Abstract

A vectorial equation that describes the Huygens principle was reported, and an expression for the secondary-spherical-wave energy density was found. With a vectorial formulation, we performed an exact calculation for the relative axial intensity of the wave diffracted by an illuminated circular aperture. The off-axis intensity in Fresnel’s and Fraunhofer’s approximations was calculated. The zone plate was also studied by vectorial formulation. We showed that with increasing number of rings, the intensity maxima magnify as (2n+2)2, their full widths decrease, their positions remain unchanged, and secondary maxima appear, in a behavior similar to that for diffraction gratings.

© 2006 Optical Society of America

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References

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  1. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).
  2. C. J. R. Sheppard and L. L. Cooper, "Fresnel diffraction by a circular aperture with off-axis illumination and its use in deconvolution of microscope images," J. Opt. Soc. Am. A 21, 540-545 (2004).
    [CrossRef]
  3. C. J. R. Sheppard, P. P. Roberts, and Min Gu, "Fresnel approximation for off-axis illumination of circular aperture," J. Opt. Soc. Am. A 10, 984-986 (1993).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).
  5. Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1, 801-808 (1984).
    [CrossRef]
  6. W. Hsu and R. Barakat, "Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems," J. Opt. Soc. Am. A 11, 623-629 (1994).
    [CrossRef]
  7. C. J. R. Sheppard and M. Hrynevych, "Diffraction by circular aperture: a generalization of Fresnel diffraction theory," J. Opt. Soc. Am. A 9, 274-281 (1992).
    [CrossRef]
  8. H. Osterberg and L. W. Smith, "Closed solution of Rayleigh's diffraction integral for axial points," J. Opt. Soc. Am. 51, 1050-1054 (1961).
    [CrossRef]

2004 (1)

1994 (1)

1993 (1)

1992 (1)

1986 (1)

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

1984 (1)

1983 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).

1961 (1)

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).

Cooper, L. L.

Gu, Min

Hrynevych, M.

Hsu, W.

Li, Y.

Osterberg, H.

Roberts, P. P.

Sheppard, C. J. R.

Smith, L. W.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

Wolf, E.

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Figures (4)

Fig. 1
Fig. 1

Thin opaque plane with a circular aperture of radius ρ o . The reference system has been taken in a way such that the origin o of coordinates coincides with the center of the aperture. The z axis is perpendicular to the plane and coincides in direction with the propagation vector of the plane wave k. Vector r ( ρ , θ ) locates the points p on the integration area of the aperture. The vector r ( ρ , θ , z ) signals the observation point p in the semispace z > 0 .

Fig. 2
Fig. 2

Comparison of the behavior of the intensity described in Eq. (17) with the Bessel function in a plane perpendicular to the axial symmetry axis of the circular aperture. Both functions have been normalized with respect to their value on the symmetry axis in each plane. (a) Behavior in a plane at a = 1600 , very close to the one that contains the first maximum in Fresnel’s area. (b) Behavior in a plane at a = 2600 , Fraunhofer’s region.

Fig. 3
Fig. 3

Thin opaque plane with n + 1 apertures in the form of circular concentric rings of radii ρ i .

Fig. 4
Fig. 4

Dependence of the relative intensity [Eq. (26)] on the axial axis of a zone plate with respect to the number of rings ( n ) : (a) n = 0 , (b) n = 4 , (c) n = 9 . With increasing number of rings, the intensity maxima grow and narrow, and secondary maxima appear. The graphs in the insets in (a) and (c) correspond to a region in close proximity to the plate. In this region the intensity of the principal maxima decreases as the maxima approach the plate.

Equations (27)

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d U ( r , r ) = z 2 π U ( x , y , 0 ) r r r r [ exp ( i k r r ) r r ] d s ,
d U x ( r , r ) ( x x ) + d U y ( r , r ) ( y y ) d U z ( r , r ) z = 0 .
d U ( r , r ) = [ d s × U ( x , y , 0 ) ] × r r r r r r [ exp ( i k r r ) 2 π r r ] ,
U ( x , y , 0 ) = U x ( x , y , 0 ) e x + U y ( x , y , 0 ) e y .
E ( z , t ) = E 0 exp { ± i [ w t k z + α ] } .
U ( x , y , 0 ) = E ( 0 , t ) ,
d E ( r , r , t ) = [ d s × E p ( t ) ] × r r r r r r [ exp ( i k r r ) 2 π r r ] ,
E p ( t ) = E 0 exp [ ± i ( w t + α ) ] .
d E ( r , r , t ) = Γ ( r ) ϵ ( r , r , t ) d s ,
ϵ e ( r , r , t ) = [ k 2 π k Γ ( r ) × E p ( t ) ] × r r r r ( i k 1 r r ) ,
ϵ e ( 0 , r , t ) = E p ( t ) 2 π Γ ( 0 ) [ k 2 + 1 r 2 ] 1 2 sin ( β ) ,
W = ϵ E p 2 8 π 2 r 2 Γ 2 ( k 2 + 1 r 2 ) [ 1 + cos 2 ( ϕ ) ] ,
E ( r , t ) = 1 2 π [ e z ( ρ o E p ) I 1 + z E p I 2 ] ,
I 1 = 2 0 π exp { i k [ ρ o 2 + r 2 2 ρ o ρ cos ( ψ ) ] 1 2 } [ ρ o 2 + r 2 2 ρ o ρ cos ( ψ ) ] 1 2 cos ( ψ ) d ψ ,
I 2 = 2 0 π r [ ρ o 2 + r 2 2 ρ o ρ cos ( ψ ) ] 1 2 R [ exp ( i k R ) R ] { 1 + ρ cos ( ψ ) [ R 2 r 2 + ρ 2 cos 2 ( ψ ) ] 1 2 } d R d ψ ,
I = 1 + z 2 z 2 + ρ o 2 2 z [ z 2 + ρ o 2 ] 1 2 cos { k [ ( z 2 + ρ o 2 ) 1 2 z ] } .
I = I 1 cos 2 ( θ ) + a 2 I 2 ,
I 1 = 1 π 0 π exp { i c [ 1 + a 2 + b 2 2 b cos ( ψ ) ] 1 2 } [ 1 + a 2 + b 2 2 b cos ( ψ ) ] 1 2 cos ( ψ ) d ψ 2 ,
t = [ l 2 + a 2 + b 2 2 b l cos ( ψ ) ] 1 2 ,
I 2 = 1 π 0 π 0 1 1 t t [ exp ( i c t ) t ] l d l d ψ 2 ,
E ( z , t ) = z E p ( t ) l = 0 2 n + 1 ( 1 ) l exp { i k [ ρ l 2 + z 2 ] 1 2 } [ ρ l 2 + z 2 ] .
I = l = 0 j = 0 2 n + 1 ( 1 ) l + j z 2 cos [ k ( w l w j ) ] w l w j ,
where w l = ( ρ l 2 + z 2 ) 1 2 .
k ( w l w j ) = ( 2 m + 1 ) π ( l + j ) , m Z .
ρ l 2 = ( l λ 2 ) 2 + ρ 0 2 + l λ ( z 0 2 + ρ 0 2 ) 1 2 , 0 l 2 n + 1 .
I ( x , y , n ) = l = 0 j = 0 2 n + 1 ( 1 ) l + j x 2 cos [ 2 π y ( w l w j ) ] w l w j ,
w l = [ ( l y 2 ) 2 + l y + x 2 ] 1 2 , 0 l 2 n + 1 .

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