Acuña-Romo equation

In geometric optics and optical engineering, the Acuña-Romo equation describes the solution to the design problem of a lens free of spherical aberration. The equation states how to shape the second surface of a lens such that the spherical aberration generated by the first refractive surface is completely corrected, for a point object on the optical axis. The equation was published in 2018 in Applied Optics by Rafael G. González-Acuña and Héctor A. Chaparro-Romo.^[1] The paper was designated the journal's Editor's pick.^[1]

Origin of spherical aberration-free lens design

Some of the most important events for the conception of the spherical aberration-free lens are:

Diocles in his work "On burning mirrors" just after demonstrating that the parabolic mirror could focus the rays to a single point, he mentions that It is possible to obtain a lens with the same property.^[2]

Ibn Sahl deals with the optical properties of mirrors and curved lenses. He has been described as the discoverer of the law of refraction (Snell's law).^[3]

Rene Descartes studies Cartesian ovals and their applications in optics.

Christiaan Huygens proposes to eliminate spherical aberration with a set of spherical lenses. Also in the preface of work Traité de la lumière he mentions that Isaac Newton and Gottfried Wilhelm Leibniz have addressed the problem.^[4]^[5]

Levi-Civita outlines the numerical solution to the design of corrective refractive surfaces.^[6]

G D. Wasserman and E. Wolf propose an aplanatic lens that is based on an integral that they solve with numerical methods.^[7]

Daniel Malacara Hernández presents an approximate design of a spherical aberration-free lens with two aspherical surfaces.^[8]

Psang Dain Lin and Chung-Yu Tsai obtains the spherical aberration-free lens design from the numerical solution of a System of non-linear equations.^[9]

Juan Camilo Valencia Estrada shows an analytical solution to the problem for certain particular cases.^[10]

Rafael G. González-Acuña and Héctor A. Chaparro-Romo present the general closed form equation for the design of a lens free of spherical aberration.^[1]^[11]^[12]^[13]^[14]^[15]

Comparison with the problem addressed by Newton and Leibniz and the Acuña-Romo equation

In the time line above it is mentioned that in the Christiaan Huygens made a reference of the problem and the interest of Newton and Leibniz in the problem.^[16]

...as has in fact occurred to two eminent Geometricians, Messieurs Newton and Leibniz, with respect to the Problem of the figure of glasses for collecting rays when one of the surfaces is given...
— Christiaan Huygens, Traité de la lumière (8 January 1690)

At that time it was common to call the lenses as glasses composed by two refractive surfaces. The problem deals with how must be the figure or shape of a refractive surface of the glass when one of the surface is given in order that the lens properly collects the rays in a point, just like the parabolic mirror. At this moment Huygens was aware of the spherical aberration.^[5] On the other hand Rafael G. González-Acuña and Héctor A. Chaparro-Romo address the same problem with any numerical approach nor paraxial approximation.^[1]

The goal is to determine the shape of the second surface
$(r_{b}, z_{b})$
, given a first surface
$(r_{a}, z_{a})$
, in order to correct the spherical aberration generated by the first surface.
— Rafael G. González-Acuña and Héctor A. Chaparro-Romo, General formula for bi-aspheric singlet lens design free of spherical aberration, Vol. 57, No. 31 Applied Optics

Essentially both problems are the same. Both problems deals a given refractive surface, how must be the shape of another refractive surface in order to collect rays properly i.e. a lens free of spherical aberration.^[1]

Mathematical derivation

The shape of the second surface of the lens $(r_{b}, z_{b})$ must be determined, given a first surface $(r_{a}, z_{a})$ , to correct the spherical aberration generated by the first surface. The origin of the cylindrical coordinate system is in the center of the input surface $z_{a} (0) = 0 .$

It is assumed that the singlet lens has a refractive index $n$ and is radially symmetric. In the center, the singlet lens has a thickness $t$ , the distance from the object to the first surface is $t_{a}$ and the distance from the second surface to the image is $t_{b}$ .

The first fundamental equation for this model is the vector form of the Snell's law, $v_{2} = \frac{1}{n} [n_{a} \times (- n_{a} \times v_{1})] - n_{a} \sqrt{1 - \frac{1}{n^{2}} (n_{a} \times v_{1}) \cdot (n_{a} \times v_{1})},$ where $v_{1}$ is the unit vector of the incident ray, $v_{2}$ is the unit vector of the refracted ray and finally $n_{a}$ is the normal vector of the first surface.

${\begin{cases} v_{1} = \frac{[r_{a}, (z_{a} - t_{a})]}{\sqrt{r_{a}^{2} + (z_{a} - t_{a})^{2}}}, \\ v_{2} = \frac{[r_{b} - r_{a}, z_{b} - z_{a}]}{\sqrt{(r_{b} - r_{a})^{2} + (z_{b} - z_{a})^{2}}}, \\ n_{a} = \frac{[{z_{a}}^{'}, - 1]}{\sqrt{1 + z_{a}'^{2}}}, \end{cases}$

where ${z_{a}}^{'}$ is the derivative with respect to $r_{a}$ of the sagitta on the first surface. By replacing the unit vectors in the vector form of Snell's law and grouping the Cartesian components one has,

${\begin{cases} r_{i} \equiv \frac{r_{b} - r_{a}}{\sqrt{(r_{b} - r_{a})^{2} + (z_{b} - z_{a})^{2}}} = \frac{(z_{a} - t_{a}) {z_{a}}^{'} + r_{a}}{n \sqrt{r_{a}^{2} + (t_{a} - z_{a})^{2}} (1 + z_{a}'^{2})} - {z_{a}}^{'} \frac{\sqrt{1 - \frac{{(r_{a} + (z_{a} - t_{a}) {z_{a}}^{'})}^{2}}{n^{2} (r_{a}^{2} + (t_{a} - z_{a})^{2}) (1 + z_{a}'^{2})}}}{\sqrt{1 + z_{a}'^{2}}}, \\ z_{i} \equiv \frac{z_{b} - z_{a}}{\sqrt{(r_{b} - r_{a})^{2} + (z_{b} - z_{a})^{2}}} = \frac{(r_{a} + (z_{a} - t_{a}) {z_{a}}^{'}) {z_{a}}^{'}}{n \sqrt{r_{a}^{2} + (t_{a} - z_{a})^{2}} (1 + z_{a}'^{2})} + \frac{\sqrt{1 - \frac{{(r_{a} + (z_{a} - t_{a}) {z_{a}}^{'})}^{2}}{n^{2} (r_{a}^{2} + (t_{a} - z_{a})^{2}) (1 + z_{a}'^{2})}}}{\sqrt{1 + z_{a}'^{2}}} . \end{cases}$

Since the singlet lens is free of spherical aberrations, the Fermat principle predicts that the optical path of any non-central beam must be equal to the optical path of the axial beam,

$- t_{a} + n t + t_{b} = - sgn (t_{a}) \sqrt{r_{a}^{2} + (z_{a} - t_{a})^{2}} + n \sqrt{(r_{b} - r_{a})^{2} + (z_{b} - z_{b})^{2}} + sgn (t_{b}) \sqrt{r_{b}^{2} + (z_{b} - t - t_{b})^{2}}$

where $sgn (t_{a})$ and $sgn (t_{b})$ are the sign function of the variable $t_{a}$ or $t_{b}$ , respectively. There is a system of equations, the two components of the vector form of Snell's law and the Fermat principle. The unique solution of the system is the Acuña-Romo equation given by its components:

${\begin{cases} z_{b} = \frac{h_{0} \pm \sqrt{\begin{array}{l} z_{i}^{2} [- 2 n f_{i} (z_{i} (z_{a} - t_{b} + t (z_{i} - 1)) + r_{a} r_{i} + t r_{i}^{2}) - (r_{i} (- z_{a} + t_{b} + t) + r_{a} z_{i})^{2} + f_{i}^{2} + h_{1} n^{2}] \end{array}}}{1 - n^{2}}, \\ r_{b} = r_{a} + \frac{r_{i} (z_{b} - z_{a})}{z_{i} + r_{a}} . \end{cases}$

The $\pm$ comes from the fact that when the refractive index is positive ie a natural material, the rays are refracted in the opposite direction when the refractive index is negative ie a metamaterial The auxiliary variables are,

${\begin{cases} f_{i} = - sgn (t_{a}) \sqrt{r_{a}^{2} + (t_{a} - z_{a})^{2}} + t_{a} - t_{b}, \\ h_{0} = n f_{i} z_{i} - n^{2} (t z_{i} + z_{a}) + r_{i}^{2} z_{a} - r_{a} r_{i} z_{i} + z_{i}^{2} (t + t_{b}), \\ h_{1} = r_{a}^{2} + 2 r_{a} r_{i} t + {(t_{b} - z_{a})}^{2} + t^{2} (r_{i}^{2} + {(- 1 + z_{i})}^{2}) - 2 t (t_{b} - z_{a}) (- 1 + z_{i}) . \end{cases}$

The conditions for the validity of the Acuña-Romo equation are: 1) the normal vector of the surface must be perpendicular to the tangent plane of the input surface at the origin and 2) the trajectories of the rays do not cross each other within of the lens.

The Acuña-Romo equations have an analogue with the parabolic mirror and the elliptical mirror since these mirrors are free of spherical aberration and the Acuña-Romo equation describes free lenses of spherical aberration.

The Acuña-Romo equation can be extended to the non-rotationally symmetric case.^[17]

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 González-Acuña and, Rafael G.; Chaparro-Romo, Héctor A. (November 2018). "General formula for bi-aspheric singlet lens design free of spherical aberration". Applied Optics. 57 (31): 9341–9345. doi:10.1364/AO.57.009341.
↑ G. J., Toomer (1976). Diocles On Burning Mirrors, Sources in the History of Mathematics and the Physical Sciences 1. New York: Springer. Search this book on
↑ Rashed, R. (1993). Géométrie et dioptrique au Xe siècle: Ibn Sahl, al-Quhi et Ibn al-Haytham. Paris: Les Belles Lettres. Search this book on
↑ Huygens, Christiaan (1690). Traité de la lumière. Leiden. Search this book on
↑ ^5.0 ^5.1 Dijksterhuis, Fokko Jan (2004). Lenses and waves: Christiaan Huygens and the mathematical science of optics in the seventeenth century. Enschede: Springer. ISBN 978-1-4020-2697-3. Search this book on
↑ Levi-Civita, T. "Complementi al teorema di Malus-Dupin. Nota I". Atti Accad. Sci. Torino. 9 (5): 185–189.
↑ Wasserman, G. D.; Wolf, E. "On the Theory of Aplanatic Aspheric Systems". Proceedings of the Physical Society. 62 (1).
↑ Malacara, Daniel. "Two Lenses to Collimate Red Laser Light". Applied Optics. 4 (12): 1652–1654. doi:10.1364/AO.4.001652.
↑ Lin, Psang Dain; Tsai, Chung-Yu. "Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays". Applied Optics. 29 (2): 174–178. doi:10.1364/JOSAA.29.000174.
↑ Valencia-Estrada, Juan Camilo; Flores-Hernández, Ricardo Benjamín.; Malacara-Hernández, Daniel. "Singlet lenses free of all orders of spherical aberration". Royal Society proceedings A. 471 (2175). doi:10.1098/rspa.2014.0608.
↑ "Julio Chacón, docente YACHAY TECH, Proyecto de Investigación de Lentes libres de aberraciones esféricas". www.elnorte.ec. Diario El Norte. December 6, 2018. Retrieved April 29, 2019.
↑ "YACHAY TECH CONTRIBUYE AL DISEÑO DE NUEVOS LENTES". www.yachaytech.edu.ec. YachayTech. December 3, 2018. Retrieved April 29, 2019.
↑ "¡Eureka! Encuentran la fórmula para resolver un antiguo problema óptico". transferencia.tec.mx. Revista Transferencia Tec. February 21, 2019. Retrieved April 29, 2019.
↑ González-Acuña, Rafael G.; Avendaño-Alejo, Maximino; Julio C., Gutiérrez-Vega. "Singlet lens for generating aberration-free patterns on deformed surfaces". Journal of the Optical Society of America A. 36 (5): 925–929. doi:10.1364/JOSAA.36.000925.
↑ Moreno, Danilo (January 1, 2019). "Nuevos lentes se diseñan en laboratorios de Yachay Tech". elnorte.ec. Diario El Norte. Retrieved April 29, 2019.
↑ Huygens, Christiaan (translated by Silvanus P. Thompson) (1912). Traité de la lumière. London: Macmillan. Search this book on
↑ González-Acuña, Rafael G.; Chaparro-Romo, Héctor A.; Julio C., Gutiérrez-Vega. "General formula to design a freeform singlet free of spherical aberration and astigmatism". Applied Optics. 58 (4): 9341–9345. doi:10.1364/AO.58.001010.

External links

Manufactura CNC de superficies ópticas correctoras - Biblioteca CIO [1] Archived 2019-05-03 at the Wayback Machine
SUPER-RESOLUTION IN OPTICAL SYSTEMS [2]
C. Huygens (translated by Silvanus P. Thompson), Treatise on Light, London: Macmillan, 1912; archive.org/details/treatiseonlight031310mbp

This article "Acuña-Romo equation" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Acuña-Romo equation. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

[Acuña-Romo-2018-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 González-Acuña and, Rafael G.; Chaparro-Romo, Héctor A. (November 2018). "General formula for bi-aspheric singlet lens design free of spherical aberration". Applied Optics. 57 (31): 9341–9345. doi:10.1364/AO.57.009341.

[Diocles-2] G. J., Toomer (1976). Diocles On Burning Mirrors, Sources in the History of Mathematics and the Physical Sciences 1. New York: Springer. Search this book on

[Sahl-3] Rashed, R. (1993). Géométrie et dioptrique au Xe siècle: Ibn Sahl, al-Quhi et Ibn al-Haytham. Paris: Les Belles Lettres. Search this book on

[Huygens-4] Huygens, Christiaan (1690). Traité de la lumière. Leiden. Search this book on

[Huygens2-5] 5.0 ^5.1 Dijksterhuis, Fokko Jan (2004). Lenses and waves: Christiaan Huygens and the mathematical science of optics in the seventeenth century. Enschede: Springer. ISBN 978-1-4020-2697-3. Search this book on

[Levi-6] Levi-Civita, T. "Complementi al teorema di Malus-Dupin. Nota I". Atti Accad. Sci. Torino. 9 (5): 185–189.

[Wolf-7] Wasserman, G. D.; Wolf, E. "On the Theory of Aplanatic Aspheric Systems". Proceedings of the Physical Society. 62 (1).

[Malacara-8] Malacara, Daniel. "Two Lenses to Collimate Red Laser Light". Applied Optics. 4 (12): 1652–1654. doi:10.1364/AO.4.001652.

[Lin-9] Lin, Psang Dain; Tsai, Chung-Yu. "Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays". Applied Optics. 29 (2): 174–178. doi:10.1364/JOSAA.29.000174.

[Camilo-10] Valencia-Estrada, Juan Camilo; Flores-Hernández, Ricardo Benjamín.; Malacara-Hernández, Daniel. "Singlet lenses free of all orders of spherical aberration". Royal Society proceedings A. 471 (2175). doi:10.1098/rspa.2014.0608.

[norte2-11] "Julio Chacón, docente YACHAY TECH, Proyecto de Investigación de Lentes libres de aberraciones esféricas". www.elnorte.ec. Diario El Norte. December 6, 2018. Retrieved April 29, 2019.

[yachay-12] "YACHAY TECH CONTRIBUYE AL DISEÑO DE NUEVOS LENTES". www.yachaytech.edu.ec. YachayTech. December 3, 2018. Retrieved April 29, 2019.

[tec-13] "¡Eureka! Encuentran la fórmula para resolver un antiguo problema óptico". transferencia.tec.mx. Revista Transferencia Tec. February 21, 2019. Retrieved April 29, 2019.

[max-14] González-Acuña, Rafael G.; Avendaño-Alejo, Maximino; Julio C., Gutiérrez-Vega. "Singlet lens for generating aberration-free patterns on deformed surfaces". Journal of the Optical Society of America A. 36 (5): 925–929. doi:10.1364/JOSAA.36.000925.

[norte-15] Moreno, Danilo (January 1, 2019). "Nuevos lentes se diseñan en laboratorios de Yachay Tech". elnorte.ec. Diario El Norte. Retrieved April 29, 2019.

[Huygens3-16] Huygens, Christiaan (translated by Silvanus P. Thompson) (1912). Traité de la lumière. London: Macmillan. Search this book on

[freeform-17] González-Acuña, Rafael G.; Chaparro-Romo, Héctor A.; Julio C., Gutiérrez-Vega. "General formula to design a freeform singlet free of spherical aberration and astigmatism". Applied Optics. 58 (4): 9341–9345. doi:10.1364/AO.58.001010.

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Acuña-Romo equation

Contents

Origin of spherical aberration-free lens design

Comparison with the problem addressed by Newton and Leibniz and the Acuña-Romo equation

Mathematical derivation

See also

References

External links

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