On-axis diffraction-limited bi-conical lenses

Juan Camilo Valencia Estrada; Jorge García Márquez

doi:10.1364/JOSAA.439565

Journal of the Optical Society of America A
Vol. 39,
Issue 1,
pp. 107-114
(2022)
•https://doi.org/10.1364/JOSAA.439565

On-axis diffraction-limited bi-conical lenses

Juan Camilo Valencia Estrada and Jorge García Márquez

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Juan Camilo Valencia Estrada and Jorge García Márquez, "On-axis diffraction-limited bi-conical lenses," J. Opt. Soc. Am. A 39, 107-114 (2022)

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Abstract

A diffraction-limited lens having both surfaces conic is shown. The analytical and numerical calculation for all possible solutions of the conical front and back surfaces is presented. Object and image distances, lens thickness, and refractive index are prescribed. The process to obtain on-axis diffraction-limited images with bi-conic lenses and the proof of the method, corroborated through an example in Oslo, are described here.

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Data Availability

No data were generated or analyzed in the presented research. This work does not have any experimental data. Code to perform the calculations is given in [10].

10. J. C. Valencia Estrada and J. García Márquez, “Code to calculate all on-axis diffraction-limited bi-aspheric lenses with conical surfaces,” figshare (2021), https://doi.org/10.6084/m9.figshare.17134403.v1.

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Equations (23)

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Conical Interfaces
Distances		Interface
Object	Image	Surface	Curvature	Constant	Condition
Real $t_{o} = - \infty$	Real $t_{i} = + \infty$	Plane	0	0
Virtual $t_{o} = R$	Real $t_{i} = R$	Spherical	$1 / R$	0	$R > 0$
Real $t_{o} = R$	Virtual $t_{i} = R$	Spherical	$1 / R$	0	$R < 0$
Virtual $t_{o} = (n + 1) R$	Real $t_{i} = (n + 1) R / n$	Spherical	$1 / R$	0	$R > 0$
Real $t_{o} = (n + 1) R$	Virtual $t_{i} = (n + 1) R / n$	Spherical	$1 / R$	0	$R < 0$
Real $t_{o} = - \infty$	Real $t_{i} > 0$	Ellipsoid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$n > 1$
Real $t_{o} = - \infty$	Virtual $t_{i} < 0$	Ellipsoid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$n > 1$
Real $t_{o} < 0$	Real $t_{i} = + \infty$	Ellipsoid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$0 < n < 1$
Virtual $t_{o} > 0$	Real $t_{i} = + \infty$	Ellipsoid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$0 < n < 1$
Real $t_{o} = - \infty$	Real $t_{i} > 0$	Hyperboloid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$0 < n < 1$
Real $t_{o} = - \infty$	Virtual $t_{i} < 0$	Hyperboloid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$0 < n < 1$
Real $t_{o} < 0$	Real $t_{i} = + \infty$	Hyperboloid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$n > 1$
Virtual $t_{o} > 0$	Real $t_{i} = + \infty$	Hyperboloid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$n > 1$

Lens	$R_{a} = 1 / c_{a}$	$K_{a}$	$R_{b} = 1 / c_{b}$	$K_{b}$	$d_{max}$
1	$- 40.3043$	$- 34.6289$	$- 7.357$	$- 2.46487$	6.82111
2	$- 0.111131$	$- 0.492398$	$- 0.355761$	$- 0.41546$	$\infty$
3	608	$- 3.09758$	$- 9.12$	$- 3.0976$	6.78321
4	0.258981	$- 0.323328$	0	$- 564.186$	$\infty$
5	0.259983	$- 0.323854$	0.001004	$- 0.458661$	$\infty$
6	$- 0.268261$	0.0375141	$- 0.49842$	$- 0.235946$	$\infty$
7	0.46227	$- 0.274371$	0.208177	$- 0.176501$	$\infty$
8	608	$- 0.902106$	$- 9.12$	$- 3.0976$	6.78321
9	9.23661	$- 0.704097$	$6.45076 \times 10^{9}$	$5.58695 \times 10^{21}$	6.62644
10	5.40373	$- 0.514104$	12.0607	$- 2.44231$	6.51212
11	2.97157	$- 0.335177$	3.89039	$- 0.036834$	6.20631
12	6.53824	$- 0.348457$	20.9092	0.181723	6.55805
13	$- 1.40885$	$- 0.636759$	$- 1.40773$	$- 0.694059$	$\infty$

Conical Interfaces
Distances		Interface
Object	Image	Surface	Curvature	Constant	Condition
Real $t_{o} = - \infty$	Real $t_{i} = + \infty$	Plane	0	0
Virtual $t_{o} = R$	Real $t_{i} = R$	Spherical	$1 / R$	0	$R > 0$
Real $t_{o} = R$	Virtual $t_{i} = R$	Spherical	$1 / R$	0	$R < 0$
Virtual $t_{o} = (n + 1) R$	Real $t_{i} = (n + 1) R / n$	Spherical	$1 / R$	0	$R > 0$
Real $t_{o} = (n + 1) R$	Virtual $t_{i} = (n + 1) R / n$	Spherical	$1 / R$	0	$R < 0$
Real $t_{o} = - \infty$	Real $t_{i} > 0$	Ellipsoid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$n > 1$
Real $t_{o} = - \infty$	Virtual $t_{i} < 0$	Ellipsoid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$n > 1$
Real $t_{o} < 0$	Real $t_{i} = + \infty$	Ellipsoid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$0 < n < 1$
Virtual $t_{o} > 0$	Real $t_{i} = + \infty$	Ellipsoid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$0 < n < 1$
Real $t_{o} = - \infty$	Real $t_{i} > 0$	Hyperboloid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$0 < n < 1$
Real $t_{o} = - \infty$	Virtual $t_{i} < 0$	Hyperboloid	$n / (n - 1) / t_{i}$	$- 1 / n^{2}$	$0 < n < 1$
Real $t_{o} < 0$	Real $t_{i} = + \infty$	Hyperboloid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$n > 1$
Virtual $t_{o} > 0$	Real $t_{i} = + \infty$	Hyperboloid	$- 1 / (n - 1) / t_{o}$	$- n^{2}$	$n > 1$

Lens	$R_{a} = 1 / c_{a}$	$K_{a}$	$R_{b} = 1 / c_{b}$	$K_{b}$	$d_{max}$
1	$- 40.3043$	$- 34.6289$	$- 7.357$	$- 2.46487$	6.82111
2	$- 0.111131$	$- 0.492398$	$- 0.355761$	$- 0.41546$	$\infty$
3	608	$- 3.09758$	$- 9.12$	$- 3.0976$	6.78321
4	0.258981	$- 0.323328$	0	$- 564.186$	$\infty$
5	0.259983	$- 0.323854$	0.001004	$- 0.458661$	$\infty$
6	$- 0.268261$	0.0375141	$- 0.49842$	$- 0.235946$	$\infty$
7	0.46227	$- 0.274371$	0.208177	$- 0.176501$	$\infty$
8	608	$- 0.902106$	$- 9.12$	$- 3.0976$	6.78321
9	9.23661	$- 0.704097$	$6.45076 \times 10^{9}$	$5.58695 \times 10^{21}$	6.62644
10	5.40373	$- 0.514104$	12.0607	$- 2.44231$	6.51212
11	2.97157	$- 0.335177$	3.89039	$- 0.036834$	6.20631
12	6.53824	$- 0.348457$	20.9092	0.181723	6.55805
13	$- 1.40885$	$- 0.636759$	$- 1.40773$	$- 0.694059$	$\infty$

Juan Camilo Valencia Estrada	https://orcid.org/0000-0003-2860-1222
Jorge García Márquez	https://orcid.org/0000-0002-6611-1614

Abstract

Data Availability

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Figures (4)

Tables (2)

Equations (23)

Journal of the Optical Society of America A