A system consists of two thin lenses in air

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 system consists of two thin lenses in air, focal lengths

f_1

and

f_2

, separated by

d>0

. Building the system matrix from the front lens to the back lens gives

A=1-d/f_1

D=1-d/f_2

and

C=-1/f_{\mathrm{eff}}

with

1/f_{\mathrm{eff}}=1/f_1+1/f_2-d/(f_1f_2)

. ((a)) Write

h_1=(D-1)/C

(front principal plane, from the front lens) and

h_2=(1-A)/C

(rear principal plane, from the back lens), and show that the signed gap by which the rear principal plane sits in front of the front one,

\mathrm{gap}=h_1-(d+h_2)

, reduces to the single closed form

\mathrm{gap}=\frac{d^2}{f_1+f_2-d}.

((b)) State the condition on

f_1

f_2

and

d

for the two principal planes to be crossed (rear in front of front), and the condition for them to keep their usual order. ((c)) Classify each of these three systems as crossed or not, and by how much: (i)

f_1=150\ \mathrm{mm}

f_2=-60\ \mathrm{mm}

d=40\ \mathrm{mm}

; (ii)

f_1=90\ \mathrm{mm}

f_2=110\ \mathrm{mm}

d=45\ \mathrm{mm}

; (iii)

f_1=60\ \mathrm{mm}

f_2=-90\ \mathrm{mm}

d=20\ \mathrm{mm}

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