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 . 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 .
Re e ences
1. Mandelb o , B.B. The F ac al Geome y o Na u e; WH F eeman and Co.: San F ancisco, CA, USA, 1982.
2. Takayasu, H. F ac al in Physical Sciences; Manches e Uni e si y P ess: Manches e , UK, 1992.
3.
Pe i, A.; Alippi, A.; Be ucci, A.; C acium, F.; Fa elly, F. Vib a ional p ope ies o a con inuous sel -simila
s uc u e. Phys. Re . B 1994,49, 15067–15075. [C ossRe ]
4.
Sapo al, B.; Haebe lé, O.; Russ, S. Acous ical p ope ies o i egula and ac al ca i ies. J. Acous . Soc. Am.
1997,102, 2014–2019. [C ossRe ]
5.
Lubniewski, Z.; S epnowski, A. Applica ion o he ac al analysis in he sea bo om ecogni ion. A ch. Acous .
1998,25, 499–511.
6.
Gibia , V.; Ba jan, A.; Cas o , K.; Chazaud, E.B. Acous ical p opaga ion in a p e ac al wa eguide. Phys. Re . E
2003,67, 066609. [C ossRe ] [PubMed]
7.
Cas iñei a-Ibáñez, S.; Rome o-Ga cía, V.; Sánchez-Pé ez, J.V.; Ga cía-Ra i, L.M. O e lapping o acous ic
bandgaps using ac al geome ies. Eu ophys. Le . 2010,92, 24007. [C ossRe ]
8.
Cas iñei a-Ibáñez, S.; Rubio, C.; Rome o-Ga cía, V.; Sánchez-Pé ez, J.V.; Ga cía-Ra i, L.M. Design,
manu ac u e and cha ac e iza ion o an acous ic ba ie made o mul i-phenomena cylind ical sca e e s
a angen in a ac al-based geome y. A ch. Acous . 2012,37, 455–462. [C ossRe ]
9.
Gomez-Lozano, V.; U is, A.; Candelas, P.; Belma , F. Acous ic ansmision h ough pe o a ed pla es wi h
ac al subwa eleng h ape u es. Solid S a e Commun. 2013,165, 11–14. [C ossRe ]
10.
Schindel, D.; Bash o d, A.; Hu chins, D. Focusing o ul asonics wa es in ai using a mic omachined F esnel
zone-pla e. Ul asonics 1997,35, 275–285. [C ossRe ]
11.
Wel e , J.T.; Sa hish, S.; Ch is ensen, D.E.; B od ick, P.G.; Heebl, J.D.; Che y, M.R. Focusing o longi udinal
ul asonic wa es in ai wi h an ape iodic la lens. J. Acous . Soc. Am.
2011
,130, 2789–2796. [C ossRe ]
[PubMed]
12.
Mole on, M.; Se a-Ga cía, M.; Da aio, C. Acous ic F esnel lenses wi h ex ao dina y ansmisión.
Appl. Phys. Le . 2014,105, 114109. [C ossRe ]
13.
Li, Y.; Yu, G.; Liang, B.; Zhou, X.; Li, G.; Cheng, S.; Cheng, J. Th ee-dimensional ul a hin plana lenses by
acous ic me ama e ials. Sci. Rep. 2014,4, 6830. [C ossRe ] [PubMed]
14.
Cal o, D.C.; Thangawng, A.L.; Nicholas, M.; Layman, C.N. Thin F esnel zone pla e lenses o ocusing
unde wa e sound. Appl. Phys. Le . 2015,107, 014103. [C ossRe ]
15. Jagga d, A.D.; Jagga d, D.L. Can o ing di ac als. Op . Commun. 1998,158, 141–148. [C ossRe ]
16.
Bo odich, F.M. F ac al Geome y. In Encyclopedia o T ibology; Wang, Q.J., Chung, Y.-W., Eds.; Sp inge :
Be lin/Heidelbe g, Ge many, 2013; Volume 2, pp. 1258–1264.
17.
Sle a, M.Z.; Hun , W.D.; B iggs, R.D. Focusing pe o mance o epoxy- and ai -backed poly inylidene
lou ide F esnel zone pla es. J. Acous . Soc. Am. 1994,96, 1627–1633. [C ossRe ]
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(CC-BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).