Abstract

The theory of Young’s interference fringes is developed with particular attention to the finite size of the sampling apertures. A spatial Fourier transform of the product of the intensity distribution of the finite-sized source and the shifted intensity impulse response of the sampling aperture allows us to define a function Ĝ whose absolute value and phase dictate the visibility and the shift of the fringes, respectively. Alternatively, the function Ĝ may be expressed as a spatial Fourier transform of the spatial-coherence function across the sampling plane times the shifted transfer function of the sampling aperture. The effect of the finiteness of the sampling aperture becomes predominant in the neighborhood of the zeros of the coherence function or in regions where the coherence function is changing fast.

© 1984 Optical Society of America

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Corrections

A. S. Marathay and D. B. Pollock, "Young’s interference fringes with finite-sized sampling apertures: erratum," J. Opt. Soc. Am. A 2, 776-776 (1985)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-2-5-776

References

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  1. B. J. Thompson, E. Wolf, “Two-beam interference with partially coherent light,”J. Opt. Soc. Am. 47, 895–902 (1957).
    [CrossRef]
  2. B. J. Thompson, “Illustration of the phase change in two-beam interference with partially coherent light,”J. Opt. Soc. Am. 48, 95–97 (1958).
    [CrossRef]
  3. B. J. Thompson, R. Sudol, “Finite aperture effects in the measurement of the degree of coherence,” J. Opt. Soc. Am. A 1, 598–604 (1984).
    [CrossRef]
  4. A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).

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Figures (1)

Fig. 1
Fig. 1

Image-forming lens. The aperture plane is shown with two small sampling apertures, each with diameter 2a and center-to-center separation 2b.

Equations (16)

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I ^ ( x , ν ) = ( λ 2 / π ) d ξ I ^ s ( ξ , ν ) A ( x - m ξ ) 2 ,
A ( x - m ξ ) = 1 λ 2 s s d α L ( α ) exp [ - i 2 π ( x - m ξ ) · ( α λ s ) ] ,
f = α / λ s ,             g = β / λ s ;             f = i ^ f + j ^ g .
A ( x - m ξ ) = ( s / s ) d f L ( λ s f ) exp [ - i 2 π ( x - m ξ ) · f ] .
L ( α ) L ( α , β ) = cyl { [ ( α - b ) 2 + β 2 ] 1 / 2 a } + cyl { [ ( α + b ) 2 + β 2 ] 1 / 2 a } ,
I ^ ( x , ν ) = 2 d ξ I ^ s ( ξ , ν ) S ( x - m ξ , ν ) × { 1 + cos [ 4 π b ( x - m ξ ) λ s ] } ,
S ( x - m ξ ) = π a 4 ( λ s s ) 2 × | Besinc { 2 π a [ ( x - m ξ ) 2 + ( y - m η ) 2 ] 1 / 2 λ s } | 2 .
cos [ 4 π b ( x - m ξ ) λ s ] = cos ( 4 π b x λ s ) cos ( 4 π b m ξ λ s ) + sin ( 4 π b x λ s ) sin ( 4 π b m ξ λ s ) ,
G ^ ( f ; x , ν ) G ^ ( f , g ; x , y , ν ) = d ξ I ^ s ( ξ , ν ) S ( x - m ξ , ν ) exp [ - i 2 π f · ξ ] .
Γ ^ ( α 1 - α 2 , ν ) = 1 π s 2 d ξ I ^ s ( ξ , ν ) exp [ - i 2 π λ s ( α 1 - α 2 ) · ξ ] .
S ( x - m ξ ) = d f T ( f ) exp [ + i 2 π f · ( x - m ξ ) ] .
G ^ ( f ; x , ν ) = ( π s 2 ) d f 12 Γ ^ ( λ s f 12 , ν ) T ( f 12 - f ) × exp [ - i 2 π x · ( f 12 - f ) ] ,
I ^ ( x , y , v ) = 2 { G ^ ( 0 , 0 ; x , y , ν ) + Re [ G ^ ( 2 b m λ s , 0 ; x , y , ν ) ] × cos ( 4 π b x λ s ) + Im [ G ^ ( 2 b m λ s , 0 ; x , y , ν ) ] sin ( 4 π b x λ s ) } .
G ^ ( f , g ; x , y , ν ) = G ^ ( f , g ; x , y , ν ) G ^ ( 0 , 0 ; x , y , ν ) ,
G ^ = G ^ e i ϕ .
I ^ ( x , y , ν ) = 2 G ^ ( 0 , 0 ; x , y , ν ) [ 1 + | G ^ ( 2 b m λ s , 0 ; x , y , ν ) | × cos ( 4 π x b λ s - ϕ ) ] .

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