Geometrical optics and ray tracing

Refraction and reflection

Refraction and reflection at a boundary

• The Snell-Descartes law or law of refraction stipulates that

$ n_i\ \sin \theta_i = n_t\ \sin \theta_t $

with θ the angle measured from the normal of the boundary, $ n $ the refractive index (which is unitless) of the medium, the subscripts $ i $ and $ t $ referring to the incident and transmitted light, respectively.

• The law of reflection states that θ_i = θ_r

Refraction and reflection at a spherical interface

With the assumptions:

• Paraxial approximation: θ ≈ sin θ ≈ tan θ
• Thin lens approximation: $ R << S_o,\ S_i $

Snell's law predicts that

$ n\ \sin \theta_1 = n'\ \sin \theta_2 $

$ \sin \theta_1 \approx \sin a + \sin b \approx {h \over S_o} + {h \over R} $

$ \sin \theta_2 \approx \sin b - \sin c \approx {h \over R} - {h \over S_i} $

$ {n \over S_o} + {n' \over S_i} = {(n'\ - n)\over R} $

Note that

• S_i does not depend on the angle $ a $.
• Light coming from a point on the filament passes through a point after refraction.
• We shall revisit these assumptions later.

a) $ S_o > {n\ R \over (n'\ - n)}\ \Rightarrow S_i > 0 $ ,	b) $ S_o = {n\ R \over (n'\ - n)}\ \Rightarrow S_i \to + \infty $,	c) $ S_o = {n\ R \over (n'\ - n)}\ \Rightarrow S_i < 0 $

Lenses

Lens maker formula

A simple lens consists of two spherical interfaces. Its focal length $ f $ is given by the lens maker formula:

$ {1 \over S_o} + {1 \over S_i} = {1 \over f} = {(n'\ - n) \over n} \left ( {1 \over R_1} - {1 \over R_2} \right ) $

Convex lens	Plano-concave lens

• Use the image from the first refraction as the object for the second.
• Note the sign convention for the second surface: $ R_2 $ < 0 for a convex lens, and $ (n'\ - n) $ has opposite sign.
• Concave lens: the lens maker formula holds for $ R_1 $ < 0.