Abstract

A vectorial theory that brings new insight into the nature of diffraction is used to obtain mathematical expressions that evaluate diffraction patterns in the near field. The equations allow us to discriminate between the contributions of the vectorial and the scalar approaches. In the near field we studied the pattern of light diffracted through a circular aperture, and it was proved that the vectorial approach is significant in a region very near the circular aperture. In spite of the obvious differences between the circular aperture and other obstacles, the present theory may also be used with other geometries.

© 2008 Optical Society of America

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References

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  1. H. L. F. v. Helmholtz, Treatise on Physiological Optics (1867) (Optical Society of America, 1924-1925).
  2. G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” reprinted in Ann. Phys. 18, 663-695 (1883).
    [CrossRef]
  3. Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259-272 (1897).
  4. A. J. W. Sommerfeld, Optics (Academic Press, 1954).
  5. A. S. Marathay and J. F. McCalmont, “Vector diffraction theory for electromagnetic waves,” J. Opt. Soc. Am. A 18, 2585-2593 (2001).
    [CrossRef]
  6. J. A. Romero and L. Hernández, “Vectorial approach to Huygens's principle for plane waves: circular aperture and zone plates,” J. Opt. Soc. Am. A 23, 1141-1145 (2006).
    [CrossRef]
  7. J. A. Romero and L. Hernández, “Huygens's secondary-sources dimensions,” J. Opt. Soc. Am. A 24, 1071-1073 (2007).
    [CrossRef]
  8. K. D. Mielenz, “Optical diffraction in close proximity to plane apertures. I. Boundary-value solutions for circular apertures and slits,” J. Res. Natl. Inst. Stand. Technol. 107, 355-362 (2002).
  9. C. J. R. Sheppard and M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274-281 (1992).
    [CrossRef]
  10. F. Depasse, M. A. Paesler, D. Courjon, and J. M. Vigoureux, “Huygens-Fresnel principle in the near field,” Opt. Lett. 20, 234-236 (1995).
    [CrossRef] [PubMed]

2007 (1)

2006 (1)

2002 (1)

K. D. Mielenz, “Optical diffraction in close proximity to plane apertures. I. Boundary-value solutions for circular apertures and slits,” J. Res. Natl. Inst. Stand. Technol. 107, 355-362 (2002).

2001 (1)

1995 (1)

1992 (1)

1954 (1)

A. J. W. Sommerfeld, Optics (Academic Press, 1954).

1897 (1)

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259-272 (1897).

1883 (1)

G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” reprinted in Ann. Phys. 18, 663-695 (1883).
[CrossRef]

Courjon, D.

Depasse, F.

Helmholtz, H. L. F. v.

H. L. F. v. Helmholtz, Treatise on Physiological Optics (1867) (Optical Society of America, 1924-1925).

Hernández, L.

Hrynevych, M.

Kirchhoff, G. R.

G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” reprinted in Ann. Phys. 18, 663-695 (1883).
[CrossRef]

Marathay, A. S.

McCalmont, J. F.

Mielenz, K. D.

K. D. Mielenz, “Optical diffraction in close proximity to plane apertures. I. Boundary-value solutions for circular apertures and slits,” J. Res. Natl. Inst. Stand. Technol. 107, 355-362 (2002).

Paesler, M. A.

Rayleigh, Lord

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259-272 (1897).

Romero, J. A.

Sheppard, C. J. R.

Sommerfeld, A. J. W.

A. J. W. Sommerfeld, Optics (Academic Press, 1954).

Vigoureux, J. M.

Ann. Phys. (1)

G. R. Kirchhoff, “Zur Theorie der Lichtstrahlen,” reprinted in Ann. Phys. 18, 663-695 (1883).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Res. Natl. Inst. Stand. Technol. (1)

K. D. Mielenz, “Optical diffraction in close proximity to plane apertures. I. Boundary-value solutions for circular apertures and slits,” J. Res. Natl. Inst. Stand. Technol. 107, 355-362 (2002).

Opt. Lett. (1)

Philos. Mag. (1)

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259-272 (1897).

Other (2)

A. J. W. Sommerfeld, Optics (Academic Press, 1954).

H. L. F. v. Helmholtz, Treatise on Physiological Optics (1867) (Optical Society of America, 1924-1925).

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Figures (2)

Fig. 1
Fig. 1

Thin opaque plane with a circular aperture of radius ρ 0 schematized. The origin of coordinates coincides with the center of the aperture. The z axis is perpendicular to the plane and coincides in direction with the vector of plane wave k. Vector r ( ρ , θ , 0 ) locates points p in the area of the aperture, and r ( ρ , θ , z ) points to observation point p in semispace z > 0 .

Fig. 2
Fig. 2

Normalized diffracted intensity in perpendicular planes to the symmetry axis versus b parameter, which is related to the distance to the axial symmetry axis. The perpendicular planes are located at distances (a) a = 0.2 , (b) a = 0.02 , and (c) a = 0.002 .

Equations (14)

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dE ( r , r , t ) = [ ds × E ( r , t ) ] × r r r r r r [ exp ( i k r r ) 2 π r r ] ,
E ( r , t ) = E x ( r , t ) e x + E y ( r , t ) e y .
dE ( r , r , t ) = [ ( r r ) ds f ( r , r ) ] E ( r , t ) [ ( r r ) E ( r , t ) f ( r , r ) ] ds ,
f ( r , r ) = 1 r r r r [ exp ( i k r r ) 2 π r r ] .
E ( r , t ) = z e x z = 0 E x ( r , t ) f ( r , r ) d s z e y z = 0 E y ( r , t ) f ( r , r ) d s e z z = 0 ( x x ) E x ( r , t ) f ( r , r ) d s e z z = 0 ( y y ) E y ( r , t ) f ( r , r ) d s .
E ( r , t ) = z [ e x + v x e z ] z = 0 E x ( r , t ) f ( r , r ) d s z [ e y + v y e z ] z = 0 E y ( r , t ) f ( r , r ) d s ,
v x = z = 0 ( x x ) E x ( r , t ) f ( r , r ) d s z = 0 z E x ( r , t ) f ( r , r ) d s ,
v y = z = 0 ( y y ) E y ( r , t ) f ( r , r ) d s z = 0 z E y ( r , t ) f ( r , r ) d s .
1 ( v x 2 + v y 2 ) 1 2 ,
E ( r , t ) = { 0 ρ = ( x 2 + y 2 ) 1 2 > ρ 0 , E 0 e x exp [ ± i ( w t + α ) ] ρ ρ 0 ,
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 ,

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