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.

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 ) $

Bi-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.

Types of spherical lenses

Ray Tracing

Ray tracing principles

	Rays passing through the optical center of a lens continue in a straight line
	Rays traveling parallel to the optical axis pass through the focal point after refraction and vice versa
	Parallel rays pass through the same point in the focal plane after refraction and vice versa