Producing acoustic ‘Frozen Waves’: Simulated experiments with diffraction/attenuation resistant beams in lossy media

$Page 1: Producing acoustic ‘Frozen Waves’: Simulated experiments with diffraction/attenuation resistant beams in lossy media$
Ultrasonics xxx (2014) xxx–xxx

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Ultrasonics

journal homepage: www.elsevier .com/locate /ul t ras

Producing acoustic ‘Frozen Waves’: Simulated experimentswith diffraction/attenuation resistant beams in lossy media

http://dx.doi.org/10.1016/j.ultras.2014.03.0080041-624X/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +55 19 35213793.E-mail address: [email protected] (M. Zamboni-Rached).

Please cite this article in press as: J.L. Prego-Borges et al., Producing acoustic ‘Frozen Waves’: Simulated experiments with diffraction/attenuation rebeams in lossy media, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.03.008

José L. Prego-Borges a, Michel Zamboni-Rached a,⇑, Erasmo Recami a,b,c, Eduardo Tavares Costa a

a Faculty of Electrical and Computer Engineering, State University of Campinas, Campinas, SP, Brazilb INFN-Sezione di Milano, Milan, Italyc Faculty of Engineering, Bergamo State University, Bergamo, Italy

a r t i c l e i n f o

Article history:Received 15 January 2014Received in revised form 12 March 2014Accepted 13 March 2014Available online xxxx

Keywords:‘‘Frozen Waves’’DiffractionAttenuationBessel beam superpositionsAnnular transducers

a b s t r a c t

The so-called Localized Waves (LW), and the ‘‘Frozen Waves’’ (FW), have raised significant attention inthe areas of Optics and Ultrasound, because of their surprising energy localization properties. The LWsresist the effects of diffraction for large distances, and possess an interesting self-reconstruction—self-healing— property (after obstacles with size smaller than the antenna’s); while the FWs, a sub-classof LWs, offer the possibility of arbitrarily modeling the longitudinal field intensity pattern inside aprefixed interval, for instance 0 6 z 6 L, of the wave propagation axis. More specifically, the FWs arelocalized fields ‘‘at rest’’, that is, with a static envelope (within which only the carrier wave propagates),and can be endowed moreover with a high transverse localization.

In this paper we investigate, by simulated experiments, various cases of generation of ultrasonic FWfields, with the frequency of f0 ¼ 1 MHz in a water-like medium, taking account of the effects of attenu-ation. We present results of FWs for distances up to L ¼ 80 mm, in attenuating media with absorptioncoefficient a in the range 70 6 a 6 170 dB=m.

Such simulated FW fields are constructed by using a procedure developed by us, via appropriate finitesuperpositions of monochromatic ultrasonic Bessel beams.

We pay due attention to the selection of the FW parameters, constrained by the rather tight restrictionsimposed by experimental Acoustics, as well as to some practical implications of the transducer design.

The energy localization properties of the Frozen Waves can find application even in many medicalapparatus, such as bistouries or acoustic tweezers, as well as for treatment of diseased tissues (inparticular, for the destruction of tumor cells, without affecting the surrounding tissues; also for kidneystone shuttering, etc.).

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The phenomena of diffraction, dispersion and attenuation arephysical effects that usually perturb the propagation of waves. Dif-fraction produces a gradual spatial spreading of the wave; whiledispersion produces a temporal spreading. Dispersion, for example,is present whenever the refraction index of the medium dependson the frequency, producing, as a final result, a progressive increasein the temporal width of the propagating pulses.

Attenuation reduces the amplitude of a propagating wave, bytransforming, roughly speaking, part of its energy into heat.

To circumvent these difficulties, which affect the developmentof any technological applications, increasing interest was paid to

the Non-Diffracting Solutions to the wave equations, also knownas ‘‘Non-Diffracting Waves’’ or Localized Waves (LW). These waveswere first theoretically predicted [see, e.g., Ref. [1]], and then —toconfine ourselves to the simplest case of the Bessel beams— exper-imentally produced in a series of papers, among which Refs. [2–4].

Indeed, the LWs are solutions to the wave equations capable ofresisting the effects of diffraction, and in some cases even ofattenuation, at least up to a certain distance (depth of field). Suchfields, being solutions to the wave equations, can find applicationin diverse technological areas, in Optics [5–9], in Acoustics[10,11,15,20,5,6], and in Geophysics [21]; besides playing interest-ing theoretical roles even in special relativity [22–25], quantummechanics [26], etc. In a very large number of theoretical andexperimental works, such soliton-like solutions to the linear waveequations have been shown to be endowed with peak-velocities Vranging from 0 to 1; even if they have been extensively studied

sistant

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[5–7] mainly for their peculiar properties, like their self-recon-struction (‘‘self-healing’’) after obstacles with size much largerthan the wavelength, provided that it be smaller than their anten-na, and not at all for their speed.

The LWs can be either Localized Beams or Localized Pulses. Forreviews, one may consult the already quoted Refs. [5,7,8,6] andRefs. therein. Rather famous became the ‘‘superluminal’’ pulses,called X-shaped waves [10,11,27–29]. [Incidentally, the superlumi-nal (or supersonic) solutions, which will not be considered in thispaper, have been shown since long to belong to ordinary standardphysics, and agree with special relativity itself [22,5,6].]

If we refer more specifically to the application of LWs in theareas of Acoustics, one has first to recall the works by Lu et al.[10,11,15–19], whose principal interest was the application ofthe ultrasonic X-shaped pulses (endowed with a supersonic peak-velocity, in this case) to medical imaging. Other research relatedwith ultrasonic localized pulses can be found —besides in pioneer-ing papers like [12–14]— also in [30–32]. In connection with singleBessel beams, one can recall for instance papers like Refs. [33–35];while, with regard to the use of annular transducers, one can recallfor example Refs. [36–41]. Among the many existing papers, let usmention Refs. [42–46] related with annular arrays; or Refs. [46–50]for segmented annular arrays; or Refs. [51–54] for Bessel beamsuperpositions suitable in the case of ultrasound wave scatteringby spherical objects.

As said before, LWs can have peak-velocities V ranging from 0 to1. There exist also the subluminal ones (properly speaking subsonic,in the case of Acoustics), which have been investigated mainly inRef. [25], after some pioneering or preliminary works [55–57].

In the ‘‘subsonic’’ area, some publications [58–61], most of themin Optics, involved superpositions of monochromatic Bessel beamswith the aim of modeling, in an arbitrary way, the shape of the LWs.Indeed, the (subluminal) localized pulses reduce to (monochro-matic) beams when their peak-velocity tends to zero ðV ! 0Þ. It isjust in the case of such waves ‘‘at rest’’ —that is, with a static enve-lope, within which only the carrier wave propagates— that it be-came possible to model the LW shape in a pre-fixed way (andinside a chosen space interval). Such peculiar localized waves werecalled Frozen Waves (FW) by us: see Refs. [58–60,62], and Ref. [23]which just appeared in Chap. 1 of the new book [6]. The FWs wereexperimentally generated for the first time, in 2012, in Optics [9];while computer simulations of the experimental production ofultrasonic FWs have recently yielded promising results [20].

We initially developed the theory of FWs for lossless media[58], but subsequently it was extended by us for absorbing media[60]: We were able to spatially model non-diffracting beams in or-der to obtain beams resistant to both diffraction and attenuation.That was performed by following the procedures exploited inRef. [60], which allowed for appropriate finite superpositions ofmonochromatic zero-order ðl ¼ 0Þ Bessel beams, with differentlongitudinal wavenumbers bm.

It is possible, of course, to conceive of superpositions of Besselbeams of higher order ðl > 1Þ, and then construct localized ‘‘tubes’’of energy along the z axis, as in Refs. [62,61]. However, this endea-vor —a priori, quite possible in Optics, by utilizing the versatility ofthe spatial light-modulators— becomes a technological challengein the ultrasound case. In fact, it would probably require a furthersubdivision of the transducer rings (that we adopted in our previ-ous paper [20], and will be considered below) into small arc seg-ments, in order to get what is called a segmented annular array[46–50]. In fact, a radiator structure like that seems to be neededbecause of the azimuthal dependence of the fields along the z axis,1

through the phase eil/. This is not a simple problem, if one bears in

1 In this paper we use cylindrical coordinates, q;/; z.

Please cite this article in press as: J.L. Prego-Borges et al., Producing acoustic ‘Frbeams in lossy media, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.20

mind that it would involve a second angular sampling process, be-sides the main sampling process, SP, required to generate the FWpatterns: See Section 6 below. [And a wrong sampled structure ofa segmented annular array could completely distort the entire FWto be generated (an effect expected to occur even in the simple caseof l ¼ 0 Bessel beams)]. To our knowledge, the study of the SP effectfor FWs is still largely unexplored. Moreover, a chosen segmentedannular transducer has to be usable, in principle, for the generationof several FW patterns: That is to say, one and the same particularsegmented annular structure of the radiator ought to be appropriatefor all the FWs to be created.

We feel, as a consequence, that questions like the use of higherorder Bessel beams must wait for a deeper understanding and con-trol of the simple l ¼ 0 case; and in this work we shall confine our-selves to zero-order Bessel beams.

The main practical issue is investigating the suitable ultrasonictransducers: This is of course a key point in order to make ouracoustic FWs realizable. Another related issue is the ‘distorting’ ef-fects possibly introduced by the responses of the individual radia-tor annuli: Namely, the effects of signal attenuation and delayproduced by the yet unknown electrical/mechanical transfer func-tion of each ring of the (piezoelectric) transducer. Once again, wedo not address this point here, since this work is chiefly devotedto the theoretical/simulation aspects of FWs. However we shallcomment on a possible solution to this problem at the end of thepaper. Finally, also the number of electronic channels involved inthe production of the FWs is an important topic during the designprocess; but, with today’s availability of cheap electronics andready-to-use multichannel electronic front ends, this problemlooks much less severe.

The Purpose of this paper is contributing to the creation of Fro-zen Waves in the ultrasound sector, by presenting a series of com-puter simulated experiments for FWs in a water-like medium. Weshall show a modeling of the non-diffracting beams to be possible,so as to overcome the effects of diffraction and attenuation allalong their depth of field. Inclusion of the effect of attenuation israther important for instance when wishing to construct acousticFWs inside the human body, having in mind one of the most usefulapplications [63].

To this aim, Section 2 introduces some generalities on attenua-tion of ultrasound in fluids, in particular in water. The modeling ofFWs in an absorbing medium is presented in Section 3 (for a reviewof the general methodology for the generation of FWs in non-atten-uating media, cf., e.g., Refs. [20,59,5,6]). Section 4 discusses theacoustic restrictions on the values for the parameters to be usedfor ultrasonic FWs: As we shall see, the main limitation in theacoustic regime is related to the low values of the ratio fre-quency/speed of sound in the medium (a problem that is not thatsevere in Optics).

A brief introduction to the impulse response (IR) method,adopted by us for the simulation of acoustic FWs, is made in Sec-tion 6. We shall describe therein the inclusion of the mediumattenuation effect via a linear model of the absorption that affectsthe initial, non-attenuated IR signal at each particular point ofspace.

In Section 7 we show the results of our simulated experimentsfor three different ultrasonic FWs, in a water-like medium (andincluding, as we said, the effect of absorption). In these ‘‘experi-ments’’, all performed with the frequency f0 ¼ 1 MHz, we actuallyexamine three absorption scenarios, with various absorption coef-ficients a in the range 70 6 a 6 170 dB=m.

This paper ends with conclusions, and some new ideas for fu-ture developments.

In an Appendix we further discuss how a proper choice ofparameters like L can help in the experimental generation of theFWs.

ozen Waves’: Simulated experiments with diffraction/attenuation resistant14.03.008


J.L. Prego-Borges et al. / Ultrasonics xxx (2014) xxx–xxx 3

2. Attenuation of ultrasound in water: generalities

The phenomenon of attenuation of ultrasound in fluids may ingeneral be associated to three types of losses or relaxation mecha-nisms: (i) heat conduction, (ii) viscosity, and (iii) internal molecu-lar processes. In polar liquids such as water, the thermal relaxationloss does not account for the excess of absorption observed in real-ity: which is in fact three times bigger than the classical absorptionvalue. Then, the main causes of the attenuation in normal water(not sea water) are due to the other causes, such as viscosity andinternal molecular losses.

Viscous losses happen whenever there is a relative motion be-tween adjacent portions of the medium. This is the case for exam-ple when a wave of sound (a longitudinal wave) producescompressions and expansions in the medium while propagating.This phenomenon causes a kind of diffusion of the momentum ofthe wave, due the molecular collisions between adjacent regionswith different velocities. The whole process can be measured bythe shear-viscosity coefficient g.

On the other hand, the losses produced by the internal molecu-lar processes may be attributed to a structural change in the fluidvolume, which occurs at a microscopic level. The theory that ex-plains this mechanism is the 1948 Hall’s theory of structural relax-ation [64]. That theory basically claims water to have two energystates: One with a lower energy (the normal state), and one, witha higher energy, in which the molecules have a more packed struc-ture. Under normal conditions, most of the molecules are in thefirst state of energy. However, the passage of a compressional waveproduces the transition of the molecular status to a more strictlypacked state. Such a process (and its reversal) lead to a relaxationmechanism and to dissipation of the wave energy. This can be ac-counted for by a non-vanishing bulk viscosity coefficient, gb.

The above-mentioned causes of attenuation in normal water(which is the case assumed in this paper) can be represented byan overall relaxation-time constant ss, on the basis of a linear anal-ysis [65] of the Navier–Stokes equation. From it the following(lossy) Helmholtz equation for the acoustic pressure can bederived:

$2�pþ �k2�p ¼ 0; ð1Þ

where

�k ¼ kþ ias ¼x

cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� i xss

p ; ð2Þ

is the complex wave vector,2 quantity c being the sound speed in theconsidered medium.

Eqs. (1) and (2) are only valid when the fluid is assumed to becontinuous, what can be summed up by imposing the conditionxss � 1. A more useful approximation for as, derived from Eq.(2), is

as � a ¼ x2 ss

2c¼ x2

2q0c3

43gþ gb

� �: ð3Þ

Here, the viscosity coefficients g (shear viscosity) and gb (bulkviscosity) are both in Pa � s. On the other hand, the complex wave-number �k and the complex speed of sound �c ¼ cR þ icI are relatedby the expression

�k ¼ kþ ia ¼ x�c

; with k � 2pk: ð4Þ

Using these quantities and assuming a monochromatic excitationx ¼ x0, a damped plane-wave solution of Eq. (1) propagating alongthe z axis can be expressed as:

2 In this paper we use overlined letters to denote complex variables.


�p ¼ P0 eið�kz�x0tÞ ð5Þ¼ P0 e�azeiðkz�x0tÞ;

where the coefficient a, measured in Np/m, finally accounts for allthe effects of attenuation of the fluid on the traveling wave.

3. Diffraction/attenuation resistant beams: Frozen Waves (FW)in absorbing media

As we discussed in the previous Section, the effects of attenua-tion of ultrasound in normal water can be represented by a decay-ing exponential term e�az in the solution, Eq. (5), of the lossyHelmholtz Eq. (1).

In order to model our acoustic FWs in an attenuating medium,we have to modify the language previously used by us for opticalFWs propagating in lossy media, by taking in mind that, given anexcitation frequency x0, in an acoustic absorbing medium a planewave possesses a complex wavenumber �k obeying relationship (4)and has the complex sound speed �c. We are therefore going tosummarize the method that was presented in Ref. [60].

An appropriate continuous superposition of such waves [5,59],with wave vectors laying on the surface of a cone with angularaperture 2h, will form a Bessel beam [5,6]:

w ¼ J0ð�kqqÞ eið�bz�x0tÞ; ð6Þ

which obeys the dispersion relation

�k2q þ �b2 � �k2 ¼ x2

0�c2 : ð7Þ

Here the transverse, �kq ¼ �k sinðhÞ, and longitudinal, �b ¼ �k cosðhÞ,spatial components of the complex wavenumber �k ¼ x0=�c ¼ kþ iasatisfy the following relations:

�kq ¼ k sinðhÞ þ i a sinðhÞ ¼ kqR þ ikqI; ð8Þ�b ¼ k cosðhÞ þ i a cosðhÞ ¼ bR þ ibI:

From them it is possible to derive that

bR

bI¼ kqR

kqI¼ k

a: ð9Þ

We can then write the expression of the Bessel beam in terms of thecomplex components of �kq and �b as

w ¼ J0½ðkqR þ ikqIÞq� eiðbRþibIÞz e�ix0t : ð10Þ

It is interesting to note that, although a plane wave has theattenuating term e�az [cf. Eq. (5)], a Bessel beam (as well as a super-position of them) has the attenuating factor e�a cosðhÞz, which in factis smaller than the former. This behavior can be seen by observingFig. 1, which illustrates the interference of the plane waves belong-ing to a Bessel beam (and whose wave vectors, as is known, stay onthe surface of a cone with axicon angle h). Notice how, in the sameamount of time, the Bessel has advanced a distance z3 � z0, whilethe corresponding plane-wave front has only traveled a distance

Fig. 1. Illustration of the interference of the plane waves forming a Bessel beam.



Fig. 2. Range and limits for the longitudinal wave numbers bRmin the case of

acoustic Frozen Waves (FW).

3 Here and in the sequel we will approximate this limit by x0=c, disregarding R½�c�,which implies a negligible difference when c (or a) is small, like in this work.


n3 ¼ ðz3 � z0Þ cosðhÞ, measured from the vertex of the cone. That is,the Bessel beam has traveled a longer distance than the singleplane wave, or, in other words, along the same distance the atten-uation of the plane wave is bigger by the factor 1= cosðhÞ than theBessel beam’s.

Let us now create a FW, endowed with the desired longitudinalintensity pattern jFðzÞj2, in the lossy medium, along the propaga-tion axis z (that is, on q ¼ 0). The envelope function FðzÞ can bearbitrarily selected inside the prefixed interval 0 6 z 6 L, providedthat the diffraction limits are respected. To such a purpose, let usconstruct the following finite superposition of co-propagating2N þ 1 Bessel beams:

Wðq; z; tÞ ¼ e�ix0tXN

m¼�N

AmJ0ð�kqmqÞ ei�bmz ð11Þ

where

�kqm � kqRm þ ikqIm ; and �bm � bRmþ ibIm

obey, for each value of m, Eqs. (7) and (9).Lets us write the real component of �bm in the form

bRm¼ Q þ 2pm

Lð12Þ

and recall that, once bRmhas been chosen, bIm

and �kqm are automat-ically defined through Eqs. (9) and (7), respectively. In Eq. (12) theparameter Q has an arbitrarily selected value provided that, withinthe range �N 6 m 6 N, it respects the condition

0 6 Q þ 2pmL6 R

x0

�c

h i¼ x0

cR 1þ c2I

c2R

� � ð13Þ

wherein the Real and Imaginary parts of �c appear.Inserting relation (12) into Eq. (11), one gets

Wðq; z; tÞ ¼ e�ix0teiQzXN

m¼�N

Am J0ððkqRm þ ikqIm ÞqÞ e�bIm z ei2pmL z: ð14Þ

To find out the limiting values of the attenuation terme�bIm z ¼ e�a cosðhmÞz, let us insert expression (12) of bRm

into Eq. (9)and get

bIm¼ Q þ 2pm

L

� �ak: ð15Þ

The minimum, maximum and central values of bImare then gi-

ven by

ðbIÞmin ¼ Q � 2pNL

� �ak

ðbIÞmax ¼ Q þ 2pNL

� �ak

ðbIÞm¼0 ¼ Qak� ~bI;

ð16Þ

respectively.The spread of the limiting values of bIm

can be expressed by

D ¼ ðbIÞmax � ðbIÞmin~bI

¼ 4pN

LQ: ð17Þ

Then, if D� 1, the values for the imaginary component of the lon-gitudinal wavenumber will be approximately equal; so thatexpð�bImzÞ � expð�~bIzÞ, and the superposition (14) can be regardedwhen q ¼ 0 as a truncated Fourier series, which can reproduce thedesired longitudinal intensity pattern jFðzÞj2 when choosing thecoefficients

Am ¼1L

Z L

0FðzÞ e~bI ze�i2pm

L zdz: ð18Þ


As we can see in the coefficients (18), the envelope function FðzÞof the FW is compensated, since the beginning, for the effects ofattenuation by the presence of the term e~bI z, which does indeedcounteract in (14) the effects of e�bIm z.

The number 2N þ 1 of terms in Eq. (14) is limited by the condi-tion (following from Eq. (13)):

N 6L

2pR

x0

�c

� �� Q

h idepending in particular on Q. The maximum possible value corre-sponds to Q ¼ x0=2c � Q0 and becomes

N 6 Nmax ¼Lx0

4pcð19Þ

where for simplicity we assume (as it often happens in the casesconsidered in this work) that cR � cI , so that Rðx0=�cÞ � x0=cR

and j�cj � cR � c. The quantity c, the sound speed in a water-likemedium, can be assumed to be approximately 1540 m/s. [Careshould be taken, however, when really adopting the limiting valueQ ¼ Q0 since it commonly leads to unpractical transducer sizes(R > 50 mm), with too many rings (Nr > 100)]. The last relationscan be easily seen at work in Fig. 2.

Another issue, we have discussed elsewhere [20,61], is that low-er values of Q imply narrower spots for the resulting FWs. Theirsize can be estimated by finding out the radius of the transversespot, as:

Dq ¼ 2:4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0=c2 � Q2q : ð20Þ

Let us conclude this Section by stressing that such a method al-lows constructing, in absorbing media, non-diffracting beamswhose longitudinal intensity pattern can be arbitrarily prefixed:In particular, one can obtain beams resistant to the diffractionand attenuation effects for long distances. This will be illustratedexplicitly in Section 6.

4. Acoustical restrictions for ultrasonic FWs

So far, we discussed the attenuation of ultrasound in water andthe way of modeling acoustic FWs in a lossy medium. In this Sec-tion our aim is to discuss the restrictions imposed by Acousticswhen selecting the appropriate values of the parameters for theultrasonic FWs. These limitations are mainly due to two causes:(i) the low values of the ratio x0=c, and (ii) the value of L, whichnormally is L < 1 m.

From Eq. (13) we can notice that, whatever the values selectedfor the real part (bRm

) of the longitudinal wavenumbers, they mustremain below the limit R½x0=�c�. Then, if one tries to create a FWwith a carrier frequency for example of f0 ¼ 2 MHz, the above men-tioned limit3 will approximately be x0=c ’ 8160 m�1.

I



Fig. 3. Relationship between the aperture radius Rmin and the chosen interval L, fordifferent values of the parameter N. The thinner lines correspond to f0 ¼ 1 MHz,while the thicker ones to f0 ¼ 2:5 MHz.

5


Compare this value with the one corresponding to optical FWs:For example, imagine the case of an optical FW constructed via ared laser with k ¼ 632 nm and with a value of c that now isc ¼ 3 108 m=s. The limiting value obtained in this case isx0c � 9:94� 106 m�1; which is more than three orders of magnitude

bigger. This is a distinct advantage for optical FWs by comparisonto their acoustical counterparts, because it allows a very large in-crease of the parameter N, for instance N ! 100, which in turnenormously enhances the details of the resulting FWs. By contrast,in the case of ultrasonic FWs the lower value of the ratio x0=c im-poses a reduction of the maximum value for the longitudinal wave-number, ðbRÞmax ¼ Q þ 2pN=L (see Fig. 2). As already mentioned,this maximum value depends on L; and also on N, which controlsthe number of the Bessel beams entering Eq. (14).

A possible solution to this problem would be increasing theoperating frequency f0, which linearly raises the mentioned limit.However, this has to be done with care, because it may require are-dimensioning of the transducer rings (width and kerf), whichshould correspondingly be decreased [this may result in technolog-ical problems if the dimensions are too small, and even in a distor-tion of the FW pattern when such dimensions do not meet therequired constraints (cf. Ref. [20])].

As we mentioned earlier, the range of the longitudinal wavenumbers Q � 2pN

L ; Q þ 2pNL

� �can be located anywhere inside the

interval ð0;x0=cÞ; however the maximum excursion is obtainedwhen Q ¼ Q0 � x0=2c (see again Fig. 2).

The terms 2p=L and D depend of course on the value of L. Oneshould also notice that, whenever the frequency is increased, theratio x0=c gets linearly augmented: This may allow increasingthe detail of the FWs by adding more terms in the superposition(14). This can be useful for example when constructing energyspots highly localized along the z axis. In the same way, whenthe frequency is lower, the values for D get higher, approachingunity [this might even violate, however, our assumption aboutthe similarity of the imaginary parts of the longitudinal wavenum-bers, i.e., bIm

� ~bI].Up to now, we have discussed only the role of the ratio x0=c as

a limiting factor for the FW creation. A second restriction for acous-tic FWs comes from the fact that L, in most cases, is in the range ofthe centimeters, i.e., L < 1 m: This makes the quotient 2pN=L in-crease. One possible solution to this problem is augmenting L asmuch as possible, so that the aforementioned quotient decays suf-ficiently, and more Bessel beams can be added into Eq. (14). Thenegative side of this option is that it might require a bigger trans-ducer for the generation of the same FW.

We can summarize our discussion about the FW parameterrestrictions, by collecting them into one expression for the deter-mination of the minimum required radius4 Rmin, a sufficient condi-tion for it being

Rmin ¼ L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0

c2b2R m¼�N

� 1

s: ð21Þ

The sufficient (but not necessary) condition (21) can be derived bysetting ðbRÞmax ¼ x0=c (see Ref. [20]). To see the dependence ofthe emitter radius Rmin on L and on parameter N, in Fig. 3 we showrelation (21) at work for two frequencies: f0 ¼ 1 MHz (thinnerlines), and f0 ¼ 2:5 MHz (thicker lines). Notice how the size requiredfor the radiators is in general bigger than that normally used in lab-oratory applications ðRmin 6 1 cmÞ. One can also notice how for cer-tain cases of N and L, the size of the transducer becomes completelyunpractical. The (not mutually exclusive) alternatives for solvingthis problem are the following: (i) increasing the operatingfrequency f0 and lowering N for the FWs, or (ii) disregarding the

4 An alternative expression can be found also in Eq. (19) of Ref. [59].


rather conservative approach represented by Eq. (21), and playingwith different interval widths L during the construction of theFWs: We shall deal with this question in the Appendix.

5. Method for the calculation of ultrasonic fields

In this Section we shall summarize the technique employed forthe calculation of the ultrasonic FWs fields in water, including theattenuation effect. To this aim, we use the well known impulse re-sponse (IR) method [66,67], in conjunction with a linear model ofthe medium attenuation [68].

The basis of the IR method is the linear system theory, which al-lows the separation of the spatial and temporal features of theacoustic field. Then, the IR function can be derived directly fromthe Rayleigh integral [69,70] by the expression:

hðr1; tÞ ¼Z

S

d t � jr1�r0 jc

� �2pjr1 � r0j

dS: ð22Þ

Eq. (22) assumes the emission aperture S to be mounted on an infi-nitely rigid baffle, with r0 denoting the location of the radiator, andr1 designating the position of the considerer field point. The speedof sound in the medium is c, as usual, while t denotes the time var-iable. This expression is nothing but the statement of Huyghens’principle, and it allows computing the acoustic field5 by addingup all the spherical wave contributions from the small elements con-stituting the radiating aperture.

Interestingly, it is also possible to reach the same results byusing the acoustic reciprocity principle. This procedure builds theh function, for a particular field point P, by finding the angularwidths (in radians) of the curve-arcs which result from the inter-section of a spherical wave emanating from P with the surface Sof the acoustic radiator. In both cases, the result is that the h func-tion depends on both the form of the emitting element, and its rel-ative position with respect to the point where the acoustic field isbeing calculated.

The original IR formulation assumes a flat, or gently curved,aperture,6 radiating into an homogeneous medium with no attenu-ation; nevertheless, the effects of the medium absorption can be in-cluded by properly modifying the IR function h. This can beperformed by applying to the Fourier transform response Hðf Þ a ‘‘fil-tering’’ function Aðf Þ which accounts for the medium attenuation,

This is the differential field pressure, relative to the static atmospheric pressureP0.

6 The aperture dimensions must be large compared to the field wavelength.




and can be chosen as a linear model [68] of the attenuation phenom-enon. The attenuated IR function will be therefore:

haðx; y; z; tÞ ¼ F�1fAðf ÞHðf Þg: ð23Þ

On using the modified IR function, the acoustic pressure can be ob-tained as

pðx; y; z; tÞ ¼ qm@vðtÞ@t haðx; y; z; tÞ; ð24Þ

where the symbol denotes the time convolution of the modifiedradiator impulse response function ha with the time derivative ofits surface signal velocity v, while qm accounts for the density ofthe medium.

As we discussed in Section 2, the losses in the medium can bedescribed by assuming a linear lossy equation, which introducesthe attenuation by means of the parameter a expressed in Np/m.

In order to simulate our acoustic FWs in an environment moreclosely resembling the human body, we set the a parameter in ourmodel to values typically found for human tissues [68,71], i.e.,0:7 6 a 6 1:7 dB=cm, with a ¼ 8:686 a. Although it is true that thissimplification may not account for all the processes occurring inreality, we consider this approach to constitute a reasonableapproximation.

It is also important to stress that during the simulations theabsorption of the wave energy will continue to occur in a normalfashion; nevertheless, the superposed beams will be able to recon-struct themselves due to the nature of their transverse fielddistribution.

We have implemented the above method into a Matlab pro-gram employing the Field II toolbox [67,72]; the latter allows theintroduction of the medium attenuation by means of a linear mod-el based on a. The annular radiators were simulated by discretizingthem into triangular elements (see Fig. 4), which are available inthe Field II package for the calculation of the haðx; y; z; tÞ function.

The steps followed by the software for the computation of theacoustic FWs fields are summarized as follows:

1. Compute the theoretical FW, Wðq; zÞ, using the chosenparameters.

2. Determine the radiator dimensions: radius ðRÞ, width ðdÞ andkerf ðDdÞ.

3. ‘‘Sample’’ the amplitude and phase of the FW function at theaperture location, i.e., of the complex function Wðq; z ¼ 0Þ.

4. Assign a sinusoidal signal vk for the ring nk, using sampledvalues.

Fig. 4. Example of annular radiator, created by the ‘‘Field II toolbox’’ package, withradius R ¼ 20 mm, and with Nr ¼ 15 rings composed by 628 triangular elements(see the text).


5. Sweep the field points ðxi;j; yi;j; zi;jÞ for the transducer ring nk.6. At point Pði;jÞ, calculate ha and pði;jÞ ¼ qm _vk ha.7. Accumulate the pressure for ring nkðp ¼ pþ pkÞ, and go back to

the fourth step.8. Store and display the p results.

6. FWs in absorbing media: results of simulated experiments

In this Section we like to demonstrate the possibilities offeredby the acoustic Frozen Waves, by presenting three different exam-ples of FWs operating in a water-like medium, including the effectof the absorption. Attenuation factors between 70 6 a 6 170 dB=mhave been tested at f0 ¼ 1 MHz, with interval widths L in the range120 6 L 6 240 mm.

Before going on, let us point out that the cases examined herewere selected as rather general in order to emphasize the capabil-ities offered by the method. We also chose a wide range of shapesfor the envelope function FðzÞ, possibly usable in practical applica-tions. In other words, our intention has been that of testing generalenough patterns, rather than some particular applications, also forbenefit, a priori, of both usefulness and simplicity.

As we mentioned in the Introduction, the use of zero-order Bes-sel beams will also result in a high degree of transverse localizationduring propagation, besides the beam resistance to diffraction andabsorption. As we know, such good properties are due to the self-reconstruction capability of the Bessel beams, as well as to theaddition of our compensating term e~bI z � eia cosðhÞz in Eq. (18)(which counteracts the attenuation effects [60,61]).

All of the simulated FW patterns have been obtained by usingthe same ultrasonic radiator, with radius R ’ 31 mm and Nr ¼ 35rings, each one having a width d ¼ 0:6 mm and a kerf ofDd ¼ 0:3 mm. As already mentioned, the simulated FWs have alsobeen produced by exploring the effects of different values of L inorder to improve our results for the fields in the presence ofabsorption.

The speed of sound inserted in the program is c ’ 1540 m=s,and the operating frequency has been fixed in all cases atf0 ¼ 1 MHz, which corresponds to a wavelength of k ’ 1:54 mm.The only parameters we varied during the simulations (apart fromthe chosen envelope FðzÞ itself, of course) are the medium attenu-ation coefficient a, expressed in Np/m [sometimes replaced by avalue in dB/cm], the value L chosen for the FW, and the value Nwhich determines the ð2N þ 1Þ number of Bessel beams in Eq. (14).

The corresponding couples of figures, shown in this Sec-tion (namely, Figs. 5–6, Figs. 8–9, and Figs. 11–12) depict: First,the theoretical FW pattern obtained in the ideal case of an infiniteaperture [i.e., the 3D plot of the function Wðq; zÞ]; and, Second, theresult of our Impulse Response (IR) simulation of its experimentalgeneration by a finite aperture, respectively. The sampling fre-quency used in the IR method has been set to fs ¼ 100 MHz.

The details in the corresponding spatial grids (q; z) had in prac-tice to be reduced, with respect to (w.r.t.) the ideal patterns. Then,suitable intervals of Dz ¼ 0:4 mm and Dq ¼ 0:1 mm were selectedfor our Field II simulated plots. Also, due to the adoption of colors(present online only) for the graphs, light-effects were added in or-der to enhance the visibility of the details also on paper.

6.1. Case 1

Our first choice of an acoustic FW to be generated consists of auniform envelope FðzÞ, with L ¼ 60 mm. This FW is defined in Eq.(25) as existing between l1 ¼ L=10 ¼ 6 mm, and l2 ¼ 9L=10 ¼54 mm, with N ¼ 6: That is, 2N þ 1 ¼ 13 Bessel beams aresuperposed. The interval width is chosen to be L ¼ 120 mm, andthe attenuation of the medium has been set either to 0.7 dB/cmor to 70 dB/cm, at f0 ¼ 1 MHz:



Fig. 9. Simulation, by the IR method, of the experimental production of the FW inthe previous figure (Case 2 of Section 6). The settings are the same, except that weare now using an annular radiator with Nr ¼ 35 rings and radius R ’ 31 mm.

Fig. 12. Simulated FW corresponding to Fig. 11, i.e. to Case 3 of Section 6. Thesettings are the same, except that now we are using a 35 rings annular radiator.

Fig. 11. The theoretical FW chosen in Case 3 of Section 6. Settings:L ¼ 240 mm; N ¼ 7; with medium attenuation 1.7 dB/cm, and f0 ¼ 1 MHz.

Fig. 8. The theoretical FW chosen in Case 2 of Section 6. Settings:L ¼ 190 mm; N ¼ 7; medium attenuation 1 dB/cm, and f0 ¼ 1 MHz.

Fig. 10. Top (scaled) view of the FW simulated in Fig. 9, corresponding to Case 2 ofSection 6. The achieved axial size of its spot is 2Dq2 ’ 3:5 mm.

Table 2Parameters for the FW corresponding to Case 2 of Section 6.

�c (m/s) hmin

(deg)hmax

(deg)zhmin

(mm)zhmax

(mm)D

1539:988� i 4:345 2.6 27.7 693 59 0.12

Fig. 5. Theoretical FW chosen in Case 1 of Section 6. Settings: L ¼ 120 mm; N ¼ 6;attenuation of the medium 0.7 dB/cm, and f0 ¼ 1 MHz.

Fig. 6. Simulation of the experimental production, by the IR Method, of the FW inthe previous figure, that is, for Case 1 of Section 6. The settings are the same asbefore, except that now we are using a Nr ¼ 35-ring annular radiator, with radiusR ’ 31 mm.

Fig. 7. Top view of the simulated intensity pattern in Fig. 6, corresponding to Case 1of Section 6. The size of the spot achieved is 2Dq1 ’ 3 mm.


�c (m/s) hmin

(deg)hmax

(deg)zhmin

(mm)zhmax

(mm)D

1539:994� i 3:042 2.6 32.3 693 49 0.17


Please cite this article in press as: J.L. Prego-Borges et al., Producing acoustic ‘Frozen Waves’: Simulated experiments with diffraction/attenuation resistantbeams in lossy media, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.03.008


Fig. 15. Comparison of the FW fields at the aperture location, i.e., jWðq; z ¼ 0Þj, forthe profiles shown in Fig. 14. The dotted line corresponds to using L ¼ 120 mm;while the continuous line comes out when L ¼ 60 mm. In both cases the sameemitter is used, operating at f0 ¼ 1 MHz with attenuation 0.7 dB/cm, and N ¼ 6.

Fig. 13. Scaled top view of the FW simulated in Fig. 12, corresponding to Case 3 ofSection 6. The size of the achieved axial spot is 2Dq3 ’ 4 mm.


�c (m/s) hmin

(deg)hmax

(deg)zhmin

(mm)zhmax

(mm)D

1539:965� i 7:387 1.15 24.5 1550 68 0.09


F1ðzÞ ¼1 for l1 6 z 6 l2

0 elsewhere:

ð25Þ

A plot of the theoretical frozen wave is shown in Fig. 5, while theresults from the impulse response method are presented in Figs. 6and 7. The last plot shows a (properly scaled) top view of Fig. 6,illustrating the achieved degrees of axial and longitudinal localiza-tion. The size of the FW spot is approximately 2Dq1 ’ 3 mm. Arounded off value of the complex speed of sound in the medium,plus some parameters of the FW, are shown in Table 1.

6.2. Case 2

The next case corresponds to two peaked regions, or localizedspots of energy, with medium absorption 1.0 dB/cm, that is,100 dB=m, at f0 ¼ 1 MHz. The envelope FðzÞ is explicitly given by:

F2ðzÞ ¼1 for l1 6 z 6 l2

1 for l3 6 z 6 l4

0 elsewhere

8><>: ð26Þ

with the limits: l1 ¼ 1:5L=10 ¼ 10:5 mm; l2 ¼ 3L=10 ¼ 21 mm; l3 ¼7:5L=10 ¼ 52:5 mm, and l4 ¼ 9L=10 ¼ 63 mm. The interval L in thiscase is L ¼ 190 mm, with N ¼ 7.

The plots in Figs. 8 and 9 correspond, once again: First, to thetheoretical FW, and: Second, to the Field II simulation of the

Fig. 14. Comparison of two Field II simulated profiles in absorbing media, for the FW coL ¼ 60 mm in the case of figure (a); and of a large value of L, namely L ¼ 120 mm, in the caNr ¼ 35 rings, operating at f0 ¼ 1 MHz with N ¼ 6, and attenuation 0.7 dB/cm.


generating experiment, respectively. Table 2 shows the parametersof this FW, in addition to the approximate value of the complexvelocity of sound in the medium. Figure 10, with regard to the sim-ulation in Fig. 9, presents a (scaled) top view of the location of itsenergy peaks, which possess an axial spot size of approximately2Dq2 ’ 3:5 mm.

6.3. Case 3

In this last example, we choose a growing intensity pattern forthe FW, operating once more at f0 ¼ 1 MHz. The absorption of themedium has been made even stronger, with a parametera ¼ 1:7 dB=cm. The FW envelope is given by a polynomial function[see Eq. (27) below, werein in the present case z can be regarded asmerely expressing the location along the propagation axis in me-ters, by a pure number] corresponding in this case to the intervalL ¼ 240 mm, with N ¼ 7 (see Figs. 11–13).

F2ðzÞ ¼z2

2 þ 3zþ 0:1 for 0 6 z 6 L

0 elsewhere

(ð27Þ

Some of the parameters of this Frozen Waves are given in Table 3,together with the complex speed of sound in the medium.

Notice the depth of field achieved in this case ðzhmin’ 1:55 mÞ

for the minimum axicon-angle Bessel beam, that is, forhmin ’ 1:15 degrees. We obtained such a value by adding another

nsidered in Case 1 of Section 6. The continuous line corresponds to the adoption ofse of figure (b). In both cases the emitter is the same, i.e., with radius R � 31 mm and



Fig. 16. Comparison of Field II non-attenuated (ideal media) profiles of the FWconsidered in Case 1, Section 6. First (dotted line), using a radiator with Nr ¼ 71rings and a radius of R ’ 63 mm calculated with Eq. (21). Second (continuous line),employing the same emitter as in Fig. 5, i.e., R ’ 31 mm and Nr ¼ 35. All casesoperate at f0 ¼ 1 MHz with L ¼ 60 mm and N ¼ 6.

Table 4Parameters for the FW profiles shown in Fig. 14(a).

Case (mm) hmin (deg) hmax (deg) zhmin (mm) zhmax (mm) D

L ¼ 60 2.6 46.3 693 30 0.36L ¼ 120 2.6 32.3 693 49 0.17


decimal digit to the constant b ¼ 0:9998 used for calculating Q, de-fined as Q ¼ b x0=c � 2pN=L.

The annular aperture employed is the same as in the previousexamples, that is, with Nr ¼ 35 rings having width d ¼ 0:6 mm,spacing Dd ¼ 0:3 mm, and radius R ¼ 31:2 mm.

7. Conclusions

The non-diffracting solutions to the wave equations are suitablesuperpositions of Bessel beams which resist efficiently the effectsof diffraction: At least up to a certain finite distance (field-depth)when emitted by finite apertures.

Particular superpositions of such beams allow the creation offields Wðq; zÞ with a previously chosen intensity envelope jFðzÞj2,and have zero peak-velocities. We called Frozen Waves (FW) thesepeculiar fields, characterized by a static envelope within whichonly the carrier wave propagates.

Experiments for producing acoustic FWs in ideal (non-lossy)media were already simulated in Ref. [20].

In this paper we have focussed our attention on the creation ofacoustic FWs in a (water-like) absorbing medium. To this purpose,we have adopted a linear model for the medium coefficient a,which depends quadratically on the frequency. This model has al-lowed us to express the solutions of the Helmholtz lossy equationin terms of exponentially damped plane waves.

In our approach, we compensate from the beginning such atten-uation effects by the addition of the term e~bI z in Eq. (18) for theFourier coefficients, with the sufficient (but not necessary) condi-tion that bIm

� ~bI , or D� 1. It is interesting to note that, even whenthe condition D� 1 is not well fulfilled, it is nevertheless possibleto create FWs successfully in a moderate absorbing environment.

By the IR method, we simulated experiments for generatinglocalized fields in three different cases, all operating atf0 ¼ 1 MHz and with the same annular aperture, by adoptingincreasing values of the attenuation, from 0.7 to 1.7 dB/cm], andvarious interval widths L.


Some problems concerning the acoustic case of FWs have beenalso discussed, by pointing out, e.g., the role of the ratio x0=c asa factor limiting the resolution achievable for the FWs patterns.This situation can be partially overcome by using a higher operat-ing frequency f0. [But, since this leads of course to a reduction ofthe wavelength k, care has to be taken in order that the dimensionsof the rings of the final piezoelectric transducer remain practicallyrealizable].

The condition that L < 1 m was also analyzed, and different(increasing) values of L considered, so as to improve our resultsfor the fields in the presence of absorption (cf. also the Appendix).A side-effect of using large values of L, however, is that it tends toincrease the amplitude of the lateral lobes: This can be observed inour figures.

Among the many possible combinations of values for the FWparameters (that is, frequency f0, interval L, number of Besselbeams 2N þ 1, and Q), the one that deserves more attention duringthe FW parameter selection is the value of Q. This value, as men-tioned before, can be anywhere inside the interval 0;x0

c

� �, as long

as the corresponding longitudinal wavenumbers jbmj also remaininside the same range.

An important observation is that, although the sizes of the FWaxial spots decrease for lower values of Q (see Eq. (20)), such valuestend to produce increasing oscillations in the fields at the aperturelocation, i.e., for Wðq; z ¼ 0Þ. And use of such resulting fields mayend in a poor sampling process, which we know to have a criticalimportance, since it eventually determines amplitudes and phasesof the sinusoidal signals that drive the rings of the radiator. Forsuch a reason, we chose values as Q ¼ b x0=c � 2pN=L, withb! 1, because in this manner the unwanted oscillations are re-duced to a minimum, and the final frozen wave results are less af-fected by such a phenomenon.

Closely related to this last point is the issue of selecting thedimensions for the annular radiator: That is, its radius R, the ringwidths d, and the spacing (or kerf, Dd) between the rings. One doesnot have a priori specific criteria for this selection, apart from the‘‘conservative’’ approach of Eq. (21) which suggests the minimumpossible radius for the aperture.

What we have found experimentally in our tests is the existenceof a relation between the wavelength k, chosen for the FW, and theaverage distance between peaks and valleys of the field at z ¼ 0[i.e., of Wðq; z ¼ 0Þ], suggesting, as expected, that lower wave-lengths correlate to reduced ring widths. On this basis, we de-signed a radiator (R � 31 mm; Nr ¼ 35 rings) whose ringdimensions (d ¼ 0:6 mm and Dd ¼ 0:3 mm) are sufficiently smallerthan the k used in the simulations. This ensures that the ring sizesdo not introduce distortions in the FW fields, during the samplingprocess of the FW magnitude and phase. At the same time, they arebig enough to reduce the number of rings, that is, of the electronicchannels, to be used.

It is interesting to note that, even when the radius of the aper-ture is smaller than half of that suggested by Eq. (21) (e.g., for case1, Rmin1 ¼ 75:7 mm), we have still been able to reasonably produceFWs in the absorbing media. Indeed, that expression, as we know,was derived on the basis of a sufficient (but not necessary)condition.

Finally, as stated in the introduction, we wish to add a fewwords concerning the practical realization of acoustic FWs, espe-cially because the main focus of this paper has been addressed tothe theoretical aspects of acoustic FWs in attenuated media, andto the restrictions imposed on them by Acoustics. When practicallyimplementing FWs in ultrasound, the main problem is of coursethe availability of suitable annular piezoelectric transducers, andof the required electronic front-end to drive the rings. They doaffect the minimum number of electronic channels necessary to




create the stationary wave fields, without distorting the originallychosen FW pattern.

A second issue arises when dealing with the transducer itself.Indeed, the Transfer Response of each individual annulus of theradiator can introduce ‘distorting’ effects in the amplitudes andphases of the signals; and attention should be paid to the effectsof signal attenuation and delay produced by the yet unknown elec-trical/mechanical transfer function of each ring of the (piezoelec-tric) transducer. To characterize each of the transducer annuli,one could scan each piezoelectric ring with a hydrophone andmeasure, at the same time, the electric input signal and the acous-tic output pressure. Then, the individual ‘‘transferences’’ for each ofthe rings can be derived, and suitable compensation factors can beadded into the amplitude and phase ideal samplings, before thesinusoidal signals can actually drive the piezoelectric rings.

We believe that the practical realization of acoustic FWs canopen the door to very interesting applications, in the field of med-icine, as well as in other valuable technological sectors.

Acknowledgments

This work was supported by CNPq, Brazil (Grants 312376/2013-8and 500364/2013-3) and FAPESP, Brazil (Grants 2011/51200-4 and2013/12025-8), besides by INFN, Italy. The authors wish to thankIoannes M. Besieris, Hugo E. Hernández-Figueroa, Jane MarchiMadureira Rached and Suzy Zamboni Rached for continuous, helpfulinterest. They are moreover quite grateful to the Editor-in-Chief andthe two anonynous Referees for their very kind generous help.

Appendix A. Effects of varying the interval widths L on theproduced acoustic FWs

Along with our discussion about the existing restrictions on theFW acoustic parameters, we have analyzed the role of the ratiox0=�c, and of the values of L, commonly smaller than 1 meter. Thislast condition imposes the restriction that, for a particular value of2pN=L, the quantity N has to be reduced as L! 0, by lowering as aconsequence the resolution of the patterns. The same condition,due to the increase of the parameter D ¼ 4pN=ðLQÞ, raises thespread of the bIm values, so that superposition (14) becomes moreand more far from a Fourier representation (when q ¼ 0).

A possible way out, is having recourse to different interval-widths L during the computation of the acoustic FW. To illustratethis point, in Fig. 14 of this Appendix we represent the intensityprofiles, along the z axis, of a f0 ¼ 1 MHz FW [see Case 1 in Sec-tion 6: Cf. Eq. (25) therein], simulated by us for an absorbing med-ium with c ’ 1540 m=s, and corresponding to L ¼ 60 andL ¼ 120 mm, respectively.

The attenuation has been set to 0.7 dB/cm, with N ¼ 6 (that is,with 2N þ 1 ¼ 13 Bessel beams), and L ¼ 60 mm. Note how, whenusing L ¼ 60 mm ( Fig. 14(a), continuous line), the intensity patternfalls down before the end of the theoretical FW (solid image); but,when increasing the value of L (Fig. 14(b), continuous line), keepingthe number of terms in the expansion, the pattern gets improved.

A second benefit of adopting larger values of L is that the fieldsat the aperture location, i.e., jWðq; z ¼ 0Þj, get significantlysmoothed. This helps the sampling process, that assigns the ampli-tudes and phases of the sinusoidal signals, which drives the ringelements. We can observe this phenomenon in Fig. 15, which cor-responds to the cases presented in Fig. 14.

The side effect of large values of L is that the original FW enve-lope, FðzÞ, gets distorted during the integrations in Eq. (18). Then, acompromise has to be sought between how much we like to pre-vent the final patterns from the attenuation effect, and how muchwe accept the original envelope FðzÞ to be altered during theprocess. To better observe this trade-off, we show in Table 4 the


values of some of the parameters used during the construction ofthe above FWs profiles.

Notice, when using for example L ¼ 60 mm, how much the va-lue of the maximum axicon angle hmax gets greater than when oneadopts larger values for L. This causes the corresponding depthzhmax ¼ R= tan hmax of the Bessel beam to be reduced, affecting the fi-nal Fourier reconstruction.

On the other hand, the hmin axicon angle and its correspondingdepth zhmin

do not get changed; this is due to the way in which weassign the value of Q, by relation Q ¼ b x0=c � 2pN=L with b! 1(e.g. b = 0.999). In this way, the values of Q are the minimum pos-sible achieved with the current values of N and L, ensuring thatjWðq; z ¼ 0Þj does not oscillate excessively.

To conclude this Appendix, we like to compare our previous re-sults, with attenuation, with those obtained when acoustic FWswere created in an ideal non-absorbing medium. To this aim,Fig. 16 illustrates the patterns obtained for the same FW inFig. 14 [see Case 1 in Section 7: Cf. Eq. (25)].

The difference, in this case, focuses on the size of the apertureused by the computer program for the creation of the FWs. The firstpattern (dotted line) uses the conservative approach of Eq. (21), i.e.,with R ’ 63 mm of radius (that is, £ > 12:5 cm of diameter!).While the second (continuous line) employs the radiusR ’ 31 mm as in Fig. 14.

In all cases (as well as in those of Figs. 14 and 15), the widthd ¼ 0:6 mm, and kerf Dd ¼ 0:3 mm of the rings were the same.

Notice how, when using the smaller radius R ’ 31 mm, the pat-terns, indicated by the black lines in Figs. 14(a) and 16, are quitesimilar. That is, both are falling down before the theoretical FW(solid image) finishes. This happens even though the first includesthe effect of attenuation, while the second does not. On the con-trary, when using the bigger radius R ’ 63 mm given by Eq. (21),the profile (dotted line) is enhanced, resembling the theoreticalFW. It is also possible to observe near z ¼ 65 mm the beginningof the repetition of the original FW pattern (solid image). This isof course the consequence of using a Fourier representation. Thisdoes not happen in Fig. 14(b), because of the use of a bigger valueof L, and also because of the effect of absorption. Moreover, it canbe observed how the non-attenuated original FW pattern in Fig. 16,with 5 peaks, is (moderately) distorted to 3 peaks as in Fig. 14.

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