Zernike annular polynomials and atmospheric turbulence

Guang-ming Dai; Virendra N. Mahajan

doi:10.1364/JOSAA.24.000139

Zernike annular polynomials and atmospheric turbulence

Guang-ming Dai and Virendra N. Mahajan

Journal of the Optical Society of America A
Vol. 24,
Issue 1,
pp. 139-155
(2007)
•https://doi.org/10.1364/JOSAA.24.000139

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Guang-ming Dai and Virendra N. Mahajan, "Zernike annular polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 24, 139-155 (2007)

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Related Topics
Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Active or adaptive optics (010.1080)
- Atmospheric turbulence (010.1330)
- Aberration compensation (220.1000)
- Aberrations (global) (220.1010)

About this Article
History
- Original Manuscript: June 7, 2006
- Manuscript Accepted: July 28, 2006

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Abstract

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.

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n	$n^{'}$	m	$g_{n^{'}} (n, m; ∊)$
0	0	0	1
1	1	1	$\frac{1}{\sqrt{1 + ∊^{2}}}$
2	0	0	$- \frac{∊^{2}}{1 - ∊^{2}}$
2	2	0	$\frac{1}{1 - ∊^{2}}$
2	2	2	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4}}}$
3	1	1	$- \frac{2 ∊^{4}}{(1 - ∊^{2}) \sqrt{1 + 5 ∊^{2} + 5 ∊^{4} + ∊^{6}}}$
3	3	1	$\frac{1 + ∊^{2}}{(1 - ∊^{2}) \sqrt{1 + 5 ∊^{2} + 5 ∊^{4} + ∊^{6}}}$
3	3	3	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6}}}$
4	0	0	$\frac{∊^{2} (1 + ∊^{2})}{{(1 - ∊^{2})}^{2}}$
4	2	0	$- \frac{3 ∊^{2}}{{(1 - ∊^{2})}^{2}}$
4	4	0	$\frac{1}{{(1 - ∊^{2})}^{2}}$
4	4	0	$- \frac{3 ∊^{6}}{1 - ∊^{6}} \sqrt{\frac{1 + ∊^{2} + ∊^{4}}{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}}$
4	4	2	$\frac{1}{1 - ∊^{2}} \sqrt{\frac{1 + ∊^{2} + ∊^{4}}{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}}$
4	4	4	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}}$
5	1	1	$\frac{∊^{4} (3 + 4 ∊^{2} + 3 ∊^{4})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 46 ∊^{4} + 46 ∊^{6} + 13 ∊^{8} + ∊^{10}}}$
5	3	1	$- \frac{4 ∊^{4} (3 + ∊^{2})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 46 ∊^{4} + 46 ∊^{6} + 13 ∊^{8} + ∊^{10}}}$
5	5	1	$\frac{1}{{(1 - ∊^{2})}^{2}} \sqrt{\frac{1 + 4 ∊^{2} + ∊^{4}}{1 + 9 ∊^{2} + 9 ∊^{4} + ∊^{6}}}$
5	3	3	$- \frac{4 ∊^{8}}{1 - ∊^{8}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 10 ∊^{8} + 4 ∊^{10} + ∊^{12}}}$
5	5	3	$\frac{1}{1 - ∊^{2}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 10 ∊^{8} + 4 ∊^{10} + ∊^{12}}}$
5	5	5	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10}}}$
6	0	0	$- \frac{∊^{2} (1 + 3 ∊^{2} + ∊^{4})}{{(1 - ∊^{2})}^{3}}$
6	2	0	$\frac{3 ∊^{2} (1 + 2 ∊^{2})}{{(1 - ∊^{2})}^{3}}$
6	4	0	$- \frac{5 ∊^{2}}{{(1 - ∊^{2})}^{3}}$
6	6	0	$\frac{1}{{(1 - ∊^{2})}^{3}}$
6	2	2	$\frac{3 ∊^{6} (2 + 3 ∊^{2} + 3 ∊^{4} + 2 ∊^{6})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 91 ∊^{4} + 339 ∊^{6} + 792 ∊^{8} + 1028 ∊^{10} + 792 ∊^{12} + 339 ∊^{14} + 91 ∊^{16} + 13 ∊^{18} + ∊^{20}}}$
6	4	2	$- \frac{5 ∊^{6} (6 + 3 ∊^{2} + ∊^{4})}{{(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 91 ∊^{4} + 339 ∊^{6} + 792 ∊^{8} + 1028 ∊^{10} + 792 ∊^{12} + 339 ∊^{14} + 91 ∊^{16} + 13 ∊^{18} + ∊^{20}}}$
6	6	2	$\frac{1}{{(1 - ∊^{2})}^{2}} \sqrt{\frac{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}{1 + 9 ∊^{2} + 45 ∊^{4} + 65 ∊^{6} + 45 ∊^{8} + 9 ∊^{10} + ∊^{12}}}$
6	4	4	$- \frac{5 ∊^{10}}{1 - ∊^{10}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 35 ∊^{8} + 20 ∊^{10} + 10 ∊^{12} + 4 ∊^{14} + ∊^{16}}}$
6	6	4	$\frac{1}{1 - ∊^{2}} \sqrt{\frac{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 35 ∊^{8} + 20 ∊^{10} + 10 ∊^{12} + 4 ∊^{14} + ∊^{16}}}$
6	6	6	$\frac{1}{\sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10} + ∊^{12}}}$

n	$n^{″}$	$h_{n^{″}} (n; ∊)$
0	0	1
1	1	ϵ
2	0	$- (1 - ∊^{2})$
2	2	$∊^{2}$
3	1	$- 2 ∊ (1 - ∊^{2})$
3	3	$∊^{3}$
4	0	$(1 - ∊^{2}) (1 - 2 ∊^{2})$
4	2	$- 3 ∊^{2} (1 - ∊^{2})$
4	4	$∊^{4}$
5	1	$∊ (1 - ∊^{2}) (3 - 5 ∊^{2})$
5	3	$- 4 ∊^{3} (1 - ∊^{2})$
5	5	$∊^{5}$
6	0	$- (1 - ∊^{2}) (1 - 5 ∊^{2} + 5 ∊^{4})$
6	2	$3 ∊^{2} (1 - ∊^{2}) (2 - 3 ∊^{2})$
6	4	$- 5 ∊^{4} (1 - ∊^{2})$
6	6	$∊^{6}$

n	m	$T_{n}^{m} (κ; ∊)$
0	0	$\frac{J_{1} (2 π κ) - ∊ J_{1} (2 π ∊ κ)}{κ}$
1	1	$- \frac{i [J_{2} (2 π κ) - ∊^{2} J_{2} (2 π ∊ κ)]}{κ \sqrt{1 + ∊^{2}}}$
2	0	$- \frac{J_{3} (2 π κ) - ∊^{3} J_{3} (2 π ∊ κ) + ∊ [∊ J_{1} (2 π κ) - J_{1} (2 π ∊ κ)]}{κ (1 - ∊^{2})}$
2	2	$- \frac{J_{3} (2 π κ) - ∊^{3} J_{3} (2 π ∊ κ) - ∊ (1 - ∊^{2}) J_{1} (2 π ∊ κ)}{κ \sqrt{1 + ∊^{2} + ∊^{4}}}$
3	1	$- \frac{i {(1 + ∊^{2}) [J_{4} (2 π κ) - ∊^{4} J_{4} (2 π ∊ κ)] + 2 ∊^{2} [∊^{2} J_{2} (2 π κ) - J_{2} (2 π ∊ κ)]}}{κ (1 - ∊^{2}) \sqrt{1 + 5 ∊^{2} + 5 ∊^{4} + ∊^{6}}}$
3	3	$\frac{i {[J_{4} (2 π κ) - ∊^{4} J_{4} (2 π ∊ κ)] - 2 ∊^{2} (1 - ∊^{2}) J_{2} (2 π ∊ κ)}}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6}}}$
4	0	$\frac{J_{5} (2 π κ) - ∊^{5} J_{5} (2 π ∊ κ) + 3 ∊^{2} [J_{3} (2 π κ) - ∊ J_{3} (2 π ∊ κ)] + ∊ (1 + ∊^{2}) [∊ J_{1} (2 π κ) - J_{1} (2 π ∊ κ)]}{κ {(1 - ∊^{2})}^{2}}$
4	2	$\frac{J_{5} (2 π κ) - ∊^{5} J_{5} (2 π ∊ κ) - 3 ∊^{3} [J_{3} (2 π κ) - ∊^{2} J_{3} (2 π ∊ κ)] - ∊ (1 - 3 ∊^{2} + 2 ∊^{4}) J_{1} (2 π ∊ κ)}{κ (1 - ∊^{2})} \sqrt{\frac{1 + ∊^{2} + ∊^{4}}{1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}}}$
4	4	$\frac{J_{5} (2 π κ) - ∊^{5} J_{5} (2 π ∊ κ) - 3 ∊^{3} (1 - ∊^{2}) J_{3} (2 π ∊ κ) - ∊ (1 - 3 ∊^{2} + 2 ∊^{4}) J_{1} (2 π ∊ κ)}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}}$
5	1	$- \frac{i {(1 + 4 ∊^{2} + ∊^{4}) [J_{6} (2 π κ) - ∊^{6} J_{6} (2 π ∊ κ)] + 4 ∊^{4} [(3 + ∊^{2}) J_{4} (2 π κ) - (1 + 3 ∊^{2}) J_{4} (2 π ∊ κ)] + ∊^{2} (3 + 4 ∊^{2} + 3 ∊^{4}) [∊^{2} J_{2} (2 π κ) - J_{2} (2 π ∊ κ)]}}{κ {(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 46 ∊^{4} + 46 ∊^{6} + 13 ∊^{8} + ∊^{10}}}$
5	3	$- \frac{i {(1 + ∊^{2} + ∊^{4} + ∊^{6}) [J_{6} (2 π κ) - ∊^{6} J_{6} (2 π ∊ κ)] + 4 ∊^{4} [∊^{4} J_{4} (2 π κ) - J_{4} (2 π ∊ κ)] - ∊^{2} (3 - 5 ∊^{2} + 5 ∊^{8} - 3 ∊^{10}) J_{2} (2 π ∊ κ)} \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6}}}{κ (1 - ∊^{8}) \sqrt{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 10 ∊^{8} + 4 ∊^{10} + ∊^{12}}}$
5	5	$- \frac{i [J_{6} (2 π κ) - ∊^{6} J_{6} (2 π ∊ κ) - 4 ∊^{4} (1 - ∊^{2}) J_{4} (2 π ∊ κ) - ∊^{2} (3 - 8 ∊^{2} + 5 ∊^{4}) J_{2} (2 π ∊ κ)]}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10}}}$
6	0	$- \frac{J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ) + 5 ∊^{2} [J_{5} (2 π κ) - ∊^{3} J_{5} (2 π ∊ κ)] + 3 ∊^{2} [(1 + 2 ∊^{2}) J_{3} (2 π κ) - ∊ (2 + ∊^{2}) J_{3} (2 π ∊ κ)] + ∊ (1 + 3 ∊^{2} + ∊^{4}) [∊ J_{1} (2 π κ) - J_{1} (2 π ∊ κ)]}{κ {(1 - ∊^{2})}^{3}}$
6	2	$- \frac{(1 + 4 ∊^{2} + 10 ∊^{4} + 4 ∊^{6} + ∊^{8}) [J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ)] + 5 ∊^{5} [∊ (6 + 3 ∊^{2} + ∊^{4}) J_{5} (2 π κ) - (1 + 3 ∊^{2} + 6 ∊^{4}) J_{5} (2 π ∊ κ)] + (6 ∊^{3} + 9 ∊^{5} + 9 ∊^{7} + 6 ∊^{9}) [∊^{3} J_{3} (2 π κ) - J_{3} (2 π ∊ κ)] - {(1 - ∊^{2})}^{5} (∊ + 3 ∊^{3} + ∊^{5}) J_{1} (2 π ∊ κ)}{κ {(1 - ∊^{2})}^{2} \sqrt{1 + 13 ∊^{2} + 91 ∊^{4} + 339 ∊^{6} + 792 ∊^{8} + 1028 ∊^{10} + 792 ∊^{12} + 339 ∊^{14} + 91 ∊^{16} + 13 ∊^{18} + ∊^{20}}}$
6	4	$- \frac{{(1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}) [J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ)] + 5 ∊^{5} [∊^{5} J_{5} (2 π κ) - J_{5} (2 π ∊ κ)] - 3 ∊^{3} (2 - 3 ∊^{2} + 3 ∊^{10} - 2 ∊^{12}) J_{3} (2 π ∊ κ) - ∊ {(1 - ∊^{2})}^{3} (1 - 2 ∊^{2} - 4 ∊^{4} - 5 ∊^{6} - 5 ∊^{8}) J_{1} (2 π ∊ κ)} \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8}}}{κ (1 - ∊^{10}) \sqrt{1 + 4 ∊^{2} + 10 ∊^{4} + 20 ∊^{6} + 35 ∊^{8} + 20 ∊^{10} + 10 ∊^{12} + 4 ∊^{14} + ∊^{16}}}$
6	6	$- \frac{J_{7} (2 π κ) - ∊^{7} J_{7} (2 π ∊ κ) - 5 ∊^{5} (1 - ∊^{2}) J_{5} (2 π ∊ κ) - 3 ∊^{3} (2 - 5 ∊^{2} + 3 ∊^{4}) J_{3} (2 π ∊ κ) - ∊ (1 - 6 ∊^{2} + 10 ∊^{4} - 5 ∊^{6}) J_{1} (2 π ∊ κ)}{κ \sqrt{1 + ∊^{2} + ∊^{4} + ∊^{6} + ∊^{8} + ∊^{10} + ∊^{12}}}$

ϵ	0	0.05	0.1	0.15	0.2	0.25	0.35	0.5	0.75
$⟨ a_{2}^{2} ⟩, ⟨ a_{3}^{2} ⟩$	0.44888	0.45000	0.45336	0.45893	0.46670	0.47660	0.50260	0.55610	0.67917
$⟨ a_{4}^{2} ⟩$	0.02322	0.02302	0.02245	0.02153	0.02033	0.01891	0.01560	0.01028	0.00292
$⟨ a_{5}^{2} ⟩, ⟨ a_{6}^{2} ⟩$	0.02322	0.02328	0.02345	0.02375	0.02418	0.02475	0.02635	0.02997	0.03888
$⟨ a_{7}^{2} ⟩, ⟨ a_{8}^{2} ⟩$	0.00619	0.00621	0.00624	0.00628	0.00630	0.00626	0.00593	0.00468	0.00162
$⟨ a_{9}^{2} ⟩, ⟨ a_{10}^{2} ⟩$	0.00619	0.00621	0.00625	0.00633	0.00645	0.00660	0.00705	0.00815	0.01132
$⟨ a_{11}^{2} ⟩$	0.00245	0.00241	0.00227	0.00207	0.00183	0.00158	0.00109	0.00052	0.00007
$⟨ a_{12}^{2} ⟩, ⟨ a_{13}^{2} ⟩$	0.00245	0.00246	0.00248	0.00251	0.00255	0.00260	0.00266	0.00247	0.00108
$⟨ a_{14}^{2} ⟩, ⟨ a_{15}^{2} ⟩$	0.00245	0.00246	0.00248	0.00251	0.00256	0.00262	0.00280	0.00326	0.00480

ϵ	0	0.05	0.1	0.15	0.2	0.25	0.35	0.5	0.75
$⟨ a_{2} a_{8} ⟩, ⟨ a_{3} a_{7} ⟩$	$- 0.01416$	$- 0.01420$	$- 0.01430$	$- 0.01444$	$- 0.01460$	$- 0.01474$	$- 0.01481$	$- 0.01392$	$- 0.00885$
$⟨ a_{4} a_{11} ⟩$	$- 0.00388$	$- 0.00381$	$- 0.00363$	$- 0.00334$	$- 0.00300$	$- 0.00262$	$- 0.00186$	$- 0.00091$	$- 0.00012$
$⟨ a_{4} a_{22} ⟩$	0.00032	0.00030	0.00024	0.00017	0.00009	0.00002	$- 0.00008$	$- 0.00011$	$- 0.00003$
$⟨ a_{5} a_{13} ⟩, ⟨ a_{6} a_{12} ⟩$	$- 0.00388$	$- 0.00389$	$- 0.00392$	$- 0.00397$	$- 0.00404$	$- 0.00412$	$- 0.00433$	$- 0.00451$	$- 0.00341$
$⟨ a_{7} a_{17} ⟩, ⟨ a_{8} a_{16} ⟩$	$- 0.00156$	$- 0.00158$	$- 0.00164$	$- 0.00172$	$- 0.00178$	$- 0.00173$	$- 0.00095$	0.00365	0.01180
$⟨ a_{11} a_{22} ⟩$	$- 0.00076$	$- 0.00073$	$- 0.00067$	$- 0.00058$	$- 0.00048$	$- 0.00038$	$- 0.00022$	$- 0.00007$	0.00000
$⟨ a_{9} a_{19} ⟩, ⟨ a_{10} a_{18} ⟩$	$- 0.00156$	$- 0.00156$	$- 0.00157$	$- 0.00159$	$- 0.00162$	$- 0.00166$	$- 0.00176$	$- 0.00196$	$- 0.00177$
$⟨ a_{14} a_{26} ⟩, ⟨ a_{15} a_{25} ⟩$	$- 0.00076$	$- 0.00076$	$- 0.00077$	$- 0.00078$	$- 0.00079$	$- 0.00081$	$- 0.00086$	$- 0.00099$	$- 0.00104$

ϵ	0	0.25	0.5	0.75
$Δ_{1}$	1.0324	1.0870	1.2461	1.5010
$Δ_{2}$	0.5835	0.6104	0.6900	0.8218
$Δ_{3}$	0.1347	0.1338	0.1339	0.1426
$Δ_{4}$	0.1115	0.1149	0.1236	0.1397
$Δ_{5}$	0.0882	0.0902	0.0937	0.1008
$Δ_{6}$	0.0650	0.0654	0.0637	0.0620
$Δ_{7}$	0.0588	0.0591	0.0590	0.0603
$Δ_{8}$	0.0526	0.0529	0.0543	0.0587
$Δ_{9}$	0.0464	0.0463	0.0462	0.0474
$Δ_{10}$	0.0402	0.0397	0.0380	0.0361
$Δ_{11}$	0.0378	0.0381	0.0375	0.0360
$Δ_{12}$	0.0353	0.0355	0.0350	0.0349
$Δ_{13}$	0.0329	0.0329	0.0325	0.0338
$Δ_{14}$	0.0304	0.0303	0.0293	0.0290
$Δ_{15}$	0.0280	0.0277	0.0260	0.0242
$Δ_{16}$	0.0268	0.0264	0.0244	0.0218
$Δ_{17}$	0.0256	0.0251	0.0229	0.0193
$Δ_{18}$	0.0244	0.0239	0.0215	0.0186
$Δ_{19}$	0.0232	0.0226	0.0201	0.0178
$Δ_{20}$	0.0220	0.0213	0.0185	0.0153
$Δ_{21}$	0.0208	0.0200	0.0169	0.0129
$Δ_{22}$	0.0202	0.0197	0.0168	0.0129
$Δ_{23}$	0.0195	0.0190	0.0159	0.0115
$Δ_{24}$	0.0189	0.0183	0.0151	0.0101
$Δ_{25}$	0.0182	0.0176	0.0142	0.0095
$Δ_{26}$	0.0176	0.0169	0.0134	0.0089
$Δ_{27}$	0.0169	0.0162	0.0125	0.0075
$Δ_{28}$	0.0162	0.0155	0.0117	0.0061

Mode	Shape Function
$d_{2}, d_{3}$	$8 ν^{2}$
$d_{4}$	$48 ν^{2} {(1 - ν)}^{2}$
$d_{5}, d_{6}$	$24 ν^{2} {(1 - ν)}^{2}$
$d_{7}, d_{8}$	$16 ν^{2} (39 ν^{4} - 84 ν^{3} + 66 ν^{2} - 24 ν + 4)$
$d_{9}, d_{10}$	$16 ν^{2} [ν^{4} + 3 {(1 - ν)}^{4}]$
$d_{11}$	$240 ν^{2} {(1 - ν)}^{2} (34 ν^{4} - 56 ν^{3} + 32 ν^{2} - 8 ν + 1)$
$d_{12}, d_{13}$	$(40 / 3) ν^{2} {(1 - ν)}^{2} (298 ν^{4} - 488 ν^{3} + 280 ν^{2} - 72 ν + 9)$
$d_{14}, d_{15}$	$80 ν^{2} {(1 - ν)}^{2} [2 ν^{4} + {(1 - ν)}^{4}]$
$d_{16}, d_{17}$	$8 ν^{2} (7265 ν^{8} - 31,220 ν^{7} + 57,750 ν^{6} - 60,020 ν^{5} + 38,087 ν^{4} - 14,892 ν^{3} + 3438 ν^{2} - 432 ν + 27)$
$d_{18}, d_{19}$	$2 ν^{2} (10,135 ν^{8} - 47,530 ν^{7} + 98,745 ν^{6} - 116,380 ν^{5} + 83,053 ν^{4} - 35,994 ν^{3} + 9231 ν^{2} - 1344 ν + 96)$
$d_{20}, d_{21}$	$8 ν^{2} [3 ν^{8} + 100 ν^{4} {(1 - ν)}^{4} + 15 {(1 - ν)}^{8}]$
$d_{22}$	$(56 / 9) ν^{2} {(1 - ν)}^{2} (144,925 ν^{8} - 500,500 ν^{7} + 747,240 ν^{6} - 636,940 ν^{5} + 338,245 ν^{4} - 111,960 ν^{3} + 21,690 ν^{2} - 2160 ν + 108)$
$d_{23}, d_{24}$	$(7 / 3) ν^{2} {(1 - ν)}^{2} (208,755 ν^{8} - 725,580 ν^{7} + 1,082,970 ν^{6} - 914,220 ν^{5} + 476,435 ν^{4} - 153,760 ν^{3} + 29,000 ν^{2} - 2880 ν + 144)$
$d_{25}, d_{26}$	$(56 / 5) ν^{2} {(1 - ν)}^{2} (9702 ν^{8} - 33,552 ν^{7} + 55,240 ν^{6} - 56,000 ν^{5} + 36,070 ν^{4} - 14,152 ν^{3} + 3192 ν^{2} - 400 ν + 25)$
$d_{27}, d_{28}$	$56 ν^{2} {(1 - ν)}^{2} [9 ν^{8} + 50 ν^{4} {(1 - ν)}^{4} + 3 {(1 - ν)}^{8}]$
$c_{2, 8}, c_{3, 7}$	$8 \sqrt{2} ν^{2} (6 ν^{2} - 6 ν + 1)$
$c_{2, 16}, c_{3, 17}$	$8 \sqrt{3} ν^{2} (45 ν^{4} - 90 ν^{3} + 61 ν^{2} - 16 ν + 1)$
$c_{5, 13}, c_{6, 12}$	$8 \sqrt{15} ν^{2} {(1 - ν)}^{2} (10 ν^{2} - 8 ν + 1)$
$c_{5, 23}, c_{6, 24}$	$8 \sqrt{21} ν^{2} {(1 - ν)}^{2} (95 ν^{4} - 160 ν^{3} + 90 ν^{2} - 20 ν + 1)$
$c_{9, 19}, c_{10, 18}$	$8 \sqrt{6} ν^{2} (60 ν^{6} - 230 ν^{5} + 399 ν^{4} - 372 ν^{3} + 183 ν^{2} - 42 ν + 3)$
$c_{11, 22}$	$8 \sqrt{35} ν^{2} {(1 - ν)}^{2} (1799 ν^{6} - 4564 ν^{5} + 4690 ν^{4} - 2560 ν^{3} + 785 ν^{2} - 120 ν + 3)$
$c_{12, 24}, c_{13, 23}$	$(4 / 3) \sqrt{35} ν^{2} {(1 - ν)}^{2} (5607 ν^{6} - 14,292 ν^{5} + 14,660 ν^{4} - 7900 ν^{3} + 2375 ν^{2} - 360 ν + 18)$
$c_{14, 26}, c_{15, 25}$	$16 \sqrt{35} ν^{2} {(1 - ν)}^{2} (56 ν^{6} - 136 ν^{5} + 185 ν^{4} - 160 ν^{3} + 75 ν^{2} - 16 ν + 1)$

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