Dynamic Stability Analysis for a Self Mixing Interferometry

Transcription

1 Univerity of Wollongong Reearch Online Univerity of Wollongong Thei Collection Univerity of Wollongong Thei Collection 16 Dynamic Stability Analyi for a Self Mixing Interferometry Yuanlong Fan Univerity of Wollongong Recommended Citation Fan, Yuanlong, Dynamic Stability Analyi for a Self Mixing Interferometry, Doctor of Philoophy thei, School of Electrical, Computer and Telecommunication, Univerity of Wollongong, Reearch Online i the open acce intitutional repoitory for the Univerity of Wollongong. For further information contact the UOW Library: reearch-pub@uow.edu.au

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3 Dynamic Stability Analyi for a Self Mixing Interferometry A thei ubmitted in fulfillment of the requirement for award of the degree Doctor of Philoophy from UNIVERSITY OF WOLLONGONG by Yuanlong Fan School of Electrical, Computer and Telecommunication Engineering February 16 I

4 Dedicated to my family II

5 Declaration Thi i to certify the work reported in thi thei wa carried out by the author, unle pecified otherwie, and that no part of it ha been ubmitted in a thei to any other univerity or imilar intitution. Yuanlong Fan February 16 III

6 Abtract In recent year, a an emerging and promiing non-contact ening technique, Self Mixing Interferomerty (SMI) ha attracted much attention of reearcher. The SMI i baed on the elf-mixing effect that occur when a mall fraction of laer light emitted by the emiconductor laer (SL) i reflected by an external target and re-enter the SL cavity. Therefore, the core component of an SMI conit of an SL, a micro-len and a target. The target form an external cavity for the laer, and when it move along the light beam, a modulation in the emitted laer power can be oberved. Thi modulated power i referred to a an SMI ignal which carrie the information of the motion of the target a well a the SL itelf. Compared to other traditional interferometric cheme, e.g., Michelon or Mach-Zender, SMI ha the advantage of a compact et-up, high enitivity and low cot. Under the condition of a table operation, an SL biaed by contant current uually emit laer light with a contant intenity. However, with the introduction of external optical feedback (EOF), the laer light can become untable. The tability ha been intenively tudied for an SL with a tationary external target. However, the reported reult motly were obtained by either ignoring the nonlinear gain or uing an approximated ytem determinant derived under certain aumption. Furthermore, the exiting tability analyi can not be directly applied on the SMI becaue the external target in an SMI move. In thi thei, firtly, in Chapter, by lifting all the aumption made in the exiting work related to the tability, an accurate ytem determinant i preented by conidering the nonlinear gain. Baed on the ytem determinant, a tability boundary i obtained and preented in a 3-parameter pace decribed by the feedback trength, the external cavity length L and the injection current J. The reult reveal more inight into the influence of thee parameter on the tability a compared to the exiting work. Secondly, the tability on an SL with a moving target, which i the cae for mot SMI, i examined in Chapter 3. For an SMI ytem, the parameter (feedback phae), IV

7 C (feedback level factor) and J decribe the operational condition for an SMI. It i very important to decribe the tability of an SMI by uing thee three parameter. By conidering a time varying, baed on the reult gained from the ytem determinant in Chapter, the dynamic tability of an SMI i preented in a plane of (, C ), from which three important region called table, emi-table and untable are recognized to characterize the feature of an SMI. We realize that the exiting SMI model i only valid in a table region, and the emi-table region ha potential application for ening and meaurement, however it need remodeling of the ytem by conidering the bandwidth of the detection component. Thirdly, in Chapter 4, the tability of an SMI i decribed in a plane of (, J ), from which a critical injection current i found, above which the SMI i alway table. Furthermore, the importance of the nonlinear gain on the tability of an SMI i revealed in thi Chapter. The reult reveal that a table SMI can be achieved by either an increae in the injection current or an increae in the external cavity length. Latly, baed on the above tability analyi, an experimental approach wa preented to determine the actual tability boundary for a practical SMI ytem, hown in Chapter 5. In that Chapter, the meaurement problem for the linewidth enhancement factor (LEF), which i an important parameter of the SL itelf, caued by an SMI operating at emi-table region i preented. Then a new method for meauring the LEF i propoed. All the reult preented in thi thei are confirmed by both imulation and experiment. The reult can be regarded a a et of general criteria for deigning a table SMI for ening and intrumentation. V

8 Acronym SMI SL LEF FP LK EOF DDE LFF MLM RK PD TC LC BS OSA EPD RO VCSEL DFB CSP SNR elf mixing interferometry emiconductor laer linwidth enhancement factor Fabry-Perot Lang-Kobayahi external optical feedback delayed differential equation low frequency fluctuation minimum linewidth mode Runge-Kutta photodiode temperature controller laer controller beam plitter optical pectrum analyzer external photodiode relaxation ocillation vertical cavity urface emitting laer ditributed feedback channeled ubtrate planar ignal noie ratio VI

9 Acknowledgement I would like to extend thank to the many people who generouly contributed to the work preented in thi thei. Special mention goe to my enthuiatic upervior, Aociate Profeor Yanguang Yu, Profeor Jiangtao Xi and Dr Qinghua Guo. My PhD ha been an amazing experience and I thank them wholeheartedly, not only for their tremendou academic upport, but alo for giving me o many wonderful opportunitie. Without their enlightened intruction, impreive kindne and patience, I could not have completed my thei. Their keen and vigorou academic obervation enlightened me not only during thi thei but I know it will continue to do o in my future tudy. Secondly, I hall extend my thank to the taff in the School of Electrical, Computer and Telecommunication Engineering (SECTE) who have helped me to develop a fundamental and eential academic competence. My incere appreciation alo goe to the teacher and tudent from the Signal Proceing for Intrumentation and Communication Reearch (SPICR) Lab, who participated in thi tudy with enthuiatic cooperation. Finally, but by no mean leat, thank go to my wife, my parent in-law and my parent, for almot unbelievable upport. They are the mot important people in my world and I dedicate thi thei to them. VII

10 Publication Journal: J1. Y. Fan, Y. Yu, J. Xi, G. Rajan, Q. Guo and J. Tong, A imple method for meauring the linewidth enhancement factor of emiconductor laer, Applied Optic, Vol. 54, No. 34, pp (15). J. Y. Fan, Y. Yu, J. Xi, and Q. Guo, "Stability limit of a emiconductor laer With optical feedback," IEEE Journal of Quantum Electronic, Vol. 51, No., pp (15). J3. Y. Fan, Y. Yu, J. Xi, and Q. Guo, "Dynamic tability analyi for a elf-mixing interferometry ytem," Optic Expre, Vol., No. 3, pp (14). J4. Y. Fan, Y. Yu, J. Xi, and Q. Guo, "Influence of injection current on the tability of a elf-mixing interferometry ytem," ubmitted to IEEE Photonic Technology Letter. International Conference: C1. Z. Liu, B. Liu, Y. Yu, Y. Fan, J. Xi, Q. Guo, K. Lin, A novel normalization method for improving the ening performance of a elf-mixing interferometry, TENCON, IEEE Region 1 Conference, 1-4 November 15. C. Y. Fan, B. Liu, Y. Yu, J. Xi, Q. Guo, J. Tong, Analyi on the tranient of a elf-mixing interferometry ening ytem, TENCON, IEEE Region 1 Conference, 1-4 November 15. C3. Y. Fan, Y. Yu, J. Xi, H. Ye, Influence of the nonlinear gain on the tability limit of a emiconductor laer with external optical feedback, Proc. SPIE, Semiconductor Laer and Application VI, 967, 9 October, 14. C4. K Lin, Y. Yu, J. Xi, Y. Fan, H. Li, Meauring Young' modulu uing a elf-mixing laer diode, Proc. SPIE 8975, Reliability, Packaging, Teting, and Characterization of MOEMS/MEMS, Nanodevice, and Nanomaterial XIII, 8975B,7 March 14; VIII

11 Table of Content Chapter 1 Introduction and Background The Mathematical Model of the SMI Derived from the Three Mirror Model Derived from the Lang and Kobayahi Model SMI Waveform Literature Review Outtanding Iue Thei Organization Chapter Determining the Stability Boundary for an SL with EOF Determine the Stability Boundary Sytem Determinant Stability Boundary Influence of External Cavity, Injection Current and Feedback Strength Influence of External Cavity Length Influence of Injection Current Influence of Feedback Strength Summary Chapter 3 Stability Analyi for an SMI The Stability Boundary Decribed by C and Critical Feedback Level Factor Determining Critical Feedback Level Factor Summary Chapter 4 Influence of the Injection Current on the Stability of an SMI Stability Boundary Decribed by J and IX

12 4. Critical Injection Current Determining Critical Injection Current Summary Chapter 5 Deigning a Stable SMI for Meauring the Linewidth Enhancement Factor Detection Problem Caued by SMI Signal in the Semi-table Region Deigning a Method for Achieving a Stable SMI Sytem A New Method for the LEF Meaurement Summary Chapter 6 Concluion and Future Work Reearch Contribution Suggeted Future Reearch Topic... 8 Reference... 8 X

13 Lit of Figure Figure 1-1: A chematic diagram of the SMI Figure 1-: Simplified chematic diagram for the three mirror model... 3 Table 1-1: Phyical meaning for the internal cavity parameter in LK equation... 6 Figure 1-3: (a) Relationhip between and, (b) relationhip between g and when C.7 and Figure 1-4: (a) A time varying, i.e., () t, caued by the target movement, (b) the correponding SMI ignal when C.7 and Figure 1-5: (a) Relationhip between and, (b) relationhip between g and when C 3 and Figure 1-6: (a) A time varying, i.e., () t, caued by the target movement, (b) the correponding SMI ignal when C 3 and Table 1-: The exiting approache for invetigating the tability boundary Figure -1: Stability boundary decribed in the 3-parameter pace of (,, J ). The untable region i above the urface Figure -: Stability boundary decribed by and for a fixed J Jth 1.3. (a) tability boundary given in thi thei on Fig. -1, (b) tability boundary given in [31] Figure -3: Laer intenity waveform of It () for the choen point,.3, ( ).5 and J Jth Figure -4: Influence of L on the tability decribed by J and when J 13.7mA Figure -5: Influence of J on the tability decribed by and L when J 13.7mA Figure -6: The relationhip of c and J... 9 Figure -7: Influence of on the tability decribed by J and L when J 13.7mA th th th R Figure -8: The relationhip of and L c XI

14 Figure 3-1: The relationhip between and Figure 3-: Influence of J and L on the tability boundary of an SMI. (a) for a fixed L.5m with different J, (b) for a fixed J 1.3J th with different L Figure 3-3: The tability boundary of an SMI when J 1.1J th and L.35m Figure 3-4: SMI ignal predicated by the LK model and the exiting SMI model repectively. (a) and (f): movement trace of the external target, (b)-(e): SMI ignal obtained by the LK model with C 1.5, C.5, C 4. and C 9. repectively. (g)-(j) SMI ignal obtained by the exiting SMI model with C 1.5, C.5, C 4. and C 9. repectively Figure 3-5: Stability boundary decribed in the 3-parameter pace of ( C critical, J, L ). The untable region i above the urface Figure 3-6: The tability boundary decribed by J and C critical for long external cavitie Figure 3-7: Experimental etup for invetigating the tability of the SMI Figure 3-8: Two optical pectra obtained with L.5m and J 1.7J th for (a) the table region, (b) the emi-table region Figure 3-9: Experimental reult with the aid of the OSA. (a) for a fixed L.5m, (b) for a fixed J J th Figure 3-1: Experimental obervation of the dynamic of the SMI ignal uing the fat ocillocope with an 8GHz bandwidth and 1.5GS/ ampling rate. (a) the target movement trace (b) the SMI ignal in table region (c)-(d) the SMI ignal in emi-table region (e) the SMI ignal in untable region Figure 3-11: Experimental obervation of the dynamic of the SMI ignal uing the detection circuit which bandwidth i 3MHz. (a)-(e) correpond to the cae hown in Fig. 3-11(a)-(e) repectively Figure 4-1: Stability region of an SMI ytem (or band table phenomenon) when C.84 and L 4.5cm. (a) tability region obtained in [4], (b) tability region obtained in thi paper uing the complete ytem determinant derived in Chapter of thi thei Figure 4-: SMI ignal correpond to the three value of P on Fig. 4-1(b). (a) the target movement trace, (b) for P.1, (c) for P.8, (d) for P Figure 4-3: The locu of on the S-plane by varying when.1n, J J 1.3 and th XII

15 .5, (b)-(d) are the correponding normalized laer intenity for 1.5,.8 and.5 repectively Figure 4-4: The tability limit of and for different value of when J Jth 1.3 (a) ( ).3, (b) ( ) R R Figure 4-5: Influence of C on the tability region of an SMI with L.15m Figure 4-6: Influence of L on the tability region of an SMI with C Figure 4-7: Simulation reult of the (a) relationhip between P and C when L.15m, and (b) c relationhip between P c and L when C... 6 Figure 4-8: Experimental reult of the (a) relationhip between P and C when L.15m, and (b) c relationhip between P c and L when C Figure 5-1: An SMI ignal obtained by the LK equation in a emi-table region when C. and 6.. (a) the movement trace of the external target, (b) the SMI ignal Figure 5-: Filtered SMI ignal after the detection component with different fcut off when C. and 6.. (a) SMI ignal in Fig. 5-1(b), (b)-(f) filtered SMI ignal of the SMI ignal in (a) Figure 5-3: Experimental SMI ignal obtained in [7] (ee Fig. in [7]) under moderate feedback. Time interval t 13 and t 4 are relevant for the determination of Figure 5-4: (a) Two filtered SMI ignal obtained in Fig. 5-(d) and (f). (b)-(c) the zoomed SMI ignal Figure 5-5: Schematic diagram of the implified SMI ytem of Fig Figure 5-6: A phyical SMI ytem Figure 5-7: Two optical pectra obtained with L.5m and J 1.7J th for (a) the table region, (b) the emi-table region Figure 5-8: Two table SMI ignal obtained from the table SMI ytem we deigned Figure 5-9: The relationhip between (a) and a well a (b) and g for a fixed value of Figure 5-1: Simulation reult of SMI ignal for two different C value when 4.. (a) the vibration trace (b) two SMI ignal for C.5 (olid line) and C. (dotted line) XIII

16 Lit of Table Table 1-1: Phyical meaning for the internal cavity parameter in LK equation... 6 Table 1-: The exiting approache for invetigating the tability boundary Table 5-1: Simulation reult of the meaurement of the LEF where the SMI ignal are uperpoed a noie ignal with SNR=dB XIV

17 Chapter 1 Introduction and Background In the lat decade, the variou application of the Self Mixing Interferometry (SMI) baed ening ytem have attracted much attention of reearcher. An SMI can be ued for meauring metrological quantitie, uch a velocity, abolute ditance, diplacement and vibration, etc. or even the parameter aociated with the laer itelf [1-19]. A chematic diagram of the SMI i hown in the following figure (Fig. 1-1). Figure 1-1: A chematic diagram of the SMI. The core part of an SMI conit of a ingle longitudinal mode Semiconductor Laer (SL) and a moving target which form an external cavity of the SL. An SMI work when a mall portion of light i back-cattered or reflected by an external target and re-enter into the laer internal active cavity. The re-entered light modulate both amplitude and frequency of the emitted SL power [, 1]. Thi modulated power i called an SMI ignal which can be ued to detect metrological quantitie aociated 1

18 with the external target and the parameter aociated with the SL itelf. The advantage of uing an SMI-baed ening ytem to perform meaurement have been preented in [3]: No optical interferometer external to the ource i required. Thi lead to a imple and compact et-up. No alignment i needed becaue the patial mode that interact with the cavity mode i filtered out patially by the laer itelf. Thi mean that detection of the diffuive target movement become poible. Senitivity of the cheme i very high (ub-nm enitivity). Due to thee advantage, the SMI ha been intenively invetigated both theoretically and experimentally for exploring all kind of ening application. In thee application, it i required that an SMI ytem operate in a table mode, in which cae the SL biaed by contant injection current uually lead to SMI ignal with ymmetric inuoidal-like fringe or aymmetric awtooth-like fringe, depending on the external optical feedback level. However, with the change of operational condition, uch a injection current and parameter aociated with the external cavity including the optical feedback level and external cavity length, the SL can alo exhibit untable behavior. In thi cae, an SMI will degrade or even loe it ening ability. Therefore it i very important to invetigate the tability of an SMI ytem with repect to it operational condition, which lead u to the main goal of thi thei, i.e., analyzing the tability of the SMI. Thi chapter give an introduction and background for thi thei. The ret of thi chapter i organized into the following ection. Section 1.1 give detailed derivation of the exiting mathematical model for decribing the SMI. A literature review for the SMI ha been given in Section 1. in term of the tability analyi. Baed on the invetigation of reviewed article, the exiting problem for invetigating the tability of the SMI are preented in Section 1.3. Section 1.4 how the tructure of thi thei.

19 1.1 The Mathematical Model of the SMI Derived from the Three Mirror Model The mathematical model of the SMI can be derived from the claical three mirror model coniting of a Fabry-Perot (FP) type laer with facet reflection coefficient r 1 and r, and the target with the reflection coefficient of r 3 []. Therefore, a implified arrangement baed on the chematic diagram hown in Fig. 1-1 can be ued for carrying out the derivation, which i hown in the following figure (Fig. 1-): Figure 1-: Simplified chematic diagram for the three mirror model. Auming rr 3 1, i.e., there i only one reflection within the external cavity, the effective reflection coefficient r eff at the laer front facet can be expreed a [, 3]: where eff 1 j 3 r r r r e (1.1) and are repectively the perturbed laer angular frequency and one roundtrip time of the light in the external cavity, and L c, where c i the peed of light. On the other hand r eff can alo be repreented in another form with repect to it amplitude A eff and phae eff a []: 3

20 r eff j eff A e (1.) eff where A 1 co( ) eff r and in( ) (1.3) eff where i the feedback trength and (1 r ) r r. A the roundtrip phae in the 3 internal cavity mut be equal to a multiple of, the phae condition of compound cavity of the three mirror model can be decribed uing the following equation (Eq. 1.4) []: ( g g ) d ( ) (1.4) L c th in eff where L correpond to a change in the round trip phae compared to q, where q i an integer. In Eq. (1.4), i called the linewidth enhancement factor (LEF) which i an important fundamental decriptive parameter of the SL becaue it determine the characteritic of SL, uch a the pectral effect, the modulation repone, the injection locking and the repone to the external optical feedback [15, 4-9]. in i the light roundtrip time in the internal cavity and i the angular frequency of the olitary laer. and without external cavity. g c and g th are repectively the threhold gain with g a d In ( rr ) (1.5) th where a repreent the lo account for any optical lo in the internal cavity. Note that g c alo mut atify the amplitude condition of the compound cavity, that i [, 3]: ( gca) d 1 eff 1 r A e (1.6) Subtituting Eq. (1.3) and (1.5) into Eq. (1.6), we can obtain the threhold gain difference: 4

21 g g co( ) (1.7) d c th Then ubtituting Eq. (1.) and (1.7) into Eq. (1.4) and conidering L, equation (1.4) change to []: (1.8) 1 in arctan( ) in If we denote, and C 1, the core part of the SMI in model can be finally derived a: C in arctan( ) (1.9) where and are repectively the light phae correponding to the perturbed and unperturbed laer angular frequency. Note that although the three mirror model explain ome intereting reult, it lack ome detail of the phyical etting of the phenomenon, e.g., the material and aociated effect for an SL [4] Derived from the Lang and Kobayahi Model Interetingly, equation (1.9) can alo be derived from the well known Lang and Kobayahi (LK) equation which are baed on Lamb equation and modified with the additional equation for the tate concentration [4]. Compared to the three mirror model, the LK equation decribe the active material and carry a decription of laer ocillator equation which yield a much more complete decription of the dynamic behaviour of a ingle mode SL with external optical feedback (EOF). The well known LK equation [3] were firt propoed in 198 and the model conit of three imultaneou Delay Differential Equation (DDE) which are hown a below: de( t) 1 1 G N t E t E t E t dt p in co ( t) ( t ) ( ), ( ) ( ) ( ) (1.1) 5

22 d( t) 1 1 E( t ) G N( t), E ( t) dt p in E( t) in ( t) ( t ) (1.11) dn( t) J N( t) dt ev G N( t), E ( t) E ( t) (1.1) where GN( t), E( t ) i the modal gain per unit of time and i expreed a [31, 3]: G N( t), E( t) GN N( t) N 1 E ( t) (1.13) and when ignoring the nonlinear effect, the modal gain can be implified a: ( ), ( ) ( ) N G N t E t G N t N (1.14) Equation (1.1)-(1.13) decribe the dynamic behavior of the three variable, namely the electric field amplitude Et (), the electric field phae () t and the carrier denity Nt () where t i the time index. () t i given by ( ) ( ) t t t, and where () t i the intantaneou optical angular frequency for an SL with EOF. The dynamic of the SL with an EOF ytem are governed by the injection current ( J ) to the SL and the parameter aociated with the external cavity including and. The other parameter in Eq. (1.1)-(1.13) are related to the olitary SL itelf, and are treated a contant for a certain SL. Thee parameter are defined in Table 1-1 [31]. Note that the value of the parameter provided in Table 1-1 are adopted from [31]. Table 1-1: Phyical meaning for the internal cavity parameter in LK equation Symbol Phyical Meaning Value G modal gain coefficient N m N carrier denity at tranparency m 3 3 nonlinear gain compreion coefficient confinement factor.3 p.5 1 m 1 photon life time in. 1 1 internal cavity round-trip time line-width enhancement factor e 19 elementary charge C 16 3 V volume of the active region 1. 1 m 9 carrier life time. 1 6

23 The core part of the SMI model i derived from the tationary olution of the above LK equation. Let E, N and repreent the tationary olution of LK equation for electric field amplitude, carrier denity and angular frequency repectively. When the ytem decribed by Eq. (1.1)-(1.13) enter into a tationary tate, we have de( t) dt, d() t dt and dn( t) dt. Subtituting E( t) E( t ) E, N() t N, and t () t into Eq. (1.1)-(1.13) and ignoring the nonlinear gain, the well known tationary olution can be obtained a below [1, 4, 1, 3, 31, 33]: (1.15) 1 in( arctan ) in N 1 co( ) N G G (1.16) p N in N E J ( ev ) N G ( N N ) N (1.17) From Eq. (1.15)-(1.17) and by conidering a moving target, the exiting SMI model can be obtained a below by introducing:, and C 1 (1.18) in Then Eq. (1.15) become: C in arctan( ) (1.19) where i aociated with the external cavity length L, i.e., 4 L, where i the unperturbed laer wavelength. Equation (1.19) i called the phae equation which i the core part of the exiting SMI model, and i the ame equation a derived by the three mirror model (ee Eq. (1.9)). By ubtituting Eq. (1.16) into Eq. (1.17), the normalized variation of the SL output 7

24 power (that i the o called SMI ignal g ) can be obtained and decribed a [1]: g co( ) (1.) Equation (1.19) and (1.) contitute the exiting SMI model which ha been widely accepted to decribe the waveform of SMI ignal [, 3, 6, 1, 3, 33-36]. A een in Eq. (1.19) and (1.), there i a traight forward procedure to contitute g, i.e., g, and alo a traight backward procedure, i.e., g, to obtain, thu retrieving the external cavity information. Therefore the knowledge of the theory of generating an SMI ignal a well a it waveform i eential to achieve good performance of the SMI SMI Waveform In the SMI model, C (called feedback level factor) i an important parameter a it characterize the waveform of an SMI ignal. When C 1(correponding to weak feedback regime), equation (1.19) preent a unique mapping from and. In thi ituation, the movement of the external target will reult in an SMI ignal waveform with a fringe tructure imilar to the traditional interference fringe, and each fringe period correpond to a phae hift that i equivalent to a diplacement of half a wavelength of the external target [35]. Figure 1-3 how the relationhip between and a well a g and when C.7 and 6. Suppoing that the external target move according to a inuoidal law with L( t) L L in( ft), where L, L and f are the initial external cavity length the vibration amplitude and frequency repectively which are choen a L.35m, L 1.5 and f 75Hz, thu leading to a time varying, i.e., ( t) 4 L( t) 6 in( ft) ( rad) which i hown in Fig. 1-4(a), and Fig. 1-4 (b) how the correponding SMI ignal when C.7 and 6. 8

25 Figure 1-3: (a) Relationhip between and when C.7 and 6., (b) relationhip between g and Figure 1-4: (a) A time varying, i.e., () t, caued by the target movement, (b) the correponding SMI ignal when C.7 and 6. In the ituation of C 1 which correpond to a moderate (1C 4.6 ) or trong ( C 4.6 ) feedback regime, equation (1.19) yield multiple poible and the SMI ignal how aymmetric hyterei and produce awtooth-like fringe. To illutrate the cenario behind the waveform of SMI ignal when C 1, figure 1-5(a) how the relationhip between and a well a g and when C 3 and 6. The actual behaviour i decribed in [1, 7, 35, 37], indicating that and g will vary along the route A1B B1 when increae, and it will however track the route of B1A A1 when decreae. Note that, the tationary olution 9

26 for within the range of, A,, B are alway untable according to [31, 38, 39], and thu will never be an ocillating mode of the SMI ignal. Figure 1-5: (a) Relationhip between and when C 3 and 6., (b) relationhip between g and Similar to the weak feedback regime cae dicued above, we preent () t, which i ame with Fig. 1-4(a), a well a the correponding SMI ignal gt () in the following figure (Fig. 1-6). Figure 1-6: (a) A time varying, i.e., () t, caued by the target movement, (b) the correponding SMI ignal when C 3 and 6. The mechanim of generating an SMI ignal a well a predicting it behaviour with repect to C ha been well-etablihed and preented in [18, 3, 33, 35, 36]. 1

27 Clearly, the exiting SMI model, i.e., Eq. (1.19) and (1.), i baed on a premie that the ytem i table, or in other word, it doe not give the condition of tability of the SMI with repect to the SL operational parameter, i.e.,, and J. The goal of thi thei i to achieve a complete and accurate tability analyi for the SMI. Before preenting our approach, a detailed literature review related to the tability analyi of an SMI i preented in the following ection. 1. Literature Review A an SMI i an SL with EOF by conidering a time varying light phae, alo called a feedback phae [4-44] or, it i neceary to dicu the tability of an SL with EOF where the feedback phae i fixed, or ay the target i tationary. Note that in a not very trict ene, it i uually convenient to conider the feedback phae, i.e.,, a an independent operation parameter with [39, 43, 45-47], becaue very tiny change of or can reult in ignificant variation on the phae. In practice, the value of can be varied by minucule change of the external cavity length (on the order of one wavelength of the laer) or by altering through lightly adjuting the injection current or temperature [45]. Under the condition of a table operation, an SL biaed by contant current uually emit a laer with a contant intenity when the target i tationary. However, with the change of operational condition, the laer output can alo exhibit untable behaviour, uch a periodic ocillation, quai-periodic ocillation, low frequency fluctuation (LFF) or chao [43, 44, 48-56]. A the tranition from table to untable tate i caued by the change in the injection current and parameter aociated with EOF, it i important to figure out when the tranition occur with repect to the value of thee parameter, or the range of thee parameter within which the SL i table or untable. Such a range i referred to a the tability limit or the tability boundary [31], or the 11

28 Hopf bifurcation [57, 58]. The firt dicuion of the tability of the SL with EOF originate in 198 [3], together with the propoal of the LK equation. By looking at the characteritic of the LK equation in a mall region near it tationary olution, a et of linear partial differential equation were obtained, from which the ytem determinant can be derived to evaluate the tability of the SL with EOF ytem. The ytem i conidered table when the ytem determinant doe not have zero in the right ide of the S-plane. Under the aumption of weak feedback trength and hort external cavity, i.e., in 1, the ytem determinant wa implified to a quadratic equation from which an expreion (Equation (9) in [3]) for identifying the untable ytem wa derived. However, the influence of the parameter aociated with SL operation on the tability boundary wa not dicued in [3]. Then baed on the work in [3], in 1984, Tromborg, et al. [45] preented reult of an analyi on the tability boundary of an SL with EOF. Three different boundarie were decribed in [45], namely boundarie A, B and C. boundarie A and B are for the bitability condition decribed in [51], [59, 6]. Boundary C i defined a a dynamic tability boundary, alo called a feedback-induced intenity pulation boundary in [45]. In thi thei, we mainly dicu the boundary C, a it i the boundary identifying the tranition from table to untable tate. In [45], by introducing further aumption, i.e., ( ) and R in R ( ) R in, the ytem determinant preented in [45] wa implified to a trancendental equation, leading to a et of coupling equation (Equation (45) and (47) in [45]) for decribing the tability boundary. Here, R and R repreent the relaxation reonance angular frequency and carrier denity relaxation time (alo called damping time [31]) repectively for the olitary laer. In [45], the tability boundary wa preented a a relationhip of the feedback phae in the range of, veru for a fixed (ee Figure 9 in [45]). In fact, uch preentation of the tability boundary, that i, in a two dimenional 1

29 plane of the feedback phae and one of the operational parameter, i ueful for invetigating the tability of the SMI in the future, a the SMI i a pecial cae of SL with EOF where the feedback phae i time varying in the range of with the movement of the target (note that i periodic, and o i according to Eq. (1.19) [61]). In the figure 9 in [45], it can be een that below a certain value of, the SL with EOF i alway table no matter what the value of the feedback phae are, therefore indicating a table operation range with repect to for the SMI. Two year later, in 1986, Oleen, et al. [39] improved the work in [45] in two repect: 1) uing the ytem determinant without the aumption in [3] and [45], and ) employing R and R of an SL with EOF rather than thoe of a olitary SL. In [39], the tability boundary wa dicued in the ame way a in [45] but for three different fixed value of, howing that the table operation range of the SMI increae with the increae of. In, another work wa reported in [46] uing the normalized LK equation for the cae of hort external cavity (.n ). Due to the ue of the normalized LK equation, the numerical analyi on LK equation i ignificantly implified [5], [46, 6, 63]. In [46], tarting from the normalized LK equation, a new ytem determinant with the normalized coefficient wa derived. The tability boundary wa numerically calculated and preented a a relationhip of the feedback phae veru for a fixed. The ame concluion with [45] can be drawn from the relationhip preented in [46]. In 9, another et of reult were reported in [61] obtained by mean of a MATLAB package DDE-BIFTOOL [64]. The tability boundary acquired in [61] wa preented a a relationhip of the feedback phae veru and the feedback phae veru J repectively for a fixed. The normalized LK equation were alo ued in [61] for the numerical calculation of the tability boundary. The underlying algorithm of the MATLAB package i the linear multi-tep method [65, 66] which can approximately calculate the location of zero of the ytem determinant. 13

30 In 1, Donati and Fathi [67] numerically preented the poible region for achieving a table SMI for both hort and long external cavitie by invetigating the tability boundary of an SL with EOF in a plane of the feedback phae and. The reult in [67] howed that the SMI can be table when i mall. More recently, in 13, Lentra [4] found that when the product of R and i equal to an integral multiple of, the SL with EOF behave like a olitary laer which i alway table no matter what the feedback phae i. The tability boundary wa preented a a relationhip of the feedback phae in the external cavity and J for a fixed. Note that in the reearch conducted in 9 [61] and 13 [4], a band table phenomenon with repect to J wa predicated theoretically. The phenomenon indicate that, in ome certain region of J, the tability i free of the influence of the feedback phae, thu leading to a table SMI, while in other region of J, the tability trongly depend on the phae. However, the tudie were only for hort cavitie and weak feedback trength ( 1 in ). An outtanding iue aociated with the reult reported in [3, 39, 4, 45, 46, 61, 67], i that, none of them conider the effect of nonlinear gain, which wa ignored in the proce of linearizing the LK equation in [3]. A a matter of fact, nonlinear gain i an important factor in decribing the dynamic behavior of an SL with EOF [3, 68, 69] and it wa reported that the tability boundary of an SL with EOF can be enhanced by the nonlinear gain [68, 7]. Hence, in order to better decribe the tability, nonlinear gain in the LK equation hould be conidered. Some preliminary work wa reported on thi problem [31, 71-73]. The work in [31] wa baed on the expreion in [45], but inert a term in the expreion of R to take into account the influence of nonlinear gain (ee Equation (19) in [45]). The tability boundary obtained in [45] i a relationhip of veru for a fixed J. Furthermore, in [31], by analyzing the characteritic point on the tability boundary, a imple analytic 14

31 expreion wa derived for the tability boundary decribed by the critical feedback trength c. Obviouly, [31] conider only partially the influence of nonlinear gain, and it i till baed on the approximation ued in [45]. In 1993, Ritter and Haug [71] preented another expreion of the ytem determinant by taking into account the nonlinear gain in the electric field amplitude equation (Equation (.1) in [71]) of LK equation. However, the influence of the gain in the electric field phae equation wa till aumed to be linear. In working out the ytem determinant, a number of term are neglected, including term proportional to ( ) R R, ( ) and in R ( R ) ( in R), and the ytem determinant wa implified to a trancendental equation. In thi work, the tability boundary wa decribed a a relationhip of veru J for a fixed. Later, in 1994, Tager and Petermann [7] preented imilar work to [71], in which the gain in the electric field amplitude equation wa conidered a nonlinear, but the gain in the electric field phae equation wa till linear. Two new coupling equation were derived from the ytem determinant under aumption different to [71], that i, and R p R ( ), where p i the contribution of nonlinear gain R in R to R in a imilar way to [31]. The two coupling equation are imilar to thoe derived in [45], but they include the influence of compound cavity mode competition on the tability of an SL with hort external cavitie. The tability boundary wa then preented a a relationhip of veru and veru the feedback phae in the external cavity for a fixed J in [7], indicating that (ee Fig. 7 in [7]) there i no region where the SL with EOF i alway table for all the value of the feedback phae with a fixed. That i to ay there i no region guaranteeing a table SMI. In 1998, Maoller and Abraham [73] invetigated the influence of on the tability limit when the SL i biaed well above the threhold. However, the nonlinear gain wa till partially conidered in the ame way a in [31]. 15

32 1.3 Outtanding Iue The reult of analyzing the above exiting literature are ummarized in Table 1- from which the outtanding reearch iue can be drawn a follow: 1. The exiting approache either ignore the nonlinear gain, which i an important factor in decribing the ytem, or make ue of an approximated expreion of the ytem determinant, thu leading to ue of incomplete ytem decription and hence inaccurate reult.. The influence of all parameter aociated with an SL with EOF operation, i.e.,, and J on the tability of an SL with EOF ha been abent in the literature, a well a the SMI, both theoretically and experimentally. Table 1-: The exiting approache for invetigating the tability boundary. Ignore the nonlinear gain Partially conider the nonlinear gain Reference Aumption made Boundary decription For fixed parameter of [3] 1 in NA NA [45] ( ), ( ) and, J R in R R in [39] NA and, J [46] Normalized LK equation and, J [61] Normalized LK equation and 1 in and, and [67] NA and, J [4] 1 in and J, [31] [71] [7] R k ( in R), R ( k in) and J, ( R R), ( in R), ( ) ( ) and J, R p R R in R, ( ) R in R J and, and [73] NA NA,, J J 3. A the exiting SMI model i built on the tationary olution of LK equation, i.e., Eq. (1.1)-(1.1), the model i only certainly valid when an SMI ytem i table. Or in other word, when the ytem enter into the untable tate, the premie for deriving the tationary olution will be no longer valid, i.e., de( t) dt, 16

33 d() t dt and dn( t) dt, thu leading to the poition that actual behaviour of the ytem cannot be decribed by the exiting SMI model. 4. The exiting SMI baed application, uch a meaurement of diplacement a well a the parameter of the SL itelf, e.g., the LEF, all aume and require that the ytem i in a table tatu. However, reearch in Chapter 3 will ubequently how that when the ytem enter into the untable tate, an untable SMI can pretend to be a table SMI, which confue reearcher due to the limited bandwidth of the detection component, thu leading to the degraded ytem performance. 1.4 Thei Organization Thi thei conit of ix chapter: Chapter 1 preent a brief introduction and background to thi reearch project. Firtly, in Section 1.1, the background of the SMI i introduced then it theoretical model originated from both the three mirror and LK model i preented. Baed on the SMI model, the characteritic of the SMI waveform a well a it mechanim related to the model are decribed. Secondly, in Section 1., the literature related to the tability analyi of an SMI i reviewed from the viewpoint of their theoretical derivation a well a the manifetation of their reult. At the end of Chapter 1, the ummarization of the outtanding problem and organization of thi thei are given. Chapter give a complete and accurate tability analyi for an SL with EOF with a particular fixed feedback phae value. In Section.1, a new and accurate ytem determinant i derived by removing the aumption or approximation made in the exiting work to enure a more accurate and complete theoretical analyi. Then by varying,, and J, the trajectory of the zero of the ytem determinant i determined by an effective numerical computation developed by thi reearcher, from which the tability boundary decribed by thee parameter can be obtained. In Section 1.3, the relationhip of each pair of the three parameter will be invetigated in detail and a lit of new and intereting dicoverie i preented. 17

34 Chapter 3 analyze the tability of the SMI. The tability of an SMI i firtly preented in a two dimenional plane of the feedback phae and the feedback level factor C in Section 3.1, baed on which three region for characterizing the behaviour of an SMI are propoed, and potential application for the three region are decribed. Furthermore, a critical optical feedback factor C critical, under which the SMI i guaranteed to be table, i approximately determined in an analytical form. Finally, in Section 3., detailed experimental procedure are deigned and decribed to verify the reult preented in Section 3.1. Chapter 4, in ection 4.1 preent, inpired by an exiting work, the tability boundary of an SMI in a two dimenional plane of and J uing the ytem determinant derived in Chapter, in order to ee how the injection current J influence the tability of an SMI. Moreover, comparing it with the exiting work, a critical injection current J critical i oberved from the plane, above which the SMI i alway table. Followed by extenive imulation performed in Section 4., the relationhip between J critical and C a well a J critical and are revealed. In Section 4.3, experiment are performed to verify the reult. Chapter 5 how a imple method for meauring the linewidth enhancement factor (LEF) uing the table SMI ignal obtained baed on the guideline deigned in the previou chapter. By invetigating the relationhip between the feedback phae and the table SMI ignal, it i found that the LEF can be meaured from the overlapping point of two table SMI ignal. Simulation and experiment verify the propoed method. Chapter 6 ummarize the reearch activitie in, and contribution made by thi thei, and at lat ugget the poible future reearch topic. Note that, regarding the citation for my publication in thi chapter. My journal paper are denoted by J1, J, J3, and J4. My conference paper are denoted by C1, C, C3 and C4. 18

35 Chapter Determining the Stability Boundary for an SL with EOF Thi chapter preent comprehenive tudie on the tability of a emiconductor laer (SL) with external optical feedback (EOF), where the external target i tationary. In particular, baed on numerical computation on the Lang and Kobayahi (LK) equation, the tability boundary under the condition of minimum linewidth i.e., p arctan( ), where p i an integer, i invetigated, revealing how it i influenced by three major parameter aociated with SL operation condition, including feedback trength ( ), external cavity length ( L ) and the injection current ( J ). In contrat to exiting work in literature, the preented removed all the approximation and aumption, hence reulting in relatively complete decription of the tability boundary. The work preented lead to a number of important and intereting dicoverie, e.g., (1) the poible table area i broader than what i decribed by the exiting work; () for long external cavitie, the tability boundary can be decribed by a linear relationhip between and L ; (3) on the tability boundary, the critical feedback trength ( c ) and critical external cavity length ( L c ) are dicovered repectively being proportional and inverely proportional to J and. 19

36 .1 Determine the Stability Boundary.1.1 Sytem Determinant The tability of a ytem i uually analyzed baed on the ytem determinant. For an SL with EOF, it ytem determinant i obtained baed on analyi of the LK equation near the tationary olution [3, 39, 43, 45, 46, 7-74] decribed by: (.1) 1 in( arctan ) in N N 1 ( )co( ) p in GN(1 E ) (.) N J ( ev ) E 1 ( )co( ) p in (.3) where E, N and repreent the tationary olution of LK equation without ignoring the nonlinear gain for electric field amplitude, carrier denity and angular frequency repectively. Note that, by ignoring the nonlinear term aociated with Et () in Eq. (1.13), i.e.,, the above tationary olution are ame a the one ued in [31, 39, 45, 46, 61], that i Eq.(1.15)-(1.17) in Chapter one of thi thei. Auming that Et (), () t and Nt () exhibit mall deviation from above tationary olution (denoted by () t, () t and () t repectively), we can obtain the E following linear differential equation from the LK Eq. (1.1)-(1.1) in matrix form: N d E () t dt E( t) E( t ) d () t A ( t) B ( t ) dt N ( t) d N () t dt (.4) where

37 1 co( ) GN ( N N) E in( ) E GN (1 E ) E in in 1 A in( ) GN ( N N) E co( ) GN (1 E ) ine in 1 GN ( N N) E(1 E ) GN (1 E ) E co( ) in( ) E in in B in( ) co( ) ine in (.5) (.6) The ytem determinant (denoted by D ()) of Eq.(.4) i derived a: where D( ) det( I A e B) 1 R 1 R 3 = co( )(1 e ) in 1 + (1 e ) ( R 1)co( ) R 1 in( ) (1 e ) R 1 R in in R R 1 + (1 e R in 1 ) ( ) co( ) in( ) (1 e ) R1 R in R (.7) R i the damping time of the relaxation ocillation of an SL with EOF, G E (1 E ), 1 1 R N R 1 R3 GN (1 E ), (1 ) R R3 E and R3 GNG( N, E ) E, where R3 i the relaxation reonance angular frequency of an SL with EOF. A E and N vary with J, o do R, R1 and R. If taking the aumption a decribed in [3, 31, 45, 7-7] and ignoring the nonlinear gain, the ytem determinant D () in Eq. (.7) become the ame a the one given in [3, 31, 45, 71, 7]. Therefore Eq. (.7) provide a more comprehenive decription on the tability property of an SL with EOF ytem..1. Stability Boundary With the ytem determinant in Eq. (.7), we are able to work out the tability boundary baed on the location of zero of Eq. (.7) on the S-plane. The ytem i 1

38 table if all the zero are located on the left hand ide of the S-plane. The zero on the imaginary axi give the tability boundary of the SL with EOF. The zero of D () are defined a the root of D ( ), which are uually complex number, that i, j. In order to work out all the zero, we inert j in Eq. (.7) and rearrange the right hand ide by eparating the real and imaginary part and et them to be zero, yielding two equation a follow: 3 ( ) a1 1 e co( ) a e in( ) b1 1 co( ) b a1 3 ( b1 c1 ) 1 e co( ) e co( ) ( b c) 1 e co( ) b3 (.8) and where ( ) a e in( ) 3 a 1 e co( ) a b 1 e co( ) e co( ) b 1 e co( ) b 1 3 ( b1 c1 ) e in( ) 1 e co( ) b c e in( ) (.9) k 1 a1 co( ) and a R 1 (.1) in R k b1 in k b ( )co( ) in( ), R1 R1 in R and 1 b 3 R1 R R (.11) 1 k c1 R in k 1 (.1) and c ( ) co( ) in( ) R1 R in R It can be een from Eq. (.8) and (.9) that the location of the zero depend on the value of the three parameter,, J and the feedback phae. However, given the complexity of Eq. (.8) and (.9), it i impoible to have analytical olution for the zero. In order to have complete knowledge on the tability of the ytem, we can only utilize numerical computation to work out the zero with repect to all the poible value of the parameter. In thi Chapter, we mainly focu on the influence of, and J, and will dicu the influence of the feedback phae in the following

39 chapter. To achieve numerically calculating the zero, we conidered that the parameter and J take point equally paced within the range [.,.1] and J [17.8 ma, 7.5 ma] repectively. Without lo of generality, we aume that wavelength of the laer i 78nm, and the external cavity length range from to.355m, implying that the external cavity length i wavelength. Hence will fall into the range rad rad 1 1 [,4173 ]. Within thi range, we take point equally paced in uch a way that 4173 m arctan( ), where m=, 1,,, 199. Such a choice of correpond to the minimal linewidth mode (MLM) which wa hown to be the mot table mode comparing to other [48, 7, 74] and the ame choice wa alo made in [31, 43, 48, 71, 7]. Such choice alo lead to rad according to Eq. (1.15). Note that uch a range correpond to [ n,.17 n]. Now, we have a parameter ample pace of combination of the three parameter value. Then the tability boundary i obtained by numerical computation following the procedure below: Step1: Chooe a et of parameter value from the above parameter ample pace. Step: Determine by olving Eq. (.8) and (.9). Note that can exhibit multiple value, and the right-mot one (denoted a ) will be elected a it determine the tability of the ytem. Step3: Repeat Step in an exhautive manner with repect to all poible combination of the parameter. Thi will yield all the zero. Step4: For each of the zero found in Step 3, check the value of. If, record the correponding value for, and J, yielding a point on the tability boundary in the three parameter pace. 3

40 Step 5: All the point found in Step 4 for will build the complete tability boundary in the three parameter pace, and J. With the procedure above, we can work out the tability boundary now. Note that all other parameter take the value in Table 1-1. In order to facilitate the comparion with exiting reult, parameter are caled in the ame way a in [31], where i normalized by multiplying a contant factor ( ), where 14.5GHz correponding to the relaxation ocillation R R frequency of the olitary laer when J Jth 1.3, where J th i the threhold current. The reult of numerical computation are depicted in Fig. -1, howing that the tability boundary i a urface with periodical fluctuation. The area above the urface i the untable region. Figure -1: Stability boundary decribed in the 3-parameter pace of (,, J ). The untable region i above the urface. In order to demontrate the phyical meaning of the reult hown in Fig. -1, we conider the relationhip between and with J a a contant. Chooing J J 1.3, we have Fig. -(a), where the ytem in the haded area i untable. For th comparion, the reult for the ame J J th obtained in [31] i hown in Fig. -(b) where the lahed area are the untable region. By comparing the two figure, we 4

41 can ee that the bottom line of the untable area, that i, the critical feedback trength (denoted a ) in Fig. -(a) ( c c ) i higher than that in Fig. -(b) ( c ). Conidering that the average range of tudied i in the order of 3 1 in exiting work [31, 39, 45, 61, 7-7] for the tability analyi of an SL with EOF, uch a difference i indeed coniderable and hould not be ignored. Figure -: Stability boundary decribed by and for a fixed J Jth 1.3. (a) tability boundary given in thi thei on Fig. -1, (b) tability boundary given in [31]. It hould be noted that the reult hown in Fig. -1 only hold for the cae of minimum linewidth mode. In the ituation where the condition i not met, the laer can till be untable within the tability boundary. Therefore, the boundary hown in Fig. -1 preent the neceary but not ufficient condition for the table operation of a laer. To further verify our reult hown in Fig. -(a), we alo tudied the dynamic of Et () in the time domain uing the 4-th order Runge-Kutta (RK) integration method on LK equation (1.1)-(1.1). We chooe an untable point from Fig. -(b) marked by with the parameter.3, ( ).5 and J J 1.3. Thi point R i determined a table hown in Fig. -(a). Other parameter take the value in Table th 1-1. The intenity of the SL output It () i calculated a I( t) E ( t) / E E where i the tationary olution of the electric filed amplitude for a olitary SL. The 5

42 waveform of It () in Fig. -3 how that a contant It () i achieved after the tranient die away. Hence, the ytem i table at thi choen point. Thi reult coincide the concluion reported in [3, 7] that nonlinear gain enhance the tability of the SL with EOF. Figure -3: Laer intenity waveform of It () for the choen point,.3, ( ).5 and J J 1.3. R th. Influence of External Cavity, Injection Current and Feedback Strength A the location of zero of the ytem determinant change with parameter, J and, it i important to invetigate the influence of, J and repectively on the tability boundary of an SL with EOF ytem. Becaue i directly dependent on the external cavity length L (that i, L c, where c i the peed of light), in the following we replace by it correponding external cavity length L which provide a more informative phyical meaning related to an SL with EOF...1 Influence of External Cavity Length Let u firtly look into the influence of L on the tability boundary. To thi end, we extract the relationhip between J and from Fig. -1 by etting ( ) at R 6

43 ix different value, including 1., 1.5,.,.5, 3. and 3.5. we have L 65 mm, 97 mm, 13 mm, 16 mm, 195 mm, 31mm So, equivalently,. The extracted relationhip are plotted in Fig. -4. Figure -4: Influence of L on the tability decribed by J and when J 13.7mA. From Fig. -4, we are able to oberve the following feature: 1. The tability boundary how a finger tructure underneath the ame aymptote for all the different value of L. A area above the aymptote i guaranteed to be table, a afe choice can be made above the aymptote which give a guideline for the election of feedback level and injection current in order to have a table ytem. For example, in order to have a table ytem, large injection current i required for the cae of high feedback trength.. Not all area below the aymptote are untable. However, with the increaing of L, the number of the finger alo increae, making the untable area to increae a well and tend to fill the area under the aymptote (that i, the haded area in Fig. -4). For the cae of relative long external cavitie (e.g., Fig. -4(f)), the tability boundary i cloe to and can be approximately decribed by the aymptote. However, for the cae of a relative hort cavity, ome area of coniderable ize under the aymptote (e.g., Fig. -4(a)) are till table. th 7

44 .. Influence of Injection Current We alo examine the influence of injection current on the tability boundary. By etting J to ix different value, the relationhip of and can alo be obtained from Fig. 1. In the relationhip, parameter i replaced by L uing L c. So, we have the relationhip between and L hown in Fig. -5 where the haded till denote the untable region. From Fig. -5, we can alo find the following feature of the tability boundary in term of and L. Figure -5: Influence of J on the tability decribed by and L when J 13.7mA. 1. An SL with very hort external cavitie (e.g., L 6.5mm in Fig. -5(b)) can alway endure a very high feedback trength, which i conitent with the reult reported in [31, 74, 75].. The tability boundary alo demontrate a finger tructure. The width of all the finger (denoted a T in Fig. -5) i nearly the ame with repect to L, but it decreae with the increae of J. th 3. The minimum value of appeared in the untable region (ee the horizontal line adhered to the tability boundary in Fig. -5), increae with the increae of J. The value of c wa alo dicued in the previou work [31, 8

45 7, 7, 76]. Figure -6 how the comparion of our reult againt the reult obtained in [31, 7, 76] where the parameter are the ame a hown in Table 1-1. It can be een, the value of c we obtain numerically i higher than thoe preented by [31, 7, 76]. Thi mean that the actual table region i larger than thoe preented by exiting literature. Note that [7] did not dicu the influence of J on c. Figure -6: The relationhip of c and J...3 Influence of Feedback Strength Similarly, we can dicu the influence of on the tability by oberving the relationhip of J and L. The tability decribed by J and L i preented in Fig. -7 with ix different value of. The haded i till the untable region. 9

46 Figure -7: Influence of on the tability decribed by J and L when J 13.7mA. We conclude the tability boundary a follow: 1. A critical injection current ratio denoted by Jc J th (hown a the horizontal line adhered to the tability boundary in Fig. -7) can be ued to determine the tability boundary in the plane of J and L. Obviouly, a high feedback th trength require a high value of J c to reach table region. For the cae with a relative high feedback trength (e.g., Fig. -7(f)), J c can be ued a the tability boundary for deigning a table ytem. However, for the cae with a relative low feedback trength, a more region under the critical J c till howing table character, we need to ue Eq. (.8) and (.9) to determine the tability.. The tability boundary exhibit ditorted finger tructure. More and more finger are oberved with increaing. 3. An SL with a hort external cavity i alway table. Let u denote the critical external cavity a L c. From Fig. -7, we can ee that L c depend on. Figure -8 give the relationhip of and L c obtained from the numerical imulation. Approximately, L c i inverely proportional to the value of, that i, 3

47 L c 4 c 8.1 (.1) (.13) R Figure -8: The relationhip of and L c..3 Summary In thi Chapter, a comprehenive analyi on the tability boundary of a ingle mode SL with a feedback phae condition p arctan( ) i preented, where p i an integer, correponding to the minimum linewidth mode. The work tart from the Lang and Kobayahi (LK) equation, following by the derivation of a ytem determinant. An effective numerical computation i then employed to analyze the location of zeroe of the ytem determinant, yielding the tability boundary with repect to three major parameter aociated with SL operation condition, including feedback trength ( ), external cavity length ( L ) and the injection current ( J ) to the SL. The reult preented in thi paper provide helpful guidance for aeing the tability of an SL with EOF, which i important for deigning SL ytem in variou application of ening and intrumentation. It hould be tated again that the tability boundarie obtained are ubject to the condition of minimum linewidth, and hence the boundarie are only able to ditinguih poible table area and untable 31

48 area. In other word, laer operating in the area within the boundarie can be table if the laer operate in the minimum width mode, and they will not be table if operating in area outide the limit. The influence of the other parameter, i.e., the feedback phae will be dicued in the following Chapter which will provide a detailed tability analyi for the SMI. 3

49 Chapter 3 Stability Analyi for an SMI A mentioned in Chapter one, when the ytem enter into the untable tate, the premie for deriving the tationary olution of the Lang and Kobayahi (LK) equation will be no longer valid, thu leading to the actual behavior of the ytem can not be decribed by the exiting SMI model. A the SMI can be conidered a an SL with a time varying EOF, or in other word, with a time varying feedback phae, in thi chapter, the tability boundary of an SMI i obtained and preented in a two dimenional plane defined by C and in order to have a clear and intuitive decription of the tability of the SMI. By tudying the feature of the boundary, a critical C (denoted a C critical ) i derived. If only an SMI i deigned with a feedback level below C critical, it ening performance can be guaranteed and the behavior of the ytem can be decribed by the exiting SMI model, otherwie by the LK model. An experimental method for determining the C critical i preented. The influence of the initial external cavity length L and the injection current J on the C critical are invetigated from both imulation and experiment which how that tability can be enhanced by increaing either L or J. Furthermore, three region on the plane of ( C, ) are propoed to characterize the behavior of an SMI, including table, emi-table and untable region. We found that the exiting SMI 33

50 model i only valid for the table region, and the emi-table region ha potential application on ening and meaurement but need re-modeling the ytem by conidering the bandwidth of the detection component. 3.1 The Stability Boundary Decribed by C and The invetigation of the tability for the SL with EOF ytem i uually baed on the ytem determinant which i obtained baed on the linearization of the LK equation near the tationary olution [3, 31, 45, 46]. For a fixed et of operational parameter, the ytem i table if all the zero of the ytem determinant are located on the left hand ide of the S-plane. In thi ituation, the correponding tationary olution are alo table. However, for the cae of an SMI ytem, a L varie with time, according to Eq. (1.18) and (1.19), alo varie with, thu leading to the variation of all the tationary olution. Therefore, in order to make ure the SMI ytem i table, it require that all the tationary olution are table during the variation of L. To better illutrate thi ituation, we recall the mechanim of the SMI decribed in Chapter 1 and preent an example in Fig. 3-1 here which how the relationhip between and according to Eq. (1.19), where C 3 and 3. The actual behavior of for a table SMI ignal ha been invetigated in [35]. It how that will vary along the route of A1 B B1 D D1 when increae, and it will however track the route of D1 C C1 A A1 when decreae. Therefore, in thi cae, to generate a table SMI ignal demand that all the tationary olution falling within the range of,, and, A1, B,, A, D,, C, D1 are table during the variation of. Note that, the tationary olution within the range of, A,, B and, C,, D are alway untable according to [31, 38, 39]. Thee untable tationary olution atify the following equation: 34

51 1 C co arctan( ) (3.1) Figure 3-1: The relationhip between and A mentioned above, the tability of a ytem i uually analyzed baed on the ytem determinant. Baed on the ytem determinant, [31, 45] howed that, to a good approximation, the table condition of the ytem a well a of the tationary olution atifie the following condition:. in( ) co( ) 1 in R R Rin ( ) (3.) for all the value of ( i defined a the imaginary part for a complex number in Laplace tranform domain. The detail can be found in [31, 45]), atifying: where R cot( ) (3.3) R R GN E p, 1 1 p R R GN (3.4) 35

52 In Eq. (3.3) and (3.4), E i the tationary electric field amplitude of the olitary laer which i determined by the injection current J [45]. R and R are called the relaxation ocillation frequency and the damping time of the olitary laer [45]. When deigning a table SMI, it i important to know how to configure the ytem in term of a proper feedback level and uitable movement range for the external target (or the feedback phae ). That i, we need to know the table boundary for the parameter C and. Hence, we propoe to decribe the tability of an SMI ytem in the plane of ( C, ). To achieve thi, let u replace by C (via C 1 ) in the table condition decribed by Eq. (3.). Note that the in amount of movement of target hould be much maller than the initial external cavity length. Then, equation (3.) can be written a: 1 C co( ) 1 in( ) R R Rin ( ) (3.5) where the equal ign correpond to the condition of tability boundary. What we want i to work out the relationhip between C and to decribe the tability for the SMI. Let u conider the parameter appeared in Eq. (3.5). Generally, i treated a a contant with the value from 3 to 6 [6, 1]. of, which can be determined by Eq. (1.19). Both i the dependent variable R and R are dependent on the injection current J according to Eq. (3.4) via E [31, 45]. i determined by J and according to Eq. (3.3). So we can ay, in Eq. (3.5) and (3.3), J and are two governing parameter that determine the tability boundary of an SMI ytem decribed by C and. Therefore, it i very important to invetigate how the two parameter influence the tability boundary. 36

53 In order to work out the tability boundarie in the plane of ( C, ), a pecific example i conidered here to demontrate how to determine the table range of the SMI ytem uing Eq. (3.5). We aume that the external target i located L.5m away from the SL which i injected with a fixed current of 17.8mA (correpond to 1.1J th ). The vibration amplitude of the target i L 1.17m (correpond to an amplitude of 1.5, where i the wavelength of the laer). With thee aumption, the variation range of will be in 6 6 [( ), ( )]n, and in 6 6 [( ), ( )]rad according to Eq. (1.18). The variation range of C i et to be [, 6]. The computation for the tability of the SMI ytem i baed on the above mentioned ituation. In the range of and C, we take 4 equally paced ample. The value for all the other parameter ued during the computation are hown in Table 1-1. The procedure performed for determining the tability of the SMI ytem are hown a follow: Step1: Chooe a et of parameter value from the above and C pace. Step: Determine by olving Eq. (1.19). Note that there can be more than one, and we dicard the one atifying Eq. (3.1). Step3: Determine by olving Eq. (3.3). Note that there can be more than one, and we chooe the one cloet to R [31]. Step4: Chooe one of the determined in Step. Step5: Subtitute C,, and into Eq. (3.5), tet if Eq. (3.5) i atified. Step6: Go back to Step 4 until all the are choen. 37

54 Step7: If all the determined in Step atify Eq. (3.5) for, C, and, record the correponding value for and C, yielding a point on the tability boundary in the two dimenional plane of and C. The above procedure will be repeated in an exhautive manner with repect to all poible combination of and C. Thu the final tability boundary can be contructed in a two dimenional plane of and C. Figure 3- how the recontructed reult. Furthermore, the reult of the tability boundarie can be obtained under different value for the parameter pair of J and. A i aociated to L via L c, intead of, we preent the influence of L on the tability boundary in order to provide a more informative phyical meaning related to the SMI. In Fig. 3-(a), the boundary i computed with three different J for a fixed L.5m. In Fig. 3-(b), the boundary i computed with three different L for a fixed J 1.3J th, where J th i the threhold injection current. The area below each boundary i the table region. In Fig. 3-, we alo indicate the different feedback regime defined by the value of C, where weak feedback regime i for C 1, moderate feedback regime for 1C 4.6 and trong feedback regime for C 4.6 [35]. Figure 3-: Influence of J and L on the tability boundary of an SMI. (a) for a fixed L.5m with different J, (b) for a fixed J 1.3J th with different L. 38

55 C From Fig. 3-, the following feature of the tability boundary can be found: 1. The tability boundary how periodic fluctuation with a period of equivalent to a half wavelength movement of the external cavity.. The ytem i alway table at a weak feedback regime and may enter untable when the feedback level i moderate or high feedback regime. 3. To achieve a table tatu at a moderate or high feedback regime, we can either increae the injection current or chooe a long external cavity. Figure 3-3 how a boundary when J 1.1J th and L.35m. In Fig. 3-3, we define three different region referred to a table, emi-table and untable repectively according to the dynamic behavior of an SMI decribed a below. A the exiting SMI decribed in Eq. (1.19) and (1.) i not able to decribe the actual behavior of an SMI when the ytem enter the region above the tability boundary, we need to tart from LK equation to invetigate the output power of an SMI, i.e., E () t. The calculation of E () t by the LK equation ue the 4-th order Runge-Kutta integration algorithm. 1 Untable region 8 Semi-table region C= Stable region C critical C= / Figure 3-3: The tability boundary of an SMI when J 1.1J th and L.35m. Now, let u tudy the feature of the SL output power (below we will call it a an SMI ignal) obtained by the LK model at the different region defined in Fig We 39

56 chooe C 1.5, C.5, C 4. and C 9. which repectively correpond to the table, emi-table, emi-table and untable region. Other parameter in the LK model take the value hown in Table 1-1. The SMI ignal gt () i calculated a the normalized E () t. Suppoing that the external target move at a inuoidal law with ame vibration trace of Fig. 1-4(a), for the purpoe of comparion, Fig. 3-4 preent the SMI ignal predicted repectively by the LK model hown from Fig. 3-4(b)-(e) and the exiting SMI model from Fig. 3-4(g)-(j). In each row of Fig. 3-4, the two SMI ignal are obtained under the ame operation condition, i.e., the ame C value. Figure 3-4: SMI ignal predicated by the LK model and the exiting SMI model repectively. (a) and (f): movement trace of the external target, (b)-(e): SMI ignal obtained by the LK model with C 1.5, C.5, C 4. and C 9. repectively. (g)-(j) SMI ignal obtained by the exiting SMI model with C 1.5, C.5, C 4. and C 9. repectively. According to the LK model, obviouly, only Fig. 3-4(b) with C 1.5 how a table SMI ignal which can alo be decribed by uing the exiting SMI model hown in Fig. 3-4(g). C 1.5 indicate the SMI i table in Fig In the region with 4

57 1.8 C 8.4, imulation uing the LK model how that the SMI ignal contain a high frequency ocillation cloe to the relaxation ocillation frequency of the olitary laer. Figure 3-4(c) and 3-4(d) give the two SMI ignal at the emi-table region, which are more complicated than Fig. 3-4(b). Hence, the behavior decribed by the LK model are different from the one by the exiting SMI model reulting from the tationary olution of the LK model. It i very intereting to oberve that, even for a complicated waveform hown in Fig. 3-4(c) and 3-4(d), the movement information of the target i till viible. Thi i why we call the region 1.8 C 8.4 a the emi-table region. With the aid of ignal proceing technology, the ytem operating at the emi-table region can alo be ued for ening and meaurement. In order to achieve thi, the SMI waveform need to be invetigated to reveal it relationhip to the movement of the target. Alo, due to the limit in the riing time of the photodiode (PD) packaged at the rear of the SL, it may not be able to detect the detail of the high frequency SMI waveform in the emi-table region, and the SMI ignal oberved will be a ditorted verion of the high frequency waveform. A complete theoretical model i required to decribe the influence of the limited bandwidth of the PD on the high frequency SMI waveform with the aim to detect the movement of the target from the ditorted SMI waveform. Obviouly, extenive work i required and could be an intereting topic for future reearch. When C 8.4, it i hard to ee the vibration information from the SMI waveform, implying that the SMI ytem may loe it ening ability. In thi ituation, the pectrum of laer i dramatically broadened, which i beyond the cope of thi paper. Fig. 3-4(e) how the SMI ignal with C 9 indicating that the SMI ytem i not uitable for ening application. Note that the SMI model i derived from the LK equation by letting de( t) dt, d() t dt and dn( t) dt. Thee condition will no longer be valid when the ytem enter emi-table or untable region, e.g., the relaxation ocillation will become undamped [77, 78]. In ummary, for the ytem working in the emi-table 41

58 or untable region, the exiting SMI model cannot be ued, but we can till ue the fundamental LK model to decribe the ytem behaviour. 3. Critical Feedback Level Factor Furthermore, from the tability boundary hown in Fig. 3- and 3-3, We noticed that from the tability boundary a critical C (denoted by C critical ) can be defined under which the ytem i guaranteed to be table. A C critical correpond to the bottom on the tability boundary, by performing differentiation with repect to on both ide of Eq. (3.5), we can obtain for C C critical a follow: arctan p R 1 (3.6) where p denote an integer. A C critical lie on the tability boundary decribed by Eq. (3.5), inerting Eq. (3.6) into Eq. (3.5), thu we can obtain: C critical L 1 1 R in R Rc L c R 1 (3.7) for all the value of atifying Eq. (3.3). Equation (3.7) can be ued to etimate C critical when deigning an SMI ytem if the value of the parameter lited in Table 1-1 are available. From Eq. (3.7), we can ee that there are three parameter influencing C critical, i.e.,, L (via and ) and J (via R and R ). Here, we mainly focu on the influence of L and J. Uing Eq. (3.7), we are able to determine tability boundary for a SMI in a 3-parameter pace with repect to C critical, L and J. To determine the boundary, the following computation procedure have been performed: 4

59 Step1: Set L [.5,.4] m and J [17.8, 37.8] ma, both with equally paced ample. All the other parameter are kept a the ame a et previouly. Note that, in thi thei, we only focu the range of L in long external cavity range. An external cavity i aid to be hort when L c ( R ), a generally aumed in literature [63, 67, 7, 74, 75]. Step: Chooe a et of parameter value from L and J. Step3: Solve Eq. (3.3) for and then calculate C critical uing Eq. (3.7). Note that there can be more than one, and we chooe the one cloet to R [31]. Step4: Record the value of C critical and correponding L and J, yielding a point on the tability boundary in a three dimenional pace. Step5: Repeat Step until all the combination of L and J are choen. Figure 3-5 how the complete tability boundary contructed uing the above computation procedure. Figure 3-5: Stability boundary decribed in the 3-parameter pace of ( C, J, L ). The untable region i above the urface. From Fig. 3-5, we can ee that for a fixed external cavity length L, the boundary how a riing finger-tructure. The number of finger increae with the increae of L. The feature of the boundary for a fixed injection current J are imilar. Figure critical 43

60 3-6 how an example of the tability boundary in the plane of J and C critical extracted from Fig. 3-5 when L.5 m,.1 m,.15 m,. m,.5 m,.3m. Figure 3-6: The tability boundary decribed by J and C critical for long external cavitie. The haded area in Fig.3-6 i the untable area. The area beneath the thin line hown in Fig. 3-6 i guaranteed to be table. Therefore a afe choice can be made under the thin line. From Fig. 3-6, it i intereting to notice that with the increae of L, the untable area tend to fill the area above the thin line. Meanwhile, the tability boundary alo tend to be decribed by the thin line. For a relatively long external cavity (i.e., L c ( R) ), R [31]. To a good approximation, we ubtitute R into Eq. (3.3) and (3.7), and Eq. (3.7) reduce to the ame form a derived in [31, 76]: C critical L c R (3.8) In order to have more inight into C critical for long external cavitie, in Eq. (3.8), we replace R uing Eq. (3.4). By doing o, Eq. (3.8) become: C L NG J N p 1 1 critical GN p c ev GN p (3.9) 44

61 which i the equation for the thin line hown in Fig Therefore, a afe choice for obtaining a table SMI ignal can be made uing (3.9) intead of Eq. (3.7). 3.3 Determining Critical Feedback Level Factor In thi ection, we preent a experimental method to determine C critical, and invetigate how L and J influence SMI etup for uch invetigation. C critical. Figure 3-7 how the experimental A.8 m band ingle mode GaAlA triple quantum well SL (HL835G) from Hitachi wa employed in the experiment. The temperature of the SL wa tabilized to within.1 C by uing a temperature controller (TC) (model TED). The injection current to the SL wa controlled by an SL controller (LC) (model LDC). The light emitted from the SL wa focued by a len and plit into three beam by a beam plitter (BS). One beam wa directed to the external target and then wa reflected back to the SL internal cavity. The econd beam wa collected by an optical pectrum analyzer (OSA) (Advantet Q8347) for monitoring the optical pectrum of the SMI. The third beam wa paed to a fat external photodiode (EPD) (PDA8GS which i provided by the Thorlab) with an 9.5GHz bandwidth in order to detect the dynamic of the laer intenity. The detected laer intenity i then tranferred into a real time ocillocope (Tektronix DSA784) with an 8GHz bandwidth and 1.5GS/ ampling rate which i high enough to capture the time erie of the laer intenity. A piece of mirror wa attached on the urface of a loudpeaker o that to achieve a trong optical feedback level. The loudpeaker wa driven by a inuoidal ignal with 75Hz generated by a ignal generator. The optical feedback level of the SMI ytem wa adjuted by an attenuator inerted in between the BS and the loudpeaker. Note that the SMI ignal can alo be detected by the PD packaged at the rear of the SL and the acquired by our deigned detection circuit. 45

62 Figure 3-7: Experimental etup for invetigating the tability of the SMI. When the SMI i in the table region, the oberved optical pectrum i clean howing the SL operating on only one ingle mode a hown in Fig. 3-8(a). When the ytem enter into emi-table region, the relaxation ocillation (RO) of the laer become undamped. In thi cae, a ubpeak correponding to the RO frequency appear near to the main peak of the optical pectrum [77, 78]. Figure 3-8(b) how the optical pectrum oberved when the ytem in the emi-table region. A our pectrum analyzer ha a relative low reolution with.nm, it i not able to eparate clearly the ubpeak from the mean peak. However, it can till tell u the appearance of the RO of the laer with frequency about -4GHz, therefore determining the tability of the ytem change from table to emi-table. One may argue that thee ubpeak may be related to the external cavity mode. However, we can confirm that the ubpeak in Fig. 3-8(b) correpond to the RO frequency by the following calculation (The principle for identifying the RO frequency i baed on Fig.3() in [77]). A an example, we conider the cae with C =4.6 to 7, which correpond to the trong feedback regime. In thi cae, there are 5 external cavity mode according to the phae equation but only 3 of them are poible ocillating mode (the other are anti-mode [31, 73, 79]). The frequency interval between two adjacent mode i f c L [77]. For the cae of Fig. 3-8(b), L.5m, thu f 6MHz. Therefore, the ditance between the main peak and it adjacent ocillating external cavity mode i 46

63 f 1.GHz. However, from Fig. 3-8(b), we can ee that the right ubpeak i around nm which correpond to the optical frequency f1 c GHz. The main peak i nm which correpond to the frequency f GHz. Therefore, the frequency interval i f f f1 3.94GHz. Obviouly according to the above calculation, the location of the ubpeak on Fig. 3-8(b) doe not correpond to the external cavity mode which ditance to the main peak i 1.GHz. Figure 3-8: Two optical pectra obtained with L.5m and J 1.7J th for (a) the table region, (b) the emi-table region. In the following experiment, we varied the feedback level from weak to trong with the aid of the attenuator, the ingle mode pectrum diplayed on the pectrum analyzer will thu change. Once the ubpeak were firt oberved from the pectrum, the SMI hould be at the point of the critically table. Then, we apply a tiny change to the attenuator by reverely rotating it degree. A table SMI ignal very cloe to the critical level can thu be obtained and we ued the ignal to calculate the parameter C by the method preented in [8]. The C calculated i approximately repreented for C critical. Baed on above experimental method for etimating C critical, the influence of J and L repectively on the C critical are alo invetigated. Figure 3-9 (a) how the 47

64 C goe up with the increae of the injection current for a fixed L.5m. critical Figure 3-9 (b) how the longer the external cavity the higher C critical for a fixed J 1.3J th. Obviouly, the experimental reult how the ame trend with the imulation analyi hown in Fig. 3-, that i, C critical can be increaed by either increae of J or L, thu leading to tability enhancement of the SMI. We note that the experimental reult obtained do not exactly agree with imulation. The reaon i that the actual value of the internal parameter for the SL ued in the experiment are different from the parameter hown in Table 1-1 for the imulation. Figure 3-9: Experimental reult with the aid of the OSA. (a) for a fixed L.5m, (b) for a fixed J Jth 1.3. Furthermore, we oberve the dynamic of the SMI in the three different region, i.e., table, emi-table and untable, uing the ocillocope in order to verify the reult hown in the left column of Fig The external target i placed.35m away from the SL and the injection current i fixed a J 1.1J th. By increaing the feedback level with the aid of the attenuator, we record the time erie of the laer intenity in Fig From Fig. 3-1, we can ee that the waveform of the SMI ignal in emi-table region i quite imilar to the imulation reult obtained uing the LK equation, which contain high frequency component cloe to the relaxation ocillation frequency of the SL. Note that we are unable to oberve the SMI ignal in 48

65 SMI ignal [1mV/div] SMI ignal [1mV/div] SMI ignal [1mV/div] SMI ignal [1mV/div] Target Movement [V/div] the table region due to low SNR of the ytem, and thu unable to determine the onet of emi-table region uing the ocillocope. (a) (b) (c) (d) (e) Figure 3-1: Experimental obervation of the dynamic of the SMI ignal uing the fat ocillocope with an 8GHz bandwidth and 1.5GS/ ampling rate. (a) the target movement trace (b) the SMI ignal in table region (c)-(d) the SMI ignal in emi-table region (e) the SMI ignal in untable region t [] Baed on the reult obtained in Fig. 3-1 (a)-(e), we alo imultaneouly recorded the SMI ignal uing the detection circuit which bandwidth i 3MHz. Figure 3-11 how the SMI ignal correpond to the cae hown in Fig. 3-1 repectively. 49

66 SMI ignal [1mV/div] SMI ignal [1mV/div] SMI ignal [1mV/div] SMI ignal [1mV/div] Target Movement [V/div] (a) (b) (c) (d) (e) Figure 3-11: Experimental obervation of the dynamic of the SMI ignal uing the detection circuit which bandwidth i 3MHz. (a)-(e) correpond to the cae hown in Fig. 3-11(a)-(e) repectively t [] From Fig (b)-(e), it i intereting to notice that the ignal oberved by the data acquiition unit in the three different region all look like the table SMI ignal decribed by the exiting SMI model. However, it hould be pointed out that the bandwidth of the PD packaged at the rear of the SL i not a high a the EPD, thu leading to the diappearance of the high frequency component of the SMI ignal hown in Fig. 3-1(b) to (e) and giving the ditorted SMI ignal or ay untrue SMI ignal hown in Fig. 3-11(b)-(e) which will negatively affect the ening and meaurement performance of the SMI, e.g., SMI baed meaurement. Therefore, a we mentioned before in Section 3.1.3, a complete theoretical model i required to decribe the influence of the limited bandwidth of the PD a well a the detection 5

67 circuit on the high frequency SMI waveform with the aim to achieve accurate ening and meaurement reult. 3.4 Summary The tability of an SMI i invetigated in thi Chapter. It i found that, to achieve a table SMI ignal for ening purpoe under moderate or trong feedback level, we can either increae the initial external cavity length or the injection current to the laer. By monitoring the pectrum of the SMI, a critical optical feedback factor C critical can be determined approximately. Under the C critical, an SMI i guaranteed to be table and the exiting SMI model can exactly decribe the waveform of an SMI ignal. Furthermore, we preented another two region on the plane of ( C, ) called emi-table and untable with boundarie correponding to the undamped relaxation ocillation and the chao tatu repectively. We found that emi-table region ha potential application on ening and meaurement but may require further ignal proceing technology. The reult preented in thi Chapter provide ueful guidance for deigning variou SMI baed ening and intrumentation. 51

68 Chapter 4 Influence of the Injection Current on the Stability of an SMI In thi Chapter, the influence of another SMI parameter, i.e., the injection current, on the tability of an SMI i both numerically and experimentally explore. In fact, the idea of work preented in the chapter i originally inpired by the two recently publihed paper [61] and [4]. In [61], Green found that, with the variation of within the range of, the ytem i alway table for ome value of the injection current J (or called band table phenomenon) when L i hort and i weak, but no explanation wa given in [61]. Later in 13, [4] predicted that the alway table region found in [61] i due to the interaction between the excitation of relaxation ocillation (RO) of the laer and L, and dicovered that the region exit when the product of RO frequency ( v RO ) and equal an integer. However, we notice that both [61] and [4] ignore the effect of nonlinear gain which i very important for decribing and modelling an SL, and can provide a good agreement between numerical reult and experimental reult. It i alo well known that the nonlinear gain ha the effect of tabilizing the dynamic of an SL with an external target. 5

69 The reult preented in thi chapter how that the table region i ignificantly wider than the region predicted previouly, and uch phenomenon i caued by the nonlinear gain inherently exited in the laer. Furthermore, the relationhip between the critical injection trength ( J c ), above which guarantee a table SMI, and the other two important ytem parameter, i.e., C and L, i invetigated. The reult preented in thi chapter are alo ueful guidance for deigning a feedback phae independent table emiconductor laer with optical feedback. 4.1 Stability Boundary Decribed by J and In thi chapter,, beginning with a reviit of the reult obtained in [4], we perform the imulation with the ame parameter value adopted in [4] by uing the complete ytem determinant derived in Chapter of thi thei, where the nonlinear gain wa included. Figure 4-1 (a) preent the reult obtained in [4] which doe not conider the nonlinear gain (ee Fig. 1 in [4]). In Fig. 4-1(a), the gray region i the alway table operation region for an SMI ytem where vro 1,,..., and the dark haded region i the untable region. The vertical axi P i the injection trength which i defined a P ( J J ) J [4]. From Fig. 4-1(a), we can ee that to achieve a th th table SMI ytem, we need to carefully chooe P to meet the condition of v 1,,.... RO Figure 4-1 (b) how the tability region obtained uing the complete ytem determinant derived in Chapter by conidering the nonlinear gain. Clearly, comparing with Fig. 4-1(a), the untable region i ignificantly uppreed, and the critical pump trength P c (dotted line in Fig. 4-1(b)), above which guarantee an table SMI ytem, i lightly lower than the condition obtained in [4], i.e., vro 1. 53

70 Figure 4-1: Stability region of an SMI ytem (or band table phenomenon) when C.84 and L 4.5cm. (a) tability region obtained in [4], (b) tability region obtained in thi paper uing the complete ytem determinant derived in Chapter of thi thei. To verify to our reult, i.e., Fig. 4-1 (b), we intentionally chooe three value of P on Fig. 4-1 (b), repectively are P, P.8, and P.1, which correpond to the cae that SMI i untable, table and untable in Fig. 4-1(a). Uing thee three value of P and keeping all the other parameter a the ame a ued in Fig. 4-1, we numerically olve the LK equation by letting the feedback phae varying a the ame trace a hown in Fig. 1-4(a), and Fig. 4- how the imulation reult. From Fig. 4-, we can ee that when P, the SMI i table which prove that our reult i correct. With the reult hown in Fig. 4-1(b), it i naturally to think about if and how P c i aociated with other two ytem parameter, i.e., C and L, becaue, for an SMI ytem, it i uually preferred to be operated under a relatively high feedback trength and long external cavity, i.e., C 1 and L c ( v RO ) [67]. However, due to the complexity of the ytem determinant i.e., Eq. (.7), it i difficult to derive the analytic expreion of P c. We have to numerically olve Eq. (.7) to ee how P c varie with C and L, which will be preented in Section

71 Figure 4-: SMI ignal correpond to the three value of P on Fig. 4-1(b). (a) the target movement trace, (b) for P.1, (c) for P.8, (d) for P. From the previou comparion hown in Fig. 4-1 a well a the reult in Fig. -, we can ee that the nonlinear gain play an important role in decribing the dynamic of the SMI, and it can greatly uppre the untable region of the SMI. Therefore, uch important factor hould not be ignored and hould be paid attention. To illutrate the importance of the nonlinear gain, we here invetigate how the nonlinear gain will influence the tability of the SL with EOF and thu revealing it importance. Let u firtly review ome background knowledge of how to determine to tability of the ytem a dicued in Chapter. With the ytem determinant in Eq. (.7), we are able to work out the location of zero of D () on the S-plane. The zero of D () are defined a the root of D ( ), which are uually complex number and can be found uing variou technique. Note that for a fixed et of parameter, Equation (.7) ha multiple root. Among the root, the right mot root (denoted a ) can be ued to determine the tability of the ytem. The ytem i table if lie in the left ide of the S-plane. The tability limit i reached when i on the imaginary axi. Figure 4-3(a) how the locu of by varying the nonlinear gain confinement factor when R ( ).5, J Jth 1.3, arctan( ) and.5. 55

72 Figure 4-3: The locu of on the S-plane by varying when.1n, J J 1.3 and.5, (b)-(d) are the correponding normalized laer intenity for 1.5,.8 and.5 repectively. The variation range of i from. to 1.5 with 31 ample. All the other parameter are adopted from Table 1-1. From Fig. 4-3(a), we can ee that from.5 to.8, the locu of move horizontally from the right half ide of the S-plane to the imaginary axi, which mean the variation of in thi range doe not change the RO frequency. Keep increaing from.8 to 1.5, move to the left half ide of the S-plane and the RO become damped, therefore leading to a table laer output. Figure 4-3(b)-(d) how the correponding normalized laer intenity output obtained by numerically olving LK equation for 1.5,.8 and.5 repectively, which verifie the correctne of the ytem determinant we derived. The method for olving LK equation i till the 4th order Runge-Kutta integration method. The intenity of the SL output It () i calculated a th the ame hown in Chapter, i.e., I( t) E ( t) / E. From Fig. 4-3, we can ee that 56

73 the nonlinear gain doe play an important role in determining the tability of the ytem. With the locu of, we are able to preent the tability limit in a two dimenional plane of and (hown a in Fig. 4-4) by oberving the locu of for two different value of, i.e., R ( ).3,.6, when J Jth 1.3. The haded area in Fig. 4-4 i the untable region. The variation range for and are choen a [.,.15], [., 1.]. For each of the parameter, we take ample by equally-paced ampling. From Fig.4-4, we can ee that, with the increae of, the value of for guaranteeing a table ytem alo increae. Thi phenomenon jut coincide with the reult how in the previou chapter a well a in [7]. When increae, the relationhip of and change from linear to ditorted while keep the ize of the untable region almot unchanged. Figure 4-4: The tability limit of and for different value of when J Jth 1.3 (a) R ( ).3, (b) R ( ) Critical Injection Current In thi ection, we firtly invetigate the band table phenomena for the cae with a fixed initial external cavity length. Figure 4-5 how the tability region on the plane of ( P, ), which i obtained by uing D () derived in Chapter with ix different 57

74 value of C but for a fixed L.15m. Note that, a i directly related to the movement of the external target for an SMI ening ytem, we are more intereted in how influence the tability. Hence we chooe to ue to repreent the x-axi in the plane rather than according to Eq. (1.19). From Fig. 4-5, it can be een that the band table phenomena occur when the ytem operate at moderate or high feedback level (e.g., C 1). Let call the band table related region a the feedback-phae-dependent table area. Obviouly, uch region occupie more and more pace in the plane of ( P, ) with the increae of C. Moreover, comparing with Fig. 4-1(a), the untable region i ignificantly uppreed due to the incluion of the nonlinear gain in D () preented in chapter. Figure 4-5: Influence of C on the tability region of an SMI with L.15m. Then, we tudy the cae with a fixed C but varying L. Figure 4-6 how the reult for ix different value of L when C i fixed to a moderate feedback level ( C ). From Fig. 4-6, we can ee that the band table related region can be compreed to a mall range of P with the increae of L. 58

75 Figure 4-6: Influence of L on the tability region of an SMI with C. Figure 4-5 and 4-6 alo indicate that there i a relative large table region located above the band table related region when the ytem operate at a moderate feedback level and with a relatively long external cavity. Hence, it i ignificant to determine the critical injection current P o that an SMI can operate in a full table region without the influence of the feedback phae. Apparently, the operation parameter C and L will determine P c. We made intenive calculation by varying the feedback level from weak ( C.5 ) to moderate ( C 4.5 ) and the cavity length from L.1m to L.4m, from which the relationhip between P c and C, and P c and L are obtained and hown in Fig From Fig. 4-7(a), we can ee that P c goe up with the increae of C. Thi mean, to achieve a table SMI at a high feedback level, a large injection current i required. Fig. 4-7 (b) how that P c goe down with the increae of L. Thi reult indicate u that we can have a table SMI ytem operate with a low injection current when the external cavity i long. 59

76 Figure 4-7: Simulation reult of the (a) relationhip between P and C when L.15m, and c (b) relationhip between P c and L when C. 4.3 Determining Critical Injection Current We make the following experiment to verify the imulation reult hown in Fig The experiment are carried out with the etup hown in Fig The SL ued in the experiment i till HL835G and it temperature i maintained at 5.1 C with Jth 4mA and maximum operating injection current of Jmax 1mA. The laer focued by a len hit a mirror urface glued on a loudpeaker. The loudpeaker i driven by a inuoidal ignal with Hz and peak-peak voltage of 4mV. Thi harmonic vibrating target can caue the feedback phae varying from to. The tability of the ytem i monitored by an optical pectrum analyzer (OSA). The detail approach for monitoring the tability can refer to Section 3. in thi thei. An attenuator i inerted in between the beam plitter (BS) and the loudpeaker in order to adjut the feedback level and thu C, and a tranlation tage i ued to hold the loudpeaker and vary the external cavity length L. The firt group of experiment for invetigating the relationhip between P c and C for a fixed L.15m follow the procedure below: Step1: Set the injection current to a high value J 1mA, that i P

77 Step: While keeping the SMI ytem table, adjut the attenuator o that C reache 4.5 (note that the value of C can be obtained from the waveform of a SMI ignal uing the method reported in [17, 35]). Step3: While keeping C unchanged baed on the oberved SMI ignal waveform, gradually decreae the injection current until the ytem jut enter the critical table tate, then record the injection current at thi moment, P c. Step4: Adjut the attenuator to decreae the C by a tep of about., and then repeat the tep from 1-3. With the procedure above, we are able to obtain a number of pair of P c and C. The reult are plotted in Fig. 4-8(a). In a imilar way we can get the relationhip between P c and L by fixing C and varying L from.1m to.4m. The reult are plotted in Fig. 4-8(b). It can be een that the experimental reult how the imilar relationhip preented in Fig Figure 4-8: Experimental reult of the (a) relationhip between P and C when L.15m, c and (b) relationhip between P c and L when C. 4.4 Summary In thi chapter, we invetigated the influence of the injection current on the tability of an SMI with a moving external cavity. Numerical calculation on the ytem determinant of the LK equation are performed. We found that the relaxation 61

78 ocillation ha a trong dependence on the feedback phae when the injection current i low and the external cavity i hort, epecially for the cae when an SMI i operated at moderate or high feedback cae. Both imulation and experiment how that there i a critical injection current P c above which the SMI can be feedback-phae-independent table. Thi critical P c i determined by C and L. The reult and the method preented in thi letter are helpful for deigning a table SMI ening ytem. 6

79 Chapter 5 Deigning a Stable SMI for Meauring the Linewidth Enhancement Factor It i well known that emiconductor laer (SL) play a key role in the emerging field of optoelectronic, uch a optical enor, optical communication and optical dic ytem. For thee application the linewidth enhancement factor (LEF), alo known a the alpha factor or -parameter, i a fundamental decriptive parameter of the SL that decribe the characteritic of SL, uch a the pectral effect, the modulation repone, the injection locking and the repone to the external optical feedback [4, 6]. Therefore, the knowledge of the value of the LEF i of great importance for SL baed application. It ha been proved that LEF exhibit a trong dependence on the combination of the refractive index, gain G and the injected carrier denity N j, and i defined by the following equation [7, 8-8]: R N I c G N j j (5.1) where i the complex electric uceptibility, upercript R and I denote the real and imaginary part of, and and c are the angular optical frequency of the SL and the peed of light repectively. 63

80 Over the pat three decade, variou technique were developed for meauring the alpha factor. Thee technique can be mainly claified [8] a: 1) the linewidth meaurement, ) the current modulation, 3) the optical injection and 4) the optical feedback technique. Uing thee technique, different type of SL were teted for the alpha value, which are ummarized a following: 1) Uing the linewidth meaurement technique: The alpha factor value from 3. to 5.3 were obtained by [8, 83] for vertical cavity urface emitting laer (VCSEL). For ditributed feedback (DFB) laer, the alpha factor wa meaured a. and 5.4 by [84] and [85] repectively. Higher value of the alpha factor were found by [4, 6] for Fabry-Perot (FP) laer which vary from 4.6 to 8.. Beide the above baic laer tructure, the value of 3.9 and 4.4 were reported repectively by [86] and [37] for channeled ubtrate planar (CSP) laer. ) Uing the current modulation technique: The alpha factor value of.5 for a VCSEL wa reported by [87]. A wide range of the alpha factor value from 1.8~6.5 for DFB laer wa reported by [88-94]. For CSP, mea trip and buried optical guide laer, the value of 1.3~. [95], [96] and [97] were obtained repectively. 3) Uing the optical injection technique: For DFB laer, the value of for the alpha factor wa obtained by [98]. In 3, alo for DFB laer, [99] calculated the value of the alpha factor a and For F-P laer, a very wide range of value from 1.~14. wa reported by [1]. The alpha factor value wa meaured a.65. for a CSP laer [11]. 4) Uing the optical feedback technique: For VCSEL, the alpha factor value wa meaured a 5..7 [8]. Different value between. and 4.9 were obtained for DFB laer [7]. A wide range from 1.8~6.8 wa obtained for FP laer [7, 1]. The above technique can alo be claified into two categorie baed on the amount of injection current to the SL. For the firt category, the injection current i below the 64

81 threhold and in thi ituation, the LEF i regarded a a material parameter and i meaured according to the definition of the LEF in Eq. (5.1). In the econd category the injection current i above or cloe to the threhold and a mathematical model for meauring the LEF wa developed from the rate equation of the SL. In thi ituation, the LEF i conidered a a model parameter or effective parameter which i detached from it phyical origin to a certain extent [9, 8]. Among the technique in the econd category, the optical feedback method, i.e., the SMI, i an emerging and promiing technique which doe not require high radio frequency or optical pectrum meaurement, thu providing eae of implementation and implicity in the ytem tructure [4, 7]. Baed on the SMI, variou method were propoed for meauring the LEF. In 4, Yu et al. [7] propoed an approach which can obtain LEF by geometrically meauring the SMI ignal waveform. However, thi approach require the SMI ignal to have zero croing point, which mean the optical feedback level C fall within a mall range, i.e., 1C 3 which i difficult to achieve for ome type of laer. Additionally, the movement trace of the target mut be away from and back to the SL at a contant peed, which i alo difficult in practice. In ubequent year, everal approache [1, 15, 19, 13] for meauring the LEF were developed, and thee approache are mainly baed on the numerical optimization for minimizing the cot function in parameter. Similarly to [7], thee method are alo retricted to certain feedback level, e.g., approache in [15, 19, 13] require a weak feedback level, i.e., C 1, and the method in [1] require a moderate feedback level, i.e., 1C 4.6. Furthermore, thee method are quite time conuming due to the large data ample to be proceed. Recently, two different approache [18, 8] were developed for meauring the LEF over a large range of C, but they till face the problem of requiring a large amount of computation time. Note that the above method [1, 15, 18, 19, 7, 8, 13] all aume that the SMI ignal they ued for meauring the LEF are true, or in other word table. However, Chapter 3 (Section 3.3) ha hown that when the SMI operate in the emi-table or 65

82 even untable (chao) region, the SMI ignal detected by the PD packaged at the rear of the SL till look like table SMI ignal. Such a phenomenon i caued by the limit in the riing time (alo known a cut-off frequency, or bandwidth) of the PD packaged at the rear of the SL, a well a the circuit ued for detection. The PD may not be able to detect the detail of the high frequency (cloe to the relaxation ocillation frequency of the SL) SMI waveform in the emi-table or untable region, and the SMI ignal oberved i actually a ditorted verion of the high frequency waveform. Therefore, if the above method [1, 15, 18, 19, 7, 8, 13] are applied to the ditorted SMI ignal to meaure the LEF, it may induce ignificant meaurement error. Furthermore, the method mentioned above are either retricted to a certain optical feedback level or are quite time conuming, which will hinder the ue of SMI for embedded and indutrial application. In thi chapter, tarting from invetigating the influence of the bandwidth of the detection component (including PD and the detection circuit) on the SMI ignal, we firtly tudy the meaurement error of the LEF at the emi-table region, to determine whether it ha a ignificant impact on the reult. Then baed on our previou tability theory (explained in Chapter 3 and 4), a et of external parameter i determined for deigning a table SMI ytem to meaure the LEF. Finally, uing the table SMI ytem we deigned, a imple method for meauring the LEF i propoed and demontrated in order to lift the above mentioned limitation of previouly propoed method. Thi imple method i baed on the relationhip between the light feedback phae and the output power from the well known LK equation. It wa found that the LEF can quite imply be meaured by the power value overlapped by two table SL output power, i.e., an SMI ignal, under two different optical feedback trength. 5.1 Detection Problem Caued by SMI Signal in the Semi-table Region Figure 5-1(b) how an SMI ignal obtained by the LK equation in the emi-table region when C. and 6.. Note that all the other parameter value are the 66

83 ame a for the parameter ued for Fig. 3-4(c). Alo note that the external target move according to inuoidal law with the ame vibration trace a for Fig. 3-4(a). That i, L( t) L L in( ft), where L, L and f are the initial external cavity length the vibration amplitude and frequency repectively, for which the following value were choen: L.35m, L 1.5 and f 75Hz, Figure 5-1: An SMI ignal obtained by the LK equation in a emi-table region when C. and 6.. (a) the movement trace of the external target, (b) the SMI ignal. From Fig. 5-1(b), it can be oberved that the emi-table SMI ignal exhibit the form of high frequency ocillation with it amplitude modulated by a low-varying ignal. Thi high frequency i cloe to the relaxation ocillation frequency of the SL (uually 1.5-4GHz). Interetingly, the low-varying envelope are imilar to the SMI ignal characterized by the ame fringe tructure. It can be een from Fig. 5-1(b), that there are nearly 6 fringe correponding to the peak-peak diplacement ( 3 ) of the target. That i, each fringe in the emi-table SMI ignal alo correpond to a target diplacement of, and hence the emi-table SMI ignal can alo be ued to meaure the diplacement with the ame reolution a the normal SMI operating in the table region. However, taking into account of a practical SMI ytem, the bandwidth (BW) of PD packaged at the rear of the SL for detection i uually under 1GHz, which i in the range of about MHz to 8MHz. Furthermore, the BW of the detection circuit i alo 67

84 uually le than 1GHz. For example, the BW of our detection circuit hown in Fig. 3-7 i 3MHz. Therefore, obviouly, the detection part of a practical SMI ytem i actually equivalent to a low pa filter on the real SMI ignal, a hown in Fig. 5-1(b). Thi will dramatically influence the waveform of the SMI ignal. Figure 5-(b)-(f) how the filtered reult by applying the detection circuit with different BW. That i, a low pa filter i ued with different cut-off frequencie, denoted a fcut off, to proce the SMI ignal in Fig. 5-1(b). The range of fcut off i from 6MHz to 3MHz, which hould cover mot practical ituation. Figure 5-: Filtered SMI ignal after the detection component with different fcut off when C. and 6.. (a) SMI ignal in Fig. 5-1(b), (b)-(f) filtered SMI ignal of the SMI ignal in (a). From the Fig. 5-(b)-(f), we can ee the following problem: 1. Sparkling noie appear in the filtered SMI ignal (Fig. 5-(b) and (c)) when fcut off are 6MHz and 4MHz repectively.. Filtered SMI waveform are different after applying a low pa filter with different fcut off. Mot meaurement approache [1, 15, 18, 19, 7, 8, 13] are performed baed on the waveform of the SMI ignal. Thu, the ditorted or filtered SMI ignal due to different BW of the detection component will work out different reult of. Here, we adopt the approach in [7] a an example to ee how the filtered SMI ignal 68

85 introduce the meaurement error. The approach in [7] can obtain by geometrically meauring the SMI ignal waveform, and thi approach require the characteritic point of the SMI ignal, i.e., zero croing point, to perform the meaurement. Figure 5-3 how a typical experimental SMI ignal, where A and B are zero croing point. A B Figure 5-3: Experimental SMI ignal obtained in [7] (ee Fig. in [7]) under moderate feedback. Time interval t 13 and t 4 are relevant for the determination of. The time interval t 13 and t 4 hown in Fig. 5-3 allow u to determine the phae difference, and thu obtain the value of baed on the following two equation: C 1 C 13 C 1 arcco( ) arctan( ) 1 C 1 C 4 C 1 arcco( ) arctan( ) 1 (5.) (5.3) 13 where 13 t 4 and 4 t T 1 T. Obviouly, the location of the characteritic point are very important. However, due to different detection BW, the waveform we oberve will be different. Figure 5-4 how an example where two filtered SMI ignal (Fig. 5-(c) and (f)) are plotted together in order to ee their ignificant impact on the meaurement reult uing the method in [7]. The olid line i the SMI ignal when the fcut off i 3MHz. 69

86 Figure 5-4: (a) Two filtered SMI ignal obtained in Fig. 5-(d) and (f). (b)-(c) the zoomed SMI ignal. From the zoomed SMI ignal, i.e., Fig. 5-4(b) and (c), we can ee that the poition of the characteritic point, i.e., Z 1-4 in Fig. 5-4(b) and (c), of the two ignal are quite different, and will thu give different meaurement reult. For the SMI ignal filtered with fcut off of 4MHz, uing the approach in [7], we obtain 5.16 wherea 5.67 when fcut off i 3MHz, which deviate from the true value of the LEF, i.e., 6.. The reult how that different fcut off of the detection component, thu leading to different SMI waveform, do largely impact on the meaurement reult. In addition, it i noted that the parkling noie ha a trong influence on the normalization of an SMI ignal. The normalization can be achieved via variou method [7, 36, 14] ued for SMI ening. We alo meaured the LEF for other filtered SMI ignal a hown in Fig. 5- uing the method decribed in [7]. The etimated LEF are repectively 4.41, 6.89, 5.51 for Fig. 5-(b) (d) and (e), which are alo quite different from the true value, i.e., 6.. Therefore, it i very important to deign a practical SMI ytem working in the table region. 5. Deigning a Method for Achieving a Stable SMI Sytem A practical table SMI ytem can be deigned baed on the method decribed in the 7

87 previou chapter. Here, a pecific example i preented, which applie the method decribed in Chapter 3, to illutrate how to deign a table SMI ytem and thu to acquire table SMI ignal. A practical SMI ytem can be repreented by the experimental etup hown in Fig. 3-7, which we re-draw here a a implified verion hown in Fig Figure 5-6 how the phyical SMI ytem correponding to Fig Figure 5-5: Schematic diagram of the implified SMI ytem of Fig Figure 5-6: A phyical SMI ytem. There are three controllable external cavity parameter in the ytem: 1. the injection current J which can be controlled by the current controller,. the optical feedback level C which can be adjuted by the attenuator, and 3. the external cavity length L. The table SMI ytem can be deigned baed on the following tep: Step1: Place the external target at a certain ditance, e.g., L.5m, from the SL. 71