A MULTIMODE SIMULATION MODEL OF MODE-COMPETITION LOW-FREQUENCY NOISE IN SEMICONDUCTOR LASERS

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1 lucuaion and oie Leer ol., o L63 L70 World cienific Publihing Comany A MULTIMODE IMULATIO MODEL O MODE-COMPETITIO LOW-REQUECY OIE I EMICODUCTOR LAER MOUTAA AHMED, MIORU YAMADA and ALAH ABDULRHMA Dearmen of Elecrical and Elecronic Engineering, Kanazawa Univeriy, Kanazawa , Jaan. Received May 00 Revied 9 eember 00 Acceed eember 00 A mulimode model i rooed o imulae he mode-comeiion low-freuency noie in emiconducor laer aing accoun of he oible mechanim of nonlinear gain auraion. A new echniue i reored o generae he Langevin noie ource ha induce onaneou-emiion flucuaion in dynamic of he laing mode. The model i alied o imulae he ineniy noie in Al- GaA laer. Agreemen of he imulaed noie reul wih he exerimenal meauremen i found. Correlaion of he noie characeriic wih dynamic of he mode comeiion i newly inroduced. The low-freuency noie i enhanced when he dynamic exhibi mode hoing or a jiering ingle mode, and i ureed when he able ingle-mode oeraion i achieved. Keyword: emiconducor laer; mulimode imulaion; mode comeiion noie; mode hoing; gain auraion; jiering ingle mode.. Inroducion Oeraion of emiconducor laer i affeced by he mode-comeiion henomena ha are inrinically caued by he nonlinear gain auraion effec []. The mode comeiion may enhance he couling of he ide mode wih he dominan mode o rongly a o generae mode hoing [ 7]. Thee henomena were alo roved o induce he oical feedbac noie when re-injecion of he laer ligh by oical feedbac exi [8]. Analyi of noie in emiconducor laer i commonly erformed by alying he aroximae mall-ignal analyi o rea he rae euaion including Langevin noie ource ha accoun for he inrinic flucuaion in he modal hoon number and elecron number [,9]. However, hi analyi i alicable for cae of mall flucuaion comared o he dc-ineniy, and i reamen become very comlicaed if a large number of mode exi. Moreover, correondence beween he noie roerie and he characeriic of he inducing mode comeiion in he ime domain i mied in ha analyi. An alernaive mehod o analyze he noie i o erform numerical inegraion of he rae euaion [5,7]. A yical meri of alying he laer mehod i o ge enough inigh of he dynam- The auhor i on leave from he Phyic Dearmen, aculy of cience, Minia Univeriy, Egy.

2 Ahmed, Yamada and Abdulrhmann ic of he mode comeiion and correlae hem wih he induced noie characeriic. However, reor on aifacory numerical reamen were limied o far becaue of wo criical roblem. The fir roblem concern wih numerical generaion of he Langevin noie ource eeing he correlaion of he injeced elecron number wih he hoon number of he laer mode. In a reviou aer, he auhor rooed a yemaic echniue o imulae correlaed Langevin noie ource in he numerical calculaion for he cae of ingle-mode oeraion []. The econd roblem i he correc inroducion of he cro-auraion on he modal gain, which are eed of he mode comeiion. In hi wor, we reor a new mulimode imulaion model of he ineniy noie by exending he reviou echniue [] o he cae of mulimode oeraion. The croauraion on he modal gain are alo aen ino accoun. The imulaion reul are in good correondence wih he exerimenal meauremen.. Mulimode imulaion Model The reen mulimode imulaion model i baed on numerical inegraion of he following rae euaion of he hoon number of M longiudinal mode and he injeced elecron number : d d [ D ] H G h = A B { b } aξ 0, = 0, ±, ±,..., ± M and d I = A. d e τ The mode =0 i aumed o cener he ecral rofile of he linear gain A [], { } b aξ A = g 0, 3 where ξ i he confinemen facor of he field ino he acive region, a and b are maerial arameer, and g i he ranaren level of. i he freuency of mode. Poiive number indicae mode on he higher-hoon energy horer wavelengh ide of he cenral mode, while negaive value aly o mode on he ooie ide. G h i he hrehold gain level, and τ i he elecron lifeime by he onaneou emiion. Incluion of he onaneou emiion, which rigger flucuaion in dynamic of he laer mode, ino he laing roce i decribed by he erm in he econd brace in. The auraion mechanim of he gain of a mode are included in erm of he coefficien B, D and H a follow. The coefficien B deermine he elf-auraion of gain, while he cro auraion by oher mode i given by D and H. The coefficien D decribe he ymmeric dierion auraion of gain, while H decribe he aymmeric auraion. Concree analye of he auraion henomenon wih he deniy-marix heory howed ha he nonlinear gain coefficien are given by [,] 9! τ in ξ B = a R 4 ε! o n o r cv, 4

3 A Mulimode imulaion model of mode-comeiion low-freuency noie 3 4 = in B D τ and g a a a H ξ τ α ξ τ ξ =. 6 R cv i he diole momen, n r i he refracive index in he acive region of volume, τ in i he inraband relaxaion ime, i an injecion level characerizing he gain auraion, i he oal hoon number, and α i he linewidh-enhancemen facor. Thi decriion of he gain auraion indicae ha couling occur among he mode, which reul in mode comeiion; an increae in he hoon number of a mode caue ureion of he gain of he oher mode o a o achieve he highe gain and dominae he laing acion. The aymmeric gain auraion funcion in romoing he gain of he mode on he longwavelengh ide while ureing he gain of he mode on he horer ide [,3]. Thi auraion i one of he oible ource of mode hoing and mode juming. The influence of he aymmeric gain auraion on he laer oeraion i enhanced a high injecion level or a large value of he α-facor, a given in 6. The erm and in and are Langevin noie ource and funcion in inducing inananeou flucuaion on and due o he onaneou emiion and he rocee of recombinaion and carrier generaion. The ource are originally defined a Poion random rocee, and are well aroximaed a Gauian one wih zero mean value. The noie ource are auo- and cro-correlaed, ' ' xy y x = δ, x,y= or, 7 where xy are he variance of he correlaion and δ i Dirac dela funcion. In rincile, he noie ource on he hoon number of differen mode are uncorrelaed, i.e.,, δ =, 8 Here, δ here i he Kronicher dela. However, The ource i cro-correlaed wih he ource becaue of he muual ineracion beween elecron and hoon during he laing roce a decribed in and. Thi roery of he cro-correlaion rereen a fundamenal roblem when generaing he ource uing he indeenden comuer random number generaor. Euaion and are ranformed ino a new e of euaion of he modal hoon number and a combined uaniy of, [ ] = h G H D B A e I A d d τ [ ] b a 0 ξ, 9 in uch a way ha he ueroing noie ource and are uncorrelaed. The orhogonaliy among he ource i arificially achieved by careful eing of he modal coefficien a

4 Ahmed, Yamada and Abdulrhmann ' = '. Thee ource can hen be generaed a each amling ime i uing heir auo-variance a i i = g and i i i = g i, i i where g and g are indeenden Gauian random deviae wih zero mean value and variance of uniy. urhermore, if hee noie ource are imagined o form an M- dimenional funcional ace, he noie ource can be rereened in hi ace by = i i i i. 3 i i Therefore, he imulaion can be done by inegraing he original rae euaion and uing and 3 or inegraing he ranformed euaion and 9 uing form -. The variance of he noie ource a a ime i are deermined by he mean value of and a he receding ime i- uoing uai-eady oluion of - d d d d 0 in he inerval = i [9,4], aξ = i i [ ] i i = aξ i i τ i, 4 and 5 {[ b ][. ] } = aξ i i 0 i g i. 6 The relaive ineniy noie RI induced by he mode comeiion i calculaed from he flucuaion in he oal hoon number δ =, wih being he mean value of, over a finie ime T wih he hel of he fa ourier ranform T a T jτ T jτ RI = [ ] = δ δ τ e dτ d δ τ e dτ. 7 T T 3. imulaion Reul In hi ecion, we aly he rooed model o inveigae he characeriic of ineniy noie in AlGaA laer. We coun a large number of 3 mode, M=5, o aure accuracy of he imulaion. The correonding 3-rae euaion of and are inegraed uing he fourh-order Runge-Kua mehod wih ime inerval of =0 over a long eriod of T=µ. Tyical value of he arameer ued in imulaion are: a=.75x - m 3 -, ξ=0., n r =3.59, =80µm -3, L=300µm, R cv =.8x -57 C m, g =.89x 8, =.53x 8, G h =5.0x -, τ in =0., τ =.79n and α=.0. The random number g and g are generaed by alying he Box-Mueller algorihm [4] o uniformly diribued

5 A Mulimode imulaion model of mode-comeiion low-freuency noie random number generaed by he comuer random ource. The RI i calculaed via 7 afer he ranien of die away. Ten ecral rofile of RI are averaged o obain a reaonable recie aiical ecral rofile. The ouu ecrum of he hoon number i deermined by averaging he modal hoon number over he inegraion ime T. ir we how he reul of alying our model in i imle cae of he ingle mode, i.e, M=0, o imulae he uanum noie wihou mode comeiion. The reul are loed in ig. a a curren I of.0 of he hrehold value Ih. The imulaion wihou gain nonlineariy, i.e., B=0 in, are alo hown o inveigae influence of he nonlinear gain on he characeriic of RI. The effec of couning he nonlinear gain i found o ure he RI daa around he ea a he relaxaion freuency. However, he lowfreuency RI i le affeced. igure alo lo he RI rofile rediced by he mallignal analyi, which wa ared by Haug [9]. Good correondence i een beween he imulaed noie daa and hoe calculaed by he mall-ignal analyi. - - ingle-mode model I =. Ih I =.36 Ih - RI [Hz ] wihou gain auraion - RI [Hz ] I =.0 Ih -4 mall-ignal analyi reen model - mode hoing mulimode -4-6 I =.8 Ih I =.80 Ih -6 jiering ingle mode able ingle mode 0 M M 0M G G reuency [Hz] ig.. The imulaed rofile of RI a I=.0Ih baed on a ingle mode model. The rofile calculaed by ignoring he gain auraion and by he mall-ignal analyi are alo loed. 0 M M 0M G G reuency [Hz] ig.. The imulaed RI rofile a four diinc oeraion. The low-freuency RI i enhanced in he mode-hoing region and i ureed in he able ingle-mode region. The ime-domain imulaion indicae ha incluion of boh he Langevin noie ource and he aymmeric gain auraion enhance he mode comeiion o induce inabiliie on he laer dynamic and affec he noie characeriic. our diinc examle of he imulaed RI a curren I=.,.8,.36 and.80ih are hown in ig.. The low-freuency comonen of he RI are almo fla whie noie and coinciden wih he uanum level when I=. and.80ih, which correond o mulimode and able ingle-mode oeraion, reecively. In he former oeraion, he inananeou mode comeiion i o rong ha mo of he mode ocillae imulaneouly. In he able ingle-mode oeraion, he inabiliie induced by couling flucuaion of he ide mode wih hoe of he main mode i o wea ha he hoon number of each mode flucuae around i dc-value. Then he induced noie correond o uanum noie in hi cae. On he oher hand, he low-freuency RI i enhanced when I=.8 and.36ih, which can be underood by examining he correonding inananeou variaion of he modal hoon number hown in ig. 3. igure 3a how random hoing of he mode =0 and -, which i een a inananeou couling of flucuaion of he hoon number of he hoing mode a well a mode wiching. Moreover, he flucuaion of he mode = are couled wih hoe of he hoing mode. uch inabiliie romoe

6 Ahmed, Yamada and Abdulrhmann Modal hoon number / = 0 mode = - a mode hoing I =.8 I h = Time [n] Modal hoon number / jiering ingle mode = 0 mode = - = - b I =.36 I h Time [n] ig. 3. imulaion reul of he modal hoon number when a I =.8I h and b I =.36I h. The figure how he induced mode hoing and jiering ingle mode. The mode number are denoed in he figure. he RI level. igure 3b how anoher ye of unable oeraion referred o a jiering ingle mode, which correond o he o-called mode ariion [ 5]. The inabiliie induced in he mode dynamic are generaed by couling he flucuaion of he idemode, =0 and -, wih hoe of he main mode, =-. The mode comeiion i no rong enough in hi cae o bring he couled mode o inananeou wiching. The erm ingle mode i aribued, in hi aer, o he high raio of he hoon number beween he dominan mode and he ronge ide mode, which exceed 0 db. We carry ou a large number of imulaion o inveigae he deendence of he lowfreuency RI on curren I. The imulaion reul a a freuency a low a Hz are loed in ig. 4. In order o correlae he imulaed reul of RI wih he mode dynamic, we examine he dynamic and he correonding ae of oeraion a each curren value. We are able o claify he RI reul ino four region a een in he figure: namely mulimode noie, mode-hoing noie, jiering ingle-mode noie, and able ingle-mode noie. Examle of he ouu ecra of he hoon number in hee region are loed a he o of he figure. ome exerimenal reul obained for a TJ AlGaA DH laer o characerize he mode-hoing noie are alo loed in he figure wih oen circle [5]. The figure indicae good correondence beween he imulaed reul and he exerimenal daa wihin he meaured range of curren. However, he meaured RI aain higher level, which may be aribued o he noie induced by diffuion of he curren from he elecrode ino he acive region in he mechanim of he curren injecion [5,6]. The ea hown around he hrehold curren I h i aribued o he maximum conribuion of he onaneou emiion o ligh amlificaion, which raidly decreae above hrehold comared wih he conribuion of he imulaed emiion []. The noie in hi region correond o he mulimode oeraion in which he ouu hoon number i diribued almo uniformly among he mode a indicaed by ecrum a. The uanum level of RI i alo aained in he able-ingle mode region,.54i < I <. 8I. In h h hi region he dynamic are free from eiher mode hoing or jiering, and he hoon number of he main mode, =-, i 0 ime higher han ha of any ide mode a hown in ecrum d. The RI level i mo enhanced in he mode-hoing region,.i < I <. 34I h h, becaue of he inabiliie induced by he mode-hoing henomenon, a dicued before. The ea of he hoing noie i aained when mode juming

7 A Mulimode imulaion model of mode-comeiion low-freuency noie < > / 0 - a b c d I =. I h I =.8 I h I =.36 I h I =.80 I h Mode number RI [Hz - ] mulimode mode hoing f ~ Hz imulaion Exerimen jiering ingle-mode able ingle mode Injecion curren I / I h ig. 4. The imulaed variaion of he low-freuency RI wih he injecion curren I. ome exerimenal reul are alo loed wih oen circle. The inveigaed ae of modal oeraion are denoed, and examle of he ime-averaged ouu ecrum in each ae are given in a d. occur. Tha i, he hoon number of he hoing ide mode, =-, balance ha of he main mode, =0. In he oher unable region of jiering ingle-mode oeraion,.34i < I <. 54I h h, he RI decreae wih curren I. However, he RI i higher han he uanum noie level becaue of he generaed exra noie. Tyical examle of he ouu ecra in he unable region of mode hoing and jiering are given in b and c of ig. 4, reecively. 4. Concluion We reored a new mulimode model of imulaing he low-freuency ineniy noie induced by he mode comeiion in emiconducor laer. A new echniue i rooed o ado generaion of he Langevin noie ource in he laer rae euaion. Correlaion of he noie characeriic wih he mode-comeiion dynamic i inroduced. The ineniy noie i enhanced when he laer exhibi mode-hoing or jiering oeraion. Acnowledgemen The auhor would lie o han he Jaan ociey for Promoion of cience JP for arly uoring he reen wor.

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