A New Method to Design Trifurcated Optical Fiber Displacement Sensors

1532 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024
A New Me hod o Design T i u ca ed Op ical
Fibe Displacemen Senso s
Go ka Zubia , G adua e S uden Membe , IEEE, Joseba Zubia , Membe , IEEE, Josu Amo ebie a ,
Go zon Aldabalde eku , and Gaizka Du ana
Abs ac —T i u ca ed op ical ibe displacemen senso s
(OFDSs) a e inc easingly used in indus ial and ae ospace
applica ions. The c i ical elemen o he senso is he ibe
bundle. Howe e , he e is no s aigh o wa d me hod o cal-
cula e i om he wo king speci ica ions o he senso , he
numbe o ibe s needed, hei size, and a angemen . This
wo k p esen s a simple ye accu a e me hod o design i-
u ca ed OFDSs. The p oposed me hod allows o de i e he
geome ical a angemen and size o he ibe s om h ee
simple equa ions, hus educing signi ican ly he di icul y
and complexi y o he OFDS design. Those h ee equa ions
depend on he wo king poin , wo king ange, equi ed sen-
si i i y, and maximum size o he bundle. In his way, he
p oposed me hod will sa e aluable ime o esea che s and
enginee s who wish o design, ab ica e, and use his ype o OFDSs. The p ocedu e is explained in de ail wi h wo
examples. The esul s p edic ed by he model a e compa ed wi h he expe imen al esul s o a bundle wi h iden ical ibe
a angemen and dimensions. The esul s show good ag eemen wi h a de ia ion o less han 1% in he wo king ange
o he senso .
Index Te ms—Design, ibe bundle, ibe op ic senso s, in ensi y modula ed op ical senso , me hod, op ical ibe
displacemen senso (OFDS), i u ca ed.
I. INTRODUCTION
HIGH p ecision me ology is essen ial in Indus y 4.0 o
moni o and imp o e he e iciency and e sa ili y o
ac o y p ocesses [1],[2]. As a esul , dis ance senso s a e used
in a wide ange o applica ions o measu e se e al pa ame e s
o in e es such as he hickness, heigh , oughness, de o ma-
Manusc ip ecei ed 2 Oc obe 2023; e ised 31 Oc obe 2023;
accep ed 25 No embe 2023. Da e o publica ion 4 Decembe 2023;
da e o cu en e sion 12 Janua y 2024. This wo k was suppo ed
in pa by he G an I+D+i/PID2021-122505OB-C31, G an TED2021-
129959B-C21, G an PDC2022-133053-C21, and G an RTC2019-
007194-4 h ough MCIN/AEI/10.13039/501100011033; in pa by he
“ERDF A way o making Eu ope”; in pa by he “Eu opean Union Nex
Gene a ion EU/PRTR”; in pa by he G an IT11452-22 and h ough
he Basque Go e nmen ; in pa by he ELKARTEK 2023 unde G an
µ4Sma -KK-2023/00016 and G an Ekohegaz II-KK-2023/00051; and
in pa by he Uni e si y o he Basque Coun y (UPV/EHU) T ans-
ligh . The wo k o Go ka Zubia was suppo ed by he Ph.D. ellowship
om he Basque Go e nmen unde G an PRE_2022_2_0269. The
associa e edi o coo dina ing he e iew o his a icle and app o ing
i o publica ion was D . Sanjee Raghuwanshi. (Co esponding au ho :
Go ka Zubia.)
Go ka Zubia and Josu Amo ebie a a e wi h he Depa men o Applied
Ma hema ics, Uni e si y o he Basque Coun y UPV/EHU, 48013
Bilbao, Spain (e-mail: [email p o ec ed]).
Joseba Zubia, Go zon Aldabalde eku, and Gaizka Du ana a e wi h
he Depa men o Communica ions Enginee ing, Uni e si y o he
Basque Coun y UPV/EHU, 48013 Bilbao, Spain.
Digi al Objec Iden i ie 10.1109/JSEN.2023.3337311
ion, dis o ion, and gap be ween su aces, o ins ance, [3],
[4],[5]. To selec he bes - i ing de ice, we ha e o conside
he applica ion, he en i onmen , and he equi ed p ecision,
as hese ac o s a e c i ical o de e mining he speci ica ions o
he senso . A p ime example o he la e can be acknowledged
in he ae onau ical u bine indus y.
In ha ield, noncon ac solu ions a e p ac ically compulso y
in o de o ca y ou any me ology measu emen s, as hea y
me allic pieces a e o a ing a high speed signi ican ly close
o each o he [6]. To ha end, he mos popula al e na i es
a e capaci i e [7], induc i e [8], ul asonic [9], d aw-wi e,
and op ical senso s [10]. Among hem, op ical ibe dis-
placemen senso s (OFDSs) o e many ad an ages such as
elec omagne ic immuni y, high esponse, speed, small size,
g ea e sa ili y, and easy ins alla ion, o example, [11],[12].
Wi hin his g oup o senso s, OFDSs based on he e lec ion
o op ical in ensi y add wo signi ican ad an ages: Fi s ly,
as an op ical ibe is used o ansmi ligh o he moni o ed
su ace, he sensing heads can be ins alled in emo e loca ions
whe e di ec lase ligh canno each, such as inside an ai c a
u bine [13]. Secondly, since hese senso s a e o en used in a
di e en ial con igu a ion [14], hey minimize se e al po en ial
e o sou ces due o e lec i e su ace oughness, empe a u e
a ia ions, luc ua ions in he emi ing ligh sou ce, and so
on [15]. This app oach is pa icula ly use ul o dynamic
© 2023 The Au ho s. This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 License.
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ZUBIA e al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1533
u bine es ing, as he blades o a e and he e lec i e su ace
is cons an ly changing [16],[17].
The wo king ange and sensi i i y o OFDSs depend on
hei geome ic s uc u e, i.e., he numbe and ans e se
a angemen o ibe s in a bundle [18],[19],[20],[21]. The
ypical OFDS con igu a ion uses wo adjacen op ical ibe s o
ansmi and ecei e he ligh signal h ough e lec ion [22].
Howe e , his a angemen limi s he amoun o e lec ed ligh ,
which educes he wo king ange. O he con igu a ions ha e
been p oposed, e.g., a pai o ben - ip op ical ibe s [23],
a hemisphe ical a angemen o ibe s, and e en andomly
dis ibu ed ansmi ing and ecei ing ibe s [24],[25], o mo e
ecen ly, an OFDS based on a mul ico e ibe wi h se en co es
hexagonally a anged [26].
Some au ho s p e e o use ibe s o bundles ha o m
an angle wi h one ano he and wi h he no mal o he
e lec ing su ace [27]. I has been ound ha se ing he
angle be ween he ansmi ing and ecei ing ibe s o 20◦
inc eases he sensi i i y up o 30 imes o e he con en ional
pa allel ibe con igu a ion. Asymme ic app oaches like his
a e well-sui ed o angle measu emen . Fo example, Sag a io
and Mead [28] designed an OFDS using a squa e s uc u e o
pe o m bo h axial and angula displacemen measu emen s;
Khia e al. [29] de eloped a ibe senso wi h ou ecei ing
ibe s a ound he ansmi ing one, o long-dis ance, high-
esolu ion measu emen s. The d awback o such ibe bundles
is ha he senso head inc eases p opo ionally wi h he angle.
Addi ionally, many o hese designs only allow he measu e-
men o uniaxial o a ion, ha is, he abili y o measu e he
il o he e lec ing su ace along jus one axis [30]. In cases
whe e he e lec ing su ace is nei he homogeneous no la ge
and also ib a es, he angle is no main ained and changes
apidly. This applies o he measu emen o ip clea ance in
ai c a u bines, whe e he e lec ing su aces a e he blades
and whose shape is wis ed [31]. In hese ha sh en i onmen s,
he angle measu emen is no ad an ageous.
The e o e, he designs used o measu e displacemen s in
u bines a e based on azimu hally symme ic a angemen s,
speci ically, bundles o ibe s a anged in concen ic ings
a ound a cen al emi ing ibe . The mos common designs a e
he i u ca ed OFDSs [32],[33]. These a e based on a ibe
bundle wi h an emi ing ibe in he cen e su ounded by wo
ibe ings a di e en adial dis ances. Typically, ibe bundles
a e cus om-designed o mee he wo king ange o sensi i i y
equi ed by he speci ic applica ion [34],[35]. Those pa ame-
e s a y based on he numbe o ibe s in each ecei ing ing,
he size and nume ical ape u e o each ibe , o he adii o he
ibe ings [36]. Va ious me hods ha e been p oposed o model
he esponse o OFDSs: Geome ic app oxima ions [25],[37],
ay acing, Mon e Ca lo calcula ions [38] o Gaussian beam-
based models. O all hese app oaches, he one p esen ed
by Cao e al. [25] o e s he wides ange o possibili ies,
albei i aces he d awback o equi ing he implemen a ion
o 8 di e en o mulas. The one based on a Gaussian o
quasi-Gaussian beam has also excellen accu acy, bu i is
much simple [39]. Anyway, designing he mos sui able ibe
bundle is a complex nume ical ask [25],[27],[38],[39] ha
he p oposed app oach in his a icle can g ea ly simpli y.
Fig. 1. Schema ic o an OFDS. The cons i uen blocks a e a ligh -
emi ing sou ce, a ibe bundle o guide he ligh , and a ecei e ci cui o
con e he collec ed ligh powe in o ol age.
Unlike p io me hods, he me hod in oduced he e allows
us o compu e he bundle design ha bes i s he wo king
speci ica ions. The p oposed me hod enables he deduc ion o
he geome ical a angemen and ibe size wi hin a bundle
om h ee simple equa ions, he eby signi ican ly educing
he di icul y and complexi y o OFDS design, which can
be ime-consuming and expensi e. The h ee equa ions only
depend on he speci ica ions de ined a each applica ion,
namely, he wo king poin and ange, he equi ed sensi i i y,
and he maximum size o he bundle. As a esul , he
p oposed me hod will sa e aluable ime o esea che s and
enginee s who wish o design, ab ica e, and use his ype
o OFDSs. This me hod can i ially be ex ended o designs
wi h ou ( e a u ca ed), i e (pen a u ca ed), o mo e ibe
ings. The only limi a ion o he me hod is ha i is es ic ed
o he design o azimu hally symme ic bundles. The a icle
is o ganized in o ou sec ions. Fi s ly, he ma hema ical
model is p esen ed. Secondly, he esul s ob ained a e
analyzed. Thi dly, he design equa ions a e de i ed. Finally,
he model is alida ed by compa ing he heo e ical esul s
wi h expe imen al ones.
II. MATHEMATICAL MODEL
The schema ic o an OFDS is shown in Fig. 1. I comp ises
a ligh sou ce, a ibe bundle, and a ecei e . The ligh emi ed
by he ligh sou ce is guided h ough he ansmi ing ibe o
he e lec ing su ace. A e e lec ion, he ligh is collec ed by
he ecei ing ibe s o he bundle. Each o hese is connec ed
o sepa a e pho odiodes, whe e he op ical powe is inally
con e ed in o ol age. The la e is a unc ion o he dis ance
be ween he bundle- ip and he e lec i e su ace. The design o
he bundle is c i ical, since he esponse o he OFDS is highly
dependen on i s geome ic a angemen , i.e., he numbe , size,
and dis ibu ion o he ansmi ing and ecei ing ibe s [15],
[20],[22],[27],[31].
The aim o his wo k is o de elop a no el app oach o
design a i u ca ed OFDS, aking he desi ed wo king poin
and ange as inpu a iables. Fo his pu pose, we ha e buil a
oy model o he esponse o he senso o unde s and he
in luence o each o he geome ic pa ame e s in o de o
ine- une i s esponse and mee he equi emen s o he speci ic
applica ion.
1534 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024
Fig. 2. S uc u e o a i u ca ed ibe bundle wi h adius R, com-
posed o a ansmi ing ibe o adius Tloca ed a he cen e , and
wo su ounding coaxial ecei ing ings. The posi ions o he ings a e
deno ed as {ρ1, ρ2}, espec i ely. Thei wid hs a e labeled {∆ρ1,∆ρ2}.
The diame e s o he ibe s {φ1, φ2}co espond o he wid hs o he
ings in he oy model, and hei posi ion equal he posi ion o he ibe s
{R1,R2}.
Fig. 2illus a es he geome y o he p oblem. I consis s o
a ci cula s uc u e composed o a ansmi ing ibe loca ed
a he cen e o he bundle and wo coaxial ings o ecei ing
ibe s su ounding i . The oy model eplaces he disc e e
ibe ings wi h con inuous homogeneous ings. This g ea ly
simpli ies he calcula ions while p ese ing all he bundle
geome y pa ame e s.
The posi ions o he ings and hei wid hs a e ρ1,ρ2,1ρ1,
and 1ρ2, espec i ely. The diame e s o he ibe s {φ1, φ2}
co espond o he wid hs o he ings in he oy model, and
hei posi ion equals he posi ions o he ibe s {R1,R2}.
O he pa ame e s a e he adii o he ansmi ing ibe Tand
he bundle R. The alidi y o he oy model is con i med in
Sec ion V.
We assume ha he ansmi ing ibe is a single mode o
educe he modal noise a he ou pu o he bundle [13]. Unde
his p emise, we can app oxima e he i adiance a he ou pu
o he ibe as a Gaussian beam, using [40]
I(ρ, z)=I0w0
w(z)2
exp−2ρ2
w2(z)
=2P
πw2(z)exp−2ρ2
w2(z)(1)
whe e w(z)is he beamwid h, ρis he adial dis ance
om he p opaga ion axis zand I0is he in ensi y o he beam
a he o igin. P e e s o he o al op ical powe ansmi ed
by he beam and w0is he wais adius a i s s e chies
poin -whe e he in ensi y d ops o 1/e2≈13.5% o i s
maximum alue. The beamwid h w(z)sp eads acco ding o
he o mula
w(z)=w0s1+z
z02
.(2)
I eaches i s minimum alue w0a he bundle ip (z=0).
z0is he Rayleigh dis ance, de ined as z0=πw2
0/λ, which
is he dis ance o e which a beam can p opaga e wi hou
Fig. 3. E olu ion o he ligh in ensi y o a Gaussian beam wi h z, aking
w0=2.4 µm and z0=27 µm.
signi ican ly di e ging. The o al ligh powe emi ed by he
ansmi ing ibe , P, can easily be es ima ed as
P=Z∞
0
I(ρ)2π d
=2πI0w0
w(z)2Z∞
0
exp−2ρ2
w2(z)ρdρ
P=π
2w2
0I0.(3)
Since he ligh powe emains cons an a di e en zdis ances
om he ansmi ing ibe - ip, bo h he peak in ensi y and he
in ensi y dis ibu ion p o ile a y wi h z. In ac , on he beam
axis, he in ensi y I(0,z)d ops o
I(0,z)=I0
z2
0
z2+z2
0
.(4)
Fig. 3shows he e olu ion o I(ρ, z)wi h z.
Fo su icien ly la ge dis ances
z≫z0H⇒ w(z)≈w0
z
z0
w(z)≈z an θ0=z an(a csin NA)(5)
being NA he nume ical ape u e o he ansmi ing ibe . The
di e gence angle θ0and w0a e pa ame e s ha solely ely on
he ligh sou ce and ansmi ing ibe cha ac e is ics. Thus,
once bo h a e selec ed, θ0and w0 emain cons an .
A a ce ain dis ance o z, he ligh beam eaches a su ace,
e lec s owa d he ecei e ings, and is collec ed a e a -
eling a o al dis ance o 2z. The in ensi y a an in ini esimal
ing o wid h dρplaced a a dis ance ρ om he ibe axis is
d I (ρ, 2z)=20P
πw2(2z)exp−2ρ2
w2(2z)2πρ dρ. (6)
The ideal e lec ance o he su ace is 0. Thus, he calcula ion
o he de ec ed i adiance in he ing loca ed a a adial dis ance
ZUBIA e al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1535
ρ1wi h wid h 21ρ1is s aigh o wa d
I(ρ1, 1ρ1,2z)=20P
πw2(2z)Zρ1+1ρ1
ρ1−1ρ1
exp−2ρ2
w2(2z)2πρ dρ
I(ρ1, 1ρ1,2z)=20Psinh4ρ11ρ1
w2(2z)exp−2ρ2
1+1ρ2
1
w2(2z).
(7)
Simila ly, he measu ed i adiance in a second ing loca ed a
a dis ance ρ2and wid h 21ρ2is
I(ρ2, 1ρ2,2z)=20Psinh4ρ21ρ2
w2(2z)exp−2ρ2
2+1ρ2
2
w2(2z).
(8)
Assuming ha he wo pho ode ec o s collec ing he ligh
en e ing he wo independen ecei ing ings ha e a ypical
quad a ic esponse [6]
Vi=ki20Psinh4ρi1ρi
w2(2z)exp−2ρ2
i+1ρ2
i
w2(2z)(9)
whe e Viis he ou pu ol age o pho ode ec o i, and kiis
a cons an ha includes he pho ode ec o esponse and o he
ac o s such as noise, de ec o ci cui gain, su ace oughness,
con amina ion, bends, e c. Consequen ly, he esponsi i y o
he senso , η(z), is de ined as he a io o he op ical in ensi ies
o i s wo ecei ing ings, i.e., V2/V1
η(z)=
k2sinh4ρ21ρ2
w2(2z)exp−2ρ2
2+1ρ2
2
w2(2z)
k1sinh4ρ11ρ1
w2(2z)exp−2ρ2
1+1ρ2
1
w2(2z)
η(z)=k
sinh4ρ21ρ2
w2(2z)
sinh4ρ11ρ1
w2(2z)exp−2ρ2
2+1ρ2
2−ρ2
1−1ρ2
1
w2(2z).
(10)
The cons an kis de ined as k=k2/k1. In he ollowing (11),
we assume ha he esponse o bo h pho ode ec o s is he
same, so k=1. Al hough he esponsi i y o he OFDS
depends on ou geome ic pa ame e s, we encompass hese
in o jus h ee: {A1,A2,q}. The eby, (10) wi h (5) compac s
o
η(z)=
sinhA2
z2 an2θ0
sinhA1
z2 an2θ0exp−q
z2(11)
whe e









A1=ρ11ρ1
A2=ρ21ρ2
q=ρ2
2+1ρ2
2−ρ2
1−1ρ2
1/2 an2θ0
w(z)≈z an θ0





Geome ical
pa ame e s.
In Fig. 4, he esponsi i y is plo ed along wi h he op ical
powe collec ed by he inne and ou e ings, P1(z)and P2(z),
espec i ely. We applied a scaling ac o o 2.5 o P1(z)and
P2(z) o isualize he h ee unc ions oge he . S ill, one can
obse e ha he esponse o each ing is signi ican ly smalle
han he combined ou pu . Fu he mo e, acco ding o Fig. 4,
he inne ing ( ed cu e) demons a es excellen pe o mance
a sho dis ances due o a na ow bu s eep linea ange, which
Fig. 4. Op ical powe collec ed by he inne - ed, P1(z)-and ou e -
g een, P2(z)- ings along wi h he esponsi i y-blue, η(z). A scale ac o o
2.5 was applied o P1(z) and P2(z) o display he h ee cu es oge he .
The able shows he linea anges and hei slopes. In yellow, some
ypical poin s o in e es o η(z) in mm: zTP =9.3, zHR =12, and
zHM =13.7.
makes he senso highly sensi i e. Con e sely, he ou e ing
(g een cu e), has a mo e ex ensi e linea ange a he cos o
a loss o sensi i i y, i.e., a slowe g ow h o i s esponse. This
is because he ecei ed ligh is b oadened and a ies adially
smoo he . Fo he same eason, he ou e ing ecei es less
powe , in his case, i e imes less han he inne ing.
Howe e , we can also obse e ha aking he a io be ween
bo h cu es esul s in a signi ican ly imp o ed esponse (blue
cu e). I is bo h g ea e in ampli ude and la ge in linea
ange. Thus, using a single- ing con igu a ion has he only
ad an age o enabling sho e dis ance measu emen s. Ne e -
heless, as we ha e seen, wo king wi h a single ing has many
disad an ages, mos ly ela ed o he inabili y o co ec he
emi ing ligh sou ce in ensi y luc ua ions, ambien ligh , e c.
Jus as an example, i we de ine he linea ange as he in e al
be ween 5% and 65% o he maximum alue o he on slope
o each cu e, we ob ain a linea ange o 1z5,65 =1.13 mm
when wo king wi h he inne ing, and 1z5,65 =3.08 mm
o he ou e . Finally, we can obse e ha he linea ange
o he esponsi i y-which combines he op ical powe o bo h
ings-ex ends up o 1z5,65 =10.77 mm.
The ypical esponsi i y o he senso is also p esen ed
in Fig. 4. I sa u a es a he limi alue, when he dis ance
o he e lec o is su icien ly la ge
lim
z→∞η(z)=
sinhA2
z2 an2θ0
sinhA1
z2 an2θ0exp−q
z2=A2
A1=ρ21ρ2
ρ11ρ1
.
(12)
Fo con enience, le us e e o his limi as p
p=lim
z→∞η(z)=ρ21ρ2
ρ11ρ1
.(13)
Acco ding o (13), he sa u a ion alue o he senso inc eases
wi h he a io o he ecei ing a eas, A2/A1.
Fig. 5illus a es how each e m o he esponsi i y unc ion
con ibu es o he esul (11), e ealing ha he hype bolic
sine e m (11) in o ange is signi ican only o small z
alues. Fo la ge z alues, i jus mul iplies he exponen-
ial by he limi p. In ac , he esponsi i y gi en by (11)
1536 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024
Fig. 5. Senso esponse η(z) in blue and S(z) in black o he ollowing
design speci ica ions (mm): ρ1=0.5, ρ2=1.5, ∆ρ2=0.1, and ∆ρ2=
0.15. Also, θ0=5◦,q=130 mm2, and k=1. Con ibu ion o he
hype bolic sines in o ange; in ed, he exponen ial one.
is indis inguishable om
η(z)=I(ρ2, 1ρ2,2z)
I(ρ1, 1ρ1,2z)≈pexp−q
z2.(14)
A dashed e ical line has been added in Fig. 5 o indica e
whe e he esponsi i y (blue line) is no longe ze o. F om
ha poin on, he hype bolic-sine- e m ba ely changes and
esembles a cons an .
Ano he key pa ame e o he senso is he sensi i i y, §,
de ined as he slope o he esponsi i y
S(z)=dη(z)
dz =2pq
z3exp−q
z2.(15)
The sensi i i y is maximum when he second de i a i e equals
ze o, i.e., sol ing o he u ning poin ,zTP, o he esponsi i y
Smax =d2η(z)
d2zz=zTP =0H⇒ zTP = 2
3q.(16)
Thus, he alue o he esponsi i y (14) a ha poin is
η(zTP)=pexp−q
z2
TP ≈0.22p.(17)
An addi ional key poin is he dis ance a which he espon-
si i y eaches hal i s maximum alue. We named i zHM
η(zHM)=1
2p.(18)
Sol ing o zHM yields o
zHM = q
ln 2 ≈1.2√q.(19)
Mo eo e , subs i u ing (16) in (19) e ealed ha he ela ion-
ship be ween zTP and zHM can be exp essed as
zHM = 3
2 ln 2 zTP =1.47zTP.(20)
In p e ious Fig. 4, we plo ed he esponsi i y along wi h
he p oposed key pa ame e s, namely: zTP,zHM, and zHR. The
la e is he dis ance poin a hal o he wo king ange o he
senso . We assumed ha he wo king ange o he senso is
be ween he [5,65]% o ηmax, which co esponds o he mos
linea egion o he esponsi i y. So, zHR will be a 35% o he
maximum esponsi i y ηmax. As we will see in Sec ion V, he
wo king ange may also be de ined by i ing he esponsi i y
o a s aigh line using Pea son’s co ela ion coe icien .
III. RESULTS AND DISCUSSION
A e in oducing he ma hema ical model in Sec ion II,
he nex s ep is o analyze he esul s. Equa ions (11) and
(14) cha ac e ize he esponsi i y o he senso . No ice ha i
depends on he geome ic pa ame e s h ough wo exp essions
ρ2
2+1ρ2
2−ρ2
1−1ρ2
1and ρ21ρ2
ρ11ρ1
.
The oles o {ρ2, 1ρ2}and {ρ1, 1ρ1}a e symme ical wi h
espec o he esponsi i y. Also, changes in {ρ2, 1ρ2}a e
equi alen , and he same is ue o {ρ1, 1ρ1}. The e o e,
we conduc ed ou ini ial analysis o he esponsi i y h ough
global changes in qand p. The dependence o he esponsi i y
on hose pa ame e s is be e exempli ied in Fig. 6.
Fig. 6(a) shows he esponsi i y as a unc ion o qand z
o di e en alues o p, om 1 o 5 in s eps o 1. The g een
su ace co esponds o p=5, and he blue a he bo om
co esponds o p=1. Inspec ing Fig. 6(a) leads us o wo
ou comes.
1) Fi s , he shape o he cu e does no change as he
alue o pinc eases. This implies ha he posi ion o
he wo king poin is independen o p, as e idenced
in (16) and (19). Howe e , i does change he scale and
slope o he cu e and, he e o e, he sensi i i y in he
neighbo hood o he wo king poin .
2) Second, he analysis o he esponsi i y e eals wo
simul aneous e ec s: on he one hand, he educ ion
o he pa ame e qshi s he u ning poin o smalle
alues; hus i we wan o wo k a sho dis ances,
he pa ame e qmus be small. This ac is suppo ed
by (16). On he o he hand, i shows ha his dec emen
o qis associa ed wi h an inc emen in he slope o
he esponsi i y; hus, he sensi i i y and qa e in e sely
p opo ional. When one inc eases, he o he dec eases.
In sho , qis a ade-o pa ame e be ween he posi ion
o he wo king poin and he sensi i i y o he senso
q↓ H⇒ zTP ↓and q↓ H⇒ Smax ↑.
As p e iously s a ed, he wo king ange is de ined
as 1z5,65 =z65 −z5, which equals o 0.94√q. This
can easily be deduced by ollowing he same p ocedu e
ha led o (16) and (20). Thus, a la ge wo king ange
o he senso implies a la ge alue o q; which, in u n,
dec eases he sensi i i y. In conclusion, he ange o
dis ances a which he esponsi i y is linea is in e sely
p opo ional o he sensi i i y
q↑ H⇒ 1z5,65 ↑so, 1z5,65 ↑ H⇒ S(z)↓.
This loss o sensi i i y can be co ec ed, a leas pa ially,
in wo ways.
a) Ex e nally, inc easing he gain o he pho ode ec-
o s.
b) Inc easing pas i is p opo ional o he sensi i -
i y (15).

ZUBIA e al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1537
Fig. 6. Responsi i y as a unc ion o qand p.(a) Responsi i y as a unc ion o {q,z} o di e en p, om p=1 o 5 in s eps o 1. The op g een
su ace co esponds o p=5 and he blue a he bo om o p=1. (b) Responsi i y as a unc ion o pand z o di e en q, om q=0.1 mm2 o
10.1 mm2in s eps o 2 mm2. The op blue su ace co esponds o q=0.1 mm2and he cyan a he bo om o q=10.1 mm2. The ed ma ke s
indica e he same poin .
The maximum and minimum alues o pa e gi en
by he size o he bundle, R≥ρ2+1ρ2, and he
size o he ansmi ing ibe , T≤ρ1−1ρ1. Fo
ha pu pose, o inc ease p, he ou e ecei ing ing
should ei he be wide , loca ed u he away, o bo h.
F om (13), widening he a ea o he ou e ing o
educing he a ea o he inne ing leads o a la ge p.
This is possible while lea ing qunchanged since hey
a e independen . The only p ac ical limi a ions o his
p ocedu e a e he maximum gain o he pho ode ec o s
[ki,(10)] and he maximum a io o he a eas o he
ecei ing ings p=A2/A1, which, in summa y, depend
on bo h he size o he bundle Rand he size o he
emi ing ibe T. Iden ical conclusions can be d awn
om Fig. 6(b), which shows he sensi i i y as a unc ion
o pand z o di e en alues o q.
In summa y, o achie e a wide wo king ange and highe
sensi i i y, we need o inc ease {p,q}in he senso design.
Mo eo e , a highe esponsi i y can be ob ained by inc eas-
ing p, see (13).
In a second analysis, we conduc ed a se o simula ions o
unde s and be e he in luence o each pa ame e by a ying
one while he es emained ixed. Fig. 7shows he esul s o
he oy model o he esponsi i y η(z)and sensi i i y, S(z).
The e e ence pa ame e s a e
ρ1=0.5 mm ρ2=1.0 mm
1ρ1=0.2 mm 1ρ2=0.2 mm θ0=5◦.
A. Va ia ion o he Posi ion o he Inne Ring ρ1
This a ia ion is equi alen o changing he posi ion o he
inne ing ibe s. Looking a Fig. 7(a), we can ha dly obse e
any change in he posi ion o he sensi i i y. This is because he
ange o alues o e which i can a y is small since he es
o he bundle limi s he inne ing. Howe e , he esponsi i y
alue d ops sha ply when we inc ease he alue o ρ1, b inging
he wo ings close . This dec ease o η(z)is accompanied
by a subs an ial d op in S(z), see Fig. 7(b), and a a ia ion
o i s maximum owa d smalle alues o z. The maximum
alue o he sensi i i y is gi en a zTP dis ance, Smax =S(zTP),
subs i u ing (16) in (15)
S(zTP)=3√3 exp−3
2ρ21ρ2 an θ0
ρ11ρ1qρ2
2+1ρ2
2−ρ2
1−1ρ2
1
.(21)
Also, he wo king ange dec eases sligh ly wi h an inc ease o
ρ1. So, as p edic ed by (16) and (21)
ρ1↑ H⇒ zTP ↓,S(zTP)↓, 1z5,65 ↓.
B. Va ia ion o he Inne Ring Wid h 1ρ1
The esponsi i y alue is e y sensi i e o he wid h o he
inne ing, as shown in Fig. 7(c). Inc easing he wid h o he
inne ing causes a signi ican dec ease in he esponsi i y.
When 1ρ1goes om [0.1,3]mm, he esponsi i y di ides by
ou . Fig. 7(d) depic s ha he same is ue o he sensi i i y,
which becomes h ee imes smalle . In sho , he esponsi i y
dec eases and la ens when he adius o he inne ibe s
inc eases. The posi ion o S(zTP), hough, emains cons an ,
meaning ha qis less sensi i e o changes in 1ρ1. Likewise,
he wo king ange is independen o 1ρ1. Summa izing
1ρ1↑ H⇒ η(z)↓,S(z)↓.
C. Va ia ion o he Posi ion o he Ou e Radius ρ2
I we modi y he ou e adius o he ibe ing, inc easing
he dis ance ρ2, as shown in Fig. 7(e), we obse e wo e ec s
on he esponsi i y. On he one hand, he cu e shi s owa d
la ge alues o z; on he o he hand, bo h i s ampli ude and
wo king ange inc ease. Howe e , based on Fig. 7( ), he e is
a small dec ease in sensi i i y as ρ2inc eases. The posi ion o
i s maximum alue zTP g ows linea ly wi h ρ2. So
ρ2↑ H⇒ q↑,p↑, η(z)↑,zTP ↑S(z)↓.
This can be unde s ood om (13) and (16). An inc ease
o ρ2 aises he alue o qand p, and, he e o e, zTP. Hence,
he maximum alue o he esponsi i y is also augmen ed. (21)
explains he sligh decay o sensi i i y as ρ2inc eases.
1538 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024
Fig. 7. All he alues om he legends a e in millime e s. Ve ical lines show he zTP poin s o e e y cu e. (a) h ough (j) show he esponsi i y
and sensi i i y o i e di e en geome ical pa ame e con igu a ions: {ρ1,∆ρ1, ρ2,∆ρ2, θ0}. Da a we e no malized wi h espec o he maximum
alue.
D. Va ia ion o he Ou e Ring Wid h 1ρ2
I we now look a he a ia ion o he senso esponse wi h
he wid h o he ou e ing, Fig. 7(g) and (h), we obse e he
opposi e beha io . This is due o he minus sign in he q ac o .
By inc easing he alue o 1ρ2in he ange [0.1,3]mm,
we iple he alue o bo h esponsi i y and sensi i i y.
As wi h 1ρ1(Sec ion III-B), he posi ion o he u ning
poin zTP ha dly a ies and can be conside ed cons an o
ZUBIA e al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1539
TABLE I
LIST OF POSSIBLE WORKING POINTS OF THE SENSOR TOGETHER WITH THE VALUE OF THEIR RESPECTIVE RESPONSIVITIES
his 1ρ2 ange. The same happens wi h he wo king ange.
Then
1ρ2↑ H⇒ η(z)↑,S(z)↑.
E. Va ia ion o he Nume ical Ape u e NA =sin(θ0)
The las pa ame e o a y is he nume ical ape u e o he
sou ce. In Fig. 7(i) and (j) we a y he NA be ween [0.1,3],
which co esponds o he accep ance angles θ0= [5,15]◦.
I we inc ease he NA, he esponsi i y inc eases in slope and
shi s owa d smalle z alues, see Fig. 7(j). I s alue is almos
quad upled and he posi ion o he maximum, i.e., he alue
o zTP, is hal ed. I is no ewo hy he huge dependence o he
wo king ange on NA; he wo king ange is indeed di ided by
h ee. Consequen ly
NA ↑ H⇒ η(z)↑,zTP ↓,S(z)↑.
F. Dead Zone
Some au ho s de ine he dead [39] o blind [20] zone o
an OFDS as he ange o dis ances in which he esponsi i y
is e y small o ze o. In Table Iwe show ha he alue z5
o which he esponsi i y is 5% o i s maximum alue
is z5=0.58√q. This pa ame e indica es he poin a which
he esponsi i y s a s o ise. I is e iden ha , in o de o
minimize he dead zone, qmus be educed.
As p e iously discussed, he e a e wo app oaches o
add ess his issue: ei he by mo ing he ecei e ibe ings,
inc easing ρ1, and dec easing ρ2; o by inc easing he NA
o he sou ce. This is easy o unde s and in bo h cases;
i we dec ease he dis ance be ween he ibe ings o he
cen e o he bundle, he e lec ed ligh eaches he wo ings
signi ican ly due o hei p oximi y o he ansmi ing ibe s.
Con e sely, inc easing he NA causes he ligh o sca e
apidly, eaching he ecei ing ings e en a a sho dis ance.
This can easily be obse ed o he case o wo equal ibe
ings 1ρ1=1ρ2
q=ρ2
2−ρ2
1
2 an2θ0=(ρ2−ρ1)(ρ2+ρ1)
2 an2(a csin NA).(22)
So
NA ↑∨(ρ2−ρ1)↓∨(ρ2+ρ1)↑ H⇒ Dead zone ↓.
The indi idual inspec ion o he pa ame e s yields he
ollowing design ips.
1) To ope a e a small dis ances, we should ei he inc ease
he NA o he sou ce, b ing he ibe ings close ,
o dec ease he ibe ings o bundle cen e dis ance.
2) Dis ances zTP,zHM,and zHR gi e us in o ma ion abou
he posi ion o he esponsi i y cu e (see Table I).
3) Sensi i i y can be imp o ed by dec easing he wid h o
he inne ing ( he adius o he inne ibe s), o equi a-
len ly, inc easing he wid h o he ou e ing ( he adius
o he ou e ibe s).
4) The posi ion wo king poin is independen o he wid h
o he ings, i.e., he diame e o he ibe s.
IV. DESIGN PROCEDURE
Al hough he ibe bundle senso has a simple p inciple and
many ou s anding ad an ages, mos o he esea ch has ocused
on he model and calcula ion o he esponse o a speci ic bun-
dle, i.e., modeling he esponse o he bundle knowing a p io i
i s geome ic pa ame e s and con igu a ion [13],[14],[15],
[16],[17],[18]. Howe e , he e is no esea ch on he design
o such bundles om gi en speci ica ions. In o he wo ds,
no p ocedu e ha ela es he a ge dis ance and he wo king
ange o he ibe pa ame e s and bundle con igu a ion.
The ul ima e goal is o de e mine {ρ1, ρ2, 1ρ1, 1ρ2} om
he design equi emen s, which a e: he size o he bundle
(ou e adius R) and he wo king poin , zWP, he sensi i -
i y, S(z), and wo king ange, 1z, o he OFDS
Inpu




R
zWP
S(zWP)
1z




| {z }
Design
equi emen s
−→



Sol e
design equa ions


−→
Ou pu
Toy model Expe imen al




ρ1
21ρ1
ρ2
21ρ2



=



R1
φ1
R2
φ2




|{z }
Geome ical
pa ame e s
.
The p ocedu e can be summa ized in he ollowing s eps.
1) Fi s , o p ac ical easons, i is usually necessa y o se
a maximum adius Ro he bundle
R=ρ2+1ρ2( i s design equa ion).(23)
1540 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024
2) Then, qis de e mined om he wo king poin . I we
conside he u ning poin o he esponsi i y as he
wo king poin , i.e., zWP =zTP =√2q/3, hen he
pa ame e qbecomes ixed
q=3
2z2
TP (second design equa ion).(24)
Any al e na i e wo king poin also se s he qpa ame e .
In gene al, and as we ha e p e iously indica ed, he
ela ionship be ween he wo king poin zWP and he
pa ame e qis o he o m zWP =h√q, being ha
nume ical cons an , see Table I.
These wo condi ions impose es ic ions on he slope
o he esponsi i y. Indeed
1ρ2=R−ρ2(25)
1ρ1=sρ2
2+(R−ρ2)2−ρ2
1−z2
WP an2θ0
h2(26)
and hen, by subs i u ing (25) and (26) in (13)
p=ρ2(R−ρ2)
ρ1qρ2
2+(R−ρ2)2−ρ2
1−z2
WP an2θ0
h2
.(27)
3) We could se ei he he sensi i i y, S(zWP), o he
maximum alue o he esponsi i y o he senso . I is
equi alen o ixing he alue o p. This is he hi d
design equa ion. Wi h he choice o {p,q} he espon-
si i y o he senso becomes ixed
η(z)=pexp−z2
WP
h2z2( hi d design equa ion).(28)
All ha emains is o de e mine om he h ee design
equa ions he geome ic alues o he wo ings, i.e., he
sizes and placemen o he ibe s. Since we ha e ou
pa ame e s {ρ1, ρ2, 1ρ1, 1ρ2}and h ee condi ions, he
p oblem is unde de e mined. We can lea e he design
based on a ee pa ame e we can ake a will, e.g., ρ2.
Exp essing he h ee design equa ions as a unc ion
o ρ2, we ob ain
BUNDLE DESIGN COOKBOOK












1ρ2=R−ρ2(DE1)
1ρ1=sρ2
2+(R−ρ2)2−ρ2
1−z2
WP an2θ0
h2(DE2)
ρ1=ρ21ρ2
p1ρ1
(DE3)












.
A simple inspec ion o he la e b ings wo clues abou
he bundle design p ocess. The smalle he adius o he inne
ing ρ1, he g ea e he in insic slope po he esponsi i y, i.e.,
he g ea e he sensi i i y. This would ecommend placing he
i s ing as close as possible o he emi e ibe , es ablishing
he new condi ion
T=ρ1−1ρ1(29)
ρ1= T
2±sR2
2− 2
T
4−z2
WP an2θ0
2h2−ρ2
2−Rρ2.(30)
Howe e , his choice is no always compa ible wi h he selec-
ion o he wo king poin o q. No ice ha we can easily aise
he chosen slope a he wo king poin , S(zWP), by jus uning
he asymme ic gain ac o k, which is de e mined om he
selec ed pho ode ec o s gains as demons a ed in (11)
dη(z)
dz zWP =S(zWP)=2kp
h2zWP
exp−1
h2.(31)
Two addi ional cons ain s mus always be sa is ied o he
design o be physically easible, namely
T≤ρ1−1ρ1(32a)
ρ2≥ρ1+1ρ1+1ρ2.(32b)
These inequali ies only e lec ha he ings in he oy model
and he ibe s in he expe imen al ibe bundle canno o e lap.
Ano he design op ion is o se a known linea esponse,
(z)=z0+mz, in a speci ied ange, [zi,z ]and ind he
esponsi i y ha bes app oxima es i wi hin ha ange. Then,
we look o he esponsi i y ha minimizes he e o in he
wo king ange o he senso [zi,z ]. Fo he sake o ease o
eading, he design equa ions o his case ha e been de eloped
in he Appendix.
We shall illus a e he p eceding p ocedu e wi h wo eal-
wo ld examples, bo h o which a e ca e ully discussed in
Sec ions V-A and V-B.
V. EXPERIMENTAL VALIDATION
The expe imen al se up has been desc ibed in de ail p e i-
ously in [34] and [35]. Rega ding he ligh sou ce, a lase
module om F ank u Componen s (HSML-0660-20-FC,
F ank u Lase Company, F ied ichsdo , Ge many) was
employed. I had a nominal ou pu powe o 20 mW a 660 nm.
An op ical isola o (IOF-660, Tho labs, New on, NJ, US) was
placed be ween he lase and he bundle o a oid e lec ions
ha could des abilize he ligh sou ce. Fo he op o-elec ical
con e sion, wo Tho labs PDA100A-EC pho ode ec o s we e
used. Finally, he esponsi i i y is calcula ed as he quo ien
o he wo ob ained ol age signals, {V1,V2} o Sec ion V-A,
and {V1,V3} o Sec ion V-B.
To alida e ou model, a e a u ca ed bundle was designed
and ab ica ed, consis ing o a single-mode ansmi ing ibe
a i s cen e and h ee concen ic ings o mul imode ecei ing
ibe s. The chosen emi ing ibe was a single-mode ibe as
easoned in Sec ion II. The ibe s o he i s wo ings ha e
he same diame e , 0.2 mm. Howe e , he ou e ing ibe s
a e 0.34 mm in diame e . The o al diame e o he bundle is
1.12 mm. A mic oscopic image o i s c oss sec ion is shown
in Fig. 8(b). The alues o he posi ion and adii o he ibe s
a e p esen ed in Table II.
This bundle allows one o alida e wice he model. Fi s ly,
we compa e he esponsi i y o wo ings ( i s and second)