".phi..function..intg..infin..infin..times..function..times..function.I.pi- ..times..times..times.d ##EQU00001## and its inverse function
".intg..infin..times..phi..function..times..function.I.pi..times..times..t- imes.d ##EQU00002## This method making use of the traditional Fourier Integrals the disclosed method need to use values of S which runs also over negative values.
"Furthermore the method disclosed in the above mentioned publication uses also a Fourier-Bessel transform. As it is well known Bessel beams constitute a base which can be used to express any function. In the disclosed method integrals running over positive values of S only since the Fourier-Bessel transform cannot accept negative values of the integration variable and integrals running also over negative values of the integration variable are mixed together. This approximation is justified as mentioned in the document only if the shift-term S could tend to infinity. However the paraxial approximation assumed in the above cited document is known as hold only for small values of S. Thus the method disclosed in the above cited publication is limited to configurations of the beam shape and dimensions for which the paraxial approximation is valid. For cases for which the paraxial condition is not met the authors are obliged to carry out numerical simulations. The approximate numerical simulation taught in the above publication aims only to the production of a flat-top beam and the document does not teach how to evaluate amplitudes and phases of the Bessel beams by a closed-form equation and not even by general numerical procedures. Only nan hope of generalisation of the disclosed approach is expressed by the authors.
"Thus principally due to the fact that the disclosed method according to the above publication is limited by the role of the negative values of S and by the condition of the paraxial approximation this disclosed method does not allow a general treatment which is based on exact analytical equations which allow a complete control of all the parameters entering into play and of their experimental effects for producing in a simple way and without any approximation a stationary localized wave-field with an a-priori determined arbitrary shape.
"A first target of the present invention is to provide a method for generating spatially confined stationary wave fields with any arbitrary (longitudinal) shape in correspondence with a segment of the type 0.ltoreq.z.ltoreq.L, quantity L being a periodicity interval which overcomes the drawbacks of the known methods.
"A second aim is to provide a method for producing a source which generates the above-said spatially confined stationary wave field.
"A further object of the present invention is to provide for an apparatus for generating a spatially confined stationary wave field."
In addition to obtaining background information on this patent, VerticalNews editors also obtained the inventors' summary information for this patent: "The invention is based on the fact that suitable superpositions of Bessel beams, can yield stationary wave fields with high transverse localization, and with an arbitrary longitudinal shape. These superpositions of Bessel beams are particularly suitable when they have the same frequency. They are new solutions of the wave equation, and may be called 'Frozen Waves' (FWs). They are created with any arbitrary shape inside a spatial region corresponding to a longitudinal segment of the type 0.ltoreq.z.ltoreq.L, and their general 'envelope' does not propagate. Moreover, their shape can be such that the stationary field is concentrated only within one or more (small) portions defined herein as sub-envelopes, L-1.ltoreq.z.ltoreq.L-2 (of the interval 0.ltoreq.z.ltoreq.L), while a negligible field is expected to exist before or after the said envelope or sub-envelopes of the stationary field. Inside the said envelope or sub-envelopes a plane carrier wave is propagating, as it will be disclosed with greater detail in the following description; while the said envelope or sub-envelopes remain stationary
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