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References

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  1. H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1802), Pt. II, pp. 187–188.
  2. N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
    [CrossRef]
  3. Rayleigh, The Theory of Sound (Dover Publications, Inc., N. Y., 1945), Vol. II, p. 106.
  4. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
    [CrossRef]
  5. L. Bergstein and E. Marom, J. Opt. Soc. Am. 56, 16 (1966).
    [CrossRef]
  6. L. Bergstein and T. Zachos, J. Opt. Soc. Am. 56, 931 (1966).
    [CrossRef]
  7. L. M. Brekhovskikh, Waves in Layered Media (Academic Press Inc., New York, 1960), p. 239.
  8. The integration and limit operations could be interchanged since the integration over u and υ is uniformly convergent in z.
  9. A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill Book Co., N. Y., 1962), p. 281.

1966 (3)

1962 (1)

Bergstein, L.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic Press Inc., New York, 1960), p. 239.

Marchand, E. W.

Marom, E.

Mukunda, N.

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill Book Co., N. Y., 1962), p. 281.

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1802), Pt. II, pp. 187–188.

Rayleigh,

Rayleigh, The Theory of Sound (Dover Publications, Inc., N. Y., 1945), Vol. II, p. 106.

Wolf, E.

Zachos, T.

J. Opt. Soc. Am. (4)

Other (5)

H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1802), Pt. II, pp. 187–188.

Rayleigh, The Theory of Sound (Dover Publications, Inc., N. Y., 1945), Vol. II, p. 106.

L. M. Brekhovskikh, Waves in Layered Media (Academic Press Inc., New York, 1960), p. 239.

The integration and limit operations could be interchanged since the integration over u and υ is uniformly convergent in z.

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill Book Co., N. Y., 1962), p. 281.

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Figures (1)

Fig. 1
Fig. 1

Huygens-principle configuration. R is the illuminated aperture in the (x,y) plane (z = 0), L is its boundary, r is the radius vector from a. point (ξ,η,0) on R to the observation point P(x,y,z).

Equations (13)

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f ( x , y , z ) = 1 2 π R f ( ξ , η ) ζ ( e i β r r ) ζ = 0 d ξ d η = β i 2 π R f ( ξ , η ) | z | r ( 1 1 i β r ) e i β r r d ξ d η ,
r 2 = ( x ξ ) 2 + ( y η ) 2 + ( z ζ ) 2 .
| z | r ( 1 1 i β r ) e i β r r = β i 2 π exp { i β [ ( x ξ ) u + ( y η ) υ + | z | w ] } d u d υ .
e i β r r = i β 2 π exp { i β [ ( x ξ ) u + ( y η ) υ + z w ] } d u d υ w , for z > 0 e i β r r = i β 2 π exp { i β [ ( x ξ ) u + ( y η ) υ z w ] } d u d υ w , for z < 0
w = { [ 1 ( u 2 + υ 2 ) ] 1 2 when u 2 + υ 2 1 , i [ ( u 2 + υ 2 ) 1 ] 1 2 when u 2 + υ 2 1.
lim z 0 | z | r ( 1 1 i β r ) e i β r r = i β 2 π exp { i β [ ( x ξ ) u + ( y η ) υ ] d u d υ = i 2 π β δ [ β ( x ξ ) ] δ [ β ( y η ) ] = ( i 2 π / β ) δ [ ( x ξ ) , ( y η ) ] .
f ( x , y , 0 ) = f ( ξ , η ) δ [ ( x ξ ) , ( y η ) ] d ξ d η .
f ( x , y , 0 ) = { f ( x , y ) inside R where f ( x , y ) is continuous , 1 2 f ( x , y ) on L , f a υ ( x , y ) at discontinuities values ( if any ) inside R , 0 outside R .
r z + [ ( x ξ ) 2 + ( y η ) 2 ] / 2 z
f ( x , y , z ) = β e i β z i 2 π z exp [ i β ( x ξ ) 2 + ( y η ) 2 2 z ] f ( ξ , η ) d ξ d η .
lim 0 [ e t 2 / e / ( π ) 1 2 ] = δ ( t ) .
lim z 0 { exp [ i β ( x ξ ) 2 / 2 z ] / z 1 2 } exp [ i β ( y η ) 2 / 2 z ] / z 1 2 = + ( 2 π i / β ) δ [ ( x ξ ) , ( y η ) ] .
lim z 0 f ( x , y , z ) = β 2 π i 2 π i β f ( ξ , η ) δ [ ( x ξ ) , ( y η ) ] d ξ d η ,