Unified analytical method for Zernike coefficient transformation of scaled, rotated, and translated pupils based on Shack’s vector multiplication

Yongfeng Zhang; Yongfeng Zhang; Shengqian Wang; Shengqian Wang; Hao Xian; Hao Xian; Hao Xian; Changhui Rao; Changhui Rao; Changhui Rao

doi:10.1364/JOSAA.427851

Journal of the Optical Society of America A
Vol. 38,
Issue 8,
pp. 1131-1139
(2021)
•https://doi.org/10.1364/JOSAA.427851

Unified analytical method for Zernike coefficient transformation of scaled, rotated, and translated pupils based on Shack’s vector multiplication

Yongfeng Zhang, Shengqian Wang, Hao Xian, and Changhui Rao

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Yongfeng Zhang, Shengqian Wang, Hao Xian, and Changhui Rao, "Unified analytical method for Zernike coefficient transformation of scaled, rotated, and translated pupils based on Shack’s vector multiplication," J. Opt. Soc. Am. A 38, 1131-1139 (2021)

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- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: April 14, 2021
- Revised Manuscript: May 26, 2021
- Manuscript Accepted: June 12, 2021
- Published: July 12, 2021

Abstract

Zernike polynomials play an essential role in characterizing and analyzing wavefront aberrations. Transformation of weighted coefficients for Zernike modes is required when pupil scaling, rotation, and/or translation exist. Here, a novel method based on Shack’s vector multiplication is first proposed to derive the transformation relation. The derived modes resulting from pupil scaling, rotation, and/or translation for each individual mode are easily indicated via this method; thus, the effect of each kind of pupil change could be studied qualitatively and quantitatively. Its remarkable computational efficiency against the direct integral is demonstrated by simulation. The method introduced here provides a generalized methodology to analyze the relationship between weighted coefficients for different description basis sets.

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Supplementary Material (2)

Name	Description
Code 1	MATLAB code for analytical Zernike Coefficient Transformation based on Shack's vector multiplication.
Supplement 1	The Zernike coefficient transformation results corresponding to the Zernike modes with radial order of 5, 6, and 7.

Data Availability

No data were generated or analyzed in the presented research.

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Maximum Radial Order	Elapsed Time for the Symbolic Integration Method (without PCT Started)/Second	Elapsed Time for the Symbolic Integration Method (with PCT Started)/Second	Elapsed Time for Our Analytical Result/Second
0	0.143544	0.955037	0.344698
1	2.153103	$2.295685 (↑ 6.62 %)$	$0.722383 (↓ 66.45 %)$
2	10.815702	$6.14013 (↓ 43.23 %)$	$3.01902 (↓ 72.09 %)$
3	34.113471	$14.530232 (↓ 57.41 %)$	$6.267011 (↓ 81.63 %)$
4	115.158416	$40.663674 (↓ 64.69 %)$	$14.745565 (↓ 87.20 %)$
5	409.950399	$114.142605 (↓ 72.16 %)$	$29.138766 (↓ 92.89 %)$
6	1759.286249	$502.953913 (↓ 71.41 %)$	$59.740078 (↓ 96.60 %)$
7	6868.977223	$1489.111141 (↓ 78.32 %)$	$109.419175 (↓ 98.41 %)$

$n$	$m$	Derived Mode	Weighted Coefficient
0	0	$Z_{0}^{0}$	$a_{0}^{0}$
1	1	$Z_{0}^{0}$	${\vec{a}}_{1}^{1} \cdot {\vec{d}}_{R_{1}}$
		${\vec{Z}}_{1}^{1}$	$s {\vec{a}}_{1}^{1} {\vec{e}}_{α}^{*}$
2	0	$Z_{2}^{0}$	$s^{2} a_{2}^{0}$
		${\vec{Z}}_{1}^{1}$	$4 a_{2}^{0} s {\vec{e}}_{α}^{*} {\vec{d}}_{R_{1}}$
		$Z_{0}^{0}$	$2 a_{2}^{0} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) + a_{2}^{0} s^{2} - a_{2}^{0}$
2	2	${\vec{Z}}_{2}^{2}$	$s^{2} {\vec{a}}_{2}^{2} {\vec{e}}_{α}^{* 2}$
		${\vec{Z}}_{1}^{1}$	$2 s {\vec{a}}_{2}^{2} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{}$
		$Z_{0}^{0}$	${\vec{a}}_{2}^{2} \cdot {\vec{d}}_{R_{1}}^{2}$
3	1	$Z_{0}^{0}$	$3 ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) ({\vec{a}}_{3}^{1} \cdot {\vec{d}}_{R_{1}}) + 3 s^{2} ({\vec{a}}_{3}^{1} \cdot {\vec{d}}_{R_{1}}) - 2 ({\vec{a}}_{3}^{1} \cdot {\vec{d}}_{R_{1}})$
		${\vec{Z}}_{1}^{1}$	$2 s^{3} {\vec{a}}_{3}^{1} {\vec{e}}_{α}^{} + 6 s ({\vec{a}}_{3}^{1} \cdot {\vec{d}}_{R_{1}}) {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}} + 3 s ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) {\vec{a}}_{3}^{1} {\vec{e}}_{α}^{} - 2 s {\vec{a}}_{3}^{1} {\vec{e}}_{α}^{}$
		$Z_{2}^{0}$	$3 s^{2} ({\vec{a}}_{3}^{1} \cdot {\vec{d}}_{R_{1}})$
		${\vec{Z}}_{2}^{2}$	$3 s^{2} {\vec{a}}_{3}^{1} {\vec{e}}_{α}^{* 2} {\vec{d}}_{R_{1}}$
		${\vec{Z}}_{3}^{1}$	$s^{3} {\vec{a}}_{3}^{1} {\vec{e}}_{α}^{*}$
3	3	$Z_{0}^{0}$	${\vec{a}}_{3}^{3} \cdot {\vec{d}}_{R_{1}}^{3}$
		${\vec{Z}}_{1}^{1}$	$3 s {\vec{a}}_{3}^{3} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{ 2}$
		${\vec{Z}}_{2}^{2}$	$3 s^{2} {\vec{a}}_{3}^{3} {\vec{e}}_{α}^{* 2} {\vec{d}}_{R_{1}}^{*}$
		${\vec{Z}}_{3}^{3}$	$s^{3} {\vec{a}}_{3}^{3} {\vec{e}}_{α}^{* 3}$
4	0	$Z_{0}^{0}$	$6 a_{4}^{0} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}})^{2} - 6 a_{4}^{0} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) + a_{4}^{0} + 2 s^{4} a_{4}^{0} + 3 s^{2} a_{4}^{0} + 12 s^{2} a_{4}^{0} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}})$
		${\vec{Z}}_{1}^{1}$	$24 s a_{4}^{0} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) {\vec{d}}_{R_{1}} {\vec{e}}_{α}^{} - 12 s a_{4}^{0} {\vec{d}}_{R_{1}} {\vec{e}}_{α}^{} + 16 s^{3} a_{4}^{0} {\vec{d}}_{R_{1}} {\vec{e}}_{α}^{*}$
		$Z_{2}^{0}$	$3 s^{4} a_{4}^{0} + 12 s^{2} a_{4}^{0} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) - 3 s^{2} a_{4}^{0}$
		${\vec{Z}}_{2}^{2}$	$12 s^{2} a_{4}^{0} {\vec{d}}_{R_{1}}^{2} {\vec{e}}_{α}^{* 2}$
		${\vec{Z}}_{3}^{1}$	$8 s^{3} a_{4}^{0} {\vec{d}}_{R_{1}} {\vec{e}}_{α}^{*}$
		$Z_{4}^{0}$	$s^{4} a_{4}^{0}$
4	2	$Z_{0}^{0}$	$4 ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) ({\vec{a}}_{4}^{2} \cdot {\vec{d}}_{R_{1}}^{2}) - 3 {\vec{a}}_{4}^{2} \cdot {\vec{d}}_{R_{1}}^{2} + 6 s^{2} ({\vec{a}}_{4}^{2} \cdot {\vec{d}}_{R_{1}}^{2})$
		${\vec{Z}}_{1}^{1}$	$8 s ({\vec{a}}_{4}^{2} \cdot {\vec{d}}_{R_{1}}^{2}) {\vec{d}}_{R_{1}} {\vec{e}}_{α}^{} + 8 s ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{} - 6 s {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{} + 8 s^{3} {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{*}$
		$Z_{2}^{0}$	$6 s^{2} ({\vec{a}}_{4}^{2} \cdot {\vec{d}}_{R_{1}}^{2})$
		${\vec{Z}}_{2}^{2}$	$12 s^{2} ({\vec{d}}_{R_{1}} \cdot {\vec{d}}_{R_{1}}) {\vec{e}}_{α}^{* 2} {\vec{a}}_{4}^{2} + 3 s^{4} {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{* 2} - 3 s^{2} {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{* 2}$
		${\vec{Z}}_{3}^{1}$	$4 s^{3} {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{}$
		${\vec{Z}}_{3}^{3}$	$4 s^{3} {\vec{d}}_{R_{1}} {\vec{e}}_{α}^{* 3} {\vec{a}}_{4}^{2}$
		${\vec{Z}}_{4}^{2}$	$s^{4} {\vec{a}}_{4}^{2} {\vec{e}}_{α}^{* 2}$
4	4	$Z_{0}^{0}$	${\vec{a}}_{4}^{4} \cdot {\vec{d}}_{R_{1}}^{4}$
		${\vec{Z}}_{1}^{1}$	$4 s {\vec{a}}_{4}^{4} {\vec{e}}_{α}^{} {\vec{d}}_{R_{1}}^{ 3}$
		${\vec{Z}}_{2}^{2}$	$6 s^{2} {\vec{a}}_{4}^{4} {\vec{e}}_{α}^{* 2} {\vec{d}}_{n}^{* 2}$
		${\vec{Z}}_{3}^{3}$	$4 s^{3} {\vec{a}}_{4}^{4} {\vec{e}}_{α}^{* 3} {\vec{d}}_{R_{1}}^{*}$
		${\vec{Z}}_{4}^{4}$	$s^{4} {\vec{a}}_{4}^{4} {\vec{e}}_{α}^{* 4}$

$n$	$m$	$Z_{n}^{m}$	$n$	$m$	$Z_{n}^{m}$
0	0	1	5	−3	$(5 ρ^{5} - 4 ρ^{3}) \sin 3 θ$
1	1	$ρ \cos θ$	5	5	$ρ^{5} \cos 5 θ$
1	−1	$ρ \sin θ$	5	−5	$ρ^{5} \sin 5 θ$
2	0	$2 ρ^{2} - 1$	6	0	$20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1$
2	2	$ρ^{2} \cos 2 θ$	6	2	$(15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) \cos 2 θ$
2	−2	$ρ^{2} \sin 2 θ$	6	−2	$(15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) \sin 2 θ$
3	1	$(3 ρ^{3} - 2 ρ) \cos θ$	6	4	$(6 ρ^{6} - 5 ρ^{4}) \cos 4 θ$
3	−1	$(3 ρ^{3} - 2 ρ) \sin θ$	6	−4	$(6 ρ^{6} - 5 ρ^{4}) \sin 4 θ$
3	3	$ρ^{3} \cos 3 θ$	6	6	$ρ^{6} \cos 6 θ$
3	−3	$ρ^{3} \sin 3 θ$	6	−6	$ρ^{6} \sin 6 θ$
4	0	$6 ρ^{4} - 6 ρ^{2} + 1$	7	1	$(35 ρ^{7} - 60 ρ^{5} + 30 ρ^{3} - 4 ρ) \cos θ$
4	2	$(4 ρ^{4} - 3 ρ^{2}) \cos 2 θ$	7	−1	$(35 ρ^{7} - 60 ρ^{5} + 30 ρ^{3} - 4 ρ) \sin θ$
4	−2	$(4 ρ^{4} - 3 ρ^{2}) \sin 2 θ$	7	3	$(21 ρ^{7} - 30 ρ^{5} + 10 ρ^{3}) \cos 3 θ$
4	4	$ρ^{4} \cos 4 θ$	7	−3	$(21 ρ^{7} - 30 ρ^{5} + 10 ρ^{3}) \sin 3 θ$
4	−4	$ρ^{4} \sin 4 θ$	7	5	$(7 ρ^{7} - 6 ρ^{5}) \cos 5 θ$
5	1	$(10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos θ$	7	−5	$(7 ρ^{7} - 6 ρ^{5}) \sin 5 θ$
5	−1	$(10 ρ^{5} - 12 ρ^{3} + 3 ρ) \sin θ$	7	7	$ρ^{7} \cos 7 θ$
5	3	$(5 ρ^{5} - 4 ρ^{3}) \cos 3 θ$	7	−7	$ρ^{7} \sin 7 θ$

$n$	$m$	${\vec{Z}}_{n}^{m} (\vec{ρ})$ or $Z_{n}^{m} (\vec{ρ})$	$n$	$m$	${\vec{Z}}_{n}^{m} (\vec{ρ})$ or $Z_{n}^{m} (\vec{ρ})$
0	0	1	5	3	$5 (\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{3} - 4 {\vec{ρ}}^{3}$
1	1	$\vec{ρ}$	5	5	${\vec{ρ}}^{5}$
2	0	$2 (\vec{ρ} \cdot \vec{ρ}) - 1$	6	0	$20 (\vec{ρ} \cdot \vec{ρ})^{3} - 30 (\vec{ρ} \cdot \vec{ρ})^{2} + 12 (\vec{ρ} \cdot \vec{ρ}) - 1$
2	2	${\vec{ρ}}^{2}$	6	2	$15 (\vec{ρ} \cdot \vec{ρ})^{2} {\vec{ρ}}^{2} - 20 (\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{2} + 6 {\vec{ρ}}^{2}$
3	1	$3 (\vec{ρ} \cdot \vec{ρ}) \vec{ρ} - 2 \vec{ρ}$	6	4	$6 (\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{4} - 5 {\vec{ρ}}^{4}$
3	3	${\vec{ρ}}^{3}$	6	6	${\vec{ρ}}^{6}$
4	0	$6 (\vec{ρ} \cdot \vec{ρ})^{2} - 6 (\vec{ρ} \cdot \vec{ρ}) + 1$	7	1	$35 (\vec{ρ} \cdot \vec{ρ})^{3} \vec{ρ} - 60 (\vec{ρ} \cdot \vec{ρ})^{2} \vec{ρ} + 30 (\vec{ρ} \cdot \vec{ρ}) \vec{ρ} - 4 \vec{ρ}$
4	2	$4 (\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{2} - 3 {\vec{ρ}}^{2}$	7	3	$21 (\vec{ρ} \cdot \vec{ρ})^{2} {\vec{ρ}}^{3} - 30 (\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{3} + 10 {\vec{ρ}}^{3}$
4	4	${\vec{ρ}}^{4}$	7	5	$7 (\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{5} - 6 {\vec{ρ}}^{5}$
5	1	$10 (\vec{ρ} \cdot \vec{ρ})^{2} \vec{ρ} - 12 (\vec{ρ} \cdot \vec{ρ}) \vec{ρ} + 3 \vec{ρ}$	7	7	${\vec{ρ}}^{7}$

Item	Zernike Representation	Item	Zernike Representation
1	${\vec{Z}}_{0}^{0} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{3}$	$\frac{1}{5} {\vec{Z}}_{5}^{3} (\vec{ρ}) + \frac{4}{5} {\vec{Z}}_{3}^{3} (\vec{ρ})$
$\vec{ρ}$	${\vec{Z}}_{1}^{1} (\vec{ρ})$	${\vec{ρ}}^{5}$	${\vec{Z}}_{5}^{5} (\vec{ρ})$
$\vec{ρ} \cdot \vec{ρ}$	$\frac{1}{2} {\vec{Z}}_{2}^{0} (\vec{ρ}) + \frac{1}{2} {\vec{Z}}_{0}^{0} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ})^{3}$	$\frac{1}{20} {\vec{Z}}_{6}^{0} (\vec{ρ}) + \frac{1}{4} {\vec{Z}}_{4}^{0} (\vec{ρ}) + \frac{9}{20} {\vec{Z}}_{2}^{0} (\vec{ρ}) + \frac{1}{4} {\vec{Z}}_{0}^{0} (\vec{ρ})$
${\vec{ρ}}^{2}$	${\vec{Z}}_{2}^{2} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ})^{2} {\vec{ρ}}^{2}$	$\frac{1}{15} {\vec{Z}}_{6}^{2} (\vec{ρ}) + \frac{1}{3} {\vec{Z}}_{4}^{2} (\vec{ρ}) + \frac{3}{5} {\vec{Z}}_{2}^{2} (\vec{ρ})$
$(\vec{ρ} \cdot \vec{ρ}) \vec{ρ}$	$\frac{1}{3} {\vec{Z}}_{3}^{1} (\vec{ρ}) + \frac{2}{3} {\vec{Z}}_{1}^{1} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{4}$	$\frac{1}{6} {\vec{Z}}_{6}^{4} (\vec{ρ}) + \frac{5}{6} {\vec{Z}}_{4}^{4} (\vec{ρ})$
${\vec{ρ}}^{3}$	${\vec{Z}}_{3}^{3} (\vec{ρ})$	${\vec{ρ}}^{6}$	${\vec{Z}}_{6}^{6} (\vec{ρ})$
$(\vec{ρ} \cdot \vec{ρ})^{2}$	$\frac{1}{6} {\vec{Z}}_{4}^{0} (\vec{ρ}) + \frac{1}{2} {\vec{Z}}_{2}^{0} (\vec{ρ}) + \frac{1}{3} {\vec{Z}}_{0}^{0} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ})^{3} \vec{ρ}$	$\frac{1}{35} {\vec{Z}}_{7}^{1} (\vec{ρ}) + \frac{6}{35} {\vec{Z}}_{5}^{1} (\vec{ρ}) + \frac{2}{5} {\vec{Z}}_{3}^{1} (\vec{ρ}) + \frac{2}{5} {\vec{Z}}_{1}^{1} (\vec{ρ})$
$(\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{2}$	$\frac{1}{4} {\vec{Z}}_{4}^{2} (\vec{ρ}) + \frac{3}{4} {\vec{Z}}_{2}^{2} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ})^{2} {\vec{ρ}}^{3}$	$\frac{1}{21} {\vec{Z}}_{7}^{3} (\vec{ρ}) + \frac{2}{7} {\vec{Z}}_{5}^{3} (\vec{ρ}) + \frac{2}{3} {\vec{Z}}_{3}^{3} (\vec{ρ})$
${\vec{ρ}}^{4}$	${\vec{Z}}_{4}^{4} (\vec{ρ})$	$(\vec{ρ} \cdot \vec{ρ}) {\vec{ρ}}^{5}$	$\frac{1}{7} {\vec{Z}}_{7}^{5} (\vec{ρ}) + \frac{6}{7} {\vec{Z}}_{5}^{5} (\vec{ρ})$
$(\vec{ρ} \cdot \vec{ρ})^{2} \vec{ρ}$	$\frac{1}{10} {\vec{Z}}_{5}^{1} (\vec{ρ}) + \frac{2}{5} {\vec{Z}}_{3}^{1} (\vec{ρ}) + \frac{1}{2} {\vec{Z}}_{1}^{1} (\vec{ρ})$	${\vec{ρ}}^{7}$	${\vec{Z}}_{7}^{7} (\vec{ρ})$

Abstract

Supplementary Material (2)

Data Availability

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Tables (5)

Equations (49)

Journal of the Optical Society of America A