Error Analysis, Design and Modeling of an Improved Heterodyne Nano-Displacement Interferometer

Transcription

1 Error Analyss, Desgn and Modelng o an mproved Heterodyne Nano-Dsplacement ntererometer S. Olyaee* and Sh. Mohammad-Nejad** Abstract: A new heterodyne nano-dsplacement wth error reducton s presented. The man errors aectng the dsplacement accuracy o the nano-dsplacement measurement system ncludng ntermodulaton dstorton error, cross-talk error, cross-polarzaton error and phase detecton error are calculated. n the desgned system, a He-Ne laser havng three-longtudnal-mode s consdered as the stablzed source. The ree spectral range o the 5cm laser cavty s about 45-MHz at 6.8-nm wavelength, whch a secondary beat requency equal to -khz s produced by combnng the reerence and measurement beams. The resoluton o the dsplacement measurement resultng rom ntermodulaton dstorton, cross-talk and cross-polarzaton errors s lmted to 8-pm. Also, the phase detecton uncertanty causes an error o only 5.9-pm n the dsplacement measurement. Furthermore, requency-path models o two- and three-longtudnal-mode laser ntererometers are modeled as the ac ntererence, ac reerence, dc ntererence and optcal power terms. A comparson study between two- and three-longtudnal-mode laser ntererometers conrms that the perormance o the desgned system s consderably mproved. Keywords: Cross-Polarzaton, Frequency-Path Model, He-Ne Laser, Heterodyne ntererometry, ntermodulaton Dstorton, Nano-Dsplacement. ntroducton Dstance or dsplacement measurement systems based on the coherent methods provde a hgh accuracy measurement n derent axes [-7]. The hgh precson dsplacement measurement s needed n many applcatons e.g. Space ntererometry Msson (SM), scheduled or lunched n 9, should be repeatedly measure the relatve angular poston o about stars to 5prad accuracy over ve years [8-9]. Ths system requres to measure dsplacement wth lnearty about pm-rms over a dstance o several meters. To reach the SM, varous errors n the dsplacement measurement system should be mnmzed. On the her hand, the abrcaton o semconductor chps necessarly mples lthographc stepper machnes n order to measure hgh accuracy dsplacement [, ]. Accuracy mprovement o the dsplacement n hgh veloctes causes the mnmum eature o the ntegrated crcuts to decrease, and thereore, the number o devces n the ranan Journal o Electrcal & Electronc Engneerng, 7. * The Author s wth the Department o Electrcal Engneerng, Shahd Rajaee Teacher Tranng Unversty, Lavzan, 6788, Tehran, ran. E-mal: s_olyaee@srttu.edu ** The Author s wth the Optoelectronc and Laser Laboratory, Department o Electrcal Engneerng, ran Unversty o Scence and Technology, Tehran, ran. E-mal: shahramm@ust.ac.r substrate wll be ncreased. The laser heterodyne ntererometer system s the best choce or these applcatons. The basc prncpals o optcal dsplacement measurement based on ntererometer have been presented by Mchelson n 88. The dynamc range and accuracy o dsplacement measurement were mproved by desgnng the heterodyne ntererometers. The bass o classcal multple-wavelength ntererometrc (MW) method was descrbed n the 97s []. Frequency and power stablzaton, nonlneartes, modelng, mplementaton and smulaton o dsplacement measurement system usng two-mode He-Ne laser were wdely presented and dscussed [- 6]. The rst nanometrc dsplacement measurement system on the bass o three-longtudnal-mode He-Ne laser was reported by Yokoyama et al. n [7] and then was mproved n 5 [8]. n ths paper, we desgn and analyze a requency-path model or two- and three-longtudnal-mode heterodyne ntererometers. Desgn and smulaton o an electronc secton o an mproved three-longtudnal-mode laser ntererometer are dscussed and the man errors ncludng ntermodulaton dstorton, cross-talk, crosspolarzaton and phase detecton errors are also calculated. A comparson study between two- and three- ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7 5

2 longtudnal-mode laser ntererometers s also presented. Prncples The source o the multple-wavelength ntererometer should produce an approprate emsson spectrum ncludng o several dscrete and stablzed wavelengths. Some eorts are stll beng deved to stablze threelongtudnal-mode He-Ne lasers [9-]. The optcal requency derences determne the range o non-ambguty o dstance and the maxmum measurable velocty. The stablty o the laser source wll lmt the absolute accuracy o the measurement. On the her hand, the maxmal absolute dstance whch can be measured by multple-wavelength s lmted by the coherence length o the source. we consder a two-wavelength ntererometry usng the optcal wavelength λ and λ, the phase sht o each wavelength wll be φ = 4 πd/λ () where d s the optcal path derence and φ s the phase sht correspondng to the wavelength λ. Thereore, the phase derence between φ and φ s gven by: ( ) φ= 4 πd /λ () And the synthetc wavelength, Λ s /λ Λ, can be expressed as: = λ = λ λ / λ λ = c/ ν ν () where ν and ν are the optcal requences correspondng to λ and λ, and c s the velocty o lght n vacuum. The He-Ne laser can be appled to MW usng derent laser lnes, e.g. at 69.4-nm and 6.8-nm n the twolongtudnal-mode ntererometer. Ths allows us to obtan a synthetc wavelength about 7-µm []. Thereore, combnng λ and λ and reachng to synthetc wavelength, Λ, the maxmal measurable absolute dstance wll be ncreased. the number o stablzed wavelengths n the gan curve s ncreased to three-longtudnal-mode, the synthetc wavelength s obtaned as: Λ = λ λ λ / λ λ λ λ λ λ = c/ (4) Thereore, the synthetc wavelength n the threelongtudnal-mode ntererometer comparng to twolongtudnal-mode system s consderably ncreased. b The ull wdth at hal maxmum (FWHM) gan prole o a He-Ne laser s about.5-ghz and the ree spectral range (FSR) or mode spacng s obtaned as: FSR = c/ L (5) where L s the cavty length. As a result, the number o standng modes n the gan prole can be adjusted by calbraton o the cavty length. Because o a small asymmetry n the gan prole, the unequal prmary (nter-mode) beat requences ( L H ) and hence the secondary beat requency, b = L H, produces. The error sources n the laser heterodyne ntererometer systems ncludes the non-lnearty n the optcal secton [, 4], laser ntensty luctuatons, laser requency nstablty, envronmental turbulences, msalgnment n the optcal nstruments, phase detecton error, crosspolarzaton and cross-talk error. Most o them are consdered as lnear errors that can be reduced or elmnated [5-]. Nano-Dsplacement Measurement Based on Twoand Three-Longtudnal-Mode Laser ntererometers The gan prole o the laser output and optcal head o the nano-dsplacement measurement system on the bass o two- and three-longtudnal-mode lasers are shown n Fg.. n the bh cases, the optcal head conssts o the base and measurement arms. Frst, the laser output s separated by a beam spltter (BS) so the base and measurement beams are produced. Then, the beam s splt nto two subsequent beams by polarzng beam spltter (PBS) and drected to each path o the ntererometers. Two relected beams are nterered to each her on the lnear polarzer. Because o orthogonally polarzed modes, the lnear polarzer should be used to nterere two beams as shown n Fg.. The stablzed multmode He-Ne lasers (havng, and requences n three-mode and havng and n two-mode) are chosen n whch the sde modes can be separated rom the center mode due to the orthogonal polarzaton o modes. But n realty, non-orthogonal and ellptcal polarzaton o beams cause each path to contan a racton o the laser beam belongng to the her path. Hence, the cross-polarzaton error s produced. n the reerence path (path.) o three-longtudnal-mode ntererometer, and are the man requences and s the leakage one, whereas n the target path (path.), s the man sgnal and the hers are as the leakages. The requency-path models o two- and threelongtudnal-mode ntererometers are shown n Fg.. n the measurement arm o three-longtudnal-mode ntererometer, there are three requency components and two paths namely the reerence and the target (the bold lnes are the man sgnal paths and the dashed lnes 54 ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7

3 are the leakage paths), whereas n two-longtudnalmode ntererometer, there are two requency components and two paths. The number o actve requency-path elements, n, s obtaned by multplyng the number o requency components by paths. Fg. 4 shows the dentcaton o the physcal orgn o each requency-path element or the measurement arms. The wave ntensty beng receved by an avalanche phodode (APD) s proportonal to the square o the tal electrcal eld and the number o dstnct ntererence terms s equal to n(n ) / =. n threelongtudnal-mode laser ntererometer, the reerence path eld s obtaned as: corner cube prsm n the reerence path (CCP.). Smlar to Eq. (6), the target path eld s descrbed by: E = cos( ωt k x ϕ ), =,, (7) where x s the mon o the corner cube prsm n the target path (CCP.). n ths system, the CCP. s xed and hence, x =. Furthermore, the wavelengths are so close that propagaton constants become almost equal to each her( k = k = k = k). By squarng the tal elds, the phocurrent o the avalanche phodode s gven as: ( ω t k x ),,, E = E cos ϕ = (6) where ϕ s the ntal phases correspondng to the electrcal eld E j, k s the propagaton constant or wave number ( π/λ ), and x s the mon o the 6 APD E = (8) CCP., b, CCP. Phodetector Phodetector (a) CCP. L H,,,,,E, E E,, CCP. Phodetector Phodetector (b) Fg. The gan prole and optcal head o the nano-dsplacement measurement system based on the two-longtudnal-mode (top) and three-longtudnal-mode (btom) He-Ne laser ntererometers. PBS: polarzng beam spltter; CCP: corner cube prsm; BS: beam spltter. ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7 55

4 The hgh requency components such as ω, ω, and ω ω j are elmnated by the avalanche phodode (, j=,,). Thereore, gnorng the hgh requences n the ully unwanted leakng ntererometers, there are dstnct ntererence terms or three-longtudnal-mode ntererometer and dstnct ntererence terms or two-mode type. The dstnct ntererence terms can be dvded nto our groups namely dc ntererence, ac ntererence, ac reerence, and optcal power []. These components n the three-longtudnal-mode ntererometer are respectvely gven by: D/K= E cos E cos A E /K= EE E cos cos E ( kx ) ( kx ) E cos( kx ) cos( ωlt kx ) (( ωh ωl) t kx ) ( ωlt kx ) cos( ωht kx ) cos( ( ωh ωl) t kx ) E cos( ω t kx ) H ( ) ( ) ( ) E E cos( ω t) (( ) ) (( ) ) /K= E E cos ω t E E cos ω t AR L H E E cos ω t L H E E cos ω ω t E E cos ω ω t L H L H OP/K= E j j= = (9) () () () small sold crcles, respectvely. On the her hand, the dameter o the crcles presents the ampltude o the sgnals. The small sold crcles are exaggerated or clarcaton. All o the dstnct ntererence terms are shown n Fg. 5 by derent lnes. E r E y θ E r snθ E t cosθ Polarzer axs Fg. Combnaton o orthogonally polarzed beams on the lnear polarzer. E t E x 4 System Desgn As shown n Fg., the ncdent optcal power s converted to phocurrent sgnal by two avalanche phododes. The schematc o the electronc secton o the desgned system s shown n Fg. 6 or threelongtudnal-mode laser ntererometer. Accordng to the gure, the phocurrent o the avalanche phododes s converted to voltage and ampled by two current to voltage converters ( VC and VC ) and low nose amplers ( A and A ), respectvely. Fgure 5 shows the combnaton o the requency-path elements n the measurement arm o two systems. The man sgnals and leakages are depcted by large and APD APD APD Fg. The requency-path model n two- and three-longtudnal-mode laser ntererometers. APD 56 ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7

5 Reerence Path Reerence Path E E E E E (a) (b) Fg. 4 dentcaton o the physcal orgn o each requency-path element n the reerence and target paths (measurement arms). (a) Two- and (b) Three-longtudnal-mode laser ntererometers. E E E E E (a) (b) Fg. 5 The combnaton graph o the requency-path elements n (a) two-longtudnal-mode and (b) three-longtudnal-mode ntererometers. K. K. LPF. Mx. APD. APD. Mx. LPF. Comp. A VC. H Hgh Voltage VC. A Comp. Base Counter. N bt Processor N bt Measurement Counter. Fg. 6 The schematc o the electronc secton o the desgned nano-dsplacement measurement system based on the threelongtudnal-mode laser ntererometer. ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7 57

6 The unwanted electrcal nducton between the base and measurement paths can be reduced by usng two solated power supples or APD basng and by separatng the grounds (analog-base, analogmeasurement and dgtal grounds). Furthermore, because o swtchng nose o dgtal secton ncludng requency and phase measurement crcuts and mcrocontroller, two hgh-speed opto-couplers solate the analog crcuts rom dgtal secton. Fg. 7 shows the desgned crcut wth cross-talk error reducton. The non-leakage base and measurement sgnals at the end o the amplers are respectvely descrbed by: v bas v = Acos mea (( ω H ω L) t) Bcos( ωht) Ccos( ω t) D = A'cos (( ωh ω) t Φ) (( ωl± ω) t Φ) C'cos( ( ω ω ) t) D' B'cos H L L () (4) where ω s the requency sht due to the Doppler ω= kx/t whch can be calculated as: eect ( ) ( ) ω= π nv/λ (5) where λ s the central wavelength, n s the reractve ndex o the medum and V s the target velocty. The represented sgnals n Eqs. (-4) wll be selmultpled by two double-balanced mxers ( Mx and Mx ). The secondary beat requency n the base and measurement arms are extracted by two low pass lters (LPF) whose cut o requency s 55kHz. The secondary beat requency s clearly much smaller than b << H L. Thereore, the output sgnals o the low pass lters or the base and measurement arms are wrtten as: the hers (, ) ( π ) Vo bas = Gcos btϕ (6) 4 n V V o mea = G cos π b± t Φ ϕ (7) λ where G and G are the tal gan o the current to voltage converter, low nose ampler (see Fg. 8a), double-balanced mxer and low pass lter or the base and measurement arms, respectvely. The above snusodal sgnals are converted to square waveorm. Then, the secondary beat requency s measured by a hgh bandwdth up/down counter. As a result, the Doppler sht requency s measured. the counted requency resultng rom the Doppler sht requency s dened as N, the dsplacement s gven by [4]: d= λ 4 Φ Φ N π (8) The ractonal term o Eq. (8) s obtaned by measurng the ntal and nal phases between the base and measurement sgnals ( Φ and Φ ) by an accurate phase detector. Fg. 9 shows the Doppler sht requency and phase sht n terms o nanometrc dsplacement. The operaton o phase detector s mportant, especally n the lower veloctes. The phase detecton method has been used n conventonal ntererometers and there are many technques or phase detecton such as averagng o the pulse wdth, trangular wave generaton, Verner scheme, etc. []. The phase measurement crcut s shown n Fg. 8(b). n ths crcut, the base and measurement sgnals are exerted to a hal exclusve-or gate and the pulse wdth s measured by a hgh speed counter. The resoluton o the phase detector s proportonal to clock pulse o the counter. Here, we suggest a unversal tme-to-dgtal converter TDC-GP as a hgh speed counter whch can be also used as a requency measurement devce. The phase derence between the base and measurement sgnals s proportonal to the output pulse wdth. For requency measurement, the sgnal must be exerted smultaneously to the stop and start nputs. The maxmum pulse wdth s / b and the phase detecton resoluton s obtaned as: δ = (9) Φ π b/ TDC where TDC = 8GHz. n summary, by measurng the shted requency ( b ± ), the velocty o the target ( CCP) wll be obtaned. Also, and the dsplacement o the target can be calculated by ntegratng the velocty or by measurng the phase Φ, ( Φ= 4 πd/λ ). 5 Error Analyss Consderng 5-cm cavty length or He-Ne laser at 6.8-nm, the ree spectral range s obtaned rom Eq. (5) as MHz. The ull wdth at hal maxmum o He-Ne laser s about.5-ghz and hence three stablzed modes can be ound n the gan curve. Assumng the prmary beat requences to be equal to 45.-MHz and 45.-MHz, the secondary beat requency becomes.-mhz [7, 8]. The maxmum measurable velocty s lmted by the secondary beat requency and rom Eq. (5) s gven as: ω= π n V /λ ω max = π b V max = b. λ / 4 n n = () 58 ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7

7 R APD. APD. R Analog GND (base) Dgtal GND Analog GND (measurement) LPF. Mx. A Base Opto. 4 Comp. R OUT VC. - Rc Hgh Voltage. Hgh Voltage. Dgtal Secton Opto. Hgh Speed Up/Down Counters Mcrocontroller H H - OUT VC. Rc Mx. A Measurement LPF. Comp. R Base CLK Accurate Phase Detector Measurement CLK Fg. 7 The schematc o the electronc secton o the desgned nano-dsplacement measurement system wth cross-talk reducton. The man sources that aect the accuracy o the threelongtudnal-mode ntererometer are as ollows: a. The cross-polarzaton error The unwanted optcal leakage o the reerence and target paths causes cross-polarzaton error. For a xed target, the ac ntererence and the ac reerence components are concdent and the mode spacng s equal to the secondary beat requency, -khz. The ac reerence terms, Eq. (), don't change wth varaton o the target velocty. The maxmum measurable velocty correspondng to Eq. () becomes mm/s. n the lower velocty, the requences o the ac and dc ntererence are slghtly shted. However, reducton o the prmary to the secondary beat requency causes the maxmum measurable velocty to be consderably reduced. b. The ntermodulaton dstorton error The amplcaton and sel-multplcaton o the sgnal produce the nonlnear components. So, n addton to the man sgnal, there wll be hgher order harmoncs. c. The phase detecton error The dsplacement measurement can be lmted by the accuracy o the phase measurement. As mentoned prevously, by usng the hgh-speed counter as shown n Fg. 8(b), the phase detecton resoluton s equal to.4 o and the dsplacement error due to the devaton o the phase detecton s lmted to 5.9-pm. d. The cross-talk error The combnng the reerence and target elds, the phase between them s shted by: Ψ ( ) = = () err θ o Ψ o arctan ol/ where θ o s the measurement phase, Ψ o s the real phase, and ol are the man sgnal and leakage ampltudes, respectvely. The ampltude o the measurement sgnal s gven by: om ( cosψ ) / = () ol ol Thereore, the phase error and hence the systematc errors can be ncreased by ncreasng the leakage ampltude as shown n Fg.. The sgnal to leakage rato s also dened by: SLR = n = ol / err () where and ol are the ampltude o the man sgnal and the th leakage, respectvely. Snce the detected phase s changed by leakage, the error n the nanometrc dsplacement measurement ( δ d) can be wrtten as: δ [ ( / )/ π] = λ/ 4 tan (4) d ol ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7 59

8 Rb L(RFC) C Vcc Doppler Sht Frequency (N) Nanometrc Dsplacement (nm) V C 4 (a) START STOP TDC-GP C CRYSTAL OSC. Vo M C R O L C D Phase Sht Derent (Deg.) Nanometrc Dsplacement (nm) Fg. 9 The Doppler sht requency and phase sht n terms o nano-dsplacement. om Φ err π Φ θ ol (b) Fg. 8 (a) The low nose ampler and (b) the phase measurement crcut. Fg. Phase sht error resultng rom leakage. Table A comparson between two- and three-longtudnal-mode He-Ne laser ntererometers. Parameter Two-Longtudnal-Mode Laser Three-Longtudnal-Mode Laser ntererometer ntererometer Unt Wavelength nm Cavty length 5 5 cm Synthetc wavelength.5 m Maxmum absolute dstance.5 5 m ntermode beat requency 6 45., 45., 87. MHz Secondary beat requency --- khz Maxmum measurable velocty.47 m/s Phase detecton accuracy (smlar crcut) pm Cross-talk and ntermodulaton dstorton error 8 pm The number o actve requency-path elements 4 6 The number o dstnct Optcal power 4 6 ntererence terms AC 6 ntererence Tal: Tal: DC ntererence AC reerence 6 6 ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7

9 Ampltude (mv) Ampltude (mv) Ampltude (mv) Ampltude (mv) 5-5 V = m/s Tme (usec) Frequency (MHz) V = m/s 4 X: Y: 9.5 V = -m/s X: 44.5 Y: 9.6 V = 5m/s 4 Frequency (MHz) (a) Ampltude (mv) Ampltude (mv) 5-5 V = -m/s Tme (usec) 5 5 V = -m/s 4 Frequency (MHz) V = -5m/s X: Y:.48 4 X: 7.58 Y: 5.4 V = m/s 4 Frequency (MHz) (b) Fg. The measured sgnals (a) beore and (b) ater the nose and cross-talk reducton (heterodyne technque). The measured sgnals beore and ater the nose and cross-talk reducton or two-longtudnal-mode ntererometer are shown n Fg.. Smlarly, the results o the smulatons ndcate that the sgnal to leakage and ntermodulaton dstorton rato beore and ater the nose and cross-talk reducton or threelongtudnal-mode ntererometer are about 7-dB and 6-dB, respectvely. As a result, accuracy o the nanometrc dsplacement due to the cross-talk error and ntermodulaton dstorton reaches about.8-nm. However, the cross-talk and ntermodulaton dstorton errors can be domnated and as a result the dsplacement resoluton can be lmted to 8-pm wthout nonlnearty consderaton. By comparng two-longtudnal-mode ntererometer wth three-longtudnal-mode ntererometer, we wll see that the dsplacement measurement resoluton s doubled. Comparson data o these systems are descrbed n Table. Although the maxmum measurable velocty n three-mode s dramatcally reduced, the dsplacement resoluton due to the phase detecton error s doubled and the beat requency s consderably decreased. Also, the maxmum absolute dstance s mproved by a actor o. 6 Concluson The requency-path models o the heterodyne nanodsplacement ntererometers based on the two- and three-longtudnal-mode He-Ne lasers have been presented. These models descrbed the ac reerence, ac ntererence, dc ntererence and optcal power components o the ntererometers. Also, the system perormance was mproved by reducton o man errors ncludng the phase detecton uncertanty, ntermodulaton dstorton, cross-polarzaton and crosstalk errors. The dsplacement resoluton lmted by ntermodulaton dstorton, cross-polarzaton and crosstalk errors was calculated as 8-pm. n addton, the phase detecton uncertanty causes 5.9-pm error n the dsplacement measurement. Fnally, a comparson between two-longtudnal-mode and three-longtudnalmode heterodyne ntererometers conrms that the dsplacement resoluton due to the phase detecton error was doubled n three-longtudnal-mode heterodyne ntererometer. Acknowledgement The authors would lke to thank ran Telecommuncaton Research Center (TRC) or nancal support. Reerences [] Chassagne L., Topcu S., Alayl Y. and Juncar P., Hghly accurate postonng control method or pezoelectrc actuators based on phase-shtng optoelectroncs, Meas. Sc. Technol., Vol. 6, pp , 6. [] Km M. and Km S., Two-way requencyconverson phase measurement or hgh-speed and hgh-resoluton heterodyne ntererometry, Meas. Sc. Technol., Vol. 5, pp. 4 48, 4. [] Yokoyama S., Ohnsh J., wasak S., Seta K., Matsumo H. and Suzuk N., Real-tme and hgh-resoluton absolute-dstance measurement usng a two-wavelength superheterodyne ntererometer, Meas. Sc. Technol., Vol., pp. 9, 999. [4] Ym N., Eom C. and Km S., Dual mode phase measurement or optcal heterodyne ntererometry, Meas. Sc. Technol., Vol., pp. 7,. ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7 6

10 [5] Ma L. S. and Hall J. L., Optcal heterodyne spectroscopy enhanced by an external optcal cavty: toward mproved workng standards, EEE J. Quantum Electron, QE-6, pp. 6-, 99. [6] Yokoyama T., Yokoyama S., Yoshmor K. and Arak T., Sub-nanometre double shearng heterodyne ntererometry or prolng large scale planar suraces, Meas. Sc. Technol., Vol. 5, pp , 4. [7] Demarest F. C., Hgh-resoluton, hgh-speed, low data age uncertanty, heterodyne dsplacement measurng ntererometer electroncs, Meas. Sc. Technol., Vol. 9, pp. 4, 998. [8] Halverson P. G. and Spero R. E., Sgnal processng and testng o dsplacement metrology gauges wth pcometre-scale cyclc nonlnearty, J. Opt. A: Pure Appl. Opt., Vol. 4, pp. S4 S,. [9] accessed January 8. [] Glsnn J., Zhou H., Damazo B., Fu J. and Slver R., Nano-lthography n ultra-hgh vacuum (UHV) or real world applcatons, Proc. Nanech 4, Boston, 4. [] Brnk M., Jasper H., Slonaker S., Wjnhoven P. and Klaassen F., Step-and-scan and step-andrepeat; a technology comparson, Proc. SPE, 76 pp. 74 5, 996. [] Eom T. B., Cho H. S. and Lee S. K., Frequency stablzaton o an nternal mrror He Ne laser by dgtal control, Rev. Sc. nstrum., Vol. 7, pp. -4,. [] Huang T. L., Chen Y. S., Shy J. T. and Lu H. P., Two-mode requency stablzaton o an nternal-mrror 6 nm He-Ne laser, Proc. Natl. Sc. Counc. ROC(A), Vol. 4, No. 4, pp ,. [4] Wu C., Perodc nonlnearty resultng rom ghost relectons n heterodyne ntererometry, Optcs Communcatons, Vol. 5, pp. 7,. [5] Olyaee S. and Mohammad Nejad S., Nonlnearty and requency-path modellng o three-longtudnal-mode nanometrc dsplacement measurement system, ET Optoelectroncs, Vol., No. 5, pp. -, 7. [6] Cosjns S. J., Hatjema H. and Schellekens P. H., Modelng and veryng non-lneartes n heterodyne dsplacement ntererometry, Precson Eng., Vol. 6, pp ,. [7] Olyaee S. and Mohammad Nejad S., Desgn and smulaton o velocty and dsplacement measurement system wth subnanometer uncertanty based on a new stablzed laser Doppler-ntererometer, The Araban Journal or Scence and Engneerng, Vol., No. C, pp , 7. [8] Yokoyama S., Yokoyama T. and Arak T., Hghspeed subnanometre ntererometry usng an mproved three-mode heterodyne ntererometer, Meas. Sc. Technol., Vol. 6, pp , 5. [9] Suh H. S., Yoon T. H., Chung M. S. and Cho O. S., Frequency and power stablzaton o a three longtudnal mode He-Ne laser usng secondary beat requency, Appl. Phys. Lett. Vol. 6, pp. 7-9, 99. [] Yeom J. Y. and Yoon T. H., Three-longtudnalmode He Ne laser requency stablzed at 6 nm by thermal phase lockng o the secondary beat requency, Appl. Opt., Vol. 44, No. -, 5. [] Olyaee S. and Mohammad Nejad S., Stablzaton o laser requency based on the combnaton o requency lockng and power balance methods, J. Appl. Sc., Vol. 7, No. 4, pp , 7. [] Dandlker R., Hug K., Poltch J. and Zmmermann E., Hgh-accuracy dstance measurements by multple-wavelength ntererometry, Opt. Eng. Vol. 4 (8), pp. 47-4, 995. [] Eom T. B., Km J. Y. and Jeong K., The dynamc compensaton o nonlnearty n a homodyne laser ntererometer, Meas. Sc. Technol. Vol., pp ,. [4] Olyaee S. and Mohammad Nejad S., Characterzaton o ellptcally polarzed lght and raton angle o PBS n the threelongtudnal-mode laser ntererometer usng the Jones matrces, J. Appl. Sc., Vol. 7, No. 9, pp. 86-8, 7. [5] Robertson D., Kllow C., Ward H., Hough J., Henzel G., Garca A., Wand V., Johann U. and Braxmaer C., LTP ntererometer-nose sources and perormance, Class. Quantum Grav., Vol., pp. 55 6, 5. [6] Wand V., Bogenstah J., Braxmaer C., Danzmann K., Garcıa A., Guzman F., Henzel G., Hough J., Jennrch O., Kllow C., Robertson D., Sodnk Z., Steer F. and Ward H., Nose sources n the LTP heterodyne ntererometer, Class. Quantum Grav. Vol., pp , 6. 6 ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7

11 [7] Chang H. F., Chou C., Teng H. K., Wu H. and Yau H., The use o polarzaton modulaton and ampltude-senstve optcal heterodyne ntererometry or lnear brerngence parameters measurement, Optcs Communcatons, Vol. 6, pp. 4 46, 6. [8] Meyers J. F., Lee J. W. and Schwartz R. J., Characterzaton o measurement error sources n Doppler global velocmetry, Meas. Sc. Technol., Vol., pp ,. [9] Sutton C. M., Nonlnearty n the length measurement usng heterodyne laser Mchelson ntererometry, J. Phys. E: Sc. nstrum., Vol., pp. 9, 987. [] Hou W. and Wlkenng G., nvestgaton and compensaton o the nonlnearty o heterodyne ntererometers, Precson Eng., Vol. 4, pp. 9 8, 99. [] Schmtz T. and Beckwth J. F., An nvestgaton o two unexplored perodc error sources n derental-path ntererometry, Precson Eng., Vol. 7, pp.,. [] Lo H. P. and Young M. S., New dgtal phase meter concept and ts applcaton, Rev. Sc. nstrum., Vol. 68, pp.894 9, 997. Shahram Mohammad-Nejad receved hs B.Sc. n Electrcal Engneerng rom Unversty o Houston, Houston, USA, n 98 and M.Sc. and Ph.D. degrees n Semconductor Materal Growth and Lasers rom Shzuoka Unversty, Shzuoka, Japan, n 99 and 99, respectvely. Proessor Mohammad-Nejad nvented the PdSrS laser or the rst tme n 99. He has publshed over 8 scentc papers and books. Currently, he s the Head o Electrcal Engneerng Department, ran Unversty o Scence and Technology, Tehran, ran. Also, he s a scentc commttee member o ranan Conerence o Electrcal Engneerng (CEE), member o nsttute o Engneerng and Technology (ET) and an ET-CEng. Hs research nterests nclude semconductor materal growth, quantum electroncs, semconductor devces, optoelectroncs, electronc devces and lasers. Saeed Olyaee was born n Mashhad, ran, n 975. He receved the B.Sc. degree n Electrcal Engneerng rom Unversty o Mazandaran, Babol, ran, n 997 and the M.Sc. and the Ph.D. degrees n Electrcal Engneerng specalzng n Optoelectroncs rom ran Unversty o Scence and Technology, Tehran, ran, n 999 and 7, respectvely. Hs doctoral dssertaton concerned nanometrc dsplacement measurement based on three-longtudnal-mode laser. Currently, he s an Assstant Proessor n Department o Electrcal Engneerng, Shahd Rajaee Unversty, Tehran, ran. Dr. Olyaee's man research nterests nclude nanodsplacement measurement, optcal nstrumentaton and optoelectronc crcuts. ranan Journal o Electrcal & Electronc Engneerng, Vol., Nos. & 4, July 7 6