Abstract

By extension of the transitional operator method developed by Wünsche, the Rayleigh–Sommerfeld and Kirchhoff solutions to the diffraction of a converging spherical (or cylindrical) wave are expressed in terms of a series of derivatives of the field estimate that follows from the Fresnel approximation. This result allows a systematic assessment of the error associated with the paraxial wave model for focused fields and offers simple corrections to this model. In particular, for simple diffracting masks, the Fresnel approximation leads to estimates of the field that have a relative error near focus that is of the order of one on the square of the f-number. The number of significant digits in the field estimate is shown to be doubled by retaining just the first of the series of corrections derived here.

© 1999 Optical Society of America

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  1. A. Wunsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  2. G. W. Forbes, D. J. Butler, R. L. Gordon, A. A. Asatryan, “Algebraic corrections for paraxial wave fields,” J. Opt. Soc. Am. A 14, 3300–3315 (1997).
    [CrossRef]
  3. M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  4. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  5. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.
  6. See Ref. 5, Sec. 3.2.4, pp. 120–124.
  7. See Ref. 5, Sec. 3.2.5, pp. 125–127.
  8. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  9. Equation (4.8) is the familiar paraxial wave equation, and it is straightforward to verify that U¯F satisfies this equation by using definition (2.8).
  10. G. W. Forbes, “Scaling properties in the diffraction of focused waves and an application to scanning beams,” Am. J. Phys. 62, 434–443 (1994).
    [CrossRef]
  11. H. Osterberg, L. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  12. The required expressions can be found either by returning to Eq. (5.2c) and taking the derivatives before evaluating at focus or by using Eqs. (4.8) and (4.11). Equivalently, since Eq. (5.5) is evaluated at z=f, we can take the partial f derivatives as ∂U¯F/∂f=dU¯F/df-∂U¯F/∂z, where Eq. (4.8) is then used for taking the partial z derivative. As an example, the first correction term given in Eq. (4.2a) can be found in this way to be   CI,1=if2k ddf ikρ ∂∂ρ ρ ∂U¯F∂ρ-dU¯Fdf=RU08f3kρ2 [8f2+4ikf(2ρ2+R2)-k2ρ2(ρ2+6R2)]ρJ1kRf ρ+4iR[2 f2-k2ρ2(ρ2+R2)]J2kRf ρexpik2 f ρ2.  
  13. G. W. Forbes, A. A. Asatryan, “Reducing canonical diffraction problems to singularity-free one-dimensional integrals,” J. Opt. Soc. Am. A 15, 1320–1328 (1998).
    [CrossRef]
  14. Such contour plots have been given recently, for example, in Ref. 2 and in G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1994 (1)

G. W. Forbes, “Scaling properties in the diffraction of focused waves and an application to scanning beams,” Am. J. Phys. 62, 434–443 (1994).
[CrossRef]

1992 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1961 (1)

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Asatryan, A. A.

Butler, D. J.

Forbes, G. W.

Gordon, R. L.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.

McBright, W. B.

M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Osterberg, H.

Smith, L.

Southwell, W. H.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.

Wunsche, A.

Am. J. Phys. (1)

G. W. Forbes, “Scaling properties in the diffraction of focused waves and an application to scanning beams,” Am. J. Phys. 62, 434–443 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Other (5)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.

See Ref. 5, Sec. 3.2.4, pp. 120–124.

See Ref. 5, Sec. 3.2.5, pp. 125–127.

Equation (4.8) is the familiar paraxial wave equation, and it is straightforward to verify that U¯F satisfies this equation by using definition (2.8).

The required expressions can be found either by returning to Eq. (5.2c) and taking the derivatives before evaluating at focus or by using Eqs. (4.8) and (4.11). Equivalently, since Eq. (5.5) is evaluated at z=f, we can take the partial f derivatives as ∂U¯F/∂f=dU¯F/df-∂U¯F/∂z, where Eq. (4.8) is then used for taking the partial z derivative. As an example, the first correction term given in Eq. (4.2a) can be found in this way to be   CI,1=if2k ddf ikρ ∂∂ρ ρ ∂U¯F∂ρ-dU¯Fdf=RU08f3kρ2 [8f2+4ikf(2ρ2+R2)-k2ρ2(ρ2+6R2)]ρJ1kRf ρ+4iR[2 f2-k2ρ2(ρ2+R2)]J2kRf ρexpik2 f ρ2.  

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