Abstract

Decomposition methods have been applied to in-plane Mueller matrix ellipsometric scattering data of the Spectralon reflectance standard. Data were measured at the wavelengths 532 nm and 1500 nm, using an achromatic optimal Mueller matrix scatterometer applying a photomultiplier tube and a high gain InGaAs detector for the two wavelengths. A parametric model with physical significance was deduced through analysis of the product decomposed matrices. It is found that when the data are analyzed as a function of the scattering angle, similar to particle scattering, the matrix elements are largely independent of incidence angle. To the first order, we propose that a Guassian lineshape is appropriate to describe the polarization index, while the decomposed diagonal elements of the retardance matrix have a form resembling Rayleigh single scattering. New models are proposed for the off diagonal elements of the measured Mueller matrix.

© 2013 OSA

1. Introduction

It is important to have a well characterized Mueller matrix of white reflectance standards, such as the Spectralon, in order to obtain correct measurements whenever applying this standard, and in particular, in cases where the polarization of light intervene in the measurement process [14]. For un-polarized and linearly polarized illumination, Spectralon white reflectance standards are well documented and often used as calibration standards for optical instruments, but for applications involving other polarization states, more documentation is needed [1]. The Spectralon is the commercial product closest to a Lambertian surface, and is a white diffuse material (based on polytetrafluoroethylene) with excellent reflection properties, produced by Labsphere, USA. It is formed as a thermoplastic resin by heat and pressure treatment. The thermoplastic materials are resin machined into flat, thermally stable, high permittivity surfaces with porosity within the top layer (generally a few tenths of a millimeter thick) that randomizes the phase of the incident light-due to internal multiple reflections-to produce diffuse reflected light [4]. Indeed, the Spectralon has the highest diffuse reflectance values of any known substance (up to 99%) in the ultraviolet, visible, and near-infrared spectral ranges [1, 2], and the reflection properties are spectrally flat between 400 and 1500 nm.

It is further fundamentally interesting to understand how a scattering Mueller matrix from a strongly diffuse medium may be interpreted. It concerns both the interpretation of Mueller matrices resulting from rough surfaces [57], powder samples, or extremely diffuse materials such as the Spectralon [2], and the important related problem of Mueller matrices resulting from biological tissue, which has recently regained considerable interest [812]. A parametric model of the Mueller matrix elements of the Spectralon diffuser, which relates to basic physical phenomena within the diffuser appears useful in order to develop the understanding of other strongly scattering media, such as biological tissue.

Two recent studies by Svensen et al. [1] and Germer et al. [2] reported the Mueller matrix measurements of the Spectralon using the wavelengths 532 nm [1] and 632.8 nm [2]. In the work of Svensen et al., the Mueller matrix was presented for several incidence angles (0°, 30°, 45° and 60°), and it was shown that the Mueller matrix could to the first order be seen as the sum of an ideal diffuser and a correction matrix [1]. Furthermore, the degree of polarization was observed to be a key parameter in order to develop a parametric model for each of the elements of the Mueller matrix. The origin of the structure in the scattering Mueller matrix elements reported in the work of Svensen et al. was not found, but in a recent study by Germer et al. [2], it was noted that particle scattering is an important process for the Mueller matrix of the Spectralon. Therefore, we have here pursued the idea that the Mueller matrix of the Spectralon appears to belong to a more general class of particle scattering problems.

Considerable effort has been made in order to increase the understanding of the Mueller matrix using various decomposition methods [8,9,1318]. Decomposition of the Mueller matrix allows to separate different polarization phenomena, and thus to identify the basic building blocks of a complicated Mueller matrix consisting of both dichroism, retardance and diffuse scattering [9]. In the current work, several decomposition methods have been applied to the in-plane Mueller matrix scattering measurements of the Spectralon. As a result, more detailed information about the sample has been extracted, and an improved parametric model for the Mueller matrix has been obtained. In particular, the parametric model was found by studying the data as a function of the Rayleigh scattering angle used in particle scattering experiments, and by using the basis functions revealed from the decomposition.

2. Decomposition theory and basis functions

Product or polar decomposition allows the decomposition of a Mueller matrix into a product of elementary Mueller matrices, describing depolarization, retardance and diattenuation [13, 14, 17]. However, a given product decomposition may not always be representing the physical phenomena taking place in the sample, particularly whenever the different polarization altering phenomena are randomly ordered or occur simultaneously. There are several possible product decomposition methods such as the symmetrical decomposition [15], and the more common forward [17] and reverse [13] decompositions which are strongly interrelated [13]. The differential decomposition [12, 16, 19] and the recently described roots decomposition [20, 21] are other possibly more appropriate alternatives. On the other hand, in the work of Svensen et al. [1], a simplified version of the sum (or spectral) decomposition [1, 14] was used to propose a parametrization of the Mueller matrix of the Spectralon.

In the Spectralon sample, many of the scattering phenomena that are altering the polarization, may be envisaged to occur simultaneously, and the imposed ordering of the polarizing phenomena in the product decomposition methods may be incorrect. As such, the forward, reverse and symmetrical polar decomposition are equally (un)valid. Unfortunately, the most natural decomposition method for such a system, the differential decomposition method, is currently strictly only valid for transmission geometries. As a result, the application of a decomposition method is in this work regarded as an empirical mathematical method in order to enhance and reveal an underlying dominating trend in order to parametrize the optical response. We choose therefore to mainly report the application of the product decomposition, due to its numerical stability in the presence of noise, in order to reveal more information about the basis functions describing the scattering Mueller matrix elements.

The measured Mueller matrix (M) is, in the forward polar decomposition, decomposed as [17]

M=MΔMRMD,
where MΔ is the depolarization matrix, MR is the retardance matrix, and MD is the diattenuation matrix. In terms of the forward polar decomposition, the depolarization, retardance and diattenuation matrices are [17]
MΔ=M11[10TpmΔ],MD=[1dTdmD],MR=[10T0mR].
The polarizance vector p =[p1  p2  p3]T and diattenuation vector d = [d1  d2  d3]T are directly available from the measured Mueller matrix, while the 3 × 3 submatrices mΔ, mR and mD are determined by the polar decomposition formalism [13, 17]. For an ideal diffuser, the MΔ is diagonal and given by M11·diag(1, α, β, γ), where α, β and γ quantify the degree of polarization of linear, elliptical and circular polarized light, respectively [14, 17]. However, in the work of Lu and Chipman [17] it was proposed that the eigenvalues of mΔ give the three principal depolarization factors, and hence the average degree of polarization can be defined by [17]
pC=13(α+β+γ).
The total retardance is calculated from the retardance matrix as [18]
R=cos1(Tr(MR)12).

A non-depolarizing matrix can in many cases be made up of the basic building blocks of a diattenuator and a retarder. The block diagonal building block, the so-called isotropic dichroic retarder, commonly encountered in specular reflection from an isotropic surface, or from Mie scattering from small particles is a particularly useful reference

MDR=M11[1m1200m2110000m33m3400m43m44].
For Mie spheres, expressions for the matrix elements are given by e.g. Bohren and Huffman [22], while for reflection from a plane isotropic surface, the matrix elements, for a {, ŝ, } right handed geometry, are given by the normal ellipsometric angles Ψ and Δ [23, 24]
m12=m21=cos(2Ψ),m33=m44=sin(2Ψ)cosΔ,m34=m43=sin(2Ψ)sinΔ.
For Rayleigh particles, the normalized Mueller matrix elements are further simplified and given by [22]
m12=m21=sin2(θR)1+cos2(θR),m33=m44=2cos(θR)1+cos2(θR),m34=m43=0.

The polarization index pD (or degree of purity) [25], which is obtained from the full Mueller matrix was selected as the key figure to model the Spectralon Mueller matrix [1]

pD=(Tr(MTM)M1123M112)1/2=(i,j=14Mij2M1123M112)1/2.
The polarization index is between 0 and 1, i.e. 0 ≤ pD ≤ 1. When pD = 0, the outgoing light is fully depolarized, and when pD = 1, the outgoing light is fully polarized. This index compares well to the average degree of polarization pc (Eq. (3)).

3. Experimental details

New Mueller matrix data were recorded for the Spectralon in the in-plane scattering geometry shown in Fig. 1. The experimental data are presented according to the right handed coordinate system {, ŷ, } where = − ,ŷ =ŝ with respect to the right handed system {ŝ, , }, see Fig. 1. All measurements were performed in the incidence plane (see Fig. 1 for the definitions of the angles). The Stokes vector is in this case given as

S=[Ix^+Iy^Ix^Iy^Ix^+y^Ix^y^IRCILC].

Mueller matrices for an illumination wavelength of λ = 532 nm were measured at every 1° for scattering angles θS (see Fig. 1) relative to the normal of the Spectralon surface from −90° to 90°, with the exception of a dead-zone of ±10° around the back-scattering angle. A rotating diffuser was used prior to the polarization state generator in order to remove speckle effects [1]. Mueller matrices at λ = 1500 nm are reported for every 2° for θS from 10° to 90°, as we will show that the Mueller matrix was found similar to the one at λ = 532 nm. It is emphasized that the data at λ = 1500 may contain speckle effects, as no rotating diffuser was available for that wavelength. It is noted that all the measured Mueller matrices were found to be physical according to the Cloude filtering criterion [26].

 

Fig. 1 Geometry for the scattering experiment, defining the incidence angle θi, the Rayleigh scattering angle θR and the Rayleigh scattering angle θS used in [1]. The figure also shows the right handed coordinate system {, ŷ, } used in the paper, where = − , ŷ =ŝ with reference to the right handed system {ŝ, , }. The arrows define positive direction.

Download Full Size | PPT Slide | PDF

A well conditioned broadband (achromatic) Mueller matrix ellipsometer [27] was used to acquire the experimental data at both wavelengths (see [1] for the experimental details at 532 nm). A 45 mW laser at 1500 nm and an InGaAs detector with variable gain 105 – 109 were used for the near infrared measurement.

4. Results and discussion

Figure 2 shows the Mueller matrices recorded at 532 nm for the angles of incidence 0°, 30°, 45°, 60° and 75°, where all elements have been normalized by M11. The M11 element is proportional to the phase function, and is shown in the top left figure. Its form is typical for a near Lambertian diffuser [1,4], although the total intensity has not been calibrated in the current work. It is particularly noted that the Mueller matrix elements have been plotted and analyzed as a function of θR. It is observed from Fig. 2 that the general trend of the normalized Mueller matrix elements appears to be independent of the incidence angle. Deviations from this trend can be observed for light scattered nearly parallel to the sample surface. It is also observed that both block diagonal and off-block diagonal elements are non-zero, particularly for angles close to the sample surface, showing that the Spectralon in these cases is far from an ideal diffuser. The diagonal elements are seen to be strongly dominating for scattering near parallel to the sample surface.

 

Fig. 2 The measured Mueller matrix data from the Spectralon surface with illumination at incidence angles θi of 0° (green x marks), 30° (purple crosses), 45° (blue circles), 60° (red stars), 75° (cyan squares) plotted as a function of the Rayleigh scattering angle θR. The full curves show the fitted parametric model in Eq. (19). All elements, except M11, are normalized to M11.

Download Full Size | PPT Slide | PDF

It is interesting to note that the amplitudes of the m24 ≈ −m42 elements oscillate with the azimuthal orientation (χ) of the sample (i.e. upon rotating the sample around its normal), and that they can be close to zero in a given orientation, see Fig. 3. Such an anisotropic effect was also observed by Germer et al. [2]. To demonstrate smaller variations in the remaining Mueller matrix elements upon sample rotation, we have also plotted Δm33m33 − 〈m33〉 and Δ(m34m43) ≡ (m34 − 〈m34〉) − (m43 − 〈 m43〉), where 〈mij〉 are the averages upon a full 360° rotation, as a function of χ. There may be a weak oscillation in the m34 and m43 elements, while the remaining elements are not observed to oscillate upon azimuthal rotation of the sample, similar to Δm33 in Fig. 3. It is thus curious that here the linear retardance between the ±45° axis is strongly oscillating, while the linear retardance along the xy axis is only weakly oscillating, when interpreted using the differential decomposition theory [12, 16]. However, the off-block diagonal elements are non-zero (except possibly m14 and m41), although only m24 and m42 are observed to be sensitive to the azimuthal orientation of the sample (χ).

 

Fig. 3 The difference of the off-diagonal elements (m24m42), Δm33m33 − 〈m33〉 and Δ(m34m43) ≡ (m34 − 〈 m34〉) − (m43 − 〈 m43〉), where 〈mij〉 are the averages upon a full 360 degrees rotation, at 532 nm. The data are plotted against the azimuthal angle of the Spectralon for incidence angle 75° and Rayleigh scattering angle 35°.

Download Full Size | PPT Slide | PDF

Figure 4 shows the recorded Mueller matrix for both wavelengths 1500 nm and 532 nm for the angle of incidence 75°. The Mueller matrix at 1500 nm has a similar form, but with small differences that will be discussed below.

 

Fig. 4 Measured Mueller matrix data from the Spectralon surface with illumination at an incidence angle of θi = 75° and wavelengths 532 nm (red triangles) and 1500 nm (blue crosses) plotted against the Rayleigh scattering angle θR. The full curve shows the fitted parametric model at 1500 nm in Eq. (19). All elements are normalized to M11.

Download Full Size | PPT Slide | PDF

4.1. Model for pD

When the data are plotted against the Rayleigh scattering angle, the depolarization indices pD are also observed to become nearly independent of the incidence angles. Previously it was proposed that the following parametric model could represent the data [1]

pD=|(θS+θi)322|=|(180θR)322|.
However, one may further argue that in the forward scattering direction, then both in the Mie and the Rayleigh scattering model, light should be highly polarized while pD should be less or equal to one. This is also the case for scattering at a grazing incidence. Therefore, we propose that a peak function, such as the Gaussian function, is a more appropriate choice to fit the data, see Fig. 5. As a result, the following Gaussian function is proposed to model pD
pD(θR)=p0+(1p0pBS)exp(0.5(θRθFwF)2)+pBS(θR),
where also a constant offset p0 is added. Furthermore, a function pBS is added in order to express the additional polarization taking place at the backscattering angle
pBS(θR)=ABSexp(0.5(|θR|180wB)2).
The increased polarization at the backscattering angle is only weakly present in the data in Fig. 5, due to the design of the current Mueller matrix apparatus, but the data of Germer et al. confirm this trend [2]. It is clear that more structure in the data may be present at the backscattering angle. It is noted that the model is strictly only valid for 90° − θiθR and θR ≤ −θi − 90°. Furthermore, the model does not capture deviations in pD for scattered light nearly parallel to the sample surface, as particularly seen from the data at 532 nm (Fig. 5(a)). Values for the fitted parameters for pD on the entire data set at 532 nm are given in Table 1, and the plot is shown in Fig. 5(a).

Tables Icon

Table 1. Fitted parameters for the polarization index (pD) and the depolarization factors, using Eq. (11) at 532 nm. The fitted data are shown in Figs. 5(a) and 5(b).

 

Fig. 5 Figures (a) and (b) show the depolarization index pD (a) and the depolarization factors α, β and γ (b) determined from forward product decomposition for five different incidence angles (θi = 0°, 30°, 45°, 60°, 75°) at λ =532 nm plotted against the Rayleigh scattering angle θR. Tentative fits are shown using the model in Eq. (11) and the parameters in Table 1. Figures (c) and (d) show the depolarization index pD (c) and the depolarization factor γ (d) determined from the forward product decomposition for two different incidence angles (θi = 30°, 75°) at λ = 1500 nm plotted against θR. The full and dotted lines in (c) show the fitted models for pD to the data at λ = 1500 nm (see Table 2) and λ = 532 nm, respectively. The full and dotted lines in (d) shows the fitted model for γ in Eq. (13) with the parameters in Table 2 for λ = 1500 nm and the model for λ = 532 nm, respectively.

Download Full Size | PPT Slide | PDF

A Gaussian lineshape was also observed for the pD data at 1500 nm, as seen in Fig. 5(c) (note that data were here only recorded for positive θR). It shows that pD from 532 nm (dashed line) is less polarized compared to 1500 nm (full line) for scattered light close to the surface plane. For the back scattering direction, 1500 nm is less polarized than 532 nm. Elsewhere, the differences appear small.

We will now develop an alternative parametric model for the Mueller matrices in Fig. 2 and in Fig. 4, by inspection of the decomposed Mueller matrices using the product decomposition, and looking for a mathematical form of the Mueller elements which may have a common physical interpretation. In particular, the forward polar decomposition allows to inspect the details of the retardance matrix MR, the diattenuation matrix MD and the elements of the depolarization matrix MΔ.

4.2. Simplified model for MΔ

The decomposed Mueller matrix was found to result in a close to diagonal mΔ. The other off-diagonal elements were therefore for simplicity neglected in the analysis. Let us inspect these diagonal components which we then for simplicity relate directly to α, β and γ of the ”diagonal” depolarization matrix at 532 nm (see Eq. (2)) in Fig. 5(b), where we also neglect the fact that we are not dealing with an ideal diffuser since the polarizance is non-zero [14]. The model in Eq. (11), was used to fit the full datasets for α, β and γ at 532 nm. The resulting parameters from the fits are given in Table 1. Again, it is noted that the parameters describing the back-scattering contribution are only indicative, since we have limited data in this region. The diagonal components α, β and γ of the depolarization matrix at 1500 nm involves another complication, as the measured matrix can be seen to change from a positive to a negative determinant as a function of the Rayleigh scattering angle. We have treated this issue using the recipe described by Ossikovski et al. [28]. However, we observe that at 1500 nm, γ has a different shape than at 532 nm, and at 1500 nm we had to use a Lorentzian peak function for a reasonable fit. As a result, the model in Eq. (11), was used to fit the full datasets for α, β at 1500 nm, while the following Lorentzian function was used to fit γ

p0+(1p0)[1+(θRθFwF)2]1.
The fitted parameters for α, β and γ for λ = 1500 nm are given in Table 2.

Tables Icon

Table 2. Fitted parameters for the depolarization factors α and β using Eq. (11) and γ using Eq. (13), at 1500 nm. The fits to pD and γ are shown in Fig. 5.

4.3. Simplified model for MR and MD matrices

Let us now analyze the retardance matrix, MR. First, inspecting the retardance in Eq. (4), one can observe a similar trend for the scattered light at 532 nm and at 1500 nm, see Fig. 6. In both cases, the retardance increases as a function of the Rayleigh scattering angle θR and crosses 90° after θR = 90°. There appears to be a definite observable difference between the two wavelengths. The jumps around 95° – 110° in the retardance at 1500 nm, are noise believed to be due to speckles.

 

Fig. 6 Retardance (given by Eq. (4)) at 532 nm (full squares) and 1500 nm (hollow triangles) with incident angle θi = 75° plotted as a function of the Rayleigh scattering angle θR. The two vertical lines indicate when the retardance crosses 90°. Note that θB (i.e. vertical lines) were determined from Fig. 7. The Figure also shows the function R ≈ arccos(C) (blue dotted line), with θB ≈ 7°.

Download Full Size | PPT Slide | PDF

Figures 7(a) and 7(c) shows the normalized mR33 and mR44 elements of the retardance matrix MR for all recorded angles of incidence. Figs. 7(b) and 7(d) shows the retardance data of mR34 and mR43 for all recorded angles of incidence. It is interesting to observe that mR33 and mR44 are nearly identical and follow the Rayleigh scattering model for all incidence angles, when analyzed as a function of θR

C=mR33=mR44=2cos(θRθB)1+cos2(θRθB),
except for an apparent shift of mR33 and mR44 of approximately θB ≈ 7° at 532 nm and θB ≈ 15° at 1500 nm as indicated by the vertical lines in Fig. 7. A similar functional form (without the shift) was used by White [29] and further proposed by Germer et al. [2] for modeling of the Spectralon. The shift θB, appears to be dispersive (i.e. a funtion of wavelength), although we find that a similar analysis of the Mueller matrix data reported by Germer et al. also gives a shift θB ≈ 7° at 632.8 nm. The shift θB is suggested related to a pseudo Brewster angle as a result of particle scattering [30]. It is also useful to observe that the total retardance R in Fig. 6 has a functional form well described by R ≈ arccos(C), particularly at 532 nm.

 

Fig. 7 Figures (a) at 532 nm and (c) at 1500 nm show the mR33 and mR44 elements from the retardance matrix MR plotted against the Rayleigh scattering angle θR. The simulated curve is given by Rayleigh scattering theory (see Eq. (14)). Figs. (b) at 532 nm and (d) at 1500 nm show the mR34 and mR43 elements from the retardance matrix MR plotted against θR. The simulated curve shows ±S (see Eq. (15)). The left Figs. (a) and (b) are data at 532 nm, with incident angles θi = 30°, 45°, 60°, 75°. The simulated data at 532 nm (full and dotted lines) use θB ≈ 7°. Figs. (c) and (d) are data at 1500 nm, with incidence angles θi = 30° and 75°, and simulated data with θB ≈ 15°. The vertical lines indicate the location of θB.

Download Full Size | PPT Slide | PDF

The mR34 and mR43 elements, which are normally zero in the Rayleigh scattering model, are proposed to have a related form

S=mR34=mR43=sin(θRθB)1+cos2(θRθB),
with a similar shift of θB ≈ 7° at 532 nm and θB ≈ 15° at 1500 nm. The origin of such a functional form could be the averaging of the Mie scattering from anisotropic shaped particles. On the other hand, one could also imagine that light scattered by particles will travel through a form-birefringent medium causing such a retardance. The retardance appears as the dominant effect, and its effect scales approximately with pDαβγ in the final Mueller matrix.

Having determined the basic functional forms for the lower right elements of the mR retardance sub-matrix, we may similarly try to parametrize the remaining elements of mR, see the experimental data in Fig. 8. Indeed, the function −0.02 + 0.17R ≈ −p0/3 + R/6, where R = arccos(C), gives a reasonable fit to the available data sets at 532 nm, for the mR23 element (note that mRij refers to the retardance matrix MR). Similarly, the function 0.02 – 0.23Np0/3 − N/4 fits the mR24 dataset, while mR22 ≈ 1. Adapting the symmetries mR23mR32 and mR24 ≈ −mR42, we obtain a reasonable model for the entire retardance sub-matrix. The experimental decomposed retardance sub-matrix at 532 nm and incidence angles θi = 30°, 45°, 60° and 75° is shown together with the simulations in Fig. 8. This model for the retardance sub-matrix appears reasonable, and a similar model also appears suitable at 1500 nm.

 

Fig. 8 The experimental retardance sub-matrix mR (symbols) and the simulated mR (full lines), both plotted against the Rayleigh scattering angle θR. The experimental data are at 532 nm, with incident angles θi = 30°, 45°, 60° and 75°. The simulated data use θB ≈ 7°, and the basis functions N, C and S in addition to p0. The notation mRij refers to the retardance matrix MR.

Download Full Size | PPT Slide | PDF

There are other interesting features in the diattenuation and the polarizance vectors, particularly in the m12 and m21 elements, of the original measured Mueller matrix, which appear to scale as pD2α2. Upon searching for the Rayleigh type functional form for the m12 and m21 elements, it was observed that

m12+p0/24pD2sin2(θRθB)1+cos2(θRθB)=N.
The offset angle θB is found to be similar, although not identical, to the corresponding angle for the retardance matrix. Although both N, S and C appear as reasonable functional forms to describe the full data sets, there are certain problems with the periodicity of these functions as a function of θR. We therefore believe that a further improved model must properly include other nonlinear effects in both θR and θB as a result of refraction, and thus as a function of incidence and the Rayleigh scattering angle. Finally, it is noted that the diattenuation sub-matrix was found to be close to the identity matrix, and any variations therein was neglected in the development of the parametric model.

4.4. Final parametric model for the Mueller matrix at 532 and 1500 nm

We now return to the original Mueller matrix, and seek a parametric model using the functional forms based on α, β and γ, in addition to N, C and S as determined in the above decomposition analysis. We propose that a simplified parametric model for the Mueller matrix for the Spectralon, can now be obtained using polar product decomposition. First, let pd[12p04α2N,0,0]T, mΔ ≈ diag(α, β, γ), mD ≈ diag(1, 1, 1) and mR[0000CS0SC], giving the MΔ, MR and MD matrices in Eq. (2). A direct application of the forward decomposition, i.e. using Eq. (1), will lead to additional terms of the order α|d|, |d|2 and smaller, and further p is no longer identical to d as initially assumed. This is an artefact due to the simplifications in the parametrisation of the decomposed matrices. We therefore argue that the observed symmetry of the experimental Mueller matrix must remain, and that the Mueller matrix should also remain physically realizable in the forward scattering direction. As a result, these additional terms can be neglected. The latter approximation gives identical results to simply letting p = 0 in MΔ, prior to the matrix multiplication, and a back-substitution of p = d into the final Mueller matrix. The latter approach will result in the following simplified Mueller matrix with a certain physical basis

MM11[112p04α2N0012p04α2Nα0000βCβS00γSγC],
where C, S and N are defined as in Eqs. (14), (15) and (16), respectively. We similarly propose a complete parametric model using the above basis functions, that will better fit also the off diagonal elements of the Spectralon Mueller matrices. From trial an error, we parametrize pd[12p04α2N,12p02α2N,13p0]T. We use mΔ ≈ diag(α, β, γ) and mD ≈ diag(1, 1, 1) as above, while the retardance sub-matrix is refined to
mR[113p0+R613p012Nsinχ13p0+R6CS13p0+12NsinχSC],
where R = arccos(C). Furthermore, the oscillation of the m24 and the −m42 elements have been introduced in the model as a function of χ. Hence, the matrix product of Eq. (1), with p = 0, followed with a back-substitution of p = d in the resulting Mueller matrix, result in the following Mueller matrix
M=M11[112p04α2N12p02α2N13p012p04α2Nαα(13p016R)α(13p012Nsinχ)13p0γ(13p0+12Nsinχ)14p0γSγC].
We get two sets of parameters, one for each wavelength. For 532 nm, the parameters for the depolarization factors are given in Table 1, with θB ≈ 7°. For 1500 nm, the parameters for the depolarization factors are given in Table 2, and θB ≈ 15°. Note that the m14 and m41 elements are negligible at 1500 nm, and that a small constant ±p0/4 has been added to the m34 and m43 elements, for a minor adjustment to the experimental data. We have used χ = 30° for the modelling of both wavelengths in Figs. 2 and 4. Similarly, one could propose a small, although more uncertain, oscillation in the m34 and m43 elements.

The Mueller matrix resulting from the parametric model in Eq. (19), is plotted in Fig. 2 together with the experimental data. It is observed, that there is a reasonable correspondence to the range of obtained experimental data. There is a distinct improvement in the correspondence between the model and the experimental results compared to previous work [1], although more parameters have been introduced in the new model. The modelled Mueller matrix for 1500 nm is shown in Fig. 4 together with the data at 532 nm. Only weak differences are observed between the two wavelengths, e.g. m14 and m41 have a much smaller offset at 1500 nm, and m44 is larger at 1500 nm. A self consistency-check was performed by recalculating the polarization index pD using Eq. (8) for both wavelengths. This recalculated polarization index was found to be in excellent agreement with both the experimental data and the Gaussian model in Eq. (11), except for the already mentioned deviation for scattering near parallel to the surface.

The parametric model indicate that Mie and Rayleigh scattering appears as an important process for the Spectralon reflectance standard. It is interesting that the complex form of the retardance matrix can be derived from the basis functions N, C and S. However, as with Rayleigh particles, little direct information about the particles themselves are projected in the current model, except for possibly the shift angle θB, which may be regarded as a pseudo-Brewster angle, and thus be related to either the average material properties or the optical properties of the scatterers [30]. The apparent dispersion of θB supports the latter hypothesis. It is striking that the m12 and the m21 elements are apparently scaling with pD2, possibly indicating a multiple scattering process. There are experimental uncertainties with respect to the constant offsets in the measured Mueller matrix (particularly m13 and m14), but we have not been able to relate these to any evident experimental artefacts. As in [1], the m14 and m41 elements have been modelled by constants.

Finally, to reduce the number of free parameters, the result in Eq. (19) can be put into a reduced form

M=M11[14pD2N2pD2N04pD2NpDpD16RpD12Nsinχ2pD2NpD16RpDCpDS0pD12NsinχpDSpDC],
where pDpCαβγ and p0 has been neglected. It is now clearly observed that the lower right 3 × 3 matrix is basically the retardance sub-matrix multiplied with the average depolarisation factor. To complete the discussion, we can now write the matrix as the sum of an ideal diffuser scaling with the average degree of depolarization, and a correction matrix scaling with the average degree of polarization
M=M11((1pD)diag(1,0,0,0)+pD[14pDN2pDN04pDN116R12Nsinχ2pDN16RCS012NsinχSC]),
i.e. in a similar form to the one previously reported [1], but with basis functions with an improved physical insight.

It appears interesting to study other well determined diffuse samples to investigate whether the above product decomposition method and in particular, the parametric model found for the Spectralon, may be applied to other strongly scattering systems, such as rough surfaces and powder samples. The exact dependency of incidence angle is expected to be revealed with a combination of more data and improved models. As a result, it is envisaged that more information about the scatterers may be extracted from the normalized Mueller matrix.

Preliminary studies of the current dataset, comparing the product decomposition and the current formulation of the differential decomposition [12, 16], give highly similar results for α, β, γ and the linear retardance in the range 0 ≤ θR ≤ 90°, while it is interesting to observe that the current formulation of the differential decomposition method must be revised for θR > 90°. Indeed, this is as expected since the current form of the differential decomposition is only valid for forward scattering. A revised version of the differential decomposition method [12, 16] could be a more appropriate tool for analysis of the scattering Mueller matrix for the Spectralon.

Finally, it has been proposed to further refine/validate the parametric model for the off-diagonal elements using a few higher order terms in the actual Mie solution. This approach would definitively result in a model based on stronger physical ground, but is out of the scope of the current work.

5. Conclusion

A new parametric model for the Mueller matrix of a Spectralon white reflectance standard diffuser has been determined through the use of polar (product) decomposition techniques. New Mueller matrix data at the wavelengths 532 nm and 1500 nm are presented and analyzed. It was found particularly useful to study the Spectralon in terms of the standard ”particle” or Rayleigh scattering angle, which made the depolarization index and the Mueller matrix elements close to independent of the incidence angle. The basic building blocks in the new model is a Gaussian parametric model for the depolarization index, multiplied with Rayleigh like functional forms deduced from the decomposed matrices. A pseudo Brewster angle appeared dispersive and characteristic of the material. The use of decomposition techniques appears promising in order to reveal the basic functional building blocks of a given scattering Mueller matrix, and it is envisaged that the methodology presented here can be used to study other strongly diffuse samples. Finally, a well characterized Mueller matrix of the Spectralon makes it also useful as a calibration standard for polarization sensitive systems.

Acknowledgment

L.M.S. Aas and J. Maria acknowledge support from The Norwegian Research Center for Solar Cell Technology (project num. 193829).

References and links

1. Ø. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and Ø. Frette, “Mueller matrix measurements and modeling pertaining to spectralon white reflectance standards,” Opt. Express 20, 15045–15053 (2012) [CrossRef]   [PubMed]  .

2. T. A. Germer and H. J. Patrick, “Mueller matrix bidirectional reflectance distribution function measurements and modeling of diffuse reflectance standards,” Proc. SPIE 8160, 81600D (2011) [CrossRef]  .

3. D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of spectralon illuminated by coherent light,” Appl. Opt. 38, 6350–6356 (1999) [CrossRef]  .

4. A. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. 50, 2431–2442 (2011) [CrossRef]   [PubMed]  .

5. T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817208 (2011).

6. I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010) [CrossRef]   [PubMed]  .

7. I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010) [CrossRef]  .

8. N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana 75, 1071–1086 (2010) [CrossRef]  .

9. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16, 110801 (2011) [CrossRef]   [PubMed]  .

10. A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE 7895, 78950E (2011) [CrossRef]  .

11. V. V. Tuchin, L. V. Wang, and D. A. Zimnyakov, Optical Polarization in Biomedical Applications (Berlin Heidelberg, 2006).

12. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics 17, 105006–105006 (2012) [CrossRef]   [PubMed]  .

13. R. Ossikovski, A. D. Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing mueller matrices,” Opt. Lett. 32, 689–691 (2007) [CrossRef]   [PubMed]  .

14. R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a) 205, 720–727 (2008) [CrossRef]  .

15. R. Ossikovski, “Analysis of depolarizing mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009) [CrossRef]  .

16. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36, 2330–2332 (2011) [CrossRef]   [PubMed]  .

17. S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996) [CrossRef]  .

18. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006) [CrossRef]   [PubMed]  .

19. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition.” Opt. lett. 36, 1942–4 (2011) [CrossRef]   [PubMed]  .

20. H. D. Noble and R. a. Chipman, “Mueller matrix roots algorithm and computational considerations.” Opt. Express 20, 17–31 (2012) [CrossRef]   [PubMed]  .

21. H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics 51, 735–744 (2012) [CrossRef]  .

22. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles Interscience (J. Wiley & Sons, New York, 1983).

23. P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci. 96, 81–107 (1980) [CrossRef]  .

24. R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

25. J. J. Gil and E. Bernabeu, “A depolarization criterion in mueller matrices,” Opt. Acta 32, 259–261 (1985) [CrossRef]  .

26. S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989) [CrossRef]  .

27. F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng. 47, 073604–073604–9 (2008) [CrossRef]  .

28. R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun. 281, 2406–2410 (2008) [CrossRef]  .

29. R. L. White, “Polarization in reflection nebulae. I - Scattering properties of interstellar grains,” Astrophys. J. 229, 954–961 (1979) [CrossRef]  .

30. A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E 73, 031406 (2006) [CrossRef]  .

References

  • View by:
  • |
  • |
  • |

  1. Ø. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and Ø. Frette, “Mueller matrix measurements and modeling pertaining to spectralon white reflectance standards,” Opt. Express20, 15045–15053 (2012).
    [CrossRef] [PubMed]
  2. T. A. Germer and H. J. Patrick, “Mueller matrix bidirectional reflectance distribution function measurements and modeling of diffuse reflectance standards,” Proc. SPIE8160, 81600D (2011).
    [CrossRef]
  3. D. A. Haner, B. T. McGuckin, and C. J. Bruegge, “Polarization characteristics of spectralon illuminated by coherent light,” Appl. Opt.38, 6350–6356 (1999).
    [CrossRef]
  4. A. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt.50, 2431–2442 (2011).
    [CrossRef] [PubMed]
  5. T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).
  6. I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104, 223904 (2010).
    [CrossRef] [PubMed]
  7. I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A81, 013806 (2010).
    [CrossRef]
  8. N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
    [CrossRef]
  9. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt.16, 110801 (2011).
    [CrossRef] [PubMed]
  10. A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
    [CrossRef]
  11. V. V. Tuchin, L. V. Wang, and D. A. Zimnyakov, Optical Polarization in Biomedical Applications (Berlin Heidelberg, 2006).
  12. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
    [CrossRef] [PubMed]
  13. R. Ossikovski, A. D. Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing mueller matrices,” Opt. Lett.32, 689–691 (2007).
    [CrossRef] [PubMed]
  14. R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
    [CrossRef]
  15. R. Ossikovski, “Analysis of depolarizing mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A26, 1109–1118 (2009).
    [CrossRef]
  16. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett.36, 2330–2332 (2011).
    [CrossRef] [PubMed]
  17. S.-Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A13, 1106–1113 (1996).
    [CrossRef]
  18. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express14, 190–202 (2006).
    [CrossRef] [PubMed]
  19. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition.” Opt. lett.36, 1942–4 (2011).
    [CrossRef] [PubMed]
  20. H. D. Noble and R. a. Chipman, “Mueller matrix roots algorithm and computational considerations.” Opt. Express20, 17–31 (2012).
    [CrossRef] [PubMed]
  21. H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics51, 735–744 (2012).
    [CrossRef]
  22. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles Interscience (J. Wiley & Sons, New York, 1983).
  23. P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci.96, 81–107 (1980).
    [CrossRef]
  24. R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).
  25. J. J. Gil and E. Bernabeu, “A depolarization criterion in mueller matrices,” Opt. Acta32, 259–261 (1985).
    [CrossRef]
  26. S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE1166, 177–185 (1989).
    [CrossRef]
  27. F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng.47, 073604–073604–9 (2008).
    [CrossRef]
  28. R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun.281, 2406–2410 (2008).
    [CrossRef]
  29. R. L. White, “Polarization in reflection nebulae. I - Scattering properties of interstellar grains,” Astrophys. J.229, 954–961 (1979).
    [CrossRef]
  30. A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E73, 031406 (2006).
    [CrossRef]

2012

Ø. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and Ø. Frette, “Mueller matrix measurements and modeling pertaining to spectralon white reflectance standards,” Opt. Express20, 15045–15053 (2012).
[CrossRef] [PubMed]

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

H. D. Noble and R. a. Chipman, “Mueller matrix roots algorithm and computational considerations.” Opt. Express20, 17–31 (2012).
[CrossRef] [PubMed]

H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics51, 735–744 (2012).
[CrossRef]

2011

R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett.36, 2330–2332 (2011).
[CrossRef] [PubMed]

N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition.” Opt. lett.36, 1942–4 (2011).
[CrossRef] [PubMed]

T. A. Germer and H. J. Patrick, “Mueller matrix bidirectional reflectance distribution function measurements and modeling of diffuse reflectance standards,” Proc. SPIE8160, 81600D (2011).
[CrossRef]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt.16, 110801 (2011).
[CrossRef] [PubMed]

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

A. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt.50, 2431–2442 (2011).
[CrossRef] [PubMed]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

2010

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104, 223904 (2010).
[CrossRef] [PubMed]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A81, 013806 (2010).
[CrossRef]

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

2009

2008

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng.47, 073604–073604–9 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun.281, 2406–2410 (2008).
[CrossRef]

2007

2006

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express14, 190–202 (2006).
[CrossRef] [PubMed]

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E73, 031406 (2006).
[CrossRef]

1999

1996

1989

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE1166, 177–185 (1989).
[CrossRef]

1985

J. J. Gil and E. Bernabeu, “A depolarization criterion in mueller matrices,” Opt. Acta32, 259–261 (1985).
[CrossRef]

1980

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci.96, 81–107 (1980).
[CrossRef]

1979

R. L. White, “Polarization in reflection nebulae. I - Scattering properties of interstellar grains,” Astrophys. J.229, 954–961 (1979).
[CrossRef]

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun.281, 2406–2410 (2008).
[CrossRef]

Antonelli, M. R.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

Arce-Diego, J. L.

Azzam, R.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

Bashara, N.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

Ben Hatit, S.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

Benali, A.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

Bernabeu, E.

J. J. Gil and E. Bernabeu, “A depolarization criterion in mueller matrices,” Opt. Acta32, 259–261 (1985).
[CrossRef]

Bhandari, A.

Bohren, C.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles Interscience (J. Wiley & Sons, New York, 1983).

Bruegge, C. J.

Buddhiwant, P.

Chipman, R.

H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics51, 735–744 (2012).
[CrossRef]

Chipman, R. a.

Cloude, S. R.

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE1166, 177–185 (1989).
[CrossRef]

De Martino, A.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

Erbe, A.

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E73, 031406 (2006).
[CrossRef]

Frette, Ø.

Garcia-Caurel, E.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

Gayet, B.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

Germer, T. A.

T. A. Germer and H. J. Patrick, “Mueller matrix bidirectional reflectance distribution function measurements and modeling of diffuse reflectance standards,” Proc. SPIE8160, 81600D (2011).
[CrossRef]

Ghosh, N.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt.16, 110801 (2011).
[CrossRef] [PubMed]

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express14, 190–202 (2006).
[CrossRef] [PubMed]

Gil, J. J.

J. J. Gil and E. Bernabeu, “A depolarization criterion in mueller matrices,” Opt. Acta32, 259–261 (1985).
[CrossRef]

Gupta, P. K.

Guyot, S.

Hamre, B.

Haner, D. A.

Hauge, P.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci.96, 81–107 (1980).
[CrossRef]

Huffman, D.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles Interscience (J. Wiley & Sons, New York, 1983).

Kildemo, M.

Kumar, S.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

Leskova, T. A.

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A81, 013806 (2010).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104, 223904 (2010).
[CrossRef] [PubMed]

Letnes, P. A.

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

Lindgren, M.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng.47, 073604–073604–9 (2008).
[CrossRef]

Lu, S.-Y.

Manhas, S.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express14, 190–202 (2006).
[CrossRef] [PubMed]

Maradudin, A. A.

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104, 223904 (2010).
[CrossRef] [PubMed]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A81, 013806 (2010).
[CrossRef]

Maria, J.

Martino, A. D.

R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun.281, 2406–2410 (2008).
[CrossRef]

R. Ossikovski, A. D. Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing mueller matrices,” Opt. Lett.32, 689–691 (2007).
[CrossRef] [PubMed]

McClain, S.

H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics51, 735–744 (2012).
[CrossRef]

McGuckin, B. T.

Muller, R.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci.96, 81–107 (1980).
[CrossRef]

Nerbø, I. S.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng.47, 073604–073604–9 (2008).
[CrossRef]

Noble, H.

H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics51, 735–744 (2012).
[CrossRef]

Noble, H. D.

Nordam, T.

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

Novikova, T.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

Ortega-Quijano, N.

Ossikovski, R.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett.36, 2330–2332 (2011).
[CrossRef] [PubMed]

R. Ossikovski, “Analysis of depolarizing mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A26, 1109–1118 (2009).
[CrossRef]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun.281, 2406–2410 (2008).
[CrossRef]

R. Ossikovski, A. D. Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing mueller matrices,” Opt. Lett.32, 689–691 (2007).
[CrossRef] [PubMed]

Patrick, H. J.

T. A. Germer and H. J. Patrick, “Mueller matrix bidirectional reflectance distribution function measurements and modeling of diffuse reflectance standards,” Proc. SPIE8160, 81600D (2011).
[CrossRef]

Pierangelo, A.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

Purwar, H.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

Sigel, R.

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E73, 031406 (2006).
[CrossRef]

Simonsen, I.

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104, 223904 (2010).
[CrossRef] [PubMed]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A81, 013806 (2010).
[CrossRef]

Singh, J.

Smith, C.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci.96, 81–107 (1980).
[CrossRef]

Soni, J.

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

Stabo-Eeg, F.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng.47, 073604–073604–9 (2008).
[CrossRef]

Stamnes, J. J.

Svensen, Ø.

Swami, M. K.

Tauer, K.

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E73, 031406 (2006).
[CrossRef]

Tuchin, V. V.

V. V. Tuchin, L. V. Wang, and D. A. Zimnyakov, Optical Polarization in Biomedical Applications (Berlin Heidelberg, 2006).

Validire, P.

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

Vitkin, I.

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

Vitkin, I. A.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt.16, 110801 (2011).
[CrossRef] [PubMed]

Wallenberg, M.

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

Wang, L. V.

V. V. Tuchin, L. V. Wang, and D. A. Zimnyakov, Optical Polarization in Biomedical Applications (Berlin Heidelberg, 2006).

White, R. L.

R. L. White, “Polarization in reflection nebulae. I - Scattering properties of interstellar grains,” Astrophys. J.229, 954–961 (1979).
[CrossRef]

Wood, M.

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

Zhao, L.

Zimnyakov, D. A.

V. V. Tuchin, L. V. Wang, and D. A. Zimnyakov, Optical Polarization in Biomedical Applications (Berlin Heidelberg, 2006).

Appl. Opt.

Appl. Optics

H. Noble, S. McClain, and R. Chipman, “Mueller matrix roots depolarization parameters,” Appl. Optics51, 735–744 (2012).
[CrossRef]

Astrophys. J.

R. L. White, “Polarization in reflection nebulae. I - Scattering properties of interstellar grains,” Astrophys. J.229, 954–961 (1979).
[CrossRef]

J. Biomed. Opt.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt.16, 110801 (2011).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Journal of Biomedical Optics

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” Journal of Biomedical Optics17, 105006–105006 (2012).
[CrossRef] [PubMed]

Opt. Acta

J. J. Gil and E. Bernabeu, “A depolarization criterion in mueller matrices,” Opt. Acta32, 259–261 (1985).
[CrossRef]

Opt. Commun.

R. Ossikovski, M. Anastasiadou, and A. D. Martino, “Product decompositions of depolarizing mueller matrices with negative determinants,” Opt. Commun.281, 2406–2410 (2008).
[CrossRef]

Opt. Eng.

F. Stabo-Eeg, M. Kildemo, I. S. Nerbø, and M. Lindgren, “Well-conditioned multiple laser mueller matrix ellipsometer,” Opt. Eng.47, 073604–073604–9 (2008).
[CrossRef]

Opt. Express

Opt. lett.

Phys. Rev. A

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: The full angular intensity distribution,” Phys. Rev. A81, 013806 (2010).
[CrossRef]

Phys. Rev. E

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E73, 031406 (2006).
[CrossRef]

Phys. Rev. Lett.

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett.104, 223904 (2010).
[CrossRef] [PubMed]

Phys. Status Solidi (a)

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing mueller matrices: how to decompose them?” Phys. Status Solidi (a)205, 720–727 (2008).
[CrossRef]

Pramana

N. Ghosh, J. Soni, M. Wood, M. Wallenberg, and I. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana75, 1071–1086 (2010).
[CrossRef]

Proc. SPIE

A. Pierangelo, S. Manhas, A. Benali, M. R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Use of mueller polarimetric imaging for the staging of human colon cancer,” Proc. SPIE7895, 78950E (2011).
[CrossRef]

T. A. Germer and H. J. Patrick, “Mueller matrix bidirectional reflectance distribution function measurements and modeling of diffuse reflectance standards,” Proc. SPIE8160, 81600D (2011).
[CrossRef]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE8172, 817208 (2011).

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE1166, 177–185 (1989).
[CrossRef]

Surf. Sci.

P. Hauge, R. Muller, and C. Smith, “Conventions and formulas for using the mueller-stokes calculus in ellipsometry,” Surf. Sci.96, 81–107 (1980).
[CrossRef]

Other

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles Interscience (J. Wiley & Sons, New York, 1983).

V. V. Tuchin, L. V. Wang, and D. A. Zimnyakov, Optical Polarization in Biomedical Applications (Berlin Heidelberg, 2006).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometry for the scattering experiment, defining the incidence angle θi, the Rayleigh scattering angle θR and the Rayleigh scattering angle θS used in [1]. The figure also shows the right handed coordinate system {, ŷ, } used in the paper, where = − , ŷ =ŝ with reference to the right handed system {ŝ, , }. The arrows define positive direction.

Fig. 2
Fig. 2

The measured Mueller matrix data from the Spectralon surface with illumination at incidence angles θi of 0° (green x marks), 30° (purple crosses), 45° (blue circles), 60° (red stars), 75° (cyan squares) plotted as a function of the Rayleigh scattering angle θR. The full curves show the fitted parametric model in Eq. (19). All elements, except M11, are normalized to M11.

Fig. 3
Fig. 3

The difference of the off-diagonal elements (m24m42), Δm33m33 − 〈m33〉 and Δ(m34m43) ≡ (m34 − 〈 m34〉) − (m43 − 〈 m43〉), where 〈mij〉 are the averages upon a full 360 degrees rotation, at 532 nm. The data are plotted against the azimuthal angle of the Spectralon for incidence angle 75° and Rayleigh scattering angle 35°.

Fig. 4
Fig. 4

Measured Mueller matrix data from the Spectralon surface with illumination at an incidence angle of θi = 75° and wavelengths 532 nm (red triangles) and 1500 nm (blue crosses) plotted against the Rayleigh scattering angle θR. The full curve shows the fitted parametric model at 1500 nm in Eq. (19). All elements are normalized to M11.

Fig. 5
Fig. 5

Figures (a) and (b) show the depolarization index pD (a) and the depolarization factors α, β and γ (b) determined from forward product decomposition for five different incidence angles (θi = 0°, 30°, 45°, 60°, 75°) at λ =532 nm plotted against the Rayleigh scattering angle θR. Tentative fits are shown using the model in Eq. (11) and the parameters in Table 1. Figures (c) and (d) show the depolarization index pD (c) and the depolarization factor γ (d) determined from the forward product decomposition for two different incidence angles (θi = 30°, 75°) at λ = 1500 nm plotted against θR. The full and dotted lines in (c) show the fitted models for pD to the data at λ = 1500 nm (see Table 2) and λ = 532 nm, respectively. The full and dotted lines in (d) shows the fitted model for γ in Eq. (13) with the parameters in Table 2 for λ = 1500 nm and the model for λ = 532 nm, respectively.

Fig. 6
Fig. 6

Retardance (given by Eq. (4)) at 532 nm (full squares) and 1500 nm (hollow triangles) with incident angle θi = 75° plotted as a function of the Rayleigh scattering angle θR. The two vertical lines indicate when the retardance crosses 90°. Note that θB (i.e. vertical lines) were determined from Fig. 7. The Figure also shows the function R ≈ arccos(C) (blue dotted line), with θB ≈ 7°.

Fig. 7
Fig. 7

Figures (a) at 532 nm and (c) at 1500 nm show the mR33 and mR44 elements from the retardance matrix MR plotted against the Rayleigh scattering angle θR. The simulated curve is given by Rayleigh scattering theory (see Eq. (14)). Figs. (b) at 532 nm and (d) at 1500 nm show the mR34 and mR43 elements from the retardance matrix MR plotted against θR. The simulated curve shows ±S (see Eq. (15)). The left Figs. (a) and (b) are data at 532 nm, with incident angles θi = 30°, 45°, 60°, 75°. The simulated data at 532 nm (full and dotted lines) use θB ≈ 7°. Figs. (c) and (d) are data at 1500 nm, with incidence angles θi = 30° and 75°, and simulated data with θB ≈ 15°. The vertical lines indicate the location of θB.

Fig. 8
Fig. 8

The experimental retardance sub-matrix mR (symbols) and the simulated mR (full lines), both plotted against the Rayleigh scattering angle θR. The experimental data are at 532 nm, with incident angles θi = 30°, 45°, 60° and 75°. The simulated data use θB ≈ 7°, and the basis functions N, C and S in addition to p0. The notation mRij refers to the retardance matrix MR.

Tables (2)

Tables Icon

Table 1 Fitted parameters for the polarization index (pD) and the depolarization factors, using Eq. (11) at 532 nm. The fitted data are shown in Figs. 5(a) and 5(b).

Tables Icon

Table 2 Fitted parameters for the depolarization factors α and β using Eq. (11) and γ using Eq. (13), at 1500 nm. The fits to pD and γ are shown in Fig. 5.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

M = M Δ M R M D ,
M Δ = M 11 [ 1 0 T p m Δ ] , M D = [ 1 d T d m D ] , M R = [ 1 0 T 0 m R ] .
p C = 1 3 ( α + β + γ ) .
R = cos 1 ( Tr ( M R ) 1 2 ) .
M D R = M 11 [ 1 m 12 0 0 m 21 1 0 0 0 0 m 33 m 34 0 0 m 43 m 44 ] .
m 12 = m 21 = cos ( 2 Ψ ) , m 33 = m 44 = sin ( 2 Ψ ) cos Δ , m 34 = m 43 = sin ( 2 Ψ ) sin Δ .
m 12 = m 21 = sin 2 ( θ R ) 1 + cos 2 ( θ R ) , m 33 = m 44 = 2 cos ( θ R ) 1 + cos 2 ( θ R ) , m 34 = m 43 = 0.
p D = ( Tr ( M T M ) M 11 2 3 M 11 2 ) 1 / 2 = ( i , j = 1 4 M i j 2 M 11 2 3 M 11 2 ) 1 / 2 .
S = [ I x ^ + I y ^ I x ^ I y ^ I x ^ + y ^ I x ^ y ^ I R C I L C ] .
p D = | ( θ S + θ i ) 3 22 | = | ( 180 θ R ) 3 22 | .
p D ( θ R ) = p 0 + ( 1 p 0 p B S ) exp ( 0.5 ( θ R θ F w F ) 2 ) + p B S ( θ R ) ,
p B S ( θ R ) = A B S exp ( 0.5 ( | θ R | 180 w B ) 2 ) .
p 0 + ( 1 p 0 ) [ 1 + ( θ R θ F w F ) 2 ] 1 .
C = m R 33 = m R 44 = 2 cos ( θ R θ B ) 1 + cos 2 ( θ R θ B ) ,
S = m R 34 = m R 43 = sin ( θ R θ B ) 1 + cos 2 ( θ R θ B ) ,
m 12 + p 0 / 2 4 p D 2 sin 2 ( θ R θ B ) 1 + cos 2 ( θ R θ B ) = N .
M M 11 [ 1 1 2 p 0 4 α 2 N 0 0 1 2 p 0 4 α 2 N α 0 0 0 0 β C β S 0 0 γ S γ C ] ,
m R [ 1 1 3 p 0 + R 6 1 3 p 0 1 2 N sin χ 1 3 p 0 + R 6 C S 1 3 p 0 + 1 2 N sin χ S C ] ,
M = M 11 [ 1 1 2 p 0 4 α 2 N 1 2 p 0 2 α 2 N 1 3 p 0 1 2 p 0 4 α 2 N α α ( 1 3 p 0 1 6 R ) α ( 1 3 p 0 1 2 N sin χ ) 1 3 p 0 γ ( 1 3 p 0 + 1 2 N sin χ ) 1 4 p 0 γ S γ C ] .
M = M 11 [ 1 4 p D 2 N 2 p D 2 N 0 4 p D 2 N p D p D 1 6 R p D 1 2 N sin χ 2 p D 2 N p D 1 6 R p D C p D S 0 p D 1 2 N sin χ p D S p D C ] ,
M = M 11 ( ( 1 p D ) diag ( 1 , 0 , 0 , 0 ) + p D [ 1 4 p D N 2 p D N 0 4 p D N 1 1 6 R 1 2 N sin χ 2 p D N 1 6 R C S 0 1 2 N sin χ S C ] ) ,

Metrics