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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 12794–12806
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Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil  »View Author Affiliations


Optics Express, Vol. 17, Issue 15, pp. 12794-12806 (2009)
http://dx.doi.org/10.1364/OE.17.012794


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Abstract

The measurement of the Mueller matrix when the probing beam is placed on the boundary between two (or more) regions of the sample with different optical properties may lead to a depolarization in the Mueller matrix. The depolarization is due to the incoherent superposition of the optical responses of different sample regions in the probe beam. Despite of the depolarization, the measured Mueller matrix has information enough to subtract a Mueller matrix corresponding to one of the regions of sample provided that this subtracted matrix is non-depolarizing. For clarity, we will call these non-depolarizing Mueller matrices of one individual region of the sample simply as the non-depolarizing components. In the framework of the theory of Mueller matrix algebra, we have implemented a procedure allowing the retrieval of a non-depolarizing component from a depolarizing Mueller matrix constituted by the sum of several non-depolarizing components. In order to apply the procedure, the Mueller matrices of the rest of the non-depolarizing components have to be known. Here we present a numerical and algebraic approaches to implement the subtraction method. To illustrate our method as well as the performance of the two approaches, we present two practical examples. In both cases we have measured depolarizing Mueller matrices by positioning an illumination beam on the boundary between two and three different regions of a sample, respectively. The goal was to retrieve the non-depolarizing Mueller matrix of one of those regions from the measured depolarizing Mueller matrix. In order to evaluate the performance of the method we compared the subtracted matrix with the Mueller matrix of the selected region measured separately.

© 2009 Optical Society of America

1. Introduction

The possibility of the retrieval of a non-depolarizing component from a depolarizing Mueller matrix has been pointed out theoretically in the framework of the theory of Mueller matrix algebra introduced by Gil and Correas [4

4. J. J. Gil, “Polarimetric characterization of light and media Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007). [CrossRef]

, 5

5. J. M. Correas, P. Melero, and J. J. Gil, “Decomposition of Mueller matrices in pure optical media,” Mon. Sem. Mat. García de Galdeano 27, 233–240 (2003).

]. Subsequent application of this approach allows subtracting theoretically up to three known non-depolarizing components from measured matrix and getting the last unknown component. Limits are only given by the internal structure of individual Mueller matrices and the presence of noise or systematic errors in experimental data. Our method is based on the same theoretical background as the decomposition of an arbitrary Mueller matrix into the linear combination of four non-depolarizing matrices published by Cloude [6

6. S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

], though there is infinite number of such matrix foursomes without direct connection to the physical matrices.

2. Theory

Polarized light can be described by four components Stokes vectors [7

7. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

] in the form:

S=[S1S2S3S4]=[Ex2+Ey2Ex2Ey22Re(ExEy)2Im(ExEy)],
(1)

where 〈x〉 represents time averaging over the measurement time and Ex and Ey denote the orthogonal field components perpendicular to the direction of propagation. The polarimetric response of any sample can be fully described by 4×4 Mueller matrix formalism [7

7. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

] describing the relation between incident and emerging Stokes vectors:

Sem=MSin=[M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44]Sin.
(2)

In practice, polarimeters usually measure normalized Mueller matrices, where all the elements are divided by the element M 11. The necessary and sufficient condition for a physically realizable Mueller matrix to represent a non-depolarizing optical system, introduced by Gil and Bernabeu [8

8. J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).

], can be written as

Tr(MTM)=i,j=14Mij2=4M112.
(3)

H=14[M11+M22+M13+M23+M31+M32M33+M44+M12+M21i(M14+M24)i(M41+M42)i(M34M43)M13+M23M11M22M33M44M31M32i(M14+M24)M12+M21i(M34+M43)i(M41M42)M31+M32+M33M44+M11M22+M13M23+i(M41+M42)i(M34+M43)M12M21i(M14M24)M33+M44M31M32+M13M23M11+M22i(M34M43)i(M41M42)i(M14M24)M12M21].
(4)

Henceforth we denote the linear transformation of a Mueller matrix M to the corresponding coherency matrix H by the linear operator 𝓗 such that H=𝓗(M). The rank of the coherency matrix H, i.e. the number of non-zero eigenvalues, determines the number of non-depolarizing components contained in measured Mueller matrix M. Moreover, for the physically realizable matrices all eigenvalues have to be non-negative since H is hermitian positive semi-definite matrix. We have developed our method for the retrieval of non-depolarizing components from measured matrices in the framework of coherency matrices. In the following, we will show how the method works applying it to the simplest case, i.e. the retrieval of a component from a depolarizing Mueller matrix produced by the sum of two non-depolarizing components provided that one of the components is known. After that we will show how to generalize the method to treat more complex cases.

A normalized depolarizing Mueller matrix M′ made of the sum of two normalized non-depolarizing matrix components M (1) and M (2) can be written in form

M=11+p(M(1)+pM(2)),
(5)

where the coefficient p denotes the relative proportion of the two non-depolarizing matrices. In this conditions the coherency matrix H′=𝓗(M′) associated to M′ has two non-zero eigenvalues (rank(H′)=2) if the matrices M (1) and M (2) are linearly independent. Our method is based on the assumption that one of the non-depolarizing matrices of Eq. (5), M (1) for instance, is known. Under this condition it is possible to find unique real number α such that [5

5. J. M. Correas, P. Melero, and J. J. Gil, “Decomposition of Mueller matrices in pure optical media,” Mon. Sem. Mat. García de Galdeano 27, 233–240 (2003).

]

rank[𝓗(M)-α𝓗(M(1))]=1.
(6)

The value of the parameter α can be obtained either using an analytical approach with the expression

α=4M112Tr(MTM)8M11M11(1)Tr(M(1)TM+MTM(1)),
(7)

based on the condition in Eq. (3), or using a numerical approach. The numerical approach is based on a least square minimization procedure that searches the value of parameter α satisfying expression in Eq. (6) by minimizing the values of three eigenvalues associated to 𝓗 (M′-αM (1)) which are supposed to be zero. Once the value of the parameter α is known, the normalized Mueller matrix M (2) can be written in terms of M′ and M (1) as

M(2)=11α(MαM(1)).
(8)

As a result, the method allows the extraction of the unknown non-depolarizing component M (2) from the original depolarizing matrix M′ by subtracting the right proportion of the known component M (1).

This method can be further generalized to subtract a non-depolarizing component from depolarizing Mueller matrices composed of up to four non-depolarizing components. The procedure is very similar to the simpler case. The depolarizing matrix constituted by the contribution of four non-depolarizing matrices can be written as

M=11+p+q+r(M(1)+pM(2)+qM(3)+rM(4)).
(9)

We are interested in the retrieval of one non-depolarizing component, which is supposed to be unknown, from depolarizing matrix M′ under the condition that the rest of non-depolarizing matrices are known. To illustrate the procedure we will take the fourth component M (4) in Eq. (9) to be unknown. The first step is to find the value of parameter α, which reduces the rank of the matrix 𝓗(M′-αM (1)) by one. Once the parameter α is known, the same algorithm is repeated to find the value of parameter β that reduces rank of the matrix 𝓗(M′-αM (1)-βM (2)). Last step is to obtain the value of parameter γ by further decreasing rank of the matrix 𝓗(M′-αM (1)-βM (2)-γM (3)). The resulting matrix M′-αM (1)-βM (2)-γM (3) is after normalization equal to the unknown matrix M (4). The procedure for the special case of the rank three depolarizing matrix is similar since we can put r=0 and reduce one step.

Necessary and sufficient condition to apply this generalized method is that the rank of the matrix M′ is equal to the number of non-depolarizing components. In the simplest case of two components this condition is equal with linear independence of the matrices. On the other hand, the linear independence is not sufficient condition to obtain rank three or rank four matrices. In following we will illustrate necessary condition for matrix of rank higher than two, which is that at most two of the non-depolarizing components can be block diagonal, the rest of the components has to have non-zero block off-diagonal elements. This condition is of relevance for practical purposes because block-diagonal matrices of isotropic samples are frequently encountered in experimental applications.

Let us consider n>2 different non-depolarizing normalized block-diagonal Mueller matrices, which can be written in the form

MB=[1cos(2ψ)00cos(2ψ)10000sin(2ψ)cos(Δ)sin(2ψ)sin(Δ)00sin(2ψ)sin(Δ)sin(2ψ)cos(Δ)],
(10)

where ψ and Δ are the ellipsometric angles [7

7. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

]. Linear combination of n matrices from Eq. (10) gives Mueller matrix M′ with the following relations between its non-zero elements:

M21=M12,M44=M33,M43=M34,M22=M11=1.
(11)

𝓗(M)=[1+M1200M33+iM3400000000M33iM34001M12].
(12)

It is immediately obvious that matrix 𝓗(M′) is of the rank two at most, because only two rows in matrix have non-zero elements. This shows that any linear combination of two or more Mueller matrices of the form in Eq. (10) leads to a coherency matrix of rank two at most, therefore only two non-depolarizing components can be in principle distinguished.

3. Experimental configuration

In order to illustrate our method we have carried out two experiments. For the first one we placed the spot between the substrate and a grating of 600 nm pitch that we will call “grating 1”. The ellipse in Fig. 1(b) labeled EC1 schematically shows the position of the beam. The matrix obtained in this way is depolarizing and has two components, one from the substrate and one from grating 1. In addition to this measurement, we have also measured the single substrate and the single grating 1 by placing the entire beam spot inside the corresponding boxes of the substrate and the grating 1, respectively. For the second experiment we placed the beam spot between the grating 1 and a second grating, with a pitch of 500 nm, that we will further call “grating 2”. The ellipse in Fig. 1(b) labeled EC2 schematically shows the position of the beam to measure the depolarizing matrix. It is worth to note that between the gratings 1 and 2 there is a uniformly etched space of the substrate. Therefore, the depolarizing matrix obtained in this way has three contribution: grating 1, grating 2 and the silicon substrate. Apart from this measurement we have additionally measured the single Mueller matrix of the grating 2 by positioning the entire beam spot inside the corresponding box.

Fig. 1. (a) Left image: photography of the patterned wafer. (b) Right image: detailed image of the grating 1 and grating 2 boxes used for the experiments. The yellow ellipses represent the positions of the beam spot during measurements of the depolarizing Mueller matrices in our experiments labeled as EC1 and EC2, respectively.

Polarimetric measurements were performed with a commercial spectroscopic Mueller matrix polarimeter (MM16 from HORIBA Jobin-Yvon), operating in the visible range (450–850 nm with a spectral step of 1.5 nm). More detailed description of this polarimeter including its operation and calibration method can be found in [10

10. E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices ,” Thin Solid Films 455–456, 120–123 (2004).

]. All measurements were done in reflection with an angle of incidence of 60°. We chose this angle to have a projected spot size on the sample surface small enough to perfectly fit inside a single box. Gratings 1 and 2 were measured in a conical configurations, where the lines of the grating are neither parallel nor perpendicular to the plane of incidence and corresponding Mueller matrices are not block-diagonal. The angle between the grating lines and the plane of incidence was for both gratings equal to 45 degrees. This configuration maximized the values of the off-diagonal elements of the matrices and the differences between the optical responses of the gratings.

4. Results and discussions

In this section we present the results of the two experiments that we conducted to illustrate the application of our method of retrieval of non-depolarizing matrix component.

4.1. Retrieval of one of two non-depolarizing components from a depolarizing Mueller matrix

The goal of this experiment was to extract one non-depolarizing component from a depolarizing Mueller matrix obtained by placing the beam spot between the grating 1 and the silicon substrate. This matrix, here called M′, is shown in Fig. 2 together with the non-depolarizing matrix of the substrate (M (1)).

Fig. 2. Spectral dependence of the measured normalized depolarizing Mueller matrix M′ (solid red line) and the normalized non-depolarizing Mueller matrix of the substrate M (1) (dashed blue line). Different boxes arranged in 4×4 array correspond to each from sixteen elements of the normalized Mueller matrices.

According to our method we have decided to extract the matrix corresponding to the grating 1 assuming that the matrix of the substrate is known. We have thus calculated the coherency matrices corresponding to M′ and M (1). Figure 3 shows the spectral dependence of the eigenvalues of the coherency matrix corresponding to M′. This matrix has two non-zero eigenvalues, which proves that the rank of this matrix is equal to two as expected. It is worth to note that there is a narrow spectral region around 525 nm in Fig. 3 where the values of one of the non-null eigenvalues become very small. This is not without some consequences in terms of the accuracy of the subtraction.

Fig. 3. Spectral dependence of the eigenvalues of the coherency matrix 𝓗(M′).
Fig. 4. Numerically obtained (solid red line) and algebraically calculated (dashed blue line) values of the parameter α depending on the wavelength of an incident light.

Beyond the fact that one of the approaches appears to be more robust than the other, both of them have limits to evaluate the value of α when one of the non-zero eigenvalues is very close to zero. In our case this situation arises because the optical behavior of M′ is very similar to the one of the non-depolarizing matrices leading to an ill-conditioned problem to find value of α. In the limiting case when M′ became non-depolarizing, the assumptions of Correas et al. [5

5. J. M. Correas, P. Melero, and J. J. Gil, “Decomposition of Mueller matrices in pure optical media,” Mon. Sem. Mat. García de Galdeano 27, 233–240 (2003).

] leading to an unique value of α, which are the basis of our subtraction method, would no longer be valid. In other words, the value of the parameter α would be indefinite. In condition of ill-conditioned problem, the presence of the experimental errors in the matrices M′ and M (1) makes the determination of α inaccurate. In our case we had to apply a Savitzky-Golay filter [11

11. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964). [CrossRef]

] to the measured matrices to decrease influence of the noise in the experimental data and to make the evaluation of α for all the wavelengths below 600 nm possible. We can conclude that use of the analytical formula in Eq. (7) should be done with caution as it may provide less accurate results than numerical approach in the presence of experimental errors. For this reason we prefer to use the numerical approach to perform matrix subtraction.

Fig. 5. Spectral values of the analytically (dashed red line) and numerically (dash-dotted blue line) retrieved Mueller matrices compared with the directly measured matrix corresponding to grating 1 (solid black line).

4.2. Retrieval of the third non-depolarizing component from a depolarizing Mueller matrix

The goal of this experiment was to retrieve one non-depolarizing component from a depolarizing matrix consisting of the combination of three different components. To measure corresponding matrix we positioned the beam between the grating 1 and grating 2 as it is demonstrated in Fig. 1 by ellipse labeled EC2. To implement our method we suppose that the non-depolarizing matrices of the substrate and grating 1 are known. Moreover, we have measured the gratings in conical diffraction in order to ensure that their matrices are not block diagonal, which is one of the necessary conditions for application of the generalized subtraction method. Figure 7 shows the spectroscopic Mueller matrices of the substrate alone, grating 1 and grating 2 alone as well as the depolarizing matrix corresponding to the “mixture” of the substrate, grating 1 and grating 2.

Fig. 6. Difference between the numerically retrieved and the experimental non-depolarizing Mueller matrices corresponding to grating 1 (both are plotted in Fig. 5).
Fig. 7. Measured normalized Mueller matrices of substrate, grating 1, grating 2, and their “mixture”. Sixteen boxes correspond to different elements of Mueller matrices.

Fig. 8. Eigenvalues of the coherency matrix obtained from measured Mueller matrix on the boundary between the three sample regions. Spectral values of all four eigenvalues are plotted in the left plot with a detailed view of the three smallest eigenvalues on the right.

Fig. 9. Spectral dependence of the resulting values of the parameters α (solid red line) and β (dashed blue line).

Once the values of α and β were known, we obtained the unknown normalized non-depolarizing matrix corresponding to grating 2 as presented before in Section 2. The accuracy of the subtraction procedure is illustrated by comparison of the retrieved Mueller matrix with the directly measured Mueller matrix in Fig. 10. Both matrices are almost identical for whole spectral range except for the mentioned narrow band between 550 and 650 nm. As previously pointed out, the origin of those differences, i.e. the lack of accuracy of the method in this spectral region, is due to the small difference existing between the smallest non-zero and the zero eigenvalue of the matrix 𝓗(M′) (see Fig. 8). The third eigenvalue is very small in the critical spectral region due to the weak anisotropic optical response of the structures resulting in small values of block off-diagonal elements of the grating Mueller matrices.

The anisotropic behavior of the optical response is revealed by the element M 22 in Fig. 7. The value of M 22 equals one for a sample with zero block off-diagonal elements in corresponding Mueller matrix. Element M 22 in Fig. 7 shows value very close to one for all Mueller matrices entering the subtraction procedure (substrate, grating 1 and grating 2) in the region between 550 and 650 nm. This situation is very similar to the pathological case of three matrices without non-zero block off-diagonal elements, discussed in Section 2, that invalidates the generalized subtraction procedure. In such case, it is not possible to distinguish accurately the three independent non-depolarizing components. As a conclusion we can say that the generalized subtraction procedure can be implemented satisfactorily except for the case where all matrices are almost block-diagonal. For the pathological case, however, the closer the matrices are to block-diagonal, the more inaccurate the decomposition becomes.

Fig. 10. Spectral dependence of the retrieved Mueller matrix (solid red line) compared with the directly measured matrix corresponding to grating 2 (dashed blue line).

5. Conclusions

This work demonstrates the feasibility of the retrieval of an unknown non-depolarizing component of a depolarizing Mueller matrix, under the condition that all other non-depolarizing components are known. The robustness and the weaknesses of the method are illustrated in two examples concerning depolarizing Mueller matrices containing two and three components, respectively. In the case of a depolarizing matrix constituted of two components, the subtraction method can be successfully applied, provided that the two components are different enough to be considered as linearly independent in numerical terms. We tested two different approaches to implement our method: the analytical and the numerical one. Despite of the fact that both approaches should be equivalent when working with ideal matrices, the numerical approach ap-peared to provide more accurate results when it was applied to experimental Mueller matrices affected by random noise and small systematic errors. It was also shown that the subtraction procedure can be carried out successfully with depolarizing matrices having three components if at least one of the non-depolarizing components is not block-diagonal.

Acknowledgments

References and links

1.

M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution,” Opt. Commun. 282, 735–741 (2009). [CrossRef]

2.

G. Subramania, Y.-J. Lee, I. Brener, T. Luk, and P. Clem, “Nano-lithographically fabricated titanium dioxide based visible frequency three dimensional gap photonic crystal,” Opt. Express 15(20), 13049–13057 (2007). [CrossRef]

3.

M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, “Experimental and theoretical characterization of a lithium niobate photonic crystal,” Appl. Phys. Lett. 87, 241101 (2005). [CrossRef]

4.

J. J. Gil, “Polarimetric characterization of light and media Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007). [CrossRef]

5.

J. M. Correas, P. Melero, and J. J. Gil, “Decomposition of Mueller matrices in pure optical media,” Mon. Sem. Mat. García de Galdeano 27, 233–240 (2003).

6.

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

7.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

8.

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).

9.

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

10.

E. Garcia-Caurel, A. De Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices ,” Thin Solid Films 455–456, 120–123 (2004).

11.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: April 20, 2009
Revised Manuscript: May 27, 2009
Manuscript Accepted: May 29, 2009
Published: July 13, 2009

Citation
M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, "Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices," Opt. Express 17, 12794-12806 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12794


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References

  1. M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, "Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size and spectral resolution," Opt. Commun. 282, 735-741 (2009). [CrossRef]
  2. G. Subramania, Y.-J. Lee, I. Brener, T. Luk, and P. Clem, "Nano-lithographically fabricated titanium dioxide based visible frequency three dimensional gap photonic crystal," Opt. Express 15(20), 13049-13057 (2007). [CrossRef]
  3. M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, "Experimental and theoretical characterization of a lithium niobate photonic crystal," Appl. Phys. Lett. 87, 241101 (2005). [CrossRef]
  4. J. J. Gil, "Polarimetric characterization of light and media Physical quantities involved in polarimetric phenomena," Eur. Phys. J. Appl. Phys. 40, 1-47 (2007). [CrossRef]
  5. J. M. Correas, P. Melero, and J. J. Gil, "Decomposition of Mueller matrices in pure optical media," Mon. Sem. Mat. García de Galdeano 27, 233-240 (2003).
  6. S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).
  7. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).
  8. J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," J. Mod. Opt. 32, 259-261 (1985).
  9. S. R. Cloude, "Conditions for the physical realisability of matrix operators in polarimetry," vol. 1166 of Proc. of SPIE, pp. 177-185 (1989).
  10. E. Garcia-Caurel, A. De Martino, and B. Drévillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004).
  11. A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964). [CrossRef]

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