Abstract

The Fraunhofer criterion defines the location of the boundary between the Fresnel and the Fraunhofer diffraction regions and thus determines the location of that region commonly referred to as the far field. The Fraunhofer criterion is usually given as an axial distance much greater than some amount relative to the maximum dimension of the aperture. By recognizing that Fresnel diffraction patterns are merely defocused Fraunhofer diffraction patterns, we show that the Fraunhofer criterion can be written precisely in terms of an allowable tolerance on defocus. This new criterion provides insight that is useful to optical designers and engineers who routinely deal with such tolerances.

© 2002 Optical Society of America

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References

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  1. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  3. J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
    [CrossRef] [PubMed]
  4. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  5. V. N. Mahajan, “Aberrations of diffracted wave fields. I. Optical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 (2000).
    [CrossRef]
  6. V. N. Mahajan, Optical Imaging and Aberrations: Part II (SPIE, Bellingham, Wash., 2001).
    [CrossRef]
  7. V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, Bellingham, Wash., 1998).
    [CrossRef]
  8. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, UK, 1950).

2000

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1978

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
[CrossRef] [PubMed]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, UK, 1950).

Mahajan, V. N.

V. N. Mahajan, “Aberrations of diffracted wave fields. I. Optical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 (2000).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations: Part II (SPIE, Bellingham, Wash., 2001).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, Bellingham, Wash., 1998).
[CrossRef]

Shack, R. V.

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Other

V. N. Mahajan, Optical Imaging and Aberrations: Part II (SPIE, Bellingham, Wash., 2001).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, Bellingham, Wash., 1998).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, UK, 1950).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (1)

Fig. 1
Fig. 1

Illustration of the regions of validity of the Fraunhofer, Fresnel, and Rayliegh-Sommerfeld diffraction integrals.

Tables (1)

Tables Icon

Table 1 Fraunhofer Criterion for Different Tolerances upon Defocus

Equations (5)

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z  k/2x12+y12max.
Wˆ=Wˆ000 (Piston Error)} Zero-+Wˆ200ρ4Piston+Wˆ020aˆ2Defocus+Wˆ111ρaˆ cosϕ-ϕLateral Magnification ErrorSecond-+Wˆ400ρ4Piston+Wˆ040aˆ4Spherical Aberration+Wˆ131ρaˆ3 cosϕ-ϕComa+Wˆ222ρ2aˆ2 cos2ϕ-ϕAstigmatism+Wˆ220ρ2aˆ2Field Curvature+Wˆ311ρ3aˆ cosϕ-ϕDistortionFourth-+ Higher-Order Terms,
Wˆ020=zˆ2dˆ2zˆ2.
z=d28W020.
ϕ=π4λz d2rad.

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