An Ultrasonic Lens Design Based on Prefractal Structures

symme y S S Le e An Ul asonic Lens Design Based on P e ac al S uc u es Se gio Cas iñei a-Ibáñez 1, Daniel Ta azó-Se ano 2, Cons anza Rubio 2,*, Pila Candelas 2 and An onio U is 2 1Depa amen o de Ingenie ía Elec ónica, Uni e si a de València, A d. de la Uni e si a s/n, Bu jasso , Valencia 46100, Spain; casise @u .es 2Cen o de Tecnologías Físicas, Uni e si a Poli ècnica de València, Camino de Ve a s/n, Valencia 46022, Spain; [email p o ec ed] (D.T.-S.); [email p o ec ed].es (P.C.); [email p o ec ed].es (A.U.) *Co espondence: c [email p o ec ed].es; Tel.: +34-963-879-521; Fax: +34-963-879-525 Academic Edi o : Palle E.T. Jo gensen Recei ed: 25 Janua y 2016; Accep ed: 15 Ap il 2016; Published: 21 Ap il 2016 Abs ac : The imp o emen in ocusing capabili ies o a se o annula sca e e s a anged in a ac al geome y is heo e ically quan i ied in his wo k by means o he ini e elemen me hod (FEM). Two di e en a angemen s o igid ings in wa e a e used in he analysis. Thus, bo h a F esnel ul asonic lens and an a angemen o igid ings based on Can o p e ac als a e analyzed. Resul s show ha he ocusing capaci y o he modi ied ac al lens is be e han he F esnel lens. This new lens is belie ed o ha e po en ial applica ions o ul asonic imaging and medical ul asound ields. Keywo ds: sound ocusing; ul asonic lens; Can o p e ac als 1. In oduc ion I has been ound ha ce ain na u al phenomena, such as snow lakes o he s uc u e o lea es in ce ain plan s, display sel -simila pa e ns. These dis inc i e ea u es can be associa ed wi h he ac al concep . F ac als a e non- egula geome ic shapes ha ha e he same deg ee o non- egula i y on all scales [ 1 ]. F ac al s uc u es ha e a ac ed he in e es o he scien i ic communi y due o hei applica ions in se e al a eas o science and echnology [2]. Among he wa e applica ions o ac al s uc u es, se e al s udies ha e been add essed in acous ics. Pe i e al. [ 3 ] analyzed he ib a ional p ope ies o a hie a chical con inuous sys em consis ing o a Can o -like sequence o piezoelec ic and esin elemen s. Sapo al e al. [ 4 ] in es iga ed nume ically he acous ical p ope ies o i egula ca i ies desc ibed by ac al shapes. They showed ha he geome ical i egula i y enhanced he low equency modal densi y and localized many o he modes a he ca i y bounda ies, and ha he damping cha ac e is ics o he ca i y we e modi ied. Lubniewski and S epnowski [ 5 ] de eloped a simple me hod o sea bo om iden i ica ion using elemen s o ac al analysis. Gibia e al. [ 6 ] epo ed he homo he ic acous ical ea u es, o bidden bands, and wa e apping phenomena o an acous ic mul isca e ing one-dimensional sys em made o cylind ical ubes o di e en diame e s, whose leng hs ollow a Can o -like s uc u e. Cas iñei a-Ibañez e al. [ 7 , 8 ] p esen ed an acous ic ba ie o noise con ol o med by igid cylinde s a anged in ac al Sie pinski iangle geome y. Gomez-Lozano e al. [ 9 ] s udied he acous ic ansmission esponse o pe o a ed pla es wi h a ac al subwa eleng h holes a ay. The ul asound ansmission spec a showed ha each i e a i e Sie pinski Ca pe has he cha ac e is ic peaks and dips o he la ice cons an o each a ay ha o med he pa e n. Ta ge ing o ul asonic beams o MHz o de a e essen ial o nonin asi e issue abla ion. Based on hese s udies, a new possibili y o cons uc ing lenses di e en om he mos common cases ha a e cons uc ed by e ac i e ma e ials wi h cu ed su aces is p oposed. Thus, in his pape , Symme y 2016,8, 28; doi:10.3390/sym8040028 www.mdpi.com/jou nal/symme y Symme y 2016,8, 28 2 o 7 a plana acous ic ac al lens is p oposed, and he ocusing p ope ies a e nume ically analyzed using he ini e elemen me hod (FEM). The esul s ob ained wi h he p oposed acous ic ac al lens a e compa ed wi h hose o a con en ional plana acous ic F esnel lens. Fo ocusing acous ic wa es in bo h ai and wa e , acous ic F esnel lenses ha e been in es iga ed [ 10 , 11 ], and laby in hine channels ha e ecen ly been used in F esnel lenses o inc ease hei e iciency [12,13]. 2. Modeling and Simula ion Analogously o op ical lenses, i is common o cons uc acous ic lenses ia e ac i e ma e ial wi h cu ed in e aces. As in op ics, di ac i e acous ic elemen s a e essen ial i he ocaliza ion o he acous ic beam is made by means o la ansduce s. F esnel zone pla es (FZPs) ocus wa es h ough cons uc i e in e e ences o di ac ed ields [ 14 ]. Hence, FZPs is di ided in o ing segmen s ha ac al e na i ely as anspa en (open-gaps) o opaque ( igid); he adii and wid h o hese segmen s a e so designed o p oduce cons uc i e in e e ence a he ocus. Figu e 1shows a gene a ion o a FZP. As a esul , he zone adii, i , which de ines he pa e n o plane wa e incidence and wa eleng h λ o a FZP is gi en by i“diλF`ˆiλ 2˙2 (1) whe e i= 1, 2, . . . , N,iis he o al numbe o zones, and Fis he ocal dis ance. Symme y 2016, 8, 28 2 o 7 pape , a plana acous ic ac al lens is p oposed, and he ocusing p ope ies a e nume ically analyzed using he ini e elemen me hod (FEM). The esul s ob ained wi h he p oposed acous ic ac al lens a e compa ed wi h hose o a con en ional plana acous ic F esnel lens. Fo ocusing acous ic wa es in bo h ai and wa e , acous ic F esnel lenses ha e been in es iga ed [10,11], and laby in hine channels ha e ecen ly been used in F esnel lenses o inc ease hei e iciency [12,13]. 2. Modeling and Simula ion Analogously o op ical lenses, i is common o cons uc acous ic lenses ia e ac i e ma e ial wi h cu ed in e aces. As in op ics, di ac i e acous ic elemen s a e essen ial i he ocaliza ion o he acous ic beam is made by means o la ansduce s. F esnel zone pla es (FZPs) ocus wa es h ough cons uc i e in e e ences o di ac ed ields [14]. Hence, FZPs is di ided in o ing segmen s ha ac al e na i ely as anspa en (open-gaps) o opaque ( igid); he adii and wid h o hese segmen s a e so designed o p oduce cons uc i e in e e ence a he ocus. Figu e 1 shows a gene a ion o a FZP. As a esul , he zone adii, i, which de ines he pa e n o plane wa e incidence and wa eleng h λ o a FZP is gi en by 2 2      += λ λ i Fi i (1) whe e i = 1, 2, …, N, i is he o al numbe o zones, and F is he ocal dis ance. Figu e 1. Schema ic sec ion diag am o he gene a ion o a F esnel zone pla e (FZP) and FZP conside ed. Due o he wa e na u e o ligh and sound, he physical phenomena de eloped in op ics could be ans e ed o acous ics. Wi h his in mind, a Can o di ac al ha is based on a polyadic Can o se has been conside ed, wi h he cons uc ion p ocedu e shown in Figu e 2. Hence, o he Can o ings, he zone adii is gi en by [15], and he pa ame e a has leng h dimensions and will be used o adjus he ing size o be compa ed wi h hose ob ained o he F esnel zone pla e: 1 1 i j s j ic j aa b γ − = = (2) whe e i = 0, 1, …, M − 1, and M is he o al numbe o ings, which can be ob ained by max 14 2 s M= (3) whe e s is he s age o g ow h o he ac al s uc u e, which, in his wo k, will be s = 0, 1, 2. Fo smax = 2, he o al numbe o ings will be M = 8; hus, he o al numbe o zones a e 2M + 1 = 19. The scaling a io be ween successi e s ages o g ow h, γ , is gi en by () 1/ gaps 1 D n γ − =+ (4) whe e ngaps is he numbe o desi ed gaps, and D is he ac al dimension. In his wo k, ngaps = 3 and D = 9/10. Fo he i s s age (s = 0), he ba leng h is L, and his leng h a ies wi h i s s age h ough he exp ession s L γ . On he o he hand, he coe icien s i j c bin Equa ion (2) depends on γ and ε ( he lacuna i y) [16]. Figu e 1. Schema ic sec ion diag am o he gene a ion o a F esnel zone pla e (FZP) and FZP conside ed. Due o he wa e na u e o ligh and sound, he physical phenomena de eloped in op ics could be ans e ed o acous ics. Wi h his in mind, a Can o di ac al ha is based on a polyadic Can o se has been conside ed, wi h he cons uc ion p ocedu e shown in Figu e 2. Hence, o he Can o ings, he zone adii is gi en by [ 15 ], and he pa ame e ahas leng h dimensions and will be used o adjus he ing size o be compa ed wi h hose ob ained o he F esnel zone pla e: ai“a s ÿ j“1 γj´1bci j(2) whe e i= 0, 1, . . . , M´1, and Mis he o al numbe o ings, which can be ob ained by M“1 24smax (3) whe e sis he s age o g ow h o he ac al s uc u e, which, in his wo k, will be s= 0, 1, 2. Fo smax = 2, he o al numbe o ings will be M= 8; hus, he o al numbe o zones a e 2M+ 1 = 19. The scaling a io be ween successi e s ages o g ow h, γ, is gi en by γ“`ngaps `1˘´1{D(4) whe e n gaps is he numbe o desi ed gaps, and Dis he ac al dimension. In his wo k, n gaps = 3 and D= 9/10. Fo he i s s age (s= 0), he ba leng h is L, and his leng h a ies wi h i s s age h ough Symme y 2016,8, 28 3 o 7 he exp ession Ls γ . On he o he hand, he coe icien s bci j in Equa ion (2) depends on γ and ε ( he lacuna i y) [16]. Symme y 2016, 8, 28 3 o 7 Figu e 2. Schema ic sec ion diag am o he gene a ion o a ac al zone pla e (FRZP) om s age 0 o s age 2. This pa ame e speci ies he dis ibu ion o he N copies in o he uni leng h segmen . In ac , he lacuna i y is essen ial o comple e he cha ac e iza ion o he ac al because s uc u es wi h di e en lacuna i y can ha e he same ac al dimension. In his wo k, ε = 44/1000. I is no ewo hy ha his s uc u e is no a egula Can o ac al. The ini ial segmen is di ided in o an odd numbe o segmen s, and he segmen loca ed in he e en posi ion is emo ed. This p ocedu e is epea ed h ough successi e s ages wi hin he same ule. Unlike he FZP, he wa eleng h does no appea in he exp ession o he Can o ings’ adii. The e o e, in his wo k, he wa eleng h dependence has been conside ed h ough pa ame e a. Figu es 1 and 3 ep esen he co esponding zone pla es wi h he FZP p o ile (Figu e 1) and he FRZP p o ile (Figu e 3) s udied he e. Figu e 3. Schema ic sec ion diag am o he gene a ion o a FRZP and FRZP conside ed. The in e ac ion o ul asound wa es wi h ul asonic lenses is a complex p oblem. The FEM seems o be an app op ia e compu a ional ool o de e mine he dis ibu ion o acous ic p essu e and he e o e he ocal posi ions and he size o he ocal spo . To dec ease he compu a ional cos o he simula ions, he geome ical p ope ies o he model ha has been implemen ed mus be aken in o accoun . Bo h ac al and F esnel lenses ha e axial symme y since hese s uc u es a ise as a esul o he o a ion o a plane a ound an axis. Fu he mo e, all he cu ing hal -planes along his axis ha e iden ical cha ac e is ics. The e o e, a hal -plane, which co esponds o he longi udinal sec ion o he semilens and makes he o a ion a ound he axis, can be implemen ed. Tha is why he nume ical calcula ion was made by means o he 2D axisymme ic me hod. The model included a pis on sou ce which consis s o an axially oscilla ing disk o dimensions equal o he axysime ic lens. A plane wa e wi h ampli ude po (IPW) impinges on he axysime ic lenses upwa d along he y di ec ion. Fo his pu pose, i is necessa y o sol e he Helmhol z equa ion gi en by 2 2 1pp c ω ρρ  ∇− ∇ =   (5) whe e ρ is he medium densi y, c is he ul asound eloci y, ω is he angula equency, and p is he acous ic p essu e. The assump ions made in he simula ions a e: (1) ha he wa eleng h o he inciden plane wa e (IPW) is la ge compa ed o he hickness o he lens; (2) ha he lens is conside ed o be acous ically igid and, he e o e, ha he Newmann bounda y condi ion (ze o sound eloci y) is applied; and (3) ha he plane wa e adia ion condi ion is applied o he bounda ies o he domain o simula e ee space and emula e he Somme eld condi ion in he nume ical solu ion o he wa e p oblem, as shown in Figu e 4. Figu e 2. Schema ic sec ion diag am o he gene a ion o a ac al zone pla e (FRZP) om s age 0 o s age 2. This pa ame e speci ies he dis ibu ion o he Ncopies in o he uni leng h segmen . In ac , he lacuna i y is essen ial o comple e he cha ac e iza ion o he ac al because s uc u es wi h di e en lacuna i y can ha e he same ac al dimension. In his wo k, ε = 44/1000. I is no ewo hy ha his s uc u e is no a egula Can o ac al. The ini ial segmen is di ided in o an odd numbe o segmen s, and he segmen loca ed in he e en posi ion is emo ed. This p ocedu e is epea ed h ough successi e s ages wi hin he same ule. Unlike he FZP, he wa eleng h does no appea in he exp ession o he Can o ings’ adii. The e o e, in his wo k, he wa eleng h dependence has been conside ed h ough pa ame e a. Figu es 1and 3 ep esen he co esponding zone pla es wi h he FZP p o ile (Figu e 1) and he FRZP p o ile (Figu e 3) s udied he e. Symme y 2016, 8, 28 3 o 7 Figu e 2. Schema ic sec ion diag am o he gene a ion o a ac al zone pla e (FRZP) om s age 0 o s age 2. This pa ame e speci ies he dis ibu ion o he N copies in o he uni leng h segmen . In ac , he lacuna i y is essen ial o comple e he cha ac e iza ion o he ac al because s uc u es wi h di e en lacuna i y can ha e he same ac al dimension. In his wo k, ε = 44/1000. I is no ewo hy ha his s uc u e is no a egula Can o ac al. The ini ial segmen is di ided in o an odd numbe o segmen s, and he segmen loca ed in he e en posi ion is emo ed. This p ocedu e is epea ed h ough successi e s ages wi hin he same ule. Unlike he FZP, he wa eleng h does no appea in he exp ession o he Can o ings’ adii. The e o e, in his wo k, he wa eleng h dependence has been conside ed h ough pa ame e a. Figu es 1 and 3 ep esen he co esponding zone pla es wi h he FZP p o ile (Figu e 1) and he FRZP p o ile (Figu e 3) s udied he e. Figu e 3. Schema ic sec ion diag am o he gene a ion o a FRZP and FRZP conside ed. The in e ac ion o ul asound wa es wi h ul asonic lenses is a complex p oblem. The FEM seems o be an app op ia e compu a ional ool o de e mine he dis ibu ion o acous ic p essu e and he e o e he ocal posi ions and he size o he ocal spo . To dec ease he compu a ional cos o he simula ions, he geome ical p ope ies o he model ha has been implemen ed mus be aken in o accoun . Bo h ac al and F esnel lenses ha e axial symme y since hese s uc u es a ise as a esul o he o a ion o a plane a ound an axis. Fu he mo e, all he cu ing hal -planes along his axis ha e iden ical cha ac e is ics. The e o e, a hal -plane, which co esponds o he longi udinal sec ion o he semilens and makes he o a ion a ound he axis, can be implemen ed. Tha is why he nume ical calcula ion was made by means o he 2D axisymme ic me hod. The model included a pis on sou ce which consis s o an axially oscilla ing disk o dimensions equal o he axysime ic lens. A plane wa e wi h ampli ude po (IPW) impinges on he axysime ic lenses upwa d along he y di ec ion. Fo his pu pose, i is necessa y o sol e he Helmhol z equa ion gi en by 2 2 1pp c ω ρρ  ∇− ∇ =   (5) whe e ρ is he medium densi y, c is he ul asound eloci y, ω is he angula equency, and p is he acous ic p essu e. The assump ions made in he simula ions a e: (1) ha he wa eleng h o he inciden plane wa e (IPW) is la ge compa ed o he hickness o he lens; (2) ha he lens is conside ed o be acous ically igid and, he e o e, ha he Newmann bounda y condi ion (ze o sound eloci y) is applied; and (3) ha he plane wa e adia ion condi ion is applied o he bounda ies o he domain o simula e ee space and emula e he Somme eld condi ion in he nume ical solu ion o he wa e p oblem, as shown in Figu e 4. Figu e 3. Schema ic sec ion diag am o he gene a ion o a FRZP and FRZP conside ed. The in e ac ion o ul asound wa es wi h ul asonic lenses is a complex p oblem. The FEM seems o be an app op ia e compu a ional ool o de e mine he dis ibu ion o acous ic p essu e and he e o e he ocal posi ions and he size o he ocal spo . To dec ease he compu a ional cos o he simula ions, he geome ical p ope ies o he model ha has been implemen ed mus be aken in o accoun . Bo h ac al and F esnel lenses ha e axial symme y since hese s uc u es a ise as a esul o he o a ion o a plane a ound an axis. Fu he mo e, all he cu ing hal -planes along his axis ha e iden ical cha ac e is ics. The e o e, a hal -plane, which co esponds o he longi udinal sec ion o he semilens and makes he o a ion a ound he axis, can be implemen ed. Tha is why he nume ical calcula ion was made by means o he 2D axisymme ic me hod. The model included a pis on sou ce which consis s o an axially oscilla ing disk o dimensions equal o he axysime ic lens. A plane wa e wi h ampli ude po(IPW) impinges on he axysime ic lenses upwa d along he ydi ec ion. Fo his pu pose, i is necessa y o sol e he Helmhol z equa ion gi en by ∇ˆ´1 ρ∇p˙“ω2 ρc2p(5) whe e ρ is he medium densi y, cis he ul asound eloci y, ω is he angula equency, and pis he acous ic p essu e. The assump ions made in he simula ions a e: (1) ha he wa eleng h o he inciden plane wa e (IPW) is la ge compa ed o he hickness o he lens; (2) ha he lens is conside ed o Symme y 2016,8, 28 4 o 7 be acous ically igid and, he e o e, ha he Newmann bounda y condi ion (ze o sound eloci y) is applied; and (3) ha he plane wa e adia ion condi ion is applied o he bounda ies o he domain o simula e ee space and emula e he Somme eld condi ion in he nume ical solu ion o he wa e p oblem, as shown in Figu e 4. Symme y 2016, 8, 28 4 o 7 Figu e 4. Schema ic diag am o he con igu a ion simula ed in he nume ical domain whe e he solu ions a e ob ained. To quan i y he acous ic ield, he sound p essu e le el is calcula ed as =20·log   (6) whe e p is he sound p essu e a a ce ain poin , and pinciden is he lens inciden p essu e. 3. Resul s and Discussion To implemen he analy ical model, lenses we e designed based on Equa ions (1) and (2). Bo h FZP and FRZP lenses ha e an ou e mos adius o 0.12233 m and eigh ings. The hickness o he blocking zone o bo h lenses was 0.003 m, and simula ions we e ca ied ou a 200 kHz. The chosen medium was wa e wi h a densi y o 1000 kg/m3 and a sound eloci y o 1500 m/s. The sol ed p oblem has 0.88 × 106 elemen s. To e alua e he ocusing capabili y o bo h he FZP and FRZP lenses, bo h buil wi hin he pa ame e s p esen ed in he p e ious sec ion, he ocal gain was calcula ed using he exp ession  =20·log     (7) whe e p is he p essu e a he ocal poin , and pinciden is he p essu e o he inciden wa e. Fo he FZP lens, he ocal gain alue was 19.4 dB, while o FRZP i was 13.1 dB. These esul s e ealed ha he ocusing e ec o FRZP lens do no imp o e he FZP lens. The e o e, he edis ibu ing sca e ing cen e s (solid ings) o he ac al lens was modi ied by mo ing he dis ance W/2 o he dispe sing elemen owa ds he o a ion axis (see Figu e 3), ai being now he dis ance om he axis o o a ion o he cen e o he ing segmen i. This new s uc u e was e e ed o as he “modi ied FRZP”. Figu e 5 shows a compa ison o he ans e se sec ion o he sound p essu e le el along he y-axes o he h ee lens conside ed. I can be seen ha only by edis ibu ing he solid ings could he acous ic ocalizing beha io imp o e conside ably. Fo he modi ied FRZP, he ocal gain was G ocus = 20.9 dB. Fu he mo e, a ema kable ea u e can be obse ed: F ac al s uc u es p oduce mul iple oci along he ans e sal axes wi h a high sound p essu e le el. This ea u e is cha ac e is ic o ac al di ac i e lenses, which is a esul o he phase sampling inhe en o hese ype o lenses [17]. Figu e 4. Schema ic diag am o he con igu a ion simula ed in he nume ical domain whe e he solu ions a e ob ained. To quan i y he acous ic ield, he sound p essu e le el is calcula ed as SPL “20¨log10 ˇˇˇˇ p pinciden ˇˇˇˇ (6) whe e pis he sound p essu e a a ce ain poin , and pinciden is he lens inciden p essu e. 3. Resul s and Discussion To implemen he analy ical model, lenses we e designed based on Equa ions (1) and (2). Bo h FZP and FRZP lenses ha e an ou e mos adius o 0.12233 m and eigh ings. The hickness o he blocking zone o bo h lenses was 0.003 m, and simula ions we e ca ied ou a 200 kHz. The chosen medium was wa e wi h a densi y o 1000 kg/m 3 and a sound eloci y o 1500 m/s. The sol ed p oblem has 0.88 ˆ106elemen s. To e alua e he ocusing capabili y o bo h he FZP and FRZP lenses, bo h buil wi hin he pa ame e s p esen ed in he p e ious sec ion, he ocal gain was calcula ed using he exp ession G ocus “20¨log10 ˇˇˇˇ p pinciden ˇˇˇˇ (7) whe e pis he p essu e a he ocal poin , and p inciden is he p essu e o he inciden wa e. Fo he FZP lens, he ocal gain alue was 19.4 dB, while o FRZP i was 13.1 dB. These esul s e ealed ha he ocusing e ec o FRZP lens do no imp o e he FZP lens. The e o e, he edis ibu ing sca e ing cen e s (solid ings) o he ac al lens was modi ied by mo ing he dis ance W/2 o he dispe sing elemen owa ds he o a ion axis (see Figu e 3), a i being now he dis ance om he axis o o a ion o he cen e o he ing segmen i. This new s uc u e was e e ed o as he “modi ied FRZP”. Figu e 5 shows a compa ison o he ans e se sec ion o he sound p essu e le el along he y-axes o he h ee lens conside ed. I can be seen ha only by edis ibu ing he solid ings could he acous ic ocalizing beha io imp o e conside ably. Fo he modi ied FRZP, he ocal gain was G ocus = 20.9 dB. Fu he mo e, a ema kable ea u e can be obse ed: F ac al s uc u es p oduce mul iple oci along he Symme y 2016,8, 28 5 o 7 ans e sal axes wi h a high sound p essu e le el. This ea u e is cha ac e is ic o ac al di ac i e lenses, which is a esul o he phase sampling inhe en o hese ype o lenses [17]. Symme y 2016, 8, 28 5 o 7 Figu e 5. T ans e se sec ion o sound p essu e le el along he y-axis o he FZP (blue), FRZP ( ed) and modi ied FRZP (black) lenses. Figu e 6 shows he spa ial dis ibu ion o he sound p essu e le el o he modi ied FRZP and he FZP lens in he XY plane. Al hough in bo h lenses some ocal zones can be conside ed, i is wo h no icing ha he modi ied FZRP p esen s sepa a ed mul i oci, while in FZP he dis ance be ween oci is negligible. This ac can be explained as an in e e ence phenomenon and is closely ela ed o he phase shi along he ings. A no mal plane wa e inciden upon a pla e lens unde goes di ac ion such ha cons uc i e in e e ence occu s a a poin . In a ac al lens, he phase is ma ched in a highe and sepa a ed a ea. On he o he hand, he ocal dis ance o FZP is 0.0875 m, while his pa ame e o FRZP is 0.137 m. (a) (b) Figu e 6. Spa ial dis ibu ion o he sound p essu e le el (dB) o (a) he modi ied FRZP; (b) he FZP s udied he e. The cu line o he ul asonic p essu e ield along he x-axis a he ocus is shown in Figu e 7a,b. Bo h lenses show simila beha io , bu a di e ence in he side lobes was obse ed. In he FRZP lens, he side lobe has mo e in ensi y bu is na owe han he FZP lens. On he o he hand, i can be obse ed ha he main lobe in he FRZP lens is wide , co esponding o he longe dep h o he ul asonic ield, compa ed o he FZP lens. The adial p essu e ield a he ocus in he inse s also illus a es his ac . The wid h o he main lobe is impo an in bo h ul asonic imaging and medical ul asonics. Figu e 5. T ans e se sec ion o sound p essu e le el along he y-axis o he FZP (blue), FRZP ( ed) and modi ied FRZP (black) lenses. Figu e 6shows he spa ial dis ibu ion o he sound p essu e le el o he modi ied FRZP and he FZP lens in he XY plane. Al hough in bo h lenses some ocal zones can be conside ed, i is wo h no icing ha he modi ied FZRP p esen s sepa a ed mul i oci, while in FZP he dis ance be ween oci is negligible. This ac can be explained as an in e e ence phenomenon and is closely ela ed o he phase shi along he ings. A no mal plane wa e inciden upon a pla e lens unde goes di ac ion such ha cons uc i e in e e ence occu s a a poin . In a ac al lens, he phase is ma ched in a highe and sepa a ed a ea. On he o he hand, he ocal dis ance o FZP is 0.0875 m, while his pa ame e o FRZP is 0.137 m. Symme y 2016, 8, 28 5 o 7 Figu e 5. T ans e se sec ion o sound p essu e le el along he y-axis o he FZP (blue), FRZP ( ed) and modi ied FRZP (black) lenses. Figu e 6 shows he spa ial dis ibu ion o he sound p essu e le el o he modi ied FRZP and he FZP lens in he XY plane. Al hough in bo h lenses some ocal zones can be conside ed, i is wo h no icing ha he modi ied FZRP p esen s sepa a ed mul i oci, while in FZP he dis ance be ween oci is negligible. This ac can be explained as an in e e ence phenomenon and is closely ela ed o he phase shi along he ings. A no mal plane wa e inciden upon a pla e lens unde goes di ac ion such ha cons uc i e in e e ence occu s a a poin . In a ac al lens, he phase is ma ched in a highe and sepa a ed a ea. On he o he hand, he ocal dis ance o FZP is 0.0875 m, while his pa ame e o FRZP is 0.137 m. (a) (b) Figu e 6. Spa ial dis ibu ion o he sound p essu e le el (dB) o (a) he modi ied FRZP; (b) he FZP s udied he e. The cu line o he ul asonic p essu e ield along he x-axis a he ocus is shown in Figu e 7a,b. Bo h lenses show simila beha io , bu a di e ence in he side lobes was obse ed. In he FRZP lens, he side lobe has mo e in ensi y bu is na owe han he FZP lens. On he o he hand, i can be obse ed ha he main lobe in he FRZP lens is wide , co esponding o he longe dep h o he ul asonic ield, compa ed o he FZP lens. The adial p essu e ield a he ocus in he inse s also illus a es his ac . The wid h o he main lobe is impo an in bo h ul asonic imaging and medical ul asonics. Figu e 6. Spa ial dis ibu ion o he sound p essu e le el (dB) o ( a ) he modi ied FRZP; ( b ) he FZP s udied he e. The cu line o he ul asonic p essu e ield along he x-axis a he ocus is shown in Figu e 7a,b. Bo h lenses show simila beha io , bu a di e ence in he side lobes was obse ed. In he FRZP lens, he side lobe has mo e in ensi y bu is na owe han he FZP lens. On he o he hand, i can be obse ed ha he main lobe in he FRZP lens is wide , co esponding o he longe dep h o he ul asonic ield, Symme y 2016,8, 28 6 o 7 compa ed o he FZP lens. The adial p essu e ield a he ocus in he inse s also illus a es his ac . The wid h o he main lobe is impo an in bo h ul asonic imaging and medical ul asonics. Symme y 2016, 8, 28 6 o 7 (a) (b) Figu e 7. T ans e se sec ion o absolu e p essu e ield along he x-axis a he ocus o (a) he F esnel zone pla e (FZP) and (b) he ac al zone pla e (FRZP) s udied he e. The inse s show he adial p essu e ields dis ibu ion. In his sense, ano he pa ame e ha is used o e alua e he ocusing capabili y o a lens is he ull wid h a hal maximum (FWHM) o he ocus. The esolu ion o he ocus is cha ac e ized by λ/2 < FWHM <λ o he analyzed equency. Thus, he FWHM alues ob ained we e 5.6 and 6.6 mm o FZP and FRZP, espec i ely. 4. Conclusions In summa y, he ul asonic modi ied FRZP lens is he e p oposed as an al e na i e o he FZP ul asonic lens. The ocusing capabili y o he FZP and modi ied FRZP lenses has been e alua ed by means o he ocal gain and he FWHM o he ocus. The ocusing capaci y o he modi ied FRZP lens is be e han he FZP lens conside ing he gain inc ease o 1.5 dB. FRZP p esen s mul i oci o conside able gain, while FZP has a single main ocus. Howe e , mo e esea ch o di e en pa ame e s o p e ac al lens is needed o disco e hei in luence on he ocaliza ion p ope ies and he e o e imp o e i s beha io . This new ype o lens can ha e he same applica ions whe e con en ional F esnel lenses a e cu en ly applied, ha is, in medical diagnosis and he apy, and acous ic imaging. Mo eo e , all imp o emen s de eloped o F esnel lenses could be applied o p e ac al lenses. Acknowledgmen s: This wo k has been suppo ed by he Gene ali a Valenciana (AICO/2015/119). Au ho Con ibu ions: Cons anza Rubio coo dina ed he heo e ical de elopmen s and pa icipa ed in he es ablishmen o he heo y p inciples used in his wo k and in he d a ing o he manusc ip . Se gio Cas iñei a-Ibañez and Daniel Ta azó-Se ano de eloped he heo y used, designed some o he compu ing asks, and pa icipa ed in he d a ing o he manusc ip . An onio U is and Pila Candelas ca ied ou he compu ing asks and pa icipa ed in he analysis o he s a e-o - he-a ma e ials and in he d a ing o he manusc ip . Con lic s o In e es : The au ho s decla e no con lic o in e es . Figu e 7. T ans e se sec ion o absolu e p essu e ield along he x-axis a he ocus o ( a ) he F esnel zone pla e (FZP) and ( b ) he ac al zone pla e (FRZP) s udied he e. The inse s show he adial p essu e ields dis ibu ion. In his sense, ano he pa ame e ha is used o e alua e he ocusing capabili y o a lens is he ull wid h a hal maximum (FWHM) o he ocus. The esolu ion o he ocus is cha ac e ized by λ /2 < FWHM < λ o he analyzed equency. Thus, he FWHM alues ob ained we e 5.6 and 6.6 mm o FZP and FRZP, espec i ely. 4. Conclusions In summa y, he ul asonic modi ied FRZP lens is he e p oposed as an al e na i e o he FZP ul asonic lens. The ocusing capabili y o he FZP and modi ied FRZP lenses has been e alua ed by means o he ocal gain and he FWHM o he ocus. The ocusing capaci y o he modi ied FRZP lens is be e han he FZP lens conside ing he gain inc ease o 1.5 dB. FRZP p esen s mul i oci o conside able gain, while FZP has a single main ocus. Howe e , mo e esea ch o di e en pa ame e s o p e ac al lens is needed o disco e hei in luence on he ocaliza ion p ope ies and he e o e imp o e i s beha io . This new ype o lens can ha e he same applica ions whe e con en ional F esnel lenses a e cu en ly applied, ha is, in medical diagnosis and he apy, and acous ic imaging. Mo eo e , all imp o emen s de eloped o F esnel lenses could be applied o p e ac al lenses. Acknowledgmen s: This wo k has been suppo ed by he Gene ali a Valenciana (AICO/2015/119). Au ho Con ibu ions: Cons anza Rubio coo dina ed he heo e ical de elopmen s and pa icipa ed in he es ablishmen o he heo y p inciples used in his wo k and in he d a ing o he manusc ip . Se gio Cas iñei a-Ibañez and Daniel Ta azó-Se ano de eloped he heo y used, designed some o he compu ing Symme y 2016,8, 28 7 o 7 asks, and pa icipa ed in he d a ing o he manusc ip . 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