Abstract

The parallelism we have previously drawn between an interferometer and an equivalent optical system enables us to study both systems similarly. The constant contrast condition along the localization surface is equivalent to the condition of isoplanatism at the system image. As the sine condition is necessary for a system to be isoplanatic, a relationship we call the equivalent sine condition must hold if the contrast is constant. Some classic interferometers do not satisfy this condition and we here analyze the Michelson interferometer. Moreover, we find that, for the record of interference fringes to be a pseudohologram, this condition must be unfulfilled.

© 1988 Optical Society of America

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References

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  1. J. M. Simon, Silvia A. Comastri, “Fringe Localization Depth,” Appl. Opt. 26, 5125 (1987)
    [CrossRef] [PubMed]
  2. S. A. Comastri, J. M. Simon, “Ray Tracing, Aberration Function and Spatial Frequencies,” Optik 66, 175 (1984).
  3. J. M. Simon, J. O. Ratto, “Modulated Interference Gratings for Image Reconstruction: A Classification for Holographic Methods and Other Techniques,” Opt. Pura Apl. 19, 93 (1986).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  5. H. H. Hopkins, “Canonical Pupil Coordinates in Geometrical and Diffraction Image Theory,” Jpn. J. Appl. Phys.Suppl. 1 4, 31 (1965).

1987

1986

J. M. Simon, J. O. Ratto, “Modulated Interference Gratings for Image Reconstruction: A Classification for Holographic Methods and Other Techniques,” Opt. Pura Apl. 19, 93 (1986).

1984

S. A. Comastri, J. M. Simon, “Ray Tracing, Aberration Function and Spatial Frequencies,” Optik 66, 175 (1984).

1965

H. H. Hopkins, “Canonical Pupil Coordinates in Geometrical and Diffraction Image Theory,” Jpn. J. Appl. Phys.Suppl. 1 4, 31 (1965).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Comastri, S. A.

S. A. Comastri, J. M. Simon, “Ray Tracing, Aberration Function and Spatial Frequencies,” Optik 66, 175 (1984).

Comastri, Silvia A.

Hopkins, H. H.

H. H. Hopkins, “Canonical Pupil Coordinates in Geometrical and Diffraction Image Theory,” Jpn. J. Appl. Phys.Suppl. 1 4, 31 (1965).

Ratto, J. O.

J. M. Simon, J. O. Ratto, “Modulated Interference Gratings for Image Reconstruction: A Classification for Holographic Methods and Other Techniques,” Opt. Pura Apl. 19, 93 (1986).

Simon, J. M.

J. M. Simon, Silvia A. Comastri, “Fringe Localization Depth,” Appl. Opt. 26, 5125 (1987)
[CrossRef] [PubMed]

J. M. Simon, J. O. Ratto, “Modulated Interference Gratings for Image Reconstruction: A Classification for Holographic Methods and Other Techniques,” Opt. Pura Apl. 19, 93 (1986).

S. A. Comastri, J. M. Simon, “Ray Tracing, Aberration Function and Spatial Frequencies,” Optik 66, 175 (1984).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Appl. Opt.

Jpn. J. Appl. Phys.

H. H. Hopkins, “Canonical Pupil Coordinates in Geometrical and Diffraction Image Theory,” Jpn. J. Appl. Phys.Suppl. 1 4, 31 (1965).

Opt. Pura Apl.

J. M. Simon, J. O. Ratto, “Modulated Interference Gratings for Image Reconstruction: A Classification for Holographic Methods and Other Techniques,” Opt. Pura Apl. 19, 93 (1986).

Optik

S. A. Comastri, J. M. Simon, “Ray Tracing, Aberration Function and Spatial Frequencies,” Optik 66, 175 (1984).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

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Figures (7)

Fig. 1
Fig. 1

Coordinate system at the interferometer: L01(P), L02(P), optical path lengths from O to P through branches 1 and 2; L1(x,y,P), L2(x,y,P), optical path lengths from Q to P through branches 1 and 2; source of area σ, total intensity IT, and wavelength λ ¯ 0 = 2π/ k ¯ 0; n,n′, refractive indices.

Fig. 2
Fig. 2

Sine condition in optical systems for an axial object; ξ ¯ and ξ ¯ , object and image coordinates; α and α′, angles subtended by the considered ray in object and image spaces, respectively.

Fig. 3
Fig. 3

Equivalent sine condition in interferometers: O, central point of the source; Q, arbitrary point of the source; O2, image of O through branch 2 in the interferometer image space; Q2, image of Q through branch 2 in the interferometer image space; P and S two points at the localization surface; P 2 and S 2 , images of P and S through branch 2 in the interferometer object space; U2 and T2, points of intersection of Q 2 S ¯ with the wavefront and its tangent, respectively; α2 and α20, angles subtended by Q 2 P ¯ and O 2 P ¯ with the normal to the localization surface. Similar considerations are valid for branch 1.

Fig. 4
Fig. 4

Fringe spacing dependence on the source point: O and Q, central and arbitrary points of the source; Q1 and Q2, images of Q through branches 1 and 2 in the interferometer image space; P, position of maximum of order m; S, position of maximum of order m − 1; α1 and α2, angles subtended by Q 1 P ¯ and Q 2 P ¯ with the normal to the observation surface.

Fig. 5
Fig. 5

Michelson interferometer. Images of the source through both branches. Parameters: D = O F ¯ ; l 1 = F H 1 ¯ ; l 2 = F H 2 ¯; z, normal to the source; , angle subtended by M1; t,ñ, thickness and index of A.

Fig. 6
Fig. 6

Michelson interferometer. Localization surface: β, angle subtended by an arbitrary incident ray with the z axis; β1 and β2, angles subtended by the emergent rays through branches 1 and 2 with the normal to the z axis; P, point on the localization surface.

Fig. 7
Fig. 7

Michelson interferometer. Equivalent sine condition: z, axis normal to the source; P, point on the localization surface; ξ′, axis along the localization surface; O,Q, source points; O1,O2, images of O through branches 1 and 2; Q1,Q2, images of Q through branches 1 and 2; O 1 K ¯ 1 and O 2 K ¯ 2, distances from O1 and O2 to the normal to the localization surface at P; Q 1 , J ¯ 1 and Q 2 J ¯ 2 distances from Q1 and Q2 to the normal to the localization surface at P.

Equations (32)

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E ( ξ ˜ , η ˜ ) = EP { G 0 ( x ˜ , y ˜ ) exp [ i k 0 W ( x ˜ , y ˜ ) ] } × exp [ - i n k 0 R ( ξ ˜ x ˜ + η ˜ y ˜ ) ] d x ˜ d y ˜ ,
I ( P ) = I 1 ( P ) + I 2 ( P ) + 2 I 1 ( P ) I 2 ( P ) × Re ( μ 12 ( P ) exp { i [ δ 12 - a 00 ( P ) ] } ) ,
a 00 ( P ) = k ¯ 0 [ L 02 ( P ) - L 01 ( P ) ] ,
μ 12 ( P ) = 1 I T σ { I ( x , y ) exp [ - i Φ ( x , y , P ) ] } × exp { - i [ a 10 ( P ) x + a 01 ( P ) y ] } d x d y ,
a 10 ( P ) = k ¯ 0 ( L 2 x | x = 0 - L 1 x | x = 0 ) , a 01 ( P ) = k ¯ 0 ( L 2 y | y = 0 - L 1 y | y = 0 ) ,
Φ ( x , y , P ) = - k ¯ 0 { [ L 02 ( P ) - L 01 ( P ) ] - [ L 2 ( x , y , P ) - L 1 ( x , y , P ) ] } - [ a 10 ( P ) x + a 01 ( P ) y ] .
μ 12 ( P ) = μ 12 ( P ) exp [ i θ 12 ( P ) ] .
L 02 ( ξ ) - L 01 ( ξ ) - [ L 2 ( x , y , ξ ) - L 1 ( x , y , ξ ) ] = L 02 ( ξ + δ ξ ) - L 01 ( ξ + δ ξ ) - [ L 2 ( x , y , ξ + δ ξ ) - L 1 ( x , y , ξ + δ ξ ) ] .
Φ ( x , y , P ) = Φ ( x , y , S ) .
W δ ξ ˜ = α 0 n ( - sin α α 0 + sin α α 0 ) ,
W δ ξ ˜ = - n ( sin α - sin α p ) + n m x ( sin α - sin α p ) .
G 2 = L 02 ( ξ ) - L 2 ( x , y , ξ ) - L 02 ( ξ + δ ξ ) + L 2 ( x , y , ξ + δ ξ ) ,
G 1 = G 2 .
G 2 = ( [ Q 2 S ] - [ Q 2 P ] ) - ( [ O 2 S ] - [ O 2 P ] ) ,
[ Q 2 S ] = [ Q 2 P ] + [ U 2 T 2 ] + [ T 2 S ] .
U 2 T 2 ¯ ( δ ξ cos α 2 ) 2 2 Q 2 P ¯ , T 2 S ¯ δ ξ sin α 2 ,
[ Q 2 S ] - [ Q 2 P ] = δ ξ sin α 2 + ( δ ξ ) 2 cos 2 α 2 2 Q 2 P ¯ ,
G 2 = δ ξ ( sin α 2 - sin α 20 ) + ( δ ξ ) 2 2 [ cos 2 α 2 Q 2 P ¯ - cos 2 α 20 Q 2 P ¯ ] .
( δ ξ ) 2 2 { [ cos 2 α 2 Q 2 P ¯ - cos 2 α 20 Q 2 P ¯ ] - [ cos 2 α 1 Q 1 P ¯ - cos 2 α 10 Q 1 P ¯ ] } = δ ξ { - [ sin α 2 - sin α 20 ] + [ sin α 1 - sin α 10 ] } .
sin α 2 - sin α 20 = sin α 1 - sin α 10 .
Δ = { [ Q Q 1 ] + [ Q 1 P ] } - { [ Q Q 2 ] + [ Q 2 P ] } .
( [ Q 1 S ] - [ Q 1 P ] ) - ( [ Q 2 S ] - [ Q 2 P ] ) = λ ¯ 0 .
[ Q 2 S ] - [ Q 2 P ] = d Q sin α 2 .
1 d Q = sin α 1 - sin α 2 λ ¯ 0 .
O 1 Y 1 = 2 l 1 cos 2 ɛ + D ( 1 - 2 sin 2 ɛ ) + t 2 n ˜ × [ 3 2 - n ˜ 2 + 2 sin 2 ɛ - 2 sin 2 ɛ ( sin ɛ + 3 cos ɛ ) ( sin ɛ + cos ɛ ) ] , O 2 Y 2 ¯ = 2 l 2 + D + t 2 n ˜ [ 1 2 - 3 n ˜ 2 ] ,
O 1 Q 1 ¯ = O 2 Q 2 ¯ = O Q ¯ = x .
β 1 = β + 2 ɛ ,             β 2 = β .
sin α 2 = O 2 K ¯ 2 - O 2 Q 2 cos θ Q 2 P ¯ , sin α 20 = O 2 K ¯ 2 O 2 P ¯ ,
sin α 2 - sin α 20 = - x cos θ O 2 P ¯ ;
O 2 P ¯ = O 2 Y ¯ 2 - P C ¯ cos β ,
sin α 2 - sin α 20 = - x cos θ cos β O 2 Y ¯ 2 - P C ¯ .
sin α 1 - sin α 10 = - x cos ( θ + 2 ɛ ) cos ( β + 2 ɛ ) O 1 Y 1 - P C ¯ .

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