Abstract

The general problem of multiple-beam interference with partially coherent quasimonochromatic light is discussed. Both one-dimensional and two-dimensional regular and irregular arrays of apertures are treated. The coherence functions that are dealt with here are those arising naturally from incoherent, quasimonochromatic slit and circular sources, i.e., sinx/x and 2J1(x)/x correlation functions. Several experimental examples are included.

© 1966 Optical Society of America

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References

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  1. A. A. Michelson, Phil. Mag. 51, 30 (1890).
  2. E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).
  3. L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).
  4. B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
    [Crossref]
  5. B. J. Thompson, J. Opt. Soc. Am. 48, 95 (1958).
    [Crossref]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1964), 2d ed., p. 505.
  7. P. Hariharan and D. Sen, J. Opt. Soc. Am. 51, 1307 (1961).
    [Crossref]
  8. W. H. Steel, in Progress in Optics V, E. Wolf, Ed. (North-Holland Pub. Co., Amsterdam, 1965), p. 147.
  9. S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. A16, 54 (1964).
  10. A. Lohmann, Optica Acta 9, 1 (1962).
    [Crossref]
  11. R. W. Ditchburn, Light (Blackie, Glasgow, 1952), p. 171.

1964 (1)

S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. A16, 54 (1964).

1962 (1)

A. Lohmann, Optica Acta 9, 1 (1962).
[Crossref]

1961 (1)

1958 (1)

1957 (1)

1954 (1)

E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).

1953 (1)

L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).

1890 (1)

A. A. Michelson, Phil. Mag. 51, 30 (1890).

Baker, L. R.

L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1964), 2d ed., p. 505.

Ditchburn, R. W.

R. W. Ditchburn, Light (Blackie, Glasgow, 1952), p. 171.

Hariharan, P.

Lohmann, A.

A. Lohmann, Optica Acta 9, 1 (1962).
[Crossref]

Michelson, A. A.

A. A. Michelson, Phil. Mag. 51, 30 (1890).

Nawata, S.

S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. A16, 54 (1964).

Sen, D.

Steel, W. H.

W. H. Steel, in Progress in Optics V, E. Wolf, Ed. (North-Holland Pub. Co., Amsterdam, 1965), p. 147.

Thompson, B. J.

Wolf, E.

B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
[Crossref]

E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1964), 2d ed., p. 505.

J. Opt. Soc. Am. (3)

Optica Acta (1)

A. Lohmann, Optica Acta 9, 1 (1962).
[Crossref]

Phil. Mag. (1)

A. A. Michelson, Phil. Mag. 51, 30 (1890).

Proc. Phys. Soc. (London) (1)

L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).

Proc. Roy. Soc. (London) (1)

E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).

Sci. Rep. Res. Inst. Tohoku Univ. (1)

S. Nawata, Sci. Rep. Res. Inst. Tohoku Univ. A16, 54 (1964).

Other (3)

W. H. Steel, in Progress in Optics V, E. Wolf, Ed. (North-Holland Pub. Co., Amsterdam, 1965), p. 147.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1964), 2d ed., p. 505.

R. W. Ditchburn, Light (Blackie, Glasgow, 1952), p. 171.

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Figures (6)

F. 1
F. 1

Multiple beam interference with partially coherent light; (a) the array, (b) coherent illumination, (c) coherence interval = S.

F. 2
F. 2

Multiple beam interference with partially coherent light; (a) the array, (b) coherence interval equal to center to center spacing of groups forming unit of array, (c) diffraction pattern of one group alone.

F. 3
F. 3

(a) two-dimensional array coherently illuminated; (b) one-dimensional array illuminated by a square, incoherent source-coherent interval slightly less than array spacing; (c) as (b) with two-dimensional array; (d) two-dimensional array of double the spacing of (c) but with same coherence as in (c).

F. 4
F. 4

Multiple-beam interference; (a) irregular array, (b) Fraunhofer diffraction pattern of (a).

F. 5
F. 5

Fraunhofer diffraction patterns of irregular arrays of lycopodium powder particles under various coherence conditions; (a) coherence interval equal to a few thousand particle diameters, (b) coherence interval equal to a few hundred particle diameters, (c) coherence interval equal to a few particle diameters.

F. 6
F. 6

Microdensitometer trace of portions of Fig. 5(a), (b), and (c).

Equations (8)

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I R = I 1 + I 2 + 2 ( I 1 I 2 ) 1 2 | γ 12 ( τ ) | cos [ β 12 ( τ ) + δ 12 ] ,
β 12 ( τ ) = 2 π ν ¯ τ + arg γ 12 ( τ ) , δ 12 = ( 2 π / λ ¯ ) ( r 2 r 1 ) ,
I R = I 1 + I 2 + I 3 + + I N + 2 ( I 1 I 2 ) 1 2 | γ 12 ( 0 ) | cos [ β 12 ( 0 ) + δ 12 ] + 2 ( I 1 I 3 ) 1 2 | γ 13 ( 0 ) | cos [ β 13 ( 0 ) + δ 13 ] + + 2 ( I n I m ) 1 2 | γ n m ( 0 ) | cos [ β n m ( 0 ) + δ n m ] m > n + + 2 ( I N 1 I N ) 1 2 | γ ( N 1 ) N ( 0 ) | cos ( β ( N 1 ) N + δ ( N 1 ) N ) .
I R = n = 1 N m = 1 N ( I n I m ) 1 2 | γ n m ( 0 ) | cos [ β n m ( 0 ) + δ n m ] .
and | γ n m ( 0 ) | = 1 when n = m | γ n m ( 0 ) | = | γ m n ( 0 ) | . }
| γ n m ( 0 ) | = 0 for n m , | γ n m ( 0 ) | = 1 for n = m ,
I R = n = 1 N I n .
| γ n , n + 1 ( 0 ) | = | γ n + 1 , n ( 0 ) | = 0 , | γ n , n + 2 ( 0 ) | = | γ n + 2 , n ( 0 ) | = 0.045 , | γ n , n + 3 ( 0 ) | = | γ n + 3 , n ( 0 ) | = 0.040 .