Abstract

If two wavetrains are to exhibit interference at some point in space they must be at least partially coherent, one with the other, at this point. This partial coherence between the two interfering wavetrains can, in most interferometers, easily be related to the mutual coherence between separated points in the single wavetrain from which the two interfering wavetrains are derived. The mutual coherence in this wave-train depends upon the distribution of the source which produces it in a manner given by the van Cittert-Zernike theorem, which in most practical cases can be economically stated in terms of source intensity as a function of direction from the point under consideration. The mutual intensity is the three-dimensional Fourier transform of this generalized source. Knowing this, we can, within the limits set by fact that the generalized source is a distribution spread upon the unit sphere, design to order source distributions to produce the mutual coherence necessary for different kinds of interferometry. Examples are given; the case of an infinitely thin annular source is worked out in detail.

© 1966 Optical Society of America

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References

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  1. G. Hansen, Zeiss Nachr., 109 (1942);Z. Angew. Phys. 6, 203 (1954);Optik 12, 5 (1955);Optik 15, 560 (1958).
  2. G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, Ltd., London, England, 1964);p. 269 et seq. translated from Beiträge zur Interferenzmikroskopie (Akademie-Verlag, Berlin, Germany1961);Also Optica Acta 11, 43, 89, and 131 (1964).
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, England, 1959).
  4. P. H. van Cittert, Physica 1, 201 (1934).
    [Crossref]
  5. F. Zernike, Physica 5, 785 (1938).
    [Crossref]
  6. C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).
    [Crossref]
  7. T. Merton, Proc. Roy. Soc. (London) 189A, 309 (1947).
  8. T. Merton, Proc. Roy. Soc. (London) 191A, 1 (1947).

1964 (1)

1947 (2)

T. Merton, Proc. Roy. Soc. (London) 189A, 309 (1947).

T. Merton, Proc. Roy. Soc. (London) 191A, 1 (1947).

1942 (1)

G. Hansen, Zeiss Nachr., 109 (1942);Z. Angew. Phys. 6, 203 (1954);Optik 12, 5 (1955);Optik 15, 560 (1958).

1938 (1)

F. Zernike, Physica 5, 785 (1938).
[Crossref]

1934 (1)

P. H. van Cittert, Physica 1, 201 (1934).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, England, 1959).

Hansen, G.

G. Hansen, Zeiss Nachr., 109 (1942);Z. Angew. Phys. 6, 203 (1954);Optik 12, 5 (1955);Optik 15, 560 (1958).

McCutchen, C. W.

Merton, T.

T. Merton, Proc. Roy. Soc. (London) 189A, 309 (1947).

T. Merton, Proc. Roy. Soc. (London) 191A, 1 (1947).

Schulz, G.

G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, Ltd., London, England, 1964);p. 269 et seq. translated from Beiträge zur Interferenzmikroskopie (Akademie-Verlag, Berlin, Germany1961);Also Optica Acta 11, 43, 89, and 131 (1964).

van Cittert, P. H.

P. H. van Cittert, Physica 1, 201 (1934).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, England, 1959).

Zernike, F.

F. Zernike, Physica 5, 785 (1938).
[Crossref]

J. Opt. Soc. Am. (1)

Physica (2)

P. H. van Cittert, Physica 1, 201 (1934).
[Crossref]

F. Zernike, Physica 5, 785 (1938).
[Crossref]

Proc. Roy. Soc. (London) (2)

T. Merton, Proc. Roy. Soc. (London) 189A, 309 (1947).

T. Merton, Proc. Roy. Soc. (London) 191A, 1 (1947).

Zeiss Nachr. (1)

G. Hansen, Zeiss Nachr., 109 (1942);Z. Angew. Phys. 6, 203 (1954);Optik 12, 5 (1955);Optik 15, 560 (1958).

Other (2)

G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, Ltd., London, England, 1964);p. 269 et seq. translated from Beiträge zur Interferenzmikroskopie (Akademie-Verlag, Berlin, Germany1961);Also Optica Acta 11, 43, 89, and 131 (1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, England, 1959).

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Figures (4)

F. 1
F. 1

Interfering wavetrains and their parent (below).

F. 2
F. 2

An incoherent source and the region it illuminates: notation.

F. 3
F. 3

A generalized source, and the Fourier-related source function I(l)d and complex degree of coherence μ(L)d which it produces in the d direction. The figure is oversimplified in that in general μ(L)d is complex.

F. 4
F. 4

The fringe system produced by two wavetrains both derived from an infinitely narrow annular source C. The width of the fringes as drawn gives an impression of their visibility. The two apparent sources seen from the X Y coordinate system would, in reality, be much further away than can be shown on the drawing. To be perfectly represented by a generalized source they would be at infinity.

Equations (27)

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x 1 = X cos ( Φ / 2 ) + Y sin ( Φ / 2 ) , y 1 = Y cos ( Φ / 2 ) X sin ( Φ / 2 ) , z 1 = Z Z 0 Δ ,
x 2 = X cos ( Φ / 2 ) Y sin ( Φ / 2 ) , y 2 = Y cos ( Φ / 2 ) + X sin ( Φ / 2 ) , z 2 = Z Z 0 ,
D x = x 1 x 2 = 2 Y sin ( Φ / 2 ) , D y = y 1 y 2 = 2 X sin ( Φ / 2 ) , D z = z 1 z 2 = Δ .
D = z Δ + 2 ( W × z ) sin ( Φ / 2 ) ,
I = V · V * .
V = V 1 + V 2 .
I = V · V * = ( V 1 + V 2 ) · ( V 1 * + V 2 * ) = V 1 · V 1 * + V 2 · V 2 * + V 1 · V 2 * + V 2 · V 1 * = I 1 + I 2 + J ( P ) 12 + J ( P ) 21 ,
I = I 1 + I 2 + 2 J ( P ) 12 r ,
I = I 1 + I 2 + 2 ( I 1 ) 1 2 ( I 2 ) 1 2 μ ( P ) 12 r .
V ( P ) = ( I max I min ) / ( I max + I min ) = 2 ( I 1 ) 1 2 ( I 2 ) 1 2 μ ( P ) 12 r / ( I 1 + I 2 ) ,
μ ( P ) 12 = μ ( P 1 , P 2 ) P ,
μ ( P ) 12 = μ ( D ) P ,
J ( P 1 , P 2 ) = σ I ( S ) [ e i k ¯ ( R 1 R 2 ) / R 1 R 2 ] d S
I ( P 1 ) = σ [ I ( S ) / R 1 2 ] d S , I ( P 2 ) = σ [ I ( S ) / R 2 2 ] d S ,
J ( P 1 , P 2 ) = J ( D ) = Ω I ( c ) exp ( i k ¯ c · D ) d Ω ,
I ( P 1 ) = I ( P 2 ) = Ω I ( c ) d Ω
μ ( D ) = J ( D ) / Ω I ( c ) d Ω .
J ( D ) = ( 2 π ) 3 2 [ Fourier transform of I ( c ) ]
J ( L ) d = 1 + 1 I ( l ) d e i k ¯ L l d l ,
J ( L ) d = ( 2 π ) 1 2 [ Fourier transform of I ( l ) d ] ,
μ ( L ) d = J ( L ) d / I = J ( L ) d / Ω I ( c ) d Ω = J ( L ) d / + I ( l ) d d l .
I ( l ) d = 2 I a ( sin 2 θ l a 2 / sin 2 α ) 1 2 , l a 2 < sin 2 α sin 2 θ I ( l ) d = 0 , l a 2 > sin 2 α sin 2 θ l a = l cos α cos θ .
μ ( L ) d = e i k ¯ L cos a cos θ J 0 ( k ¯ L sin α sin θ ) ,
L sin θ = D x = 2 Y sin ( Φ / 2 ) L cos θ = D y = 2 X sin ( Φ / 2 ) .
μ ( P ) 12 = exp [ ( 4 π i / λ ¯ ) X cos α sin ( Φ / 2 ) ] × J 0 [ ( 4 π / λ ¯ ) Y sin α sin ( Φ / 2 ) ] ,
I = I 1 + I 2 + 2 ( I 1 ) 1 2 ( I 2 ) 1 2 cos [ ( 4 π X / λ ¯ ) cos α sin ( Φ / 2 ) ] × J 0 [ ( 4 π Y / λ ¯ ) sin α sin ( Φ / 2 ) ] .
μ ( P x ) 12 = exp ( 2 π i L l 0 / λ ¯ ) [ sin ( π L a / λ ¯ ) / π L a / λ ¯ ] .