabricated by nanoimprint lithography,” Appl. Phys., A Mater. Sci. Process. 87(2), 143–150 (2007). [CrossRef]

Appl. Surf. Sci.

C. Chen, M. W. Horn, S. Pursel, C. Ross, and R. W. Collins, “The ultimate in real-time ellipsometry: Multichannel Mueller matrix spectroscopy,” Appl. Surf. Sci. 253(1), 38–46 (2006). [CrossRef]

Eur. Phys. J. B

M. Becchi and P. Galatola, “Berreman-matrix formulation of light propagation in stratified anisotropic chiral media,” Eur. Phys. J. B 8(3), 399–404 (1999). [CrossRef]

J. Electrochem. Soc.

M. Losurdo, M. M. Giangregorio, P. Capezzuto, G. Bruno, G. Malandrino, I. L. FragalÌ€, L. Armelao, D. Barreca, and E. Tondello, “Structural and optical properties of nanocrystalline Er2O 3 thin films deposited by a versatile low-pressure MOCVD approach,” J. Electrochem. Soc. 155(2), G44 (2008). [CrossRef]

J. Nanopart. Res.

M. Losurdo, M. Bergmair, G. Bruno, D. Cattelan, C. Cobet, A. de Martino, K. Fleischer, Z. Dohcevic-Mitrovic, N. Esser, M. Galliet, R. Gajic, D. Hemzal, K. Hingerl, J. Humlicek, R. Ossikovski, Z. V. Popovic, and O. Saxl, “Spectroscopic ellipsometry and polarimetry for materials and systems analysis at the nanometer scale: state-of-the-art, potential, and perspectives,” J. Nanopart. Res. , 1–34 (2009).

J. Opt. Soc. Am.

R. W. Collins and J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. 16(8), 1997–2006 (1999). [CrossRef]
D. W. Berreman, “Optics in stratified and anisotropic media - 4X4 matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972). [CrossRef]

J. Phys. D

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999). [CrossRef]

N. J. Phys.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007). [CrossRef]

Nat Nano

E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat Nano 2(1), 25–32 (2007). [CrossRef]

Opt. Commun.

W.-Q. Zhang, “New phase shift formulas and stability of waveplate in oblique incident beam,” Opt. Commun. 176(1-3), 9–15 (2000). [CrossRef]

Opt. Lett.

Phys. Rev. B

M. Gilliot, A. E. Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76(4), 045424 (2007). [CrossRef]
Z. M. Huang, J. Q. Xue, Y. Hou, J. H. Chu, and D. H. Zhang, “Optical magnetic response from parallel plate metamaterials,” Phys. Rev. B 74(19), 193105 (2006). [CrossRef]
A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]

Phys. Rev. Lett.

S. Chandola, K. Hinrichs, M. Gensch, N. Esser, S. Wippermann, W. G. Schmidt, F. Bechstedt, K. Fleischer, and J. F. McGilp, “Structure of Si(111)-in Nanowires determined from the Midinfrared Optical Response,” Phys. Rev. Lett. 102(22), 226805 (2009). [CrossRef] [PubMed]
J. A. Fagan, J. R. Simpson, B. J. Landi, L. J. Richter, I. Mandelbaum, V. Bajpai, D. L. Ho, R. Raffaelle, A. R. H. Walker, B. J. Bauer, and E. K. Hobbie, “Dielectric response of aligned semiconducting single-wall nanotubes,” Phys. Rev. Lett. 98(14), 147402 (2007). [CrossRef] [PubMed]
A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901–153904 (2006). [CrossRef] [PubMed]

Phys. Status Solidi A

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205(4), 743–747 (2008). [CrossRef]

Proc. R. Soc. Lond.

E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. Lond. 253(1274), 349–357 (1959). [CrossRef]
B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959). [CrossRef]

Science

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

Surf. Sci.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980). [CrossRef]

Thin Solid Films

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, “Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle,” Thin Solid Films 455–456, 14–23 (2004). [CrossRef]
A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455–456, 112–119 (2004). [CrossRef]
G. E. Jellison., “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314(1-2), 33–39 (1998). [CrossRef]
A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and A. Gioti, “Mueller matrix spectroscopic ellipsometry: formulation and application,” Thin Solid Films 455–456, 43–49 (2004). [CrossRef]
D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films 455–456, 3–13 (2004). [CrossRef]
R. A. Synowicki, J. N. Hilfiker, and P. K. Whitman, “Mueller matrix ellipsometry study of uniaxial deuterated potassium dihydrogen phosphate (DKDP),” Thin Solid Films 455–456, 624–627 (2004). [CrossRef]
J. N. Hilfiker, B. Johs, C. M. Herzinger, J. F. Elman, E. Montbach, D. Bryant, and P. J. Bos, “Generalized spectroscopic ellipsometry and Mueller-matrix study of twisted nematic and super twisted nematic liquid crystals,” Thin Solid Films 455–456, 596–600 (2004). [CrossRef]
E. Compain, B. Drevillon, J. Huc, J. Y. Parey, and J. E. Bouree, “Complete Mueller matrix measurement with a single high frequency modulation,” Thin Solid Films 313–314(1-2), 47–52 (1998). [CrossRef]

Other

C. Brosseau, “Interaction of radiation with linear media,” in Fundamentals of polarized light: a statistical optics approach (John Wiley and Sons, Inc., New York, 1998), Chap. 4.
W. Osten, ed., Optical inspection of microsystems (CRC Press, 2006).
R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw Hill, Inc., New York, 1995), Chap. 22.
H. Fujiwara, “Ellipsometry of anisotropic materials,” in Spectroscopic ellipsometry: principles and application (John Wiley and Sons Ltd, West Sussex, 2007), Chap. 6.
L. Novotny, and B. Hecht, “Propagation and focusing of optical fields,” in Principles of nano-optics (Cambridge University Press, New York, 2007), Chap. 3.
R. M. A. Azzam, and N. M. Bashara, “Reflection and transmission of polarized light by stratified planar structure,” in Ellipsometry and polarized light (North Holland, Amsterdam, 1989), Chap. 4.

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