Proof that spot diagram borders are always caustic curves and/or
marginal rays II: annular (circular and elliptical) and single
hexagonal exit pupils

Jorge González-García; Alberto Cordero-Dávila; Diego Gabriel Reyes-Olguín

doi:10.1364/AO.449270

Applied Optics
Vol. 61,
Issue 8,
pp. 1876-1884
(2022)
•https://doi.org/10.1364/AO.449270

Proof that spot diagram borders are always caustic curves and/or marginal rays II: annular (circular and elliptical) and single hexagonal exit pupils

Jorge González-García, Alberto Cordero-Dávila, and Diego Gabriel Reyes-Olguín

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Jorge González-García, Alberto Cordero-Dávila, and Diego Gabriel Reyes-Olguín, "Proof that spot diagram borders are always caustic curves and/or marginal rays II: annular (circular and elliptical) and single hexagonal exit pupils," Appl. Opt. 61, 1876-1884 (2022)

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Abstract

In a previous paper [Appl. Opt. 61, C20 (2022) [CrossRef] ], it was proven that for a circular exit pupil and any optical path differences, the border of any spot diagram is integrated by the caustic surface and/or marginal rays. In this paper, the previous results are extended to annular (circular an elliptical) as well as hexagonal (single and segmented) exit pupils. Several examples of wavefronts affected by linear combinations of orthonormal Zernike aberrations are shown.

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Number	Meaning	Orthonormal Zernike Polynomial
1	Piston	1
2	Tilt about y	$2 x / \sqrt{1 + ϵ^{2}}$
3	Tilt about x	$2 y / \sqrt{1 + ϵ^{2}}$
4	Defocus	$\sqrt{3} (2 x^{2} + 2 y^{2} - 1 - ϵ^{2}) / (1 - ϵ^{2})$
5	Primary astigmatism 45°	$2 \sqrt{6} x y / \sqrt{1 + ϵ^{2} + ϵ^{4}}$
6	Primary astigmatism 0°	$\sqrt{6} (x^{2} - y^{2}) / \sqrt{1 + ϵ^{2} + ϵ^{4}}$
7	Primary coma about y	$\sqrt{8} (δ_{71} - δ_{72}) / δ_{73}$ where $δ_{71} = (1 - ϵ^{2}) (x^{2} y + y^{3})$ $δ_{72} = 2 y (1 + ϵ^{2} + ϵ^{4})$ $δ_{73} = (1 - ϵ^{2}) \sqrt{(1 + ϵ^{2}) (1 + 4 ϵ^{2} + ϵ^{4})}$
8	Primary coma about x	$\sqrt{8} (δ_{81} - δ_{82}) / δ_{73}$ where $δ_{81} = 3 (1 + ϵ^{2}) (x^{3} + x y^{2})$ $δ_{82} = 2 x (1 + ϵ^{2} + ϵ^{4})$
9	Trefoil about y	$\sqrt{8} (3 x^{2} y - y^{3}) / \sqrt{(1 + ϵ^{2} + ϵ^{4} + ϵ^{6})}$
10	Trefoil about x	$\sqrt{8} (x^{3} - 3 x y^{2}) / \sqrt{(1 + ϵ^{2} + ϵ^{4} + ϵ^{6})}$
11	Primary spherical	$\sqrt{5} (δ_{111} + δ_{112}) / δ_{113}$ where $δ_{111} = 6 (x^{2} + y^{2})^{2} + (1 + 4 ϵ^{2} + ϵ^{4})$ $δ_{112} = - 6 (1 + ϵ^{2}) (x^{2} + y^{2})$ $δ_{113} = \sqrt{(1 + ϵ^{2} + ϵ^{4} + ϵ^{6})}$
12	Secondary astigmatism at 0°	$\sqrt{10} (δ_{121} - δ_{122}) / δ_{126}$ where $δ_{121} = 4 (x^{4} - y^{4})$ $δ_{122} = 3 (1 - ϵ^{8}) (x^{2} - y^{2}) / (1 - ϵ^{6})$ $δ_{123} = [1 / (1 - ϵ^{2})]$ $δ_{124} = [16 (1 - ϵ^{10})]$ $δ_{125} = 15 (1 - ϵ^{8})^{2} / (1 - ϵ^{6})$ $δ_{126} = \sqrt{δ_{123} (δ_{124} - δ_{125})}$
13	Secondary astigmatism at 45°	$\sqrt{10} (δ_{131} - δ_{132}) / δ_{126}$ where $δ_{131} = (4 x^{3} y + 4 x y^{3})$ $δ_{132} = 3 (1 - ϵ^{8}) x y / (1 - ϵ^{6})$
14	Vertical quatrefoil	$\sqrt{10} (x^{4} + y^{4} - 6 x^{2} y^{2}) / δ_{14}$ where $δ_{14} = \sqrt{1 + ϵ^{2} + ϵ^{4} + ϵ^{6} + ϵ^{8}}$
15	Oblique quatrefoil	$4 \sqrt{10} (x^{3} y - x y^{3}) / δ_{14}$

Number	Meaning	Orthonormal Zernike Polynomial
1	Piston	1
2	Tilt about y	$2 x$
3	Tilt about x	$2 y / b$
4	Defocus	$[\sqrt{3} / \sqrt{3 - 2 b^{2} + 3 b^{4}}] (- 1 - b^{2} + 4 ρ^{2})$
5	Primary astigmatism 45°	$[2 \sqrt{6} / b] x y$
6	Primary astigmatism 0°	$E_{61} [x^{2} - y^{2} + \frac{1}{4} (- 1 + b^{2}) - E_{62}]$ where $E_{61} = [\sqrt{9 + 9 b^{4} - 6 b^{2}} / (\sqrt{2} b^{2})]$ $E_{62} = \frac{3 (- 1 + b^{4}) (1 - 4 ρ^{2} + b^{2})}{4 (3 - 2 b^{2} + 3 b^{4})}$ $ρ = \sqrt{x^{2} + y^{2}}$
7	Primary coma about y	$[4 / (b \sqrt{5 - 6 b^{2} + 9 b^{4}})] [- (1 + 3 b^{2}) + 6 ρ^{2}] y$
8	Primary coma about x	$[4 / \sqrt{9 - 6 b^{2} + 5 b^{4}}] [- (3 + b^{2}) + 6 ρ^{2}] x$
9	Trefoil about y	$E_{91} [E_{92} x^{2} - E_{93} y^{2} + E_{94}] y$ where $E_{91} = [4 / (b^{3} \sqrt{5 - 6 b^{2} + 9 b^{4}})]$ $E_{92} = 3 b^{2} (3 b^{2} - 1)$ $E_{93} = (5 - 3 b^{2})$ $E_{94} = 3 b^{2} (1 - b^{2})$
10	Trefoil about x	$E_{101} [E_{102} x^{2} - E_{103} y^{2} + E_{104}] x$ where $E_{101} = [4 / (b^{2} \sqrt{9 - b^{2} + 5 b^{4}})]$ $E_{102} = 5 b^{2} (5 b^{2} - 3)$ $E_{103} = 3 (3 - b^{2})$ $E_{104} = 3 b^{2} (1 - b^{2})$
11	Primary spherical	$E_{111} [6 ρ^{4} - \frac{3}{4} (2 b^{2} x^{2} + 2 y^{2} + 6 x^{2} + 6 b^{2} y^{2}) + \frac{1}{8} (3 + 2 b^{2} + 3 b^{4})]$ where $E_{111} = \frac{8 \sqrt{5}}{\sqrt{45 - 60 b^{2} + 94 b^{4} - 15 b^{8}}}$
12	Secondary astigmatism at 0°	$E_{121} {[x^{2} - y^{2}] [4 ρ^{2} - 3] + E_{122} [E_{123} ρ^{4} - E_{124} x^{2} - E_{125} y^{2} + E_{126}]}$ where $E_{121} = [\sqrt{10} α / (b^{2} γ)]$ $α = \sqrt{45 - 60 b^{2} + 94 b^{4} - 60 b^{5} + 45 b^{8}}$ $γ = \sqrt{35 - 60 b^{2} + 114 b^{4} - 60 b^{6} + 35 b^{8}}$ $E_{122} = [\sqrt{5} / (2 \sqrt{2} b^{2} β)]$ $β = \sqrt{β_{1} + β_{2} + β_{3}}$ $β_{1} = 1575 - 4800 b^{2} + 12020 b^{4}$ $β_{2} = - 17280 b^{6} + 21066 b^{8} - 17280 b^{10}$ $β_{3} = 12020 b^{12} - 4800 b^{14} + 1575 b^{16}$ $E_{123} = 240 (- 3 + 2 b^{2} - 2 b^{6} + 3 b^{8})$ $E_{124} = 60 (- 9 + 3 b^{2} + 2 b^{4} - 6 b^{6} + 7 b^{8} + 3 b^{10})$ $E_{125} = 24 (15 - 70 b^{2} + 92 b^{4} - 82 b^{6} + 45 b^{8})$ $E_{126} = 12 b^{2} (- 5 + 2 b^{2} - 2 b^{6} + 5 b^{8})$
13	Secondary astigmatism at 45°	$[2 \sqrt{10} / b δ] [8 ρ^{2} - 3 b^{2} - 3] x y$ where $δ = \sqrt{5 - 6 b^{2} + 5 b^{4}}$
14	Vertical quatrefoil	$E_{141} [E_{142} x^{4} + E_{143} y^{4} + E_{144} x^{2} y^{2} - E_{145} y^{2} - E_{146} x^{2} + E_{147}]$ where $E_{141} = [\sqrt{10} / (b^{4} γ)]$ $γ = \sqrt{35 - 60 b^{2} + 114 b^{4} - 60 b^{6} + 35 b^{8}}$ $E_{142} = b^{4} (35 b^{4} - 30 b^{2} + 3)$ $E_{143} = (35 - 30 b^{2} + 3 b^{4})$ $E_{144} = 6 b^{2} (5 - 18 b^{2} + 5 b^{4})$ $E_{145} = 6 b^{2} (5 - 6 b^{2} + b^{4})$ $E_{146} = 6 b^{4} (1 - 6 b^{2} + 5 b^{4})$ $E_{147} = 3 b^{4} (1 - b^{2})$
15	Oblique quatrefoil	$E_{151} [x^{2} - y^{2} + (\frac{3 b^{2} - 3}{8})] x y$ where $E_{151} = [4 \frac{4 \sqrt{10}}{b \sqrt{5 + 6 b^{2} + 5 b^{4}}}]$

Number	Meaning	Orthonormal Zernike Polynomial
1	Piston	1
2	Tilt about y	$2 \sqrt{6 / 5} x$
3	Tilt about x	$2 \sqrt{6 / 5} y$
4	Defocus	$[\sqrt{5 / 43}] [12 (x^{2} + y^{2}) - 5]$
5	Primary astigmatism 45°	$[4 \sqrt{15 / 7}] x y$
6	Primary astigmatism 0°	$[2 \sqrt{15 / 7}] (x^{2} - y^{2})$
7	Primary coma about y	$[4 \sqrt{42 / 3685}] [25 (x^{2} + y^{2}) - 14] y$
8	Primary coma about x	$[24 \sqrt{42 / 3685}] [25 (x^{2} + y^{2}) - 14] x$
9	Trefoil about y	$[(4 / 3) \sqrt{10}] (3 x^{2} - y^{2}) y$
10	Trefoil about x	$[4 \sqrt{70 / 103}] (x^{2} - 3 y^{2}) x$
11	Primary spherical	$[3 / \sqrt{1072205}] δ_{h}$ where $δ_{h} = 6020 (x^{2} + y^{2})^{2} - 5140 (x^{2} + y^{2}) + 737$

Number	Meaning	Orthonormal Zernike Polynomial
1	Piston	1
2	Tilt about y	$2 x / \sqrt{1 + ϵ^{2}}$
3	Tilt about x	$2 y / \sqrt{1 + ϵ^{2}}$
4	Defocus	$\sqrt{3} (2 x^{2} + 2 y^{2} - 1 - ϵ^{2}) / (1 - ϵ^{2})$
5	Primary astigmatism 45°	$2 \sqrt{6} x y / \sqrt{1 + ϵ^{2} + ϵ^{4}}$
6	Primary astigmatism 0°	$\sqrt{6} (x^{2} - y^{2}) / \sqrt{1 + ϵ^{2} + ϵ^{4}}$
7	Primary coma about y	$\sqrt{8} (δ_{71} - δ_{72}) / δ_{73}$ where $δ_{71} = (1 - ϵ^{2}) (x^{2} y + y^{3})$ $δ_{72} = 2 y (1 + ϵ^{2} + ϵ^{4})$ $δ_{73} = (1 - ϵ^{2}) \sqrt{(1 + ϵ^{2}) (1 + 4 ϵ^{2} + ϵ^{4})}$
8	Primary coma about x	$\sqrt{8} (δ_{81} - δ_{82}) / δ_{73}$ where $δ_{81} = 3 (1 + ϵ^{2}) (x^{3} + x y^{2})$ $δ_{82} = 2 x (1 + ϵ^{2} + ϵ^{4})$
9	Trefoil about y	$\sqrt{8} (3 x^{2} y - y^{3}) / \sqrt{(1 + ϵ^{2} + ϵ^{4} + ϵ^{6})}$
10	Trefoil about x	$\sqrt{8} (x^{3} - 3 x y^{2}) / \sqrt{(1 + ϵ^{2} + ϵ^{4} + ϵ^{6})}$
11	Primary spherical	$\sqrt{5} (δ_{111} + δ_{112}) / δ_{113}$ where $δ_{111} = 6 (x^{2} + y^{2})^{2} + (1 + 4 ϵ^{2} + ϵ^{4})$ $δ_{112} = - 6 (1 + ϵ^{2}) (x^{2} + y^{2})$ $δ_{113} = \sqrt{(1 + ϵ^{2} + ϵ^{4} + ϵ^{6})}$
12	Secondary astigmatism at 0°	$\sqrt{10} (δ_{121} - δ_{122}) / δ_{126}$ where $δ_{121} = 4 (x^{4} - y^{4})$ $δ_{122} = 3 (1 - ϵ^{8}) (x^{2} - y^{2}) / (1 - ϵ^{6})$ $δ_{123} = [1 / (1 - ϵ^{2})]$ $δ_{124} = [16 (1 - ϵ^{10})]$ $δ_{125} = 15 (1 - ϵ^{8})^{2} / (1 - ϵ^{6})$ $δ_{126} = \sqrt{δ_{123} (δ_{124} - δ_{125})}$
13	Secondary astigmatism at 45°	$\sqrt{10} (δ_{131} - δ_{132}) / δ_{126}$ where $δ_{131} = (4 x^{3} y + 4 x y^{3})$ $δ_{132} = 3 (1 - ϵ^{8}) x y / (1 - ϵ^{6})$
14	Vertical quatrefoil	$\sqrt{10} (x^{4} + y^{4} - 6 x^{2} y^{2}) / δ_{14}$ where $δ_{14} = \sqrt{1 + ϵ^{2} + ϵ^{4} + ϵ^{6} + ϵ^{8}}$
15	Oblique quatrefoil	$4 \sqrt{10} (x^{3} y - x y^{3}) / δ_{14}$

Number	Meaning	Orthonormal Zernike Polynomial
1	Piston	1
2	Tilt about y	$2 x$
3	Tilt about x	$2 y / b$
4	Defocus	$[\sqrt{3} / \sqrt{3 - 2 b^{2} + 3 b^{4}}] (- 1 - b^{2} + 4 ρ^{2})$
5	Primary astigmatism 45°	$[2 \sqrt{6} / b] x y$
6	Primary astigmatism 0°	$E_{61} [x^{2} - y^{2} + \frac{1}{4} (- 1 + b^{2}) - E_{62}]$ where $E_{61} = [\sqrt{9 + 9 b^{4} - 6 b^{2}} / (\sqrt{2} b^{2})]$ $E_{62} = \frac{3 (- 1 + b^{4}) (1 - 4 ρ^{2} + b^{2})}{4 (3 - 2 b^{2} + 3 b^{4})}$ $ρ = \sqrt{x^{2} + y^{2}}$
7	Primary coma about y	$[4 / (b \sqrt{5 - 6 b^{2} + 9 b^{4}})] [- (1 + 3 b^{2}) + 6 ρ^{2}] y$
8	Primary coma about x	$[4 / \sqrt{9 - 6 b^{2} + 5 b^{4}}] [- (3 + b^{2}) + 6 ρ^{2}] x$
9	Trefoil about y	$E_{91} [E_{92} x^{2} - E_{93} y^{2} + E_{94}] y$ where $E_{91} = [4 / (b^{3} \sqrt{5 - 6 b^{2} + 9 b^{4}})]$ $E_{92} = 3 b^{2} (3 b^{2} - 1)$ $E_{93} = (5 - 3 b^{2})$ $E_{94} = 3 b^{2} (1 - b^{2})$
10	Trefoil about x	$E_{101} [E_{102} x^{2} - E_{103} y^{2} + E_{104}] x$ where $E_{101} = [4 / (b^{2} \sqrt{9 - b^{2} + 5 b^{4}})]$ $E_{102} = 5 b^{2} (5 b^{2} - 3)$ $E_{103} = 3 (3 - b^{2})$ $E_{104} = 3 b^{2} (1 - b^{2})$
11	Primary spherical	$E_{111} [6 ρ^{4} - \frac{3}{4} (2 b^{2} x^{2} + 2 y^{2} + 6 x^{2} + 6 b^{2} y^{2}) + \frac{1}{8} (3 + 2 b^{2} + 3 b^{4})]$ where $E_{111} = \frac{8 \sqrt{5}}{\sqrt{45 - 60 b^{2} + 94 b^{4} - 15 b^{8}}}$
12	Secondary astigmatism at 0°	$E_{121} {[x^{2} - y^{2}] [4 ρ^{2} - 3] + E_{122} [E_{123} ρ^{4} - E_{124} x^{2} - E_{125} y^{2} + E_{126}]}$ where $E_{121} = [\sqrt{10} α / (b^{2} γ)]$ $α = \sqrt{45 - 60 b^{2} + 94 b^{4} - 60 b^{5} + 45 b^{8}}$ $γ = \sqrt{35 - 60 b^{2} + 114 b^{4} - 60 b^{6} + 35 b^{8}}$ $E_{122} = [\sqrt{5} / (2 \sqrt{2} b^{2} β)]$ $β = \sqrt{β_{1} + β_{2} + β_{3}}$ $β_{1} = 1575 - 4800 b^{2} + 12020 b^{4}$ $β_{2} = - 17280 b^{6} + 21066 b^{8} - 17280 b^{10}$ $β_{3} = 12020 b^{12} - 4800 b^{14} + 1575 b^{16}$ $E_{123} = 240 (- 3 + 2 b^{2} - 2 b^{6} + 3 b^{8})$ $E_{124} = 60 (- 9 + 3 b^{2} + 2 b^{4} - 6 b^{6} + 7 b^{8} + 3 b^{10})$ $E_{125} = 24 (15 - 70 b^{2} + 92 b^{4} - 82 b^{6} + 45 b^{8})$ $E_{126} = 12 b^{2} (- 5 + 2 b^{2} - 2 b^{6} + 5 b^{8})$
13	Secondary astigmatism at 45°	$[2 \sqrt{10} / b δ] [8 ρ^{2} - 3 b^{2} - 3] x y$ where $δ = \sqrt{5 - 6 b^{2} + 5 b^{4}}$
14	Vertical quatrefoil	$E_{141} [E_{142} x^{4} + E_{143} y^{4} + E_{144} x^{2} y^{2} - E_{145} y^{2} - E_{146} x^{2} + E_{147}]$ where $E_{141} = [\sqrt{10} / (b^{4} γ)]$ $γ = \sqrt{35 - 60 b^{2} + 114 b^{4} - 60 b^{6} + 35 b^{8}}$ $E_{142} = b^{4} (35 b^{4} - 30 b^{2} + 3)$ $E_{143} = (35 - 30 b^{2} + 3 b^{4})$ $E_{144} = 6 b^{2} (5 - 18 b^{2} + 5 b^{4})$ $E_{145} = 6 b^{2} (5 - 6 b^{2} + b^{4})$ $E_{146} = 6 b^{4} (1 - 6 b^{2} + 5 b^{4})$ $E_{147} = 3 b^{4} (1 - b^{2})$
15	Oblique quatrefoil	$E_{151} [x^{2} - y^{2} + (\frac{3 b^{2} - 3}{8})] x y$ where $E_{151} = [4 \frac{4 \sqrt{10}}{b \sqrt{5 + 6 b^{2} + 5 b^{4}}}]$

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Applied Optics