Spiral beams, coherent states and wavelets

It is well known that successful and constructive solving of a problem is essentially depends on the used tools. For example, Hermite-Gauss beams are convenient for a physical problems with rectangular symmetry, and Laguerre-Gauss beams are preferable for the problems with a rotational symmetry. Another way is the usage of coherent states. It is turned out that the coherent states are very effective for solving some problems of coherent optics, such as the synthesis of light fields with predetermined intensity distribution, or spiral beams. In [1] it was shown that there are structurally stable light fields (spiral beams) that have intensity distribution in the shape of an arbitrary planar curve. The properties of spiral beams are similar to the properties of coherent states in quantum mechanics. Moreover, the specific astigmatic transformation of spiral beams shaped like a planar curve leads to a synthesis of complex wavelets [2]. This possibility is based on the following property: spiral beams are orthogonal ones if and only if the corresponding astimagtic transformed functions (wavelets) are orthogonal ones. The structure of these wavelets depends on the shape of an appropriate generating curve. In this work the approach for the wavelet synthesis, based on the spiral beams, and some examples are presented.