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In theory the two end mirrors must be highly parallel to each other.
However, this condition is almost impossible to maintain in practice
because expansion and contraction of the mirror supports occur
as the laser is heated and cooled during normal operation.
Instead of using flat mirrors that are parallel to each other, modern
helium-neon lasers use concave mirrors spaced in a semi-confocal
arrangement.
Also it is important to add that the end mirrors are not the common
silver-backed cosmetic mirrors but are made of transparent glass
with thin film quarter-wavelength front-surface dielectric coatings
that reflect photons of the desired wavelength and effectively
absorb those of other wavelengths.