Abstract

In this research, the polar decomposition (PD) method is applied to experimental Mueller matrices (MMs) measured on two-dimensional microstructured surfaces. Polarization information is expressed through a set of parameters of easier physical interpretation. It is shown that evaluating the first derivative of the retardation parameter, δ, a clear indication of the presence of defects either built on or dug in the scattering flat surface (a silicon wafer in our case) can be obtained. Although the rule of thumb thus obtained is established through PD, it can be easily implemented on conventional surface polarimetry. These results constitute an example of the capabilities of the PD approach to MM analysis, and show a direct application in surface characterization.

© 2011 Optical Society of America

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References

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  1. G. Videen and W. S. Bickel, “Light-scattering mueller matrix for a rough fiber,” Appl. Opt. 31, 3488–3492 (1992).
    [CrossRef] [PubMed]
  2. S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
    [CrossRef]
  3. Y. Cui and R. M. A. Azzam, “Applications of the normal-incidence rotating-sample ellipsometer to high- and low-spatial-frequency gratings,” Appl. Opt. 35, 2235–2238 (1996).
    [CrossRef] [PubMed]
  4. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its mueller matrix,” Optik 76, 67–71 (1987).
  5. 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]
  6. J. Chung, W. Jung, M. J. Hammer-Wilson, P. Wilder-Smith, and Z. Chen, “Use of polar decomposition for the diagnosis of oral precancer,” Appl. Opt. 46, 3038–3045 (2007).
    [CrossRef] [PubMed]
  7. M. K. Swami, S. Manhas, P. Buddhiwant, N. Ghosh, A. Uppal, and P. K. Gupta, “Polar decomposition of 3×3 mueller matrix: a tool for quantitative tissue polarimetry,” Opt. Express 14, 9324–9337 (2006).
    [CrossRef] [PubMed]
  8. C. Collet, J. Zallat, and Y. Takakura, “Clustering of mueller matrix images for skeletonized structure detection,” Opt. Express 12, 1271–1280 (2004).
    [CrossRef] [PubMed]
  9. 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,” Opt. Commun. 282, 735–741 (2009).
    [CrossRef]
  10. M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 41, 2488–2493(2002).
    [CrossRef] [PubMed]
  11. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  12. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  13. J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
    [CrossRef]
  14. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  15. M. Foldyna, E. García-Caurel, R. Ossikovski, A. D. Martino, and J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing mueller matrices,” Opt. Express 17, 12794–12806 (2009).
    [CrossRef] [PubMed]
  16. J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
    [CrossRef]
  17. H. Hulst, Light Scattering by Small Particles (Dover, 1981).
  18. J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
    [CrossRef]

2009 (3)

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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

M. Foldyna, E. García-Caurel, R. Ossikovski, A. D. Martino, and J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

2004 (2)

C. Collet, J. Zallat, and Y. Takakura, “Clustering of mueller matrix images for skeletonized structure detection,” Opt. Express 12, 1271–1280 (2004).
[CrossRef] [PubMed]

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

2002 (1)

1999 (2)

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[CrossRef]

1996 (2)

1992 (1)

1987 (1)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its mueller matrix,” Optik 76, 67–71 (1987).

1986 (1)

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

1981 (1)

H. Hulst, Light Scattering by Small Particles (Dover, 1981).

1978 (1)

Albella, P.

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

Azzam, R. M. A.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its mueller matrix,” Optik 76, 67–71 (1987).

Bickel, W. S.

Buddhiwant, P.

Chen, Z.

Chipman, R. A.

Chung, J.

Cloude, S. R.

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

Collet, C.

Cui, Y.

de la Peña, J.

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

de la Peña, J. L.

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

M. Foldyna, E. García-Caurel, R. Ossikovski, A. D. Martino, and J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[CrossRef] [PubMed]

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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

García-Caurel, E.

Ghosh, N.

Gil, J.

Gil, J. J.

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

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its mueller matrix,” Optik 76, 67–71 (1987).

González, F.

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[CrossRef]

Gupta, P. K.

Hammer-Wilson, M. J.

Hulst, H.

H. Hulst, Light Scattering by Small Particles (Dover, 1981).

Jung, W.

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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

Lu, S. Y.

Manhas, S.

Martino, A. D.

Mishchenko, I. A.

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

Mishchenko, L. T.

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

Moreno, F.

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[CrossRef]

Muttiah, R. S.

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

Oberemok, Y. A.

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

M. Foldyna, E. García-Caurel, R. Ossikovski, A. D. Martino, and J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[CrossRef] [PubMed]

Saiz, J.

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

Saiz, J. M.

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[CrossRef]

Sanz, J. M.

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

Savenkov, S. N.

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

Smith, M. H.

Swami, M. K.

Takakura, Y.

Uppal, A.

Valle, P.

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

Valle, P. J.

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[CrossRef]

Videen, G.

Wilder-Smith, P.

Zallat, J.

Appl. Opt. (4)

Eur. Phys. J. Appl. Phys. (1)

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

J. Appl. Phys. (1)

J. L. de la Peña, F. González, J. M. Saiz, F. Moreno, and P. J. Valle, “Sizing particles on substrates. a general method for oblique incidence,” J. Appl. Phys. 85, 432–438(1999).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (2)

S. N. Savenkov, L. T. Mishchenko, R. S. Muttiah, Y. A. Oberemok, and I. A. Mishchenko, “Mueller polarimetry of virus-infected and healthy wheat under field and microgravity conditions,” J. Quant. Spectrosc. Radiat. Transfer 88, 327–343 (2004).
[CrossRef]

J. M. Sanz, P. Albella, F. Moreno, J. M. Saiz, and F. González, “Application of the polar decomposition to light scattering particle systems,” J. Quant. Spectrosc. Radiat. Transfer 110, 1369–1374 (2009).
[CrossRef]

Opt. Commun. (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,” Opt. Commun. 282, 735–741 (2009).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Optik (2)

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

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its mueller matrix,” Optik 76, 67–71 (1987).

Part. Part. Syst. Charact. (1)

J. de la Peña, J. Saiz, P. Valle, F. González, and F. Moreno, “Tracking scattering minima to size metallic particles on flat substrates,” Part. Part. Syst. Charact. 16, 113–118 (1999).
[CrossRef]

Other (1)

H. Hulst, Light Scattering by Small Particles (Dover, 1981).

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Figures (8)

Fig. 1
Fig. 1

(a) Experimental setup: Positioning (incidence on sample and synchronous waveplates rotations) and measurements are computer controlled. (b) Scattering configuration for a two-rib sample.

Fig. 2
Fig. 2

SEM image of a two-rib structure with dimensions h = 1 μm , w = 2 μm , and d = 6 μm .

Fig. 3
Fig. 3

Scattering pattern ( m 00 ): Comparison with (a) diffraction minima (single structures: h = 1 μm and w = 3 μm ) and (b) interference minima (double structures: h = 1 μm , w = 3 μm , and d = 4 μm ).

Fig. 4
Fig. 4

MM elements versus scattering angle ( 90 ° to 90 ° ) for a single Si rib ( h = 1 μm and w = 1 μm ) and groove ( h = 1 μm and w = 3 μm ).

Fig. 5
Fig. 5

PD parameters for single equivalent structures on Si: (a) rib, h = 1 μm and w = 1 μm ; (b) groove, h = 1 μm and w = 3 μm . Top row (diattenuation parameters: α, β, and t), center row (retardance parameters: φ, δ, and ρ), bottom row (depolarization parameters: d i , a i and z i ).

Fig. 6
Fig. 6

MM elements versus scattering angle ( 90 ° to 90 ° ) for a single Si rib ( h = 1 μm and w = 1 μm ): pure and depolarizer matrix.

Fig. 7
Fig. 7

δ slope for a single rib/groove obtained through PD. The dominant sign of the slope is shown in black bars (increment Δ δ corresponds to Δ θ = 1 ° ).

Fig. 8
Fig. 8

Values of δ and its slope sign for different geometries and materials. Upper cases, Au-sputtered single rib (left) and groove (right) of h = 1 μm and w = 3 μm . Bottom cases, two parallel Si ribs/grooves of h = 1 μm , w = 3 μm , and d = 7 μm .

Tables (1)

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Table 1 Values of ϒ R G for 2D Square-Profiled Structures h = 1 μm a

Equations (7)

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M = M Δ M R M D ,
M D ( t 1 , t 2 , α , β ) = m 00 · M D ( t , α , β ) .
M R ( ϕ , δ , ρ ) = M R ( ϕ , δ ) · ( 1 0 0 0 0 cos ( 2 ρ ) sin ( 2 ρ ) 0 0 sin ( 2 ρ ) cos ( 2 ρ ) 0 0 0 0 1 ) .
tan ( δ ) = m 23 m 32 m 22 + m 33 .
M Δ ( d i , a i , z i ) = ( 1 0 0 0 z 1 d 1 a 1 a 2 z 2 a 1 d 2 a 3 z 3 a 2 a 3 d 3 ) = ( 1 0 T P Δ m Δ ) .
t r ( M T M ) 4 m 00 2 .
ϒ R G = θ > 0 Δ δ i | Δ δ i | θ < 0 Δ δ i | Δ δ i | N ,

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