Adaptive sampling in convergent beams

Julián Espinosa; David Mas; Jorge Pérez; Carlos Illueca

doi:10.1364/OL.33.001960

Optics Letters
Vol. 33,
Issue 17,
pp. 1960-1962
(2008)
•https://doi.org/10.1364/OL.33.001960

Adaptive sampling in convergent beams

Julián Espinosa, David Mas, Jorge Pérez, and Carlos Illueca

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Julián Espinosa, David Mas, Jorge Pérez, and Carlos Illueca, "Adaptive sampling in convergent beams," Opt. Lett. 33, 1960-1962 (2008)

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Abstract

Numerical calculation of convergent Fresnel patterns through fast Fourier transform usually requires a large number of samples to fulfill the Nyquist sampling condition around the focus. From polynomial decomposition of the wavefront it is possible to determine which polynomial orders are the main contributors to the number of samples. This information can be used to properly modify the initial wavefront and relax the Nyquist condition thus giving a more efficient numerical algorithm.

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Equations (19)

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Wavefront	Samples
Wavefront	$N_{4}$	$N_{5}$	$N_{12}$	$N_{13}$	$N_{14}$	$N_{24}$	$N_{25}$	$N_{26}$	$N_{else}$
A	7324	0	236	0	0	13	0	0	0
B	7324	1461	236	282	5	13	68	3	0
C	1500	1461	236	282	5	13	68	3	0

j	n	m	$c_{n, m} Z_{n, m} (ρ, θ)$	$γ_{j} G_{j} (ρ, θ)$
0	0	0	$c_{0, 0}$	$(c_{0, 0} - \sqrt{3} c_{2, 0} + \sqrt{5} c_{4, 0} - \sqrt{7} c_{6, 0} + \dots)$
1	1	$- 1$	$2 c_{1, - 1} ρ \sin (θ)$	$(2 c_{1, - 1} - 2 \sqrt{8} c_{3, - 1} + 3 \sqrt{12} c_{5, - 1} - 16 c_{7, - 1} + \dots) ρ \sin (θ)$
2	1	1	$2 c_{1, 1} ρ \cos (θ)$	$(2 c_{1, 1} - 2 \sqrt{8} c_{3, 1} + 3 \sqrt{12} c_{5, 1} - 16 c_{7, 1} + \dots) ρ \cos (θ)$
3	2	$- 2$	$\sqrt{6} c_{2, - 2} ρ^{2} \sin (2 θ)$	$(\sqrt{6} c_{2, - 2} - 3 \sqrt{10} c_{4, - 2} + 6 \sqrt{14} c_{6, - 2} + \dots) ρ^{2} \sin (2 θ)$
4	2	0	$\sqrt{3} c_{2, 0} (2 ρ^{2} - 1)$	$(2 \sqrt{3} c_{2, 0} - 6 \sqrt{5} c_{4, 0} + 12 \sqrt{7} c_{6, 0} + \dots) ρ^{2}$
5	2	2	$\sqrt{6} c_{2, 2} ρ^{2} \cos (2 θ)$	$(\sqrt{6} c_{2, 2} - 3 \sqrt{10} c_{4, 2} + 6 \sqrt{14} c_{6, 2} + \dots) ρ^{2} \cos (2 θ)$

Wavefront	Samples
Wavefront	$N_{4}$	$N_{5}$	$N_{12}$	$N_{13}$	$N_{14}$	$N_{24}$	$N_{25}$	$N_{26}$	$N_{else}$
A	7324	0	236	0	0	13	0	0	0
B	7324	1461	236	282	5	13	68	3	0
C	1500	1461	236	282	5	13	68	3	0

j	n	m	$c_{n, m} Z_{n, m} (ρ, θ)$	$γ_{j} G_{j} (ρ, θ)$
0	0	0	$c_{0, 0}$	$(c_{0, 0} - \sqrt{3} c_{2, 0} + \sqrt{5} c_{4, 0} - \sqrt{7} c_{6, 0} + \dots)$
1	1	$- 1$	$2 c_{1, - 1} ρ \sin (θ)$	$(2 c_{1, - 1} - 2 \sqrt{8} c_{3, - 1} + 3 \sqrt{12} c_{5, - 1} - 16 c_{7, - 1} + \dots) ρ \sin (θ)$
2	1	1	$2 c_{1, 1} ρ \cos (θ)$	$(2 c_{1, 1} - 2 \sqrt{8} c_{3, 1} + 3 \sqrt{12} c_{5, 1} - 16 c_{7, 1} + \dots) ρ \cos (θ)$
3	2	$- 2$	$\sqrt{6} c_{2, - 2} ρ^{2} \sin (2 θ)$	$(\sqrt{6} c_{2, - 2} - 3 \sqrt{10} c_{4, - 2} + 6 \sqrt{14} c_{6, - 2} + \dots) ρ^{2} \sin (2 θ)$
4	2	0	$\sqrt{3} c_{2, 0} (2 ρ^{2} - 1)$	$(2 \sqrt{3} c_{2, 0} - 6 \sqrt{5} c_{4, 0} + 12 \sqrt{7} c_{6, 0} + \dots) ρ^{2}$
5	2	2	$\sqrt{6} c_{2, 2} ρ^{2} \cos (2 θ)$	$(\sqrt{6} c_{2, 2} - 3 \sqrt{10} c_{4, 2} + 6 \sqrt{14} c_{6, 2} + \dots) ρ^{2} \cos (2 θ)$

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Equations (19)

Optics Letters