A ray refracts at a single spherical surface separating air ( ) from glass ( )

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Geometrical optics and imaging

What an Instrument Can See, Ch The paraxial system as an operator

Problem

A ray refracts at a single spherical surface separating air (n1=1.00n_1=1.00) from glass (n2=1.50n_2=1.50), radius of curvature R=+50 mmR=+50~\text{mm} (centre of curvature to the right). Using the plain-angle ray column (y,θ)(y,\theta), the refraction matrix is M=(10−(n2−n1)/(n2R)n1/n2).\mathbf{M}=\begin{pmatrix}1 & 0\\ -(n_2-n_1)/(n_2R) & n_1/n_2\end{pmatrix}. Someone computes C=−(0.50)/(1.50×50 mm)=−0.00667 mm−1C=-(0.50)/(1.50\times 50~\text{mm})=-0.00667~\text{mm}^{-1} and D=1/1.5=0.667D=1/1.5=0.667, then argues: "every ray transfer matrix has determinant 1, so something is wrong here, because AD−BC=1×0.667−0×C=0.667≠1AD-BC=1\times0.667-0\times C=0.667\ne1." ((a)) Compute det⁡M\det\mathbf{M} for this matrix and state whether it equals 1. ((b)) Explain the flaw in the reasoning quoted above, distinguishing the (y,nθ)(y,n\theta) and (y,θ)(y,\theta) columns.

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