Abstract

In interferometry and optical testing, system wave-front measurements that are analyzed on a restricted subdomain of the full pupil can include predictable systematic errors. In nearly all cases, the measured rms wave-front error and the magnitudes of the individual aberration polynomial coefficients underestimate the wave-front error magnitudes present in the full-pupil domain. We present an analytic method to determine the relationships between the coefficients of aberration polynomials defined on the full-pupil domain and those defined on a restricted concentric subdomain. In this way, systematic wave-front measurement errors introduced by subregion selection are investigated. Using vector and matrix representations for the wave-front aberration coefficients, we generalize the method to the study of arbitrary input wave fronts and subdomain sizes. While wave-front measurements on a restricted subdomain are insufficient for predicting the wave front of the full-pupil domain, studying the relationship between known full-pupil wave fronts and subdomain wave fronts allows us to set subdomain size limits for arbitrary measurement fidelity.

© 2001 Optical Society of America

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References

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  1. D. J. Fischer, J. T. O’Bryan, R. Lopez, H. P. Stahl, “Vector formulation for interferogram surface fitting,” Appl. Opt. 32, 4738–4743 (1993).
    [CrossRef] [PubMed]
  2. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  3. M. Carpio, D. Malacara, “Closed Cartesian representation of the Zernike polynomials,” Opt. Commun. 110, 514–516 (1994).
    [CrossRef]
  4. S. D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed. (McGraw-Hill, New York, 1980), Chap. 6.
  5. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992), p. 126.
  6. C. R. Hayslett, W. H. Swantner, “Wave-front derivation by three computer programs,” Appl. Opt. 19, 161–163 (1980).
    [CrossRef]
  7. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [CrossRef] [PubMed]
  8. J. S. Loomis, FRINGE User’s Manual (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1976).
  9. “FRINGE Zernikes,” in Code V Reference Manual, R. C. Juergens, ed. (Optical Research Associates, Pasadena, Calif., 1998), Vol. 1, version 8.30, pp. 2A-595–2A-596.
  10. “Fitting interferograms using Zernike polynomials,” in FAST! V/AI Operations Manual (Phase Shift Technology, Tucson, Ariz., 1990), pp. 2-3–2-5.
  11. J. Spanier, K. B. Oldham, “The unit-step and related functions,” in An Atlas of Functions (Hemisphere, Washington, D.C., 1987), Chap. 8, pp. 63–69.

1994 (2)

M. Carpio, D. Malacara, “Closed Cartesian representation of the Zernike polynomials,” Opt. Commun. 110, 514–516 (1994).
[CrossRef]

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
[CrossRef] [PubMed]

1993 (1)

1980 (2)

Bevington, P. R.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992), p. 126.

Carpio, M.

M. Carpio, D. Malacara, “Closed Cartesian representation of the Zernike polynomials,” Opt. Commun. 110, 514–516 (1994).
[CrossRef]

Conte, S. D.

S. D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed. (McGraw-Hill, New York, 1980), Chap. 6.

de Boor, C.

S. D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed. (McGraw-Hill, New York, 1980), Chap. 6.

Fischer, D. J.

Hayslett, C. R.

Loomis, J. S.

J. S. Loomis, FRINGE User’s Manual (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1976).

Lopez, R.

Mahajan, V. N.

Malacara, D.

M. Carpio, D. Malacara, “Closed Cartesian representation of the Zernike polynomials,” Opt. Commun. 110, 514–516 (1994).
[CrossRef]

O’Bryan, J. T.

Oldham, K. B.

J. Spanier, K. B. Oldham, “The unit-step and related functions,” in An Atlas of Functions (Hemisphere, Washington, D.C., 1987), Chap. 8, pp. 63–69.

Robinson, D. K.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992), p. 126.

Silva, D. E.

Spanier, J.

J. Spanier, K. B. Oldham, “The unit-step and related functions,” in An Atlas of Functions (Hemisphere, Washington, D.C., 1987), Chap. 8, pp. 63–69.

Stahl, H. P.

Swantner, W. H.

Wang, J. Y.

Appl. Opt. (4)

Opt. Commun. (1)

M. Carpio, D. Malacara, “Closed Cartesian representation of the Zernike polynomials,” Opt. Commun. 110, 514–516 (1994).
[CrossRef]

Other (6)

S. D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed. (McGraw-Hill, New York, 1980), Chap. 6.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, Boston, 1992), p. 126.

J. S. Loomis, FRINGE User’s Manual (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1976).

“FRINGE Zernikes,” in Code V Reference Manual, R. C. Juergens, ed. (Optical Research Associates, Pasadena, Calif., 1998), Vol. 1, version 8.30, pp. 2A-595–2A-596.

“Fitting interferograms using Zernike polynomials,” in FAST! V/AI Operations Manual (Phase Shift Technology, Tucson, Ariz., 1990), pp. 2-3–2-5.

J. Spanier, K. B. Oldham, “The unit-step and related functions,” in An Atlas of Functions (Hemisphere, Washington, D.C., 1987), Chap. 8, pp. 63–69.

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Figures (1)

Fig. 1
Fig. 1

Centered circular subdomain of the unit circle, with radius p.

Tables (4)

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Table 1 Matrix of Fit Coefficients for Single Input Zernike Polynomial Terms of Unit Magnitudea

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Table 2 Approximate Shrink Matrix H(p)a

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Table 3 Matrix of Fit Coefficients for Single Input Zernike Polynomial Terms of Unit Magnitude with p = 99% and p = 98% Subdomain Radiia

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Table 4 Approximate Nonzero Elements of the Shrink Matrix on a Subdomain of Radius p with q = 1 - p a

Equations (33)

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Unm(r, θ)=Rnm(r)cos mθ,m0Rnm(r)sin mθ,m>0,
Rnm(r)s=0n(-1)sns n-snrn-2s
= s=0n(-1)s(n-s)!s!n-m2-s!n+m2-s! rn-2s,
nn-|m|2=n-m2,m>0n+m2,m0.
aba!b!(a-b)!.
Cnm(s)(-1)sns n-sn=(-1)sn-|m|2s n-sn-|m|2
= (-1)s(n-s)!s!(n-s)!(n-s-n)!
= (-1)s(n-s)!s!n-|m|2-s!n+|m|2-s!
= (-1)s(n-s)!s!n-m2-s!n+m2-s!.
Rnm(r)=s=0nCnm(s)rn-2s.
Rnm(pr)=s=0nCnm(s)(pr)n-2s
=s=0nCnm(s)pn-2srn-2s.
W=n1man1mUn1m=n1an1mUn1m.
s=0nCnm(s)pn-2srn-2s=n1an1mRn1m=n1an1ms=0nCn1m(s)rn1-2s.
Cnm(0)000Cnm(1)Cn-2m(0)00Cnm(2)Cn-2m(1)Cn-4m(0)0 an,man-2,man-4,m
=pnCnm(0)pn-2Cnm(1)pn-4Cnm(2). 
Γijnm=Cn-2(i-1)m(j-i),ji0,j<i.
pj=pn-2(j-1)Cnm(j-1).
anm=pn,
an-2,m=Cnm(1)Cn-2m(0) pn-2(1-p2)
=-pnCnm(1)Cn-2m(0)+pn-2Cnm(1)Cn-2m(0),
an-4,m=Cnm(2)Cn-4m(0) pn-4(1-p4)-Cn-2m(1)Cnm(1)Cn-4m(0)Cn-2m(0) pn-2(1-p2)
=pn1Cn-4m(0)Cn-2m(1)Cnm(1)Cn-2m(0)-Cnm(2)-pn-2Cn-2m(1)Cnm(1)Cn-4m(0)Cn-2m(0)+pn-4Cnm(2)Cn-4m(0).
anmn0m0=δmm0[nn0]j=1L[(Γn0m0)-1]ijpj.
anmn0m0=δmm0[nn0]×j=1L[(Γn0m0)-1]ijpn0-2(j-1)Cn0m0(j-1).
W=jwjUj=Uw=UTw.
wp=H(p)w.
W=Uwp=U[H(p)w].
U40(r, θ)=6r4-6r2+1.
w=w00w11w1-1w20w22w2-2w31w3-1w40=H(p)w=w00-2qw20-2qw40pw11-4qw31pw1-1-4qw3-1p2w20-6qw40p2w22p2w2-2p3w31p3w3-1p4w40.
=a(1-pn)anq.
p>1-an.

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