Abstract

Two-beam interference with partially coherent light is discussed. A new proof of the general interference law for partially coherent fields is obtained and is illustrated by means of simple and direct experiments. Photographs are given which show the changes in the visibility of the fringes as the degree of coherence varied and the results are compared with the predictions of the theory.

© 1957 Optical Society of America

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References

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  1. F. Zernike, Physica 5, 785 (1938).
    [CrossRef]
  2. (a)H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951); (b)Proc. Roy. Soc. (London) A217, 408 (1953).
  3. (a)E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954); (b)Proc. Roy. Soc. (London) A230, 246 (1955); (c)Nuovo cimento 12, 884 (1954); (d)Proceedings of the Symposium on Astronomical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam1956), p. 177; (e)Phil. Mag. 2 (8), 351 (1957).
  4. A. Blanc-Lapierre and P. Dumontet, Rev. opt. 34, 1 (1955).
  5. (a)D. Gabor, Proceedings of the Symposium on Astromical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam, 1956), p. 59; (b)Proceedings of the Third London Symposium on Information Theory, edited by C. Cherry (Butterworths Scientific Publications, London, 1956), p. 26.
  6. A. T. Forrester, Am. J. Phys. 24, 192 (1956).
    [CrossRef]
  7. (a)R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956); (b)Nature 178, 1046 (1956).
  8. (a)P. H. Van Cittert, Physica 1, 201 (1934); (b)Physica 6, 1129 (1939).
  9. L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).
  10. Arnulf, Dupy, and Flamant, Rev. opt. 32, 529 (1953).
  11. (a) Taylor, Hinde, and Lipson, Acta Cryst. 4, 261 (1951); (b) Hanson, Lipson, and Taylor, Proc Roy. Soc. (London) A218, 371 (1953); (c)W. Hughes and C. A. Taylor, J. Sci. Instr. 30, 105 (1953).

1956 (2)

A. T. Forrester, Am. J. Phys. 24, 192 (1956).
[CrossRef]

(a)R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956); (b)Nature 178, 1046 (1956).

1955 (1)

A. Blanc-Lapierre and P. Dumontet, Rev. opt. 34, 1 (1955).

1954 (1)

(a)E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954); (b)Proc. Roy. Soc. (London) A230, 246 (1955); (c)Nuovo cimento 12, 884 (1954); (d)Proceedings of the Symposium on Astronomical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam1956), p. 177; (e)Phil. Mag. 2 (8), 351 (1957).

1953 (2)

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

Arnulf, Dupy, and Flamant, Rev. opt. 32, 529 (1953).

1951 (2)

(a) Taylor, Hinde, and Lipson, Acta Cryst. 4, 261 (1951); (b) Hanson, Lipson, and Taylor, Proc Roy. Soc. (London) A218, 371 (1953); (c)W. Hughes and C. A. Taylor, J. Sci. Instr. 30, 105 (1953).

(a)H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951); (b)Proc. Roy. Soc. (London) A217, 408 (1953).

1938 (1)

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

1934 (1)

(a)P. H. Van Cittert, Physica 1, 201 (1934); (b)Physica 6, 1129 (1939).

Arnulf,

Arnulf, Dupy, and Flamant, Rev. opt. 32, 529 (1953).

Baker, L. R.

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

Blanc-Lapierre, A.

A. Blanc-Lapierre and P. Dumontet, Rev. opt. 34, 1 (1955).

Dumontet, P.

A. Blanc-Lapierre and P. Dumontet, Rev. opt. 34, 1 (1955).

Dupy,

Arnulf, Dupy, and Flamant, Rev. opt. 32, 529 (1953).

Flamant,

Arnulf, Dupy, and Flamant, Rev. opt. 32, 529 (1953).

Forrester, A. T.

A. T. Forrester, Am. J. Phys. 24, 192 (1956).
[CrossRef]

Gabor, D.

(a)D. Gabor, Proceedings of the Symposium on Astromical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam, 1956), p. 59; (b)Proceedings of the Third London Symposium on Information Theory, edited by C. Cherry (Butterworths Scientific Publications, London, 1956), p. 26.

Hanbury Brown, R.

(a)R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956); (b)Nature 178, 1046 (1956).

Hinde,

(a) Taylor, Hinde, and Lipson, Acta Cryst. 4, 261 (1951); (b) Hanson, Lipson, and Taylor, Proc Roy. Soc. (London) A218, 371 (1953); (c)W. Hughes and C. A. Taylor, J. Sci. Instr. 30, 105 (1953).

Hopkins, H. H.

(a)H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951); (b)Proc. Roy. Soc. (London) A217, 408 (1953).

Lipson,

(a) Taylor, Hinde, and Lipson, Acta Cryst. 4, 261 (1951); (b) Hanson, Lipson, and Taylor, Proc Roy. Soc. (London) A218, 371 (1953); (c)W. Hughes and C. A. Taylor, J. Sci. Instr. 30, 105 (1953).

Taylor,

(a) Taylor, Hinde, and Lipson, Acta Cryst. 4, 261 (1951); (b) Hanson, Lipson, and Taylor, Proc Roy. Soc. (London) A218, 371 (1953); (c)W. Hughes and C. A. Taylor, J. Sci. Instr. 30, 105 (1953).

Twiss, R. Q.

(a)R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956); (b)Nature 178, 1046 (1956).

Van Cittert, P. H.

(a)P. H. Van Cittert, Physica 1, 201 (1934); (b)Physica 6, 1129 (1939).

Wolf, E.

(a)E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954); (b)Proc. Roy. Soc. (London) A230, 246 (1955); (c)Nuovo cimento 12, 884 (1954); (d)Proceedings of the Symposium on Astronomical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam1956), p. 177; (e)Phil. Mag. 2 (8), 351 (1957).

Zernike, F.

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

Acta Cryst. (1)

(a) Taylor, Hinde, and Lipson, Acta Cryst. 4, 261 (1951); (b) Hanson, Lipson, and Taylor, Proc Roy. Soc. (London) A218, 371 (1953); (c)W. Hughes and C. A. Taylor, J. Sci. Instr. 30, 105 (1953).

Am. J. Phys. (1)

A. T. Forrester, Am. J. Phys. 24, 192 (1956).
[CrossRef]

Nature (1)

(a)R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956); (b)Nature 178, 1046 (1956).

Physica (2)

(a)P. H. Van Cittert, Physica 1, 201 (1934); (b)Physica 6, 1129 (1939).

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

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

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

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

(a)H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951); (b)Proc. Roy. Soc. (London) A217, 408 (1953).

(a)E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954); (b)Proc. Roy. Soc. (London) A230, 246 (1955); (c)Nuovo cimento 12, 884 (1954); (d)Proceedings of the Symposium on Astronomical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam1956), p. 177; (e)Phil. Mag. 2 (8), 351 (1957).

Rev. opt. (2)

A. Blanc-Lapierre and P. Dumontet, Rev. opt. 34, 1 (1955).

Arnulf, Dupy, and Flamant, Rev. opt. 32, 529 (1953).

Other (1)

(a)D. Gabor, Proceedings of the Symposium on Astromical Optics, edited by Z. Kopal (North-Holland Publishing Company, Amsterdam, 1956), p. 59; (b)Proceedings of the Third London Symposium on Information Theory, edited by C. Cherry (Butterworths Scientific Publications, London, 1956), p. 26.

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Figures (6)

Fig. 1
Fig. 1

Illustrating the interference of two partially coherent beams.

Fig. 2
Fig. 2

Illustrating the van Cittert-Zernike theorem.

Fig. 3
Fig. 3

The diffractometer.

Fig. 4
Fig. 4

Two-beam interference with partially coherent light. (a) Observed patterns (b) Theoretical intensity curves computed from formula (4.7). The chain lines represent the curves Imax and Imin. In G(b), for 2h=2.5, read 2h=2.5 cm.

Fig. 5
Fig. 5

Calculation of the intensity distribution in the focal plane F.

Fig. 6
Fig. 6

The degree of coherence, as function of the separation 2h of the apertures at P1 and P2.

Equations (28)

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V P ( t ) = K 1 V 1 ( t - t 1 ) + K 2 V 2 ( t - t 2 ) ,
t 1 = r 1 / c ,             t 2 = r 2 / c ,
I P = V P ( t ) V P * ( t ) = K 1 K 1 * V 1 ( t - t 1 ) V 1 * ( t - t 1 ) + K 2 K 2 * + V 2 ( t - t 2 ) V 2 * ( t - t 2 ) + K 1 K 2 * V 1 ( t - t 1 ) V 2 * ( t - t 2 ) + K 2 K 1 * V 2 ( t - t 2 ) V 1 * ( t - t 1 ) ,
τ = t 2 - t 1 = r 2 - r 1 c
I P = K 1 2 I 1 + K 2 2 I 2 + 2 K 1 K 2 J 12 ( τ ) ,
J 12 ( τ ) = V 1 ( t + τ ) V 2 * ( t ) ,
γ 12 ( τ ) = J 12 ( τ ) ( J 11 ( 0 ) ) 1 2 ( J 22 ( 0 ) ) 1 2 = J 12 ( τ ) ( I 1 ) 1 2 ( I 2 ) 1 2 ,
I P = I P ( 1 ) + I P ( 2 ) + 2 ( I P ( 1 ) ) 1 2 ( I P ( 2 ) ) 1 2 γ 12 ( τ ) .
γ 12 ( τ ) = γ 12 ( τ ) exp { - i [ ω ¯ τ - α 12 ( τ ) ] } .
τ c ~ 1 / Δ ω .
γ 12 ( 0 ) = g 12 ,             α 12 ( 0 ) = arg γ 12 ( 0 ) = β 12 ,
I P = I P ( 1 ) + I P ( 2 ) + 2 ( I P ( 1 ) ) 1 2 ( I P ( 2 ) ) 1 2 g 12 cos [ β 12 - δ ] ,
δ = ω ¯ c ( r 2 - r 1 ) = 2 π λ ¯ ( r 2 - r 1 ) ,
I = g 12 [ I P ( 1 ) + I P ( 2 ) + 2 ( I P ( 1 ) ) 1 2 ( I P ( 2 ) ) 1 2 cos ( β 12 - δ ) ] + ( 1 - g 12 ) [ I P ( 1 ) + I P ( 2 ) ] .
I incoh I coh = 1 - g 12 g 12 .
I P = 2 I P ( 0 ) [ 1 + g 12 cos ( β 12 - δ ) ] .
V = I max - I min I max + I min = g 12 ,
γ 12 ( 0 ) = g 12 e i β 12 = 1 ( I 1 ) 1 2 ( I 2 ) 1 2 S j ( x , y ) exp | i ω ˜ c ( R 1 - R 2 ) | R 1 R 2 d x d y ,
I P ( 0 ) = ( 2 J 1 ( v ) v ) 2 ,
v = 2 π λ a sin ϕ .
γ 12 ( 0 ) = g 12 e i β 12 = 2 J 12 ( u ) u ,
g 12 = | 2 J 12 ( u ) u | , β 12 = 0             when             2 J 12 ( u ) u > 0 , = π             when             2 J 1 ( u ) u < 0 ,
u = 2 π λ ¯ r 1 sin 2 θ = 2 π λ ¯ r 1 2 h f 1 .
where             δ = 2 π λ ¯ ( 2 h sin ϕ ) = C u v , C = λ ¯ f 1 / 2 π r 1 a .
I ( ϕ , h ) = 2 ( 2 J 1 ( v ) v ) 2 { 1 + | 2 J 1 ( u ) u | cos [ β 12 ( u ) - C u v ] } .
and             I max ( ϕ , h ) = 2 ( 2 J 1 ( v ) v ) 2 { 1 + | 2 J 1 ( u ) u | } , I min ( ϕ , h ) = 2 ( 2 J 1 ( v ) v ) 2 { 1 - | 2 J 1 ( u ) u | } .
g 12 = | 2 J 1 ( u ) u | = | 2 J 1 ( 2 π λ ¯ r f 1 2 h ) 2 π λ ¯ r f 1 2 h | ,
γ 12 ( τ ) = 1 ( I 1 ) 1 2 ( I 2 ) 1 2 0 S j ( x , y ; ω ) exp ( - i ω [ τ - R 1 - R 2 c ] ) R 1 R 2 d x d y .