Difference between revisions of "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.

Refraction at a spherical interface: (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 $