Abstract

Exponential versions of the Jones and Mueller–Jones matrices are derived by use of the theory of semigroup transformations in conjunction with Kronecker products of matrices.

© 1996 Optical Society of America

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References

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  1. R. C. Jones, “A new calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  2. N. Go, “Optical activity of anisotropic solutions,” J. Phys. Soc. Jpn. 23, 88–93 (1967).
    [CrossRef]
  3. E. L. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1964).
  4. R. Azzam, N. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  5. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones matrix of polarized theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  6. T. Troxell, H. Scheraga, “Electric dichroism and polymer conformation. I. Theory of the optical properties of anisotropic media and method of measurement,” Macromolecules 4, 519–527 (1971).
    [CrossRef]
  7. H. Jensen, J. Schellman, T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192–200 (1978).
    [CrossRef]
  8. J. Schellman, H. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987) contains an extensive bibliography of references to the chemical literature on polarized light.
    [CrossRef]
  9. R. Barakat, “Theory of the coherency matrix of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
    [CrossRef]
  10. E. Hille, R. Phillips, Functional Analysis and Semi-groups, revised ed. (American Mathematical Society, Providence, R.I., 1957).
  11. P. Lax, R. Phillips, Scattering Theory (Academic, New York, 1967), Chap. 3 and App. 1.
  12. W. Schurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  13. R. Horn, C. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
    [CrossRef]
  14. P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).
  15. F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
    [CrossRef]
  16. R. Simon, “The relationships between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  17. D. Anderson, R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305–2319 (1994).
    [CrossRef]
  18. A. Richmond, “Expansions for the exponential of a sum of matrices,” in M. Gover, S. Barnett, eds., Applications of Matrix Theory (Clarendon, Oxford, 1989), pp. 283–290.
  19. J. Weddeburn, Lectures on Matrix Theory (American Mathematical Society, Providence, R.I., 1934).
  20. H. Fox, R. Barakat, “Trace-orthogonal hermitian basis matrices of artibtrary dimension,” J. Comp. Phys. 21, 326–332 (1976).
    [CrossRef]

1994 (1)

1987 (1)

J. Schellman, H. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987) contains an extensive bibliography of references to the chemical literature on polarized light.
[CrossRef]

1982 (1)

R. Simon, “The relationships between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

1981 (1)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones matrix of polarized theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

1978 (1)

1976 (1)

H. Fox, R. Barakat, “Trace-orthogonal hermitian basis matrices of artibtrary dimension,” J. Comp. Phys. 21, 326–332 (1976).
[CrossRef]

1971 (1)

T. Troxell, H. Scheraga, “Electric dichroism and polymer conformation. I. Theory of the optical properties of anisotropic media and method of measurement,” Macromolecules 4, 519–527 (1971).
[CrossRef]

1967 (1)

N. Go, “Optical activity of anisotropic solutions,” J. Phys. Soc. Jpn. 23, 88–93 (1967).
[CrossRef]

1963 (1)

1948 (1)

1942 (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

1929 (1)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).

Anderson, D.

Azzam, R.

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

Barakat, R.

D. Anderson, R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305–2319 (1994).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones matrix of polarized theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

H. Fox, R. Barakat, “Trace-orthogonal hermitian basis matrices of artibtrary dimension,” J. Comp. Phys. 21, 326–332 (1976).
[CrossRef]

R. Barakat, “Theory of the coherency matrix of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
[CrossRef]

Bashara, N.

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

Fox, H.

H. Fox, R. Barakat, “Trace-orthogonal hermitian basis matrices of artibtrary dimension,” J. Comp. Phys. 21, 326–332 (1976).
[CrossRef]

Go, N.

N. Go, “Optical activity of anisotropic solutions,” J. Phys. Soc. Jpn. 23, 88–93 (1967).
[CrossRef]

Hille, E.

E. Hille, R. Phillips, Functional Analysis and Semi-groups, revised ed. (American Mathematical Society, Providence, R.I., 1957).

Horn, R.

R. Horn, C. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

Jensen, H.

J. Schellman, H. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987) contains an extensive bibliography of references to the chemical literature on polarized light.
[CrossRef]

H. Jensen, J. Schellman, T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192–200 (1978).
[CrossRef]

Johnson, C.

R. Horn, C. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

Jones, R. C.

Lax, P.

P. Lax, R. Phillips, Scattering Theory (Academic, New York, 1967), Chap. 3 and App. 1.

O’Neill, E. L.

E. L. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1964).

Perrin, F.

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

Phillips, R.

P. Lax, R. Phillips, Scattering Theory (Academic, New York, 1967), Chap. 3 and App. 1.

E. Hille, R. Phillips, Functional Analysis and Semi-groups, revised ed. (American Mathematical Society, Providence, R.I., 1957).

Richmond, A.

A. Richmond, “Expansions for the exponential of a sum of matrices,” in M. Gover, S. Barnett, eds., Applications of Matrix Theory (Clarendon, Oxford, 1989), pp. 283–290.

Schellman, J.

J. Schellman, H. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987) contains an extensive bibliography of references to the chemical literature on polarized light.
[CrossRef]

H. Jensen, J. Schellman, T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192–200 (1978).
[CrossRef]

Scheraga, H.

T. Troxell, H. Scheraga, “Electric dichroism and polymer conformation. I. Theory of the optical properties of anisotropic media and method of measurement,” Macromolecules 4, 519–527 (1971).
[CrossRef]

Schurcliff, W.

W. Schurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Simon, R.

R. Simon, “The relationships between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Soleillet, P.

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).

Troxell, T.

H. Jensen, J. Schellman, T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192–200 (1978).
[CrossRef]

T. Troxell, H. Scheraga, “Electric dichroism and polymer conformation. I. Theory of the optical properties of anisotropic media and method of measurement,” Macromolecules 4, 519–527 (1971).
[CrossRef]

Weddeburn, J.

J. Weddeburn, Lectures on Matrix Theory (American Mathematical Society, Providence, R.I., 1934).

Ann. Phys. (1)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).

Appl. Spectrosc. (1)

Chem. Rev. (1)

J. Schellman, H. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987) contains an extensive bibliography of references to the chemical literature on polarized light.
[CrossRef]

J. Chem. Phys. (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

J. Comp. Phys. (1)

H. Fox, R. Barakat, “Trace-orthogonal hermitian basis matrices of artibtrary dimension,” J. Comp. Phys. 21, 326–332 (1976).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. Soc. Jpn. (1)

N. Go, “Optical activity of anisotropic solutions,” J. Phys. Soc. Jpn. 23, 88–93 (1967).
[CrossRef]

Macromolecules (1)

T. Troxell, H. Scheraga, “Electric dichroism and polymer conformation. I. Theory of the optical properties of anisotropic media and method of measurement,” Macromolecules 4, 519–527 (1971).
[CrossRef]

Opt. Commun. (2)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones matrix of polarized theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

R. Simon, “The relationships between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Other (8)

A. Richmond, “Expansions for the exponential of a sum of matrices,” in M. Gover, S. Barnett, eds., Applications of Matrix Theory (Clarendon, Oxford, 1989), pp. 283–290.

J. Weddeburn, Lectures on Matrix Theory (American Mathematical Society, Providence, R.I., 1934).

E. L. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1964).

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

E. Hille, R. Phillips, Functional Analysis and Semi-groups, revised ed. (American Mathematical Society, Providence, R.I., 1957).

P. Lax, R. Phillips, Scattering Theory (Academic, New York, 1967), Chap. 3 and App. 1.

W. Schurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

R. Horn, C. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

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Equations (66)

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Φ 0 ( ω ) = J ( ω ) Φ i ( ω ) J * ( ω ) ,
J ( ω ) = [ j 11 ( ω ) j 12 ( ω ) j 21 ( ω ) j 22 ( ω ) ] .
Φ 0 ( ω ) = 1 2 [ S 0 ( ω ) + S 1 ( ω ) S 2 ( ω ) - i S 3 ( ω ) S 2 ( ω ) + i S 3 ( ω ) S 0 ( ω ) - S 1 ( ω ) ] , Φ i ( ω ) 1 2 [ $ 0 ( ω ) + $ 1 ( ω ) $ 2 ( ω ) - i $ 3 ( ω ) $ 2 ( ω ) + i $ 3 ( ω ) $ 2 ( ω ) - $ 1 ( ω ) ] ,
J ( ω , z ) ,             Φ i ( ω , z ) ,             Φ 0 ( ω , z ) .
J ( ω , z 2 + z 1 ) = J ( ω , z 2 ) J ( ω , z 1 ) ,
J ( ω , z ) = exp [ z K 1 ( ω ) ] ,
K 1 ( ω ) = lim z 0 J ( ω , z ) - I z .
J ( ω , z 2 + z 1 ) = exp [ ( z 2 + z 1 ) K 1 ( ω ) ] .
2 E z 2 = 1 c 2 Q 2 E t 2
E = | E 1 ( t , z ) E 2 ( t , z ) |
d 2 d z 2 E ( ω , z ) + ( ω c ) 2 Q E ( ω , z ) = 0.
E ( ω , z ) U E ( ω , z )
UQU - 1 = Λ [ δ 1 0 0 δ 2 ] ,
δ 1 , 2 = ½ Tr Q ± ½ [ ( Tr Q ) 2 - 4 det Q ] 1 / 2 ,
d 2 d z 2 E ( ω , z ) + ( ω c ) 2 Λ E ( ω , z ) .
E ( ω , z ) = | E 1 ( ω , 0 ) exp ( - i ω c δ 1 1 / 2 z ) E 2 ( ω , 0 ) exp ( - i ω c δ 2 1 / 2 z ) | .
E ( ω , z ) = J ( ω , z ) E ( ω , 0 ) ,
J ( ω , z ) = [ e 1 ( ω z ) ( cos θ ) 2 + e 2 ( ω z ) ( sin θ ) 2 ½ [ e 1 ( ω z ) - e 2 ( ω z ) ] sin 2 θ ½ [ e 1 ( ω z ) - e 2 ( ω z ) ] sin 2 θ e 1 ( ω z ) ( sin θ ) 2 + e 2 ( ω z ) ( cos θ ) 2 ] .
e l ( ω z ) exp ( - i ω c δ l 1 / 2 z ) .
det U = 1 = u 11 u 22 - u 12 u 21 , u 11 2 + u 21 2 = u 12 2 + u 22 2 = 1 , u 11 u 21 + u 12 u 22 = 0.
u 11 = cos θ ,             u 12 = sin θ , u 21 = - sin θ ,             u 22 = cos θ .
K 1 ( ω ) = [ δ 1 1 / 2 ( cos θ ) 2 + δ 2 1 / 2 ( sin θ ) 2 ½ ( δ 1 1 / 2 - δ 2 1 / 2 ) sin 2 θ ½ ( δ 1 1 / 2 - δ 2 1 / 2 ) sin 2 θ δ 1 1 / 2 ( sin θ ) 2 + δ 2 1 / 2 ( cos θ ) 2 ] .
Φ 0 ( ω , z ) = J ( ω , z ) Φ i ( ω ) J * ( ω , z ) = exp ( z K 1 ( ω ) Φ i ( ω ) exp [ z K 1 ( ω ) ] .
exp ( z K 1 ) = 1 ( δ 1 - δ 2 ) { [ exp ( z δ 1 ) - exp ( z δ 2 ) ] K 1 - [ z δ 2 exp ( z δ 1 ) - z δ 1 exp ( z δ 2 ) ] I }
exp ( z K 1 ) = exp ( z δ ) K 1 - ( δ - 1 ) I
exp ( z K 1 ) = A K 1 + B I ,
Φ 0 ( ω , z ) = A ( ω , z ) 2 K 1 ( ω ) Φ i ( ω ) K 1 * ( ω ) + B ( ω , z ) 2 Φ i ( ω ) + A ( ω , z ) B * ( ω , z ) K 1 ( ω ) Φ i ( ω ) + A * ( ω , z ) B ( ω ) Φ i ( ω ) K 1 * ( ω ) .
$ ( ω ) = | $ 0 ( ω ) $ 1 ( ω ) $ 2 ( ω ) $ 3 ( ω ) | ,             S ( ω ) | S 0 ( ω ) S 1 ( ω ) S 2 ( ω ) S 3 ( ω ) |
S ( ω ) = M ( ω ) $ ( ω ) .
M ( ω ) = A [ J ( ω ) J ˜ ( ω ) ] A - 1
A = [ 1 0 0 1 1 0 0 - 1 0 1 1 0 0 i - i 0 ] ,
A - 1 = ½ A * .
M * ( ω ) GM ( ω ) = det M ( ω ) G ,
G [ 1 O - 1 - 1 O - 1 ] .
M ( ω , z ) = exp [ z N 1 ( ω ) ] ,
M ( ω , z 2 + z 1 ) = A [ J ( ω , z 2 + z 1 ) J ˜ ( ω , z 2 + z 1 ) ] A - 1 .
( ab ) ( cd ) = ( a c ) ( b d ) ,
M ( ω , z 2 + z 1 ) = A [ J ( ω , z 2 ) J ˜ ( ω , z 2 ) ] [ J ( ω , z 1 ) J ˜ ( ω , z 1 ) ] A - 1 = A [ J ( ω , z 2 ) J ˜ ( ω , z 2 ) ] A - 1 A [ J ( ω , z 1 ) J ˜ ( ω , z 1 ) ] A - 1 = M ( ω , z 2 ) M ( ω , z 1 ) .
J ˜ = exp ( K ˜ ) .
J J ˜ = ( J I 2 ) ( I 2 J ˜ ) ,
J I 2 = exp ( K I 2 ) , I 2 J ˜ = exp ( I 2 K ˜ ) .
J J ˜ = exp ( K I 2 ) exp ( I 2 K ˜ ) .
J J ˜ = exp ( K I 2 + I 2 K ˜ ) .
M = exp [ A ( K I 2 + I 2 K ˜ ) A - 1 ]
M ( ω , z ) = exp [ z N 1 ( ω ) ] ,
N 1 ( ω ) A [ K 1 ( ω ) I 2 + I 2 K ˜ 1 ( ω ) ] A - 1 .
N ( ω ) = 1 2 [ ( k 11 + k 11 * + k 22 + k 22 * ) ( k 11 + k 11 * - k 22 - k 22 * ) ( k 12 + k 12 * + k 21 + k 21 * ) i ( k 12 * - k 12 + k 21 - k 21 * ) ( k 11 + k 11 * - k 22 - k 22 * ) ( k 11 + k 11 * + k 22 + k 22 * ) ( k 12 + k 12 * - k 21 - k 21 * ) i ( k 12 * - k 12 + k 21 * - k 21 ) ( k 12 + k 12 * + k 21 + k 21 * ) - ( k 12 + k 12 * - k 21 - k 21 * ) ( k 11 + k 11 * + k 22 + k 22 * ) i ( k 11 - k 11 * + k 22 - k 22 * ) i ( k 12 * - k 12 + k 21 - k 21 * ) i ( k 12 - k 12 * + k 21 - k 21 * ) i ( k 11 * - k 11 + k 22 - k 22 * ) ( k 11 + k 11 * + k 22 + k 22 * ) ]
k j k = R j k + i I j k ,
N 1 [ R 11 + R 22 R 11 - R 22 R 12 + R 21 I 12 - I 21 R 11 - R 22 R 11 + R 22 R 12 - R 21 I 12 - I 21 R 12 + R 21 - ( R 12 - R 21 ) R 11 + R 22 - ( I 11 - I 22 ) I 12 - I 21 - ( I 12 + I 21 ) ( I 11 - I 22 ) R 11 + R 22 ] .
σ 0 = [ 1 0 0 1 ] ,             σ 1 = [ 1 0 0 - 1 ] , σ 2 = [ 0 1 1 0 ] ,             σ 3 = [ 0 - i - i 0 ] .
Tr σ j σ k = 2 δ j k .
J = 1 2 l = 0 3 j l σ l = 1 2 [ j 0 + j 1 j 2 - i j 3 j 2 + i j 3 j 0 - j 1 ] ,
j l = Tr ( J σ l )
j 0 = j 11 + j 22 , j 1 = j 11 - j 22 , j 2 = j 12 + j 21 , j 3 = i j 12 - i j 21 .
j l = j l exp ( i θ l ) exp ( - k δ l ) exp ( i θ l ) ,
J = l = 0 3 J l ,
J l exp ( - k δ l ) exp ( i θ l ) σ l .
J = [ cos ψ sin ψ sin ψ - cos ψ ] ,
J = [ cos ½ α - sin ½ α sin ½ α cos ½ α ] .
J = [ 0 - 1 1 0 ] = - i σ 3 .
K = 1 2 l = 0 3 k l σ l = 1 2 [ k 0 + k 1 k 2 - i k 3 k 2 + i k 3 k 0 - k 1 ] ,
k 0 = k 11 + k 22 = ( R 11 + R 22 ) + i ( I 11 + I 22 ) , k 1 = k 11 - k 22 = ( R 11 - R 22 ) + i ( I 11 - I 22 ) , k 2 = k 12 + k 21 = ( R 12 + R 21 ) + i ( I 12 + I 21 ) , k 3 = i k 12 - i k 21 = - ( I 12 - I 21 ) + i ( R 12 - R 21 ) .
N = [ Re k 0 Re k 1 Re k 2 Re k 3 Re k 1 Re k 0 Re k 3 Im k 2 Re k 2 - Re k 3 Re k 0 - Im k 3 Re k 3 - Im k 2 Im k 3 Re k 0 ] .
J I 2 = n = 0 1 n ! K n I 2 .
I 2 = I 2 n , K n I 2 n = ( K I 2 ) n ,
J I 2 = n = 0 1 n ! ( K I 2 ) n , = exp ( K I 2 ) .

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