Abstract

Consider a generally aberrated one-dimensional (1D) optical pupil P illuminated by quasi-monochromatic light of mean wavelength λ¯. In past work it was found that, if the pupil’s intensity point-spread function (psf) is multiply convolved with itself, as in an imaging relay system, and then ideally (stigmatically) demagnified, the resulting psf s(x) approaches a fixed Cauchy form s(x)=Δx(π2x2+Δx2)-1, which is independent of the aberrations of the pupil. Here Δx is the Nyquist sampling interval given by Δx=λ¯f/2 with f the f/number of the pupil. This Cauchy form for this intensity psf s(x) also manifestly lacks sidelobes. The overall questions that we examine are how far do these effects carry over to the case of a circular, two-dimensional (2D) pupil, and to what extent do practical imaging considerations compromise the theoretical results? It is found that, in the presence of spherical aberration of all orders, the resulting theoretical psf of a large number of self-convolutions approaches a “circular” Cauchy form, S(r)=2Δr[π2r2+(4Δr/π)2]-3/2, where Δr is the Nyquist sampling interval λ¯f/2 with f the f/number of the (now) circular pupil. Thus, for these aberrations the 1D effect does carry over to the 2D case: The output psf does not depend on the aberrations and completely lacks sidelobes. However, when all aberrations are generally present, the output psf s(r, θ) does depend on the aberrations, although its azimuthal average over θ still preserves the circular Cauchy form, as a superposition of Cauchy functions. Imaging requirements for achieving these ideal effects are briefly discussed as well as probability laws for photons that are implied by the above-mentioned PSF’s s(x) and S(r). Real-time super resolution is not attained, since the stigmatic imaging demanded of the demagnification step requires the use of a larger-apertured lens. Rather, the approach achieves significant aberration suppression.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. R. Frieden, Probability, Statistical Optics and Data Testing, 3rd ed. (Springer-Verlag, Berlin, 2001).
  2. B. R. Frieden, “Polynomial expansion in classical aberrations and spatial frequency for the wave-aberrated optical transfer function,” Opt. Acta 11, 33–41 (1964) (this author’s first published paper).
    [Crossref]
  3. H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–576 (1956).
    [Crossref]
  4. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  6. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 29–186.
  7. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 194–195.
  8. R. J. Glauber, “Photon statistics,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 1–43.
  9. T. S. McKechnie, “Speckle reduction,” in Laser Speckle, Vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), 123–170; see in particular p. 126.
  10. H. Kiemle, U. Wolff, “Application de cristaux liquides en holographie optique,” Opt. Commun. 3, 26–28 (1971).
    [Crossref]
  11. A. S. Marathay, L. Heiko, J. L. Zuckerman, “Study of rough surfaces by light scattering,” Appl. Opt. 9, 2470–2476 (1970).
    [Crossref] [PubMed]
  12. Ref. 1, p. 64.
  13. Ref. 1, p. 199.

1971 (1)

H. Kiemle, U. Wolff, “Application de cristaux liquides en holographie optique,” Opt. Commun. 3, 26–28 (1971).
[Crossref]

1970 (1)

1964 (1)

B. R. Frieden, “Polynomial expansion in classical aberrations and spatial frequency for the wave-aberrated optical transfer function,” Opt. Acta 11, 33–41 (1964) (this author’s first published paper).
[Crossref]

1956 (1)

H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–576 (1956).
[Crossref]

Frieden, B. R.

B. R. Frieden, “Polynomial expansion in classical aberrations and spatial frequency for the wave-aberrated optical transfer function,” Opt. Acta 11, 33–41 (1964) (this author’s first published paper).
[Crossref]

B. R. Frieden, Probability, Statistical Optics and Data Testing, 3rd ed. (Springer-Verlag, Berlin, 2001).

Glauber, R. J.

R. J. Glauber, “Photon statistics,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 1–43.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 194–195.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Heiko, L.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–576 (1956).
[Crossref]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 29–186.

Kiemle, H.

H. Kiemle, U. Wolff, “Application de cristaux liquides en holographie optique,” Opt. Commun. 3, 26–28 (1971).
[Crossref]

Marathay, A. S.

McKechnie, T. S.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle, Vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), 123–170; see in particular p. 126.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 29–186.

Wolff, U.

H. Kiemle, U. Wolff, “Application de cristaux liquides en holographie optique,” Opt. Commun. 3, 26–28 (1971).
[Crossref]

Zuckerman, J. L.

Appl. Opt. (1)

Opt. Acta (1)

B. R. Frieden, “Polynomial expansion in classical aberrations and spatial frequency for the wave-aberrated optical transfer function,” Opt. Acta 11, 33–41 (1964) (this author’s first published paper).
[Crossref]

Opt. Commun. (1)

H. Kiemle, U. Wolff, “Application de cristaux liquides en holographie optique,” Opt. Commun. 3, 26–28 (1971).
[Crossref]

Proc. Phys. Soc. London Sect. B (1)

H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 69, 562–576 (1956).
[Crossref]

Other (9)

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 29–186.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 194–195.

R. J. Glauber, “Photon statistics,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 1–43.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle, Vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), 123–170; see in particular p. 126.

B. R. Frieden, Probability, Statistical Optics and Data Testing, 3rd ed. (Springer-Verlag, Berlin, 2001).

Ref. 1, p. 64.

Ref. 1, p. 199.

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 (2)

Fig. 1
Fig. 1

Curves of the characteristic function ϕX¯(ω) for indicated values of N. Once N exceeds ∼500, the curves become indistinguishable from the indicated exponential law with β0π/(λ¯f), λ¯ the mean wavelength and f the f/number. Since the Fourier transform of an exponential law is a Cauchy law, this confirms that the Fourier transforms of the characteristic functions, the PDF’s pX¯(x), approach a Cauchy law.

Fig. 2
Fig. 2

S(r) is the circular Cauchy psf due to a large number of self-convolutions of the psf resulting from a generally spherically aberrated pupil. It is shown along with the Airy disk psf S0(r) for the same pupil P. These curves also represent probability laws on the size of the arithmetic mean position of a large number of photons. To foster comparison of shapes, both curves are normalized to unity at the origin. The abscissa r/Δr is radial distance r in units of the Nyquist sampling interval Δr defined by Eq. (25). The Cauchy psf completely lacks sidelobes and is independent of the spherical aberrations of the pupil. However, the apparent gain in resolution would require the additional use of a lens whose pupil much exceeds that of P, so superresolution is not achieved in practice by the system.

Equations (36)

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

x¯1Nn=1Nxn,
exp(jωx¯)=expjω 1Nn=1Nxn=n=1Nexpj ωN xn,j-1,
expj ωN xN,-ω,
pX¯(x)=Δx(π2x2+Δx2).
x(x, y)x,xn(xn, yn)xn,x¯(x¯, y¯)x¯,dxdxdydx.
x¯1Nn=1Nxn,
exp(jω  x¯)=expjω  1Nn=1Nxn=n=1Nexpj ωNxn
expj ωNxN forall|ω|
ϕX¯(ω)exp(jω  x¯)=dx pX¯(x)exp(jω  x)forall|ω|.
pX¯(x)=1(2π)2dωϕX¯(ω)exp(-jω  x).
exp(jω  x)ϕX(ω)dx pX(x)exp(jω  x),
pX(x)=s(x).
ϕX(ω)=dx s(x)exp(jω  x)τ*(ω),
ϕX¯(ω)=expj ωNxN=ϕxωNN=τ*ωNN.
ω=(ω, α),|ω|ω,0ω
ϕX¯(ω, α)=τ*ωN, αN=τ*(ω, α)|ω++ωNτ*(ω, α)ωω++ω22N22τ*(ω, α)ω2ω++N.
ln ϕX¯(ω, α)=N ln1+ωNτ*(ω, α)ωω++ω22N22τ*(ω, α)ω2ω++.
ln(1+η)=η-η2/2+,
ln ϕX¯(ω, α)=NωNτ*(ω, α)ωω++ω22N22τ*(ω, α)ω2ω+-N2ωNτ*(ω, α)ωω++ω22N22τ*(ω, α)ω2ω+2+.
limN ln ϕX¯(ω, α)=ωτ*(ω, α)ωω+.
limN ϕX¯(ω, α)=expωτ*(ω, α)ωω+.
pX¯(r, θ)=1(2π)20dωω-ππdα ×expωτ*(ω, α)ωω+×exp[-jωr cos(θ-α)],
τ(ω, α)=τ(ω),0ω.
pX¯(r, θ)pR¯(r)=12π0dωωJ0(ωr)expωτ*(ω)ωω+.
τ*(ω)ωω+=-2πρ0=-4π2 Δr,
Δr=π/2ρ0=λ¯f/2,
pR¯(r)=12πb21[1+(r/b)2]3/2,b4π2 Δr.
p¯(r)-ππdθ pX¯(r, θ)
p¯(r)=1(2π)20dωω-ππdα expωτ*(ω, α)ωω+×-ππdθ exp[-jωr cos(θ-α)]=-ππdα 12π0dωωJ0(ωr)×expωτ*(ω, α)ωω+.
p¯(r)=Re-ππdα pR¯(r; b(α)),
pR¯(r; b(α))12πb2(α)1[1+(r/b(α))2]3/2,
b(α)τ*(ω, α)ωω+.
τ(ω)=τ1(ω)τ2(ω)τN(ω).
pR¯(r)=S(r)=12πb21[1+(r/b)2]3/2,b4π2 Δr.
t(x)t*(x)=exp{-k2σ2[1-ρ(x,x)]}.
S0(r)=2J1(πr/2Δr)(πr/2Δr)2

Metrics