Chapter 11 read the Fabry-Perot etalon as a linear shift-invariant system by asking…

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What an Instrument Can See, Ch The lens as a Fourier transformer

Problem

Chapter 11 read the Fabry-Perot etalon as a linear shift-invariant system by asking the filter template's four questions: what is the system, what is its impulse response, what is its transfer function, and what bandwidth does that transfer function set. Its transfer function turned out to be

H(\nu) = t_1t_2/(1-r_1r_2e^{i2\pi\nu T})

, built from the cavity's round-trip time

T

and its round-trip amplitude factor

r_1r_2

. Apply the same four questions to free space, with a field's angular spectrum amplitude

A(\nu_x,\nu_y;0)

as the input and

A(\nu_x,\nu_y;z)

as the output. ((a)) Name this chapter's transfer function

G(\nu_x,\nu_y)

, and state the spatial-frequency analogue of the etalon's bandwidth question: up to what value of

\nu\equiv\sqrt{\nu_x^2+\nu_y^2}

|G|=1

? ((b)) A field is decomposed into two angular-spectrum components, at

\nu_1 = 800

cycles per millimetre and

\nu_2 = 2000

cycles per millimetre, propagating through air at

\lambda = 550\ \mathrm{nm}

over

z = 5.0\ \mu\mathrm{m}

. Find

|G|

for each component, and state which one behaves like a signal inside the etalon's passband (unattenuated, only phase-shifted) and which behaves like one well outside it.

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