Abstract

A general analytical method using the formalisms of polarization coherency and Jones’s matrices is provided for the evaluation of all polarization effects in fourier spectroscopy. The method applies to any incident state of arbitrary (complete, random, or partial) polarization. Inversely, it may also be used for determining the intensity and state of polarization of the source of light. TE- and TM-mode reflectivity and transmissivity for beam splitters and the dependence of these quantities on the incident polarization are obtained. It is demonstrated that three different efficiencies for these optical components must be introduced. Interferometer efficiency expressions for the source beam and the detector beam are also derived and shown to be essentially different from the previous efficiencies. Polarization effects of beam splitters, reflectors, and their composite combinations (interferometers) are investigated in detail. General conditions for complete or restricted polarization compensation are derived. Theoretical SNR expressions for both the source beam and the detector beam are also obtained; these formulas specifically account for the incident state of polarization, the polarization effects of the interferometer, and make use of the exact expressions for the appropriate interferometer efficiency. In an Appendix, a brief comparison is made between some usual representations of the state of wave polarization.

© 1972 Optical Society of America

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  1. Such an experiment has in fact been carried out by Loewenstein and Engelsrath [J. Phys. 28, C2-153 (1967)] using a far ir beam splitter consisting of an unsupported thin film of Mylar (polyethylene terephthalate). These authors succeeded in obtaining an almost completely polarized beam, perpendicular to the plane of incidence onto the beam splitter, despite the fact that Mylar displays both birefringence and optical activity. Other similar experiments using polarizing beam splitters are discussed by W. H. Steel, Interferometry (Harvard U.P., Cambridge, 1967), pp 101–102.
  2. Some exceptions to this are the works of W. H. Steel, Opt. Acta 11, 211, 1964; G. Roland, Thesis, Université de Liège, Belgium, 1965; A. G. Tescher, J. Opt. Soc. Am. 56, 554 (1966); Proc. Aspen Int. Conf. on Fourier Spectroscopy, 1970, AFCRL 71-0019, Special Report 114, G. A. Vanasse, A. T. Stair, D. J. Baker, Eds. (1971); and E. V. Loewenstein, T. Engelsrath [J. Phys. 28, C2-153 (1967)]. It should be emphasized, however, that our approach is more general and systematic. It is capable of handling arbitrary states of polarization (random, partial, or complete). It provides a theory, applicable to any two-beam interferometer, that includes the effects from all interferometer components and that also considers all four polarization parameters.
    [CrossRef]
  3. A. L. Fymat, K. D. Abhyankar, Appl. Opt. 9, 1075 (1970); Aspen Int. Conf. on Fourier Spectroscopy, 1970, AFCRL 71-0019, Special Report 114 (1971), p. 377; NASA Technical Brief 70-10405 (November1970).
    [CrossRef] [PubMed]
  4. J. S. Hall, J. Opt. Soc. Am. 41, 963 (1951).
    [CrossRef]
  5. E. P. Clancy, J. Opt. Soc. Am. 42, 357 (1952).
    [CrossRef]
  6. R. C. Jones, J. Opt. Soc. Am. 31, 488, 493 (1941); J. Opt. Soc. Am. 32, 486 (1942); J. Opt. Soc. Am. 37, 107, 110 (1947); J. Opt. Soc. Am. 38, 671 (1948).
    [CrossRef]
  7. W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962), Appendix 2.
  8. E. Wolf, Nuovo Cimento (Ser. X) 13, 1165 (1959).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chaps. 1 and 10.
  10. L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  11. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 10.
  12. A. S. Marathay, in Introduction to Statistical Optics, E. L. O’Neill (Addison-Wesley, Reading, Mass., 1963), Chap. 9.
  13. A. L. Fymat, Appl. Opt. 10, 2499 (1971).
    [CrossRef] [PubMed]
  14. A. L. Fymat, Appl. Opt. 10, 2709 (1971).
  15. D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).
  16. W. H. Steel, Interferometry (Harvard U.P., Cambridge, 1967), p. 37.
  17. J. Terrien, Rev. Opt. 38, 29 (1957).
  18. M. V. R. K. Murty, J. Opt. Soc. Am. 50, 83 (1960).
    [CrossRef]
  19. H. Sakai, in Proc. Aspen Int. Conf. on Fourier Spectroscopy, AFCRL 71-0019, Special Report 114 (1971), p. 19.
  20. A. L. Fymat (submitted for publication).
  21. G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 259.
    [CrossRef]
  22. S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950), Chap. 1.
  23. M. A. F. Thiel, Beitr. Radioastron. Report Max Planck Institut für Radioastronomie, Bonn, 1, 114 (1970).
  24. G. B. Parrent, P. Roman, Nuovo Cimento 15, 370 (1960).
    [CrossRef]
  25. A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).

1971 (2)

A. L. Fymat, Appl. Opt. 10, 2709 (1971).

A. L. Fymat, Appl. Opt. 10, 2499 (1971).
[CrossRef] [PubMed]

1970 (1)

1967 (1)

Such an experiment has in fact been carried out by Loewenstein and Engelsrath [J. Phys. 28, C2-153 (1967)] using a far ir beam splitter consisting of an unsupported thin film of Mylar (polyethylene terephthalate). These authors succeeded in obtaining an almost completely polarized beam, perpendicular to the plane of incidence onto the beam splitter, despite the fact that Mylar displays both birefringence and optical activity. Other similar experiments using polarizing beam splitters are discussed by W. H. Steel, Interferometry (Harvard U.P., Cambridge, 1967), pp 101–102.

1965 (2)

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).

1964 (1)

Some exceptions to this are the works of W. H. Steel, Opt. Acta 11, 211, 1964; G. Roland, Thesis, Université de Liège, Belgium, 1965; A. G. Tescher, J. Opt. Soc. Am. 56, 554 (1966); Proc. Aspen Int. Conf. on Fourier Spectroscopy, 1970, AFCRL 71-0019, Special Report 114, G. A. Vanasse, A. T. Stair, D. J. Baker, Eds. (1971); and E. V. Loewenstein, T. Engelsrath [J. Phys. 28, C2-153 (1967)]. It should be emphasized, however, that our approach is more general and systematic. It is capable of handling arbitrary states of polarization (random, partial, or complete). It provides a theory, applicable to any two-beam interferometer, that includes the effects from all interferometer components and that also considers all four polarization parameters.
[CrossRef]

1960 (2)

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

M. V. R. K. Murty, J. Opt. Soc. Am. 50, 83 (1960).
[CrossRef]

1959 (1)

E. Wolf, Nuovo Cimento (Ser. X) 13, 1165 (1959).
[CrossRef]

1957 (1)

J. Terrien, Rev. Opt. 38, 29 (1957).

1952 (1)

1951 (1)

1946 (1)

D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).

1941 (1)

R. C. Jones, J. Opt. Soc. Am. 31, 488, 493 (1941); J. Opt. Soc. Am. 32, 486 (1942); J. Opt. Soc. Am. 37, 107, 110 (1947); J. Opt. Soc. Am. 38, 671 (1948).
[CrossRef]

Abhyankar, K. D.

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 10.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chaps. 1 and 10.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950), Chap. 1.

Clancy, E. P.

Fymat, A. L.

Gabor, D.

D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).

Hall, J. S.

Jones, R. C.

R. C. Jones, J. Opt. Soc. Am. 31, 488, 493 (1941); J. Opt. Soc. Am. 32, 486 (1942); J. Opt. Soc. Am. 37, 107, 110 (1947); J. Opt. Soc. Am. 38, 671 (1948).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Marathay, A. S.

A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).

A. S. Marathay, in Introduction to Statistical Optics, E. L. O’Neill (Addison-Wesley, Reading, Mass., 1963), Chap. 9.

Murty, M. V. R. K.

Parrent, G. B.

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

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 10.

Roman, P.

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

Sakai, H.

H. Sakai, in Proc. Aspen Int. Conf. on Fourier Spectroscopy, AFCRL 71-0019, Special Report 114 (1971), p. 19.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 259.
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962), Appendix 2.

Steel, W. H.

Some exceptions to this are the works of W. H. Steel, Opt. Acta 11, 211, 1964; G. Roland, Thesis, Université de Liège, Belgium, 1965; A. G. Tescher, J. Opt. Soc. Am. 56, 554 (1966); Proc. Aspen Int. Conf. on Fourier Spectroscopy, 1970, AFCRL 71-0019, Special Report 114, G. A. Vanasse, A. T. Stair, D. J. Baker, Eds. (1971); and E. V. Loewenstein, T. Engelsrath [J. Phys. 28, C2-153 (1967)]. It should be emphasized, however, that our approach is more general and systematic. It is capable of handling arbitrary states of polarization (random, partial, or complete). It provides a theory, applicable to any two-beam interferometer, that includes the effects from all interferometer components and that also considers all four polarization parameters.
[CrossRef]

W. H. Steel, Interferometry (Harvard U.P., Cambridge, 1967), p. 37.

Terrien, J.

J. Terrien, Rev. Opt. 38, 29 (1957).

Thiel, M. A. F.

M. A. F. Thiel, Beitr. Radioastron. Report Max Planck Institut für Radioastronomie, Bonn, 1, 114 (1970).

Vanasse, G. A.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 259.
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

E. Wolf, Nuovo Cimento (Ser. X) 13, 1165 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chaps. 1 and 10.

Appl. Opt. (3)

J. Inst. Elec. Eng. (London) (1)

D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).

J. Opt. Soc. Am. (5)

M. V. R. K. Murty, J. Opt. Soc. Am. 50, 83 (1960).
[CrossRef]

J. S. Hall, J. Opt. Soc. Am. 41, 963 (1951).
[CrossRef]

E. P. Clancy, J. Opt. Soc. Am. 42, 357 (1952).
[CrossRef]

R. C. Jones, J. Opt. Soc. Am. 31, 488, 493 (1941); J. Opt. Soc. Am. 32, 486 (1942); J. Opt. Soc. Am. 37, 107, 110 (1947); J. Opt. Soc. Am. 38, 671 (1948).
[CrossRef]

A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).

J. Phys. (1)

Such an experiment has in fact been carried out by Loewenstein and Engelsrath [J. Phys. 28, C2-153 (1967)] using a far ir beam splitter consisting of an unsupported thin film of Mylar (polyethylene terephthalate). These authors succeeded in obtaining an almost completely polarized beam, perpendicular to the plane of incidence onto the beam splitter, despite the fact that Mylar displays both birefringence and optical activity. Other similar experiments using polarizing beam splitters are discussed by W. H. Steel, Interferometry (Harvard U.P., Cambridge, 1967), pp 101–102.

Nuovo Cimento (1)

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

Nuovo Cimento (Ser. X) (1)

E. Wolf, Nuovo Cimento (Ser. X) 13, 1165 (1959).
[CrossRef]

Opt. Acta (1)

Some exceptions to this are the works of W. H. Steel, Opt. Acta 11, 211, 1964; G. Roland, Thesis, Université de Liège, Belgium, 1965; A. G. Tescher, J. Opt. Soc. Am. 56, 554 (1966); Proc. Aspen Int. Conf. on Fourier Spectroscopy, 1970, AFCRL 71-0019, Special Report 114, G. A. Vanasse, A. T. Stair, D. J. Baker, Eds. (1971); and E. V. Loewenstein, T. Engelsrath [J. Phys. 28, C2-153 (1967)]. It should be emphasized, however, that our approach is more general and systematic. It is capable of handling arbitrary states of polarization (random, partial, or complete). It provides a theory, applicable to any two-beam interferometer, that includes the effects from all interferometer components and that also considers all four polarization parameters.
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Rev. Opt. (1)

J. Terrien, Rev. Opt. 38, 29 (1957).

Other (10)

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 10.

A. S. Marathay, in Introduction to Statistical Optics, E. L. O’Neill (Addison-Wesley, Reading, Mass., 1963), Chap. 9.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chaps. 1 and 10.

W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962), Appendix 2.

H. Sakai, in Proc. Aspen Int. Conf. on Fourier Spectroscopy, AFCRL 71-0019, Special Report 114 (1971), p. 19.

A. L. Fymat (submitted for publication).

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 259.
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950), Chap. 1.

M. A. F. Thiel, Beitr. Radioastron. Report Max Planck Institut für Radioastronomie, Bonn, 1, 114 (1970).

W. H. Steel, Interferometry (Harvard U.P., Cambridge, 1967), p. 37.

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Tables (1)

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Table I Comparison of Polarization

Equations (118)

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E ( σ , t ) = [ E x ( σ , t ) E y ( σ , t ) ]
E j ( σ , t ) = A j ( σ , t ) exp ( - i τ ) ,
= lim t 1 2 t - t t d t .
μ x y = J x y ( J x x ) 1 2 ( J y y ) 1 2 = μ x y exp ( i γ x y ) ,
μ x y = 0 : random polarization ( if , in addition , J x x = J y y ) , 0 < μ x y < 1 : partial polarization , μ x y = 1 : complete polarization .
J ( σ ) = E ( σ , t ) × E ( σ , t ) = ( E x E x * E x E y * E y E x * E y E y * ) ,
[ J i j ( σ ) ] = ( J x x J y y J y x J y y ) .
P = ( 1 - { 4 D ( J ) / [ T ( J ) ] 2 } ) 1 2 .
P R = ( J x x - J y y ) / ( J x x + J y y ) = P ( J x y = 0 ) .
E = KE ,
J = E × E = KJK .
K ( α ) = ( cos α - sin α sin α cos α ) .
J = 1 2 ( 1 - i i 1 )
J = ( cos α - sin α sin α cos α ) 1 2 ( 1 - i i 1 ) ( cos α sin α - sin α cos α ) = 1 2 ( 1 - i i 1 ) .
J = ( cos α - sin α sin α cos α ) ( 1 0 0 0 ) ( cos α sin α - sin α cos α ) = ( cos 2 α cos α sin α cos α sin α sin 2 α ) .
K = n K n ,             n = 1 , 2 , , N .
K = ( cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ ) ( exp ( i ) 0 0 exp ( - i ) ) = ( exp ( i ) cos 2 θ exp ( - i ) cos θ sin θ exp ( i ) cos θ sin θ exp ( - i ) sin 2 θ ) ,
J = [ exp ( i ) cos 2 θ exp ( - i ) cos θ sin θ exp ( i ) cos θ sin θ exp ( - i ) sin 2 θ ] 1 2 ( 1 1 1 1 ) × [ exp ( - i ) cos 2 θ exp ( - i ) cos θ sin θ exp ( i ) cos θ sin θ exp ( i ) sin 2 θ ] = ( cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ ) ( 1 + sin 2 θ cos 2 ) .
I ( θ , ) = 1 + sin 2 θ cos 2 .
J ( i ) = ( J i j i ) = c ν 1 4 π ( A 2 - A A * - A A * A 2 ) cos θ 1 ,
J ( r ) = ( J i j r ) = c ν 1 4 π ( R 2 - R R * - R R * R 2 ) cos θ 1 ,
J ( t ) = ( J i j t ) = c ν l 4 π ( T 2 - T T * - T T * T 2 ) cos θ l ,
r = R / A , t = T / A , r = R / A , r = R / A , t = T / A , t = T / A ,
J ( r ) = S ( r ) J ( i ) S ( r ) = ( r 2 J x x i r r * J x y i r r * J y x i r 2 J y y i ) ,
J ( t ) = p l p 1 S ( t ) J ( i ) S ( t ) = p l p 1 ( t 2 J x x i t t * J x y i t t * J y x i t 2 J y y i ) ,
TE mode reflectivity :             R = J x x r J x x i = r 2 ,
TE mode transmissivity :             J = J x x t J x x i = p l p 1 t 2 ,
TM mode reflectivity :             R = J y y r J y y i = r 2 ,
TM mode transmissivity :             J = J y y t J y y i = p l p 1 t 2 ,
Total reflectivity :             R = J x x r + J y y r J x x i + J y y i = R J x x i + R J y y i J x x i + J y y i = r 2 ,
Total transmissivity :             J = J x x t + J y y t J x x i + J y y i = J J x x i + J J y y i J x x i + J y y i = ( p l p 1 ) t 2 .
J x x r + J x x t = J x x i ,             J y y r + J y y t = J y y i , ( J x x r + J y y r ) + ( J x x t + J y y t ) = J x x i + J y y i ,
R + J = 1 = r 2 + p l p 1 t 2 ,
R + J = 1 = r 2 + p l p 1 t 2 ,
R + J = 1 = r 2 + p l p 1 t 2 .
P ( r ) = [ 1 - 4 R R ( J x x i J y y i - J x y i J y x i ) ( R J x x i + R J y y i ) 2 ] 1 2 ,
P ( t ) = [ 1 - 4 J J ( J x x i J y y i - J x y i J y x i ) ( J J x x i + J J y y i ) 2 ] 1 2 .
P ( r ) = R J x x i - R J y y i R J x x i + R J y y i ,
P ( t ) = J J x x i - J J y y i J J x x i + J J y y i .
η ¯ = 4 R J .
S k ( r ) = ( r 0 0 r ) k ,             S k ( t ) = ( t 0 0 t ) k ,             k = 0 , 1 , 2.
N l = N ¯ l = 2 S l ( r ) S l ( t ) , N 2 = N ¯ 2 = 2 S 2 ( r ) S 2 ( t ) , l = 0 , 1
N k = ( n 0 0 n ) k ,
N l = 2 ( r t 0 0 r t ) i ,             N 2 = 2 ( r t 0 0 r t ) 2 ,
N k = ( η 0 0 η ) k = ( p l p 1 ) N k N k ,             k = 0 , 1 , 2.
TE mode efficiency :             η , l = 4 R , l J , l ,             η , 2 = 4 R , 2 J , 2 ,
TM mode efficiency :             η , l = 4 R , l J , l ,             η , 2 = 4 R , 2 J , 2 ,
Total efficiency :             η k = η , k + η , k .
J source = [ ( J i j ) source ] = F source J ( i ) F source ,
F source = F source , 1 + F source , 2 ,
F source , 1 = R 1 S 1 ( r ) M 1 S 1 ( r ) ,
F source , 2 = R 2 S 2 ( t ) M 2 S 2 ( t ) .
( J x x ) source = a 2 J x x i + b 2 J y y i + 2 R e ( a b * J x y i ) , ( J y y ) source = c 2 J x x i + d 2 J y y i + 2 R e ( c d * J x y i ) , ( J x y ) source = ( J y x ) * source = a c * J x x i + b d * J y y i + a d * J x y i + b c * J y x i , x = x 1 + x 2 , x = a , b , c , d ,
Source beam : a 1 = exp ( - i δ / 2 ) M ( 1 ) r 2 , 1 , a 2 = exp ( i δ / 2 ) M ( 2 ) t , 2 t , 2 , b 1 = exp ( - i δ / 2 ) M 12 ( 1 ) r , 1 r , 1 , b 2 = exp ( i δ / 2 ) M 12 ( 2 ) t , 2 t , 2 , c 1 = exp ( - i δ / 2 ) M 21 ( 1 ) r 1 , r , 1 , c 2 = exp ( i δ / 2 ) M 21 ( 2 ) t , 2 t , 2 , d 1 = exp ( - i δ / 2 ) M 22 ( 1 ) r 2 , 1 , d 2 = exp ( i δ / 2 ) M 22 ( 2 ) t , 2 t , 2 ,
( J x x ) source = a 2 J x x i , ( J y y ) source = d 2 J y y i , ( J x y ) source = ( J y x ) * source = a d * J x y i .
J det = [ ( J i j ) det ] = F det J ( i ) F det ,
F det , 1 = R 1 S 1 ( t ) M 1 S 1 ( r )
F det , 2 = R 2 S 2 ( r ) M 2 S 2 ( t ) .
Detector beam : a 1 = exp ( - i δ / 2 ) M 11 ( 1 ) t , 1 r , 1 , a 2 = exp ( i δ / 2 ) M 11 ( 2 ) r , 2 t , 2 , b 1 = exp ( - i δ / 2 ) M 12 ( 1 ) t , 1 r , 1 , b 2 = exp ( i δ / 2 ) M 12 ( 2 ) r , 2 t , 2 c 1 = exp ( - i δ / 2 ) M 21 ( 1 ) t , 1 r , 1 , c 2 = exp ( i δ / 2 ) M 21 ( 2 ) r , 2 t , 2 , d 1 = exp ( - i δ / 2 ) M 22 ( 1 ) t , 1 r , 1 , d 2 = exp ( i δ / 2 ) M 22 ( 2 ) r , 2 t , 2 .
G 1 = 2 F source , 1 ( δ = 0 ) = 2 S 1 ( r ) M 1 S 1 ( r ) ,
G 2 = 2 ( p 1 / p 1 ) F source , 2 ( δ = 0 ) = 2 ( p l / p 1 ) S 2 ( t ) M 2 S 2 ( t ) ,
G = G 1 + G 2 ,
H 1 = 2 F det , 1 ( δ = 0 ) = 2 S 1 ( t ) M 1 S 1 ( r ) ,
H 2 = 2 F det , 2 ( δ = 0 ) = 2 S 2 ( r ) M 2 S 2 ( t ) ,
H = H 1 + H 2 .
G m = G m G m ,             m = 1 , 2 ,
H m = ( p l / p 1 ) H m H m .
Subbeam 1             { TE mode efficiency : g , 1 = 4 [ R ( R M 11 2 + R M 12 2 ) ] 1 , TM mode efficiency : g , 1 = 4 [ R ( R M 21 2 + R M 22 2 ) ] 1 , Total efficiency : g 1 = 4 [ ( R 2 M 11 2 + R 2 M 22 2 ) + R R ( M 12 2 + M 21 2 ) ] 1 ,
Subbeam 2             { TE mode efficiency : g , 2 = 4 [ J ( J M 11 2 + J M 12 2 ) ] 2 , TM mode efficiency : g , 2 = 4 [ J ( J M 21 2 + J M 22 2 ) ] 2 , Total efficiency : g 2 = 4 [ ( J J M 11 2 + J J M 22 2 ) + ( J J M 12 2 + J J M 21 2 ) ] 2 ,
Source beam             { TE mode efficiency : g = g , 1 + g , 2 , TM mode efficiency : g = g , 1 + g , 2 , Total efficiency : g = g + g = g 1 + g 2 ,
Subbeam 1             { TE mode efficiency : h , 1 = 4 ( R J M 11 2 + R J M 12 2 ) 1 , TM mode efficiency : h , 1 = 4 ( R J M 22 2 + R J M 21 2 ) 1 , Total efficiency : h 1 = 4 [ ( R J M 11 2 + R J M 22 2 ) + ( R J M 12 2 + R J M 21 2 ) ] 1 ,
Subbeam 2             { TE mode efficiency : h , 2 = 4 ( R J M 11 2 + R J M 12 2 ) 2 , TM mode efficiency : h , 2 = 4 ( R J M 21 2 + R J M 22 2 ) 2 , Total efficiency : h 2 = 4 [ ( R J M 11 2 + R J M 22 2 ) + ( R J M 12 2 + R J M 21 2 ) ] 2 ,
Detector beam             { TE mode efficiency : h = h , 1 + h , 2 , TM mode efficiency : h = h , 1 + h , 2 , Total efficiency : h = h + h = h 1 + h 2 .
h , 1 = ( M 11 2 ) 1 η , 1
h , 1 = ( M 22 2 ) 1 η , 1 .
h , 2 = ( M 11 2 ) 2 η , 2
h , 2 = ( M 22 2 ) 2 η , 2 .
F source , 1 J ( i ) F source , 1 = F source , 2 J ( i ) F source , 2 ,
S 1 ( r ) M 1 S 1 ( r ) J ( i ) S 1 ( r ) M 1 S 1 ( r ) S 2 ( t ) M 2 S 2 ( t ) J ( i ) S 2 ( t ) M 2 S 2 ( t )
[ F x x ( 1 ) 2 - F x x ( 2 ) 2 ] J x x i + [ F x y ( 1 ) 2 - F x y ( 2 ) 2 ] J y y i + 2 R e { [ F x x ( 1 ) F x y ( 1 ) * - F x x ( 2 ) * F x y ( 2 ) * ] J x y i } = 0 , [ F y x ( 1 ) 2 - F y x ( 2 ) 2 ] J x x i + [ F y y ( 1 ) 2 - F y y ( 2 ) 2 ] J y y i + 2 R e { [ F y x ( 1 ) F y y ( 1 ) * - F y x ( 2 ) F y y ( 2 ) * ] J x y i } = 0 , [ F x x ( 1 ) * F y x ( 1 ) - F x x ( 2 ) * F y x ( 2 ) ] J x x i + [ F x y ( 1 ) * F y y ( 1 ) - F x y ( 2 ) * F y y ( 2 ) ] J y y i + [ F x x ( 1 ) * F y y ( 1 ) - F x x ( 2 ) * F y y ( 2 ) ] J y x i + [ F x y ( 1 ) * F y x ( 1 ) - F x y ( 2 ) * F y x ( 2 ) ] J x y i = 0 ,
F source , 1 F source , 2 ,
x m ( δ 0 ) = 0 ,             m = 1 , 2 ,
x 1 ( δ = 0 ) - x 2 ( δ = 0 ) = 0 ;
J source = ( F source , 1 + F source , 2 ) J ( i ) ( F source , 1 + F source , 2 ) ,
T ( J source ) = T 1 ( J source ) + T 2 ( J source ) ,
T 1 ( J source ) { T [ F source , 1 J ( i ) F source , 1 ] + T [ F source , 2 J ( i ) F source , 2 ] } ,
T 2 ( J source ) = { T [ F source , 1 J ( i ) F source , 2 + T [ F source , 2 J ( i ) F source , 1 ] } .
T 1 ( J source ) - T 2 ( J source ) = T [ ( F source , 1 - F source , 2 ) J ( i ) ( F source , 1 - F source , 2 ) ] = 0.
( F x x 2 + F y x 2 ) J x x i + ( F x y 2 + F y y 2 ) J y y i + 2 R e [ ( F x x F x y * + F y x F y y * ) J x y i ] = 0 ,
Source beam : F x x = a 1 - a 2 , F x y = b 1 - b 2 , F y x = c 1 - c 2 , F y y = d 1 - d 2 .
F det , 1 J ( i ) F det , 1 F det , 2 J ( i ) F det , 2 ,
S 1 ( t ) M 1 S 1 ( r ) J ( i ) S 1 ( r ) M 1 S 1 ( t ) = S 2 ( r ) M 2 S 2 ( t ) J ( i ) S 2 ( t ) M 2 S 2 ( r ) .
F det , 1 F det , 2 ,
R 1 S 1 ( t ) M 1 S 1 ( r ) R 2 S 2 ( r ) M 2 S 2 ( t ) .
S ( t ) M S ( r ) S ( r ) M S ( t ) .
M = R ( γ ) R F ( C ) R ( β ) R F ( B ) R ( α ) R F ( A ) ,
M = R ( β ) R F ( B ) R ( α ) R F ( A ) ,
T 1 ( J det ) - T 2 ( J det ) = T [ ( F det , 1 - F det , 2 ) J ( i ) ( F det , 1 - F det , 2 ) ] = 0 ;
I source ( τ ; σ ) = Trace ( J source ) = ( a 2 + c 2 ) J x x i + ( b 2 + d 2 ) J y y i + 2 R e [ ( a b * + c d * ) J x y i ] ,
I source ( τ ; σ ) = s 1 ( σ ) + s 2 ( σ ) cos δ + s 3 ( σ ) sin δ ,
s 1 ( σ ) = 1 2 { ( α 1 2 + α 3 2 + β 1 2 + β 3 2 ) J x x i + ( α 2 2 + α 4 2 + β 2 2 + β 4 2 ) J y y i + 2 R e [ ( α 1 α 2 * + α 3 α 4 * + β 1 β 2 * + β 3 β 4 * ) J x y i ] } ,
s 2 ( σ ) = 1 2 { ( α 1 2 + α 3 2 ) J x x i + ( α 2 2 + α 4 2 ) J y y i + 2 R e [ ( α 1 α 2 * + α 3 α 4 * - β 1 β 2 * - β 3 β 4 * ) J x y i ] } ,
s 3 ( σ ) = R e [ i ( β 1 α 2 * - α 1 β 2 * + β 3 α 4 * - α 3 β 4 * ) J x y i ] ,
α 1 = M 11 ( 1 ) ( r , 1 ) 2 + M 11 ( 2 ) t 1 , 2 t , 2 ,             β 1 = M 11 ( 2 ) t , 2 t , 2 - M 11 ( 1 ) ( r , 1 ) 2 , α 2 = M 12 ( 1 ) r , 1 r , 1 + M 12 ( 2 ) t , 2 t , 2 ,             β 2 = M 12 ( 2 ) t , 2 t , 2 - M 12 ( 1 ) r , 1 r , 1 , α 3 = M 21 ( 1 ) r , 1 r , 1 + M 21 ( 2 ) t , 2 t , 2 ,             β 3 = M 21 ( 2 ) t , 2 t , 2 - M 21 ( 1 ) r , 1 r , 1 , α 4 = M 22 ( 1 ) ( r , 1 ) 2 + M 22 ( 2 ) t , 2 t , 2 ,             β 4 = M 22 ( 2 ) t , 2 t , 2 - M 22 ( 1 ) ( r , 1 ) 2 .
s 2 ( σ ) = ρ s ( σ ) cos ψ s ( σ ) , s 3 ( σ ) = ρ s ( σ ) sin ψ s ( σ ) ,
I source ( τ t σ ) = s 1 ( σ ) + ρ s ( σ ) cos [ δ - ψ s ( σ ) ] .
ψ s ( σ ) = tan - 1 [ s 3 ( σ ) / s 2 ( σ ) ]
Var . [ I source ( 0 ; σ ) ] = ρ s ( σ ) cos ψ s ( σ ) ;
( S / N ) source = ( η T ) 1 2 ( π A / R ) g ( σ ) ρ s ( σ ) cos ψ s ( σ ) δ σ / N . E . P . ,
I det ( τ ; σ ) = Trace ( J det ) = ( a 2 + c 2 ) J x x i + ( b 2 + d 2 ) J y y i + 2 R e [ ( a b * + c d * ) J x y i ] ,
I det ( τ ; σ ) = d 1 ( σ ) + ρ d ( σ ) cos [ δ - ψ d ( σ ) ] ,
α 1 = M 11 ( 1 ) r , 1 t , 1 + M 11 ( 2 ) r , 2 t , 2 , β 1 = M 11 ( 2 ) r , 2 t , 2 - M 11 ( 1 ) r , 1 t , 1 , α 2 = M 12 ( 1 ) r , 1 t , 1 + M 12 ( 2 ) r , 2 t , 2 , β 2 = M 12 ( 2 ) r , 2 t , 2 - M 12 ( 1 ) r , 1 t , 1 , α 3 = M 21 ( 1 ) r , 1 t , 1 + M 21 ( 2 ) r , 2 t , 2 , β 3 = M 21 ( 2 ) r , 2 t , 2 - M 21 ( 1 ) r , 1 t , 1 , α 4 = M 22 ( 1 ) r , 1 t , 1 + M 22 ( 2 ) r , 2 t , 2 , β 4 = M 22 ( 2 ) r , 2 t , 2 - M 22 ( 1 ) r , 1 t , 1 .
( S / N ) det = ( η T ) 1 2 ( π A / R ) h ( σ ) ρ d ( σ ) cos ψ d ( σ ) δ σ / N . E . P . ,
C = E × E * = ( J x x , J x y , J y x , J y y ) ,
S = TC = [ J x x + J y y , J x x - J y y , J x y + J y x , - i ( J x y - J y x ) ] ,
U = 1 2 T = 1 2 ( 1 0 0 1 1 0 0 - 1 0 1 1 0 0 - i i 0 ) .
S = T C = [ J x x , J y y , J x y + J y x , - i ( J x y - J y x ) ] ,
T = ( 1 0 0 0 0 0 0 1 0 1 1 0 0 - i i 0 ) .

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