Chapter 1 Baic Decription of Laer Diode Dynamic by Spatially Averaged Rate Equation: Condition of Validity A laer diode i a device in which an electric current input i converted to an output of photon. The time-dependent relation between the input electric current and the output photon i commonly decribed by a pair of equation decribing the time evolution of photon and carrier denitie inide the laer medium. Thi pair of equation, known a the laer rate equation, i ued extenively in the following chapter. It i, therefore, appropriate, in thi firt chapter, to ummarize the reult of Moreno [] regarding the condition under which the rate equation are applicable. 1.1 The Local Rate Equation The tarting point for the analyi of laer kinetic involve the coupled rate equation, which are baically local photon and injected carrier conervation equation [3]: @X C C c @X C @t @z @X @X c @t @z dn D A.N N tr /X C C ˇ N ; D A.N N tr /X C ˇ N ; D J ed N A.N N tr /.X C C X /; (1.1a) (1.1b) (1.1c) where z i the patial dimenion along the length of the laer, with reflector of (power) reflectivitie R placed at z D L=, X C and X are the forward and backward propagating photon denitie (which are proportional to the light intenitie), N i the local carrier denity, N tr i the electron denity where the emiconductor medium become tranparent, c i the group velocity of the waveguide mode, A i the gain contant in 1 /(unit carrier denity), ˇ i the fraction of pontaneou emiion entering the laing mode, i the pontaneou recombination lifetime of the carrier, z i the ditance along the active medium with z D at the center of the laer, J i the pump current denity, e i the electronic charge, and d the thickne K. Lau, Ultra-high Frequency Linear Fiber Optic Sytem, Springer Serie in Optical Science 159, DOI 1.17/978-3-64-16458-3 1, Springer-Verlag Berlin Heidelberg 11 3
4 1 Baic Decription of Laer Diode Dynamic of the active region in which the carrier are confined. For the remaining of thi chapter, it i aumed that N tr D, the only conequence of which i a DC hift in the electron denity, which i of ignificance only in conidering laing threhold. In addition, the following implifying aumption are made in writing down (1.1): (a) The quantitie X decribe the local photon number denitie of a longitudinal mode of the paive laer cavity at a given (longitudinal) poition in the laer cavity, at time t, integrated over the laing linewih of the longitudinal mode, whichiaumeobemuchnarrowerthan the homogeneouly broadened laer gain pectrum. (b) The gain coefficient (AN ) i a linear function of the injected carrier denity N (A i popularly known a the differential optical gain coefficient and i hown in later chapter to play a key role in determining direct modulation bandwih of laer diode). (c) Variation of the carrier and photon denitie in the lateral dimenion are neglected. (d) Diffuion of carrier i ignored. Aumption 1 and are very reaonable aumption that can be derived from detailed analyi [4 6]. The repreentation of the emiconductor laer a a homogeneouly broadened ytem can alo be derived from baic conideration [7]. Tranvere modal and carrier diffuion effect, ignored in aumption 3 and 4, can lead to modification of the dynamic behavior of laer [8, 9]. Equation (1.1) are to be olved ubject to the boundary condition L L X D RX C ; (1.a) L L X C D RX : (1.b) The teady-tate olution of (1.1) give the tatic photon and electron ditribution inide the laer medium and ha been olved analytically in [4]. The olution i ummarized a follow, where the zero ubcript denote teady-tate quantitie: X C.z/ D aeu.z/ ˇ Ac ; (1.3a) X.z/ D ae u.z/ ˇ Ac ; (1.3b) where a i a quantity given by the following trancendental equation:.1 ˇ/ C a inh D gl ; (1.4)
1. Spatially Averaged Rate Equation and their Range of Validity 5 where D 1.R 1/ ˇ.Ra/ C 4 R C.R 1/ ˇ Ra (1.5) and g D AJ ed i the unaturated gain, and u.z/ i given trancendentally by The electron denity N.z/ i given by.1 ˇ/u.z/ C a inh u.z/ D gz: (1.6) AcN.z/ D g 1 C a coh u.z/ ˇ : (1.7) Figure 1.1 how plot of X C.z/, X.z/,andg.z/ D AcN.z/ for a 3 m laer with three value of end-mirror reflectivitie. (a).3, (b).1, and (c).9. The high nonuniformity in the ditribution become apparent at low reflectivitie. 1. Spatially Averaged Rate Equation and their Range of Validity Equation (1.1) contitute a et of three coupled nonlinear differential equation in two variable that do not lend themelve to eay olution. Coniderable implification can be made if the longitudinal patial variable (z) i integrated over the length of the laer. Such implification i valid only when the end-mirror reflectivity i ufficiently large, A more precie definition of the range of validity of uch an aumption i given in the following, ummarizing the approach of []. To begin, (1.1a)and(1.1b) are integrated in the z variable, reulting in dx C dx L L C c X C X C D A.NX C / C ˇ N ; L L c X X D A.NX / C ˇ N ; Z L= where denote the patial average L= dp C c.1 R/P.L=/ L.1 C R/ dz. Adding (1.8a)and(1.8b), L D A.NP/ C ˇ N (1.8a) (1.8b) ; (1.9)
6 1 Baic Decription of Laer Diode Dynamic a PHOTON..16.1.8 Gain X + X 5 4 3 GAIN (cm 1 ).4 1 b 15 75.5.4 75 15 z (µm) 1 Gain 8 PHOTON.3. X X + 6 4 GAIN (cm 1 ).1 c 15 75. 75 15 z (µm) 1.16 X + 8 PHOTON.1.8 X Gain 6 4 GAIN (cm 1 ).4 15 75 75 15 z (µm) Fig. 1.1 Steady-tate photon and electron-denity ditribution inide laer diode with mirror reflectivitie of (a).3,(b).1,and(c).9
1. Spatially Averaged Rate Equation and their Range of Validity 7 where P D X C C X i the total local photon denity and the boundary condition (1.) have been ued. Equation (1.1c) integrate traightforwardly to dn D J ed N A.NP/ (1.1) where a uniform pump current of denity J i aumed. A i known a the differential optical gain. It i hown in later chapter to play a key role in determining direct modulation bandwih of laer diode. Introducing factor f 1 and f a follow: f 1 D.NP/ N P ; (1.11) f D P.L=/ P.1 C R/ ; (1.1) one can write the patially averaged rate equation (1.9)and(1.1) in the following form: dp D Af 1 N P P c.1 R/f L C ˇ N ; (1.13) dn D J ed N Af 1 N P ; (1.14) which are recognized a the commonly ued rate equation [1,11] if the condition f 1 D 1; (1.15) f D 1 ln R (1.16) 1 R are atified. The firt of thee condition require, for the quantitie N and P,that the patial average of the product equal the product of the patial average. Thi condition i not atified in general, but it will be true if the electron denity N i uniform, a in the cae when R approache unity, which i apparent from Fig. 1.1c. The econd condition require the photon lo rate in (1.13) to be inverely proportional to the conventional photon lifetime. It will alo be atified if R i very cloe to unity, ince both (1.1)and(1.16)convergeto:5 at thi limit. A more precie delineation of the range of the applicability of condition (1.15) and (1.16) i obtained by calculating f 1 and f from exact teady-tate olution (1.3) (1.7), and comparing them with (1.15)and(1.16). From (1.3)and(1.7), f 1 D Z Z L P 1 C A P dz Z ; (1.17) P dz dz 1 C A P f D LX C.L=/ R P dz ; (1.18)
8 1 Baic Decription of Laer Diode Dynamic Fig. 1. Variation of f 1 and 1 f with R when ˇ 1 3 and gl > 1. 1.8 1.6 1.4 (1.15), (1.16) EXACT - calculated from(1.11), (1.1) f 1,1/f 1. 1..8 f 1.6 1/f.4...1.1 1. R where the integral are evaluated over the length of the laer. Thee integral can be numerically evaluated uing (1.3) (1.7), and the reult are hown in Fig. 1.. Figure 1. how numerically computed plot (olid line) of f 1 and 1=f a a function of end-mirror reflectivity R; the calculation wa done with the laer biaed above threhold. The dotted line are the ideal value of f 1 and f given by (1.15) and (1.16). The figure indicate that the uual rate equation are reaonably accurate for R larger than approximately. valid for laer diode contructed from III V material, which have facet reflectivitie of :3. The above reult lead to the concluion that the imple rate equation, expreed in (1.19) and(1.) (where the N and P now denote averaged quantitie, in the longitudinal patial dimenion): dn dp D J ed N ANP (1.19) D ANP P p C ˇ N (1.) (1= p D c=.l/ln.1=r/ i the claical photon lifetime and A D c) are reaonable repreentation if the end-mirror reflectivity i above. and the laer i above threhold. The pontaneou emiion factor ˇ in (1.) i a factor of two higher than that defined in (1.1) due to the incluion of photon propagating in both direction. Common GaA or quaternary laer, with the mirror formed by the cleaved crytal facet, have a reflectivity of :3 and are thu well within the cope of (1.19) and (1.).In AppendixD,the exactmall ignal verion of (1.1)) i olved numerically, and it i found that (1.19) and(1.) can very accurately decribe the mall ignal frequency repone of the laer for end-mirror reflectivitie a low a 1 3.Thii
1. Spatially Averaged Rate Equation and their Range of Validity 9 certainly not expected from a phyical tandpoint and erve a a urprie bonu for thi implification. Another factor that can render the patially uniform aumption invalid i when fat phenomena, occurring on the time cale of a cavity tranit time, are being conidered. It i obviou that the concept of cavity lifetime and that of cavity mode, appearing in (1.), are no longer applicable on that time cale. In common emiconductor laer where the cavity length i approximately 3 m, the cavity tranit time i about 3.5 p. The uual rate equation are, therefore, not applicable in decribing phenomena horter than about 5 p, or at modulation frequencie higher than 6 GHz. Modulation regime in the millimeter wave frequencie can take advantage of thi cavity round-trip effect and i known a reonant modulation, dicuedindetailinpartii of thi book. In the following chapter, (1.19)and(1.) are ued extenively and erve a the bai for mot of the analyi of the direct modulation characteritic of laer.
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