Full-aperture wavefront reconstruction from annular subaperture
interferometric data by use of Zernike annular polynomials and a matrix method for
testing large aspheric surfaces
Xi Hou, Fan Wu, Li Yang, Shibin Wu, and Qiang Chen
Xi Hou, Fan Wu, Li Yang, Shibin Wu, and Qiang Chen, "Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces," Appl. Opt. 45, 3442-3455 (2006)
We propose a more accurate and efficient reconstruction method used in testing large
aspheric surfaces with annular subaperture interferometry. By the introduction of the
Zernike annular polynomials that are orthogonal over the annular region, the method
proposed here eliminates the coupling problem in the earlier reconstruction algorithm
based on Zernike circle polynomials. Because of the complexity of recurrence
definition of Zernike annular polynomials, a general symbol representation of that in
a computing program is established. The program implementation for the method is
provided in detail. The performance of the reconstruction algorithm is evaluated in
some pertinent cases, such as different random noise levels, different subaperture
configurations, and misalignments.
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Orthogonal Zernike Annular Polynomials Zi (ρ, θ, ε)a
i
n
m
j
p
q
Zernike Annular Polynomials
Meaning
1
0
0
0
0
0
1
Piston
2
1
1
0
1
1
x Tilt
3
1
−1
0
1
1
y Tilt
4
2
0
1
1
0
Defocus
5
2
2
0
2
2
Astigmatism
6
2
−2
0
2
2
Astigmatism
7
3
1
1
2
1
Coma
8
3
−1
1
2
1
Coma
9
4
0
2
2
0
Spherical
10
3
3
0
3
3
Trefoil
11
3
−3
0
3
3
Trefoil
aThe indices i, n,
m, and j are defined as the polynomial
number, radial degree, azimuthal frequency, and degree of polynomial
Qjm(ρ2), respectively. The indices p
and q are introduced for the convenience of representing
polynomials inside the computing program, where n =
2p − q and . The polynomials are ordered in ascending values of
p.
Table 2
Area and Misalignment Coefficients of the Five Simulated Subapertures
Subaperture (Configuration 1)
Subaperture Area
Zernike Coefficients of Misalignments
Piston
x Tilt
y Tilt
Defocus
1
0.1 ≤ P1 ≤ 0.35
−0.009
1.091
0.406
−0.200
2
0.35 ≤ P2 ≤ 0.55
−0.003
1.012
0.706
−0.521
3
0.55 ≤ P3 ≤ 0.75
−0.019
−1.091
−0.406
−0.200
4
0.75 ≤ P4 ≤ 0.9
−0.030
1.512
1.706
−0.221
5
0.9 ≤ P5 ≤ 1
0.009
0.091
0.406
−0.290
Table 3
Comparison of Annular Zernike Coefficients Reconstructed from Simulated
Subaperture Data with Different Random Noise Levels
Term
Initial Value
Ratio of Relative Random Noise
σ = 0
σ = 0.1
σ = 0.15
σ = 0.2
σ = 0.25
σ = 0.3
σ = 0.35
σ = 0.4
5
1
1
0.995
1.000
1.002
0.999
1.006
0.991
1.003
6
0
0
−0.001
−0.008
−0.003
0.003
−0.002
0.002
0.017
7
0
0
−0.006
−0.003
−0.016
0.003
0.044
−0.050
0.039
8
0
0
−0.012
0.030
0.021
0.008
−0.027
0.040
0.040
9
0.5
0.5
0.494
0.548
0.508
0.478
0.544
0.433
0.553
10
0
0
0
−0.002
−0.004
−0.011
0.006
0.025
−0.014
11
0
0
−0.001
−0.001
−0.006
0.007
0.005
0.015
0.004
12
0
0
0
0.003
0.017
0.002
0.019
0.007
0.003
13
0
0
−0.002
0.004
−0.001
−0.005
−0.017
0.012
−0.019
14
−0.63
−0.63
−0.622
−0.646
−0.604
−0.627
−0.675
−0.677
−0.619
15
0
0
−0.026
0.041
0.006
−0.054
0.025
0.057
−0.003
16
0
0
−0.083
−0.058
−0.146
0.010
−0.057
−0.087
−0.282
Table 4
Comparison of Reconstructed Full-Aperture Wavefronts for Different Subaperture
Configurations
Subaperture Configuration
Noise-to-Signal Ratio σ
Reconstructed Full-Aperture Wavefront
PV (Wave)
rms (Wave)
Δ
I
0
2.715
0.498
0
0.1
2.614
0.495
0.067
II
0
2.715
0.498
0
0.1
2.648
0.485
0.051
III
0
2.715
0.498
0
0.1
2.737
0.500
0.017
Table 5
Comparison of Reconstructed Full-Aperture Wavefronts for Different Subaperture
Defocuses
Configuration I
Misalignments of Five Subapertures (Wave)
σ
PV (Wave)
rms (Wave)
Δ
Piston
X Tilt
Y Tilt
Defocus
1
0
1, 2, 3, 4, 5
0
2.715
0.498
0
2
1, 2, 3, 4, 5
0.1
2.772
0.504
0.034
3
1, 2, 3, 4, 5
0.1
2.746
0.503
0.024
4
−5, −2, 3, 5, 8
0
2.715
0.498
0
5
0, 0, 2, 0, −0.3
0.5, 0.1, 0.4, 1.5, 0
0, 0.5, −0.6, 0.2, 1.3
−5, −2, 3, 5, 8
0.1
2.628
0.504
0.096
Tables (5)
Table 1
Orthogonal Zernike Annular Polynomials Zi (ρ, θ, ε)a
i
n
m
j
p
q
Zernike Annular Polynomials
Meaning
1
0
0
0
0
0
1
Piston
2
1
1
0
1
1
x Tilt
3
1
−1
0
1
1
y Tilt
4
2
0
1
1
0
Defocus
5
2
2
0
2
2
Astigmatism
6
2
−2
0
2
2
Astigmatism
7
3
1
1
2
1
Coma
8
3
−1
1
2
1
Coma
9
4
0
2
2
0
Spherical
10
3
3
0
3
3
Trefoil
11
3
−3
0
3
3
Trefoil
aThe indices i, n,
m, and j are defined as the polynomial
number, radial degree, azimuthal frequency, and degree of polynomial
Qjm(ρ2), respectively. The indices p
and q are introduced for the convenience of representing
polynomials inside the computing program, where n =
2p − q and . The polynomials are ordered in ascending values of
p.
Table 2
Area and Misalignment Coefficients of the Five Simulated Subapertures
Subaperture (Configuration 1)
Subaperture Area
Zernike Coefficients of Misalignments
Piston
x Tilt
y Tilt
Defocus
1
0.1 ≤ P1 ≤ 0.35
−0.009
1.091
0.406
−0.200
2
0.35 ≤ P2 ≤ 0.55
−0.003
1.012
0.706
−0.521
3
0.55 ≤ P3 ≤ 0.75
−0.019
−1.091
−0.406
−0.200
4
0.75 ≤ P4 ≤ 0.9
−0.030
1.512
1.706
−0.221
5
0.9 ≤ P5 ≤ 1
0.009
0.091
0.406
−0.290
Table 3
Comparison of Annular Zernike Coefficients Reconstructed from Simulated
Subaperture Data with Different Random Noise Levels
Term
Initial Value
Ratio of Relative Random Noise
σ = 0
σ = 0.1
σ = 0.15
σ = 0.2
σ = 0.25
σ = 0.3
σ = 0.35
σ = 0.4
5
1
1
0.995
1.000
1.002
0.999
1.006
0.991
1.003
6
0
0
−0.001
−0.008
−0.003
0.003
−0.002
0.002
0.017
7
0
0
−0.006
−0.003
−0.016
0.003
0.044
−0.050
0.039
8
0
0
−0.012
0.030
0.021
0.008
−0.027
0.040
0.040
9
0.5
0.5
0.494
0.548
0.508
0.478
0.544
0.433
0.553
10
0
0
0
−0.002
−0.004
−0.011
0.006
0.025
−0.014
11
0
0
−0.001
−0.001
−0.006
0.007
0.005
0.015
0.004
12
0
0
0
0.003
0.017
0.002
0.019
0.007
0.003
13
0
0
−0.002
0.004
−0.001
−0.005
−0.017
0.012
−0.019
14
−0.63
−0.63
−0.622
−0.646
−0.604
−0.627
−0.675
−0.677
−0.619
15
0
0
−0.026
0.041
0.006
−0.054
0.025
0.057
−0.003
16
0
0
−0.083
−0.058
−0.146
0.010
−0.057
−0.087
−0.282
Table 4
Comparison of Reconstructed Full-Aperture Wavefronts for Different Subaperture
Configurations
Subaperture Configuration
Noise-to-Signal Ratio σ
Reconstructed Full-Aperture Wavefront
PV (Wave)
rms (Wave)
Δ
I
0
2.715
0.498
0
0.1
2.614
0.495
0.067
II
0
2.715
0.498
0
0.1
2.648
0.485
0.051
III
0
2.715
0.498
0
0.1
2.737
0.500
0.017
Table 5
Comparison of Reconstructed Full-Aperture Wavefronts for Different Subaperture
Defocuses