Abstract

An alternative approach to the theory of polarization optical devices, based on the Dirac-algebraic formalism, is presented. The advantages of this treatment over the widespread Jones-matrix formalism are discussed. The operators of some basic homogeneous optical devices are expanded in their eigenbases and in the Cartesian basis. The global and spectral properties of some cascades of polarization devices are analyzed.

© 2003 Optical Society of America

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References

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  1. T. Tudor, “Spectral analysis of device operators: the rotating birefringent plate,” J. Opt. Soc. Am. A 18, 926–931 (2001).
    [CrossRef]
  2. T. Tudor, “Spectral analysis of the device operators in polarization dynamics,” J. Mod. Opt. 48, 1669–1689 (2001).
    [CrossRef]
  3. I Tarassov, Physique Quantique et Opérateurs Linéaires (Mir, Moscow, 1984).
  4. A Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. 1.
  5. C Cohen-Tannoudji, B Diu, F Laloë, Quantum Mechanics (Wiley, New York, 1977), Vol. 1.
  6. P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1947).
  7. C Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).
  9. S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  10. T Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).
  11. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  12. M. Richartz, H.-Y. Hsü, “Analysis of elliptical polarization,” J. Opt. Soc. Am. 39, 136–157 (1949).
    [CrossRef]

2001 (2)

T. Tudor, “Spectral analysis of device operators: the rotating birefringent plate,” J. Opt. Soc. Am. A 18, 926–931 (2001).
[CrossRef]

T. Tudor, “Spectral analysis of the device operators in polarization dynamics,” J. Mod. Opt. 48, 1669–1689 (2001).
[CrossRef]

1994 (1)

1949 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).

Brosseau, C

C Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

Chipman, R. A.

Cohen-Tannoudji, C

C Cohen-Tannoudji, B Diu, F Laloë, Quantum Mechanics (Wiley, New York, 1977), Vol. 1.

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1947).

Diu, B

C Cohen-Tannoudji, B Diu, F Laloë, Quantum Mechanics (Wiley, New York, 1977), Vol. 1.

Hsü, H.-Y.

Kato, T

T Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).

Laloë, F

C Cohen-Tannoudji, B Diu, F Laloë, Quantum Mechanics (Wiley, New York, 1977), Vol. 1.

Lu, S.-Y.

Messiah, A

A Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. 1.

Richartz, M.

Shurcliff, W. A.

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

Tarassov, I

I Tarassov, Physique Quantique et Opérateurs Linéaires (Mir, Moscow, 1984).

Tudor, T.

T. Tudor, “Spectral analysis of device operators: the rotating birefringent plate,” J. Opt. Soc. Am. A 18, 926–931 (2001).
[CrossRef]

T. Tudor, “Spectral analysis of the device operators in polarization dynamics,” J. Mod. Opt. 48, 1669–1689 (2001).
[CrossRef]

J. Mod. Opt. (1)

T. Tudor, “Spectral analysis of the device operators in polarization dynamics,” J. Mod. Opt. 48, 1669–1689 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Other (8)

T Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).

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

I Tarassov, Physique Quantique et Opérateurs Linéaires (Mir, Moscow, 1984).

A Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1965), Vol. 1.

C Cohen-Tannoudji, B Diu, F Laloë, Quantum Mechanics (Wiley, New York, 1977), Vol. 1.

P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, Oxford, UK, 1947).

C Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, 1996).

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Equations (45)

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M J ( ϕ ,   Ω ,   t )
= cos   ϕ 1 0 0 1 + 1 2 sin   ϕ i 1 1 - i exp ( 2 i Ω t ) + 1 2 sin   ϕ i - 1 - 1 - i exp ( - 2 i Ω t ) ,
D ( ϕ ,   Ω ,   t ) = cos   ϕ I + i   sin   ϕ | L R | exp ( 2 i Ω t ) + i   sin   ϕ | R L | exp ( - 2 i Ω t ) ,
| E out = D ( ϕ ,   Ω ,   t ) | E in = cos   ϕ I | E × exp ( i ω t ) + i   sin   ϕ | L R | E × exp [ i ( ω + 2 Ω ) t ] + i   sin   ϕ | R L | E × exp [ i ( ω - 2 Ω ) t ] .
D = λ 1 | S 1 S 1 | + λ 2 | S 2 S 2 | ,
λ 1 = exp ( i δ / 2 ) , λ 2 = exp ( - i δ / 2 ) ,
R | E M ( δ ) = | E M E M | exp ( i δ / 2 ) + | E m E m | exp ( - i δ / 2 ) .
R | P θ ( δ ) = | P θ P θ | exp ( i δ / 2 ) + | P θ + 90 ° × P θ + 90 ° | exp ( - i δ / 2 ) .
R | R ( δ ) = | R R | exp ( i δ / 2 ) + | L L | exp ( - i δ / 2 ) ,
P | E = | E E | .
P | P x = | P x P x |
P | P θ = | P θ P θ |
Q | E l ( α ) = | E l E l | exp ( α / 2 )   + | E h E h | exp ( - α / 2 ) .
| R = 1 2   ( | P x + i | P y ) ,
| L = 1 2   ( | P x - i | P y ) .
| P x = 1 2   ( | R   + | L ) ,
| P y = 1 2   i ( | L - | R ) .
| P θ = cos   θ | P x + sin   θ | P y .
P | P θ = ( cos   θ | P x + sin   θ | P y ) ( cos   θ P x | + sin   θ P y | ) = cos 2   θ | P x P x | + sin   θ   cos   θ ( | P x P y | + | P y P x | ) + sin 2   θ | P y P y | .
R | P θ ( δ ) = [ cos 2   θ   exp ( i δ / 2 ) + sin 2   θ   exp ( - i δ / 2 ) ]   × | P x P x | + 2 i   sin   θ   cos   θ   sin δ 2   ( | P x P y | + | P y P x | ) + [ sin 2   θ   exp ( i δ / 2 ) + cos 2   θ   exp ( - i δ / 2 ) ] | P y P y | = cos δ 2 + i   cos   2 θ   sin δ 2 | P x P x | + i   sin   2 θ   sin δ 2   ( | P x P y | + | P y P x | ) + cos δ 2 - i   cos   2 θ   sin δ 2 | P y P y | ,
R | R ( δ ) = cos δ 2   ( | P x P x | + | P y P y | ) + sin δ 2   ( | P x P y | - | P y P x | ) = cos δ 2 + sin δ 2   ( | P x P y | - | P y P x | ) ,
R | L ( δ ) = cos δ 2   ( | P x P x | + | P y P y | ) - sin δ 2   ( | P x P y | - | P y P x | ) = cos δ 2 - sin δ 2   ( | P x P y | - | P y P x | ) ,
D = i = 1 2 j = 1 2 S i | D | S j | S i S j | ,
C = R | P x ( π / 2 ) P | P 45 ° .
C = [ | P x P x | exp ( i π / 4 ) + | P y P y | exp ( - i π / 4 ) ] × 1 2 ( 1 + | P x P y | + | P y P x | ) = 1 2 [ | P x P x | exp ( i π / 4 ) + | P y P y | exp ( - i π / 4 ) + | P x P y | exp ( i π / 4 ) + | P y P x | exp ( - i π / 4 ) ] = 1 2 | P x ( P x | + P y | ) exp ( i π / 4 ) + 1 2 | P y ( P x | + P y | ) exp ( - i π / 4 ) = 1 2 [ | P x + | P y exp ( - i π / 2 ) ] ( P x | + P y | ) exp ( i π / 4 ) = 1 2   ( | P x - i | P y )   1 2   ( P x | + P y | ) exp ( i π / 4 ) ,
C l | L P 45 ° | ,
R | P 45 ° ( π ) R | P x ( π ) = i ( | P x P y | + | P y P x | ) × i ( | P x P x | - | P y P y | ) = | P x P y | - | P y P x | ,
R | P 45 ° ( π ) R | P x ( π ) = R | R ( π ) = - R | L ( π ) .
R | P θ ( π ) R | P x ( π ) .
R | P θ ( π ) R | P x ( π ) = - R | L ( 4 θ ) .
R | P x ( π ) R | P - θ ( π ) .
R | P x ( π ) R | P - θ ( π ) = - R | L ( 4 θ ) .
R | P θ / 2 ( π ) P | P x = i   cos   θ | P x P x | + i   sin   θ | P y P x | ,
P | P x R | P - θ / 2 ( π )
= | P x P x | ( i   cos   θ | P x P x | - i   sin   θ | P x P y | - i   sin   θ | P y P x | - i   cos   θ | P y P y | ) = i   cos   θ | P x P x | - i   sin   θ | P x P y | ;
R | P θ / 2 ( π ) P | P x = i ( cos   θ | P x + sin   θ | P y ) P x | | P θ P x | .
| P θ , with λ 1 = P x | P θ = cos   θ ,
| P y , with λ 2 = 0 .
P | P θ P | P x = cos   θ | P θ P x | .
| P θ , with λ 1 = cos   θ P x | P θ = cos 2   θ ,
| P y , with λ 2 = 0 ,
P | P x P | P - θ = cos   θ | P x P - θ | .
| P x , with λ 1 = cos   θ P - θ | P x = cos 2   θ ,
| P 90 ° - θ , with λ 2 = 0 .
η = | P θ | P y | = | sin   θ | .

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