Abstract

A Zernike expansion over a circle is given for an arbitrary function of a single linear spatial coordinate. The example of a half-plane mask (Hilbert filter) is considered. The expansion can also be applied to cylindrical aberrations over a circular pupil. A product of two such series can thus be used to expand an arbitrary separable function of two Cartesian coordinates.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993).
  2. D. Malacara, Optical Shop Testing (Wiley, New York, 1992).
  3. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  4. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  5. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, New York, 1972).
  6. A. B. Bhatia, E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
    [CrossRef]
  7. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  8. B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica (Amsterdam) 13, 605–620 (1947).
    [CrossRef]
  9. W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
    [CrossRef]
  10. R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 7, 262–264 (1982).
    [CrossRef] [PubMed]
  11. G. Conforti, “Zernike aberration coefficients from Seidel and higher-order power-series coefficients,” Opt. Lett. 8, 407–408 (1983).
    [CrossRef] [PubMed]
  12. D. K. Hamilton, C. J. R. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. (Oxford) 133, 27–39 (1984).
    [CrossRef]
  13. A. I. Mahan, C. V. Bitterei, S. M. Cannon, “Far-field diffraction patterns of single and multiple apertures bounded by arcs and radii of concentric circles,” J. Opt. Soc. Am. 54, 721–732 (1964).
    [CrossRef]
  14. A. Barna, “Fraunhofer diffraction by semi-circular apertures,” J. Opt. Soc. Am. 67, 122–123 (1977).
    [CrossRef]
  15. J. R. Izatt, “Diffraction patterns produced by apertures bounded by arcs and radii of concentric circles,” J. Opt. Soc. Am. 55, 106–107 (1965).
    [CrossRef]
  16. G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and its Applications (Cambridge U. Press, Cambridge, UK, 1999).

1988 (1)

1984 (1)

D. K. Hamilton, C. J. R. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. (Oxford) 133, 27–39 (1984).
[CrossRef]

1983 (1)

1982 (1)

1977 (2)

W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
[CrossRef]

A. Barna, “Fraunhofer diffraction by semi-circular apertures,” J. Opt. Soc. Am. 67, 122–123 (1977).
[CrossRef]

1976 (1)

1974 (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

1965 (1)

1964 (1)

1954 (1)

A. B. Bhatia, E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

1947 (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica (Amsterdam) 13, 605–620 (1947).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, New York, 1972).

Andrews, G. E.

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and its Applications (Cambridge U. Press, Cambridge, UK, 1999).

Askey, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and its Applications (Cambridge U. Press, Cambridge, UK, 1999).

Barna, A.

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Bhatia, A. B.

A. B. Bhatia, E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Bitterei, C. V.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993).

Cannon, S. M.

Conforti, G.

Hamilton, D. K.

D. K. Hamilton, C. J. R. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. (Oxford) 133, 27–39 (1984).
[CrossRef]

Izatt, J. R.

Mahan, A. I.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica (Amsterdam) 13, 605–620 (1947).
[CrossRef]

Noll, R.

Roddier, F.

Roy, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and its Applications (Cambridge U. Press, Cambridge, UK, 1999).

Sheppard, C. J. R.

D. K. Hamilton, C. J. R. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. (Oxford) 133, 27–39 (1984).
[CrossRef]

Stegun, I.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, New York, 1972).

Tango, W. J.

W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
[CrossRef]

Tyson, R. K.

Wolf, E.

A. B. Bhatia, E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993).

Appl. Opt. (1)

Appl. Phys. (1)

W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
[CrossRef]

J. Microsc. (Oxford) (1)

D. K. Hamilton, C. J. R. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. (Oxford) 133, 27–39 (1984).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Lett. (2)

Physica (Amsterdam) (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica (Amsterdam) 13, 605–620 (1947).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

A. B. Bhatia, E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Sov. J. Opt. Technol. (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Other (4)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993).

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, New York, 1972).

G. E. Andrews, R. Askey, R. Roy, Special Functions, Vol. 71 of Encyclopedia of Mathematics and its Applications (Cambridge U. Press, Cambridge, UK, 1999).

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 (1)

Fig. 1
Fig. 1

Expansion of a signum function sgn(x) in terms of Zernike polynomials. The unit circle cuts off at x = cos θ. The curves correspond to the sums of the first 3, 5, 11, and 99 nonzero terms, respectively.

Tables (1)

Tables Icon

Table 1 Some Low-Order Chebyshev Polynomials

Equations (31)

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

Unm=Rnmρcos mθ,Un-m=Rnmρsin mθ,
Rnmρ=s=0n-m/2-1sn-s!s!n+m2-s!n-m2-s!×ρn-2s.
01 RnmρRnmρρdρ=12n+1 δnn,
Ltρ0Rnmρ=-1n-m/2n+m/2!m!n-m/2! ρm.
Rnmρ=-1n-m/2ρmPn-m/2m,01-2ρ2,
Rnmρ=ρmPn-m/20,m2ρ2-1.
Rn0ρ+2 m=2,m evenn Rnmρcos mθ=Unx, n even,
2 m=1,m oddn Rnmρcos mθ=Unx, n odd,
Rn0ρ+2 m=2,m evenn-1m/2Rnmρsin mθ=Uny, n even,
2 m=1,m oddn-1m-1/2Rnmρcos mθ=Uny, n odd,
x=ρ cos θ, y=ρ sin θ.
-11 UnxUnx1-x21/2dx=π2 δnn.
Un1=n+1,
Ltx0Unx=-1n/2, n even,=-1n-1/2n+1/2!n-1/2! 2x, n odd.
sgnx=4πm=1,m oddnn=1,n odd-1n-1/2n+1nn+2 ×Rnmρcos mθ.
01 Rnmρρdρ=-1m-1/2m2nn+2.
sgnx=4πn=1,n odd-1n-1/2n+1nn+2 Unx.
01 Unx1-x21/2dx=π4, n=0,=-1n-1/2n+1nn+2, n odd,=0, n0 and even.
02π01 Rnm exp-iνρ cos θρdρdθ=2π-1n/2Jn+1ννcos mθ.
Skl=n=0m=0n aklmnAnm,
a0000=1,aklnm=-1n-k/2n+k2!1+δm01/2n-k2!k+m2!k-m2!×1+δm02l-1, m=l,aklnm=-1n-k/2n+k2!1+δm01/2n-k2!k+m2!k-m2!×-1m-l/22lmm+l2-1!m-lm-l2-1!l!, ml.
Φn=2k=0n ak0n0ρk+2 k=lnl=2nm=l,m evenk aklnmρk cosl θ.
Skln=m=lk aklnm.
ν=m-l2,
w=k-l2,
S=c ν=0w-1ν2ν+lν+l-1!ν!w-ν!ν+w+l!, w0,
c=-1n-k/2n+k2!n-k2!2l-1l!.
S=c ν=0w-1ννν+l-1!ν!w-ν!ν+w+l!+c ν=0w-1νν+l!ν!w-ν!ν+w+l!=c ν=0w-1νν+l-1!ν-1!w-ν!ν+w+l!+c ν=0w-1νν+l!ν!w-ν!ν+w+l!=c ν=0w-1νν+l!ν!w-ν-1!ν+w+l+1!+c ν=0w-1νν+l!ν!w-ν!ν+w+l!=-c l!w-1!w+l+1!ν=0w-1-1νν!ν+l!l!×w-1!w-ν-1!w+l+1!ν+w+l+1! +c l!w!w+l!ν=0w-1νν!ν+l!l!×w!w-ν!w+l!ν+w+l!=-c l!w-1!w+l+1!2F1l+1,-w+1w+l+2;1 +c l!w!w+l!2F1l+1,-ww+l+1;1,
2F1-n,ac;1=c-a+n-1!c-a-1!c-1!c+n-1!,
S=-c l!w-1!w+l+1!2w-1!w!w+l+1!2w+l! +c l!w!w+l!2w-1!w-1!w+l!2w+l! =0.
Φn=k=0,k evenn-1n-k/2n+k2!2kn-k2!k! ρk cosk θ.

Metrics