A frequency-metrology laboratory must resolve the two hyperfine components of the…

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

Optics and Photonics, Ch Fraunhofer diffraction from apertures and gratings

Problem

A frequency-metrology laboratory must resolve the two hyperfine components of the

632.8\ \text{nm}

line from a stabilised helium-neon laser, separated by

\Delta\lambda=2.0\times10^{-3}\ \text{nm}

. The instrument only ever needs to display a window

0.050\ \text{nm}

wide around the line. Two candidates are available: a diffraction grating ruled at

1200

lines per millimetre, and an air-spaced Fabry-Perot etalon of finesse

\mathcal{F}=30

whose plate spacing

d

can be chosen freely. The grating is used in first order; for the etalon, the order

m

and free spectral range are related, as for any such device, by

\Delta\lambda_{\rm FSR}=\lambda/m

, and the plate spacing sets the order through

m=2d/\lambda

at normal incidence. ((a)) Find the resolving power

\lambda/\Delta\lambda

the instrument must supply. ((b)) For the grating, find the number of illuminated lines

N

needed in first order and the corresponding minimum illuminated width, and comment on whether this is a practical size. ((c)) For the etalon, use its resolving power

m\mathcal{F}

to find the order

m

needed, then the plate spacing

d=m\lambda/2

, and check that the resulting free spectral range

\lambda^2/(2d)

is at least as large as the

0.050\ \text{nm}

window that must be displayed. ((d)) State which device should be used.

Related theory

Interference and diffraction

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