Abstract

A method is described which allows the determination of the ellipticity of light (i.e., the degree of polarization as well as the phase difference of the mutually perpendicular electric field components) by intensity measurements behind two reflection polarizers. The method yields simultaneously the complex reflection coefficient of the first polarizer and its optical constants if it consists of only one mirror. The method is especially suited for the extreme ultraviolet where neither transmission polarizers nor simple phase shifters are available.

© 1971 Optical Society of America

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References

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  1. R. N. Hamm, R. A. MacRae, E. T. Arakawa, J. Opt. Soc. Amer. 55, 1460 (1965).
    [CrossRef]
  2. J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967), Chap. 9.
  3. K. Rabinovitch, L. R. Canfield, R. P. Madden, Appl. Opt. 4, 1005 (1965).
    [CrossRef]
  4. G. Rosenbaum, B. Feuerbacher, R. P. Godwin, M. Skibowski, Appl. Opt. 7, 1917 (1968).
    [CrossRef] [PubMed]
  5. J. R. Beattie, G. K. T. Conn, Phil. Mag. 46, 222 (1955).
  6. All complex quantities will be denoted by bold type.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965), p. 24.
  8. M. Schledermann, Diplomarbeit, Universität Hamburg, 1969; M. Schledermann, M. Skibowski, to be published.
  9. A. A. Sokolov, I. M. Ternov, Synchrotron Radiation (Akademie-Verlag, Berlin, 1968), Chap. I, Sec. 3.
  10. K. C. Westfold, Astrophys. J. 130, 231 (1959); M. P. C. Legg, K. C. Westfold, Astrophys. J. 154, 499 (1968).
    [CrossRef]
  11. P. Joos, Phys. Rev. Lett. 4, 558 (1960).
    [CrossRef]
  12. G. K. T. Conn, G. K. Eaton, J. Opt. Soc. Amer. 44, 546 (1954).
    [CrossRef]
  13. B. Feuerbacher, R. P. Godwin, M. Skibowski, Rev. Sci. Instrum. 40, 305 (1969).
    [CrossRef]
  14. M. Skibowski, W. Steinmann, J. Opt. Soc. Amer. 57, 112 (1967).
    [CrossRef]
  15. L. R. Canfield, G. Hass, W. Hunter, J. Phys. (Paris) 25, 124 (1964).
    [CrossRef]

1969 (1)

B. Feuerbacher, R. P. Godwin, M. Skibowski, Rev. Sci. Instrum. 40, 305 (1969).
[CrossRef]

1968 (1)

1967 (1)

M. Skibowski, W. Steinmann, J. Opt. Soc. Amer. 57, 112 (1967).
[CrossRef]

1965 (2)

R. N. Hamm, R. A. MacRae, E. T. Arakawa, J. Opt. Soc. Amer. 55, 1460 (1965).
[CrossRef]

K. Rabinovitch, L. R. Canfield, R. P. Madden, Appl. Opt. 4, 1005 (1965).
[CrossRef]

1964 (1)

L. R. Canfield, G. Hass, W. Hunter, J. Phys. (Paris) 25, 124 (1964).
[CrossRef]

1960 (1)

P. Joos, Phys. Rev. Lett. 4, 558 (1960).
[CrossRef]

1959 (1)

K. C. Westfold, Astrophys. J. 130, 231 (1959); M. P. C. Legg, K. C. Westfold, Astrophys. J. 154, 499 (1968).
[CrossRef]

1955 (1)

J. R. Beattie, G. K. T. Conn, Phil. Mag. 46, 222 (1955).

1954 (1)

G. K. T. Conn, G. K. Eaton, J. Opt. Soc. Amer. 44, 546 (1954).
[CrossRef]

Arakawa, E. T.

R. N. Hamm, R. A. MacRae, E. T. Arakawa, J. Opt. Soc. Amer. 55, 1460 (1965).
[CrossRef]

Beattie, J. R.

J. R. Beattie, G. K. T. Conn, Phil. Mag. 46, 222 (1955).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965), p. 24.

Canfield, L. R.

K. Rabinovitch, L. R. Canfield, R. P. Madden, Appl. Opt. 4, 1005 (1965).
[CrossRef]

L. R. Canfield, G. Hass, W. Hunter, J. Phys. (Paris) 25, 124 (1964).
[CrossRef]

Conn, G. K. T.

J. R. Beattie, G. K. T. Conn, Phil. Mag. 46, 222 (1955).

G. K. T. Conn, G. K. Eaton, J. Opt. Soc. Amer. 44, 546 (1954).
[CrossRef]

Eaton, G. K.

G. K. T. Conn, G. K. Eaton, J. Opt. Soc. Amer. 44, 546 (1954).
[CrossRef]

Feuerbacher, B.

B. Feuerbacher, R. P. Godwin, M. Skibowski, Rev. Sci. Instrum. 40, 305 (1969).
[CrossRef]

G. Rosenbaum, B. Feuerbacher, R. P. Godwin, M. Skibowski, Appl. Opt. 7, 1917 (1968).
[CrossRef] [PubMed]

Godwin, R. P.

B. Feuerbacher, R. P. Godwin, M. Skibowski, Rev. Sci. Instrum. 40, 305 (1969).
[CrossRef]

G. Rosenbaum, B. Feuerbacher, R. P. Godwin, M. Skibowski, Appl. Opt. 7, 1917 (1968).
[CrossRef] [PubMed]

Hamm, R. N.

R. N. Hamm, R. A. MacRae, E. T. Arakawa, J. Opt. Soc. Amer. 55, 1460 (1965).
[CrossRef]

Hass, G.

L. R. Canfield, G. Hass, W. Hunter, J. Phys. (Paris) 25, 124 (1964).
[CrossRef]

Hunter, W.

L. R. Canfield, G. Hass, W. Hunter, J. Phys. (Paris) 25, 124 (1964).
[CrossRef]

Joos, P.

P. Joos, Phys. Rev. Lett. 4, 558 (1960).
[CrossRef]

MacRae, R. A.

R. N. Hamm, R. A. MacRae, E. T. Arakawa, J. Opt. Soc. Amer. 55, 1460 (1965).
[CrossRef]

Madden, R. P.

Rabinovitch, K.

Rosenbaum, G.

Samson, J. A. R.

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967), Chap. 9.

Schledermann, M.

M. Schledermann, Diplomarbeit, Universität Hamburg, 1969; M. Schledermann, M. Skibowski, to be published.

Skibowski, M.

B. Feuerbacher, R. P. Godwin, M. Skibowski, Rev. Sci. Instrum. 40, 305 (1969).
[CrossRef]

G. Rosenbaum, B. Feuerbacher, R. P. Godwin, M. Skibowski, Appl. Opt. 7, 1917 (1968).
[CrossRef] [PubMed]

M. Skibowski, W. Steinmann, J. Opt. Soc. Amer. 57, 112 (1967).
[CrossRef]

Sokolov, A. A.

A. A. Sokolov, I. M. Ternov, Synchrotron Radiation (Akademie-Verlag, Berlin, 1968), Chap. I, Sec. 3.

Steinmann, W.

M. Skibowski, W. Steinmann, J. Opt. Soc. Amer. 57, 112 (1967).
[CrossRef]

Ternov, I. M.

A. A. Sokolov, I. M. Ternov, Synchrotron Radiation (Akademie-Verlag, Berlin, 1968), Chap. I, Sec. 3.

Westfold, K. C.

K. C. Westfold, Astrophys. J. 130, 231 (1959); M. P. C. Legg, K. C. Westfold, Astrophys. J. 154, 499 (1968).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965), p. 24.

Appl. Opt. (2)

Astrophys. J. (1)

K. C. Westfold, Astrophys. J. 130, 231 (1959); M. P. C. Legg, K. C. Westfold, Astrophys. J. 154, 499 (1968).
[CrossRef]

J. Opt. Soc. Amer. (3)

R. N. Hamm, R. A. MacRae, E. T. Arakawa, J. Opt. Soc. Amer. 55, 1460 (1965).
[CrossRef]

G. K. T. Conn, G. K. Eaton, J. Opt. Soc. Amer. 44, 546 (1954).
[CrossRef]

M. Skibowski, W. Steinmann, J. Opt. Soc. Amer. 57, 112 (1967).
[CrossRef]

J. Phys. (Paris) (1)

L. R. Canfield, G. Hass, W. Hunter, J. Phys. (Paris) 25, 124 (1964).
[CrossRef]

Phil. Mag. (1)

J. R. Beattie, G. K. T. Conn, Phil. Mag. 46, 222 (1955).

Phys. Rev. Lett. (1)

P. Joos, Phys. Rev. Lett. 4, 558 (1960).
[CrossRef]

Rev. Sci. Instrum. (1)

B. Feuerbacher, R. P. Godwin, M. Skibowski, Rev. Sci. Instrum. 40, 305 (1969).
[CrossRef]

Other (5)

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967), Chap. 9.

All complex quantities will be denoted by bold type.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1965), p. 24.

M. Schledermann, Diplomarbeit, Universität Hamburg, 1969; M. Schledermann, M. Skibowski, to be published.

A. A. Sokolov, I. M. Ternov, Synchrotron Radiation (Akademie-Verlag, Berlin, 1968), Chap. I, Sec. 3.

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Figures (2)

Fig. 1
Fig. 1

Sketch of the double polarizer system: P1 first polarizer, P2 second polarizer, D detector. The x,y,z, x1,y1,z1, and x2,y2,z2 systems are fixed to the laboratory, P1 and P2, respectively, z,z1,z2 being the direction of light propagation and simultaneously the axes of rotation of P1 and P2. The y1-z1 plane and y2-z2 plane are the planes of incidence of the mirrors used in the construction of P1 and P2. The definition of the rotation angles ψ and ϕ is shown in the lower part.

Fig. 2
Fig. 2

Photograph of the double polarizer system constructed for polarization studies in the extreme ultraviolet. Light is incident from the lower left through the ball bearing. It is reflected at a plate and passes through a four-mirror polarizer before being detected by a multiplier.

Equations (23)

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( E x 2 E y 2 ) = r s 1 r s 2 ( 1 0 0 ϱ 2 ) ( cos φ sin φ - sin φ cos φ ) ( 1 0 0 ϱ 1 ) × ( cos ψ sin ψ - sin ψ cos ψ ) ( E x E y ) .
J ψ , φ = F ( E x 2 2 + E y 2 2 ) d F = K i = 1 9 A i B i C i ,
A 1 = cos 2 ψ cos 2 φ , B 1 = 1 + U ϱ 1 2 ϱ 2 2 , C 1 = C 2 = C 3 = C 4 = 1 , A 2 = sin 2 ψ cos 2 φ , B 2 = U + ϱ 1 2 ϱ 2 2 , A 3 = cos 2 ψ sin 2 φ , B 3 = ϱ 2 2 + U ϱ 1 2 , A 4 = sin 2 ψ sin 2 φ , B 4 = ϱ 1 2 + U ϱ 2 2 , A 5 = sin 2 ψ sin 2 φ , B 5 = 1 2 ϱ 1 ( ϱ 2 2 - 1 ) ( 1 - U ) , C 5 = cos δ 1 , A 6 = sin 2 ψ cos 2 φ , B 6 = 1 - ϱ 1 2 ϱ 2 2 , C 6 = C 7 = V , A 7 = sin 2 ψ sin 2 φ , B 7 = ϱ 2 2 - ϱ 1 2 , A 8 = cos 2 ψ sin 2 φ , B 8 = ϱ 1 ( 1 - ϱ 2 2 ) , C 8 = V cos δ 1 , A 9 = sin 2 φ , B 9 = ϱ 1 ( ϱ 2 2 - 1 ) , C 9 = W sin δ 1 ,
N = F E x 2 d F , U = N - 1 F E y 2 d F , V = N - 1 F E x E y cos θ d F , W = N - 1 F E x E y sin θ d F ,
K = N r s 1 r s 2 2 .
J ψ , φ = K f ( ψ , φ , U , V , W , ϱ 1 , ϱ 2 , δ 1 ) .
U = E x - 2 E y 2 , V = E x - 1 E y cos θ = U 1 2 cos θ , W = E x - 1 E y sin θ = U 1 2 sin θ .
J = K ( cos 2 ψ cos 2 ϕ + ϱ 1 2 sin 2 ψ sin 2 ϕ - 1 2 ϱ 1 cos δ 1 sin 2 ψ sin 2 ϕ ) .
J 0 , 0 = K ( 1 + U ϱ 1 2 ϱ 2 2 ) , J π / 2 , 0 = K ( U + ϱ 1 2 ϱ 2 2 ) , J 0 , π / 2 = K ( ϱ 2 2 + U ϱ 1 2 ) , J π / 2 , π / 2 = K ( ϱ 1 2 + U ϱ 2 2 ) .
R 1 J 0 , 0 - 1 J π / 2 , 0 = ( U + ϱ 1 2 ϱ 2 2 ) ( 1 + U ϱ 1 2 ϱ 2 2 ) - 1 , R 2 J 0 , 0 - 1 J 0 , π / 2 = ( ϱ 2 2 + U ϱ 1 2 ) ( 1 + U ϱ 1 2 ϱ 2 2 ) - 1 , R 3 J 0 , 0 - 1 J π / 2 , π / 2 = ( ϱ 1 2 + U ϱ 2 2 ) ( 1 + U ϱ 1 2 ϱ 2 2 ) - 1 .
U = A ± ( A 2 - 1 ) 1 2 , ϱ 1 2 = a ± ( a 2 - 1 ) 1 2 , ϱ 2 2 = a ¯ ± ( a ¯ 2 - 1 ) 1 2 ,
A = 1 2 ( 1 + R 1 2 - R 2 2 - R 3 2 ) ( R 1 - R 2 R 3 ) - 1 , a = 1 2 ( 1 - R 1 2 - R 2 2 + R 3 2 ) ( R 3 - R 1 R 2 ) - 1 , a ¯ = 1 2 ( 1 - R 1 2 + R 2 2 - R 3 2 ) ( R 2 - R 1 R 3 ) - 1 .
J ψ , 0 = K [ ( 1 + U ϱ 1 2 ϱ 2 2 ) cos 2 ψ + ( U + ϱ 1 2 ϱ 2 2 ) sin 2 ψ + V ( 1 - ϱ 1 2 ϱ 2 2 ) sin 2 ψ ] ,
J π / 4 , 0 = K [ 1 2 ( ϱ 1 2 ϱ 2 2 + 1 ) ( 1 + U ) + V ( 1 - ϱ 1 2 ϱ 2 2 ) ] .
J 0 , φ = K [ ( 1 + U ϱ 1 2 ϱ 2 2 ) cos 2 φ + ( ϱ 2 2 + U ϱ 1 2 ) sin 2 φ + ( W sin δ 1 - V cos δ 1 ) ϱ 1 ( ϱ 2 2 - 1 ) sin 2 φ ] ,
J π / 2 , φ = K [ ( U + ϱ 1 2 ϱ 2 2 ) cos 2 φ + ( ϱ 1 2 + U ϱ 2 2 ) sin 2 φ + ( W sin δ 1 + V cos δ 1 ) ϱ 1 ( ϱ 2 2 - 1 ) sin 2 φ ] .
J = J ψ , φ φ = K ( ϱ 2 2 - 1 ) i = 1 6 A i B i C i ,
A 1 = cos 2 ψ sin 2 φ , B 1 = 1 - U ϱ 1 2 , C 1 = C 2 = 1 , A 2 = sin 2 ψ sin 2 φ , B 2 = U - ϱ 1 2 , A 3 = sin 2 ψ cos 2 φ , B 3 = ϱ 1 ( 1 - U ) , C 3 = cos δ 1 , A 4 = sin 2 ψ sin 2 φ , B 4 = ϱ 1 2 + 1 , C 4 = V , A 5 = cos 2 ψ cos 2 φ , B 5 = - 2 ϱ 1 , C 5 = V cos δ 1 , A 6 = cos 2 φ , B 6 = 2 ϱ 1 , C 6 = W sin δ 1 .
tan 2 φ m = Z M - 1 f m ( ψ , U , V , W , ϱ 1 , δ 1 ) ,
Z = ϱ 1 ( U - 1 ) cos δ 1 sin 2 ψ + 2 ϱ 1 V cos δ 1 cos 2 ψ - 2 ϱ 1 W sin δ 1 , M = ( 1 - U ϱ 1 2 ) cos 2 ψ + ( U - ϱ 1 2 ) sin 2 ψ + ( ϱ 1 2 + 1 ) V sin 2 ψ .
ψ = 0 :             tan 2 φ m = 2 ϱ 1 ( 1 - U ϱ 1 2 ) - 1 ( V cos δ 1 - W sin δ 1 ) , ψ = π 2 :             tan 2 φ m = - 2 ϱ 1 ( U - ϱ 1 2 ) - 1 ( V cos δ 1 + W sin δ 1 ) .
tan 2 φ m = 2 ϱ 1 cos δ 1 ( ϱ 1 2 tan ψ - cot ψ ) - 1 .
tan 2 φ m = 2 ϱ 1 cos δ 1 ( U - 1 ) [ ( U - ϱ 1 2 ) tan ψ + ( 1 - U ϱ 1 2 ) cot ψ ] - 1 .

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