Snell's Law

Reference derivation for Reflection and Refraction in a Raytracer

Overview

Snell's law is a an ancient empircal result, but it can be recovered a priori by a variety of paths (no pun intended)

  • Ray model & Fermat's principle with calculus
  • Wavefront model & Huygen's princple with geometry

Ray model & Fermat's principle with calculus

Fermat's principle definition

The path taken by a ray between two given points is the path that can be traversed in the least amount of time.

This is something that is taken for granted now, but it does give pause to think how strange it is to to ascribe behavior to nature and it was quite controversial in its time.

Problem Setup

setup1.png

Figure 1: Refraction of light at the interface between two media of different refractive indices

Preliminary values to be used (\(d_i\) is the length of the ray):

\[d_1 = \sqrt{a^2 + {x_1}^2}\] \[d_2 = \sqrt{b^2 + {x_2}^2}\] \[\sin{\alpha} = \frac{x_1}{d_1} = \frac{x_1}{\sqrt{a^2 + {x_1}^2}}\] \[\sin{\beta} = \frac{x_1}{d_2} = \frac{x_2}{\sqrt{b^2 + {x_2}^2}}\]

Speed of ray (where \(\eta\) is the respective index of refraction of the medium): \[s_i = \frac{c}{\eta_i}\]

The total time taken for the ray: \[T_{total} = t_1 + t_2\]

Derivation

Definition of speed: \[s_i = \frac{d_i}{t_i} = \frac{c}{\eta_i}\] \(\implies\) \[t_i = \frac{\eta_i}{c} d_i\] \(\implies\) \[T_{total} = \frac{\eta_1}{c} d_1 + \frac{\eta_2}{c} d_2\] \[T_{total} = \frac{\eta_1}{c} \sqrt{a^2 + {x_1}^2} + \frac{\eta_2}{c} \sqrt{b^2 + {x_2}^2}\]

To invoke Fermat's principle, we want to minimize this function (we find a critical point which may or may not be a minimum) and we need to make a small substituion so that we can take a whole derivative with respect to the distance

\[T_{total} = \frac{\eta_1}{c} \sqrt{a^2 + {x_1}^2} + \frac{\eta_2}{c} \sqrt{b^2 + {\left(l - x_1 \right)}^2}\]

Taking a derivative with respect to distance and setting it to zero to find its critical points: \[ 0 = \frac{1}{2} {\left( a^2 + {x_1}^2 \right)}^{-\frac{1}{2}} 2x_1 \frac{\eta_1}{c} + \frac{1}{2} {\left( b^2 + {\left(l - x_1 \right)}^2 \right)}^{-\frac{1}{2}} \left( {2x_1 - 2l} \right) \frac{\eta_1}{c} \] \(\implies\) \[ 0 = \cancel{\frac{1}{2}} {\left( a^2 + {x_1}^2 \right)}^{-\frac{1}{2}} \cancel{2}x_1 \frac{\eta_1}{c} + \cancel{\frac{1}{2}} {\left( b^2 + {\left(l - x_1 \right)}^2 \right)}^{-\frac{1}{2}} \cancel{2}\left( {x_1 - l} \right) \frac{\eta_1}{c} \]

\[ 0 = \frac{x_1}{\sqrt{a^2 + {x_1}^2}} \frac{\eta_1}{\cancel{c}} - \frac{ \left( l - x_1 \right) }{\sqrt{b^2 + {\left(l - x_1 \right)}^2}} \frac{\eta_2}{\cancel{c}} \] Referencing our preliminary trig expressions, substitute with \(\sin{\alpha}\) and \(\sin{\beta}\)

\[ 0 = \sin{\alpha} \eta_1 - \sin{\beta} \eta_2 \] \(\implies\) \[ \eta_1 \sin{\alpha} = \eta_2 \sin{\beta} \]

Wavefront model & Huygen's princple with geometry

It feels slightly out of place to bring in Huygen's principle (in the context of geometric optics for computer graphics) as it occupies some middle ground between ray and wave representations of light, but intuition and trust in something is built by reaching the same result from many, disparate paths.

It's also very elegant in that Snell's law pops out as a purely geometrical argument.

Huygen's principle definition

Every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms the wavefront.

Here is a picture of a wavefront as originally conceived by Huygen from Wikipedia

Derivation

Here is a well explained and animated derivation, check it out.