A spectrometer slit has a rectangular response of height and width

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Interference and diffraction

Mathematical Methods for Physics, Ch Fourier transforms, spectra and convolution

Problem

A spectrometer slit has a rectangular response of height

1

and width

w = 0.40\,\mathrm{nm}

, so

\tilde g(k) = 2\sin(kw/2)/k

and

\tilde g(0) = w

. Two equal, narrow spectral lines sit at

x = \pm d/2

. Each line's own transform stays within 2 per cent of its value at

k = 0

for

|k| < 16\,\mathrm{nm^{-1}}

, so treat it as constant there. The transform of the record,

\tilde f(k)\tilde g(k)

, is measured with an error of fixed size at every

k

, at most 1 per cent of the record's value at

k = 0

. The true transform

\tilde f(k)

is recovered by dividing the record by

\tilde g(k)

. ((a)) Show that the error in the recovered

\tilde f(k)

, as a fraction of

\tilde f(0)

, is at most

0.01\big/\left|\dfrac{\sin(kw/2)}{kw/2}\right|

. Find the largest

k_{\max}

such that this stays within 10 per cent for every

k

from

0

k_{\max}

. ((b)) Show that the pair of lines has

\tilde f(k) = \tilde f(0)\cos(kd/2)

for

|k| < 16\,\mathrm{nm^{-1}}

, while a single line would give a constant. For

d = 0.20\,\mathrm{nm}

, evaluate at

k = 10.0\,\mathrm{nm^{-1}}

the error bound from part (a) and the fall

1 - \tilde f(k)/\tilde f(0)

. Can this record tell the pair from a single line? ((c)) Repeat part (b) for lines

0.05\,\mathrm{nm}

apart, and decide whether the record can separate them.