A long glass rod of index terminates in a single spherical surface of radius (its…

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

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

Problem

A long glass rod of index

n_1 = 1.50

terminates in a single spherical surface of radius

R = -20\ \mathrm{mm}

(its centre of curvature lies

20\ \mathrm{mm}

inside the rod, to the left of the surface); beyond the surface is air,

n_2 = 1.00

. A point object sits on the axis inside the rod,

s_o = 80\ \mathrm{mm}

to the left of the surface. ((a)) Build the system matrix from the object plane to the image plane using the general reduced-angle translation matrix

\begin{pmatrix}1 & d/n\\0&1\end{pmatrix}

for a gap

d

in a medium of index

n

, and the refraction matrix

\begin{pmatrix}1&0\\-(n_2-n_1)/R&1\end{pmatrix}

. Impose

B=0

and show that the conjugate condition takes the form

n_1/s_o + n_2/s_i = (n_2-n_1)/R

. ((b)) Use that condition to find the image distance

s_i

. ((c)) Read the transverse magnification off the matrix element

A

, and state whether the image is real or virtual, and upright or inverted.