Abstract

The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.

© 2010 Optical Society of America

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References

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  1. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, 3rd ed., M.Bass, V.N.Mahajan, and E.Van Stryland, eds. (McGraw-Hill , 2010), Vol. II, pp. 11.3–11.41.
  2. V. N. Mahajan, Optical Imaging and Aberrations Part II: Wave Diffraction Optics (SPIE, 2001).
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  4. S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
    [CrossRef]
  5. X. Hou, F. Wu, L. Yang, and Q. Chen, “Comparison of annular wavefront interpretation with Zernike circle and annular polynomials,” Appl. Opt. 45, 8893–8901 (2006).
    [CrossRef] [PubMed]
  6. M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241–248 (1993).
    [CrossRef]
  7. C.-J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21, 4521–4525 (1982).
    [CrossRef] [PubMed]
  8. V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [CrossRef]
  9. G.-m. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: Error analysis,” Appl. Opt. 47, 3433–3445 (2008).
    [CrossRef] [PubMed]
  10. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 547–666.
    [CrossRef]
  11. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
    [CrossRef]

2008 (1)

2007 (1)

2006 (1)

2003 (1)

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

1993 (1)

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241–248 (1993).
[CrossRef]

1982 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

Chen, Q.

Dai, G.-m.

DiVittorio, M.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

Gilbreath, C.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

Hou, X.

Kim, C.-J.

Mahajan, V. N.

G.-m. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: Error analysis,” Appl. Opt. 47, 3433–3445 (2008).
[CrossRef] [PubMed]

V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
[CrossRef]

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, 3rd ed., M.Bass, V.N.Mahajan, and E.Van Stryland, eds. (McGraw-Hill , 2010), Vol. II, pp. 11.3–11.41.

V. N. Mahajan, Optical Imaging and Aberrations Part II: Wave Diffraction Optics (SPIE, 2001).

Melozzi, M.

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241–248 (1993).
[CrossRef]

Mozurkewich, D.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

Pezzati, L.

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241–248 (1993).
[CrossRef]

Restaino, S. R.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

Teare, S. W.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Wu, F.

Yang, L.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42, 2491–2495 (2003).
[CrossRef]

Proc. SPIE (1)

M. Melozzi and L. Pezzati, “Interferometric testing of annular apertures,” Proc. SPIE 1781, 241–248 (1993).
[CrossRef]

Other (5)

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
[CrossRef]

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, 3rd ed., M.Bass, V.N.Mahajan, and E.Van Stryland, eds. (McGraw-Hill , 2010), Vol. II, pp. 11.3–11.41.

V. N. Mahajan, Optical Imaging and Aberrations Part II: Wave Diffraction Optics (SPIE, 2001).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (8)

Fig. 1
Fig. 1

Orthonormal annular coefficients a j and Zernike circle coefficients b ^ j for a four-polynomial expansion.

Fig. 2
Fig. 2

Ratio of the orthonormal annular co efficients a j and Zernike circle coefficients b ^ j for an 11-polynomial expansion.

Fig. 3
Fig. 3

Orthonormal annular coefficients a j and Zernike circle coefficients b ^ j , illustrating how the latter change as the number of polynomials changes from 4 to 11.

Fig. 4
Fig. 4

Standard deviation as obtained from the orthonormal annular coefficients a j and Zernike circle coefficients b ^ j of a 4- and 11-polynomial expansion.

Fig. 5
Fig. 5

Contours of (a) Seidel aberration function of Eq. (37) for a circular pupil with A t = A d = A a = 1 , A c = 2 , and A s = 3 in waves. (b) Same Seidel aberration function, but for an annular pupil with obscuration ratio ϵ = 0.5 .

Fig. 6
Fig. 6

Contours of an annular Seidel aberration function for ϵ = 0.5 fit with only four polynomials, as in (a) Eq. (38) or (43), and (b) Eq. (54).

Fig. 7
Fig. 7

Contours of the residual aberration function after removing the interferometer setting errors. (a) W RA of Eq. (47) using annular polynomials (b) W RC b ^ of Eq. (48) using circle polynomials correctly, and (c) W RC b of Eq. (55) using circle polynomials incorrectly.

Fig. 8
Fig. 8

Contours of the difference or the error function (a) Eq. (49) and (b) obtained by subtracting Eq. (47) from Eq. (55).

Tables (5)

Tables Icon

Table 1 Orthonormal Annular Polynomials A j ( ρ , θ ; ϵ ) for an Obscuration Ratio ϵ

Tables Icon

Table 2 Annular Polynomials A j ( ρ , θ ; ϵ ) in Terms of the Zernike Circle Polynomials Z j ( ρ , θ ) a

Tables Icon

Table 3 Nonzero Elements of 11 × 11 Conversion Matrix M for Obtaining the Annular Polynomials A j ( ρ , θ ; ϵ ) from the Zernike Circle Polynomials Z j ( ρ , θ )

Tables Icon

Table 4 Nonzero Elements c j j of 11 × 11 Matrix C Z Z of the Zernike Circle Polynomials over an Annular Pupil of Obscuration Ratio ϵ, where c j j = c j j

Tables Icon

Table 5 Nonzero Elements d j j of 11 × 11 Matrix C Z F of the Zernike Circle Polynomials over an Annular Pupil of Obscuration Ratio ϵ

Equations (92)

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W ^ ( x , y ) = j = 1 J a j F j ( x , y ) ,
1 A pupil F j ( x , y ) F j ( x , y ) d x d y = δ j j ,
a j = 1 A pupil W ( x , y ) F j ( x , y ) d x d y .
W ^ = a 1 ,
σ W ^ 2 = W ^ 2 ( x , y ) W ^ ( x , y ) 2
= j = 2 J a j 2 ,
F j ( x , y ) = i = 1 J M j i Z i ( x , y ) ,
{ F j } = M { Z j } ,
W ^ ( x , y ) = j = 1 J b ^ j Z j ( x , y ) ,
1 π x 2 + y 2 1 Z j ( x , y ) Z j ( x , y ) d x d y = δ j j ,
1 π 0 1 0 2 π Z j ( ρ , θ ) Z j ( ρ , θ ) ρ d ρ d θ = δ j j .
W ^ ( x , y ) = j = 1 J a j i = 1 j M j i Z i ( x , y ) = j = 1 J i = j J a i M i j Z j ( x , y ) .
b ^ j = i = j J a i M i j .
b ^ = M T a ,
a = ( M T ) 1 b ^ .
S ^ = Z b ^ ,
b ^ = Z 1 S ^ ,
b j = 1 A pupil W ( x , y ) Z j ( x , y ) d x d y .
b j = j = 1 J a j 1 A pupil Z j ( x , y ) F j ( x , y ) d x d y = j = 1 J a j Z j | F j ,
b = C Z F a ,
j = 1 J b ^ j Z j ( x , y ) = j = 1 J a j F j ( x , y ) ,
j = 1 J b ^ j Z j | Z j = j = 1 J a j Z j | F j ,
C Z Z b ^ = C Z F a = b ,
C Z F = C Z Z M T .
c j j = 1 A pupil Z j ( x , y ) Z j ( x , y ) d x d y ,
d j j = 1 A pupil Z j ( x , y ) F j ( x , y ) d x d y .
A even j ( ρ , θ ; ϵ ) = 2 ( n + 1 ) R n m ( ρ ; ϵ ) cos m θ , m 0 ,
A odd j ( ρ , θ ; ϵ ) = 2 ( n + 1 ) R n m ( ρ ; ϵ ) sin m θ , m 0 ,
A j ( ρ , θ ; ϵ ) = n + 1 R n 0 ( ρ ; ϵ ) , m = 0 ,
1 π ( 1 ϵ 2 ) ϵ 1 0 2 π A j ( ρ , θ ; ϵ ) A j ( ρ , θ ; ϵ ) ρ d ρ d θ = δ j j .
{ A j } = M { Z j } ,
W ^ ( ρ , θ ; ϵ ) = j = 1 J a j A j ( ρ , θ ; ϵ ) ,
a j = 1 π ( 1 ϵ 2 ) ϵ 1 0 2 π W ( ρ , θ ; ϵ ) A j ( ρ , θ ; ϵ ) ρ d ρ d θ .
( b ^ 1 b ^ 2 b ^ 3 b ^ 4 ) = ( 1 0 0 3 ϵ 2 ( 1 ϵ 2 ) 1 0 ( 1 + ϵ 2 ) 1 / 2 0 0 0 0 ( 1 + ϵ 2 ) 1 / 2 0 0 0 0 ( 1 ϵ 2 ) 1 ) ( a 1 a 2 a 3 a 4 ) = ( a 1 3 ϵ 2 ( 1 ϵ 2 ) 1 a 4 ( 1 + ϵ 2 ) 1 / 2 a 2 ( 1 + ϵ 2 ) 1 / 2 a 3 ( 1 ϵ 2 ) 1 a 4 ) ,
b ^ 1 = a 1 3 ϵ 2 ( 1 ϵ 2 ) 1 a 4 ,
b ^ 2 = ( 1 + ϵ 2 ) 1 / 2 a 2 ,
b ^ 3 = ( 1 + ϵ 2 ) 1 / 2 a 3 ,
b ^ 4 = ( 1 ϵ 2 ) 1 a 4 .
b ^ 1 = a 1 3 ϵ 2 ( 1 ϵ 2 ) 1 a 4 + 5 ϵ 2 ( 1 + ϵ 2 ) ( 1 ϵ 2 ) 2 a 11 ,
b ^ 2 = ( 1 + ϵ 2 ) 1 / 2 a 2 ( 2 2 ϵ 4 / B ) a 8 ,
b ^ 3 = ( 1 + ϵ 2 ) 1 / 2 a 3 ( 2 2 ϵ 4 / B ) a 7 ,
b ^ 4 = ( 1 ϵ 2 ) 1 a 4 15 ϵ 2 ( 1 ϵ 2 ) 2 a 11 ,
b ^ 5 = ( 1 + ϵ 2 + ϵ 4 ) 1 / 2 a 5 ,
b ^ 6 = ( 1 + ϵ 2 + ϵ 4 ) 1 / 2 a 6 ,
b ^ 7 = [ ( 1 + ϵ 2 ) / B ] a 7 ,
b ^ 8 = [ ( 1 + ϵ 2 ) / B ] a 8 ,
b ^ 9 = ( 1 + ϵ 2 + ϵ 4 + ϵ 6 ) 1 / 2 a 9 ,
b ^ 10 = ( 1 + ϵ 2 + ϵ 4 + ϵ 6 ) 1 / 2 a 10 ,
b ^ 11 = ( 1 ϵ 2 ) 2 a 11 ,
B = ( 1 ϵ 2 ) [ ( 1 + ϵ 2 ) ( 1 + 4 ϵ 2 + ϵ 4 ) ] 1 / 2 .
W ^ ( x , y ) = a 1 A 1 + a 2 A 2 + a 3 A 3 + a 4 A 4
= a 1 + 2 ( 1 + ϵ 2 ) 1 / 2 a 2 x + 2 ( 1 + ϵ 2 ) 1 / 2 a 3 y + 3 ( 1 ϵ 2 ) 1 a 4 [ ϵ 2 + ( 2 ρ 2 1 ) ] .
W ^ ( x , y ) = b ^ 1 Z 1 + b ^ 2 Z 2 + b ^ 3 Z 3 + b ^ 4 Z 4
= b ^ 1 + 2 b ^ 2 x + 2 b ^ 3 y + 3 b ^ 4 ( 2 ρ 2 1 ) .
W ( ρ , θ ; ϵ ) = A t ρ cos θ + A d ρ 2 + A a ρ 2 cos 2 θ + A c ρ 3 cos θ + A s ρ 4 , ϵ ρ 1 ,
W ^ ( ρ , θ ; ϵ ) = a 1 A 1 + a 2 A 2 + a 4 A 4 ,
a 1 = ( 1 + ϵ 2 ) ( 2 A d + A a ) / 4 + ( 1 + ϵ 2 + ϵ 4 ) A s / 3 ,
a 2 = ( 1 + ϵ 2 ) 1 / 2 A t / 2 + ( 1 + ϵ 2 + ϵ 4 ) ( 1 + ϵ 2 ) 1 / 2 A c / 3 ,
a 4 = ( 1 ϵ 2 ) ( 2 A d + A a ) / 4 3 + ( 1 ϵ 4 ) A s / 2 3 .
σ W ^ 2 = a 2 2 + a 4 2 .
W ( ρ , θ ; ϵ ) = a 1 A 1 + a 2 A 2 + a 4 A 4 + a 6 A 6 + a 8 A 8 + a 11 A 11 ,
a 6 = 1 2 6 ( 1 + ϵ 2 + ϵ 4 ) 1 / 2 A a ,
a 8 = 1 ϵ 2 6 2 ( 1 + 4 ϵ 2 + ϵ 4 1 + ϵ 2 ) 1 / 2 A c ,
a 11 = ( 1 ϵ 2 ) 2 6 5 A s .
σ W 2 = a 2 2 + a 4 2 + a 6 2 + a 8 2 + a 11 2 .
W ^ ( ρ , θ ; ϵ ) = b ^ 1 Z 1 + b ^ 2 Z 2 + b ^ 4 Z 4 ,
b ^ 1 = ( 2 A d + A a ) / 4 + [ 1 ϵ 2 ( 1 + ϵ 2 ) / 2 ] A s / 3 ,
b ^ 2 = a 2 / ( 1 + ϵ 2 ) 1 / 2 ,
b ^ 4 = a 4 / ( 1 ϵ 2 ) .
W ( ρ , θ ; ϵ ) = b ^ 1 Z 1 + b ^ 2 Z 2 + b ^ 4 Z 4 + b ^ 6 Z 6 + b ^ 8 Z 8 + b ^ 11 Z 11 ,
b ^ 1 = ( 2 A d + A a ) / 4 + A s / 3 ,
b ^ 2 = A t / 2 + A c / 3 ,
b ^ 4 = ( 2 A d + A a ) / 4 3 + A s / 2 3 ,
b ^ 6 = A a / 2 6 ,
b ^ 8 = A c / 6 2 ,
b ^ 11 = A s / 6 5 .
W RA ( ρ , θ ; ϵ ) = a 6 A 6 + a 8 A 8 + a 11 A 11 .
W RC b ^ ( ρ , θ ; ϵ ) = b ^ 6 Z 6 + b ^ 8 Z 8 + b ^ 11 Z 11 = ( A a / 2 6 ) Z 6 + ( A c / 6 2 ) Z 8 + ( A s / 6 5 ) Z 11 .
Δ W R b ^ ( ρ , θ ; ϵ ) = 1 6 ϵ 2 ( 4 + ϵ 2 ) A s + 2 3 ϵ 4 1 + ϵ 2 A c ρ cos θ + ϵ 2 A s ρ 2 ,
a 6 b ^ 6 = ( 1 + ϵ 2 + ϵ 4 ) 1 / 2 ,
a 8 b ^ 8 = ( 1 ϵ 2 ) ( 1 + 4 ϵ 2 + ϵ 4 1 + ϵ 2 ) 1 / 2 ,
a 11 b ^ 11 = ( 1 ϵ 2 ) 2 .
W ( ρ , θ ; ϵ ) = b 1 Z 1 + b 2 Z 2 + b 4 Z 4 + b 6 Z 6 + b 8 Z 8 + b 11 Z 11 + ,
b j = 1 π ( 1 ϵ 2 ) ϵ 1 0 2 π W ( ρ , θ ; ϵ ) Z j ( ρ , θ ) ρ d ρ d θ .
b 1 = a 1 ,
b 2 = ( 1 + ϵ 2 ) 1 / 2 a 2 ,
b 4 = 1 4 3 ( 1 + ϵ 2 + 4 ϵ 4 ) ( 2 A d + A a ) + 1 2 3 ( 1 + ϵ 2 + ϵ 4 + 3 ϵ 6 ) A s ,
b 6 = ( 1 + ϵ 2 + ϵ 4 ) 1 / 2 a 6 ,
b 8 = 2 ϵ 4 A t + 1 6 2 ( 1 + ϵ 2 + ϵ 4 + 9 ϵ 6 ) A c ,
b 11 = 5 4 ϵ 4 ( 3 ϵ 2 1 ) ( 2 A d + A a ) + 1 6 5 ( 1 + ϵ 2 + ϵ 4 9 ϵ 6 + 36 ϵ 8 ) A s ,
W ^ ( ρ , θ ; ϵ ) = b 1 Z 1 + b 2 Z 2 + b 4 Z 4 .
W RC b ( ρ , θ ; ϵ ) = b 6 Z 6 + b 8 Z 8 + b 11 Z 11 .

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