Abstract

The exact calculation of the field at a point due to a single Fresnel zone is carried out by using the Maggi–Rubinowicz contour integral. The result agrees with the Fresnel theorem in the limit for k very large, k being the propagation constant of the incident wave. The results obtained suggest a new interpretation of the physical meaning of the Maggi–Rubinowicz contour integral in diffraction theory as representing a contribution of elementary or Fresnel zones, in exactly the same manner that the Kirchhoff integral does when considered as an expression of Huygens’ principle.

© 1969 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 373.
  2. B. A. Lippmann, J. Opt. Soc. Am. 55, 360 (1965).
    [Crossref]
  3. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer Verlag, Berlin, 1967).
  4. R. W. Ditchburn, Light (Blackie and Son Ltd, London, 1963), pp. 167, 172.
  5. A calculation using the principle of interference of elementary vibrations is given by R. M. Shourcri, Thèse de Maîtrise, Université Laval, Québec, p. 36 (1968). The answer is identical to Eq. (9). The factor −2i is omitted in the result given by R. W. Ditchburn.
  6. A. Rubinowicz, in Progress in Optics, IV, E. Wolf Ed., (North–Holland Publ. Co., Amsterdam, 1965). See also Ref. 3, p. 88.
  7. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
    [Crossref]
  8. A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec; Gauthier-Villars, Paris, 1964), p. 446.

1966 (1)

1965 (1)

Boivin, A.

A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec; Gauthier-Villars, Paris, 1964), p. 446.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 373.

Ditchburn, R. W.

R. W. Ditchburn, Light (Blackie and Son Ltd, London, 1963), pp. 167, 172.

Lippmann, B. A.

Marchand, E. W.

Rubinowicz, A.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer Verlag, Berlin, 1967).

A. Rubinowicz, in Progress in Optics, IV, E. Wolf Ed., (North–Holland Publ. Co., Amsterdam, 1965). See also Ref. 3, p. 88.

Shourcri, R. M.

A calculation using the principle of interference of elementary vibrations is given by R. M. Shourcri, Thèse de Maîtrise, Université Laval, Québec, p. 36 (1968). The answer is identical to Eq. (9). The factor −2i is omitted in the result given by R. W. Ditchburn.

Wolf, E.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 373.

J. Opt. Soc. Am. (2)

Other (6)

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer Verlag, Berlin, 1967).

R. W. Ditchburn, Light (Blackie and Son Ltd, London, 1963), pp. 167, 172.

A calculation using the principle of interference of elementary vibrations is given by R. M. Shourcri, Thèse de Maîtrise, Université Laval, Québec, p. 36 (1968). The answer is identical to Eq. (9). The factor −2i is omitted in the result given by R. W. Ditchburn.

A. Rubinowicz, in Progress in Optics, IV, E. Wolf Ed., (North–Holland Publ. Co., Amsterdam, 1965). See also Ref. 3, p. 88.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 373.

A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec; Gauthier-Villars, Paris, 1964), p. 446.

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Figures (8)

Fig. 1
Fig. 1

Intersections of the Fresnel ellipsoids with a spherical surface S. O is the point source, P the observation point. The projection of the intersection circles, of the 1st, 2nd, ⋯nth ellipsoids with the spherical surface S are denoted respectively by e1e1′, e2e2′, ⋯enen′.

Fig. 2
Fig. 2

Diffraction by an aperture whose radius is smaller than the radius of the first Fresnel zone considered on a spherical surface. f<1. ρ1 is the radius of the contour of the first zone.

Fig. 3
Fig. 3

The diffracting aperture of the plane A coincides with the nth Fresnel ellipsoid defined by coshu = 1+/kR, R = OP = 2c.

Fig. 4
Fig. 4

Variation of y = sin2v/(sinh2u+sin2v) with respect to sinv. The figure also shows the dependence of the curve y on the propagation constant k; k>k>k0.

Fig. 5
Fig. 5

Intersections of the Fresnel ellipsoids with a plane A at a distance x from the mid-point of OP.

Fig. 6
Fig. 6

Determination of the Fresnel zones over a plane A with respect to the observation point P. The lines 1, 2, 3, ⋯ are respectively the directrices of the 1st, 2nd, 3rd Fresnel parabolas considered in the plane of the figure.

Fig. 7
Fig. 7

Observation point P in the geometrical shadow. The singularity of the vector potential is the intersection point D of the line OP with the surface G.

Fig. 8
Fig. 8

Inclined plane the normal to which makes an angle α with the axis OP of the Fresnel ellipsoids. The perpendicular line from P to the inclined plane does not coincide necessarily with the ellipsoid shown in the figure.

Equations (31)

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φ zone ( n ) = φ ell ( n - 1 ) - φ ell ( n ) , φ ell ( 0 ) = exp i k R / R ,
φ tot ( N ) = n = 1 N φ zone ( n ) = φ ell ( 0 ) - φ ell ( N ) ,
φ ell ( n ) = ( - 1 ) n e i k R R × ( 1 + n π / 2 k R ) - ( r 0 / R ) ( r 0 / R ) - ( n π / 2 k R ) r 0 / R ( 1 + n π / k R - r 0 / R ) ,
φ zone ( 1 ) = e i k R R × [ 1 + ( 1 + π / 2 k R ) - ( r 0 / R ) ( r 0 / R ) - π / ( 2 k R ) r 0 / R ( 1 + π / k R ) - ( r 0 / R ) ] .
φ zone = e i k R / R - φ ell .
φ zone = ( e i k R / R ) ( 1 - e i n π ) .
n λ = ( f ρ 1 ) 2 ( 1 r 0 + 1 r ρ ) ( f ρ 1 ) 2 ( 1 r 0 + 1 r ρ ) = f 2 λ , n = f 2 , φ zone = e i k R R ( 1 - e i f 2 π ) = e i k R R exp ( i f 2 π 2 ) [ exp ( - i f 2 π 2 ) - exp ( i f 2 π 2 ) ] ,
φ zone = - 2 i e i k R R sin ( f 2 π 2 ) exp ( i f 2 π 2 ) ,
φ tot ( N ) = U k ( P ) = e i k R R [ 1 - ( - 1 ) n sin 2 v sinh 2 u + sin 2 v ] .
φ ell ( n ) = ( - 1 ) n e i k R R R 2 / 4 ( 1 + n π / k R ) 2 - x 2 R 2 / 4 ( 1 + n π / k R ) 4 - x 2 ,
φ zone ( 1 ) = e i k R R [ 1 + R 2 / 4 ( 1 + π / k R ) 2 - x 2 R 2 / 4 ( 1 + π / k R ) 4 - x 2 ] .
φ ell ( n ) = ( - 1 ) n e i k R R [ ( 1 + n π k R ) 4 + ( π n R k h R ) 2 ( 1 + n π k R ) 2 ] - 1 2 ,
φ par ( n ) = ( - 1 ) n exp [ i k ( r + r p ) ] r p + n π / 2 k r p + n π / k .
φ zone ( 1 ) = exp [ i k ( r + r p ) ] ( 1 + r p + π / 2 k r p + π / k ) < 2 exp [ i k ( r + r p ) ] .
U k ( P ) = φ ell ( n ) .
W = 1 4 π exp ( i k r 0 ) r 0 exp ( i k r p ) r p r 0 × r p r 0 r p + r 0 · r p
W = 1 4 π exp ( i k 2 c cosh u ) sin v c 2 sinh u ( cosh 2 u - cos 2 v ) .
2 π r = 2 π c sinh u     sin v : φ ell ( n ) = ( - 1 ) n e i k R R sin 2 v cosh 2 u - cos 2 ν .
cosh u = 1 + n π k R
cos v = r 0 - r p R = r 0 - ( R + n π / k - r 0 ) R = 2 r 0 R - ( 1 + n π k R )
cos v = x c cosh u = x R / 2 [ 1 + ( n π / k R ) ] ,
φ ell ( n ) = 1 4 π exp ( i k 2 c cosh u ) 2 c 0 2 π sin 2 v sinh 2 u + sin 2 v d θ .
z = - x cot α .
sin θ ( v ) = - cot α coth u cot v .
tan v 1 = - cot α coth u ,
cot v = sin θ / tan v 1
sin 2 v = tan 2 v 1 / ( tan 2 v 1 + sin 2 θ ) .
I = 0 2 π sin 2 v sinh 2 u + sin 2 v d σ = 0 2 π tan 2 v 1 tan 2 v 1 ( 1 + sinh 2 u ) + sinh 2 u sin 2 θ d θ .
I = 2 π / ( cosh 4 u + sinh 4 u tan 2 α ) 1 2 .
I = 2 π [ ( 1 + n π k R ) 4 + ( n π R k R h ) 2 ( 1 + n π 2 k R ) 2 ] - 1 2 ,
φ ell ( n ) = ( - 1 ) n e i k R R [ ( 1 + n π k R ) 4 + ( n π R k R h ) 2 ( 1 + n π 2 k R ) 2 ] - 1 2 .