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.

Guang-ming Dai and Virendra N. Mahajan Appl. Opt. 47(19) 3433-3445 (2008)

References

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Matrix of Fit Coefficients for Single Input Zernike Polynomial Terms of Unit Magnitude^{a}

Fit Zernike Terms on a Subdomain

Input Zernike Terms of Unit Magnitude

j

0

1

2

3

4

5

6

7

8

n

0

1

1

2

2

2

3

3

4

m

0

1

−1

0

2

−2

1

−1

0

0

0

0

1

0

0

−(1 − p^{2})

0

0

0

0

−(1 − p)(2p −

1

1

1

p

0

0

0

0

−2p(1 − p^{2})

0

0

2

1

−1

p

0

0

0

0

−2p(1 − p^{2})

0

3

2

0

p^{2}

0

0

0

0

−3p^{2}(1 − p

4

2

2

p^{2}

0

0

0

0

5

2

−2

p^{2}

0

0

0

6

3

1

p^{3}

0

0

7

3

−1

p^{3}

0

8

4

0

p^{4}

Note that the diagonal elements equal p^{n}. This matrix, H( p), is called the shrink matrix. Identically zero terms of higher than n th order are om from each column.

Here q is equal to 1 − p, where p is the maximum radius of the subdomain on the full unit circle. Only the lowest-ordered q-dependent components are shown.

Table 3

Matrix of Fit Coefficients for Single Input Zernike Polynomial Terms of Unit Magnitude with p = 99% and p = 98% Subdomain Radii^{a}

Fit Zernike Terms

Input Zernike Terms of Unit Magnitude

p = 99% Subdomain Radii

p = 98% Subdomain Radii

j

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

n

0

1

1

2

2

2

3

3

4

0

1

1

2

2

2

3

3

4

m

0

1

−1

0

2

−2

1

−1

0

0

1

−1

0

2

−2

1

−1

0

0

0

0

1

0

0

−0.020

0

0

0

0

−0.019

1

0

0

−0.040

0

0

0

0

−0.036

1

1

1

0.99

0

0

0

0

−0.039

0

0

0.98

0

0

0

0

−0.078

0

0

2

1

−1

0.99

0

0

0

0

−0.039

0

0.98

0

0

0

0

−0.078

0

3

2

0

0.98

0

0

0

0

−0.059

0.96

0

0

0

0

−0.11

4

2

2

0.98

0

0

0

0

0.96

0

0

0

0

5

2

−2

0.98

0

0

0

0.96

0

0

0

6

3

1

0.97

0

0

0.94

0

0

7

3

−1

0.97

0

0.94

0

8

4

0

0.96

0.92

Terms are rounded to two significant figures.

Table 4

Approximate Nonzero Elements of the Shrink Matrix on a Subdomain of Radius p with $q=1-p$^{
a
}

For a given input aberration with unit magnitude of a single Zernike polynomial term ${U}_{n}^{m},$
the components of the fit wave front are shown. The results shown assume that $p\approx 1$
and are thus valid for small values of q
only. Only the lowest-ordered q
-dependent components are shown.

Tables (4)

Table 1

Matrix of Fit Coefficients for Single Input Zernike Polynomial Terms of Unit Magnitude^{a}

Fit Zernike Terms on a Subdomain

Input Zernike Terms of Unit Magnitude

j

0

1

2

3

4

5

6

7

8

n

0

1

1

2

2

2

3

3

4

m

0

1

−1

0

2

−2

1

−1

0

0

0

0

1

0

0

−(1 − p^{2})

0

0

0

0

−(1 − p)(2p −

1

1

1

p

0

0

0

0

−2p(1 − p^{2})

0

0

2

1

−1

p

0

0

0

0

−2p(1 − p^{2})

0

3

2

0

p^{2}

0

0

0

0

−3p^{2}(1 − p

4

2

2

p^{2}

0

0

0

0

5

2

−2

p^{2}

0

0

0

6

3

1

p^{3}

0

0

7

3

−1

p^{3}

0

8

4

0

p^{4}

Note that the diagonal elements equal p^{n}. This matrix, H( p), is called the shrink matrix. Identically zero terms of higher than n th order are om from each column.

Here q is equal to 1 − p, where p is the maximum radius of the subdomain on the full unit circle. Only the lowest-ordered q-dependent components are shown.

Table 3

Matrix of Fit Coefficients for Single Input Zernike Polynomial Terms of Unit Magnitude with p = 99% and p = 98% Subdomain Radii^{a}

Fit Zernike Terms

Input Zernike Terms of Unit Magnitude

p = 99% Subdomain Radii

p = 98% Subdomain Radii

j

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

n

0

1

1

2

2

2

3

3

4

0

1

1

2

2

2

3

3

4

m

0

1

−1

0

2

−2

1

−1

0

0

1

−1

0

2

−2

1

−1

0

0

0

0

1

0

0

−0.020

0

0

0

0

−0.019

1

0

0

−0.040

0

0

0

0

−0.036

1

1

1

0.99

0

0

0

0

−0.039

0

0

0.98

0

0

0

0

−0.078

0

0

2

1

−1

0.99

0

0

0

0

−0.039

0

0.98

0

0

0

0

−0.078

0

3

2

0

0.98

0

0

0

0

−0.059

0.96

0

0

0

0

−0.11

4

2

2

0.98

0

0

0

0

0.96

0

0

0

0

5

2

−2

0.98

0

0

0

0.96

0

0

0

6

3

1

0.97

0

0

0.94

0

0

7

3

−1

0.97

0

0.94

0

8

4

0

0.96

0.92

Terms are rounded to two significant figures.

Table 4

Approximate Nonzero Elements of the Shrink Matrix on a Subdomain of Radius p with $q=1-p$^{
a
}

For a given input aberration with unit magnitude of a single Zernike polynomial term ${U}_{n}^{m},$
the components of the fit wave front are shown. The results shown assume that $p\approx 1$
and are thus valid for small values of q
only. Only the lowest-ordered q
-dependent components are shown.