This problem has two independent parts

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

Optics and Photonics, Ch Image formation by mirrors and lenses

Problem

This problem has two independent parts, both using the ray transfer matrices built in this chapter for the paraxial ray column (y,nθ)(y, n\theta), with reduced angle written u=nθu = n\theta. ((a)) A single refracting surface of power D=(n2−n1)/R\mathcal{D} = (n_2-n_1)/R has ray transfer matrix (10−D1).\begin{pmatrix}1&0\\-\mathcal{D}&1\end{pmatrix}. Two paraxial rays, labelled 1 and 2, arrive at the surface with heights y1,y2y_1, y_2 and reduced angles u1,u2u_1, u_2. For a pair of rays at any plane define H=y1u2−y2u1.\mathcal{H} = y_1u_2 - y_2u_1. Apply the refraction matrix to each ray separately to find y1′,u1′y_1', u_1' and y2′,u2′y_2', u_2' just after the surface, substitute into the definition of H\mathcal{H} evaluated after the surface, and show that it equals the value of H\mathcal{H} evaluated before the surface. ((b)) A thin lens of index nln_l sits in air (n=1n = 1 on both sides) and has two spherical surfaces of radii R1R_1 and R2R_2, separated by a gap of zero axial length (the thin-lens limit, in which the lens has no thickness). Write down the refraction matrix at each surface and the translation matrix for the zero-length gap between them, multiply the three matrices in the order convention CV2 requires, and show that the system matrix is (10−1/f1),1f=(nl−1)(1R1−1R2),\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}, \qquad \frac{1}{f} = (n_l-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right), the lensmaker's equation for a lens in air.

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