Abstract

Two partially coherent and perpendicularly polarized vibrations induce, in a polarization sensitive photographic emulsion H, an anisotropy that is continuously varying across the plate. When the two incident amplitudes are assumed to be constant all over H, those variations are shown to be periodic functions of the effective phase shift φ. After exposure H is observed between crossed polarizers. It reconstructs an interferogram whose contrast is maximum whatever may be the degree of partial coherence and the relative amplitudes of the two vibrations. In plane rotation of the plate, H causes changes in the spatial frequency of the interferogram and localization of its dark fringes. When the direction of the polarizer is that of one of the original vibrations, the distance in terms of phase shift between any two consecutive dark fringes reaches π, and these fringes characterize the lines of equal phase φ = . Similar phenomena may be retrieved from that last setting by rotation of the analyzer initially crossed with the polarizer.

© 1980 Optical Society of America

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References

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  1. F. Weigert, Verb. Dtsch. Phys. Ges. 21, 479 (1919).
  2. Sh. D. Kakichashvili, Opt. Spectrosc. USSR 33, 171 (1972).
  3. J. M. C. Jonathan, R. Kinany, Opt. Commun. 27, 61 (1978).
    [CrossRef]
  4. J. M. C. Jonathan, M. May, Opt. Commun. 28, 30 (1979).
    [CrossRef]
  5. J. M. C. Jonathan, M. May, Opt. Commun. 28, 295 (1979).
    [CrossRef]
  6. J. M. C. Jonathan, M. May, Opt. Commun. 29, 7 (1979).
    [CrossRef]
  7. J. Upatnieks, C. Leonard, Appl. Opt. 8, 85 (1969).
    [CrossRef] [PubMed]
  8. D. H. McMahon, W. T. Maloney, Appl. Opt. 9, 1363 (1970).
    [CrossRef] [PubMed]
  9. S. Tcherdyncev, Sci. Ind. Photogr. 6, 285 (1935).
  10. A. E. Cameron, A. M. Taylor, J. Opt. Soc. Am. 24, 316 (1934).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 544.

1979 (3)

J. M. C. Jonathan, M. May, Opt. Commun. 28, 30 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 28, 295 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 29, 7 (1979).
[CrossRef]

1978 (1)

J. M. C. Jonathan, R. Kinany, Opt. Commun. 27, 61 (1978).
[CrossRef]

1972 (1)

Sh. D. Kakichashvili, Opt. Spectrosc. USSR 33, 171 (1972).

1970 (1)

1969 (1)

1935 (1)

S. Tcherdyncev, Sci. Ind. Photogr. 6, 285 (1935).

1934 (1)

1919 (1)

F. Weigert, Verb. Dtsch. Phys. Ges. 21, 479 (1919).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 544.

Cameron, A. E.

Jonathan, J. M. C.

J. M. C. Jonathan, M. May, Opt. Commun. 28, 295 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 29, 7 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 28, 30 (1979).
[CrossRef]

J. M. C. Jonathan, R. Kinany, Opt. Commun. 27, 61 (1978).
[CrossRef]

Kakichashvili, Sh. D.

Sh. D. Kakichashvili, Opt. Spectrosc. USSR 33, 171 (1972).

Kinany, R.

J. M. C. Jonathan, R. Kinany, Opt. Commun. 27, 61 (1978).
[CrossRef]

Leonard, C.

Maloney, W. T.

May, M.

J. M. C. Jonathan, M. May, Opt. Commun. 29, 7 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 28, 295 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 28, 30 (1979).
[CrossRef]

McMahon, D. H.

Taylor, A. M.

Tcherdyncev, S.

S. Tcherdyncev, Sci. Ind. Photogr. 6, 285 (1935).

Upatnieks, J.

Weigert, F.

F. Weigert, Verb. Dtsch. Phys. Ges. 21, 479 (1919).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 544.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Commun. (4)

J. M. C. Jonathan, R. Kinany, Opt. Commun. 27, 61 (1978).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 28, 30 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 28, 295 (1979).
[CrossRef]

J. M. C. Jonathan, M. May, Opt. Commun. 29, 7 (1979).
[CrossRef]

Opt. Spectrosc. USSR (1)

Sh. D. Kakichashvili, Opt. Spectrosc. USSR 33, 171 (1972).

Sci. Ind. Photogr. (1)

S. Tcherdyncev, Sci. Ind. Photogr. 6, 285 (1935).

Verb. Dtsch. Phys. Ges. (1)

F. Weigert, Verb. Dtsch. Phys. Ges. 21, 479 (1919).

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 544.

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Figures (9)

Fig. 1
Fig. 1

Uniform anisotropic plate, characterized by its neutral lines Ox′ and Oy′, splits the incident rectilinear vibration P into a directly transmitted one t and a doubly refracted one r.

Fig. 2
Fig. 2

Vibration t is removed by analyzer A crossed with P while r is transmitted with maximum irradiance. There is no more light at the emergence of the analyzer A(π/2 + ɛ) rotated through ɛ from its setting A(π/2) and perpendicular to the vibration r + t

Fig. 3
Fig. 3

Ex and EY are two perpendicular rectilinear vibrations partially coherent with an effective phase shift φ. They generate a partially polarized light vibration whose polarized part is only due to ωx and ωy or to ωx and ωy in quadrature.

Fig. 4
Fig. 4

Anisotropic plate characterized by its optic axis Ox′ is illuminated by rectilinear vibration P inclined at an angle v on Ox. It is observed through analyzer A whose transmission direction makes an angle β with P.

Fig. 5
Fig. 5

Variations of αM vs φM.

Fig. 6
Fig. 6

Variations of AM vs φM.

Fig. 7
Fig. 7

Mi (i = 1,2,…,8) are the respective centers of different elementary areas of H corresponding to different values of φi included between 0 and 2π. Vectors Mix′ represent the direction of the different induced optic axis, and their lengths are proportional to ai.

Fig. 8
Fig. 8

Variations vs φ and for different values of v of the irradiance of H observed between crossed polarizers.

Fig. 9
Fig. 9

(a) Low-contrasted sinuso i ¨ dal fringes generated by a polarization interferometer illuminated by an extended source and observed between crossed polarizers. (b) Maximum contrast interferogram reconstructed by H illuminated by a rectilinear vibration parallel to Ex (v = 0) and observed between crossed polarizers. (c) and (d) Maximum contrast fringes observed in the plane of H when the analyzer is no longer crossed with the polarizer. The dark fringes represent the lines: (c) φ = 2K π; (d) φ = (2K + 1)π.

Equations (33)

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r { x r = a cos u , y r = k a sin u ,
J = ( J x x J x y J y x J y y ) with J x x = E x E x * , J y y = E y E y * < J x y = J y x * = E x E y * < ,
J = ( A O O A ) + ( ω x x J x y J y x ω y y )
ω x x ω y y - J x y J y x = 0.
tan 2 α = ω x y + ω y x ω x x - ω y y = J x y + J y x J x x - J y y .
ω x x = ω x ω x * = 1 2 [ ( J x x - J y y ) 2 + 4 J x y J y x ] 1 / 2 + 1 2 J x x - J y y cos 2 α ω y y = ω y ω y * = 1 2 [ ( J x x - J y y ) 2 + 4 J x y J y x ] 1 / 2 + 1 2 J x x - J y y cos 2 α } .
ω x y = ω x ω y * = ω y x * = 1 2 ( J x y - J y x ) = ± 1 2 exp [ j ( π / 2 ) J x y - J y x ,
J x x > J y y - π 4 < α < π 4 J x x < J y y π 4 < α < 3 π 4 } .
V ω ( t ) = ω x x 0 + ω y y 0 ,
ω x = ɛ ( ω x x ) 1 / 2 cos Ω t ω y = ± ɛ ( ω y y ) 1 / 2 sin Ω t } ,
V ω ( t ) = ɛ { ( ω x x ) 1 / 2 - ( ω y y ) 1 / 2 ] cos Ω t x 0 + ( ω y y ) 1 / 2 ( c o s Ω t x 0 ± s i n Ω t y 0 ) } .
V ( t ) = V ω ( t ) = ɛ [ ( ω x x ) 1 / 2 - ( ω y y ) 1 / 2 ] cos Ω t x 0 .
A = V ( t ) 2 = ω x x + ω y y - 2 ( ω x x ) 1 / 2 ( ω y y ) 1 / 2 ,
A = [ ( J x x - J y y ) 2 + 4 J x y J y x ] 1 / 2 - J x y - J y x .
P { x p = cos ( α - v ) , y p = - sin ( α - v ) ,
r { x r = A cos ( α - v ) , y r = - k A sin ( α - v ) .
U = ( A + t ) cos β + A ( 1 - k ) sin ( α - v ) sin ( β + v - α ) .
I = 1 4 ( 1 - k ) 2 { [ ( J x x - J y y ) 2 + 4 J x y J y x ] 1 / 2 - J x y - J y x } 2 × [ ( J x y + J y x ) cos 2 v - ( J x x - J y y ) sin 2 v ] 2 ( J x x - J y y ) 2 + ( J x y + J y x ) 2 .
{ E x = a exp ( j Ω t ) , E y = b exp [ j ( Ω t + φ t ) ] .
E x E y * = J x y = J y x * = γ a b exp ( - j φ ) .
tan 2 α M = 2 γ a M b M cos φ M a M 2 - b M 2 and a M 2 - b M 2 cos 2 α M > 0 ,
A M = [ ( a M 2 - b M 2 ) 2 + 4 γ 2 a M 2 b M 2 ] 1 / 2 - 2 γ a M b M sin φ M .
α ( φ M ) = α ( 2 π - φ M ) .
tan 2 α max = 2 γ a b a 2 - b 2
A max = [ ( a - b ) 2 + 4 γ 2 a 2 b 2 ] 1 / 2 , A min = [ ( a - b ) 2 + 4 γ 2 a 2 b 2 ] 1 / 2 - 2 γ a b . }
I ( M ) = ¼ ( 1 - k ) 2 × { [ ( a 2 - b 2 ) 2 + 4 γ 2 a 2 b 2 ] 1 / 2 - 2 γ a b sin φ M } 2 × [ ( a 2 - b 2 ) sin 2 v - 2 γ a b cos φ M cos 2 v ] 2 ( a 2 - b 2 ) 2 + 4 γ 2 a 2 b 2 cos 2 φ M ,
α M = v , i . e . , tan 2 v = 2 γ a b a 2 - b 2 cos φ M .
U ( β 0 ) = ( A 0 + t ) cos β 0 + A 0 ( 1 - k ) sin α 0 sin ( β 0 - α 0 ) = 0.
I β 0 ( φ M ) = ¼ ( 1 - k ) 2 ( A 0 + t ) 2 { [ A M ( A + 0 t ) sin 2 α M - A 0 ( A M + t ) sin 2 α 0 ] sin β 0 - [ A M ( A 0 + t ) ( 1 - cos 2 α M ) - A 0 ( A M + t ) ( 1 - c o s 2 α 0 ) ] cos β 0 } 2 .
I β 0 ( φ 0 + 2 K π ) = 0 ,
I β 0 [ φ 0 + ( 2 K + 1 ) π ] = ( 1 - k ) 2 A 0 2 s i n 2 2 α 0 s i n 2 β 0 ,
τ = a 2 + b 2 + 2 γ a b cos φ , or τ = a 2 + b 2 - 2 γ a b cos φ ,
I ( M ) = ( 1 - k ) 2 γ 2 a 4 ( 1 - sin φ M ) 2 cos 2 2 v .

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