Stigmatic Optics. Rafael G González-Acuña

2.1 From the wave equation, through Helmholtz equation to end with the eikonal equation

       2.2 The eikonal equation

       2.3 The ray equation

       2.3.1 n as constant

       2.3.2 n(r⃗) as a function

       2.4 The Snell law from eikonal

       2.5 The Fermat principle from eikonal

       2.6 End notes

       Further reading

       3 Calculus of variations

       3.1 Calculus of variations

       3.2 The Euler equation

       3.3 Newton’s second law

       3.4 End notes

       Further reading

       4 Optics of variations

       4.1 Introduction

       4.2 Lagrangian and Hamiltonian optics

       4.3 Law of reflection

       4.4 Law of refraction

       4.5 The Fermat principle and Snell’s law

       4.6 Malus–Dupin’s theorem

       4.7 End notes

       Further reading

       5 Stigmatism and stigmatic reflective surfaces

       5.1 Introduction

       5.2 Aberrations

       5.3 Conic mirrors

       5.4 Elliptic mirror

       5.5 Circular mirror

       5.6 Hyperbolic mirror

       5.7 Parabolic mirror

       5.8 End notes

       Further reading

       6 Stigmatic refractive surfaces: the Cartesian ovals

       6.1 Introduction

       6.2 Stigmatic surfaces

       6.2.1 Case I: ro=ri=0,zo→−∞ and zi=f

       6.2.2 Case II: ro=ri=0,zo=f and zi→−∞

       6.3 Analytical stigmatic refractive surfaces

       6.3.1 Case A: ro=ri=0, zo→−∞ and zi=f

       6.3.2 Case B: ro=ri=0,zo=f and zi→−∞

       6.3.3 Case C: ro=ri=0,zo=∓f and zi=±f

       6.3.4 Case D: ro=ri=0,zo=−αf and zi=+f

       6.3.5 Case E: ro=ri=0,zo=αf and zi=−f

       6.4 Conclusions

       Further reading

       7 The general equation of the Cartesian oval

       7.1 From Ibn Sahl to Rene Descartes

       7.2 A generalized problem

       7.3 Mathematical model

       7.4 Illustrative examples

       7.5 Collimated input rays

       7.6 Illustrative examples

       7.7 Collimated output rays

       7.8 Illustrative examples

       7.9 Reflective surface

       7.9.1 Parabolic mirror

       7.10 Illustrative examples