Abstract

A method of computing the diffraction integral has been developed for application to the on-axis irradiance of rectangular focused light beams with central and noncentral rectangular obscurations. Numerical methods for the generation of quantitative results are presented.

© 1992 Optical Society of America

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References

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  1. V. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  2. H. Osterberg, L. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  3. C. Rivolta, “Annular circular aperture: intensity maxima and minima on the optical axis,” Appl. Opt. 27, 922–925 (1988).
    [CrossRef] [PubMed]
  4. R. E. English, N. George, “Diffraction patterns in the shadows of disks and obstacles,” Appl. Opt. 27, 1581–1587 (1988).
    [CrossRef] [PubMed]
  5. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), p. 307.

1988 (2)

1983 (1)

1961 (1)

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Figures (4)

Fig. 1
Fig. 1

(a) Definition of variables for an example of a square aperture with a dimension of 2a with the noncentered obscuration located at A and B of the 2∊a dimension where ∊ < 1. s defines the distance from an aperture area element to the focus R, and l defines the distance to the range of interest z. (b) 1–4 in the rectangular aperture indicate the four areas over which the diffraction integral must be evaluated. A and B are the coordinates of the center of the obscuration.

Fig. 2
Fig. 2

Comparison of the on-axis irradiance of the square and circular centrally obscured apertures. The circular aperture is 0.3 m. The square aperture is 0.265 m2 so that the two apertures have the same area. In each case the fractional obscuration is 0.25.

Fig. 3
Fig. 3

Comparison of the normalized on-axis irradiance as a function of the range for three examples of obscuration fractions for square apertures with centralized square obscurations. The aperture dimensions are as in Fig. 2.

Fig. 4
Fig. 4

Comparison of the normalized on-axis irradiance for three square aperture sizes with scaled central obscurations. The dimensions represent half of the edge length. The focus is at a 3000-m range in each case. The obscuration is 25% of the aperture size in all cases.

Equations (35)

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U ( z ) = z A 0 2 π ρ= a 0 θ= 0 2 π 1 l exp ( i k s ) s l [ exp ( i k l ) / l ] ρ d ρ d θ ,
l = ( ρ 2 + z 2 ) 1 l 2 ,
s = ( ρ 2 + R 2 ) 1 l 2 ,
l ( 1 + ρ 2 2 z 2 ) z , 1 / l 1 / z , s ( 1 + ρ 2 2 R 2 ) R , 1 / s 1 / R .
I ( z ) = U ( z ) U * ( z ) = I 0 R 2 k 2 ( 1 z 2 + k 2 ) 4 ( R z ) 2 sin 2 ( Q 2 ) , z R ,
Q = ( 1 2 ) k 2 ( 1 z 1 R ) a 2 .
U ( z ) = A 0 R 2 ( 1 R i k ) ρ= a a ρ d ρ = A 0 R 2 ( 1 R i k ) ( a 2 2 2 a 2 2 ) , I ( z ) = U ( z ) U * ( z ) = A 0 2 a 2 4 R 4 ( 1 2 ) 2 ( 1 R 2 + k 2 ) A 0 2 a 2 4 R 4 ( 1 2 ) 2 ( 2 π λ ) 2 = A 0 2 A 2 R 4 λ 2 = P A R 2 λ 2 ,
U ( z ) = z A 0 2 π R z exp ( i k l ) exp ( i k s ) ( i k l 1 l 2 ) d x d y ,
U ( z ) = ( z A 0 2 π R z 3 z i k A 0 2 π R z 2 ) exp [ i k ( l s ) ] d x d y .
U ( z ) = z A 0 2 π R z 2 ( 1 z i k ) exp [ i k ( z R ) ] exp [ i k ρ 2 2 ( 1 z 1 R ) ] d x d y .
U ( z ) = z A 0 2 π R z 2 ( 1 z i k ) exp [ i k ( z R ) ] exp [ α ( x 2 + y 2 ) ] d x d y ,
α = i k 2 ( 1 z 1 R ) ·
U ( z ) = z A 0 8 α R z 2 ( 1 z i k ) exp [ i k ( z R ) ] [ 2 π x exp ( t 2 ) d t ] [ 2 π y exp ( t 2 ) d t ] .
U 1 = z A 0 8 α R z 2 ( 1 z i k ) exp [ i k ( z R ) ] [ erf ( α a ) erf ( α a ) ] [ erf ( α a ) + erf ( α a ) ] ,
U 2 = B [ erf ( α a ) + erf ( α a ) ] [ erf ( α a ) erf ( α a ) ] , U 3 = B [ erf ( α a ) erf ( α a ) ] [ erf ( α a ) + erf ( α a ) ] , U 4 = B [ erf ( α a ) + erf ( α a ) ] [ erf ( α a ) + erf ( α a ) ] ,
B = z A 0 8 α R z 2 ( 1 z i k ) exp [ i k ( z R ) ] .
U ( z ) = U 1 + U 2 + U 3 + U 4 = 4 B [ erf 2 ( α a ) erf 2 ( α a ) ] .
U ( z ) = B open ( 1 ) n + 1 erf ( α x ) erf ( α y ) B obsured ( 1 ) n erf ( α x ) erf ( α y ) .
U ( z ) = 3 B [ erf 2 ( α a ) ] B { erf 2 [ α a ( 2 1 ) ] } 2 B { erf ( α a ) erf [ α a ( 2 1 ) ] } .
I ( z ) = 16 B B * ( X 2 + Y 2 ) ,
I ( z ) = I 0 R 2 k 2 ( 1 z 2 + k 2 ) ( R z ) 2 ( X 2 + Y 2 ) ,
X 2 = ( { Re [ erf ( α a ) ] } 2 { Im [ erf ( α a ) ] } 2 { Re [ erf ( α a ) ] } 2 + { Im [ erf ( α a ) ] } 2 ) 2 ,
Y 2 = ( 2 { Re [ erf ( α a ) ] Im [ erf ( α a ) ] { Re [ erf ( α a ) ] Im [ erf ( α a ) ] } ) 2 ,
I ( z ) = P A R 2 λ 2
α = [ k 4 ( 1 R 1 z ) ] 1 / 2 ( 1 + i ) if R < z , α = [ k 4 ( 1 z 1 R ) ] 1 / 2 ( 1 + i ) if R > z .
erf ( q ) = 1 exp ( y 2 x 2 ) [ cos ( 2 x y ) i sin ( 2 x y ) ] w * ( y + i x ) ,
w ( q ) = exp ( q 2 ) erfc ( i q ) ,
w ( q ) = exp ( 2 i p 2 ) { 1 + ( i 1 ) [ C ( 2 π p ) + i S ( 2 π p ) ] } .
[ ± k 4 ( 1 R 1 z ) ] 1 / 2 ,
F = a 2 λ ( 1 R 1 z ) = 2 π p 2 ,
Re { w [ ( 1 + i ) p ] } = cos ( 2 p 2 ) [ 1 C ( 2 π p ) S ( 2 π p ) ] + sin ( 2 p 2 ) [ C ( 2 π p ) S ( 2 π p ) ] ,
Im { w [ ( 1 + i ) p ] } = cos ( 2 p 2 ) [ C ( 2 π p ) S ( 2 π p ) ] + sin ( 2 p 2 ) [ C ( 2 π p ) + S ( 2 π p ) 1 ] ,
C ( x ) = 1 / 2 + f ( x ) sin ( π x 2 / 2 ) g ( x ) cos ( π x 2 / 2 ) , S ( x ) = 1 / 2 f ( x ) cos ( π x 2 / 2 ) g ( x ) sin ( π x 2 / 2 ) .
Re { w [ ( 1 + i ) p ] } = f ( 2 π p ) + g ( 2 π p ) , Im { w [ ( 1 + i ) p ] } = f ( 2 π p ) g ( 2 π p ) .
C ( x ) x , S ( x ) π x 3 6 .

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