Frequency-Modulated Optical Feedback Interferometry for Nanometric Scale Vibrometry

“© © 2016 IEEE. Pe sonal use o his ma e ial is pe mi ed. Pe mission om IEEE mus be ob ained o all o he uses, in any cu en o u u e media, including ep in ing/ epublishing his ma e ial o ad e ising o p omo ional pu poses, c ea ing new collec i e wo ks, o esale o edis ibu ion o se e s o lis s, o euse o any copy igh ed componen o his wo k in o he wo ks.” 1 F equency modula ed op ical eedback in e e ome y o nanome ic scale ib ome y Aji Jha1,2, F ancisco J. Azcona1,S uden Membe , IEEE, San iago Royo1,Membe , IEEE, Abs ac —We demons a e a no el me hod which makes e - icien use o lase nonlinea dynamics when subjec o op ical sel injec ion o sub-wa eleng h displacemen sensing pu poses. The p oposed me hodology combines wo di e en phenomena aking place inside he lase ca i y: op ical sel injec ion, which esul s in op ical eedback in e e ence, and lase con inuous wa e equency modula ion, gi ing ise o a wa eleng h sweeping e ec in he lase ’s emission. We p esen a combina ion o hese phenomena o measu e ib a ion ampli udes below λ/2wi h esolu ions o a ew nanome e s, bandwid h dependen upon he dis ance o ex e nal a ge , ampli ude and equency o cu en modula ion. The basic heo e ical de ails and a ma hema ical model a e p esen ed o he de eloped measu emen p inciple. Expe imen al esul s, wi h he sys em wo king as a ib ome e o measu e a a ge ib a ion o ampli ude λ/5(137.5 nm) wi h mean peak o peak e o o 2.4 nm jus by poin ing he lase diode on o he a ge and applying some signal p ocessing is also demons a ed. Index Te ms—Op ical eedback, Dopple equency, op ical senso s, nano displacemen sensing, injec ion locking. I. INTRODUCTION AS desc ibed by Lang and Kobayashi [1], when he cohe - en ligh e lec ed om an ex e nal a ge is sel -injec ed in o he lase ’s ca i y, hen bea ing o ime delayed op ical ield om dis an a ge and ield inside i s ca i y cause he lase o en e in o mul i s able s a es. Depending upon he eedback ield sel -injec ed in o he ca i y, se e al applica ions ha e been de eloped in he las decade. When he eedback s eng h is high, he lase diode (LD) eaches a chao ic egion, which has been used o enc yp ed op ical communica ions [2]. On he o he hand, unde mode a e o weak eedback condi ions, he LD ou pu powe expe iences pe iodic luc ua ions wi h a pe iod equi alen o hal emission wa eleng h (λ/2), which can be associa ed o in e e ence inges, hus leading o he concep o op ical eedback in e e ome y (OFI). Classic OFI me hods a e limi ed o he measu emen o dis- placemen s la ge han λ/2. In i s o igins, he inge coun ing me hod enabled a λ/2 esolu ion, as desc ibed by Dona i in [3]. Du ing he las wo decades, he inc ease o accu acy and esolu ion has been a d i e o he esea ch in he ield [4]–[6]. The capabili y o measu e ib a ion displacemen s smalle han λ/2was ecen ly demons a ed o he i s ime [7] using mechanical modula ion and a double lase se up. 1Cen e o Senso s, Ins umen s and Sys ems De elopmen , UPC- Ba celonaTech, Rambla S . Neb idi 10, 08222, Te assa, Spain. 2Ins i u e o Measu emen and Con ol Enginee ing, Ka ls uhe Ins i u e o Technology, Engle -Bun e-Ring 21, 76131 Ka ls uhe, Ge many. Co esponding au ho - [email p o ec ed]. This wo k is unded by Eu opean Union (G an No. 159224-1-2009-1-FR- ERA MUNDUS-EMJD), Spanish Minis y o Science and Inno a ion (P ojec no. DPI2014-56881-R) and Agau (G an no. 2012FI BI 00240). Lase and pho odiode D i e TIA Piezo ( a ge ) Signal gene a o Powe supply Fig. 1: Expe imen al se -up o p oposed me hodology. (TIA: T ans-Impedance Ampli ie ). In ha app oach, he bandwid h o he senso was dependen upon he p oduc o he equency and ampli ude o he linea mechanical s age used o p oduce a e e ence OFI signal. The use o a mechanical modula o wi h a linea esponse inc eased he senso ’s cos and size while limi ing he measu emen bandwid h. To o e come hese p oblems, we p opose a no el app oach ha akes ad an age o he ini e non-ze o linewid h en- hancemen ac o (α) ha couples he lase ’s ampli ude and equency modula ions, in o de o eplace he mechanical modula ion depic ed in [7] by an elec onic modula ion. The de eloped me hod in oduces wo main ad an ages o e [7]. Fi s , he se up is simpli ied since he use o a second lase and o ex e nal mechanical elemen s o p o ide he modula ion is no longe necessa y. This educes he cos and size o he senso while making i compa able o classic OFI applica ions. The use o only one lase also educes he possibili y o bias in he measu emen in oduced by he di e ence o he physical p ope ies be ween each LD o i s ci cui y. Second, he bandwid h and esolu ion o he p oposed me hod a e dependen only upon he LD elec onic and op ical pa ame e s, in con as o he mechanical scheme desc ibed be o e. This, in u n, allows o each la ge bandwid hs since he me hod is no longe limi ed by mechanical cons ain s. In he ollowing sec ions we desc ibe he wo king p inciple o a p oposed op ical senso based on OFI e ec s induced by con inuous wa e equency modula ion (CWFM-OFI), includ- ing he me hodology o calcula e he a ge ib a ion wa e o m - ampli ude and he equency bandwid h o senso . II. SYSTEM DESCRIPTION AND THEORY The p oposed concep o he expe imen is shown in Fig.1. Fi s , a pe iodic cu en modula ion is applied while he a ge is kep s a iona y. When he cu en eeding he lase is mod- ula ed, no only he in ensi y o he emi ed signal is changed, bu also i s emission wa eleng h, because o he coupling be ween ampli ude and equency modula ion, deno ed by α pa ame e . To a ain linea induced wa eleng h changes, a 2 iangula wa e is used as modula ion signal. Unde hese condi ions, he lase can be hough as a mul i-wa eleng h sou ce whose wa eleng h changes wi h ime p opo ionally o he injec ed cu en . The ligh emi ed by he lase hen hi s he s a iona y a ge and pa o he emi ed ligh is back e lec ed in o he ca i y, whe e i in e e es wi h he s anding wa e al- eady p esen in he ca i y, p oducing consecu i e in e e ence inges in he emi ed powe wi h a phase di e ence o 2π be ween hem. The op ical eedback signal ob ained consis s o a se ies o small ipples, caused by he bea ing o he ime delayed e lec ed elec ic ield wi h he emi ed ield inside he ca i y, supe imposed on o he amp o he powe signal modula ed by he in ensi y as shown in Fig. 2(a). The signal is hen subjec ed o a di e en ia ion which sepa a es he ipples caused by in e e ence om he amp caused by modula ion. I has been shown ha he numbe o inges which appea on he amp depend upon he ound ip ime delay (dis ance be ween he lase diode and he a ge ) and upon he wa eleng h peak o peak change caused by he modula ion cu en [6]. We call his he e e ence case and he inges p oduced e e ence inges. In expe imen al signals, a e applying di e en ia ion, he ime o occu ence o each inge will be no ed and eco ded. We know ha each λ/2 a ge displacemen p oduces an in e e ence inge in he signal [1], [3]. Thus, i he a ge ib a ion ampli ude is smalle han λ/2, no addi ional inges a e c ea ed as a esul o he a ge mo ion. E en hough he o al numbe o inges emains cons an , he a ge mo ion changes he equency seen by he lase due o he change in op ical pa h di e ence, in he equi alen o a Dopple shi in he equency domain. Thus, when compa ed o he o me case, he new se o inges will be shi ed in ime p opo - ionally o he op ical pa h change. F om his poin onwa d, we e e o his case as ib a ion case, and o i s inges as ib a ion inges. I will be shown ha i is possible o ex ac he comple e in o ma ion o he a ge displacemen by compa ing he e e ence and ib a ion cases. Ma hema ically, all pa ame e s ela ed o he e e ence case will be deno ed by subsc ip and he ib a ion case by subsc ip . The excess phase φxand he emi ed powe Px(wi h x= [ , ]) equa ions ha go e n bo h cases a e de ined by [8] φ ( ) = 2πτex ( c −( h +im( )Ω )) +Csin(2π c τex + an−1α) = 0,(1) P ( ) = P ocos(2π c ( )τex ),(2) φ ( ) = 2πτex ( )( c −( h +im( )Ω )) +Csin(2π c τex ( ) + an−1α) = 0,(3) P ( ) = P ocos(2π c ( )τex ( )),(4) whe e Lex is he s a iona y dis ance om he lase o he a ge , and Lex ( ) = Lex +a( ), wi h a( ) = A cos(2π ) desc ibes a a ge mo ion wi h peak ampli ude and equency A and , espec i ely. τex = 2Lex /c is he ex e nal ound ip ime o he e e ence case, τex ( ) = 2Lex ( )/c he ound ip ime o he ib a ion case, c and c a e he emission equency o he lase a e eedback o he e e ence and ib a ion case espec i ely, h is he s andalone lase equency a i s ope a ing poin when no eedback is p esen , TABLE I: Simula ion pa ame e s Pa ame e s Value Dis ance o ex e nal a ge om lase (Lex ) ( o wa d pa h) 0.3 m Peak o peak modula ing cu en (Im(pp)) 1.5 mA Modula ion equency ( m) 100 Hz Ta ge ib a ion ampli ude (pp) (Am)λ 10 Ta ge ib a ion equency ( ) 200 Hz F equency modula ion coe icien (Ω ) [8] -3 GHz/mA Feedback s eng h (C) 0.9 Emission wa eleng h o lase (λ h) 692 nm line wid h enhancemen ac o (α) 3 0 0.005 0.01 24 24.5 25 25.5 26 Ampli ude(mW) ime (sec) 0 0.005 0.01 −4 −2 0 2 4x 10−3 ime (sec) Ampli ude(a.u) 0 0.002 0.004 0.006 0.008 0.01 −0.1 −0.05 0 0.05 0.1 Ampli ude/λ 0 0.002 0.004 0.006 0.008 0.01 24 24.5 25 25.5 26 0 0.002 0.004 0.006 0.008 0.01 24 24.5 25 25.5 26 Powe (mW) ime (sec) 2 2.5 3 x 10−3 24.8 25 ime (sec) 2 2.5 x 10−3 0 1 2 3 x 10−3 ime (sec) m=100Hz =200Hz (a) (b) (c) Fig. 2: Simula ion esul s. (a) Emi ed powe a ia ions in e e ence (blue solid) and ib a ion ( ed dash) cases (inse gi es he magni ied iew). (b) Shi in inges in bo h cases. (c) Ta ge wa e o m econs uc ed (black s a ) compa ed wi h he e e ence ( ed squa e). A ows explains he o ma ion o sampling poin s om inges. By symme y o modula ion signal, p ocessing any one amp gi es in o ma ion abou a ge ib a ion, making o he amp edundan . im( )is he iangula AC modula ion cu en d i ing he lase , Ω is he equency coe icien o lase , Cis he eedback s eng h, Pis he emission powe o he lase unde eedback, wi h a o al ampli ude o P o and P o o he e e ence and ib a ion cases. Simula ions o pe o mance based on Eq. (1) - (4) we e ca ied ou using he lase diode pa ame e s lis ed in Table I. The goal o he simula ion was o p o e he appea ance o powe shi s in ime be ween he e e ence and ib a ion cases. Nonlinea equa ions Eq. (1) and Eq. (3) we e sol ed nume i- cally o ind c and c espec i ely. Then he co esponding powe luc ua ions we e calcula ed using Eq. (2) and Eq. (4). Simula ion esul s a e p esen ed in Fig. 2(a) and 2(b), which show a signi ican shi o he emission powe ela ed o he a ge mo ion. A. Displacemen Calcula ions Now we will calcula e he a ge displacemen ou o he he ime shi be ween ib a ion and e e ence inges. I is o be no ed ha he phase di e ence be ween each consecu i e inge is equi alen o a 2πphase change, which in OFI can be ela ed o λ/2i a displacemen is p esen . Thus, he ac ion o ime shi on he ib a ion inges ela i e o he e e ence inges mul iplied by λ/2is equi alen o he a ge displacemen . Le n , n , n , n−1 be ec o s con aining he eco ded alues o he ime o occu ence o he e e ence and ib a ion inges, he ime di e ence be ween co esponding 3 inges, and he ime di e ence be ween consecu i e e e ence inges ob ained in hal modula ion pe iod. whe e he supe - sc ip is he numbe o elemen s in he conside ed ec o , xk is he momen o occu ence o he k h inge. Thus, he a ge displacemen A , ec can be compu ed simply using A , ec = n−1 n−1 ×λ/2(5) Using Eq. (5), he a ge wa e o m is econs uc ed , shown in Fig. 2(c). I is wo h no ing, ha , since he a ge ib a ion is wice he modula ion equency, he en i e a ge wa e o m lie in one amp (hal pe iod) o he modula ion signal wi h a ow illus a ing ha each inge co esponds o a measu emen poin . So i is desi able o p ocess one o he amp o a ge ib a ion ela ed in o ma ion. B. F equency Bandwid h Le m=1 Tm be he modula ion equency o cu en o he lase . In ime Tm/2(conside ing only one amp o modula ion signal; u he explained in sec ion III), we ha e N inges gi ing N−1sample poin s. Thus he sampling a e o a ge econs uc ion is gi en by s, ec = (2N m−1). F om Nyquis heo em, he maximum equency ha can be econs uc ed is hal he sampling a e and using 2N m>> 1, can be app oxima ed as , ec ≃N m. We see ha he maximum equency o a ge ib a ion ha can be econs uc ed using he p oposed senso is dependen upon he p oduc o modula- ion equency and he numbe o inges. Hence keeping he numbe o inges Ncons an , he equency o a ge ib a ion ha can be de ec ed is di ec ly p opo ional o he modula ion equency i.e. , ec ∝ m. Since, a lase can be modula ed a la ge equencies (MHz) keeping signi ican equency de ia ion [9] as compa ed o mechanical ib a ion o lase in [7], his me hod gi es wide bandwid h as compa ed o [7]. This also demons a es he ac ha he senso ’s bandwid h ha was limi ed by he bandwid h o piezo in o me is now solely de e mined by he p ope ies (maximum modula ion equency) o lase used and hence ex ending he bandwid h o senso as compa ed o o me . III. EXPERIMENT AND RESULTS Expe imen ally, a Hi achi HL6501 0.65 µm band AlGaAsP lase diode (LD) wi h a mul i-quan um well (MQW) s uc u e was used o es he me hod. The emission wa eleng h was measu ed wi h Ins umen Sys em’s SPECTRO 320(D) R5 uni . The lase was hen di ec ly modula ed o ob ain he desi ed equency chi p. Since unde his ope a ing condi ions he LD may show mode hopping, a de ailed analysis o i s spec al cha ac e is ics as a unc ion o he injec ion cu en was pe o med o loca e he op imal ope a ing poin as shown in Fig. 3, whose measu emen s we e pe o med epe i i ely in s able wo king condi ions. The LD ope a ing poin was selec ed as I h = 48 mA wi h λ h = 692.5 nm. The lase is kep powe ed on and unde s able wo king condi ions when acqui ing he e e ence and ib a ion measu emen s. To p oduce he equency chi p, he LD is ed using a iangula cu en modula ion wi h alue a he ope a ing poin (I h) peak o peak ampli ude o 250 mV (0.2 mA) and a equency o m= 100 Hz. The LD is ocused on he a ge using a 45 50 55 60 692 692.5 693 693.5 694 694.5 bias,(mA) Wa eleng h,(nm) Linea , egion Linea , egion Ope a ing poin ,48,mA, 692.5nm Mode,hop Linea , egion Mode,hop Fig. 3: Va ia ion o emission wa eleng h wi h bias cu en . Tho labs lens 352240 wi h ocal leng h o 8 mm and nume ical ape u e o 0.5 placed a a dis ance o 3.5 mm om he lase . A piezoelec ic linea s age PI-LISA (P-753.3CD) placed a a dis ance ( o wa d pa h) o 12 cm om he lase is used as a ge ; he s age includes an embedded capaci i e senso wi h a esolu ion o 0.2 nm [10] which is used o compa ing he ob ained esul s. P e ious o each measu emen , he lase is se o wo k in he weak eedback egion and i is allowed o a ain a s able s a e. To dissipa e he hea p oduced by he LD, he lase is moun ed on an aluminium pla o m. In a i s s ep, a measu emen o he e e ence case is acqui ed and denoised using a sym6 wa ele ans o m [11] and, hen, he ime o occu ence o he inges is no ed. Nex , he a ge is se in o ib a ion by applying o he AC sinusoidal signal wi h ampli ude 55 mV and equency 200 Hz (so ha en i e a ge ib a ion wa e o m lie in one amp o modula ion signal) o he piezoelec ic s age using a signal gene a o , esul ing in a 137.5 nm displacemen measu ed by he embedded capaci i e senso [10]. The choice o a ge ib a ion equency o 200 Hz was chosen as a p oo o concep accompanied by simula ions and heo e ical p oo and limi ed by he piezo. Due o he a ge mo ion, he op ical pa h di e ence be ween lase and a ge changes esul ing in a shi o he inges when compa ed o he e e ence case. As in he e e ence case, he signal is acqui ed, denoised and he ime o occu ence o he inges is eco ded. An example o ypical expe imen al esul s is shown in Fig. 4. The inge shi is easily app ecia ed and will be used o eco e he a ge ampli ude and equency o ib a ion. Once he ime o occu ence o each inge is compu ed o he e e ence and ib a ion cases, he a ge displacemen is calcula ed using Eq. (5). Fig. 5 shows he expe imen al a ge displacemen e ie ed co esponding o he de ec ed inges esul ing om a single oscilla ion o he a ge p esen ed in Fig. 4, compa ed wi h he da a ob ained om he embedded capaci i e senso . I should be no ed ha only one amp is p ocessed o ge he ull pe iod o a ge wa e o m (as he a ge ib a ion equency is wice ha o modula ion equency). The expe imen was pe o med nine imes unde equi alen condi ions. The mean e o (pp) was 2.4 nm. Fo conciseness, only i e measu emen s a e lis ed in Table II. Howe e , he - es was ca ied on Ns= 9 samples wi h he deg ee o eedom d = 2Ns−2= 16 and signi icance le el αs= 0.05. Unde hese condi ions, he p obabili y o ob aining sample da a i null hypo hesis we e ue is p= 0.92. 4 0 2 4 6 x 10−3 0 0.02 0.04 ime(sec) Ampli ude(V) 0 2 4 6 x 10−3 0 0.02 0.04 ime(sec) Ampli ude(V) 0 2 4 6 x 10−3 0 0.02 0.04 ime(sec) Ampli ude(V) 0 2 4 6 x 10−3 0 0.5 1 ime(sec) Ampli ude(a.u) (d) (b) (c) (a) Fig. 4: Expe imen al esul s. (a) OFI signal, e e ence case, a e di e en ia ion o sepa a ing he inges om he amp in he acqui ed signal; (b) OFI signal, ib a ion case, also a e di e en ia ion (c) Denoised and di e en ia ed OFI inges o e e ence (solid line) and ib a ion (b oken line) cases; (d) Shi in ime occu ence o inges in he ib a ion cases wi h espec o e e ence case ha o ms he basis o displacemen calcula ion. 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10−3 −80 −60 −40 −20 0 20 40 60 80 ime (sec) Ampli ude (nm) e e ence i OFI i e e ence measu ed OFI measu ed Fig. 5: Expe imen al esul s (Exp . 1 in Table II). P oposed OFI based a ge displacemen compa ed agains piezoelec ic s age embedded capaci i e senso . Since p > αs, we conclude ha he da a samples in OFI and Re (Table II) ha e same mean a 95% con idence le el. Fo illus a ion pu poses, he i s case, which had an e o (pp) o 1.5 nm is shown in Fig. 5. In o de o u he p o e he easibili y o he p oposed me hodology, i was applied o de ec a bi a y wa e o m shapes. Resul s o expe imen s o measu e Gaussian pulse shapes o wid h 4 ms and ampli ude 63.8 nm a e p esen ed in Fig. 6, showing an a e age e o (pp) o 4.05 nm measu ed unde compa able condi ions o ha o Fig. 5. The ela i ely high e o in Gaussian wa e o m is due o he limi ed numbe o sample poin s o ollow apid changes in he slope o he displacemen gi ing ise o apid phase changes (as compa ed o o me ). IV. CONCLUSION We ha e p oposed and expe imen ally demons a ed a me hodology o make e icien use o di ec lase injec ion cu en modula ion o induce CWFM and nonlinea dynamics e ec s in a LD subjec ed o op ical eedback o measu e nanome ic ampli ude displacemen s. The key con ibu ion o his wo k is he use o single lase diode o de ec sub TABLE II: Expe imen al esul s. Exp . Re (nm) OFI (nm) E o (nm) 1 137.5 139.0 1.5 2 155.55 155.18 0.37 3 126.08 124.58 1.5 4 132.95 132.41 0.54 5 159.1 156.3 2.8 0 0.002 0.004 0.006 0.008 0.01 0.012 −10 0 10 20 30 40 50 60 70 ime (sec) Ampli ude (nm) e e ence i OFI i e e ence measu ed OFI measu ed Fig. 6: Expe imen al esul s. Gaussian pulse de ec ed com- pa ed agains piezoelec ic s age embedded capaci i e senso . wa eleng h a ge ib a ions wi h nanome ic accu acies, using he modula ion o he injec ion cu en and expanding he bandwid h o o me DSMI senso s [7] o a leas he MHz equency ange in de ec ion. The mean peak o peak e o o he p oposed senso ou o nine di e en measu emen o a single oscilla ion is 2.4 nm; a Gaussian pulse o wid h 4 ms and ampli ude 63.8 nm is also de ec ed wi h mean e o (pp) o 4.08 nm. The maximum a ge ib a ion de ec able is limi ed o λ/2, and he bandwid h o he senso is p opo ional o he numbe o inges and equency o he lase cu en modula ion. REFERENCES [1] R. Lang and K. 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