"According to the invention, other eigenfunctions of the transverse laplacian operator can be used different from Bessel beams. For instance, stationary fields (FWs) can be obtained by the superpositions furnished by the disclosed method, provided that the said eigenfunctions yield a constant value (the same for all of them) along the z-axis. In such a case, the method allows defining a Fourier series for .alpha.=0 with the same coefficients A.sub.n furnished by the method in the Bessel Beams case.
"As it will become evident form the following description, such waves can be used in a great variety of applications, ranging from optical or electromagnetic tweezers to laser-type surgery and optical or ultrasound lancets, from micro-lithography to ultrasound or electromagnetic thermo-therapy of tumors, from jamming electromagnetic regions (balls) to a new kind of effective holography, from ultrasound kidney stone fragmentation to other ultrasound applications, etc.
"The present invention refers to a method for producing a stationary localized wave field with an a-priori predetermined arbitrary longitudinal shape, inside a periodicity interval 0.ltoreq.z.ltoreq.L, and with high transverse localization, and the apparatuses accordingly designed. Namely, from a general point of view, this invention provides a method for producing a stationary localized wave field of an a-priori predetermined arbitrary shape, comprising the following steps:
"a) defining at least a volume being limited in the direction of the axis z of propagation of a beam, along a certain longitudinal interval 0.ltoreq.z.ltoreq.L;
"b) defining a priori within the said longitudinal interval an intensity pattern .mu.F(z).mu..sup.2 describing the desired localized and stationary wave field which function F(z) is represented by means of a discrete Fourier Series or by a similar expansion in terms of (trigonometric) orthogonal functions;
"c) providing a discrete, generic superposition of Bessel beams or other highly transversally confined beams;
"d) calculating the maximum allowed number of Bessel beams to be superposed;
"e) calculating the amplitudes, and the phase velocities and the transverse and longitudinal wavenumbers of each Bessel beam of the superposition, needed to obtain the predefined STATIONARY intensity pattern (within the predefined longitudinal interval);
"f) recognizing and controlling the effect of each of the said parameters under point e) for controlling the longitudinal shape of the stationary localized wave field.
"According to the present method also a partial control of the transversal shape of the stationary wave field is carried out by superposing for instance high order Bessel beams.
"Thanks to the fact that the above method is based on exact analytical equations the method according to the invention allows a complete control of all 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 predetermined arbitrary shape. The method according to the invention allows to exercise a complete control of the longitudinal shape and at least a partial control of the transverse shape of the stationary wave-field.
"The method according to the invention has the great advantage that it can be translated in a experimental device with which each relevant parameter for controlling the longitudinal and transverse shape of the stationary wave field can be influenced by simple settings of the constructive aspects of the said device.
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