Abstract

The road maps presented provide several options to detect and identify biological and chemical materials based on the appropriate measurements of specific Mueller matrix elements. The optical activity of these chiral materials (birefringence, retardation, diattenuation, optical rotation, circular dichroism, and cross polarization) are represented by ellipses associated with the like and cross polarized components of the reflected waves due to vertical and horizontal linearly polarized incident waves. It is also represented by ellipses associated with optical rotation and circular dichroism. These measurements can be made in situ over large areas of the samples. Only one pair of measurements is necessary to provide the data to identify the biological or chemical materials. However the redundancy in the series of measurements that can be made can be used to confirm the veracity of the results and to minimize the occurrences of false alarms. This research has numerous applications in biomedicine, biochemistry, defense, security, and industry.

© 2009 Optical Society of America

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References

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  1. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).
  2. L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge U. Press, 2004).
    [CrossRef]
  3. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994).
    [CrossRef]
  4. V. Sankaron, J. T. Walsh, and D. J. Matland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. 7, 300-306 (2002).
    [CrossRef]
  5. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices on polar decomposition,” J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  6. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29, 2234-2236 (2004).
    [CrossRef] [PubMed]
  7. A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
    [CrossRef]
  8. E. Bahar, “Mueller matrices for waves reflected and transmitted through chiral materials, waveguide modal solutions and applications,” J. Opt. Soc. Am. B 24, 1610-1619 (2007).
    [CrossRef]
  9. E. Bahar, “Review of full wave solutions for rough surface scattering and depolarization: comparison with geometric and physical optics, perturbation and two scale hybrid solutions,” J. Geophys. Res. 92, 749-759 (1988).
  10. E. Bahar, “The relationship between optical rotation and circular dichroism and elements of the Mueller matrix for natural and artificial chiral materials,” J. Opt. Soc. Am. B 25, 218-222 (2008).
    [CrossRef]
  11. M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
    [CrossRef]
  12. S. Manhas, M. K. Swami, P. Buddhiwant, N. Gosh, P. K. Gupta, and K. Sing, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscatter geometry,” Opt. Express 14, 190-202 (2006).
    [CrossRef] [PubMed]
  13. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).
  14. E. Bahar and R. D. Kubik, “Description of a versatile optical polarimetric scatterometer that measures all sixteen elements of the Mueller matrix for reflection and transmission: application to measurements of scatter cross sections, ellipsometric parameters, optical activity and the complex chiral parameters,” Opt. Eng. 47, 093603 (2008).
    [CrossRef]
  15. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
    [CrossRef]
  16. C. A. Browne and F. W. Zerban, “Physical and chemical methods of sugar analysis,” (Wiley, 1941).
  17. C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
    [CrossRef]
  18. E. Bahar, “Characterization of natural and artificial optical activity by the Mueller matrix for oblique incidence, total internal reflection and Brewster angle,” J. Opt. Soc. Am. B 25, 1294-1302 (2008).
    [CrossRef]
  19. A. H. Carrieri, “Neural network pattern recognition by means of differential absorption Mueller matrices spectroscopy,” Appl. Opt. 38, 3759-3766 (1999).
    [CrossRef]
  20. A. H. Carrieri, J. R. Bottinger, D. J. Owens, and E. S. Roese, “Differential absorption Mueller matrix spectroscopy and the infrared detection of crystalline organics,” Appl. Opt. 37, 6550-6557 (1998).
    [CrossRef]
  21. G. W. Kattawar and D. J. Grey, “Mueller matrix imaging of targets in turbid media: effect of volume scattering function,” Appl. Opt. 36, 7225-7230 (2003).
    [CrossRef]
  22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).

2008

E. Bahar, “The relationship between optical rotation and circular dichroism and elements of the Mueller matrix for natural and artificial chiral materials,” J. Opt. Soc. Am. B 25, 218-222 (2008).
[CrossRef]

E. Bahar and R. D. Kubik, “Description of a versatile optical polarimetric scatterometer that measures all sixteen elements of the Mueller matrix for reflection and transmission: application to measurements of scatter cross sections, ellipsometric parameters, optical activity and the complex chiral parameters,” Opt. Eng. 47, 093603 (2008).
[CrossRef]

E. Bahar, “Characterization of natural and artificial optical activity by the Mueller matrix for oblique incidence, total internal reflection and Brewster angle,” J. Opt. Soc. Am. B 25, 1294-1302 (2008).
[CrossRef]

2007

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

E. Bahar, “Mueller matrices for waves reflected and transmitted through chiral materials, waveguide modal solutions and applications,” J. Opt. Soc. Am. B 24, 1610-1619 (2007).
[CrossRef]

2006

2004

2003

G. W. Kattawar and D. J. Grey, “Mueller matrix imaging of targets in turbid media: effect of volume scattering function,” Appl. Opt. 36, 7225-7230 (2003).
[CrossRef]

2002

V. Sankaron, J. T. Walsh, and D. J. Matland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef]

2001

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

1999

1998

1997

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

1996

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

M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
[CrossRef]

1988

E. Bahar, “Review of full wave solutions for rough surface scattering and depolarization: comparison with geometric and physical optics, perturbation and two scale hybrid solutions,” J. Geophys. Res. 92, 749-759 (1988).

1986

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

1852

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Angelia, A.

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Azzam, R. M. A.

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

Backman, V.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Badizadigan, K.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Badoz, J.

M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
[CrossRef]

Bahar, E.

E. Bahar, “The relationship between optical rotation and circular dichroism and elements of the Mueller matrix for natural and artificial chiral materials,” J. Opt. Soc. Am. B 25, 218-222 (2008).
[CrossRef]

E. Bahar and R. D. Kubik, “Description of a versatile optical polarimetric scatterometer that measures all sixteen elements of the Mueller matrix for reflection and transmission: application to measurements of scatter cross sections, ellipsometric parameters, optical activity and the complex chiral parameters,” Opt. Eng. 47, 093603 (2008).
[CrossRef]

E. Bahar, “Characterization of natural and artificial optical activity by the Mueller matrix for oblique incidence, total internal reflection and Brewster angle,” J. Opt. Soc. Am. B 25, 1294-1302 (2008).
[CrossRef]

E. Bahar, “Mueller matrices for waves reflected and transmitted through chiral materials, waveguide modal solutions and applications,” J. Opt. Soc. Am. B 24, 1610-1619 (2007).
[CrossRef]

E. Bahar, “Review of full wave solutions for rough surface scattering and depolarization: comparison with geometric and physical optics, perturbation and two scale hybrid solutions,” J. Geophys. Res. 92, 749-759 (1988).

Barron, L.

L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

Bashara, N. M.

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

Bigio, I. J.

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Bottinger, J. R.

Browne, C. A.

C. A. Browne and F. W. Zerban, “Physical and chemical methods of sugar analysis,” (Wiley, 1941).

Buddhiwant, P.

Carrieri, A. H.

Chipman, R. A.

Cloude, S. R.

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

Dasari, R.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Eick, E. A.

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Feld, M. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Freyer, J. P.

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Georgakoudi, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Gosh, N.

Goudail, F.

Grey, D. J.

G. W. Kattawar and D. J. Grey, “Mueller matrix imaging of targets in turbid media: effect of volume scattering function,” Appl. Opt. 36, 7225-7230 (2003).
[CrossRef]

Gupta, P. K.

Gurjar, R. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Hielsher, A. H.

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Itzkan, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Kattawar, G. W.

G. W. Kattawar and D. J. Grey, “Mueller matrix imaging of targets in turbid media: effect of volume scattering function,” Appl. Opt. 36, 7225-7230 (2003).
[CrossRef]

Kubik, R. D.

E. Bahar and R. D. Kubik, “Description of a versatile optical polarimetric scatterometer that measures all sixteen elements of the Mueller matrix for reflection and transmission: application to measurements of scatter cross sections, ellipsometric parameters, optical activity and the complex chiral parameters,” Opt. Eng. 47, 093603 (2008).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994).
[CrossRef]

Li, L. W.

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

Liong, M. S.

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

Lu, S.-Y.

Manhas, S.

Matland, D. J.

V. Sankaron, J. T. Walsh, and D. J. Matland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef]

Morio, J.

Mourant, J. R.

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Owens, D. J.

Perelman, L. T.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Qui, C. W.

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

Roese, E. S.

Sankaron, V.

V. Sankaron, J. T. Walsh, and D. J. Matland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef]

Silverman, M. P.

M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
[CrossRef]

Sing, K.

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Strange, W.

M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
[CrossRef]

Swami, M. K.

Vitkin, I. A.

M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
[CrossRef]

Walsh, J. T.

V. Sankaron, J. T. Walsh, and D. J. Matland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef]

Yao, H. Y.

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

Zerban, F. W.

C. A. Browne and F. W. Zerban, “Physical and chemical methods of sugar analysis,” (Wiley, 1941).

Zouhdi, S.

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

Appl. Opt.

J. Biomed. Opt.

V. Sankaron, J. T. Walsh, and D. J. Matland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. 7, 300-306 (2002).
[CrossRef]

J. Geophys. Res.

E. Bahar, “Review of full wave solutions for rough surface scattering and depolarization: comparison with geometric and physical optics, perturbation and two scale hybrid solutions,” J. Geophys. Res. 92, 749-759 (1988).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microwave Opt. Technol. Lett.

C. W. Qui, H. Y. Yao, S. Zouhdi, L. W. Li, and M. S. Liong, “On the constitutive relations of G chiral media and the potential to realize negative index media,” Microwave Opt. Technol. Lett. 48, 2534-2538 (2007).
[CrossRef]

Nat. Med.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadigan, I. Itzkan, R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 245-1249 (2001).
[CrossRef]

Opt. Commun.

M. P. Silverman, W. Strange, J. Badoz, and I. A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. 132, 410-416 (1996).
[CrossRef]

Opt. Eng.

E. Bahar and R. D. Kubik, “Description of a versatile optical polarimetric scatterometer that measures all sixteen elements of the Mueller matrix for reflection and transmission: application to measurements of scatter cross sections, ellipsometric parameters, optical activity and the complex chiral parameters,” Opt. Eng. 47, 093603 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

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

Proc. SPIE

A. H. Hielsher, E. A. Eick, A. Angelia, J. R. Mourant, J. P. Freyer, and I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” Proc. SPIE 2976, 298-305 (1997).
[CrossRef]

Trans. Cambridge Philos. Soc.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Other

L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994).
[CrossRef]

C. A. Browne and F. W. Zerban, “Physical and chemical methods of sugar analysis,” (Wiley, 1941).

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

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Figures (2)

Fig. 1
Fig. 1

Reflection flow graph.

Fig. 2
Fig. 2

Optical activity flow graph.

Equations (43)

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

D = ε ( E + β × E ) ,
B = μ ( H + β × E ) .
R C = R C O + k 1 β 1 R C O .
R 10 C O = [ R 10 R R 0 0 R 10 L L ] = 1 2 T 01 H H T 10 V V tan 2 θ 1 [ 1 0 0 1 ] .
sin θ 1 = ( k o k 1 ) sin θ o = 1 n 1 sin θ o .
R L = A 1 R C A = R L O + k 1 β 1 R L O .
R 10 L 0 = ( R 10 V V 0 0 R 10 H H ) , k 1 β 1 R L 0 = j k 1 β 1 ( 0 R 10 L L R 10 R R 0 ) = ( 0 R 10 V H R 10 H V 0 ) ,
A = ( 1 j 1 j ) .
T V H = T H V ( Z 1 Z 0 ) = j k 1 β 1 R L L = R V H = R H V .
Z 0 = ( μ 0 ε 0 ) 1 2 , Z 1 = ( μ 1 ε 1 ) 1 2 .
S f = M S i = [ M T L M T R M B L M B R ] S i .
M T L = 1 2 [ R H H 2 + R V V 2 R H H 2 R V V 2 R H H 2 R V V 2 R H H 2 + R V V 2 ] = [ m 11 m 12 m 21 m 22 ] ,
M B R = [ Re [ R V V R H H * ] Im [ R V V R H H * ] Im [ R V V R H H * ] Re [ R V V R H H * ] ] = [ m 33 m 34 m 43 m 44 ] ,
M B L = [ Re [ ( R H H R V V ) R H V * ] Re [ ( R H H + R V V ) R H V * ] Im [ ( R H H + R V V ) R H V * ] Im [ ( R H H R V V ) R H V * ] ] = [ m 31 m 32 m 41 m 42 ] ,
M T R = [ Re [ ( R H H R V V ) R H V * ] Im [ ( R H H + R V V ) R H V * ] Re [ ( R H H + R V V ) R H V * ] Im [ ( R H H R V V ) R H V * ] ] = [ m 13 m 14 m 23 m 24 ] .
( R H H + R V V ) R H V * = ( m 23 + j m 14 ) = m 32 + j m 41 ,
( R H H R V V ) R H V * = ( m 13 + j m 24 ) = m 31 + j m 42 .
R H V = R V H = ( m 23 j m 14 ) ( R H H + R V V ) * = j k 1 β 1 R 10 R R .
R H V = j m 14 ( R H H + R V V ) .
E v i = Re [ ( R V V a ¯ v + R H V a ¯ H ) e j ω t ] = R V V cos ( ω t + ϕ V V ) a ¯ V + R H V cos ( ω t + ϕ H V ) a ¯ H .
E H i = R V H cos ( ω t + ϕ V H ) a ¯ v + R H H cos ( ω t + ϕ H H ) a ¯ H .
ρ = R V V R H H = R V V R H H exp [ j ( ϕ V V ϕ H H ) ] = tan ( Ψ ) exp ( i Δ ) .
tan 2 α = tan 2 Ψ cos Δ .
b a = tan χ .
cos 2 χ = cos 2 Ψ cos 2 α = sin 2 Ψ cos Δ sin 2 α .
cos 2 Ψ = m 12 m 11 ,
tan Δ = m 43 m 33 .
n 2 = ε r = tan 2 ϕ 0 [ 1 4 ρ ( 1 + ρ ) 2 sin 2 ϕ 0 ] ,
R P P = cos θ 0 Q cos θ 1 cos θ 0 + Q cos θ 1 .
R = R H V R V V = R H V R V V exp [ j ( ϕ H V ϕ V V ) ] = tan ζ exp ( j δ ) .
tan 2 η = tan 2 ζ cos δ .
B A = tan ξ , cos 2 ξ = cos 2 ζ cos 2 η = sin 2 ζ cos δ sin 2 η .
γ 1 = k 1 ( 1 k 1 β 1 ) = k 0 ( n 1 j n 1 ) = k 0 n 1 ,
γ 2 = k 1 ( 1 + k 1 β 1 ) = k 0 ( n 2 j n 2 ) = k 0 n 2 ,
k 1 = ω μ 1 ε 1 = n k 0 = ( n j n ) k 0 , k 0 = ω μ 0 ε 0 .
v 1 = c n 1 , v 2 = c n 2 ,
d 1 = ω n 1 c = n 1 k 0 , d 2 = ω n 2 c = n 1 k 0 .
OR + j CD = ( γ 1 γ 2 ) l 2 = [ Re ( γ 1 γ 2 ) + j Im ( γ 1 γ 2 ) ] l 2 .
OR + j CD = k 1 2 β 1 = k 0 2 ( n j n ) 2 ( β + j β ) = k 0 [ ( n 1 n 2 ) j ( n 1 n 2 ) ] 2 .
cos [ k 0 ( n 1 n 2 ) l 2 ] a ¯ x sin [ k 0 ( n 1 n 2 ) l 2 ] a ¯ y = cos ( OR ) a ¯ x sin ( OR ) a ¯ y .
sin [ [ k 0 ( n 1 n 2 ) l 2 ] ] a ¯ x + cos ( n 1 n 2 ) l 2 a ¯ y = sin ( OR ) a ¯ x + cos ( OR ) a ¯ y .
CD = ( n 2 n 1 ) k 0 l 2 = ( d 2 d 1 ) l 2 .
OR + j CD = k 1 2 β 1 = k 1 ( m 14 + j m 23 ) ( R H H + R V V ) * R 10 L L .

Metrics