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

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References

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  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]

2009

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]

2007

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]

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]

2005

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]

2004

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).

2003

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).

1986

S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).

1985

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

1964

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

Baida, F. I.

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]

Bernabeu, E.

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

Bernal, M.-P.

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]

Brener, I.

Clem, P.

Cloude, S. R.

S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).

Correas, J. M.

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).

Courjal, N.

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]

De Martino, A.

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]

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).

Drévillon, B.

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).

Foldyna, M.

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]

Garcia-Caurel, E.

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]

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).

Gil, J. J.

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]

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).

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

Golay, M. J. E.

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

Lee, Y.-J.

Licitra, C.

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]

Luk, T.

Melero, P.

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).

Ossikovski, R.

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]

Roussey, M.

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]

Savitzky, A.

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

Subramania, G.

Anal. Chem.

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

Appl. Phys. Lett.

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]

Eur. Phys. J. Appl. Phys.

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]

J. Mod. Opt.

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

Mon. Sem. Mat. García de Galdeano

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).

Opt. Commun.

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]

Opt. Express

Optik (Stuttgart)

S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).

Thin Solid Films

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).

Other

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

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

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Figures (10)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
Fig. 3.

Spectral dependence of the eigenvalues of the coherency matrix ��(M′).

Fig. 4.
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.

Fig. 5.
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).

Fig. 6.
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.
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.
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.
Fig. 9.

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

Fig. 10.
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).

Equations (12)

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S=[S1S2S3S4]=[Ex2+Ey2Ex2Ey22Re(ExEy)2Im(ExEy)],
Sem=MSin=[M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44]Sin.
Tr (MTM)=i,j=14Mij2=4M112.
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].
M=11+p (M(1)+pM(2)) ,
rank[𝓗(M)-α𝓗(M(1))]=1.
α=4M112Tr(MTM)8M11M11(1)Tr(M(1)TM+MTM(1)),
M(2)=11α (MαM(1)).
M=11+p+q+r (M(1)+pM(2)+qM(3)+rM(4)) .
MB=[1cos(2ψ)00cos(2ψ)10000sin(2ψ)cos(Δ)sin(2ψ)sin(Δ)00sin(2ψ)sin(Δ)sin(2ψ)cos(Δ)],
M21=M12,M44=M33,M43=M34,M22=M11=1 .
𝓗 (M)=[1+M1200M33+iM3400000000M33iM34001M12].

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