Abstract

We present an approach to calculating the complex amplitude of a three-dimensional (3D) diffracted light field in the paraxial approximation based on a 3D Fourier transform. Starting from the Huygens–Fresnel principle, the method is first developed for the computation of the light distribution around the focus of an apertured spherical wave. The method, with modification, is then extended to treat the 3D diffraction of an aperture with an arbitrary transmittance function.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Schnars and W. Jüptner, Appl. Opt. 33, 179 (1994).
    [CrossRef] [PubMed]
  2. K. Matsushima and T. Shimobaba, Opt. Express 17, 19662 (2009).
    [CrossRef] [PubMed]
  3. K. Matsushima, H. Schimmel, and F. Wyrowski, J. Opt. Soc. Am. A 20, 1755 (2003).
    [CrossRef]
  4. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, Opt. Express 13, 9935 (2005).
    [CrossRef] [PubMed]
  5. S. J. Jeong and C. K. Hong, Appl. Opt. 47, 3064(2008).
    [CrossRef] [PubMed]
  6. M. Paturzo and P. Ferraro, Opt. Express 17, 20546(2009).
    [CrossRef] [PubMed]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  9. C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).
    [CrossRef]
  10. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, Opt. Express 14, 11277 (2006).
    [CrossRef] [PubMed]
  11. Y. Li and E. Wolf, J. Opt. Soc. Am. A 1, 801 (1984).
    [CrossRef]
  12. C. J. R. Sheppard, Opt. Lett. 25, 1660 (2000).
    [CrossRef]
  13. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

2009 (2)

2008 (1)

2006 (1)

2005 (1)

2003 (1)

2000 (1)

1994 (1)

1984 (1)

1964 (1)

Alfieri, D.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).

De Nicola, S.

Ferraro, P.

Finizio, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hong, C. K.

Jeong, S. J.

Jüptner, W.

Lasser, T.

Leitgeb, R. A.

Leutenegger, M.

Li, Y.

Matsushima, K.

McCutchen, C. W.

Paturzo, M.

Pierattini, G.

Rao, R.

Schimmel, H.

Schnars, U.

Sheppard, C. J. R.

Shimobaba, T.

Stamnes, J. J.

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

Wolf, E.

Y. Li and E. Wolf, J. Opt. Soc. Am. A 1, 801 (1984).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).

Wyrowski, F.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

General geometry for 3D diffraction calculation.

Fig. 2
Fig. 2

Geometry for calculating the field distribution around the focal region.

Fig. 3
Fig. 3

3D field distribution of a focused wave obtained by the proposed method: (a), (b)  z = 0 and (c), (d)  y = 0 .

Fig. 4
Fig. 4

Focused field distribution along different lines given by the closed-form expression (black solid curve), the proposed 3D method (red asterisks), and McCutchen’s method (blue circles). The output data points are downsampled for a clear display in the two numerical methods: (a), (b)  y = z = 0 and (c), (d)  x = y = 0 .

Fig. 5
Fig. 5

3D field distribution of an astigmatic Gaussian beam propagating in free space: (a), (d)  z = 0 , (b), (e)  y = 0 , and (c), (f)  x = 0 .

Fig. 6
Fig. 6

Field distribution of an astigmatic free-space Gaussian beam along different lines: (a), (b)  y = z = 0 , (c), (d)  x = z = 0 , and (e), (f)  x = y = 0 . The results are calculated with the 2D angular spectrum (black solid curve) and 3D direct Fourier transform (red asterisks).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

U ( P ) = A i λ Σ U ( Q ) exp ( i k r ) r cos θ d S ,
U ( P ) = i λ A exp ( i k f ) f W U ( Q ) exp ( i k r ) r d S ,
r z + f 2 f + z + x 2 + y 2 2 ( f + z ) x ξ + y η + z γ f + z ,
U P ( x , y , z ) i A λ f z f 3 exp [ i k x 2 + y 2 + 2 z 2 2 ( f z ) ] × V U Q ( ξ , η , γ ) P ( ξ , η , γ ) e i k x ξ + y η + z γ f d ξ d η d γ ,
P ( ξ , η , γ ) = δ ( ξ 2 + η 2 + γ 2 f ) ,
U P ( x , y , z ) = i A λ d z d 3 exp [ i k x 2 + y 2 + 2 z 2 2 ( d z ) ] × V U Q ( ξ , η ) P ( ξ , η , γ ) e i k ξ 2 + η 2 2 d e i k x ξ + y η + z γ d d ξ d η d γ ,

Metrics