Estimating field-dependent nodal aberration theory coefficients from Zernike full-field displays by utilizing eighth-order astigmatism

Eric M. Schiesser; Aaron Bauer; Jannick P. Rolland

doi:10.1364/JOSAA.36.002115

Journal of the Optical Society of America A
Vol. 36,
Issue 12,
pp. 2115-2128
(2019)
•https://doi.org/10.1364/JOSAA.36.002115

Estimating field-dependent nodal aberration theory coefficients from Zernike full-field displays by utilizing eighth-order astigmatism

Eric M. Schiesser, Aaron Bauer, and Jannick P. Rolland

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Eric M. Schiesser, Aaron Bauer, and Jannick P. Rolland, "Estimating field-dependent nodal aberration theory coefficients from Zernike full-field displays by utilizing eighth-order astigmatism," J. Opt. Soc. Am. A 36, 2115-2128 (2019)

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- Geometrical Optics
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About this Article
History
- Original Manuscript: August 22, 2019
- Revised Manuscript: October 23, 2019
- Manuscript Accepted: October 29, 2019
- Published: November 27, 2019

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Abstract

When using freeform surfaces in optical design, the field dependence of the aberrations can become quite complex, and understanding these aberrations facilitates the design process. Here we calculate the field dependence of low-order Zernike astigmatism (Z5/6) up to the eighth order in nodal aberration theory (NAT). Expansion of NAT astigmatism terms to the eighth order facilitates a more accurate fit to the Zernike astigmatism data. We then show how this estimated field dependence can be used to quantitatively analyze a freeform telescope design. This analysis tool adds to the optical designer’s arsenal when up against the challenge of designing with freeform optics.

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NAT Coefficient	$Z_{5}$ Field Dependence	$Z_{6}$ Field Dependence
$W_{222}$	$(H_{y}^{2} - H_{x}^{2})$	$2 H_{x} H_{y}$
$\| {\vec{A}}_{222} \|$	$- 2 H_{y}$	$- 2 H_{x}$
$\| {\vec{B}}_{222}^{2} \|$	1	0
$W_{422}$	$(H_{y}^{2} - H_{x}^{2}) H^{2}$	$2 H_{x} H_{y} H^{2}$
$\| {\vec{A}}_{422} \|$	$- 4 H_{y}^{3}$	$- 2 H_{x} (H_{x}^{2} + 3 H_{y}^{2})$
$\| {\vec{B}}_{422} \|$	$3 H^{2}$	0
$\| {\vec{C}}_{422}^{3} \|$	$- H_{y}$	$H_{x}$
$W_{622}$	$(H_{y}^{2} - H_{x}^{2}) H^{4}$	$2 H_{x} H_{y} H^{4}$
$\| {\vec{A}}_{622} \|$	$2 H_{y} (H_{x}^{2} - 3 H_{y}^{2}) H^{2}$	$- 2 H_{x} (H_{x}^{2} + 5 H_{y}^{2}) H^{2}$
$\| {\vec{B}}_{622}^{2} \|$	$7 H_{x}^{4} + 7 H_{y}^{4} + 6 H_{x}^{2} H_{y}^{2}$	$4 H_{x} H_{y} (H_{y}^{2} - H_{x}^{2})$
$\| {\vec{C}}_{622}^{3} \|$	$- 4 H_{y} (H_{x}^{2} + 5 H_{y}^{2})$	$- 4 H_{x} (H_{x}^{2} + 5 H_{y}^{2})$
$\| {\vec{D}}_{622}^{4} \|$	$(H_{y}^{2} - H_{x}^{2})$	$- 2 H_{x} H_{y}$

	Iteration 1	Iteration 2
Starting $B_{222}$ Value (waves)	130.911	−0.018
$B_{222, M 1}^{α}$ (waves/micrometers)	5.59	5.63
$Z_{5, M 1} \|_{B_{222} = 0}$ (micrometers)	−23.4261	−23.4229
Final $B_{222}$ Value (waves)	−0.018	8.9e-008

		Z8 Proportionality Constants and Ratio
Overall Z8		Values (Waves/Micrometers)
Aberrations (Waves)			M1	M2	M3
$_{G} A_{222}$	63.34	$A_{222, j}^{α}$	−0.02141	−1.649	1.491
$_{G} A_{131}$	−62.64	$A_{131, j}^{α}$	−3.019	1.267	−0.4658
Ratio	−1.01	Ratio	0.00709	−1.30	−3.20
			Predicted Surface Coefficients (Micrometers)
			M1	M2	M3
		$M 1 + M 2$	4.59	−38.48	--
		$M 2 + M 3$	--	−56.95	−20.50
		$M 1 + M 3$	14.16	--	42.69

NAT Coefficient	$Z_{5}$ Field Dependence	$Z_{6}$ Field Dependence
$W_{222}$	$(H_{y}^{2} - H_{x}^{2})$	$2 H_{x} H_{y}$
$\| {\vec{A}}_{222} \|$	$- 2 H_{y}$	$- 2 H_{x}$
$\| {\vec{B}}_{222}^{2} \|$	1	0
$W_{422}$	$(H_{y}^{2} - H_{x}^{2}) H^{2}$	$2 H_{x} H_{y} H^{2}$
$\| {\vec{A}}_{422} \|$	$- 4 H_{y}^{3}$	$- 2 H_{x} (H_{x}^{2} + 3 H_{y}^{2})$
$\| {\vec{B}}_{422} \|$	$3 H^{2}$	0
$\| {\vec{C}}_{422}^{3} \|$	$- H_{y}$	$H_{x}$
$W_{622}$	$(H_{y}^{2} - H_{x}^{2}) H^{4}$	$2 H_{x} H_{y} H^{4}$
$\| {\vec{A}}_{622} \|$	$2 H_{y} (H_{x}^{2} - 3 H_{y}^{2}) H^{2}$	$- 2 H_{x} (H_{x}^{2} + 5 H_{y}^{2}) H^{2}$
$\| {\vec{B}}_{622}^{2} \|$	$7 H_{x}^{4} + 7 H_{y}^{4} + 6 H_{x}^{2} H_{y}^{2}$	$4 H_{x} H_{y} (H_{y}^{2} - H_{x}^{2})$
$\| {\vec{C}}_{622}^{3} \|$	$- 4 H_{y} (H_{x}^{2} + 5 H_{y}^{2})$	$- 4 H_{x} (H_{x}^{2} + 5 H_{y}^{2})$
$\| {\vec{D}}_{622}^{4} \|$	$(H_{y}^{2} - H_{x}^{2})$	$- 2 H_{x} H_{y}$

	Iteration 1	Iteration 2
Starting $B_{222}$ Value (waves)	130.911	−0.018
$B_{222, M 1}^{α}$ (waves/micrometers)	5.59	5.63
$Z_{5, M 1} \|_{B_{222} = 0}$ (micrometers)	−23.4261	−23.4229
Final $B_{222}$ Value (waves)	−0.018	8.9e-008

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