What should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with a 4f system: discussion

Justo Arines

doi:10.1364/JOSAA.414521

Journal of the Optical Society of America A
Vol. 38,
Issue 3,
pp. 437-439
(2021)
•https://doi.org/10.1364/JOSAA.414521

What should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with a 4f system: discussion

Justo Arines

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Justo Arines, "What should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with a 4f system: discussion," J. Opt. Soc. Am. A 38, 437-439 (2021)

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Abstract

The aim of this paper is to describe and demonstrate what should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with a ${4}f$ system. I provide theoretical and experimental evidence. The experimental setup consisted of two ${4}f$ systems of magnifications 1 and 1/3 with their corresponding wavefront sensors at their ends. Spherical and cylindrical trial lenses were measured. In addition, I measured a phase plate with high-order aberrations. I show that the Zernike coefficients of the wavefront expansion at two planes conjugated by a ${4}f$ system are related independently of the magnification of the ${4}f$ system by the following equation: ${b_i} = {(- 1)^{n}}{a_i}$, with $n$ being the order of the radial Zernike polynomial.

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Zernike	Arm 2: HASO ( $d i a m = 1. 5 m m$ )		Arm 3: WFS30-14AR ( $d i a m = 4. 5 m m$ )
Polyn.	2, −2	2,0	2, −2	2,0
$+ 1$ ESF		$0.77 \pm 0.02$		$0.72 \pm 0.02$
$+ 2.5$ ESF		$1.80 \pm 0.05$		$1.74 \pm 0.05$
−1 ESF		$- 0.73 \pm 0.02$		$- 0.73 \pm 0.02$
−2.5 ESF		$- 1.89 \pm 0.05$		$- 1.80 \pm 0.05$
$+ 1$ CIL	$- 0.48 \pm 0.02$	$0.36 \pm 0.02$	$- 0.48 \pm 0.02$	$0.36 \pm 0.02$
−1 CIL	$0.52 \pm 0.02$	$- 0.42 \pm 0.02$	$0.49 \pm 0.02$	$- 0.41 \pm 0.02$

	Theoretical	Arm 2: HASO ( $d i a m = 1. 5 m m$ )	Arm 3: WFS30-14AR ( $d i a m = 4. 5 m m$ )
$+ 1$ ESF	−1.000	$- 0.107 \pm 0.004$	$- 1.005 \pm 0.004$
$+ 2.5$ ESF	−0.400	$- 0.045 \pm 0.004$	$- 0.419 \pm 0.004$
−1 ESF	1.000	$0.110 \pm 0.004$	$1.025 \pm 0.004$
−2.5 ESF	0.400	$0.043 \pm 0.004$	$0.403 \pm 0.004$

$n$	$m$	Plane $Π_{1}$	Plane $Π_{3}$
2	−2	$0.56 \pm 0.02$	$0.45 \pm 0.02$
2	0	$0.32 \pm 0.02$	$0.36 \pm 0.02$
2	2	$- 0.63 \pm 0.02$	$- 0.55 \pm 0.02$
3	−3	$0.22 \pm 0.02$	$- 0.22 \pm 0.02$
3	−1	$0.08 \pm 0.02$	$- 0.08 \pm 0.02$
3	1	$- 0.19 \pm 0.02$	$0.19 \pm 0.02$
3	3	$- 0.23 \pm 0.02$	$0.24 \pm 0.02$
4	−4	$- 0.03 \pm 0.02$	$- 0.02 \pm 0.02$
4	−2	$0.01 \pm 0.02$	$0.02 \pm 0.02$
4	0	$- 0.03 \pm 0.02$	$- 0.03 \pm 0.02$
4	2	$- 0.01 \pm 0.02$	$- 0.02 \pm 0.02$
4	4	$0.01 \pm 0.02$	$0.02 \pm 0.02$

Zernike	Arm 2: HASO ( $d i a m = 1. 5 m m$ )		Arm 3: WFS30-14AR ( $d i a m = 4. 5 m m$ )
Polyn.	2, −2	2,0	2, −2	2,0
$+ 1$ ESF		$0.77 \pm 0.02$		$0.72 \pm 0.02$
$+ 2.5$ ESF		$1.80 \pm 0.05$		$1.74 \pm 0.05$
−1 ESF		$- 0.73 \pm 0.02$		$- 0.73 \pm 0.02$
−2.5 ESF		$- 1.89 \pm 0.05$		$- 1.80 \pm 0.05$
$+ 1$ CIL	$- 0.48 \pm 0.02$	$0.36 \pm 0.02$	$- 0.48 \pm 0.02$	$0.36 \pm 0.02$
−1 CIL	$0.52 \pm 0.02$	$- 0.42 \pm 0.02$	$0.49 \pm 0.02$	$- 0.41 \pm 0.02$

	Theoretical	Arm 2: HASO ( $d i a m = 1. 5 m m$ )	Arm 3: WFS30-14AR ( $d i a m = 4. 5 m m$ )
$+ 1$ ESF	−1.000	$- 0.107 \pm 0.004$	$- 1.005 \pm 0.004$
$+ 2.5$ ESF	−0.400	$- 0.045 \pm 0.004$	$- 0.419 \pm 0.004$
−1 ESF	1.000	$0.110 \pm 0.004$	$1.025 \pm 0.004$
−2.5 ESF	0.400	$0.043 \pm 0.004$	$0.403 \pm 0.004$

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Journal of the Optical Society of America A