Optics Communications

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1 Optics Communictions 8 (8) Contents lists ville t ScienceDirect Optics Communictions journl homepge: Femtosecond pulse propgtion in silicon wveguides: Vritionl pproch nd its dvntges Smudr Roy, Shyml K. Bhdr, *, Govind P. Agrwl Fier Optics Lortory, Centrl Glss nd Cermic Reserch Institute, CSIR, 96 Rj S.C. Mullick Rod, Kolkt-7 3, West Bengl, Indi Institute of Optics, University of Rochester, Rochester, NY 467, USA rticle info strct Article history: Received 5 August 8 Received in revised form 9 August 8 Accepted 9 August 8 We investigte the propgtion chrcteristics of ultrfst pulses inside silicon wveguides considering frequency chirp ssocited with ech input pulse. Effects of liner losses, two-photon sorption, nd free-crrier dynmics re included nlyticlly within the frmework of modified vritionl formlism nd the results re vlidted y compring them with full numericl simultions. It is found tht n initil chirp helps in mintining the pulse shpe nd spectrum in the nomlous-dispersion regime, therey resulting in soliton-like propgtion of ultrshort opticl pulses. Ó 8 Elsevier B.V. All rights reserved.. Introduction Silicon photonics hs ttrcted gret del of ttention in the recent yers ecuse of its rod ppliction domin covering optoelectronic integrtion to iosensing [,]. Silicon hs excellent liner nd nonliner properties in the mid-infrred spectrl region tht re useful for vriety of pplictions relted to emerging photonics devices. The opticl mode inside silicon-on-insultor (SOI) wveguides is tightly confined to ri-like structure ecuse of the high refrctive index of silicon (3.45) compred to the ir or silic cting s cld. The high vlues of the Kerr coefficient (nerly times lrger thn tht of silic glss) nd Rmn gin coefficient (nerly times lrger thn tht of silic glss) led to efficient nonliner interction with opticl fields t reltively low power levels [3,4]. For this reson, SOI wveguides hve een used to produce rodnd mplifier nd tunle lsers exploiting the nonliner effects such s Rmn mplifiction [5 7] nd four-wve mixing [8]. They hve lso een used for spectrl rodening of ultrshort opticl pulses through self phse modultion [9] nd supercontinuum genertion []. The possiility of forming stle opticl soliton inside SOI wveguide is lso eing investigted ecuse the pulse width cn then e mintined close to its input vlue [,]. Experimentlly, it is oserved tht solitons cn e formed inside 5 mm long SOI wveguide y lunching femtosecond pulses with su-picojoule energy []. However, the liner loss, two-photon sorption (TPA), nd free-crrier sorption (FCA) influence the pulse shpe nd spectrum significntly []. Indeed, these loss mechnisms re considered to e mjor ostcle for soliton formtion in SOI wveguides. * Corresponding uthor. E-mil ddress: skhdr@cgcri.res.in (S.K. Bhdr). The modeling of pulse propgtion inside SOI wveguides requires numericl solution of generlized nonliner Schrödinger eqution. Although numericl scheme is eventully necessry for vlidting the experimentl dt, its exclusive use often limits physicl insight into the nonliner processes tht govern the propgtion process. In the present pper, we exploit the vritionl formlism for studying the prmetric effects on femtosecond pulse inside SOI wveguide. The min limittion of the vritionl technique is tht it requires the functionl form of the pulse shpe to remin the sme even though the prmeters of the pulse such s its mplitude, width, phse, nd chirp re llowed to chnge during propgtion. In the cse of soliton-like propgtion, the pulse dynmics cn well e treted with the help of vritionl process since the pulse shpe is expected to remin close to tht of the input pulse during propgtion. An dvntge of this method is tht, the perturing effects of liner loss, TPA, nd FCA cn e treted y introducing Ryleigh s dissiption function (RDF) [3 5]. Such semi-nlyticl pproch leds to set of ordinry differentil equtions for the pulse prmeters such s mplitude, width, phse, nd chirp. We stress tht the time-dependent free-crrier dynmics is included in our derivtion to the vritionl equtions. These equtions not only provide considerle physicl insight, they cn lso e solved rpidly over lrge rnge of the prmeter spce. We find the regime of vlidity of the vritionl technique y compring its predictions with the numericl solution otined y solving the generlized nonliner Schrödinger eqution with the stndrd split-step lgorithm [6].. Theoreticl model The propgtion of n opticl pulse through SOI wveguide is governed y the extended nonliner Schrödinger eqution [] given y 3-48/$ - see front mtter Ó 8 Elsevier B.V. All rights reserved. doi:.6/j.optcom.8.8.3

2 589 S. Roy et l. / Optics Communictions 8 (8) ou oz þ i o u ot 3 o 3 u 6 ot ¼ ic j 3 uj u C j uj u l u r N Cu; ðþ where u; ; 3 ; c; C; l ; r, nd N C represent the slowly vrying field mplitude, second-order dispersion coefficient, third-order dispersion coefficient, nonliner Kerr coefficient, TPA coefficient, liner loss prmeter, FCA coefficient, nd free crrier density, respectively. Since the TPA-induced free-crrier density N C hs profound effect on the pulse mplitude, the dynmic nture of N C is included y solving the rte eqution [3], dn C dt ¼ TPA hm eff j uðz; tþj 4 N C s C ; where s C is the crrier life time, hm is the photon energy t the incident wvelength, eff is the effective mode re, nd TPA ¼ C eff is the usul TPA prmeter. For short opticl pulse (t s C ), one cn ignore s C, s crriers do not hve enough time to recomine over the pulse durtion. In this sitution, it is possile to solve Eq. () nlyticlly for given opticl field. We ssume tht the sech-type shpe of the input pulse remins unchnged during propgtion ut llows its prmeter to evolve with the propgtion distnce z. In this cse, suitle form of the opticl field is uðz; tþ ¼Asech t " ( )# exp i / C t ; ð3þ t where A, t, u, nd C represent the mplitude, width, phse, nd chirp, respectively, nd ll of them vry with z. For this pulse shpe, we cn solve Eq. () nlyticlly, nd the crrier density is found to e N C ¼ TPAA 4 t hm eff 3 þ tnh t t t 3 tnh3 t t ðþ : ð4þ To solve Eq. () with the vritionl technique, we first find the Lgrngin nd the RDF ssocited with it. They re given y L ¼ ðuu z u u z Þþ i! j u tj 3 3! ðu t u tt u t u Þþic tt j uj4 nd ð5þ R ¼ Cðuu z u u z Þjuj þ ðuu z u u z Þþ rn Cðuu z u u z Þ: The reduced form of Lgrngin nd RDF is otined y integrting them over time [4]: L g ¼ Z Ldt R g ¼ Z Rdt: With the help of Eqs. (3) (7), we otin the following explicit expressions of L g nd R g : L g ¼ ia o/ p t oc t oz 6 oz C ot " # þ i A þ C p oz 3t 4 þ i 3 ca4 t ð8þ "!# R g ¼ð iþ o/ 4 oz 3 CA4 t þ t A þ 3 r TPAA 6 t hm eff þ i t oc oz C ot "!# ðp 6Þ CA 4 þ p oz 9 A þ p 8 r TPAA 6 t hm eff ð9þ The finl step is to employ the Euler Lgrnge eqution in the form d dt ol g oq z ol g oq þ or g oq z ¼ ; ð6þ ð7þ ðþ where q = A, t, u or C nd the suffix z indictes the corresponding derivtives. For ech q, we otin n ordinry differentil eqution, resulting in the following set of four coupled equtions: oa oz ¼ l A C t ot oz ¼ C oc oz ¼ 8 p CA C þ t! A C p þ A 3 r 3 3 CA 5 t hm eff! ðþ þ 4 t p CA t ðþ 4 p þ C þ 4 p ¼ þ 5 3t 6 ca : ð4þ The vritionl Eqs. () (4) show clerly how the pulse prmeters chnge during the propgtion inside SOI wveguide nd how they re coupled with ech other. More to the point, they show which liner nd nonliner process ffect prticulr pulse prmeter. Considerle physicl insight is gined y noting tht the pulse width in Eq. () is ffected directly not only y the dispersion prmeter (s expected) ut lso y the TPA (somewht unexpected). Moreover, wheres dispersion my led to pulse rodening or compression depending on the sign of, TPA lwys leds to the pulse rodening. It my e noted tht 3 does not pper in Eqs. () (4). It is well known [6] tht 3 introduces reltively smll temporl shift of the pulse center tht we ignore here. Also, if the crrier frequency of the pulse shifts, it chnges the effective vlue of. Since no such frequency shift occurs in our cse, 3 does not ffect the mjor pulse prmeters. Eq. () shows tht wveguide dispersion cn introduce pulse compression when the pulse is suitly chirped. More specificlly, pulse compression occurs if the condition C < is mintined throughout the propgtion distnce. Similrly, s expected, the pulse mplitude in Eq. () is ffected y the three loss mechnisms (liner loss, TPA, nd FCA). However, somewht surprisingly, it is lso ffected y dispersion when pulse is chirped. The chirp itself is ffected y the Kerr nonlinerity, wveguide dispersion, nd y the TPA process. In the next section we discuss the effect of input chirping on pulse evolution nd show tht n initil chirp of the pulse cn help in propgting it s soliton. We ignore the phse eqution in the following discussion ecuse the opticl phse does not ffect the three most relevnt pulse prmeters (A, t, nd C). 3. Results nd discussion To investigte the possiility of soliton formtion, we egin y focusing on femtosecond pulses propgting in the nomlous-dispersion region of silicon wveguide. Tle shows the vlues of device nd pulse prmeters employed in our study. We solve the set of three coupled equtions, Eqs. () (3), using Mtl softwre nd compre in Fig. the results for input pulses tht re either initilly unchirped (dshed lines) or chirped suitly (solid Tle Vlues of the prmeters used for numericl clcultion Prmeter nme Symol Vlue Wveguide length L cm Liner loss l db/m Effective re eff.38 lm Group velocity dispersion ±.56 ps /m Nonliner coefficient c 47 m W Nonliner loss C 6.5 m W Wvelength k 55 nm Input pek power P 4.76 Wtt Pulse width t () 5 fs

3 S. Roy et l. / Optics Communictions 8 (8) Normlised Width.. C () = C () =.4 Chirp Normlised Width.5 β >, C< β <, C> Chirp Normlised Amplitude Normlised Energy.8.7 Normlised Amplitude Normlised Energy Fig.. () Evolution of pulse, width, chirp, mplitude, nd energy for initilly chirped (solid line) nd unchirped (dshed line) pulses. () Effect of norml (dshed lines) nd nomlous (solid lines) dispersion on pulse prmeters. Input pulses re chirped in oth cses such tht C <. The used chirp vlues re.4 (solid lines) nd.4 (dshed lines), respectively. lines) to ensure tht its width nerly remins unchnged t the output end. The importnt point to note is tht the pulse width increses monotoniclly for unchirped pulses ecuse of the mplitude decy resulting from liner nd nonliner losses. In contrst, when the input pulse is slightly chirped (C =.4), the width first decreses efore incresing such tht the output pulse width is nerly equl to its input vlue. These results suggest tht solitonlike propgtion in SOI wveguide cn e relized in spite of losses y optimizing the input chirp. We compre in Fig. the prmetric evolution of chirped pulse in the norml nd nomlous dispersion regions. In oth cses, the pulse is suitly chirped so tht the condition C < is vlid t z =. Since C < initilly, the pulse egins to compress, s evident from the width plot. However, this condition is violted soon fter for pulses propgting in the norml-dispersion regime nd their width egins to increse rpidly. In contrst, pulse continues to compress until z exceeds 5 mm in the cse of nomlous dispersion. As result, the pulse width is nerly the sme s tht of the input pulse when it exits the wveguide. This difference is consequence of the fct tht the SPM-induced chirp dds to the dispersion-induced chirp in the cse of norml dispersion ut sutrcts when dispersion is nomlous [6] nd is ehind of soliton formtion in the ltter cse. This conclusion is lso supported y Fig. where we compre the temporl nd spectrl profiles of the input (dotted curves) nd output (solid curves) pulses in the regime of nomlous dispersion. As seen there, in the cse of chirped input pulses, oth the pulse shpes nd spectr lmost coincide. In contrst, when input pulses re unchirped, the output pulse (dshed curve) rodens considerly ecuse of liner nd nonliner losses. We next investigte how trustworthy is the vritionl pproch used in otining the results shown in Figs. nd. For this purpose, we solve the originl nonliner Schrödinger eqution given in Eq. () with the split-step Fourier method [6] nd compre Z = cm Z = cm t (fs) λ (μm) Fig.. Comprison of input (dotted curve) nd output pulse shpe () nd spectrum () for initilly chirped (solid line) nd unchirped (dshed line) pulse.

4 589 S. Roy et l. / Optics Communictions 8 (8) C () = C () = z = cm C () = t ( fs ) Fig. 3. () Evolution of the normlized mplitude for input pulses with nd without n initil chirp. The solid lines represent the vritionl solution wheres the open circles re the corresponding numericl dt. () Comprison of the output pulse shpe; the dshed nd solid curves represent the numericl nd vritionl solutions, respectively. Fig. 4. () Soliton-like propgtion nture of n initilly chirped pulse nd () the dynmic evolution of the free crrier density over the propgtion distnce. the results with those otined using the vrition Eqs. () (3). Fig. 3 shows the comprison in the cse of nomlous dispersion for oth the chirped nd unchirped input pulses. The pek intensity of the pulse displyed in Fig. 3 shows tht the pulse mplitude in ll cses is reduced considerly ecuse of liner loss nd nonliner losses resulting from TPA nd FCA. The importnt point to note is tht the vritionl results gree with the numericl dt to within few percent. The predicted output pulse in the two cses is compred in Fig. 3. Agin, the numericl output (dshed curve) grees well with the vritionl prediction (solid curve). As mentioned erlier, we were le to include the temporl vritions of the TPA-generted crrier density N C through the nlytic expression given in Eq. (4). Of course, N C lso vries with z ecuse it depends on the locl vlue of the pulse intensity. Fig. 4 shows the soliton-like propgtion of chirped pulse together with the corresponding chnges in the crrier density. As expected, the crrier density uilds up continuously over the pulse durtion nd cquires its mximum vlue nering the triling edge of the pulse. The mximum vlue of crrier density does not remin constnt long the wveguide length ut decys grdully with incresing distnce. As is evident from Eq. (4), the genertion of crrier density is minly governed y the intensity of the propgting pulse which itself decys with distnce owing to liner nd nonliner losses. However, the pulse mintins its shpe nd width in spite of its reduced pek mplitude ecuse of chirp-induced pulse compression. In conclusion, the propgtion dynmics of ultrshort opticl pulses inside silicon wveguide cn e descried resonly well in the soliton regime with modified vritionl technique. The resulting set of dynmic equtions cn e solved reltively esily to study the evolution of importnt pulse prmeters such s the width, chirp, nd pek intensity. Our procedure is cple of including the crrier density dynmics quite ccurtely. We hve shown tht the input chirp plys n impotent role in mintining the pulse shpe nd spectrum nd is useful for relizing solitonlike propgtion of opticl pulses. We hve employed the splitstep Fourier method to solve the extended nonliner Schrödinger eqution directly nd to verify the ccurcy of the vritionl results. Acknowledgements Authors wish to thnk Dr. H. S. Miti, Director, CGCRI, for his continuous encourgement, guidnce, nd support in this work. They lso wish to thnk the stff memers of the Fier Optic Lortory t CGCRI for their unstinted coopertion nd help.

5 S. Roy et l. / Optics Communictions 8 (8) One of the uthors (S.R.) is indeted to Council of Scientific nd Industril Reserch (CSIR) for finncil support in crrying out this work. References [] B. Jlli, S. Fthpour, IEEE J. Lightwve Technol. 4 (6) 46. [] R. Soref, IEEE J. Sel. Top. Quntum Electron. (6) 678. [3] L. Yin, G.P. Agrwl, Opt. Lett. 3 (7) 3. [4] Q. Lin, O.J. Pinter, G.P. Agrwl, Opt. Express 5 (7) 664. [5] R. Clps, D. Dimitropoulos, V. Rghunthn, Y. Hn, B. Jlli, Opt. Express (3) 73. [6] H. Rong, A. Liu, R. Jones, O. Cohen, D. Hk, R. Nicolescu, A.W. Fng, M.J. Pnicci, Nture 433 (5) 9. [7] H. Rong, R. Jones, A. Liu, O. Cohen, D. Hk, A.W. Fng, M.J. Pnicci, Nture 433 (5) 75. [8] H. Fukud, K. Ymd, T. Shoji, M. Tkhshi, T. Tsuchizw, T. Wtne, J. Tkhshi, S. Itshi, Opt. Express 3 (5) 469. [9] H.K. Tsng, C.S. Wong, T.K. Ling, Appl. Phys. Lett. 8 () 46. [] L. Yin, Q. Lin, G.P. Agrwl, Opt. Lett. 3 (7) 39. [] L. Yin, Q. Lin, G.P. Agrwl, Opt. Lett. 3 (6) 95. [] J. Zhng, Q. Lin, G. Piredd, R.W. Boyd, G.P. Agrwl, P.M. Fuchet, Opt. Express 5 (7) 768. [3] H. Goldstein, Clssicl Mechnics, second ed., Nros Pulishing House,, p. 4. [4] D. Anderson, Phys. Rev. A 7 (983) 335. [5] S. Roy, S.K. Bhdr, Physic D Nonliner Phenomen 3 () (7) 3. [6] G.P. Agrwl, Nonliner Fier Optics, fourth ed., Acdemic Press, Boston, 7.