<title>Hybrid method for calculating diffractive optical elements (DOEs) focusing into radial focal domain</title>

Abstract

The problem of focusing into an off-axis radial domain is reduced to the 1-1) task of focusing into a segment. A new iterative method is proposed assuming the calculation of two 1-D Fourier transforms per each iteration and enabling us to generate diffractive optical elements (DOEs) intended for focusing into a radial off-axis focal domain A phase calculated within ray-tracing approach was used as a starting point for iterative procedure.

Technological applications of focusing optical elements

“…Revealing the most effective solution is especially important for basic (the most widespread) problems of focusing. For instance, solving of problems of focusing into a transverse segment [13][14][15] and a ring 6, 16 is basic for focusators directed at letters [17][18] and solving of problems of focusing into a rectangle is basic for design of focusators for planar domains [19][20] . It is interesting that solutions obtained for laser radiation focusing appeared to be useful for focusing of surface electromagnetic waves 21-24 . In some cases (for example, complex forms of distribution of intensity or focal region) it is impossible to obtain an analytic expression for the focusator's phase function.…”

Iterative calculation and technological realization of DOE laser-beam focusing into nonaxial radially symmetrical domains

“…In [5,6J, in view of (2), (4), (5), and (6) it has been proposed that the phase function of a DOE focusing into a radial domain of intensity 1(r), r >>0, r EEl r2 ] should be computed as the phase function of a 1D DOE focusing a beam with the intensity distribution of (7) into a linesegment with the intensity distribution given by Jj(,)=J()r, (8) Note that according to (6), the term w2'r,,) corresponds to the field complex amplitu4e upon focusing into a linesegment J (r), -r2 r -r1 and does not affect the field structure at r [r , r2].…”

“…Thus, the procedure proposed in [5,6] can apply to the iterative design of a DOE focusing the input beam of an arbitrary radially symmetrical structure into a radial off-axis focal domain. In [5,61 the numerical simulation is used to show that the constructed iterative procedure allows one to design the elements that can generate from a uniform illuminating beam the focal distribution in the form of widened ring with a 89-92% energy efficiency and a 7-13% error.…”

“…In this paper we apply the procedure proposed in [5,6] to the design of an element aimed at focusing the Gaussian beam with intensity di3tribution…”

“…The vitality of the problem becomes clearer when we take into account that the relaxation parameters' optimization based on the adaptive modification of the error reduction [21 calls for the multiple iterations of the element calculation and simulation. The design of radial DOEs with the use of the iterative Fienup algorithm has been discussed in [1, 3,4,5,6]. The recalculation of field between the DOE plane and the focal plane in the Fresnel-Kirchhoff approximation corresponds to the computation of the zero-order Hankel transform.…”