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  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;
Bondarescu et al.  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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