Feedback interferometry with frequency modulation

Transcription

1 Feedbak inerferomery wih frequeny modulaion ior. obolev, Galina A. Kashheeva Insiue of Auomaion and Eleromery (IAE) iberian Branh of he Russian Aademy of ienes (B RA) 1, Kopuga prospe, Novosibirsk, 639, Russia Tel.: + 7 [383] Fax: +7 [383] sobolev@iae.nsk.su Absra Mehods of laser inerferomery based on he opial feedbak and frequeny-modulaed laser diode are desribed. An inerferene signal is analyzed for problems onerned wih range finding, displaemen esimaion, and measuremen of veloiies and vibraion parameers. Relaionships beween he obje movemen haraerisis and he inerferene signal parameers are found. Keywords: elf-mixing inerferomery, opial feedbak, frequeny modulaion 1. Inroduion The inerferomeri mehods find now he wides appliaion in sienifi experimens and indusries, however he appropriae equipmen is very ompliaed, expensive and does no always mees he users needs. This repor is devoed o some quesions of he self-mixing inerferomery wih opial feedbak and frequeny modulaion [1] and o he ways of realizaion is advanages. The disussed mehod is based on using of he semionduor laser whih is fed by ramp injeion urren as i is shown on he Fig. 1. Thus he laser frequeny varies linearly in a ime. The laser beam is direed owards he obje under invesigaion. A baksaered radiaion reeners ino he laser aviy where i is amplified and inerferes wih he primary one. A phoodiode inorporaed in he laser ase produes an inerferomeri signal. These feaures allow o reae he simple and heap devies for he ranging, mirodisplaemens, veloiy and vibraions measuremens. This mehod is of ineres beause radiional laser inerferomeers do no always mee he ondiions of presen-day sienifi experimen and measuremens in he indusry. Conrary o lassial inerferomeers ha onain a grea number of opial elemens and require horough alignmen, he design of an opial feedbak, frequeny-modulaed inerferomeer is simple. I onsiss of a semionduor laser and a ollimaing lens, and does no require adjusmen. Hene, i is muh less expensive han a sandard Mihelson inerferomeer. The high gain of he aive medium of a semionduor laser ensures obaining an inerferene signal even from disan (abou 3-5 m) and weakly saering objes. A laser beam an be foused o a spo of he order of wavelengh, herefore, besides he high sensiiviy, he inerferomeer makes i possible o invesigae wihou perurbaions he movemen parameers of very small objes. One of he ways of exending he laser inerferomery apabiliies, simplifying he design, and lowering he os of he devies is o inegrae he effes of opial feedbak and frequeny modulaion in he same devie. In his repor, we desribe universal ehnique of measuring he displaemen and disane parameers, whih is based on a frequenymodulaed laser. In his siuaion, he feedbak raio an vary over a wider range, e.g., from.1 o 4.6. Our goal is o represen resuls of deailed analysis of he measuremen mehod and find relaionships beween he inerferene signal parameers and he disane o he obje, is displaemens, veloiy, and vibraions. 1-33

2 . Feedbak inerferomery wih frequeny modulaion The desribed mehod of measuring he disane o he objes and heir displaemen parameers is based on he propery of semionduor lasers o hange he radiaion frequeny under he aion of hanging injeion urren. When he urren hanges are insignifian (o 5%), he dependene beween hem and he radiaion frequeny is linear, and if he urren is periodially modulaed, he radiaion frequeny and power hange. The modulaed ligh beam is direed o he obje as is shown in Fig. 1a. The saered radiaion omes bak o he laser aive medium, where i is amplified and inerferes wih he iniial radiaion. ine he ligh veloiy value is finie, he saered radiaion omes o he laser wih a delay (Fig. 1b), hus, he radiaion frequeny and he frequeny generaed by he laser a he momen do no mah. As a resul, an inerferomeri signal appears a he laser phoodiode oupu. The signal parameers onain useful informaion abou he obje refleane, disane, and displaemen. The posiive effe of inegraing he frequeny modulaion and opial feedbak is ha he laer realizes an inerferomeer whose properies are quie equivalen o he Mihelson inerferomeer, whereas he frequeny modulaion enables o reae a arrier frequeny for he inerferene signal, and his makes i possible o deermine he direion of obje movemen. Phoodiode Laser diode Collimaor (a) Obje (b) Laser radiaion frequeny Differene frequeny Delay τ Toal deviaion Iniial radiaion frrequeny -Т/ +Т/ Fig. 1. Feedbak inerferomeer wih frequeny modulaion. When he laser injeion urren hanges linearly, he radiaion frequeny may be desribed as T T f ( ) f in ondiion (1) where f is he iniial laser radiaion frequeny, is he slope of frequeny hanging in ime under he modulaing urren, is he urren ime, and T is he sawooh modulaion period. If he obje is spaed from he laser a he disane, he saered field arrives a he laser afer he delay, where is he ligh veloiy. The phoourren of he laser phoodiode] is defined as: 1-34

3 E E E E E E os i Z R Z R Z R () E Z is he probing field inensiy, where E R is he inensiy of he saered field amplified by he laser (wih regard o he feedbak oeffiien), E Z and E R are heir ampliude values, is he laser radiaion irular frequeny, and is he phoodeeor sensiiviy. We will onsider he inerferene signal behavior when he obje begins o move wih he veloiy, saring from a disane a =. From () i follows ha is oal phase d 4 4 d 4 d d 4 I was regarded ha d 4 d 4, (3) с, (4) where λ is he laser radiaion wavelengh. Le us onsider how he obje displaemen affes he signal frequeny defined as a ime derivaive of signal phase: 1 d( ) ( ) f ( ) d d 4 4 Analyzing (3) and (5) we see ha he phase and frequeny of he inerferene signal depend on he disane (range) o he saering obje, is displaemens, and veloiy. Thus, i beomes possible o measure hese obje haraerisis. I is also imporan o noie he presene of he onsan frequeny omponen (he seond erm (5)), whih is he arrier. 3. Range measuremens If he obje does no move, hen as i follows from (5) he inerferene signal yli frequeny f does no depend on ime and is proporional o he spaing beween he laser and he ligh-refleing or ligh-saering obje, i.e., is proporional o he range. Thus, if we esimae he signal frequeny, hen he range o he obje for he known oeffiien α an be alulaed from he expression f. (6) Unforunaely, esimaion of f is no easy beause he signal is no a oninuous proess, is phase inerrups on eah modulaion period. Le us onsider his siuaion in more deail. Aording o (3), a ( ), he signal phase looks as d (5) 1-35

4 Frequeny,kHz (7) Analyzing (6) we noie ha, firsly, he ime is no a oninuous value: sine he inerferene signal is a resul of frequeny modulaion, he value of hanges periodially from o T; seondly, he iniial signal phase is a onsan: i depends only on he spaing beween he laser and he obje, and is prinipal value may ake on any value beween and π. Hene, he obained signal is a periodi proess wih he period T and represens sinusoid piees repeaing wih he modulaion frequeny. The signal Fourier sperum is a se of frequeny harmonis. The signal has one harmoni omponen only if he modulaion period onains an ineger number of is periods. To measure he frequeny f i is neessary o form on eah period of modulaion some impulse onaining he ineger number N of he signal periods, o measure duraion of his pulse and hen o evaluae he signal frequeny, as N f, (8) T where T is duraion of he menioned impulse. A known value of f range an be deermined from (6). A plo in Fig. validaes he linear range-frequeny dependene (6). The rms deviaions of range esimaes are below.5 mm Disane, mm Fig.. Range-frequeny dependene. 4. Displaemen measuremen The variable phase par aused by displaemen, as i follows from (3), is desribed by: ( ) 4 ( ), (9) where ( ) ( ) d. The analysis of he expression (9) has shown ha he basi onribuion in he phase inremen is given wih is firs member. In view of his we shall define he displaemen () as (1) 4 Experimens on he model inerferomeer showed ha he raio of rms value of noise n o signal ampliude A is abou %, i.e., А.. n 1-36

5 5. eloiy measuremen When he obje is displaed from some iniial disane from he laser, he Doppler shif frequeny f D, aording o (5), is added or subraed o he inerferene signal arrier frequeny.. Thus, he problem of finding he direion is easily solved. The analysis of he expression (5) has shown ha if he obje moves wih speed () he final signal frequeny may be desribed as: f ( ). (11) с Thus, he frequeny aquires no only he Doppler inremen proporional o he veloiy, bu also an inremen proporional o he pah of he moving obje. Probably, his peuliariy of he inerferene signal frequeny will find some exoi appliaion. 6. Measuremen of vibraion parameers Le he diffusing obje vibrae aording o he harmoni law wih he vibraion veloiy os. (1) Then, aording o (11) he signal frequeny will look like f ( ) os The variable par of insananeous frequeny (13) is proporional o vibroveloiy, and he onsan par of (13) plays a role of arrier frequeny. If an operaion of frequeny demodulaion will be exeue hen we reeive he signal proporional o insan values of vibroveloiy on he demodulaor oupu. The funional sheme of vibromeer is presened in he Fig. 3. Phoodiode urren C laser wih phoodiode The obje of invesigaion (13) Curren soure Modulaor ibrodisplaemen signal Amplifier Demodulaor Inerferene signal Fig. 3. The funional sheme of vibromeer. 7. Conlusion The desribed analysis of self-mixing frequeny-modulaed laser inerferomery mehods has validaed is wide appliabiliy in he indusry and sienifi experimens. The obained resuls make i possible o alulae and design devies for esimaing a range, displaemens, veloiy, and vibraion parameers of diffusing surfae objes. Referenes 1. Th. Bosh, N. ervagen, and. Donai. Opial Feedbak Inerferomery for ensing Appliaion. Op. Eng. 1, 4 (1), p