Self-mixing interferometry and its applications

Transcription

1 Univeity of Wollongong Reeach Online Faculty of Engineeing and Infomation Science - Pape: Pat A Faculty of Engineeing and Infomation Science 16 Self-mixing intefeomety and it application Yanguang Yu Univeity of Wollongong, yanguang@uow.edu.au Yuanlong Fan Univeity of Wollongong, yf555@uowmail.edu.au Bin Liu Univeity of Wollongong, bl987@uowmail.edu.au Publication Detail Y. Yu, Y. Fan & B. Liu, "Self-mixing intefeomety and it application," in Optical Deign and Teting VII, 16, pp. 11U-1-11U-13. Reeach Online i the open acce intitutional epoitoy fo the Univeity of Wollongong. Fo futhe infomation contact the UOW Libay: eeach-pub@uow.edu.au

2 Self-mixing intefeomety and it application Abtact Thi pape eview the elf-mixing intefeence (SMI) in tem of it opeation pinciple, the featue of SMI ignal and it configuation. SMI efe to a phenomenon that occu when a mall faction of the light emitted by a lae i backcatteed o eflected by an extenal taget and e-ente the lae active cavity, thu leading to the modulation of the lae output powe. Thi i a emakably univeal phenomenon, occuing in lae egadle of type. A few application example ae peented baed on the eeach wok done in ou goup, including SMI ening fo diplacement meauement, mateial paamete and lae paamete. An SMI with the lae opeating at the elaxation ocillation i intoduced which ha potential fo achieving moe enitive ening. Dicipline Engineeing Science and Technology Studie Publication Detail Y. Yu, Y. Fan & B. Liu, "Self-mixing intefeomety and it application," in Optical Deign and Teting VII, 16, pp. 11U-1-11U-13. Thi confeence pape i available at Reeach Online:

3 Self-mixing intefeomety and it application Yanguang Yu* a, Yuanlong Fan b and Bin Liu a a School of Electical, Compute and Telecommunication Engineeing, Univeity of Wollongong, Nothfield Ave, Wollongong, NSW, 5, Autalia b School of Phyic, Univeity of Wollongong, Nothfield Ave, Wollongong, NSW, 5, Autalia ABSTRACT Thi pape eview the elf-mixing intefeence (SMI) in tem of it opeation pinciple, the featue of SMI ignal and it configuation. SMI efe to a phenomenon that occu when a mall faction of the light emitted by a lae i backcatteed o eflected by an extenal taget and e-ente the lae active cavity, thu leading to the modulation of the lae output powe. Thi i a emakably univeal phenomenon, occuing in lae egadle of type. A few application example ae peented baed on the eeach wok done in ou goup, including SMI ening fo diplacement meauement, mateial paamete and lae paamete. An SMI with the lae opeating at the elaxation ocillation i intoduced which ha potential fo achieving moe enitive ening. Keywod: elf-mixing intefeomety, optical feedback, lae diode 1. INTRODUCTION In ecent yea, a an emeging and pomiing non-contact ening technique, Self Mixing Intefeomety (SMI) ha attacted much attention of eeache 1,. The SMI i baed on the elf-mixing ect that occu when a mall faction of lae light emitted by the emiconducto lae (SL) i eflected by an extenal taget and e-ente the SL intenal cavity. Compaed to othe taditional intefeometic cheme, e.g., Michelon o Mach-Zende, SMI ha the following advantage 1 : No optical intefeomete extenal to the ouce i equied. Thi lead to a imple and compact et-up. No alignment i needed becaue the patial mode that inteact with the cavity mode i filteed out patially by the lae itelf. Thi mean that detection of the diffuive taget movement become poible. Senitivity of the cheme i vey high (ub-nm enitivity). Due to the above advantage, vaiou application of the SMI ha been developed, uch a meauement of metology, lae paamete, and phyical quantitie 1. Moe pecifically, thee meauement include diplacement, vibation, velocity, ditance, linewidth, alpha facto, thickne, efaction index, mechanical eonance, and te/tain hyteei, etc. Geneally, it i deied that an SMI opeate in a table mode, in which cae the hape of SMI ignal i chaacteized by a paamete called Optical Feedback Facto (OFF) 3 C which i aociated with extenal cavity length, linewidth enhancement facto (LEF) of the lae and the amount of light eflected by the taget. At weak feedback egime with <C<1, an SMI ignal contain inuoidal-like finge tuctue. At modeate feedback egime, which i fo C>1, an SMI ignal how aymmetic hyteei and poduce awtooth-like finge tuctue. Thee table SMI ignal can be ued fo vaiou application of ening and intumentation a mentioned above 1,. Howeve, when the SMI opeate in an untable mode, the SL output powe exhibit a high fequency ocillation (cloe to the elaxation-ocillation (RO) fequency of the lae 5, 6, uually -4GHz) with it amplitude modulated by a low time vaying envelop elated to the movement of the extenal taget. In thi ituation, the lae output i efeed to a a elaxation-ocillation elf-mixing intefeence (RO-SMI) ignal. In thi pape, we aim to povide a compehenive eview of the table SMI fom the view point of it exiting model and application. In thi eview, it fitly how that the theoetical model of the SMI can be deived fom eithe the thee mio model 7 o the Lang and Kobayahi (LK) 8 model, baed on which the chaacteitic of the SMI ignal a well a it mechanim ae decibed. Then, the exiting SMI baed application on meauing diplacement, Young modulu and LEF ae eviewed. Finally, a highly enitive SMI eno with the lae opeating at the RO i peented. * yanguang@uow.edu.au; phone

4 .1 Opeating pinciple. SMI OPERATING PRINCIPLE AND ITS FEATURES Figue 1 peent a chematic diagam of an SMI, which how that the coe component of an SMI only conit of an SL, a photodiode (PD) and a taget. The taget fom an extenal cavity fo the lae, and when it move along the light beam, a modulation in the emitted lae powe can be detected by the PD packaged at the ea of the SL and then obeved. Thi modulated powe i efeed to a an SMI ignal which caie the infomation of the motion of the taget a well a the SL itelf. Uually, the SMI model can be deived fom the claical thee mio model coniting of a Faby-Peot (FP) type lae 7 with facet eflection coicient 1 and, and the taget eflection coicient of 3, a hown in Fig. 1. Figue 1. A chematic diagam of an SMI. Auming 3 = 1, i.e., thee i only one eflection within the extenal cavity, the ective eflection coicient at the lae font facet can be expeed a 7, 9 : whee cavity, and ( ) = + e ω (1) 1 j 3 ω and ae epectively the petubed lae angula fequency and one oundtip time of the light in the extenal = L c, whee L i the extenal cavity length and c i the peed of light. On the othe hand, can alo be epeented in anothe fom with epect to it amplitude whee A and phae φ a 7, 1 : j = A e φ () [ ] A 1 co( ) = + κ ω = κ in( ω ) (3) whee κ i the feedback tength and κ = (1 ). A the oundtip phae in the intenal cavity mut be equal to a 3 multiple of π, the phae condition of compound cavity of the thee mio model can be decibed uing the following equation 7 : φ = α ( g g ) d + ( ω ω ) + φ (4) L c th in whee coepond to a change in the ound tip phae compaed to π q, whee q i an intege. In Eq. (4), α i φl called the linewidth enhancement facto (LEF) 4, 11, 1 which i an impotant fundamental deciptive paamete of the SL becaue it detemine the chaacteitic of SL, uch a the pectal ect, the modulation epone, the injection locking and the epone to the extenal optical feedback 13, 14. in i the light oundtip time in the intenal cavity and ω i the angula fequency of the olitay lae. g c and g th ae epectively the thehold gain with and without extenal cavity. g = a + d In ( ) (5) th whee a epeent the lo account fo any optical lo in the intenal cavity. Note that g c alo mut atify the amplitude condition of the compound cavity, that i 7, 9, 1 : A e [( g a ) d ] c 1 1 Subtituting Eq. (3) and (5) into Eq. (6), we can obtain the thehold gain diffeence: = (6)

5 κ gc gth = co( ω ) (7) d Then ubtituting Eq. () and (7) into Eq. (4) and conideing =, equation (4) change to 7, 9, 1 : φ L κ ω = ω + α ω + α (8) [ ] 1 in actan( ) in If we denote φ = ω, φ = ω and C κ = 1+, the coe pat of the SMI model can be finally deived a: in α [ ] φ = φ C in φ + actan( α ) (9) whee φ ae epectively the light phae coeponding to the petubed and unpetubed lae angula fequency. Note that although the thee mio model explain ome inteeting eult, it lack ome detail of the phyical etting of the phenomenon, e.g., the mateial and aociated ect fo an SL 1. Inteetingly, equation (9) can alo be deived fom the well known Lang and Kobayahi (LK) equation which ae baed on Lamb equation and modified with the additional equation fo the tate concentation 1. Compaed to the thee mio model, the LK equation decibe the active mateial and cay a deciption of lae ocillato equation which yield a much moe complete deciption of the dynamic behaviou of a ingle mode SL with extenal optical feedback (EOF). The well known LK equation 8 wee fit popoed in 198 and the model conit of thee imultaneou Delay Diffeential Equation (DDE) which ae hown a below: de( t) 1 1 κ = G [ N( t), E( t) ] E( t) + E( t ) co [ ω + φ( t) φ( t )] (1) dt p in dφ( t) 1 1 κ E( t ) = α G N t E t ω + φ t φ t dt p in E( t) [ ( ), ( )] in [ ( ) ( )] (11) dn( t) J N( t) = G[ N( t), E( t) ] E ( t) (1) dt ev whee G[ N ( t), E( t )] i the modal gain pe unit time and i expeed a 15 : [ ] = [ ] εγ G N( t), E( t) GN N( t) N 1 E ( t) (13) Equation (1)-(13) decibe the dynamic behavio of the thee vaiable, namely the electic field amplitude E( t ), the electic field phae φ ( t) and the caie denity N( t ), whee t i the time index. ( t) φ i given by φ( ) [ ω( ) ω ] t = t t, and whee ω ( t) i the intantaneou optical angula fequency fo an SL with EOF. The dynamic of the SL with an EOF ytem ae govened by the injection cuent ( J ) to the SL and the paamete aociated with the extenal cavity including κ and. The othe paamete in Eq. (1)-(13) ae elated to the olitay SL itelf, and ae teated a contant fo a cetain SL. Thee paamete ae defined in Table 1 6, 16. Note that the value of the paamete povided in Table 1 ae adopted fom 16. The coe pat of the SMI model i deived fom the tationay olution of the above LK equation. Let E, N and ω epeent the tationay olution of LK equation fo electic field amplitude, caie denity and angula fequency epectively. When the ytem decibed by Eq. (1)-(13) ente into a tationay tate, we have de( t) dt =, dφ( t) dt = ω ω and dn( t) dt =. Subtituting E( t) = E( t ) = E, N( t) N φ( t) = ω ω t into Eq. (1)-(13) and ignoing the nonlinea gain, the well known tationay olution can be obtained a below 8 : κ ω = ω 1+ α in( ω + actan α) (14) in p N in N =, and ( ) 1 co( ) N = N + κ ω G G (15)

6 E J ( ev ) N = G ( N N ) N (16) Table 1: Phyical meaning fo the intenal cavity paamete in LK equation Symbol Phyical Meaning Value G modal gain coicient N m N caie denity at tanpaency m ε nonlinea gain compeion coicient m Γ confinement facto.3 photon life time 1 p in. 1 1 intenal cavity ound-tip time α line-width enhancement facto e elementay chage C 16 3 V volume of the active egion 1. 1 m 9 caie life time. 1 Fom Eq. (14)-(16) and by conideing a moving taget, the exiting SMI model can be obtained a below by intoducing: κ φ = ω, φ = ω and C = 1+ α (17) Then Eq. (14) become the ame equation a deived by the thee mio model (ee Eq. (9)), whee φ i aociated with the extenal cavity length L, i.e., φ = 4π L λ, whee λ i the unpetubed lae wavelength. By ubtituting Eq. (15) into Eq. (16), the nomalized vaiation of the SL output powe (that i the o called SMI ignal g ) can be obtained and decibed a 17 : g = co( φ ) (18) Equation (9) and (18) contitute the exiting SMI model which ha been widely accepted to decibe the wavefom of SMI ignal 17. A een in Eq. (9) and (18), thee i a taight fowad pocedue to contitute g, i.e., φ φ g, and alo a taight backwad pocedue, i.e., g φ φ, to obtain φ, thu etieving the extenal cavity infomation. Appaently, the knowledge of the theoy of geneating an SMI ignal a well a it wavefom i eential to achieve good pefomance of the SMI.. Featue of SMI ignal SMI ignal can be chaacteized by the value of facto C 3, 18- which i an impotant paamete fo SMI ytem. The following SMI egime have been well ecognized in liteatue: Regime 1 (weak feedback): whee C<1. Equation (9) peent a unique mapping fom φ which i hown in Figue (a). In thi ituation, the movement of the extenal taget will eult in an SMI ignal wavefom with a finge tuctue imila to a taditional intefeence finge, and each finge peiod coepond to a phae hift that i equivalent to a diplacement of half a wavelength of the extenal taget 17. Figue how the elationhip between φ (Fig. (a)) a well a g (Fig. (b)) when C =.7 and α = 6 accoding to Eq. (9) and (18). Suppoing that the extenal taget move accoding to a inuoidal law with L( t) = L + L in( π ft), whee L, L and f ae the initial extenal cavity length the vibation amplitude and fequency epectively which ae choen a L =.35m, L = 1.5λ and f = 75Hz, thu leading to a time vaying φ, i.e., φ ( t) = 4 π L( t) λ = 6π in( π ft) ( ad) which i hown in Fig. (c), and Fig. (d) how the coeponding SMI. Fom Fig. (d), it can be een that, in the weak feedback egime, the SMI ignal how ymmetical inuoidal finge hape which i imila to the tadition intefeence finge. in

7 Figue : (a) Relationhip between φ, (b) elationhip between g, (c) a time vaying φ, i.e., φ( t), caued by the taget movement, (d) coeponding SMI ignal when C =.7 and α = 6. Regime (modeate feedback): whee 1<C<4.6. In thi ituation, equation (9) yield thee poible φ. To illutate the cenaio behind the wavefom of SMI ignal when 1 < C < 4.6, figue 3 how the elationhip between φ (Fig. 3(a)) a well g (Fig. 3(b)) when C = 3 and α = 6. The actual behaviou i decibed in 1, indicating that φ and g will vay along the oute A1 B B1 when φ inceae, and it will howeve tack the oute of B1 A A1 when φ deceae. Note that, the tationay olution fo φ within the ange of φ, A, φ, B ae alway untable accoding to 1, and thu will neve be an ocillating mode of the SMI ignal. Simila to the weak feedback egime cae dicued above, we peent φ ( t), which i ame with Fig. (c), a well a the coeponding SMI ignal g (Fig.3(d)) in the following figue 3. Fom Fig. 3(d), it can be een that the SMI ignal how aymmetic hyteei and poduce awtooth-like finge. Futhemoe, by looking at both Fig. 3 (c) and (d), it can be een that the lope of each finge coepond to the moving diection of the extenal taget, that i a negative lope if the taget move towad the SL, and a poitive lope if the taget move away fom the SL. Figue 3: (a) Relationhip between φ, (b) elationhip between g, (c) a time vaying φ, i.e., φ( t), caued by the taget movement, (d) coeponding SMI ignal when C = 3 and α = 6. Regime 3 (tong feedback): whee C>=4.6. Equation (9) yield even poible φ when C=9. Simila to Regime 1 and, the cenaio behind the wavefom of SMI ignal when C > 4.6 i illutated in Fig. 4. Figue 4(a) and (b) epectively how the elationhip between φ, g when C = 9 and α = 6. Thi time, φ and g will vay along the oute B B1 when φ inceae, and it will howeve tack the oute of A A1 when φ deceae. Note that, the tationay olution fo φ wa elected baed on the ule of chooing the one with leat minimal linewidth 1. Fom

8 Fig. 4(d), it can be een that the SMI ignal till how aymmetic hyteei and poduce awtooth-like finge which i imila to Fig. 3(d). Howeve, it i inteeting to notice that ome finge ae lot which i due to tong hyteei occued when feedback i tong 1. Figue 4: (a) Relationhip between φ, (b) elationhip between g, (c) a time vaying φ, i.e., φ( t), caued by the taget movement, (d) coeponding SMI ignal when C = 9 and α = 6. Moe infomation about the mechanim of geneating an SMI ignal a well a pedicting it behaviou with epect to C can be found in 17, 1,. 3.1 Diplacement meauement 3. SMI BASED APPLICATIONS Diplacement meauement i one of impotant application of an SMI-baed ening. Uing an SMI to meaue diplacement can tace it hitoy back to 1995, a finge counting method 17 i developed baed on the fact that each finge on an SMI ignal coepond to a half-wavelength hift of a moving taget. Although the eolution i low (jut half-wavelength), thi method i imple and the poblem of ambiguity in taditional intefeomety ha been eliminated with the aid of the SMI. In the following decade, vaiou method, e.g., phae hifting method, phae modulating method, phae unwapping method etc., ae developed with the aim to impove the meauement accuacy. Table ummaize thee method and thei coeponding accuacie and etiction in tem of C. Table : Summay of exiting method fo meauing diplacement uing the SMI. Method Feedback level Accuacy (eolution) AM and FM Demodulation Method 3 Modeate N/A Fequency Tuning Method 4 Weak 1-6 nm Finge Counting Technique 17 Modeate 47 nm ( λ / ) Linea Intepolation Method 5 Weak o Modeate 65 nm ( λ /1 ) Phae Shifting Method 6 Weak 65 nm ( λ /1 ) Speckle Tacking Technique 7 Modeate 6 Few pat in 1 (faction-ofwavelength) Fouie Tanfom Method 8 Weak 8 nm ( λ / 5 ) Sinuoidal Phae Modulating Method 9 Weak Le than 1 nm ( λ / 8 ) Phae Unwapping Method 14, 3, 31 Weak o Modeate Ten of nanomete Among thee method, the phae unwapping uing advanced ignal poceing i a vey pomiing technique fo SMI baed diplacement ening. It i baed on the fact that a π phae gap coepond to a finge on an SMI ignal. Thi technique i moe appealing becaue it doe not equie any exta optical o electical element to be added to the baic SMI ening tuctue. Futhemoe, thi method can achieve highe eolution a it tie to etablih a unique mapping fom an SMI ignal and the lae phae, and the late will give an accuate diplacement. The pinciple of phae unwapping baed diplacement meauement can be ummaized a following tep:

9 Step1 Obtain φ accoding to Eq. (18) by applying invee coine function to g, that i, φ = acco( g) which i wapped within the ange of [,π ]. Step Phae unwap φ in ode to obtain the tue phae φ accoding to: a. When φ eache π inceaingly, a tep of π i added to the eult of unwapping opeation. b. When φ eache deceaingly, a tep of π hould be ubtacted to the eult of unwapping opeation. Step3 Calculate φ uing Eq. (9), and then obtain diplacement L uing L = φλ (4 π ). Obviouly, detection of citical point of an SMI ignal i vey impotant a it ignificantly impact on the accuacy fo etimating φ and hence the diplacement a well. The phae unwapping method wa fit mentioned in The econtuction accuacy of thi wok i in the ode of ten of nanomete. Baed on 3, Be et al. 31 developed a ytematic way to econtuct diplacement unde modeate feedback level by intoducing an optimization algoithm which jointly etimate C, LEF and the diplacement. Howeve, thi optimization appoach i enitive to noie contained in an SMI ignal becaue the citeion of optimization depend on the intantaneou powe of the econtucted ignal dicontinuitie 31. Late, in 9, baed on the wok in 31, an impoved algoithm i popoed by Zabit et al. 3 by intoducing the adaptive tanition detection, which can automatically conveging to the optimal thehold fo detecting finge. In 11, Fan, et al. 14 found that none of the exiting phae unwapping method i able to pefectly econtuct diplacement fom an SMI ignal due to an inheent meauement eo in thee method. By invetigating the featue of the finge hape of an SMI ignal in diffeent cenaio, 14 noticed that the eo i actually induced due to inaccuate egmentation of each finge of an SMI ignal duing the phae unwapping. Fo an SMI ignal in modeate feedback, it featue can be chaacteized by fou point (maked by R, P, V, and J in Fig. 5) and egment of P to J ae ignoed by peviou method, thu intoducing the eo. Figue 5: Accuate egmentation of an SMI ignal fo phae unwapping 14. R i called evee point, a it i the moment when the taget change it movement diection. P and V ae called the peak point and valley point, epectively, denoting the maximum and minimum point on each finge. J i efeed to a the jumping point, on which an SMI ignal exhibit a udden change. Baed thee chaacteitic point, an accuate algoithm i developed baed on the following equation: φ ( ) ( 1), ( ) ( ) n + π k when v k n < p k φ ( n) = φ ( n) + π ( k fo ight-inclined pat (19) 1), when p( k) n < j( k) φ ( ) ( 1), ( ) ( ) 1 n π K k + when v k n < p k + φ ( n) = φ ( n) π ( K k + 1), when p( k) + 1 n < j( k) fo left-inclined pat () whee k i the finge numbe. k = 1,,..., K whee K i the total finge numbe in each half egment of an SMI ignal. The popoed algoithm in 14 i veified by both imulation and expeiment and the eult how that the accuacy i impoved in contat to exiting technique. 3. Mateial paamete--young modulu Young modulu i one of fundamental mechanical paamete fo decibing vaiou kind of mateial behaviou, uch a pooity, textue and integanula phae 33. Theefoe, it i of ignificant inteet to know the value of thi

10 paamete ince knowledge of Young modulu i equied both fo analyi and deign. Significant amount of method have been devoted to detemining Young modulu expeimentally Howeve, thee method uually need a dedicated tet etup and might not be feaible to cay out in a time and cot ective way. Becaue of the eae of pecimen pepaation and a vaiety of tet pecimen hape, SMI baed impule excitation appoach 38 i a contactle optical technique, and i expeimentally appoved with high accuacy and epeatability than taditional method. A Young modulu (denoted by Y) affect the vibation behaviou of mateial tuctue, by applying the evee of thi idea, the vibation behaviou of a pecific pecimen can povide the mateial Young modulu infomation. Impule excitation method i one kind of thi technique baed on eonant fequency (denoted by f RO ) meauement in tem of longitudinal o flexual vibation of the tet pecimen with imple geomety 39. Fo a ectangula pecimen (L: length, b: width, h: thickne, m: ma), accoding to the tandad eleaed by ASTM E18761, the calculation fomula of Y i expeed a below while L / h : 3 mf ROL Y = ( h L) 3 bh (1) Figue 6 how the fibe-coupled SMI ytem fo captuing the vibation ignal y( t ) fom the tet pecimen and obtaining f 38 RO. The ytem mainly conit of a lae diode (LD), coupling fibe and the teted pecimen. The LD i at DC biaed with the LD contolle. The tempeatue contolle i ued to tabilize the tempeatue of the LD. The emitting lae fom the LD i focued onto the left end of pecimen. A mall potion of the light will be back-catteed o eflected by the pecimen and e-ente the LD intenal cavity. Both the amplitude and fequency of the LD powe ae modulated by the movement of the pecimen. Thi modulated LD powe (denoted by g ) i efeed to a an SMI ignal which i detected by the PD packaged in the ea of the LD and amplified by a tan-impedance amplifie, then ecoded by an ocillocope o collected by peonal compute via analog digital data acquiition (DAQ) cad. Figue 7(a) how an SMI ignal obtained via expeiment when the pecimen i an aluminum alloy 661 with L=13.43mm, b= 1.4mm, h=.mm and m= 8.7g. The eonant fequency of the pecimen i hown in Fig. 7(b) which i 599Hz, thu giving Young modulu of Y=7.GPa accoding to Eq. (1). Figue 6: Schematic diagam of fibe-coupled SMI fo meauing Young modulu 38. (a) Expeiment SMI SMS ignal of of aluminum alloy 1.4 (b) FFT fo expeiment SMS ignal of aluminum P(t) SMI (V) ignal (V) Spectum of FFT X: 599 Y: t () Fequency (Hz) Figue 7: Expeimental eult fo meauing Young modulu when the pecimen i an aluminum alloy 38.

11 3.3 alpha meauement The linewidth enhancement facto (LEF) (alo efeed to a the alpha facto, the α paamete, Heny facto, the chip facto, o the phae-amplitude coupling facto) i one of fundamental paamete fo emiconducto lae (SL). The paamete wa intoduced by Heny in It value i vey impotant fo decibing many apect of lae behaviou, uch a pectal ect, modulation epone, injection locking and the epone to extenal optical feedback 1. Theefoe, it i of ignificant inteet to know the value of thi paamete ince knowledge of alpha i equied both fo analyi and deign. Vaiou method have been developed fo expeimental detemination of the alpha paamete, which can be categoized a linewidth meauement method, cuent modulation method, optical injection and optical feedback 1, 4. Among thee method, the optical feedback method, i.e., the SMI, i an emeging and pomiing technique which doe not equie high adio fequency o optical pectum meauement, thu poviding eae of implementation and implicity in the ytem tuctue 1, 13. In 4, Yu et al. 13 fitly popoe the LEF can be obtained uing SMI by geometically meauing the SMI ignal wavefom, and thi appoach equie the chaacteitic point of the SMI ignal, i.e., zeo coing point, to pefom the meauement, which mean C fall within a mall ange, i.e., 1 < C < 3.5. Figue 8 how a typical expeimental SMI ignal, whee A and B ae zeo coing point. A B Figue 8: An expeimental SMI ignal obtained in 13 (ee Fig. in 13 ) unde modeate feedback. Time inteval t 13 and t 4 ae elevant fo the detemination of α. The time inteval t 13 and t 4 hown in Fig. 7 allow u to detemine the phae diffeence, and thu obtain the value of α baed on the following two equation: φ C 1 C 13 = C acco( ) actan( ) + φ 1+ α C 1 C 4 = C 1 + acco( ) + actan( ) 1+ α α α π π () (3) 13 whee 13 t 4 φ = π 4 = π t. T 1 T A the above method i limited to the modeate feedback only, in fact, it can jut wok when 1<C<3, a method in 41 wa peented to pefom Fouie Tanfom on an SMI ignal to calculate α. The appoach in 41 eliminate the feedback level etiction ued in the peviou method and povide a fat and eay meauement technique fo the LEF. 4. SMI WITH THE LASER OPERATING AT THE RELAXATION OSCILLATION It i hown in Section.1 that the exiting mathematical model of the SMI can be deived fom teady-tate olution of the LK equation 8. Theefoe, the model i valid unde the aumption that both electic field E( t ) and caie denity N( t ) in an SL with a tationay extenal cavity can each a contant tate afte a tanient peiod. Howeve, the wok in 5, 16, 4 how that the elaxation ocillation (RO) o chao may occu fo an SL unde cetain opeation condition. In thi

12 cae, E( t ) and N( t ) exhibit complicated ocillation athe than a contant level. Depending on the value of C, a ecent wok in 43 how that thee opeational egion (efeed to a table, emi-table and untable which ae hown in Fig. 9(a)) can be ued to decibe the behaviou of an SMI. The table egion i whee the exiting SMI opeate. The untable egion i whee lae exhibit untable output and hence not uitable fo SMI ening. It i inteeting to note that when an SL opeate in the emi-table egion, the wavefom of the lae intenity E ( t ) exhibit a table ocillation and thu i capable of ening the movement of the extenal taget. By olving the L-K equation, hown in Eq. (1)- (1), we get the lae intenity E ( t ) (thi i SMI ignal) and the eult ae hown in Fig. 9(c)-(g). The paamete value ued fo imulation ae hown in Table 1. Figue 9: (a) Thee opeational egion decibing the behaviou of an SMI. (b) the taget diplacement, (c)-(g) the SMI ignal coeponding to diffeent C value hown in (a). Thee opeational egion can be ecognized fom the imulation eult in Fig. 9, and the featue of SMI ignal at thee egion can be ummaized a following: 1. The ignal exhibit the fom of high fequency ocillation with it amplitude modulated by a low-vaying ignal. Inteetingly, the low-vaying envelope ae imila to the SMI ignal chaacteized by the ame finge tuctue.. The Peak-Peak (P-P) value of an SMI ignal (in Fig. 9(g)) located in the table egion i aound.1 while the P-P value of the RO-SMI ignal in emi-table egion ae about 6.64, 5.4 and 4.1 epectively a hown in Fig. 9(d)-(f). Hence the RO-SMI ignal ae much tonge (moe than 3 time tonge) than that of the uual SMI ignal. 3. When the SMI ytem ente the untable egion hown in Fig. 9(c), the lae output will be untable. In thi cae thee i not an obviou elationhip between the taget movement and the lae output, and hence the ytem i not uitable fo uch wavefom baed ening. By uing an extenal fat photo-detecto to obeve uch SMI ignal, we obtained expeimental ignal hown in Fig. 1. In the expeiment, the lae ouce i a commecial emiconducto lae (Hitachi, HL835G), which i a ingle mode quantum well lae with the emitting wavelength of 83nm and maximum output powe (P) of 4mW. We et it initial extenal cavity to be 14.5cm and the injection cuent 53mA. The taget i a PZT contolled by a ignal with an 3.V DC offet upe-poitioned with a inuoidal voltage ignal (Hz and 3.9V P-P). The coeponding diplacement (P-P) geneated by the PZT i.8 µ m. The attenuato i adjuted to change the feedback level. It can be een that the peakpeak value of an RO-SMI ignal i much highe than that of the SMI ignal in Fig. 1(b). The finge numbe in an RO- SMI i the ame a the one in the SMI ignal. Theefoe, we confim that uch SMI ha a ame diplacement eolution with uual SMI ignal but a lage ignal amplitude.

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