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## Patent Issued for Method and Apparatus for Producing Stationary Intense Wave Fields of Arbitrary Shape

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".psi..function..rho..times..times..theta..times..times..function..omega..- times..times..times..theta..rho..times.eI.times..omega..times..times..thet- a..times..times..times..theta..times. ##EQU00003## On using the usual variables (.omega., .beta., k.sub..rho.), where .beta.e k.sub..rho. are the longitudinal and transverse wavenumbers, and .omega.=c.beta./cos .theta., the Bessel beam writes

".psi..function..rho..function..rho..times..rho..times.eI.beta..times..tim- es..times.eI.omega..times..times..rho..omega..beta..gtoreq..omega..beta.&g- t; ##EQU00004##

"The conditions .omega./.beta.>0 and k.sub..rho..sup.2.gtoreq.0 ensure forward propagation, with no evanescent waves, and physical behaviour to the Bessel function J.sub.o respectively. In the case of electromagnetic waves, quantity c represents the ordinary speed of light in vacuum.

"Now, let us consider the following superposition of Bessel beams with the same frequency .omega..sub.0:

".PSI..function..rho.eI.omega..times..times..times..times..function..rho..- times..times..times..rho..times.eI.beta..times. ##EQU00005## where n are integer numbers, A.sub.n are constant coefficients, and .beta..sub.n, k.sub..rho.n are the longitudinal and transverse wavenumbers (still to be determined), respectively; where, for each n, the parameters (.omega..sub.0, k.sub..rho.n, .beta..sub.n) must satisfy Eq. (3).

"Because of conditions (3) and (4), we must have

".ltoreq..beta..ltoreq..omega. ##EQU00006##

"Now, our goal is using Eq. (5) to obtain a predetermined longitudinal intensity pattern within the interval 0.ltoreq.z.ltoreq.L.

"Let us suppose that the desired pattern in the interval 0.ltoreq.z.ltoreq.L is given by a function F(z). We know that in that interval we can expand function F in a Fourier series

".function..infin..infin..times..times..times.eI.times..times..times..time- s..pi..times..times..times..times..intg..times..function..times.eI.times..- times..times..times..pi..times..times..times.d ##EQU00007##

"It would be natural to use .beta..sub.n=2.pi.n/L in Eq. (5), with A.sub.n=B.sub.n. HOWEVER this choice would imply negative values of .beta..sub.n (backward waves), what is forbidden by our condition (4), because .omega..sub.0 is obviously positive.

"To overcome this problem we write .beta..sub.n as:

".beta..times..times..pi..times. ##EQU00008## where Q>0 is a value chosen depending on the conditions of the given experimental situation. According to Eq. (6), we have

".ltoreq..+-..times..times..pi..times..ltoreq..omega. ##EQU00009##

"The in equation (10) determines the maximum value of n, that we call N, once we have chosen Q, L and .omega..sub.0. in this way, to obtain a longitudinal pattern of intensity approximately equal to the desired one, F(z), in the interval 0.ltoreq.z.ltoreq.L, the solution (5) should be written as:

".PSI..function..rho.eI.times..times..omega..times..times.eI.times..times.- .times..times..times..times..times..times.eI.times..times..pi..times..time- s..times..times..times..times..intg..times..function..times.eI.times..time- s..times..times..pi..times..times..times.d ##EQU00010##

"Obviously, we get only an approximation of the desired longitudinal pattern because the trigonometric series (11) has been truncated. The number of terms is defined, once the values of Q, L and .omega..sub.0 are chosen.

"When .rho..noteq.0, .PSI.(.rho., z, t) is

".PSI..function..rho.eI.times..times..omega..times..times.eI.times..times.- .times..times..times..times..times..times..function..rho..times..times..ti- mes..rho..times.eI.times..times..times..pi..times..times..times..times..ti- mes..rho..times..times..omega..times..times..pi..times..times. ##EQU00011##

"The coefficients A.sub.n will give the amplitudes and the relative phases of each Bessel beam in the superposition.

"Because we are adding together zero order Bessel functions, we can expect a high field concentration around .rho.=0."

For more information, see this patent: Abate, Valerio; Dartora, Cesar Augusto; Figueroa, Hugo Enrique Hernandez; Mattiuzzi, Marco; Nobrega, Kleber Zuza; Rached, Michel Zamboni; Recami, Erasmo. Method and Apparatus for Producing Stationary Intense Wave Fields of Arbitrary Shape. U.S. Patent Number 8509928, filed May 23, 2005, and published online on August 13, 2013. Patent URL: http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=28&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1378&f=G&l=50&co1=AND&d=PTXT&s1=20130813.PD.&OS=ISD/20130813&RS=ISD/20130813

Keywords for this news article include: Bracco Imaging S.P.A.

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Source: Journal of Engineering

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