International Journal of Scientific & Engineering Research, Volume 7, Issue 11, November ISSN

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1 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 6 ISSN Numerical Analyi of Intenity and Phae Noie of Solitary AlGaA Semiconductor Laer Oeratin in Multimode Sazzad M.S. Imran, Farjana Ferdou Tonni and Sumona Ilam Sumi Abtract In thi aer, both intenity and hae noie of olitary emiconductor laer oeratin in multimode ha been analyzed. A elfconitent numerical model i ued for imulation of the rate equation for hoton number, it hae and carrier number for tyical 78 nm AlGaA laer. Self-ureion coefficient and both ymmetric and aymmetric cro ain ureion coefficient are conidered in the analyi. We have alo conidered Lanevin noie ource for the hoton number, it hae and carrier number which induce intantaneou fluctuation in hoton and carrier number. We have decribed the noie effect throuh numerical imulation of the relative intenity noie (RIN) and frequency or hae noie (FN). The hoton number and carrier number variation with injection current have been obtained. Laer linewidth ha been calculated for different injection current value. The frequency ectrum of the intenity noie i calculated with the hel of fat Fourier tranform (FFT). The tranient behavior a well a the teady-tate reone of emiconductor laer to varyin current inut i alo analyzed becaue the tranient reone i inificant in determinin the noie characteritic of the laer outut. Reult how that both intenity and hae noie decreae with the increae of the injection current denity. Linewidth have been decreain ubtantially with increain injection current a well. Index Term Rate equation, tranient reone, relative intenity noie, hae noie, linewidth, emiconductor laer, Lanevin noie ource, elf-ureion coefficient, ain ureion. INTRODUCTION EMICONDUCTOR laer are amon the mot imortant enerated by emiconductor laer, includin their S otoelectronic device which are the mot common tye correlation, well. [], [], [3], [4], [5], [6], [7], [8], [9], [] Thee of laer roduced. They rovide chea and comact ize model, baed on the decrition of noie throuh ower laer that have a wide rane of alication in fiber otic ectral denitie, are extremely ueful in modelin the effect communication, barcode reader, laer ointer, that linear rocee have on emiconductor laer liht. CD/DVD/Blu-ray readin and recordin, laer rintin, However, the effect of device or rocee that are either cannin and increainly directional liht ource. They nonlinear (e.. hih ower tranmiion on certain tye of conit of comlex multi-layer tructure requirin ilica fiber) or not time-invariant (e.. amlitude and/or nanometer cale accuracy and an elaborate dein. The erformance of a emiconductor laer i adverely affected by the reence of noie []. Under certain oeratin ituation, however, emiconductor laer often how undeired erformance characteritic uch a intability of lain mode and eneration of extra noie. Their theoretical decrition i imortant not only from a fundamental oint of view, but alo in order to enerate new and imroved dein. The decrition can be done at variou level of frequency modulation) may not alway be decribed in the frequency domain. Our firt miletone i the calculation of D. Marcue, who reorted a model of intenity fluctuation in which the Lanevin noie ource on the hoton and carrier number are enerated with defined auto- and crocorrelation []. In hi aer [3], D. Laaoa et al. reented an alorithm deined to roduce imroved imulation of amlitude and hae fluctuation reent in the liht outut of emiconductor laer. The alorithm, known a imroved accuracy and effort, reultin in different level of time reolved imulation, i baed on the hyical rincile undertandin. The noie reent in the outut of laer diode limit their reliability when alied a liht ource in otical communication ytem, otical dic, etc. Thu analyi of the noie become crucial in imrovin the efficiency of emiconductor laer. There are ome theoretical model exit that decribe the random amlitude and hae fluctuation in the liht of oeration of emiconductor laer, and take fully into account the correlation between amlitude and hae fluctuation, a well a mode-artition noie at the laer outut. Reult obtained, uin thi alorithm, are then hown to aree with noie decrition baed on the ower ectral denity. The imulation reult aree very well with theoretical rediction ince the alorithm mimic the hyical rocee inherent to laer oeration. Sazzad M.S. Imran i currently an Aociate Profeor in the Deartment of Electrical and Electronic Enineerin at the Univerity of Dhaka, Intenity and hae noie on the outut of laer diode limit their reliability when alied a liht ource in otical communication ytem, otical dic, otical meaurin, etc. Banladeh. azzadmi@du.ac.bd The quantum noie correond to intrinic fluctuation in Farjana Ferdou Tonni and Sumona Ilam Sumi are currently uruin mater deree roram in the Det. of Electrical and Electronic the hoton number, carrier number and hae that are Enineerin at the Univerity of Dhaka, Banladeh. enerated durin the quantum interaction rocee of the tonni9@yahoo.com and umonaumi99@yahoo.com lain field with the injected chare carrier. Exce noie i 6 htt://

2 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 7 ISSN enerated when other effect, uch a the re-injection of liht dθ αaξ by otical feedback, amlify the intrinic fluctuation. = ( N N ) F dt V + θ Analyi of the laer noie i neceary for further (.) imrovement of the device erformance. In the aer [4], M. For injected carrier (electron) number: Ahmed et al. reorted a elf-contained numerical model to dn N I = A S + + FN (t) analyze the intenity and hae noie and broadenin of the dt τ e (.3) line hae. Author demontrated a ytematic technique to where Gtho i the threhold ain of olitary laer iven by, enerate the Lanevin noie ource on the hoton number, carrier number and hae while keein their auto- and c Gtho = k + ln cro-correlation atified. The technique could be nr L R f Rb (.4) undertood a a eneralization of the method by Marcue []. and G i the ain of mode whoe wavelenth i λ. G i The time variation of fluctuatin hoton number, carrier defined uch that the ureion by mode itelf and both number and hae are analyzed and their tatitic a well. the ymmetric ain ureion (SGS) and aymmetric ain Frequency ectra of both intenity and hae noie are ureion (AGS) by other mode q are taken into calculated with the hel of the fat Fourier tranform (FFT). account. [4], [5] The noie reult are comared with thoe redicted by the mall-inal analyi [5], [6], [7], [8], [9], [], [], [], [3], G = A B S ( D ( q) + H ( q) ) S q q [4]. Author ueted that their rooed model can be (.5) alied to analyze comlicated henomena under otical Here A i the linear ain, B i the coefficient of elfureion, and D(q) and H(q) are coefficient of SGS and feedback with uitable extenion of the model. In thi aer, a elf-conitent numerical model i to be AGS, reectively. Thee coefficient are iven by, alied to analyze the intenity and hae noie of 78 nm aξ A [ ( ) ] = N N bv λ λ AlGaA laer. The rate equation model for hoton number, V (.6) carrier number and otical hae for emiconductor laer 9 ω ξτ in oeratin in multimode are determined coniderin B = arcv ( N N ) nonlinear ain coefficient. The arameter of the rate 4 ε nr V (.7) equation for tyical AlGaA are ued for numerical 4 B D imulation to demontrate the noie characteritic. The noie ( ) = (.8) q 3 ( πcτ in λ ) ( λ λq ) + effect i decribed throuh relative intenity noie (RIN) and frequency or hae noie (FN). The variation of linewidth for 3λ aξ α( N N ) H ( q) = different injection current i exlained. Time varyin rofile 8πc V λq λ (.9) for the hoton number, carrier number and hae fluctuation Other arameter are: or frequency variation are demontrated. The averae hoton a = differential ain coefficient, number and averae carrier number variation with injection ξ = field confinement factor, current are alo hown. V = volume of the active reion, The followin ection lay down the theoretical model of L = lenth of the active reion, our analyi. Thi art form the bai of the work to be done α = linewidth enhancement factor, in thi aer. It talk about the rate equation for hoton number and carrier number and alo about the introduction N = time averae value of N(t), of noie ource. After the develoment of the alorithm for τ = averae electron lifetime, numerical imulation in Matlab uin Rune-Kutta method I = injection current, of the fourth order, we dicu the variou reult of the e = electron chare, imulation for the laer rate equation and analyze them in k = internal lo of the laer cavity, the third ection. Finally, the aer end u with ome N = electron number at tranarency, concludin remark on the reult. ħ = reduced Planck contant, ω = πc/λ i the anular frequency of mode, τin = electron intraband relaxation time, THEORETICAL MODEL OF ANALYSIS Rcv = diole moment and. Laer Rate Equation for Photon Number, It NS = electron number characterizin elf-ureion Phae and Carrier Number coefficient. The reent noie analyi of the olitary emiconductor The function FS(t), Fθ(t) and FN(t) are Lanevin noie laer i baed on the followin multimode rate equation ource for hoton number, it hae and carrier number, model of the injected carrier number N(t) and modal hoton reectively. They are added to the rate equation to account number S(t) and hoton hae θ(t) with =, ±, ±,, ±6. for intrinic fluctuation of the laer aociated with quantum For hoton number: tranition of electron between the valence and conduction ds ξ an V = ( G Gtho ) S + + F (.) band [6]. S dt [ ( λ λ ) δλ] + Thee noie ource are iven by, For hoton hae: 6 htt://

3 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 8 ISSN VSS F S =. V SS θ =.. F θ ( S + ) (.) (.) VNN + K VSN F N =. N K FS (.) The variance are defined a, ξa ξan VSS = ( N + N ) + Gth S + V V (.3) a VSN = ξ {( N + N ) S + N} V (.4) ξa N I VNN = ( N + N ) S + + V τ e (.5) VSN K = VSS (.6), θ and N are random number eneration in rane of,, θ and N Δt i the time-te of the calculation. The central mode = with wavelenth λ i aumed to lie at the eak of the ain ectrum, and the wavelenth of the other mode i defined a, λ λ = λ + λ = λ + =, ±, ±,... nr L (.7). Noie and Sectral Linewidth The frequency content of the mode fluctuation i x 5.8 meaured in term of RIN, which i calculated from the.6 fluctuation δ S( t) = S( ti ) S in the total hoton number..4. where S ) = S ) with S S =. t.8 The noie i calculated from the Fourier frequency comonent [7]. The RIN and FN ectrum can be oriinally defined a the Fourier tranform of the autocorrelation function written a, RIN = FFT[ δs( t )] (.8) i S T FN [ ( )] = FFT ν t (.9) i T where variation of the otical hae i iven a, dθ θ ) ν ) = = (.) π dt π and θ ) = θ ) θ ) (.) Laer linewidth i the full-width at half-maximum (FWHM) ectrum of the laer oeratin in inle-mode. It i determined from the low frequency comonent of the frequency or hae noie a, egth ( + α ) F = 4πFN = ω= I π I ( I I th ) I th (.) 6 htt:// where I evn = = current at tranarency. τ 3 NUMERICAL SIMULATION AND DISCUSSION The aim of the author i to obtain the hoton number S(t), hae number θ(t) and carrier number N(t) and correondin noie term throuh numerical imulation. The fourth order Rune-Kutta [], [3], [4], [5] alorithm ha been ued to olve the rate equation to obtain the reult. For the numerical interation, a hort time interval of Δt=5 ha been ued. Such a mall value of Δt roduce noie ource that can aroximately decribe a white noie ectrum u to a frequency of GHz. Thi i o taken uch that the behavior of the laer both before and after relaxation frequency can be tudied. Value of the imulation arameter for 78 nm AlGaA emiconductor laer are- a =.75x - m 3 -, b = 3x 9 m 3 A -, Rcv =.8x -57 C m, δλ = 3 nm, α =.6, ξ =., τin =., τ =.79 n, N =.7x 8, N =.x 8, V = μm 3, d =. μm, L = 3 μm, nr = 3.6, k = /cm, Rf =., Rb = Fluctuation of Photon and Carrier Number The averae number of hoton for variou mode, calculated at an injection current of.7 time the threhold value, i lotted in Fi. 3.. From the fiure it can be een that the averae hoton number i maximum for mode number m = +. Thi indicate that mode no. m = +, with wavelenth λ+ = 78.3 nm, i the dominant mode. Dominant mode hift from λ to λ+ due to nonlinear ain ureion. Time Averae Photon Number.6.4. Total mode= Mode Number, m Fi. 3.: Averae number of hoton for variou mode for injection current I =.7. Throuh the numerical imulation, it wa oible to obtain and lot the time varyin rofile of the hoton number S(t) for variou mode, total hoton number S(t) and the carrier number N(t). Fi. 3., 3.3 and 3.4 how thee rofile for an injection current I equal to.7 time the threhold value Ith. Fi. 3. and 3.3 how the rofile before termination of tranient for hoton number variation and fi. 3.4 how carrier number variation. Fi. 3. and 3.3 how the rofile for teady tate reone after termination of the tranient for hoton number fluctuation and fi. 3.4 how carrier number fluctuation. It i imortant to analyze total hoton number variation, becaue combined hoton number ive u the laer outut ower and ractically we cannot earate different mode at the laer outut.

4 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 9 ISSN Modal Photon Number, S (t) x 5 Total mode=3 Mode m= Mode m=+ Mode m=+ time even after the tranient reone ha ended. It can alo be een that the rofile for teady tate reone ive u a dominant mode m = + and the hoton number i maximum for thi mode no. +. The other mode are well ureed by elf and mutual ain ureion mechanim..9 x x 5 Carrier Number N(t) Modal Photon Number, S (t).5.5 Total mode=3 Mode m= Mode m=+ Mode m= x Fi. 3.: Time varyin rofile of hoton number. Durin tranient for different mode. After termination of tranient for different mode. Total Photon Number S(t) Total Photon Number S(t) 4 x Total mode x I/I th Total mode Fi. 3.3: Time varyin rofile of total hoton number durin tranient. Time varyin rofile of total hoton number after termination of tranient. From Fi. 3. and, it can be een that initially the fluctuation in hoton number ha a tranient reone and we don t et any dominant mode; but after the termination of tranient, all the mode fluctuate cloe to their averae value. Thee fluctuation do not die away, but continue with 6 htt:// Carrier Number N(t) Fi. 3.4: Time varyin rofile of carrier number durin tranient. Time varyin rofile of carrier number after termination of tranient. The ame henomena can be oberved for the time varyin rofile of the total hoton number and carrier (electron) number. Initially, time varyin hoton number and carrier number how tranient reone and after few n tranient die away and both total hoton number and the carrier number fluctuate around their averae value. To enable comarion and to oberve the effect of chane in injection current on the time varyin rofile of hoton number and electron number, Fi. 3.5, (c) and (d) were lotted for three different injection current, I =.3Ith, I =.5Ith and I =.7Ith. The behavior of thee hyical quantitie before and after the tranient reone i hown earately. Fi. 3.5(c) and (d) how that althouh the averae carrier number remain ame there i a decreae in the manitude of the fluctuation with increain injection current I. Thi decreae alo reflect in the fluctuation of the total hoton number [a hown in fi. 3.5], that i, intenity noie decreae with increain current injection. Fi. 3.5 how that the averae hoton number increae with increain injection current; reetition of the hoton number fluctuation become fater a well, which indicate a hift of the relaxation ocillation frequency toward the hiher frequency.

5 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 3 ISSN Total Photon Number S(t) 4 x Total mode 3 =.3 =.5 S av 3 x =.6 A Total mode = x 5 =.3; ; Total mode x 8 I.45 th =.6 A Total Photon Number S(t) Carrier Number N(t) Carrier Number N(t) x 5 =.5; ; Total mode x 5 ; ; Total mode x x 8 =.3; x 8 =.5; Time(n).76 x 8 ; (d) Fi. 3.5: Time varyin rofile of total hoton and carrier number durin and after tranient reone for I =.3, I =.5, I =.7. Tranient reone of the hoton number. Steady-tate reone of the hoton number. (c) Tranient reone of carrier number. (d) Steadytate reone of the carrier number. 3. Averae Photon and Averae Carrier Number It i oible to et time averae hoton number and carrier number for different injection current I/Ith =.9,.,., throuh numerical imulation a hown in Fi. 3.6 and. 6 htt:// N av Fi. 3.6: Profile of the averae hoton number for different value of =.3 injection current. Profile of the averae carrier number for different =.5 value of injection current. From Fi. 3.6, it can be tated that before the threhold value of injection current I < Ith, only few hoton were enerated due to ontaneou emiion only and the averae hoton number wa near to. But after croin the threhold value of the injection current I > Ith, timulated emiion tart and the number of hoton enerated linearly deend on the carrier injection. Thu the averae hoton (c) number increae in a linear manner with the injection current I. From Fi. 3.6, it can be een that before reachin the threhold value, the averae carrier number increae very raidly in a linear manner. After croin the threhold, timulated emiion tart and a we increae carrier injection more and more carrier recombine to enerate hoton. Thu averae carrier number maintain a contant value. 3. Fluctuation of Otical Phae To enable comarion and to oberve the effect of chane in injection current on the time varyin rofile of frequency fluctuation, Fi. 3.7 and are lotted for three different injection current I =.3Ith, I =.5Ith and I =.7Ith for the dominant mode m = +. The behavior of thi hyical quantity before and after the tranient reone i hown earately. From the fiure, it i een that fluctuation of the ocillatin hae are ureed and become reular with increain injection current I. Sureion of hae fluctuation occur becaue when the current i far from threhold, the contribution of the random ontaneou tranition to the emitted liht can be nelected when comared to the timulated tranition and hence, the emitted liht become more coherent.

6 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 3 ISSN We conider only dominant mode to calculate the fluctuation of the otical hae of the laer outut. Other mode how qualitatively ame fluctuation characteritic. Thi i becaue for multimode laer we have to conider hae noie for different mode earately for the otical hae may not even be defined for a laer ocillatin on multile reonator mode. [8] low frequency reime. That how that the noie characteritic of the laer are affected by the natural reonance of electron and hoton oulation. Other ureed mode how qualitatively ame FN rofile thouh the noie level are relatively hiher due to more fluctuation of the otical hae. Otical Phae Variation (Hz).5 x Mode no. m=+ Total mode=3 =.3 =.5 Relative Intenity Noie (Hz - ) Total mode= Frequency (Hz) 5 x 8 =.3; ; Mode no. m=+ 8 Mode no. m=+ Otical Phae Variation (Hz) x 8 =.5; ; Mode no. m= x 8 ; ; Mode no. m= Frequency (Hz) Fi. 3.7: Time varyin rofile of otical hae fluctuation for different value of injection current. Durin tranient tate. After tranient tate. Phae Noie (Hz) 7 Fi. 3.8: Frequency ectrum of quantum RIN for I =.7. Frequency ectrum of hae noie (FN) for I = Relative Intenity Noie (RIN) & Phae Noie (FN) In the outut of all tye of laer, noie fluctuation are reent that are reater than the natural quantum noie of the hoton tream. Thi i becaue the quantum fluctuation in the electron and hoton oulation are amlified in the otical reonator. Thi in turn may be attributed to the dicrete and random nature of the emiion and recombination rocee. The effect of fluctuation in the hoton and carrier oulation on the outut of the laer i imilar to that in which would be roduced by deliberate modulation of the two oulation. Comlementary rocee of hoton emiion or abortion and electron recombination or eneration are correlated in time. All thee comlementary rocee are taken into conideration in our laer rate equation while analyzin the overall effect. The reone of the laer to the fluctuation deend on the umin current. There i a reonant interaction that manifie the noie over a certain band of frequencie around the relaxation frequency. Fi. 3.8 and how the ectrum of quantum RIN and FN (for the dominant mode m = +) for injection current I =.7Ith. From the fiure we ee that both RIN and FN how a ronounced eak around their relaxation ocillation frequency. Sectrum of the RIN and FN are quite flatter at Relative Intenity Noie (Hz - ) Phae Noie (Hz) =.3 =.5 Total mode Frequency (Hz) =.3 =.5 Mode no. m= Frequency (Hz) Fi. 3.9: RIN ectrum for different value of injection current. FN ectrum for different value of injection current. 6 htt://

7 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 3 ISSN For a comarative tudy of the effect of injection current on the RIN and FN ectrum, Fi. 3.9 and are obtained for three different value of injection current, for I =.7Ith, I =.5Ith and I =.3Ith. From the fiure it i een that the increae of the reetition of the hoton and carrier fluctuation with increain injection current I correond to a hift of the relaxation ocillation frequency of the RIN and FN ectrum toward the hiher frequency ide. On the other hand, the ureion of the hoton and carrier fluctuation with increain I lead to a correondin decreae in the level of the RIN ectrum. 3.3 Laer Linewidth Laer linewidth can be calculated from the equation (.) uin FN at very low frequencie due to flatne of FN at lower frequencie. Uin hae noie at frequency khz, laer linewidth for the mode number m = + at variou injection current were calculated and the rofile of linewidth variation with different injection current wa hown in the fi. 3.. Linewidth (Hz) Other mode how qualitatively ame linewidth variation. The fiure rove the raid narrowin of linewidth Δf with increain current I near threhold. After timulated emiion or lain tart, the laer maintain almot a contant linewidth with increain current injection. Thi decreae of the linewidth correond to the decreae of the hae fluctuation or frequency noie a evident in Fi CONCLUSION The obtained reult can be ummarized a follow. ) With increain injection current, there i a hift of the relaxation ocillation frequency toward the hiher frequency. ) It i found that before the threhold value of injection current, only few hoton were enerated due to ontaneou emiion only and the averae hoton number wa near to. But after croin the threhold value of the injection current, timulated emiion tart and the number of hoton enerated linearly deend on the carrier injection. 3) It i een that before reachin the threhold value, the averae carrier number increae very raidly in linear 6 htt:// manner. After croin the threhold, averae carrier number maintain a contant value. 4) It ha been oberved that both RIN and FN how a ronounced eak around their relaxation ocillation frequency. Sectrum of the RIN and FN are quite flatter at low frequency reime. The noie characteritic of the laer are affected by the natural reonance of electron and hoton oulation. 5) With increain the injection current near threhold, raid narrowin of laer linewidth i oberved. After timulated emiion or lain tart, the laer maintain almot a contant linewidth with increain current injection. REFERENCES [] Minoru Yamada, Theory of Semiconductor Laer From Bai of Quantum Electronic to Analye of the Mode Cometition Phenomena and Noie, Sriner Serie in Otical Science, vol. 85, 5. [] D. Marcue, Comuter imulation of laer hoton fluctuation: theory of inle-cavity laer, IEEE J. Quantum Electron., vol. QE- 4.5 x 7, no., , 984. I 4 th = A mode no, m = -6 Mode no. m=+ [3] D. Laaoa, M. Vea-Leal and C. Fanana, Imroved time-reolved 3.5 imulation of amlitude and hae fluctuation in emiconductor 3 laer liht, Journal of Ot. Quant. Electron., vol. 4, ,.5 8. [4] M. Ahmed, M. Yamada and M. Saito, Numerical modelin of.5 intenity and hae noie in emiconductor laer, IEEE J. Fittin curve Oriinal curve Quantum Electron., vol. 37, no.,. 6-6,..5 [5] D. E. McCumber, Intenity fluctuation in the outut of CW laer ocillator, Phy. Rev., vol. 4,. 36 3, ( ) [6] H. Hau, Quantum-mechanical rate equation for emiconductor Fi. 3.: Profile of linewidth variation with different injection current for laer, Phy. Rev., vol. 84, , 969. mode no m = +. [7] T. Paoli, Near-threhold behavior of the intrinic reonant frequency in a emiconductor laer, IEEE J. Quantum Electron., vol. QE-5,. 87 8, 979. [8] G. Arnold and K. Petermann, Intrinic noie of emiconductor laer in otical communication ytem, Ot. Quantum Electron., vol.,.7 9, 98. [9] C. H. Henry, Theory of the hae noie and ower ectrum of a inle mode injection laer, IEEE J. Quantum Electron., vol. QE-9, , 985. [] C. H. Henry, Phae noie in injection laer, IEEE J. Lihtwave Technol., vol. LT-4,. 98 3, 986. [] M. Yamada, Theory of mode cometition noie in emiconductor injectionlaer, IEEE J. Quantum Electron., vol. QE-,. 5 59, 986. [] M. Yamada, Theoretical analyi of line-broadenin due to modecometitionand otical feedback in emiconductor injection laer, Tran. IEICE, vol. E7,. 5 6, 988. [3] G. P. Arawal and G. R. Gray, Intenity and hae noie in microcavityurface-emittin emiconductor laer, Al. Phy. Lett., vol. 59, , 99. [4] M. Yamada, Variation of intenity noie and frequency noie with the ontaneou emiion factor in emiconductor laer, IEEE J. Quantum Electron., vol. 3,. 5 59, 994. [5] M. Ahmed, Numerical characterization of intenity and frequency fluctuation aociated with mode hoin and inle-mode jitterin in emiconductor laer, Phyic D, vol. 76,., 3.

8 International Journal of Scientific & Enineerin Reearch, Volume 7, Iue, November-6 33 ISSN [6] Paul L. Evrie and Javier E. Habun, A firt coure in comutational hyic, Second Edition, Jone and Bartlett Publiher,. [7] C. Serrat and C. Maoller, Modelin atial effect in multilonitudinal-mode emiconductor laer, Phyical Review A, vol. 73,. 438 (-6), 6. [8] R. Pachotta, H.R. Telle and U. Keller, Noie of Solid State Laer, Solid-State Laer and Alication, CRC Pre, Boca Raton FL, Chater, , 7. 6 htt://