Abstract

Rayleigh’s diffraction integral is solved in closed form as regards all axial points when a divergent or a convergent spherical wave is specified as the electromagnetic disturbance incident upon a circular aperture or obstacle. Diffraction of divergent waves is treated briefly. The method is applied more fully to the diffraction of convergent waves by circular apertures. It is shown that the axial “focal point” of the converged spherical wave falls inside, at, or outside the geometrical focal point according as the angular semiaperture θm of the lens is less than, equal to, or greater than a particular angle that falls near 70.5°. The magnitudes of the departures of the focal point from the geometrical focal point are illustrated by examples for both the radar and optical regions.

© 1961 Optical Society of America

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References

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  1. C. J. Bouwkamp, Repts. Progr. Phys. 17, (1954), Eq. (2.7), 39.
    [Crossref]
  2. R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, Rhode Island, 1944), p. 356.
  3. See reference 1, pp. 49 and 50.
  4. Consequently, the authors have a strong preference for Rayleigh’s diffraction integral.
  5. See reference 2, pp. 361–363.
  6. See reference 2, p. 363.
  7. G. W. Farnell, J. Opt. Soc. Am. 48, 643–47 (1958).
    [Crossref]
  8. See reference 7, p. 644.

1958 (1)

1954 (1)

C. J. Bouwkamp, Repts. Progr. Phys. 17, (1954), Eq. (2.7), 39.
[Crossref]

Bouwkamp, C. J.

C. J. Bouwkamp, Repts. Progr. Phys. 17, (1954), Eq. (2.7), 39.
[Crossref]

Farnell, G. W.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, Rhode Island, 1944), p. 356.

J. Opt. Soc. Am. (1)

Repts. Progr. Phys. (1)

C. J. Bouwkamp, Repts. Progr. Phys. 17, (1954), Eq. (2.7), 39.
[Crossref]

Other (6)

R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, Rhode Island, 1944), p. 356.

See reference 1, pp. 49 and 50.

Consequently, the authors have a strong preference for Rayleigh’s diffraction integral.

See reference 2, pp. 361–363.

See reference 2, p. 363.

See reference 7, p. 644.

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Figures (5)

Fig. 1
Fig. 1

(a) Notation with respect to the diffraction of a divergent spherical wave by a circular aperture. Spherical waves diverge from the source S and are diffracted toward an axial point P by the circular aperture of radius R. (b) Diffraction of a divergent spherical wave by a circular obstacle. Corresponding elements have the same notation as in (a).

Fig. 2
Fig. 2

Notation with respect to the diffraction of a convergent spherical wavefront W by a circular aperture of radius R. Rays from S are converged upon the geometrical image S′ located at distance d0 from the aperture. The resulting scalar disturbance u(z0) or u(z) is to be determined at the axial point P.

Fig. 3
Fig. 3

Plot of the normalized irradiance |u(z)|2 against distance z in wavelengths λ from the geometrical focus for θm=10° and 20° with R fixed at 20 wavelengths λ where λ=λ0/n is the wavelength in the image space. The irradiance is normalized to assume the value unity at z=0. Distance z is negative when the point of observation falls between the diffracting aperture and the geometrical focal point.

Fig. 4
Fig. 4

Plot of the normalized irradiance against z for θm=45°, 60°, 70.53°, and 85° with the radius R of the aperture fixed at 20 wavelengths. When θm=70.53°, the peak value of the irradiance (and hence the expected focal point) falls at the geometrical focal point where z=0. The inset reveals more clearly the distribution of the irradiance near z=0.

Fig. 5
Fig. 5

Variation of the actual irradiance is |u(0)|2 times λ2 with the angular semiaperture θm at the geometrical focal point z=0.

Tables (1)

Tables Icon

Table I Departures z of the focal point from the geometrical focal point as a function of θm and R.

Equations (42)

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2 π u ( x , y , z 0 ) = - z 0 u ( ζ , η , 0 ) 1 r r × ( exp ( i k n r ) r ) d ζ d η ,
[ ( x - ζ ) 2 + ( y - η ) 2 + z 0 2 ] 1 2 ,
u ( ζ , η , 0 ) = exp ( i k n r 0 ) / r 0
u a ( z 0 ) = - z 0 0 R exp ( i k n r 0 ) r 0 1 r r ( exp ( i k n r ) r ) ρ d ρ ;
u 0 ( z 0 ) = - z 0 R exp ( i k n r 0 ) r 0 1 r r ( exp ( i k n r ) r ) ρ d ρ ;
r 0 = ( d 0 2 + ρ 2 ) 1 2 ;             r = ( z 0 2 + ρ 2 ) 1 2 .
exp ( i k n r 0 ) r 0 1 r r ( exp ( i k n r ) r ) ρ d ρ = 1 r exp [ i k n ( r 0 + r ) ] r 0 + r .
u a ( z 0 ) = exp [ i k n ( d 0 + z 0 ) ] / ( d 0 + z 0 ) + cos α m exp ( i π ) exp ( i k n d ) / d ;
u 0 ( z 0 ) = cos α m exp ( i k n d ) / d ;
d = ( d 0 2 + R 2 ) 1 2 + ( z 0 2 + R 2 ) 1 2
cos α m = z 0 / ( z 0 2 + R 2 ) 1 2
u a ( z 0 ) 2 = 1 ( d 0 + z 0 ) 2 { 1 - 2 cos α m d 0 + z 0 d × cos [ k ( Δ 0 + Δ ) ] + cos 2 α m ( d 0 + z 0 ) 2 d 2 } ;
Δ 0 = n [ ( d 0 2 + R 2 ) 1 2 - d 0 ] ;
Δ = n [ ( z 0 2 + R 2 ) 1 2 - z 0 ] .
d 0 2 u a ( z 0 ) 2 1 - 2 cos α m cos ( k Δ ) + cos 2 α m ,
u ( z 0 ) = - z 0 exp ( i k C ) 0 R exp ( - i k n r 0 ) r 0 1 r r × ( exp ( i k n r ) r ) ρ d ρ ,
exp ( - i k n r 0 ) r 0 1 r r ( exp ( i k n r ) r ) ρ d ρ = - 1 r exp [ i k n ( r - r 0 ) ] r - r 0 .
u ( z 0 ) = - exp ( i k C ) [ exp ( i k n z ) / z - cos α m exp ( i k n d 1 ) / d 1 ] ,
z = z 0 - d 0 ;
d 1 = ( z 0 2 + R 2 ) 1 2 - ( d 0 2 + R 2 ) 1 2 .
u ( d 0 ) = - d 0 exp ( i k C ) d 0 ( d 0 2 + R 2 ) 1 2 ( i n k / r 2 - 1 / r 3 ) d r ,
u ( d 0 ) = - exp ( i k C ) [ i n k ( 1 - cos θ m ) - sin 2 θ m / 2 d 0 ] ,
cos θ m = d 0 / ( R 2 + d 0 2 ) 1 2 .
u ( z 0 ) u ( 0 ) = exp ( i k C ) exp ( - i k n d 0 ) / d 0 ,
u ( z ) = u ( z 0 ) ,
u ( z ) = - i k n exp ( i k C ) 0 θ m exp ( i k n z cos θ ) sin θ d θ ,
u ( z ) L = - exp ( i k C ) [ exp ( i k n z ) / z - exp ( i k n z cos θ m ) / z ]
d 1 = Z 0 [ 1 + ( 2 z d 0 + z 2 ) / Z 0 2 ] 1 2 - Z 0 ,
[ 1 + ( 2 z d 0 + z 2 ) / Z 0 2 ] 1 2 1 + z d 0 / Z 0 2 .
d 1 = z d 0 / Z 0 = z cos θ m
λ 2 u ( z ) 2 = 1 z 2 + [ z + d 0 Z ( Z - Z 0 ) ] 2 - 2 ( z + d 0 ) z Z ( Z - Z 0 ) × cos [ 2 π ( Z - Z 0 - z ) ]
λ 2 u ( z ) 2 = [ Z ( Z - Z 0 ) - z ( z + d 0 ) z Z ( Z - Z 0 ) ] 2 + 4 ( z + d 0 ) z Z ( Z - Z 0 ) × sin 2 [ π ( Z - Z 0 - z ) ] ,
Z = [ ( z + d 0 ) 2 + R 2 ] 1 2 ;
Z 0 = [ d 0 2 + R 2 ] 1 2 .
λ 2 u ( z ) 2 = [ g d 0 + z ( 1 - g ) ] 2 + 4 ( d 0 + z ) d 0 + z ( 1 - g ) [ sin ( π z ϕ ) z ] 2 ;
2 g = 1 - v 2 + v 3 w - 5 v 4 w 2 / 4 + 7 v 5 w 3 / 4 - 21 v 6 w 4 / 8 + 33 v 7 w 5 / 8 ;
v = ( 2 d 0 + z ) / 2 Z 0 ;             w = z / Z 0 ;             ϕ = ( Z 0 - d 0 - z g ) / Z 0 .
D z d d z ( λ 2 u ( z ) 2 ) ,
D z = - 0.75 ( 1 - σ 2 ) ( 1 - σ ) d 0 - 3 [ σ 3 + σ 2 + ( 24 π 2 d 0 2 + 1 ) σ / 3 - ( 8 π 2 d 0 2 - 1 ) / 3 ]
σ = cos θ m = d 0 / ( d 0 2 + R 2 ) 1 2 > 0.
σ 3 + σ 2 + ( 24 π 2 d 0 2 + 1 ) σ / 3 - ( 8 π 2 d 0 2 - 1 ) / 3 = 0.
8 π 2 d 0 2 = ( 1 + σ + 3 σ 2 + 3 σ 3 ) / ( 1 - 3 σ ) .