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 refers to a mode of width parameter . In words, we add elementary Gaussian waves whose propagation axes generate a cone of small aperture , having its axis along the direction and its vertex at . We assume that this is the situation at the middle of a 3 km cavity, and are interested in the amplitude at the end (or input) mirror. A sum of elementary Gaussian beams may be propagated by propagating each component separately and summing up at the end. We get, up to some phase factors irrelevant for expressing the intensity, the following amplitude for the mode at any abscissa ; where and (as above). is a normalization factor. We have the standard relation . This formula makes it clear that this solution is a Bessel mode truncated by a Gaussian envelope. We therefore call these “Gauss–Bessel” modes. The intensity pattern of such modes depend obviously on the width parameter and on the aperture angle . Combinations of these exist such that the intensity pattern is spread on the mirror surface, apart from a central peak. An example is shown in Figure 5. The wavefront is nearly conical (see Figure 6).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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