Abstract

Techniques for analysis and synthesis of linear optical systems involving polarization effects are related and extended through fuller exploitation of the algebra of complex 2 × 2 matrices.

© 1971 Optical Society of America

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References

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  1. F. E. Wright, J. Opt. Soc. Am. 20, 529 (1930).
    [Crossref]
  2. U. Fano, J. Opt. Soc. Am. 39, 859 (1949).
    [Crossref]
  3. M. Richartz and Hsien-yu Hsü, J. Opt. Soc. Am. 39, 136 (1949).
    [Crossref]
  4. G. A. Deschamps, Proc. IRE 39, 540 (1951).
    [Crossref]
  5. G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [Crossref]
  6. H. G. Jerrard, J. Opt. Soc. Am. 44, 634 (1954).
    [Crossref]
  7. S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 247 (1956).
  8. S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 398 (1956).
  9. H. C. Ko, Proc. IRE 50, 1950 (1962).
    [Crossref]
  10. W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N. J., 1964).
  11. G. G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852); reprinted in Mathematical and Physical Papers(Cambridge U. P., New York, 1901), Vol. III, p. 233.
  12. G. B. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).
    [Crossref]
  13. R. W. Schmieder, J. Opt. Soc. Am. 59, 297 (1969).
    [Crossref]
  14. B. Rossi, Optics (Addison–Wesley, Reading, Mass., 1957).
  15. A. Blanc-Lapierre and P. Dumontet, Rev. Opt. 34, 1 (1955).
  16. H. A. Tolhoek, Rev. Mod. Phys. 28, 277 (1956).
    [Crossref]
  17. E. Wolf, Nuovo Cimento 13, 1165 (1959).
    [Crossref]
  18. F. Perrin, J. Chem. Phys. 10, 415 (1942).
    [Crossref]
  19. H. Poincaré, Théorie Mathematique de la Lumière, Vol. 2 (Carré, Paris, 1892).
  20. E. Collett, Am. J. Phys. 36, 713 (1968).
    [Crossref]
  21. R. Barakat, J. Opt. Soc. Am. 53, 317 (1963).
    [Crossref]
  22. P. Rastall, Rev. Mod. Phys. 36, 820 (1964).
    [Crossref]
  23. N. G. Parke, J. Math. Phys. 28, 131 (1949).
  24. P. Soleillet, Ann. Phys. (10),  12, 33 (1929).
  25. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  26. H. Hurwitz and R. C. Jones, J. Opt. Soc. Am. 31, 493 (1941).
    [Crossref]
  27. R. C. Jones, J. Opt. Soc. Am. 31, 500 (1941).
    [Crossref]
  28. R. C. Jones, J. Opt. Soc. Am. 32, 486 (1942).
    [Crossref]
  29. R. C. Jones, J. Opt. Soc. Am. 37, 107 (1947).
    [Crossref]
  30. R. C. Jones, J. Opt. Soc. Am. 37, 110 (1947).
    [Crossref]
  31. R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948).
    [Crossref]
  32. R. C. Jones, J. Opt. Soc. Am. 46, 126 (1956).
    [Crossref]
  33. R. Menasse, Ph.D. thesis, MIT, 1955.
  34. P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (Van Nostrand, New York, 1958).
  35. I. M. Gelfand, Lectures on Linear Algebra (Wiley–Interscience, New York, 1961).
  36. A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).
  37. E. T. Whittaker, Analytic Dynamics (Cambridge U. P., New York, 1964).

1969 (1)

1968 (1)

E. Collett, Am. J. Phys. 36, 713 (1968).
[Crossref]

1965 (1)

1964 (1)

P. Rastall, Rev. Mod. Phys. 36, 820 (1964).
[Crossref]

1963 (1)

1962 (1)

H. C. Ko, Proc. IRE 50, 1950 (1962).
[Crossref]

1960 (1)

G. B. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).
[Crossref]

1959 (1)

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[Crossref]

1956 (4)

H. A. Tolhoek, Rev. Mod. Phys. 28, 277 (1956).
[Crossref]

S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 247 (1956).

S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 398 (1956).

R. C. Jones, J. Opt. Soc. Am. 46, 126 (1956).
[Crossref]

1955 (1)

A. Blanc-Lapierre and P. Dumontet, Rev. Opt. 34, 1 (1955).

1954 (1)

1952 (1)

1951 (1)

G. A. Deschamps, Proc. IRE 39, 540 (1951).
[Crossref]

1949 (3)

1948 (1)

1947 (2)

1942 (2)

1941 (3)

1930 (1)

1929 (1)

P. Soleillet, Ann. Phys. (10),  12, 33 (1929).

1852 (1)

G. G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852); reprinted in Mathematical and Physical Papers(Cambridge U. P., New York, 1901), Vol. III, p. 233.

Ballard, S. S.

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N. J., 1964).

Barakat, R.

Blanc-Lapierre, A.

A. Blanc-Lapierre and P. Dumontet, Rev. Opt. 34, 1 (1955).

Collett, E.

E. Collett, Am. J. Phys. 36, 713 (1968).
[Crossref]

Deschamps, G. A.

G. A. Deschamps, Proc. IRE 39, 540 (1951).
[Crossref]

Dumontet, P.

A. Blanc-Lapierre and P. Dumontet, Rev. Opt. 34, 1 (1955).

Fano, U.

Gelfand, I. M.

I. M. Gelfand, Lectures on Linear Algebra (Wiley–Interscience, New York, 1961).

Halmos, P. R.

P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (Van Nostrand, New York, 1958).

Hsü, Hsien-yu

Hurwitz, H.

Jerrard, H. G.

Jones, R. C.

Ko, H. C.

H. C. Ko, Proc. IRE 50, 1950 (1962).
[Crossref]

Marathay, A. S.

Menasse, R.

R. Menasse, Ph.D. thesis, MIT, 1955.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 247 (1956).

S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 398 (1956).

Parke, N. G.

N. G. Parke, J. Math. Phys. 28, 131 (1949).

Parrent, G. B.

G. B. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).
[Crossref]

Perrin, F.

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

Poincaré, H.

H. Poincaré, Théorie Mathematique de la Lumière, Vol. 2 (Carré, Paris, 1892).

Ramachandran, G. N.

Ramaseshan, S.

Rastall, P.

P. Rastall, Rev. Mod. Phys. 36, 820 (1964).
[Crossref]

Richartz, M.

Roman, P.

G. B. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).
[Crossref]

Rossi, B.

B. Rossi, Optics (Addison–Wesley, Reading, Mass., 1957).

Schmieder, R. W.

Shurcliff, W. A.

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N. J., 1964).

Soleillet, P.

P. Soleillet, Ann. Phys. (10),  12, 33 (1929).

Stokes, G. G.

G. G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852); reprinted in Mathematical and Physical Papers(Cambridge U. P., New York, 1901), Vol. III, p. 233.

Tolhoek, H. A.

H. A. Tolhoek, Rev. Mod. Phys. 28, 277 (1956).
[Crossref]

Whittaker, E. T.

E. T. Whittaker, Analytic Dynamics (Cambridge U. P., New York, 1964).

Wolf, E.

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[Crossref]

Wright, F. E.

Am. J. Phys. (1)

E. Collett, Am. J. Phys. 36, 713 (1968).
[Crossref]

Ann. Phys. (10) (1)

P. Soleillet, Ann. Phys. (10),  12, 33 (1929).

J. Chem. Phys. (1)

F. Perrin, J. Chem. Phys. 10, 415 (1942).
[Crossref]

J. Math. Phys. (1)

N. G. Parke, J. Math. Phys. 28, 131 (1949).

J. Opt. Soc. Am. (16)

Nuovo Cimento (2)

G. B. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).
[Crossref]

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[Crossref]

Proc. Indian Acad. Sci. (2)

S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 247 (1956).

S. Pancharatnam, Proc. Indian Acad. Sci. 44A, 398 (1956).

Proc. IRE (2)

H. C. Ko, Proc. IRE 50, 1950 (1962).
[Crossref]

G. A. Deschamps, Proc. IRE 39, 540 (1951).
[Crossref]

Rev. Mod. Phys. (2)

H. A. Tolhoek, Rev. Mod. Phys. 28, 277 (1956).
[Crossref]

P. Rastall, Rev. Mod. Phys. 36, 820 (1964).
[Crossref]

Rev. Opt. (1)

A. Blanc-Lapierre and P. Dumontet, Rev. Opt. 34, 1 (1955).

Trans. Cambridge Phil. Soc. (1)

G. G. Stokes, Trans. Cambridge Phil. Soc. 9, 399 (1852); reprinted in Mathematical and Physical Papers(Cambridge U. P., New York, 1901), Vol. III, p. 233.

Other (7)

B. Rossi, Optics (Addison–Wesley, Reading, Mass., 1957).

H. Poincaré, Théorie Mathematique de la Lumière, Vol. 2 (Carré, Paris, 1892).

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N. J., 1964).

E. T. Whittaker, Analytic Dynamics (Cambridge U. P., New York, 1964).

R. Menasse, Ph.D. thesis, MIT, 1955.

P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (Van Nostrand, New York, 1958).

I. M. Gelfand, Lectures on Linear Algebra (Wiley–Interscience, New York, 1961).

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Figures (4)

Fig. 1
Fig. 1

States of + and − helicity, x and y linear polarization, and arbitrary polarization on a modified Poincaré sphere.

Fig. 2
Fig. 2

Synthesis of phase shifter.

Fig. 3
Fig. 3

Alternate synthesis of phase shifter.

Fig. 4
Fig. 4

Synthesis of general polarizer.

Equations (46)

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σ 0 = [ 1 0 0 1 ] ,
σ = { σ 1 , σ 2 , σ 3 } ,
σ 1 = [ 0 1 1 0 ] , σ 2 = [ 0 - i i 0 ] , σ 3 = [ 1 0 0 - 1 ] .
S = s 0 σ 0 + s · σ .
s = s ŝ ,
ψ / 2 = tilt of ellipse from x axis tan [ 1 2 ( 1 2 π - θ ) ] = minor axis / major axis .
S = s 0 2 - s 2
S σ 0 = S S = S S ,
S = s 0 σ 0 - s · σ .
S = V S V .
V = V N V N - 1 V 1 .
V 1 = k exp ( v · σ ) ,
V = k N exp ( N v · σ ) .
V 1 = v 0 ( σ 0 + v ˆ · σ )
V = ( 2 v 0 ) N - 1 V 1 .
V = U H ,
( A B ) = B A
U = U - 1
H = H
( a · σ ) ( b · σ ) = a · b σ 0 + i ( a × b ) · σ .
W = V V = w 0 σ 0 + w · σ ,
W = H 2 = ( h 0 2 + h 2 ) σ 0 + 2 h 0 h · σ .
h 0 = 1 2 [ ( w 0 + w ) 1 2 + ( w 0 - w ) 1 2 ]
h = 1 2 [ ( w 0 + w ) 1 2 - ( w 0 - w ) 1 2 ] .
U H = ( u 0 h 0 + u · h ) σ 0 + ( u 0 h + h 0 u + i u × h ) · σ
u 0 = Re ( v 0 ) / h 0 ,             u = Im ( v ) / h 0 ,
s 0 = ( h 0 2 + h 2 ) s 0 + 2 h 0 h · s ,
S = U ± = σ 0 ± u ˆ · σ ,             or             H ± = σ 0 ± ĥ · σ .
U = exp [ ( - i ϕ / 2 ) u ˆ · σ ˆ ] .
H = h 0 σ 0 + h · σ .
D = 4 S / [ Tr ( S ) ] 2 = 1 - P 2 ,
D / D = { H 2 [ Tr ( S ) ] 2 } / { [ Tr ( H 2 S ) ] 2 } .
s / s 0 < - h / h 0
S = H ( H ) 2 H
s 0 = ( h 0 2 + h 2 ) s 0 + 2 h 0 h · s
s = [ s 0 - ( h 0 2 + h 2 ) s 0 ] / 2 h 0 h .
H = U r H l U r - 1 .
H = U ret 1 H l U ret 1 - 1 ,
U U ret 1 = U ret 2 U rot .
V = U ret2 U rot H l U ret1 - 1 .
U U ret1 = U ret2 U ret3 ,
V n = v n 0 σ 0 + v n · σ ,
V = U H = ( u 0 h 0 + u · h ) σ 0 + ( u 0 h + h 0 u + i u × h ) · σ .
( π - ϕ 1 ) / 2 < θ < ( π + ϕ 1 ) / 2.
π / 2 - ϕ < θ < π / 2 + ϕ .
1 2 Tr ( Σ i V i S V i ) = 1 2 Tr [ Σ i ( V V ) i S ] = 1 2 Tr ( S ) ,