Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 9 / b / ii

Place the interface at

y = 0

, with source

(x_{A}, h_{1})

and observer

(x_{B}, - h_{2})

in the two uniform media. Let

(s, 0)

be the crossing point. The optical length is

S (s) = μ_{1} (s - x_{A})^{2} + h_{1}^{2} + μ_{2} (x_{B} - s)^{2} + h_{2}^{2} .

Differentiating with respect to

s

and setting the result to zero gives

μ_{1} \frac{s - x _{A}}{( s - x _{A} ) ^{2} + h _{1}^{2}} = μ_{2} \frac{x _{B} - s}{( x _{B} - s ) ^{2} + h _{2}^{2}} .

Recognizing the ratios as the sines of the angles measured from the interface normal proves Snell's law:

μ_{1} sin θ_{1} = μ_{2} sin θ_{2} .

The stationarity condition is the continuity of the tangential optical momentum across the planar interface.

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