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Campus Monterrey

School of Engineering and Sciences

The general equation of the stigmatic lens

A thesis presented by

Rafael Guillermo Gonz´alez Acu ˜na

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of Philosophiæ Doctor Degree

in

Nanotechnology

Monterrey, Nuevo Le´on, December, 2020

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I, Rafael Guillermo Gonz´alez Acu˜na, declare that this thesis titled, ”The general equation of the stigmatic lens” and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research degree at this University.

• Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this dissertation is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.

Rafael Guillermo Gonz´alez Acu˜na Monterrey, Nuevo Le´on, December, 2020

2020 by Rafael Guillermo Gonz´alez Acu˜nac All Rights Reserved

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A mis padres por su incansable apoyo.

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Infinite thanks to God for giving me wisdom and understanding to be able to reach the end of my PhD, to provide me with everything I need to get out forward and for all that he has given me. For being my creator, the engine of my life, for not having let me give in any moment and enlighten me to get ahead and thanks God for this beautiful world full of majesty and mystery.

I want like to express my sincere gratitude to my thesis advisor Dr. Julio Guti´errez Vega. His continuous motivation, patience and support were crucial during my bachelor and graduate studies to complete this work. I deeply appreciate the opportunity of pursuing a research career in optics under your guidance and example. In addition, I would like to extend my gratitude to the rest of my thesis committee Dr. Servando L´opez Aguayo, Dr. Ra´ul Hern´andez Aranda, Dr. Maximino Avenda˜no Alejo, and Dr. Dorili´an L´opez Mago, for their support, knowledge, comments and feedback during my graduate studies. My decision of becoming a researcher were highly influenced by their teaching and advice.

H´ector Alejandro Chaparro Romo, no tengo palabras! gracias por el apoyo y la man- cuerna, camarada!

I would also like to thank Gustavo Medina, Yoshio Castillejos, Mora, Ileana Paulette, Max, Eliel, Mike, Joel, Homero, Mawa, Rojo, Feri, Tamayo, Chapa, Yepiz, Erick, Balderas, Rohan, Mateusz, Julian, Luis, Job Mendoza, Benjas, Ferrer, Mabel, Arturo, Vera, Vivi, Barry R. Johnson, Simon Thinbault, Bernardino, Cuevas, Abundio, Reinhard Klette, Alois Herkom- mer, Andrea, Michelle, la familia de Chaparro, to IOP, to YachayTech University, to Wolfram Research, to ITO at Stuttgart University, to UNDAM and many more! Thank you very much!

Moreover, I acknowledge to CONACyT for the financial support given and to the Tec- nol´ogico de Monterrey for the tuition support.

Finally, I want to thank my whole familiy for their support along this complicated years, particularly to my parents Carmen Leticia Acu˜na Medell´ın and Rogelio Gonz´alez Cant´u for their help.

Gracias mam´a, pap´a, lo logramos!

I want to start the following thesis with the Pater noster,

Pater noster, qui es in caelis: sanctificetur Nomen Tuum; adveniat Regnum Tuum; fiat voluntas Tua, sicut in caelo, et in terra. Panem nostrum cotidianum da nobis hodie; et dimitte nobis debita nostra, sicut et nos dimittimus debitoribus nostris; et ne nos inducas in tenta- tionem; sed libera nos a Malo, Amen.

Rafael Guillermo Gonz´alez Acu˜na

Instituto Tecnol´ogico y de Estudios Superiores de Monterrey Dec 2020

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by

Rafael Guillermo Gonz´alez Acu˜na Abstract

I present a solution to a problem proposed by Diocles, in ancient Greece. A problem with more than two millennia without an analytical close form solution, the design of the stigmatic lens. Stigmatism refers to the image-formation property of an optical system which focuses a single point source in object space into a single point in image space. Two such points are called a stigmatic pair of the optical system. A stigmatic lens is a lens such that the point object and the point image are stigmatic.

In this doctoral thesis, the general equation of the stigmatic lenses was found. The most important implications of this equation are the uniqueness of stigmatism and a new methodol- ogy for designing stigmatic lenses in a totally analytical way, free of iterations and numerical approximations. This methodology is the basis for designing more complex stigmatic systems with a freeform shape, an arbitrary number of refractive surfaces, telescopes etc.

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1.1 Reflection of a ray, the incident angle θ1 is equal to the reflected angle θ2. . . 4

1.2 Refraction of ray in a flat surface. . . 4

1.3 Parabolic mirror. O is the point object and I is the point image . . . 7

1.4 Spherical mirror. . . 7

1.5 Elliptical mirror. . . 8

1.6 Hyperbolic mirror. . . 9

1.7 Cartesian Oval. . . 9

1.8 Spherical aberration in spherical surface. D is the maximum diameter. . . 10

1.9 Cartesian Oval has coma. . . 11

1.10 An astigmatism presented in a lens. . . 12

1.11 Pincushion distortion. . . 12

1.12 Barrel distortion. . . 12

1.13 Petzval field curvature in Cartesian Oval. . . 13

2.1 On-axis stigmatic singlet. The first surface is given by (za, ra) , and the second surface is given by (zb, rb). The gap between the first surface and the object is ta, the central is t, and the gap between the second surface and the image is tb. ~v1 is unit vector of the incident ray, ~v2 is the unit vector of the refracted ray inside the lens, ~v3 is the output ~nais the normal vector of the first surface and ~nbis the normal vector of the second surface. . . 15

2.2 An on-axis stigmatic lens for finite object and finite image, proposed by Huy- gens in Trait´e de la lumi`ere. . . 22

2.3 An aspherical lens similar to that proposed by Huygens in Trait ´e de la lumi `e reConfiguration settings: n = 1.5, ta = −55mm, t = 29mm, tb = 30mm, za = 29 +p2962 − ra2 and zb = Eq. (2.26). . . 23

2.4 Configuration settings: n = 1.5, ta = −30mm, t = 4mm, tb = 30mm, za = −ra2/60 and zb = Eq. (2.26). . . 23

2.5 Configuration settings: n = 1.5, ta = −30mm, t = 4mm, tb = 20mm, za = ra2/60 and zb = Eq. (2.26). . . 23

2.6 Configuration settings: n = 1.5, ta= −30mm, t = 4mm, tb = 20mm, za= 0 and zb = Eq. (2.26). . . 24

2.7 Configuration settings: n = −1.5, ta = −55mm, t = 10mm, tb = 30mm, za = 90 −p902− ra2and zb = Eq. (2.26). . . 24

2.8 Configuration settings: n = 1.7, ta = 30mm, t = 7mm, tb = 35mm, za = r2a/40 and zb = Eq. (2.26). . . 25

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2.10 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, tb = −15mm, za = 0 and zb = Eq. (2.26). . . 26 2.11 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, tb = −15mm,

za = ra2/90 and zb = Eq. (2.26). . . 26 2.12 Configuration settings: n = 1.5, ta = 30mm, t = 5mm, tb = −20mm,

za = −ra2/60 and zb = Eq. (2.26). . . 27 2.13 Configuration settings: n = 1.5, ta = 30mm, t = 5mm, tb = −20mm,

za = ra2/60 and zb = Eq. (2.26). . . 27 2.14 Configuration settings: n = 1.5, t = 5mm, tb = 30mm, za = −r2a/100 and

zb = Eq. (2.32). . . 29 2.15 Configuration settings: n = 1.5, t = 5mm, tb = 30mm, za = ra2/100 and

zb = Eq. (2.32). . . 30 2.16 Configuration settings: n = 1.5, t = 5mm, tb = −40mm, za = −ra2/100 and

zb = Eq. (2.32). . . 30 2.17 Configuration settings: n = 1.5, t = 5mm, tb = −40mm, za = r2a/100 and

zb = Eq. (2.32). . . 31 2.18 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, za = −ra2/100 and

zb = Eq. (2.39). . . 32 2.19 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, za = r2a/100 and

zb = Eq. (2.39). . . 33 2.20 Configuration settings: n = 1.5, ta = 30mm, t = 5mm, za = −r2a/100 and

zb = Eq. (2.39). . . 33 2.21 Configuration settings: n = 1.5, ta = 30mm, t = 5mm, za = ra2/120 and

zb = Eq. (2.39). . . 33 2.22 Configuration settings: n = 1.5, t = 5mm, za= −r2a/16 and zb = Eq. (2.42). 35 3.1 On-axis freeform stigmatic singlet. The first surface is given by (xa, ya, za)

, and the second surface is given by (xb, yb, zb). The length between the first surface and the object is ta, the thickness at the centre of the lens is t, and the length between the second surface and the image is tb. ~v1is unit vector of the incident ray, ~v2is the unit vector of the refracted ray inside the lens, ~v3is the output ~nais the normal vector of the first surface and ~nb is the normal vector of the second surface. . . 39 3.2 Configuration settings: n = 1.7, ta = −50mm, t = 10mm, tb = 70mm,

za = (x2a+ 4y2a)/130 and zb = Eq. (3.20). . . 43 3.3 Configuration settings: n = 1.9 ta = −50mm, t = 10mm, tb = 60mm,

za = (x2a+ (ya− 1)2)/100 and zb = Eq. (3.20). . . 43 3.4 Configuration settings: n = 1.7 ta = −60mm, t = 10mm, tb = 70mm,

za = (x2a+ 1.52ya2)/130 and zb = Eq. (3.20). . . 43 3.5 Configuration settings: n = 2 ta = 80mm, t = 10mm, tb = 70mm, za =

(x2a+ 1.52ya2)/100 and zb = Eq. (3.20). . . 44 3.6 Configuration settings: n = 1.9 ta = 50mm, t = 10mm, tb = −80mm,

za = (x2a+ 4y2a)/100 and zb = Eq. (3.20). . . 44

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3.8 Configuration settings: n = 1.5 ta = −95mm, t = 15mm, za = −90 + p902− x2a− 16ya2 and zb = Eq. (3.34). . . 49 3.9 Configuration settings: n = 1.5 t = 5mm, za = cos(0.3xa + 0.35ya) and

zb = Eq. (3.37). . . 51 4.1 The original system is composed by N refractive surfaces. An extra corrective

surface N + 1 can be obtained get an on-axis stigmatic system. . . 52 4.2 The focuses of first parabolic surfaces are with given by f1 = −50, f2 = −50

and f3 = −90. . . 59 4.3 The first three surfaces are parabolas with focuses given by f1 = 50, f2 = −80

and f3 = 50. . . 60 5.1 Telescopic system composed by N lenses with a aspherical corrective mirror

(zN +1, rN +1) such as the system is on-axis stigmatic. . . 62 5.2 The focuses of first parabolic surfaces are f1 = −80, f2 = 90 and f3 = 120. . 66 5.3 The first three surfaces are parabolas with focuses given by f1 = 50, f2 = −90

and f3 = −30. . . 67 6.1 Diagram of a on-axis stigmatic singlet. The first surface is (za, ra) , and the

second surface is (zb, rb). ~v1 is unit vector of the incident ray, ~v2 is the unit vector of the refracted ray inside the lens, ~v3 is the output ~na is the normal vector of the first surface and ~nbis the normal vector of the second surface. . 69 6.2 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, tb = 30mm,

ha = −5mm, hb = 3mm, za= −r2a/30 and zb = Eq. (6.22). . . 73 6.3 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, tb = 30mm,

ha = −5mm, hb = 3mm, za= 0 and zb = Eq. (6.22). . . 73 6.4 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, tb = 30mm,

ha = 8mm, hb = −5mm, za= −r2a/80 and zb = Eq. (6.22). . . 73 6.5 Configuration settings: n = 1.5, ta = −30mm, t = 5mm, tb = 30mm,

ha = 5mm, hb = −5mm, za= r2a/120 and zb = Eq. (6.22). . . 74 7.1 Aplanatic collector lens. . . 78 7.2 Configuration settings: n = 1.505595, w = −0.5, ts = 120.727mm, tc =

124.013mm, da = ±31.305mm, tb = 401.617mm, hb = ∓156.525mm za = 55 −p(ra/2)2+ 552, zbs = Eq. (7.30) and zbc = Eq. (7.9). . . 86 7.3 Configuration settings: n = 1.505595, w = −0.5, ts = 102.409mm, tc =

105.296mm, da= ±26.8328mm, tb = 189.571mm, hb = ∓71.5542mm za = 50 −p(ra/2)2+ 502, zbs = Eq. (7.30) and zbc = Eq. (7.9). . . 86 7.4 Configuration settings: n = 1.505595, w = −0.5, ts = 115.011mm, tc =

120.448mm, da= ±35.7771mm, tb = 444.904mm, hb = ∓214.663mm za = 0, zbs = Eq. (7.30) and zbc = Eq. (7.9). . . 87 7.5 Configuration settings: n = 1.505595, w = −0.5, ts = 115.011mm, tc =

120.448mm, da = ±35.7771mm, tb = 300.75mm, hb = ∓143.108mm za = 0, zbs = Eq. (7.30) and zbc = Eq. (7.9). . . 87

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−40 +pra2/4 + 402, zbs = Eq. (7.30) and zbc = Eq. (7.9). . . 88 7.7 Configuration settings: n = 1.505595, w = −0.5, ts = 102.534mm, tc =

109.166mm, da= ±35.7771mm, tb = 325.731mm, hb = ∓178.885mm za =

−90 +pra2/4 + 902, zbs = Eq. (7.30) and zbc = Eq. (7.9). . . 88

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4.1 The image is located at z = 7 + 5 + 9 + 90 = 111mm. . . 59

4.2 The image is located at z = 8 + 4 + 9 + 90 = 111mm. . . 60

5.1 The image is located at z = 4 + 5 + 120 − 50 = 79mm. . . 65

5.2 The image is located at z = 10 + 5 + 120 − 50 = 85mm. . . 66

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Abstract v

List of Figures ix

List of Tables x

1 Geometrical optics 1

1.1 Introduction . . . 1

1.2 Fermat’s principle . . . 2

1.3 Reflection . . . 3

1.4 Refraction . . . 3

1.5 Relation between the Snell’s law and Fermat’s Principle . . . 5

1.6 Stigmatism . . . 6

1.7 Aberrations . . . 8

1.8 End notes . . . 12

2 On-axis stigmatic lens 14 2.1 Introduction . . . 14

2.2 Finite image-object . . . 15

2.2.1 Mathematical model . . . 15

2.2.2 Illustrative examples . . . 22

2.3 The lens, the parabola and Diocles . . . 22

2.3.1 Mathematical model . . . 25

2.3.2 Illustrative examples . . . 29

2.4 The collimator lens . . . 29

2.4.1 Mathematical model . . . 30

2.4.2 Illustrative examples . . . 32

2.5 The single-lens telescope . . . 34

2.5.1 Mathematical model . . . 34

2.5.2 Illustrative examples of single-lens telescopes and single-beam expander 35 2.6 End notes . . . 35

3 On-axis Stigmatic freeform lens 37 3.1 Introduction and Descartes prediction . . . 37

3.2 Finite image-object . . . 38

3.3 Mathematical model . . . 38 xi

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3.4.1 Illustrative examples . . . 46

3.5 The freeform collimator lens . . . 47

3.5.1 Illustrative examples . . . 49

3.6 The beam-sheaper . . . 49

3.6.1 Illustrative example . . . 50

3.7 End notes . . . 50

4 On-axis stigmatic optical systems 52 4.1 Introduction . . . 52

4.1.1 Mathematical model . . . 53

4.2 Snell’s law . . . 54

4.2.1 Surfaces expressed in terms of the refracted rays . . . 57

4.2.2 Illustrative examples . . . 58

4.3 End notes . . . 59

5 On-axis stigmatic reflective telescope 61 5.1 Introduction . . . 61

5.1.1 Mathematical model . . . 61

5.2 Example . . . 64

5.3 End notes . . . 66

6 Off-axis stigmatic lens 68 6.1 Introduction . . . 68

6.2 Mathematical model . . . 68

6.2.1 Snell’s law . . . 68

6.2.2 Illustrative examples . . . 72

6.2.3 A non symmetric solution . . . 72

6.3 Mathematical implications of a non-symmetric solution . . . 74

6.4 End notes . . . 76

7 Aplanatic singlet lens: General setting & design 77 7.1 Introduction . . . 77

7.2 Collector off-axis stigmatic lens . . . 79

7.3 Collector on-axis stigmatic lens for an arbitrary reference path . . . 81

7.4 The merge of two solutions . . . 85

7.5 Examples . . . 86

7.6 End notes . . . 89

A Algorithms 90

B Conclusions 97

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Geometrical optics

1.1 Introduction

Geometrical optics, also named ray optics, is one of the oldest sciences. It studies the light through geometry. Euclid proposed the first premise used in geometrical optics in his book Optics. The premise is that light propagates in straight lines, making a connection between the paths of light and the geometry proposed in its masterpiece, Elements, since Euclid’s Elements focus on the nature of straight lines. a

In geometric optics, it is common to describe the propagation of light in terms of rays.

The ray is a useful abstraction to approximate the paths along which light propagates under certain circumstances. Euclid’s premise on the propagation of light in straight lines is only valid when light travels around a homogeneous medium. The refraction index of a homo- geneous medium is a dimensionless number that describes how fast light travels through the medium. The refraction index is defined by

n = c

v, (1.1)

where, c is the speed of light in vacuum and v is the phase velocity of light in the medium.

Geometric optics does not deal with specific optical effects, such as diffraction, polar- ization and interference. However, it deals with reflection and refraction, which are the two main phenomena studied in geometrical optics. The reflection of light is the phenomenon of returning the rays of light that fall on the surface of an object; commonly, these objects are mirrors. Refraction is the redirection of a light ray when it enters on a medium where its speed is different. Refraction occurs when the refractive index of the input medium is different from the refractive index of the output medium.

Ignoring diffraction and interference is useful in practice when the wavelength is small compared to the size of the optical elements in which the light interacts. This paradigm is particularly useful for describing geometric aspects of images, including, mirrors, lenses,

aElementsis a mathematical and geometric treatise that consists of thirteen books. In the books lay the foundation of the Euclidean geometry.

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axicons and other optical devices. Geometrical optics is the leading theory used in optical design. The art of designing cameras, microscopes, telescopes and optical systems, in general, is called optical design.

Geometric optics is a very challenging science, where most of the results are particular cases obtained using sophisticated optimization algorithms, for example, the design of a spe- cific camera or telescope. The hardness of geometrical optics happens because the equations that model the light rays and their interactions with the different media can be quite long, complicated, and in many cases, nonlinear. Nonlinearity has led optical design to resemble art rather than a scientific discipline. The paraxial approximation, also called small-angle approximation, simplifies geometrical optics. When the angles are small, the mathematical behaviour of the optical systems becomes linear.

A ray that forms a small angle θ concerning the optical axis of the system and it is located near the axis along the optical system is called paraxial. The paraxial approximation allows three relevant approximations for the paraxial ray, which mathematically speaking can be written as

sin θ ∼= θ, (1.2)

tan θ ∼= θ, (1.3)

cos θ ∼= 1. (1.4)

In some cases, the second-order approximation is also called paraxial. The approximations above for sine and tangent do not change for the second-order paraxial approximation, while for cosine the second-order approximation is,

cos θ ∼= 1 − θ2

2. (1.5)

In this thesis are not implemented the paraxial approximation, second-order approximation, third-order approximation, or any other order approximation. The only approximations im- plemented in this thesis are the premises of geometrical optics.

Here in this chapter, we are going to explore the two main physical phenomena in geo- metrical optics, reflection and refraction and two main physical principles related to them.

1.2 Fermat’s principle

Fermat’s principle says that the path taken between two points by a ray of light is the path that can be passed in the shortest time. Fermat’s principle can be stated as,

”The optical length of the path followed by light between two fixed different points is the global minima. The optical length is the physical length multiplied by the refractive index of the medium”

Please pay attention to the word global minimum. Remember that the global minimum is the smallest overall value of a set. So, imagine that we have a set, such as its elements are all the possible optical paths from one point to another. These paths have their respective

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optical length. What the Fermat principle says is that the only physically valid way of our set is the one that has the smallest value of an optical path length.

Mathematically the Fermat’s principle can be described as the time T a point of the ray needs to cover a path between the points A and B, i.e,

T = Z t1

t0

dt = 1 c

Z t1

t0

c v

ds

dtdt = 1 c

Z B A

nds, (1.6)

Remember that c is the speed of light in vacuum, ds an infinitesimal displacement along the ray, v = ds/dt the speed of light in a medium and n = c/v the refractive index of that medium, t0 is the starting time (the ray is in A), t1 is the arrival time at point B. The optical path length of a ray from a A to B is defined by following integral

S = Z B

A

nds, (1.7)

which is related to the travel time by S = cT . The optical path length is a purely geometrical quantity since time is not considered in its calculation. The global minimum in the light travel time between two points A and B is equivalent to the global minimum of the optical path length between A and B.

1.3 Reflection

The reflection of light is the phenomenon of returning the rays of light that affect the surface of an object, commonly this surface is glossy and in geometrical optics are called by the name of mirrors. Reflection of light is predictable, and geometry describes it. The incident angle and the reflected angle are equal. This equality is known as the Law of Reflection.

In Fig. 1.1 can be seen as a simple diagram of a reflection taking place on a flat mirror, the incident angle θ1 is equal to the reflected angle θ2, in other means θ1 = θ2. Note that mirrors with curved surfaces can be modelled by ray tracing and using the law of reflection at each point on the surface.

1.4 Refraction

Refraction occurs when light travels through a region that has a changing index of refraction.

The simplest case of refraction occurs when there is an interface between a uniform medium with an index of refraction n1 and another medium with an index of refraction n2. In such situations, the Snell’s Law, also called Law of Refraction, describes the resulting deflection of the light ray. The Snell’s Law is given by,b

n1sin(θ1) = n2sin(θ2), (1.8)

where, θ1and θ2are the angles for the incident and refracted rays, respect to the normal vector of the surface.

bAlthough popularly known as Snell’s Law in honour of Willebrord Snellius (1580-1626), Ibn Sahl (9401000) was the first person to find the law of refraction in 984 AC.

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Figure 1.1: Reflection of a ray, the incident angle θ1is equal to the reflected angle θ2.

Figure 1.2: Refraction of ray in a flat surface.

Fig. 1.2 is a simple diagram of a reflection taking place on a flat interface between two

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mediums with constant refraction indexes. Like the mirrors, interfaces with curved surfaces should be modelled by ray tracing and using the law of reflection at each point on the surface.

When the light rays are in a homogeneous medium, their paths are straight lines. So, in these conditions, the light rays can be represented by vectors. Remember that vectors are mathematical entities that have magnitude and direction. In the law of reflection and the law of refraction, we are only interested in the course of the rays. Then, we can express these laws with unit vectors.

Therefore, Snell’s law can be written in terms of unit vectors. For that, we use two im- portant results. The first one is the Snell’s law in the usual form, Eq. (1.8). Second important result is the property of the cross product,

ˆ

n × ˆs1 = |ˆn||ˆs1| sin θ1ζ,ˆ (1.9) where ˆs1 is a unit vector of the incident ray, ˆn is the normal vector of the surface and ˆζ is a vector perpendicular to both ˆs1and ˆn. Now we use the trigonometric identity,

cos2θ2 = 1 − sin2θ2, (1.10)

and the scalar form of Snell’s law, Eq. (1.8), we can get an expression for ˆs2, which is a unit vector of the refracted ray. To achieve that, we can look for the component of the refracted ray along the surface normal,

−ˆn · ˆs2 = |ˆn||ˆs2| cos θ2, (1.11) we add that to the component the surface itself in the plane of incidence,

sin θ2(ˆn × ˆζ), (1.12)

after some algebraic manipulation, we finally get,

ˆs2 = n1 n2

[ˆn × (−ˆn × ˆs1)] − ˆn s

1 − n1 n2

2

(ˆn × ˆs1) · (ˆn × ˆs1). (1.13) Above’s equation, which looks more complex than the usual form of Snell’s law, but it is advantageous. Since it describes the unit vector with the direction of the refracted ray in terms of the normal surface vector and the unit incident vector.

1.5 Relation between the Snell’s law and Fermat’s Principle

Fermat’s principle states that light travels along the path that takes the least time. Consider the refraction index as a measure of the speed of light in a material. Then you can use Fermat’s principle to derive the same angular relationship between the incident and refracted rays as the Snell’s law. The Fermat’s principle and the Snell’s law share information in common, but they are not the same.

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1.6 Stigmatism

Imagine that you have a solid connected body with a refractive index different than the envi- ronment where the body is located. This body is a lens if it has the function of focusing or dispersing rays through the refraction that arises from the difference between the refraction index of the mentioned body and the medium. Lenses are fundamental elements of study in geometric optics.

If the body has the function of redirecting the rays such that there is no dispersion or focus on them, then the element in question is a Prism. If the body does not have a function, then it is just a translucent rock.

We have mentioned that the lenses have the function of focusing the light or scattering it. It is also true that mirrors have such features.

The law of reflection and the law of refraction depend on the normal of the surface.

Therefore, the shape of the lens/mirror is essential to fulfilling its predetermined function.

When it comes to focusing the light, what is wanted, in principle, is that the rays that come from a point object converge into a point image. Therefore, what is wanted is to have stigmatic lenses and stigmatic mirrors. Stigmatism refers to the image-formation property of an optical system which focuses a point object into a point image. Such points are called a stigmatic pair of the optical system.

Stigmatic lenses and stigmatic mirrors need to have a very particular shape. In the case of the mirrors, the reflective surfaces that form the stigmatic mirrors are the conic sections.c

• For mirrors with parabolic surfaces, parallel rays that hit the mirror produce reflected rays that converge into a common focus. See Fig. 1.3.d

• For mirrors with spherical surfaces, all rays that emerge from a point object located at a finite distance from the reflecting surface are reflected in the same point. If and only if the object is situated in the centre of the circumference. See Fig. 1.4.

• For mirrors with elliptic surfaces, all rays that emerge from a point object are reflected in another point, the point image. See Fig. 1.5.

• For mirrors with hyperbolic surfaces, all rays that arise from a point object are reflected in a single virtual point image. See Fig. 1.6.

The beauty of conical mirrors lies in the intrinsic geometric properties of conic sections.

Other curved surfaces may also focus light, but not in a single point. The stigmatic refractive surface is the Cartesian Oval, which is a fourth-order function. In other means, the Cartesian oval is a surface such that all the rays that emerge from a point object are focused on a single image point, once they are refracted. See Fig. 1.7.

The conic mirrors and the Cartesian oval are results of interest that have been preserved and will be preserved over time by their analytic nature.

cConic sections are all those obtained by cutting a cone with a plane. The Greek mathematician Apollonius of Perge (262-190 BC) was the first to study in detail the conic sections. Apollonius classified the conics in fourth types: ellipses, hyperbolas, circles, and parables.

dDiocles (240 BC - ca. 180 BC), in his work Burning Mirrors, was the first person that reported this property of the parabolic mirror.

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Figure 1.3: Parabolic mirror. O is the point object and I is the point image

Figure 1.4: Spherical mirror.

In the case of stigmatic lenses, it took more than two thousand years to have a general equation that describes their surfaces. The general equation that describes the stigmatic lenses is not trivial, and it is the central theme of this thesis.e

At this point, it is convenient to define what is analytical optical design. It is the design

eIn the book Burning Mirrors, just after demonstrating the parabolic property of the mirror property, Diocles mentions that it is possible to obtain a lens with the same property. A fact, that he does not show, but opens the conjecture. In 2018, Rafael G. Gonzlez-Acua and Hctor A. Chaparro-Romo presented a closed-form analytic solution that solves the conjecture [1].

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Figure 1.5: Elliptical mirror.

of an optical system based exclusively on the premises of geometrical optics without counting any paraxial approximation and without having any optimization process. The relevance of the results given by the analytical optical design is that they are preserved over time since they are general and not particular cases. Examples of analytical optical design results are conical mirrors, Cartesian oval and stigmatic lenses.

1.7 Aberrations

When a system is not stigmatic for all points of the object, then the system has aberrations.

In this section, we will show the terminology implemented throughout this thesis on optical aberrations.

If the optical system has aberration, the point object is projected in a region in the image space instead of at a single point. The nature of the region of space where the image is formed depends on the type of aberration. The optical aberrations of the optical system distort the

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Figure 1.6: Hyperbolic mirror.

Figure 1.7: Cartesian Oval.

image formed by the optical system.

Aberrations fall into two classes: chromatic and monochromatic. The variation of a lens’s refractive index concerning the wavelength causes the chromatic aberrations.

The geometry of the optical system causes the monochromatic aberrations. In general, it occurs both when light is reflected or refracted so the reflection law, the Snell’s law, and the Fermat’s principle are involved in the phenomenon. They have information about the geometry of the imaging system. Therefore, the shape of the optical system is crucial to vanish the monochromatic aberrations. The five basic types of monochromatic aberrations are the spherical aberration, coma, astigmatism, field curvature, and image distortion.

Spherical aberration is the phenomenon that exists in an optical system when a point

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object located on the optical axis does not have a stigmatic correspondence with a point image.

In other words, the rays that leave the point object on the optical axis do not converge on a point image on the optical axis. It is called spherical aberration because the spherical lens has this phenomenon. There are lenses called aspherical because their shape is different from the sphere. In most cases, their main goal is to reduce spherical aberration. In the following chapters, We, will see that a stigmatic lens is aspherical.

Figure 1.8: Spherical aberration in spherical surface. D is the maximum diameter.

In Fig. 1.8 there is an example of spherical aberration generated by a spherical surface

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with constant refraction index n along with the material.

Coma

Coma aberration in an optical system refers to the aberration suffered by the image of a point object outside the axis. Coma makes that the image appears distorted, with a tail, like a coma or a comet. In other words, an optical system with coma has no stigmatic relationship between a point object outside the optical axis and a point image, since this image is not a point but a region.

Figure 1.9: Cartesian Oval has coma.

Another way to explain how the coma looks like, in Fig. 1.9 we put an interface with rays that come from an off-axis object, then the rays cross the surface of index refraction n, and they are refracted. The refraction causes an inversion of some of the rays, as it can be seen in the same figure. This inversion in the photos looks like a coma or a comet.

Astigmatism

The astigmatism of a point object takes place when two perpendicular planes have different image points. In Fig. 1.10, we put a lens with astigmatism.

Image distortion

The distortion in a forming image system is measured with a rectilinear projection. The rectilinear projection is passed through the system, a projection in which straight lines in a scene remain straight in the image if there is no distortion in the system. The most common distortions are in Figs. 1.11 and 1.12.

Field curvature

Petzval field curvature, named for Joseph Petzval, describes the optical aberration in which a flat object normal to the optical axis is focused as a curved image. In Fig. 1.13 is the field curvature in Cartesian Oval.

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Figure 1.10: An astigmatism presented in a lens.

Figure 1.11: Pincushion distortion.

Figure 1.12: Barrel distortion.

1.8 End notes

This thesis is aimed primarily at the needs of a reader primarily interested in the elementary mathematical properties of stigmatic lenses. Therefore, we do not mention in detail how aberrations are fought through optimization algorithms.

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Figure 1.13: Petzval field curvature in Cartesian Oval.

We focused on the previous necessary concepts to be able to understand and design stigmatic lenses. These concepts are the Fermat Principle and Snell’s Law in its vector form.

The study of optical aberrations and their different representations are beyond the scope of this thesis.

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On-axis stigmatic lens

2.1 Introduction

A lens is a particular device that can manipulate the trajectory of the light that passes through it. The lenses are translucent bodies with a refraction index different from the medium; typ- ically the medium is air which its refraction index is one, nair = 1. In this thesis, we shall use n as the refraction index of our singlet lenses. The singlet lenses are lenses with just two refractive surfaces, the first refractive surface and the second refractive surface. The geometry and shape of these two surfaces can be model by mathematical equations in a vast of ways giving different performances and applications to the lenses. Here in this chapter, we focus on radially symmetric lenses. When the light passes through almost every lens, the spherical aberration appears. The rays that strike the border converges first than the rays that strike at the centre.

In this chapter, the goal is to design a stigmatic singlet lens for a point object and a point image. This problem was first proposed by Diocles two thousand years ago. Many renowned scientists tried to solve analytically, but they failed. There are many numerical solutions based on mathematical optimization strategies. But there is only one analytic solution, and in this chapter, we are going to find it. In words of Ren´e Descartes the problem is,

”I might go farther and show how, if one surface of a lens is given and is neither entirely plane nor composed of conic sections or circles, the other surface can be so determined as to transmit all the rays from a given point to another point, also given. This is no more difficult than the problems I have just explained:

indeed. it is much easier since the way is now open; I prefer, however, to leave this for others to work out, to the end that they may appreciate the more highly the discovery of those things here demonstrated, through having themselves to meet some difficulties.”

The quote is from De la nature des lignes courbes. For sure, Ren´e Descartes is referring to a numerical solution, since the analytical one it took more than 300 years since Descartes wrote these letters.

14

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2.2 Finite image-object

In this section, we are going to get the general formulae for a single lens when the object- image distance is finite.

2.2.1 Mathematical model

The goal is to determine the shape of the second surface (zb, rb), given a first surface (za, ra), in order to get an on-axis stigmatic lens. Therefore, the objective is to find how is (zb, rb) given (za, ra), where rais the only independent variable, zb, rb and zaare functions of ra. The origin of the coordinate system is located at the center of the input surface za(0) = 0. The sign conversion of the unit vectors is denoted by the origin of the coordinate.

The singlet lens has refraction index n and is radially symmetric. At the centre, the singlet-lens has a thickness of t. The distance from the object to the first surface is ta. The distance from the second surface to the image is tb, as it can be seen in Fig. 2.1.

Figure 2.1: On-axis stigmatic singlet. The first surface is given by (za, ra) , and the second surface is given by (zb, rb). The gap between the first surface and the object is ta, the central is t, and the gap between the second surface and the image is tb. ~v1is unit vector of the incident ray, ~v2 is the unit vector of the refracted ray inside the lens, ~v3is the output ~nais the normal vector of the first surface and ~nbis the normal vector of the second surface.

Fermat Principle

For a spherical aberration-free singlet lens, the optical path of any non-central ray must be equal to the optical path of the axial ray. It does not matter if the point object and point image are real or virtual. We need to combine all these scenarios in a single equation that express the equality of an optical path of the axial ray and the optical path of a nonaxial ray. The axial ray is our reference ray.

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In the following equations, the left side is for the optical path of the axial ray, and the right side is for the optical path of the nonaxial ray. See Fig. 2.1.

We start with real object and real image, ta< 0 and tb > 0,

−ta+ nt + tb = p

r2a+ (za− ta)2+ np

(rb− ra)2+ (zb− za)2 +

q

rb2+ (zb − t − tb)2, (2.1)

With virtual object and real image we have: ta> 0 and tb > 0,

−ta+ nt + tb = −p

r2a+ (za− ta)2+ np

(rb− ra)2+ (zb− za)2 +

q

rb2+ (zb − t − tb)2, (2.2)

Then, with real object and virtual image: ta < 0 and tb < 0, we have

−ta+ nt + tb = p

r2a+ (za− ta)2+ np

(rb− ra)2+ (zb− za)2

− q

r2b + (zb− t − tb)2, (2.3)

and finally with virtual object and virtual image: ta> 0 and tb < 0, we have

−ta+ nt + tb = −p

r2a+ (za− ta)2+ np

(rb− ra)2+ (zb− za)2

− q

r2b + (zb− t − tb)2, (2.4)

If we associate the four cases of Fermat’s principle we can have a single equation that explains the phenomenon for the four cases, this equation is given by

−ta+ nt + tb = −sgn(ta)p

r2a+ (za− ta)2+ np

(rb− ra)2+ (zb− za)2 +sgn(tb)

q

rb2+ (zb − t − tb)2. (2.5) Above’s equation has information about how the light travels in our optical system. The optical path is constant for all the rays.a

Snell’s law

Now, we focus on Snell’s law, which is the equation that information has when light crosses from one medium to another. The vector form of the Snell’s law at the first surface is,

ˆ

v2 = 1

−sgn(ta)n[ˆna× (−ˆna× ˆv1)] − ˆna

r 1 − 1

n2(ˆna× ˆv1) · (ˆna× ˆv1), (2.6) where ˆv1 is the unit vector of the incident ray, ˆv2 is the unit vector of the refracted ray, ˆv3 is the unit vector of output ray. ˆna is the normal vector of the first surface, ˆnb is the normal vector of the second surface. Finally, −sgn(ta) comes from the fact that the object can be real

aThe function sgn(·) is the sign of its argument. It can be only one or minus one. But if the argument is zero sgn(0) is no defined.

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or virtual. Seen in Fig. 2.1.

The unit vectors at the first surface are,























 ˆ

v1 = [ra, (za− ta), 0 ] pra2+ (za− ta)2,

ˆ

v2 = [rb− ra, zb− za, 0 ] p(rb− ra)2+ (zb− za)2,

ˆ

na = [z0a, −1, 0 ] p1 + za0 2 ,

(2.7)

where za0 is the derivative respect to raof the sagitta of the first surface.

Replacing Eq. (2.7) in Eq. (2.6) and separating the Cartesian components we get the following expressions,

rb− ra

q

(zb− za)2+ (rb− ra)2

= (za− ta)za0 + ra

−sgn(ta)np

ra2+ (ta− za)2 1 + za0 2 (2.8)

− za0 s

1 − (ra+ (za− ta)za0)2 n2[ra2+ (ta− za)2] 1 + za0 2

p1 + za0 2 , and,

zb− za q

(zb− za)2+ (rb− ra)2

= (ra+ (za− ta)za0) za0

−sgn(ta)np

ra2+ (ta− za)2 1 + z0 2a  (2.9)

+ s

1 − (ra+ (za− ta)za0)2 n2[ra2+ (ta− za)2] 1 + za0 2

p1 + za0 2 ,

In the left side of Eqs. (2.8) and (2.9) are the unknowns zb, rb. Please notice that the right side of Eqs. (2.8) and (2.9) dependent only on parameters that we know, za0, za, t, and n. The right side of Eq. (2.8) and (2.9) are the cosine directors of the refracted ray. Let be ℘z the cosine director of the z direction and let be ℘rthe cosine director of the r direction, then ℘2r+℘2z = 1.

Thus,

r = (za− ta)za0 + ra

−sgn(ta)np

r2a+ (ta− za)2 1 + za0 2 − za0 s

1 − (ra+ (za− ta)z0a)2 n2[r2a+ (ta− za)2] 1 + z0 2a 

p1 + za0 2 , (2.10)

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and,

z = (ra+ (za− ta)za0) za0

−sgn(ta)np

r2a+ (ta− za)2 1 + za0 2 + s

1 − (ra+ (za− ta)za0)2 n2[r2a+ (ta− za)2] 1 + za0 2

p1 + za0 2 , (2.11) We can assign a name to the distance that each ray needs to pass inside the lens,

ϑ = q

(zb− za)2+ (rb− ra)2. (2.12) If we express Eq. (2.8) and (2.9) in terms of the cosine directors and the length ϑ, we get,

zb − za

ϑ = ℘z, (2.13)

and, rb− ra

ϑ = ℘r. (2.14)

Then, we can solve for zb and rb,

zb = za+ ϑ℘z, (2.15)

and,

rb = ra+ ϑ℘r. (2.16)

The above’s expression is the solution that we want to get. It describes, point by point, how must the second surface of the lens such as it is stigmatic. The only problem in this equation is that we do not know the length of ϑ. But, we can find the solution mixing this result with the Fermat principle.

Solution

In the last section, we get how must zb and rb be, but in terms of ϑ. Therefore we need an extra equation, and that equation is the Fermat principle that relates the optical path an axial ray with the optical path of a non-axial ray. We recall it, Eq. (2.5)b

−ta+ nt + tb = −sgn(ta)p

r2a+ (za− ta)2+ np

(rb− ra)2+ (zb− za)2 + sgn(tb)

q

r2b + (zb− t − tb)2. (2.5)

In the left, side last equation are inserted the distance ϑ and the unknowns zb and rb. We can replace them using Eq. (2.12) (2.15), and (2.16),

−ta+ nt + tb = −sgn(ta)p

ra2+ (za− ta)2+ nϑ +sgn(tb)p

(ra+ ϑ℘r)2+ (za+ ϑ℘z− t − tb)2. (2.17)

bIt is essential to remark, that we have, just two unknowns zb and rb but we have a system with three equations. Two equations are given by the Snell’s law at the first surface another one provided by the Fermat principle. There is nothing wrong here since the equations granted by the Snell’s law are not independent.

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Now, we still have a big equation which is bothering to manipulate. So, we can assign some variables to simplify it. Let’s define these variables,c

f ≡ −ta+ nt + tb+ sgn(ta)p

r2a+ (za− ta)2, (2.18) and,

τ ≡ za− t − tb. (2.19)

We replace f and τ in Eq. (2.17). Then we square it, we get an expression that is very similar to one of the most famous theorems in the history of humankind, Pythagoras theorem:

(f − nϑ)2 = (ra+ ϑ℘r)2+ (τ + ϑ℘z)2, (2.20) Let’s expand the above’s square binomials,

f2− 2f nϑ + n2ϑ2 = r2a+ 2rarϑ + ℘2rϑ2+ τ2+ 2τ ℘rϑ + ℘2rϑ2. (2.21) Now, we collect the terms that are multiplied by ϑ,

ϑ2(1 − n2) + ϑ[2(f n + rar+ τ ℘z)] + (r2a+ τ2− f2) = 0. (2.22) The equation above has the form of an ancient friend of elementary algebra, the quadratic equation. Then, the solution is given by the quadratic formula,d

ax2+ bx + c = 0, x = −b ±√

b2− 4ac

2a . (2.23)

Using the quadratic formula, we can find the solution for ϑ,

ϑ = −[2(f n + rar+ τ ℘z)] ±p[2(fn + rar+ τ ℘z)]2− 4(1 − n2)(ra2+ τ2− f2)

2(1 − n2) .

(2.24) Simplifying, we can remove the number two,

ϑ = −(f n + rar+ τ ℘z) ±p(fn + rar+ τ ℘z)2 − (1 − n2)(ra2+ τ2− f2)

(1 − n2) .

(2.25)

cThe first variable is called f because it has all the elements that do not have the unknowns of the equation given by the Fermat principle, f is for Fermat. The second variable is named τ because it has information on the thickness of the segment from the second surface to the image, τ is for thickness.

dIt is fair to solve a two-thousand-year-old problem with a two-thousand-year-old formula. The quadratic formula was first discovered by Babylonians two thousand years ago. Euclid used geometric approaches to determine quadratic equations in Book 2 of his magnum opus, Elements. This prominent mathematical treatise is the foundation of geometry, which is the ruler in geometrical optics. For this reason, it is not a surprise to find the Pythagoras theorem [2].

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Finally, we can use the solution to the problem, Eq. (2.26), (zb = za+ ϑ℘z,

rb = ra+ ϑ℘r. (2.26)

Eq. (2.26) is the most important equation in the chapter. It tells how must be the second sur- face, point by point, such as we get a stigmatic lens. It is significant to observe that Eq. (2.26) only works for singlet lenses such as the rays inside them do not cross each other. In other terms, Eqs. (2.15) and (2.16) tell that for a point of the first surface, there is a unique point in the second surface for it, to get stigmatism.

Also, it is necessary to remark that in the expression of ϑ, Eq. (2.25), there is no expres- sion for sgn(tb). In other words, the second surface adapts its shape according to where the image is located; it does not matter if the image is real or virtual.

Another essential remark about Eq. (2.25) is that it has a plus-less sign ± multiplying the square root, the plus-less sign led us to two solutions. The problem is, which is the solution that works for a given case? Well, it depends on the sign of the refraction index n, (n could be negative or positive) and if the object/image is virtual or real. In the following sections, we are going to show several illustrative examples.

The process to get Eq. (2.26) looks very easy if you read this chapter from the beginning to here. eHowever, for centuries, people tried to get the general analytical closed-form solu- tion, but they failed [?, 3, 4]. The secret is that in the whole process, we do not use any angle.

The usual form of Snell’s law complicates everything since there is no clear relation between the angles and the optical paths. With Snell’s law in its vector form, it is very easy to see the association between the optical paths and the director cosines.

Note that we simply did not obtain the general analytical solution in a closed way for the problem; we also discovered that the solution is unique.

Christiaan Huygens knew about the problem we just solved. In the preface of his mag- num opus, Trait de la Lumire, Huygens mentioned that Sir Isaac Newton and Gottfried Wil- helm Leibniz were interested in the problem. In Huygens words [?],

”I write, and not for the intention of decreasing from the merit of those who, without having seen anything that I have written, may be found to have attended of like matters: as has, in truth, happened to two prominent Geometricians, Messieurs Newton and Leibniz, with regard to the enigma of the shape of glasses for collecting rays when one of the surfaces is provided.”

Huygens showed a particular interest in the problem and referred to Descartes. In chap- ter 1 of Trait de la Lumire, Huygens mentions the following passage,

”And lastly, I shall approach the several shapes of transparent and reflecting forms by which rays are collected at a point or are directed aside in various ways. From this, it will be seen with what tools, following our new theory, we find not only the Ellipses, Hyperbolas, and other curves which Mr. Descartes

eThe process can be easy, but the equation is not, it is gigantic if you expand it takes more than nine pages.

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has ingeniously developed for this purpose;but also those which the surface of a glass lens ought to hold when its other surface is provided as spherical or plane, or of any other shape that may be.”

Also from Trait de la Lumire we have,

”Let us now turn to our way and let us observe how it leads without challenge to the finding of the curves which one side of the glass requires when the other side is of a given figure; a figure, not only plane or spherical, or made by one of the conic sections (which is the limitation with which Descartes introduced this problem, giving the answer to those who should come after him) but gen- erally any figure whatever: that is to say, one created by the revolution of any given curved line to which one must simply know how to draw straight lines as tangents.”

Note Huygens’ reference to Descartes in bold. Huygens mentioned the same quote from Descartes that we presented at the beginning of this chapter.

Huygens went further and solved the problem, drawing with rule and compass the solu- tion. The following passage is from chapter 6 of Trait de la Lumire.

”Let the given figure be that made by the revolution of some curve such as AK regarding the axis AV, and that this side of the glass receives rays coming from the point L. Moreover, let the thickness AB of the centre of the glass be given, and the point F at which one aspires the rays to be all perfectly focused, whatever be the initial refraction happening at the surface AK.”

”I announce that for this the single condition is that the outline BDK which com- poses the other surface shall be such that the path of the light from the point L to the surface AK, and from thence to the surface BDK, and from thence to the point F, shall be traversed wherever in equal times, and in each case in a time equal to that which the light uses to advance along the straight line LF of which the part AB is within the glass.

Let LG be a ray descending on the arc AK. Its refraction GV will be provided by means of the tangent which will be carried at the point G. Now, in GV the point D need be determined such that FD together with f of DG and the straight line GL, may be equal to FB together with f of BA and the straight line AL ; which, as is plain, make up a given length. Or rather, by deducing from each the length of LG, which is also provided, it will merely be needful to adjust FD up to the straight line VG in such a way that FD together with -f of DG is equal to a provided straight line, which is a quite obvious plane problem: and the point D will be one of those through which the curve BDK ought to pass. And similarly, having drawn another ray LM, and found its refraction MO, the point N will be found in this line, and so on as many occasions as one wants.”

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