Pump-to-Stokes relative intensity noise transfer and analytical modeling of mid-infrared silicon Raman lasers

Transcription

1 Pum-to-Stoke relative tenity noie tranfer and analytical modelg of mid-frared ilicon aman laer J. Ma 1 and S. Fathour 1,2,* 1 CEOL, The College of Otic and Photonic, Univerity of Central Florida, Orlando, Florida, 32816, USA 2 Deartment of Electrical Engeerg and Comuter Science, Univerity of Central Florida, Orlando, Florida, 32816, USA *fathour@creol.ucf.edu Abtract: An analytical model for mid-frared (mid-) ilicon aman laer (SL) i develoed. The relative tenity noie (N) tranfer from the um to the Stoke the laer i alo vetigated. The analytical model can be ued a a veratile and efficient tool for analyi, deign and otimization of mid- SL. t i hown that converion efficiency of 70% i attaable and the low-frequency N tranfer may be ureed to below 1 db by umg low-lo waveguide at high tenitie Otical Society of America OCS code: ( ) aman effect; ( ) ntegrated otic device. eference and lk 1. B. Jalali and S. Fathour, Silicon hotonic, J. Lightwave Technol. 24(12), (2006). 2. B. Jalali, V. aghunathan,. Shori, S. Fathour, D. Dimitrooulo, and O. Stafudd, Proect for ilicon mid- aman laer, EEE J. Sel. To. Quantum Electron. 12(6), (2006). 3. V. aghunathan, D. Borlaug,.. ice, and B. Jalali, Demontration of a Mid-frared ilicon aman amlifier, Ot. Exre 15(22), (2007). 4. D. Borlaug, S. Fathour, and B. Jalali, Extreme value tatitic ilicon hotonic, EEE Photon. J. 1(1), (2009). 5. T. Baehr-Jone, A. Sott,. lic, A. Sott, B. Penkov, W. Aher, and M. Hochberg, Silicon-on ahire tegrated waveguide for the mid-frared, Ot. Exre 18(12), (2010). 6.. Soref, Mid-frared hotonic ilicon and germanium, Nat. Photonic 4(8), (2010). 7. X. Liu,. M. Ogood, Y. A. Vlaov, and W. M. J. Green, Mid-frared otical arametric amlifier ug ilicon nanohotonic waveguide, Nat. Photonic 4(8), (2010). 8. S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio,. B. Divlianky, N. Alic, S. Mookherjea, and S. adic, Mid-frared wavelength converion ilicon waveguide ug ultracomact telecom-band-derived um ource, Nat. Photonic 4(8), (2010). 9. G. Z. Mahanovich, M. M. Milošević, M. Nedeljkovic, N. Owen, B. Xiong, E. J. Teo, and Y. Hu, Low lo ilicon waveguide for the mid-frared, Ot. Exre 19(8), (2011). 10. Z. Cheng, X. Chen, C. Y. Wong, K. Xu, C. K. Y. Fung, Y. M. Chen, and H. K. Tang, Mid-frared gratg couler for ilicon-on-ahire waveguide, EEE Photon. J. 4(1), (2012). 11. O. Boyraz and B. Jalali, Demontration of a ilicon aman laer, Ot. Exre 12(21), (2004). 12. H. ong,. Jone, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, A contuou-wave aman ilicon laer, Nature 433(7027), (2005). 13. A. Liu, L. Liao, and H. ong, ecent develoment ilicon hotonic: 2.5 Gb/ ilicon otical modulator and ilicon aman laer, Proc. SPE 5730, (2005). 14. X. Chen, N. C. Panoiu, and. M. Ogood, Theory of aman-mediated uled amlification ilicon-wire waveguide, EEE J. Quantum Electron. 42(2), (2006). 15. M. Kraue,. Draheim, H. enner, and E. Brkmeyer, Cacaded ilicon aman laer a mid-frared ource, Electron. Lett. 42(21), (2006). 16. M. Kraue, H. enner, and E. Brkmeyer, Theory of ilicon aman amlifier and laer, Silicon Photonic for Telecommunication and Biomedice, S. Fathour and B. Jalali, ed. (CC Pre, 2012), D. ukhlenko, M. Premaratne, and G. P. Agrawal, Nonlear ilicon hotonic: analytical tool, EEE J. Sel. To. Quantum Electron. 16(1), (2010). 18. H. ong, S. Xu, O. Cohen, O. aday, M. Lee, V. Sih, and M. Paniccia, A cacaded ilicon aman laer, Nat. Photonic 2(3), (2008). 19. M. Kraue, S. Cierullie, H. enner, and E. Brkmeyer, Pum-to-Stoke N tranfer aman fiber laer and it imact on the erformance of co-umed aman amlifier, Ot. Commun. 260(2), (2006). 20. X. Sang, D. Dimitrooulo, and B. Jalali, nfluence of um-to-ignal N tranfer on noie figure ilicon aman amlifier, EEE Photon. Technol. Lett. 20(24), (2008). # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17962

2 21.. D. ukhlenko,. Udagedara, M. Premaratne, and G. P. Agrawal, Effect of free carrier on um-to-ignal noie tranfer ilicon aman amlifier, Ot. Lett. 35(14), (2010). 22. X. Liu, X. Sang, B. Yan, K. Wang, C. Yu, and W. Dou, nfluence of um-to-stoke N tranfer on the gle order ilicon aman laer, J. Otoelectron. Adv. Mater. 4, (2010) Cla, D. Dimitrooulo, V. aghunathan, Y. Han, and B. Jalali, Obervation of timulated aman amlification ilicon waveguide, Ot. Exre 11(15), (2003). 24. S. Pearl, N. otenberg, and H. M. van Driel, Three-hoton abortion ilicon for nm, Al. Phy. Lett. 93(13), (2008). 25. F. Lelgard, C. Martelli, S. Borne, L. Lorcy, D. Bayart, F. Catella, P. Chartier, and E. Faou, Modelg of multiwavelength aman fiber laer ug a new and fat algorithm, EEE Photon. Technol. Lett. 16(12), (2004). 26. J. Zhou, J. Chen, X. Li, G. Wu, and Y. Wang, Exact analytical olution for aman fiber laer, EEE Photon. Technol. Lett. 18(9), (2006). 27. Z. Q, X. Zhou, Q. Li, H. Wu, and Z. Zhou, An imroved theoretical model of nth-order cacaded aman fiber laer, J. Lightwave Technol. 25(6), (2007). 28. S. A. Bab, D. V. Churk, and E. V. Podivilov, ntenity teraction cacade of a two-tage aman fiber laer, Ot. Commun. 226(1-6), (2003). 29. C. Huang, Z. Cai, C. Ye, H. Xu, and Z. Luo, Otimization of dual-wavelength cacaded aman fiber laer ug an analytic aroach, Ot. Commun. 272(2), (2007). 30. K. Huang, X. Zhou, Z. Q, H. Wu, and Z. Zhou, A novel fat numerical algorithm for cacaded aman fiber laer ug the analytic aroximate olution, Ot. Commun. 271(1), (2007). 31. H. ong, Y.-H. Kuo, S. Xu, A. Liu,. Jone, M. Paniccia, O. Cohen, and O. aday, Monolithic tegrated aman ilicon laer, Ot. Exre 14(15), (2006). 1. ntroduction Silicon i now an etablihed otical material for aive and active tegrated otic and otoelectronic the near-frared (near-) regime [1]. More recently, ilicon hotonic ha been urued the 3-5 µm or mid-wave frared (MW or mid-) regime with alication chemical and biological eng, tiue hotoablation, environmental monitorg and free-ace communication [2 10]. The key advantage of the mid- wavelength range, a comared with near-, i the abence of two-hoton abortion (TPA) and free-carrier abortion (FCA) at wavelength above ~2.2 µm [2]. Otical aman amlification at 3.4 µm [3,4], four-wave mixg and arametric amlification at ~2.2 µm [7,8], ilicon-on-ahire (SOS) waveguide at 4.5 µm [5], ilicon-on-ulator waveguide at 3.39 µm [9] and SOS gratg couler at 2.75 µm [10] are ome of the recent develoment the emergg field of mid- ilicon hotonic. SL were firt demontrated the near- regime [11,12]. The work were followed by everal imulation on aman laer and amlifier [13 17]. The longet wavelength exerimentally reorted the near- i a cacaded laer oeratg at µm [18]. Cacaded aman laer u to 3 µm and umed at 1.55 µm have been tudied baed on fully numerical method [15]. No exerimental work on mid- SL ha yet been reorted to the bet of our knowledge. t i, nonethele, mentioned that demontration of 3.4 µm mid- SL ha been attemted by the reent author mirror-coated 1-ch thick ilicon got and ug a etu imilar to that decribed ef [4]. Although ontaneou aman emiion wa certaly oberved and the bulk device wa erha at or lightly above threhold, the low average ower and the oor beam quality of the emloyed um ource (VBANT Model 2731 otical arametric ocillator by OPOTEK, nc.) revented defite and undiutable confirmation of lag. t i our belief that otical waveguidg would alleviate ome of thee iue rovided that low-lo mid- ilicon waveguide [9] and efficient coulg cheme [10] are imultaneouly emloyed. Thi aer reort an analytical model for mid- SL for the firt time. The develoed analytical model can be ued to avoid time-conumg fully numerical imulation the deign and analyi of the device. The model i validated by comarg it with numerical olution of couled-wave equation, and i ued to redict the erformance of the laer. The ician ditribution of um amlitude fluctuation ha been hown to have a ignificant imact on the ule-to-ule ga tatitic of mid- aman amlifier [4]. Similarly, the N tranferred from the tability of the um ource to the outut Stoke can have a ignificant imact on the erformance of mid- SL. The N tranfer ha been # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17963

3 theoretically and exerimentally tudied aman fiber laer [19], and ha been numerically imulated near- ilicon aman amlifier and laer [20 22]. n thi aer, the imact of N tranfer on the erformance of mid- SL i vetigated for the firt time. 2. Methodology There are everal nonlear effect ilicon beide aman catterg that may fluence the erformance of SL. Neceary aumtion hould be made order to imly the model without any lo of accuracy. Stimulated Brillou Scatterg (SBS) reent a eriou roblem aman fiber laer (FL). However, SBS can be ignored SL a the Brillou catterg coefficient for ilicon i two order of magnitude maller than the aman ga coefficient [23]. Three-hoton abortion (3PA) and aociated free-carrier effect are alo negligible becaue the correondg coefficient for ilicon i very mall [24]. Degenerate four-wave mixg between the um and the generated Stoke wave can be dicarded a the hae matchg condition cannot be atified due to the large difference of the teractg wavelength. Therefore, only timulated aman catterg (SS) i conidered. Thi aumtion i conitent with reviou work on SL [13,15,16]. With the abence of TPA and FCA at above 2.2 µm, the couled-wave equation for aman lag lend themelve to analytical olution, a develoed here. Thi i contrat to near- wavelength where achievg accurate analytical olution i difficult, if not imoible, and hence fully numerical imulation are uually emloyed [13 17]. The model i alicable to variou ilicon waveguide configuration a well a bulk ilicon, coated or uncoated with dielectric or tegrated mirror, rovided that the redicted um tenitie are achievable ractice. Figure 1(a) how the chematic of the SL analyzed thi work which the ut um i jected from the left-hand ide (LHS) and the outut Stoke i from the right-hand ide (HS). A dicued later, a device which the outut i from the LHS wa alo tudied but exhibited very imilar erformance. n either cae, the device conit of a ilicon waveguide of length L, whoe facet are coated with multilayer dielectric film. A contuou wave (CW) um laer () at wavelength λ i couled to the LHS (l) of the cavity and the outut Stoke () wavelength λ i exited from the HS (r) via timulated aman catterg. The reflectivitie of the left and right mirror at λ and λ are l, r, l and r, reectively. Fig. 1. (a) Schematic of the tudied mid- ilicon aman laer; (b) Otical mode rofile (TE) at um (to) and the Stoke (bottom) wavelength the SOS waveguide with air tocladdg havg rib width of 2 µm, rib height of 2 µm and lab height of 1µm. The calculation are obtaed from a commercial numerical mode olver (BeamPOP by Soft). A tyical micron-ize ridge waveguide with the geometry decribed the cation of Fig. 1 wa analyzed. Figure 1(b) how the TE otical mode roagatg the SOS waveguide at both the um and the Stoke wavelength. The effective core area of the waveguide i ~3 µm 2. The overla of the two mode, Γ, i calculated to be cloe to unity (99.75%). The overla tegral, Γ, i cluded the followg model (Eq. (1) and (2)) for comletene. # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17964

4 However, it value i aumed to be 1 for the reent micron-ize ridge waveguide. t i noted that Γ could be coniderably maller than unity ubmicron waveguide. A mentioned, TPA and FCA are negligible at mid- wavelength [2]. By alo neglectg ontaneou aman catterg at and above threhold, the evolution of the forward ( + ) and backward ( ) roagatg um and Stoke tenitie are governed by the followg couled-wave equation [13] d λ + = α Γg ( + ), dz λ (1a) d + = α +Γg ( + ) (1b) dz where g i the aman ga coefficient, α and α are the lear roagation loe at λ and λ, reectively. The correondg boundary condition are + (0) = (1 ) + (0), l l + ( L ) = ( L ), r (0) = (0), + l + ( L ) = ( L ), r where i the ut um tenity. Equation (1) and (2) might be conidered to be imilar to the couled-wave equation and boundary condition of near- FL. Although everal analytical and numerical model for FL have been ublihed [25 30], each ha it own hortcomg for the reent cae, a follow. F. Lelgard et al. imlified the numerical algorithm for olvg the equation by tranformg the two-ot boundary value roblem to an itial value roblem, but the olution wa till fully numerical [25]; The analytical olution develoed by J. Zhou et al. aumed gle a um, i.e., anti-reflection coated mirror [26]; Other analytical model develoed for FL aume zero left-mirror reflectivity at the um wavelength (a imlifyg valid aumtion becaue of the low dex of ilica) [27 30]; Z. Q et al. made the further imlifyg aumtion of zero reidual um ower reflected back to the ut end [27]; S. A. Bab et al. not only aumed zero left-mirror reflectivity but alo aumed that the outut ower creae learly with the ut [28]. However, none of thee model are alicable to ilicon aman laer becaue a coniderable amount of um ower reflect back and forth between the right- and left-hand ide mirror to the cavity. ndeed, a variety of dielectric coated mirror, with different reflectivitie at um and Stoke wavelength are commonly conidered ilicon aman laer. Therefore, thi aer, nonzero left- and right- mirror reflectivitie at both um and Stoke wavelength are cluded the modelg of aman laer, for the firt time, to account for ilicon and other high-dex material cae (the imlet examle may be uncoated air-ilicon terface with reflectivity of ~30% at both um and Stoke). Thi will require develog a more comlicated mathematical treatment of the roblem, a reented here. The above differential equation with the boundary condition can be olved numerically by collocation. However, an itial gue i uually required for numerical method. n our cae, thi gue i difficult to fd becaue zero Stoke tenity i alway a oible olution even when the um tenity i above lag threhold. Therefore, an analytical olution to thi boundary-value roblem i develoed. The reented olution can be utilized not only a an itial gue for numerical olver but alo a a fully-analytical model. 1/ 2 + 1/2 Firt, the geometric mean tenity = ( ) and the ga factor +,,,,,, G ( z ) = 1/ 2ln ( z ) / ( z ) are defed [25 30]. A roved Aendix A, and (2) # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17965

5 are contant, i.e., they are deendent of z. A a reult, Eq. (1) and (2) can then be rewritten term of, and G, ( z ) (Eq. (A2)-(A4)). A lear deendence of G ( z ) on z i alo aumed, i.e., G ( z ) = G (0) + z [ G ( L ) G (0)] / L. (3) Thi lear aumtion imlie that the um delete exonentially both forward and backward direction. The validity of thi lear deendency aumtion wa confirmed by comarion with fully numerical olution (Fig. 2). Baed on the above, the threhold tenity of the ilicon aman laer i obtaed a 2δ δ δ αδ le e le th = αl αl g l e + re (1 )( ), (1 )(1 )(1 ) (4) where δ = α L+ 1/ 2ln(1/ ), (5a) r δ = α L+ 1/ 2ln(1/ ) (5b) l r are lo factor of the um and Stoke wave due to lear roagation lo and mirror tranmiion loe. A detailed derivation of Eq. (4) i reented Aendix A. By defg above threhold, can be olved from h[ G, ( z )] h[ G, (0)] L, ( z ) = L, G ( L ) G (0),, 1/ 2 1 (6) =δ / [2 g L ( L )]. (7) Conequently, the geometric mean tenity and ga factor of the Stoke wave are obtaed from = λ [ G (0) δ ] / [2 g λ L ( L )], (8) 1/2 0 G z G α z g L z (9) 1/2 ( ) = (0) + 2 ( ). G, ( z ) and G, ( L ) Eq. (3) and Eq. (5)-(7) are eaily obtaed from Eq. (2). Fally, the tenity ditribution for the um and Stoke wave are ( z ) = ex[ G ( z )], (10) 1/2,,, and the outut of the laer at the Stoke wavelength on the HS of the waveguide Fig. 1(a) i fally + out = (1 r ) ( L ). (11) An equation imilar to Eq. (11) can be eaily obtaed if the laer outut beam i at the LHS, i.e., the cae where the laer outut and ut beam are counterroagatg. Unlike the above model for the light-light characteritic, the governg equation for N tranfer SL do not lend themelve to analytical olution. To etimate the effect of N tranfer from the um to the Stoke outut, the noie comonent at angular frequency Ω the um noie ectrum i conidered. The tenity fluctuation of the um and the Stoke wave, normalized to the average tenitie, are rereented by m ( z ) and m ( z ), reectively: # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17966

6 where ( z ) and ( z ) ( z, t ) = ( z )[1 + m ( z )ex( iω t )], (12a) ( z, t ) = ( z )[1 + m ( z )ex( iω t )], (12b) are time-deendent average tenitie. m ( z ), m ( z ) comlex value atifyg m ( z ), m ( z ) << 1. The ut um i aumed to be modulated by a uoidal function at angular frequency Ω, i.e., = [1+ m ex( iω t )], where m i a mall real number. The different value of um and Stoke grou velocitie, v and v, hould be accounted for imilar to aman amlifier [20,21]. Thi lead to reduced N tranfer: are 1 λ + + = α g ( + ), z v t λ (13a) = α + g ( + ). (13b) z v t Subtitutg Eq. (12) to Eq. (13) and (2) and neglectg higher order fluctuation term, a total of eight couled differential equation and eight boundary condition can be obtaed. Thi i a omewhat more comlicated roblem comared to the only four [20] or ix (to account for carrier denity noie [21]) equation and boundary condition that ought to be olved for ilicon aman amlifier. The equation for teady tate oeration and correondg boundary condition can be obtaed by relacg, Eq. (1) and (2) with, and olved ug the analytical model mentioned above. The other four couled equation and four boundary condition that decribe the mall fluctuation on the um and Stoke wave along tranmiion length z are derived a: dm dz iω λ + + = m g ( m + m ), v λ (14a) dm dz iω + + = m g ( m + m ), (14b) v + + (0) m (0) = (1 ) m + (0) m (0), l l m ( L ) = m ( L ), m (0) = m (0), m ( L ) = m ( L ). Equation (14) and (15), together with the teady-tate equation, can be numerically olved ug the collocation method, from which the N tranferred from the um to the Stoke i calculated a 3. eult and dicuion T N 2 2 (15) + m ( L, Ω) ( Ω ) =. (16) m The above general model wa alied to ecific examle. n all the followg numerical and analytical olution, a um wavelength of 2.88 µm i ued [3,4]. The correondg Stoke wavelength i 3.39 µm accordg to ilicon otical honon energy. The exerimentally etimated aman ga coefficient g of 9 cm/gw at thee wavelength wa emloyed [3]. A non-coated 2-cm long-cavity wa firt analytically modeled by aumg that reflectivitie at both um and Stoke wavelength were 30%. Fully numerical imulation to # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17967

7 couled-wave equation wa alo carried out for thi ecial cae by ug the reult from the analytical method a guee for the itial olution. Figure 2(a) reent the comarion between the analytical and numerical olution. The tenity ditribution of the um and Stoke wave the laer cavity for an ut tenity of = 200 MW/cm 2 are lotted ug both method. Such um tenitie can be attaed ractice by olid-tate mid- laer (e.g., otical arametric ocillator) [3,4]. Alo hown the et of Fig. 2(a) i the utoutut (light-light) characteritic of the laer. t i clearly evident that the reult have excellent agreement. The validity of the model wa rigorouly teted under other boundary condition examle not reented here. Meanwhile, the analytical model i roved to be much fater than the traditional numerical way of olvg thi et of equation. For examle, the time conumed for lottg the et of Fig. 2(a) i 200 time fater than numerical imulation. Therefore, our analytical model can be confidently ued a a convenient and efficient tool deign and otimization of mid- SL. The reult of Fig. 2(a) alo ugget that at mid- wavelength, where TPA and FCA are negligible, it i oible to um a noncoated CW SL above threhold with a reaonable um tenity of around 100 MW/cm 2. t i remded that CW near- SL are not achievable at any um tenity without ug aroriate mirror coatg on to of emloyg the carrier wee-out technique to reduce the carrier lifetime [12]. (a) (b) Fig. 2. (a) ntenity ditribution of um and Stoke wave with = 200 MW/cm 2. l = r = l = r = 30%, α = α = 0.5 db/cm and L = 2 cm were aumed. The et how the utoutut characteritic of the aman laer. (b) Threhold tenity veru cavity length for variou outut facet reflectivitie and roagation loe and for l = 10%, r = l = 90%. Ug the decribed analytical model, it i eay to analyze mid- SL and otimize their deign arameter. ndeed, achievg all the followg reult would have been extremely challengg and time-conumg baed on fully numerical model. Figure 2(b) how the threhold tenity a a function of L for different right facet reflectivitie and two different lear roagation loe of 0.5 and 2.0 db/cm. Unlike near- SL that have no lag threhold outide a limited range of length [16], mid- ilicon waveguide cavitie can lae for any given length if enough um ower i available. Alo, it i evident that for fixed reflectivitie, there i an otimum length, where the lag threhold reache a mimum. Thi i more remarkable at the higher tudied roagation lo (2.0 db/cm), a an otimum length of < ~1 cm can be recognized. Figure 3 how the fluence of the cavity arameter, length and facet reflectivitie, on the energy converion efficiency of the laer, defed a out /. Generatg each 3D lot Fig. 3 wa achieved about 10 mute with a tyical dekto PC (with a 3 GHz ntel() Core(TM)2 Duo CPU), while it can take day to make imilar lot baed on fully numerical method. However, our analytical model offer an efficient way to otimize the deign of mid- SL. n thi cae, L and r could be otimized under certa um tenitie. For lear roagation lo of α = α = 0.5 db/cm, maximum converion efficiencie of 55.8% and 45.1% are obtaed at ut tenitie of = 200 and 100 MW/cm 2, reectively (Fig. 3(a) and (c)). Such high converion efficiency have been reviouly etimated baed on fully # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17968

8 numerical imulation and dicate that ilicon aman laer the mid- can atta erformance comarable to near- fiber aman laer [15]. The laer efficiency, however, dro quickly for device with the higher lo, i.e., α = α = 2.0 db/cm. The maximum converion efficiency obtaed are 30.5% and 13.5% at ut tenitie of = 200 and 100 MW/cm 2, reectively (Fig. 3(b) and (d)). The otimum length are below 0.4 cm thee two cae. Further creag the length will reult higher lag threhold and lower loe efficiency at the ame time. Nonethele, thee rediction dicate a key advantage of mid- laer, a comact laer cavitie can be demontrated. n comarion, u to 5 cm length are required at near- wavelength [12]. t i noted that even more comact device can be enviioned ug rg reonator aman laer, a reorted the near- [31]. However, the tudy of uch laer i beyond the coe of thi work. Fig. 3. Converion efficiency veru cavity length L and outut facet reflectivity r for two roagation lo value and two um tenitie. The maximum converion efficiencie achievable for four roagation lo value are lotted veru um tenity Fig. 4. For device with the lowet lo, i.e., α = α = 0.1 db/cm, the maximum oible converion efficiency i ~73%. Thi could be conidered a a ractical limit for the efficiency of a mid- SL aumg extremely low-lo ilicon waveguide. Pumg the laer with tenitie above 200 MW/cm 2 i unneceary thi cae a it could hardly imrove the converion efficiency. For device with higher lo, i.e., α = α = 1.0 or 2.0 db/cm, the converion efficiency ha not yet reached aturation at an tenity of 500 MW/cm 2. The mimum achievable lag threhold for the four roagation lo value are recognizable by the terection of the curve with the x-axi. t i noted that unlike aman amlifier whoe ga deend on whether a co- or counterroagatg cheme i emloyed [16], our tudie on mid- SL ugget that there i little deendency of the device converion efficiency on the roagation direction of the um and the outut Stoke wave. Thi difference between aman amlifier and laer can be attributed to the fact that the aymmetric imact of the nonuniform um ditribution along the waveguide i more ronounced amlifier whoe Stoke ignal i tyically aed a gle time through the waveguide a ooed to laer whoe Stoke outut wave exerience everal roundtri the cavity. # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17969

9 Fig. 4. Maximum converion efficiency veru um tenity for four different roagation lo value. Fally, the N tranfer a SL with a lear waveguide lo of 0.5 db/cm and for an otimized um tenity of 100 MW/cm 2, i.e., L = 1.20 cm, r = 38% (aumg l = 10% and r = l = 90%), wa numerically evaluated. Noie frequencie rangg from zero to ten of gigahertz were cluded. The grou velocity at the um wavelength Eq. (13a) and 14(a) wa calculated a v = c/n eff, where c i the eed of light vacuum and n eff i the effective dex of the ilicon waveguide at um wavelength. The grou velocity at the Stoke wavelength wa obtaed from v = 1/[(λ λ )D + 1/v ], where D = 120 /(nm.km) i the local grou-velocity dierion ilicon calculated by the Sellmeier equation. Thi material dierion domate the waveguide dierion the tudied large cro-ection waveguide. Thi wa validated by Soft calculation and i conitent with reviou work [20 22]. Fally, m = 0.01 i aumed our imulation but even higher value for thi quantity change the followg calculation ignificantly. Figure 5(a) reent the um-to-stoke N tranfer ectrum for um tenitie of 50, 75, 100 and 200 MW/cm 2. t i evident that the N tranfer rema contant at low frequencie, then tart to ocillate at the free ectral range (FS) of the laer cavity, i.e., υ = c/(2n eff L) = 3.6 GHz. The oberved trong ocillation at higher frequencie ugget that laer ource with N ectra no wider than a few GHz are required for umg mid- laer with cavity length of ~1 cm. The low-frequency tranferred N, a well a the magnitude of the high-frequency ocillation, dro a the um ower creae. Thi i conitent with theoretical and exerimental N tranfer ectrum of aman fiber laer [19]. The low-frequency N tranfer for the SL could be above 12 db when umed at 50 MW/cm 2, and dro to below 1 db at a um tenity of 200 MW/cm 2. eult very cloe to Fig. 5(a) were obtaed for the cae which the laer outut and the ut um are counterroagatg. Thi differ from aman amlifier that how a higher N bandwidth for the coroagation cheme a comared with the counterroagation cheme [20]. # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17970

10 (a) (b) Fig. 5. (a) N tranfer ectra for mid- SL umed at 50, 75, 100 and 200 MW/cm 2 and (b) the device low frequency N tranfer veru um tenity for four roagation lo value. Modal arameter: L = 1.20 cm, r = 38%, α = 0.5 db/cm (otimized deign for = 100 MW/cm 2 ), D = 120 /(nm km). Figure 5(b) ummarize the low-frequency N tranfer veru um tenity under different roagation lo value. The N tranfer oberved goe to fity right below the lag threhold and decreae with creag um tenity. Alo evident i the noticeable creae the low-frequency N tranfer with creag roagation lo. Unlike near- SL, which the N tranfer i trongly affected by FCA [21], the N tranfer mid- laer i maly determed by the lear roagation lo and the um tenity. Therefore, umg at well-above lag threhold and reducg the lear lo of ilicon waveguide are two crucial requirement for decreag the N tranfer mid- SL. 4. Concluion n ummary, an analytical model i reented to redict the erformance of mid- SL. The model ha excellent agreement with numerical olution, and hence it can be conveniently ued a a veratile deign and otimization tool or to fd good itial guee for numerical method. The effect of cavity arameter on the converion efficiency of the laer were tudied. The um-to-stoke N tranfer of mid- SL i alo numerically tudied. The reult how that the low frequency N tranfer i a trong function of um tenity and lear roagation lo. For a lear roagation lo of 0.5 db/cm and a um tenity of 200 MW/cm 2, the otimized cavity length and outut facet reflectivity are 1.04 cm and 22%, reectively. The maximum oible converion efficiency i ~56%, and the um-to-stoke N tranfer i ~1 db. For a lear roagation lo of 2.0 db/cm and ame um tenity, the correondg value will be 0.34 cm (length), 58% ( r ), 31% (converion efficiency) and 4 db (N tranfer). The reult of thi tudy redict trong roect for mid- ilicon aman laer for high-erformance biochemical and communication alication rovided that low-noie mid- um ource with high beam-quality and efficient waveguidg cheme exit. Aendix A The derivation of the threhold tenity (Eq. (4)) i reented here. The deendence of and on roagation ditance z i roved firt. Subtitutg Eq. (1) to give = ( ) 1/ 2 + 1/2 + d d d + = + dz dz dz + λ + λ + + = [ α + g ( + )] + [ α g ( + )] = 0. λ λ (17) # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17971

11 Similarly, it can be roved that d / dz = 0. Equation (1a) and (1b) are then rewritten a follow: dg dz dg dz λ = α 2g co h( G ) λ (18) = α + 2g co h( G ) (19) The boundary condition Eq. (2) are alo rewritten term of, and G, ( z ), i.e., G (0) = ln (1 l ) + (1 l ) + 4 l G ( L ) = ln, G (0) = ln, G ( L ) = ln r l r The teady-tate condition for the um and the Stoke wave can be acquired by tegratg Eq. (18) and (19): (20) λ G (0) g [2 co h( G )] dz, L = δ + λ 0 (21) L g [2 co h( G )] dz = δ. (22) 0 Below or at threhold, i.e., th, = 0, the um ga factor can be obtaed from Eq. (18) and (21): From the firt boundary condition Eq. (20), G ( z ) = δ α z. (23) (1 l ) =. ex( δ ) ex( δ ) l Under the aumtion that G ( z ) i learly deendent on z, the tegration Eq. (22) can be aroximated a 0 (24) L h[ G ( z )] h[ G (0)] co h[ G ( z )] dz = L = L ( z ). (25) G ( L ) G (0) At or above threhold, i.e., th, Eq. (22) and (25) yield: δ = 2 g L ( L ). (26) when =, the mean um tenity th 1/ 2 mut atify both Eq. (24) and (26), i.e., th (1 l ) δ = =. ex( δ ) ex( δ ) 2 g L ( L ) l The threhold tenity Eq. (4) i then olved from Eq. (27). (27) # $15.00 USD eceived 5 Ar 2012; revied 15 Jun 2012; acceted 13 Jul 2012; ublihed 23 Jul 2012 (C) 2012 OSA 30 July 2012 / Vol. 20, No. 16 / OPTCS EXPESS 17972