The search for weakly diffracting beams leads naturally to nondiffractive beams. There is an obvious solution to Helmholtz’s equation in cylindrical coordinates;
where is an arbitrary integer,
a Bessel function, and
and
are numbers such that
(
). This was first noted (in the case
) by Durnin [16] and is thus called a
“Durnin” beam by some. We feel that for clarity it is more appropriate to call it the “Bessel” beam. The
case of
is particularly interesting in the context of hyper-resolution, for instance, due to the
sharp central peak when
is large, but it is forbidden in our case for the same reason. The
transverse structure of such a wave is independent on
and similar to a wave guided in the core
of a cylindrical fiber (in the cladding, there is a different solution smoothly matched and of
finite extension). However, the energy carried by a Bessel beam is infinite, exactly as in the
case of a plane wave. In fact, the wavefront is flat in the case of
. The impossibility of
generating waves of infinite extension leads to truncated waves having diffractive behavior and
consequent clipping losses. The result depends on the method of truncation. The wavefront of such a
truncated Bessel wave after propagation is hardly compatible with a reasonable mirror shape
anyway.
The best way of truncating a Bessel beam is to make the following construction;
where Bondarescu et al. [4] have carried out an optimization of coating thermal noise by combining LG modes.
Using the better series of coefficients, they reach a wave analogous to a Gauss–Bessel mode and with a
conical wavefront of the same kind. We intend to include these kinds of modes in an update to
this review. To be specific, we give, in the section related to coating thermal noise (8.3.2), the
figure of merit of the mode described in Figures 5
and 6
, which is not optimal, but already
exhibits a good value, regarding coating thermal noise in the infinite mirror approximation (see
Section 8.3).
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