Study on the orthogonality property of Zernike modes in light beams undergoing free space propagation

Akanshu Chauhan; Bosanta R. Boruah

doi:10.1364/JOSAA.481741

Journal of the Optical Society of America A
Vol. 40,
Issue 5,
pp. 961-968
(2023)
•https://doi.org/10.1364/JOSAA.481741

Study on the orthogonality property of Zernike modes in light beams undergoing free space propagation

Akanshu Chauhan and Bosanta R. Boruah

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Akanshu Chauhan and Bosanta R. Boruah, "Study on the orthogonality property of Zernike modes in light beams undergoing free space propagation," J. Opt. Soc. Am. A 40, 961-968 (2023)

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Table of Contents Category
- Scattering and Propagation
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About this Article
History
- Original Manuscript: November 24, 2022
- Revised Manuscript: March 22, 2023
- Manuscript Accepted: March 31, 2023
- Published: April 21, 2023

Abstract

This paper presents a study to investigate how the orthogonality property of Zernike modes gets modified as a light beam carrying the mode in its phase propagates through free space. We use a numerical simulation using scalar diffraction theory to generate propagated light beams carrying the commonly encountered Zernike modes. We present our results in terms of the inner product and orthogonality contrast matrix at propagation distances ranging from near field to far field regions. Our study will help in the understanding of how far the Zernike modes describing the phase profile of a light beam in a given plane remain approximately orthogonal to one another on propagation.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	$Z_{4}$	$Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$	$Z_{9}$	$Z_{10}$	$Z_{11}$
$Z_{4}$	0.9865	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0260
$Z_{5}$	0.0000	0.9947	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
$Z_{6}$	0.0000	0.0000	0.9781	0.0000	0.0000	0.0000	0.0000	0.0000
$Z_{7}$	0.0000	0.0000	0.0000	0.9799	0.0000	0.0107	0.0000	0.0000
$Z_{8}$	0.0000	0.0000	0.0000	0.0000	0.9799	0.0000	0.0107	0.0000
$Z_{9}$	0.0000	0.0000	0.0000	0.0107	0.0000	0.9796	0.0000	0.0000
$Z_{10}$	0.0000	0.0000	0.0000	0.0000	0.0107	0.0000	0.9796	0.0000
$Z_{11}$	0.0260	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.9737

	$Z_{4}^{p}$	$Z_{5}^{p}$	$Z_{6}^{p}$	$Z_{7}^{p}$	$Z_{8}^{p}$	$Z_{9}^{p}$	$Z_{10}^{p}$	$Z_{11}^{p}$
$Z_{4}^{p}$	0.4887	0.0294	0.0433	0.5878	0.5864	0.0490	0.0490	0.1853
$Z_{5}^{p}$	0.0294	0.7385	0.0369	0.0359	0.0351	0.0385	0.0385	0.0971
$Z_{6}^{p}$	0.0433	0.0369	0.7233	0.3080	0.2076	0.0446	0.0284	0.0920
$Z_{7}^{p}$	0.5878	0.0359	0.3080	1.2326	0.5799	0.0376	0.0670	0.2030
$Z_{8}^{p}$	0.5864	0.0351	0.2076	0.5799	1.2229	0.0681	0.1159	0.2031
$Z_{9}^{p}$	0.0490	0.0385	0.0446	0.0376	0.0681	0.5207	0.0408	0.0815
$Z_{10}^{p}$	0.0490	0.0385	0.0284	0.0670	0.1159	0.0408	0.5207	0.0815
$Z_{11}^{p}$	0.1853	0.0971	0.0920	0.2030	0.2031	0.0815	0.0815	0.3377

	$Z_{4}^{p}$	$Z_{5}^{p}$	$Z_{6}^{p}$	$Z_{7}^{p}$	$Z_{8}^{p}$	$Z_{9}^{p}$	$Z_{10}^{p}$	$Z_{11}^{p}$
$Z_{4}^{p}$	0.0040	0.0220	0.0194	0.0216	0.0216	0.0172	0.0172	0.0205
$Z_{5}^{p}$	0.0220	0.2707	0.1841	0.1883	0.1883	0.1672	0.1672	0.1891
$Z_{6}^{p}$	0.0194	0.1841	0.2462	0.1788	0.1727	0.1575	0.1582	0.1776
$Z_{7}^{p}$	0.0216	0.1883	0.1788	0.2001	0.1794	0.1573	0.1590	0.1802
$Z_{8}^{p}$	0.0216	0.1883	0.1727	0.1794	0.1998	0.1590	0.1603	0.1801
$Z_{9}^{p}$	0.0172	0.1672	0.1575	0.1573	0.1590	0.1456	0.1431	0.1608
$Z_{10}^{p}$	0.0172	0.1672	0.1582	0.1590	0.1603	0.1431	0.1456	0.1608
$Z_{11}^{p}$	0.0205	0.1891	0.1776	0.1802	0.1801	0.1608	0.1608	0.1815

	$Z_{4}$	$Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$	$Z_{9}$	$Z_{10}$	$Z_{11}$
$Z_{4}$	0.9865	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0260
$Z_{5}$	0.0000	0.9947	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
$Z_{6}$	0.0000	0.0000	0.9781	0.0000	0.0000	0.0000	0.0000	0.0000
$Z_{7}$	0.0000	0.0000	0.0000	0.9799	0.0000	0.0107	0.0000	0.0000
$Z_{8}$	0.0000	0.0000	0.0000	0.0000	0.9799	0.0000	0.0107	0.0000
$Z_{9}$	0.0000	0.0000	0.0000	0.0107	0.0000	0.9796	0.0000	0.0000
$Z_{10}$	0.0000	0.0000	0.0000	0.0000	0.0107	0.0000	0.9796	0.0000
$Z_{11}$	0.0260	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.9737

	$Z_{4}^{p}$	$Z_{5}^{p}$	$Z_{6}^{p}$	$Z_{7}^{p}$	$Z_{8}^{p}$	$Z_{9}^{p}$	$Z_{10}^{p}$	$Z_{11}^{p}$
$Z_{4}^{p}$	0.4887	0.0294	0.0433	0.5878	0.5864	0.0490	0.0490	0.1853
$Z_{5}^{p}$	0.0294	0.7385	0.0369	0.0359	0.0351	0.0385	0.0385	0.0971
$Z_{6}^{p}$	0.0433	0.0369	0.7233	0.3080	0.2076	0.0446	0.0284	0.0920
$Z_{7}^{p}$	0.5878	0.0359	0.3080	1.2326	0.5799	0.0376	0.0670	0.2030
$Z_{8}^{p}$	0.5864	0.0351	0.2076	0.5799	1.2229	0.0681	0.1159	0.2031
$Z_{9}^{p}$	0.0490	0.0385	0.0446	0.0376	0.0681	0.5207	0.0408	0.0815
$Z_{10}^{p}$	0.0490	0.0385	0.0284	0.0670	0.1159	0.0408	0.5207	0.0815
$Z_{11}^{p}$	0.1853	0.0971	0.0920	0.2030	0.2031	0.0815	0.0815	0.3377

Abstract

Data availability

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Tables (3)

Equations (13)

Journal of the Optical Society of America A