Difference between revisions of "Geometrical optics and ray tracing"
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* We shall revisit these assumptions later. | * We shall revisit these assumptions later. | ||
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− | [[Image: 20.309 130819 RefSphere2.png|center| | + | !colspan="3"|[[Image: 20.309 130819 RefSphere2.png|frameless|center|750px]] |
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− | + | |a) <math> S_o > {n\ R \over (n'\ - n)}\ \Rightarrow S_i > 0</math> , | |
− | + | |b) <math> S_o = {n\ R \over (n'\ - n)}\ \Rightarrow S_i \to + \infty</math>, | |
− | + | |c) <math> S_o = {n\ R \over (n'\ - n)}\ \Rightarrow S_i < 0</math> | |
− | + | |} | |
{{Template:20.309 bottom}} | {{Template:20.309 bottom}} |
Revision as of 16:52, 19 August 2013
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 θi = θr
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
- Si 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 $ |