Rate Equations for modeling dispersive nonlinearity in Fabry-Perot semiconductor optical amplifiers

Transcription

1 Rate Equations for modelin disersive nonlinearity in Fabry-Perot semiconductor otical amlifiers Michael D. Sánchez, Penyue Wen, Matthias Gross, Sadik C. Esener University of California San Dieo, La Jolla, CA USA Abstract: We model the non-linear ain characteristics of a Fabry-Perot semiconductor otical amlifier usin a modified hoton density rate equation. Good areement is found with exerimental results, with the simulation accurately reroducin all the major characteristics of the amlifier. To our knowlede, this is the first calculation usin only the rate equations that accurately redicts the ain and nonlinear behavior of FPSOAs. 003 Otical Society of America OCIS codes: ( ) Laser amlifiers; ( ) Laser theory; ( ) Semiconductor otical amlifiers; (30.430) Nonlinear otical devices; ( ) Semiconductor nonlinear otics includin MQW References and Links 1. D. Wiedenmann, B. Moeller, R. Michalzik, and K.J.Ebelin, Performance characteristics of vertical-cavity semiconductor laser amlifiers, Electron. Lett. 3, (1996). C. Tombin, T. Saitoh and T. Mukai, Performance rediction for vertical-cavity semiconductor laser amlifiers, IEEE J. Quantum Electron. 30, (1994) 3. J. Pirek, S. Bjorlin and E. Bowers, Desin and analysis of vertical-cavity semiconductor otical amlifiers, IEEE J. Quantum Electron. 37, (001) 4. Daniel T. Cassidy, Comarison of rate equation and Fabry-Perot aroaches to modelin a diode laser Al. Ot., (1983) 5. Adams, M.J., Collins, J.V., and Hennin, I.D., Analysis of semiconductor laser otical amlifiers, IEE Proc. J Otoelectron. 13, 58-63, (1985). 6. G. P. Arawal and N. K. Dutta, Semiconductor lasers, (Kluwer Academic, 1993) 7. M.J. Adams, Time Deendent Analysis of Active and Passive Otical Bistability in Semiconductors, IEE Proceedins J Otoelectron. 13, , (1985). 8. P. Wen, M. Sánchez, M. Gross, O. Kibar, S. Esener, New hoton density rate equation for Fabry-Perot semiconductor otical amlifiers (FP SOAs), in Physics and Simulation of Otoelectronic Devices X, Proc. SPIE 4646, 43-50, (00) 9. Royo, P; Koda, R; Coldren, L.A., Vertical cavity semiconductor otical amlifiers: comarison of Fabry- Perot and rate equation aroaches., IEEE J. Quantum Electron. 38, 79-84, (00). 10. M. Sánchez, P. Wen, M. Gross, S. Esener, Nonlinear ain in vertical-cavity semiconductor otical amlifiers. IEEE Phot. Tech. Lett. 15, 507-9, (003). 11. P. Wen, M. Sánchez, M. Gross, S. Esener, Vertical-cavity otical AND ate, Ot. Commun. 19, , (003) 1. T.E. Sale, Vertical Cavity Surface Emittin Lasers, (Research Studies Press, Somerset, Enland, 1995) 13. Coldren, L, Corzine, S., Diode Lasers and Photonic Interated Circuits, (Wiley-Interscience, New York, NY, 1995) 14. Shin, J.H., Hwan, J.K, Ha, H, Lee, Y.H., Anamalous above-threshold sontaneous emission in ainuided vertical-cavity surface-emittin lasers, Al. Phys. Lett. 68, , (1996) 15. Ha, KH, Lee, YH, Determiniation of Cavity Loss in Proton Imlanted Vertical-Cavity Surface Emittin Lasers, Jn. J. Al. Phys. 37, L37-L374, (1998) Introduction Fabry-Perot semiconductor otical amlifiers (FPSOAs) have been a subject of research for more than two decades. As one tye of FPSOAs, vertical-cavity semiconductor otical (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 689

2 amlifiers (VCSOAs) have been investiated in recent years [1-3]. They have at least two advantaes over traditional ede-emittin semiconductor otical amlifiers (EESOAs). First, the hih coulin loss from the narrow, asymmetric ain reion of EESOAs is reduced in VCSOAs by virtue of the relatively lare (~10 um), circular inut / outut structure. Second, the mode-artitionin noise is suressed, due to sinle lonitudinal mode oeration of VCSOAs. Furthermore, -D arrays of VCSOAs are easily fabricated, and thus attractive for arallel alications such as otical information rocessin, otical interconnects and remote imain systems []. A difficulty in the modelin of FPSOAs is that the standard laser rate equations fail to accurately redict the behavior of Fabry-Perot (FP) lasers and amlifiers subjected to external otical injection [4]. The Fabry-Perot equations are usually used instead of the rate equations when considerin this class of amlifier [3,5]. However, an accurate solution based on the rate equations is desirable for a number of reasons. For one, device oeration very near and throuh the lasin threshold can be redicted, whereas the Fabry-Perot equations have a sinularity at the lasin threshold. An additional advantae is that the sontaneous emission of the amlifier is included in a self-consistent manner. Finally, the rate equations can be used to model multi-mode oeration and noise dynamics [6]. Several authors have suested hybrid analyses [3,7] or modifications to the rate equations [8,9] in order to more accurately model FPSOAs. Gain saturation has been of articular concern in these reorts, both as a eneral modelin issue and because of the resultin nonlinear behavior. The nonlinearity is a result of the coulin between the carrier concentration and index of refraction in the cavity. This henomenon is articularly imortant in resonant amlifier structures, since the resonant wavelenth shifts as the inut injection is increased and the ain saturates. Under roer conditions, the shift in resonance results in differential ain-like and bistable inut/outut (I/O) [10]. VCSOAs with these characteristics may be useful as nonlinear otical elements, erformin simle otical loic or reeneration [11]. The oal of this work is to redict the non-linear ain behavior usin the modified rate equations. The rate equations iven in [6], with the hoton injection term in [8], are used as a basis, with additional modifications introduced to account for disersive nonlinearity in the device. The equations are numerically solved and the redictions comared to exerimental results from a VCSOA. Good areement is found, with the effects of the nonlinearity clearly reroduced. Theory Thouh the rate equations allow a multi-mode simulation, the VCSOAs measured oerate in a sinle lonitudinal and transverse mode near the threshold, and also usually have a dominant olarization state. Thus, the simulation detailed here is limited to the sinle-mode rate equations, and a sinle olarization. To bein with, the rate equations are listed in eneralized form (as in [6]): dn dt η I q dn e e = γ ene GN, = ( G γ ) dt These equations reresent the time rate of chane in the carrier (N e ) and hoton (N ) number in the laser cavity, with several substitutions to simlify the equations. In the carrier rate equation, there are three terms. The first reresents the electrical injection of carriers, via the bias current I, differential efficiency η e, and chare of an electron, q. The middle term contains a recombination rate term γ e, which is the inverse of the electron lifetime, τ e : N P + R s + C γ e = 1 τ e = A + Bn + Cn () The three coefficients A,B,C reresent non-radiative, radiative and Auer recombination, resectively, and n is the carrier density. The last term of the carrier rate equation is the N inj (1) (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 690

3 consumtion of carriers by stimulated emission, which is directly roortional to the ain rate (G) and the hoton number. Next, consider the hoton equation: γ is the hoton loss rate, and R s is the sontaneous emission rate. N inj is the number of hotons externally injected into the cavity, and C is a coulin coefficient. The ain is aroximated usin a linear exression, since only a narrow rane of bias currents is to be considered: G = Γv, = a( n n0 ), v = c µ (3) Here, Γ is the lateral confinement factor, a is the differential ain coefficient, n is the carrier density, n 0 the transarency carrier density, µ is the rou refractive index, and c is the seed of liht in vacuum. Thus v is the rou velocity of liht in the laser cavity. This value is a weak function of the carrier density, but in the first order aroximation used here, it can be considered a constant. The hoton loss rate is defined as: ( 1 L) ln[ 1 R ] γ = 1 τ = v ( α + α ), α R (4) m i m = The mirror loss is α m, and α i is the internal loss, with τ reresentin the hoton lifetime. The mirror loss is aroximated as a distributed loss over the lenth of the cavity (L), and is a function of the mirror reflectivites R 1 and R, which are the front and rear mirrors, resectively. The next term of the hoton equation in Eq. (1) reresents the sontaneous emission hotons: 1 R s = β BnN = β Bn V, V = Γ V (5) s e s l The sontaneous emission factor is β s, and V is the volume of the ain medium. For ede emittin FPSOAs, the entire laser cavity is the ain medium, but in a quantum well VCSOA, the ain medium is only a fraction of the laser cavity. This fact is accounted for by the use of a lonitudinal confinement factor, which is desinated Γ l. To this oint, the rate equations listed here follow the standard form in the literature. The modification that allows accurate modelin of FPSOAs is the use of an alternate hoton injection coulin term, as roosed by Wen, et al. in [8]. The coulin term used is this aer is very similar: C R N = inj = η R R 1 in ( 1+ G )( 1 R Cosφ ) + ( R G kl) G s s (1 R ) s (1 R ) Sin(φ ) (1 R1 ) N + 4R Sin φ τ The wavenumber is k, the hoton roundtri time is τ RT, and η in is a fittin arameter. The sinle ass amlitude ain is iven by G s and φ reresents the sinle ass hase chane. Equation. (6) is derived from the summation of the otical field and its multile reflections inside the laser cavity. This term differs from the exression in [8] by the addition of a fittin arameter η in, and a factor of (1-R 1 ). This factor results from the fact that in [8], the coulin term is exressed as a function of the hoton number injected into the cavity. Here, N inj reresents the hotons delivered to the front face of the amlifier, and so a roortionality factor is required. The fittin arameter is justified as follows: Note that the injection term is made into a rate by dividin by the roundtri time (τ RT ). Some correction is required to account for the reality that the actual rate at which hotons are added to the cavity is somewhat less than the term above, since it takes more than one roundtri to build u the interference. Such a correction is included in the fittin arameter η in, which also accounts for hysical coulin loss in the otical system. RT inj, (6) (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 691

4 The deendence of the index of refraction on carrier density is included in the sinle ass hase chane. The hase term is derived from the steady state solution of the hase rate equation [6]. The sinle ass ain and hase are iven by: G s = e ( ΓΓl L α i L) ( β ) ΓΓ La( n n ), φ = φ0 c l 1 (7) In the hase exression, β c is the linewidth enhancement factor, and φ 0 is the detunin from the resonant frequency. The carrier density without otical injection is iven in the arameter n 1. Note the lonitudinal confinement is resent in these equations because the ain and index chane occur only within the active reion. The second term of the hase equation has been included in the calculations of this aer in order to model the nonlinear amlifier behavior. That term coules the otical hase to the carrier density in the amlifier, and Eq. (6) is a sensitive function of the hase, φ. Thus, Eqs. (6) and (7) imlicitly form an additional coulin between the hoton density and carrier rate equations that is not catured by the traditional rate equations. Now, with substitution of equations -7 into equation 1, one can solve the couled equations to model the behavior of a FPSOA, includin non-linear carrier consumtion and index of refraction. The calculated hoton number is converted to a measurable quantity usin well-known Eq. (6). An outut coulin coefficient η out is included in this conversion, to account for losses in the otical system before the detector. Equations (1-7) define all the exressions necessary to this simulation, but do not have a eneral analytic solution, so redictions are obtained numerically, usin Mathematica (Wolfram Research). These equations are alied to a VCSOA in the next section, and the redictions comared to measured results. Exeriment results To test the redictions of these equations, a VCSOA oerated in reflection mode has been used. A eneral schematic of an electrically umed, reflection mode VCSOA structure is shown in Fi. 1, with the transverse and lonitudinal otical intensity distributions suerimosed on the structural diaram. Fi. 1. General schematic of VCSOA, showin the transverse (left) and lonitudinal (riht) otical intensity rofile. (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 69

5 The VCSOA consists of two multi-layer Bra reflectors rown vertically on the substrate, with a short cavity and quantum well (QW) ain layer in between. Proton imlantation or oxidation confine the electrical current and otical field. The VCSOA can be modeled as a FPSOA if the front and rear mirrors are aroximated as a air of hard mirrors laced at the enetration deth of the lonitudinal otical field. The confinement factors, Γ and Γ l are defined by the overla of the otical mode with the active ain reion (QW). The VCSOA used is a roton imlanted VCSEL manufactured by Emcore. The aerture size is 8 µm. Further details of the device and the exerimental setu can be found in [10]. One difficulty of the rate equations is the introduction of many arameters, most of which are not directly measurable from a comleted device. Most of the required arameters have been taken from the literature, and a few could be directly measured. Table 1 lists all the arameters and the values used in the calculation. Table 1: Rate equation arameters for VCSOA Fixed Variable Parameter Value Parameter Value λ nm α [10,15] cm -1 µ [1] 3.3 η e 0.4 L 3λ 0 /µ a [15] cm Area π(4 µm) n 0 [15] cm -3 Γ [1] 1.0 β s [6] Γ l [14,15] 0.1 β c [6].7 R 1, R 0.99, η in, η out 0.0, 0.7 A[14,15] B [14] cm 3 C [14] cm 6 The arameters listed as Fixed, are measured or taken from the literature without modification. The "Variable arameters are fittin arameters. For those values with references listed, the literature has been used to identify an aroximate value and reasonable rane, then fine-tuned by fittin to measured data. The cavity lenth and mirror reflectivites are calculated usin a SEM icture of the laser die to count the mirror airs, then alyin standard equations for reflectivity and enetration deth [13]. All arameters excet β c, α i, η in and, η out are fixed by fittin the L-I curve of the laser. The internal loss is determined by fittin to the width of the ain window of the device at low inut ower (100nW), as in [10], and linewidth enhancement factor is adjusted to fit the measured inut/outut (I/O) characteristic of a device at a sinle bias current. Coulin loss can be estimated from characterization of the otical system, but bears some deree of uncertainty. Thus, some variation of η in and η out is also allowed when fittin the I/O characteristic. The value of η in resented here comares favorably (factor of ) to our estimates based on calculation of the hoton buildu time. The first ste is fittin the steady state solution, without otical injection, to the measured L-I curve of a device. The VCSOA used reaches the lasin threshold with about 6.1 ma of umin current. The fit near threshold is shown in Fi.. (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 693

6 100 Outut ower (µw) Ibias (ma) Fi.. L-I curve. Dots are measured data, solid line is calculation After achievin a reasonable fit to the L-I curve, the trends of the simulation were investiated, to see if it relicated our exerimental observations. One test of the simulation is the calculation of ain vs. detunin. Fi. 3 shows the comarison: nw µw 10 µw nw µw 10 µw Gain 40 Gain Detunin (m) Detunin (m) Fi. 3. Gain vs. detunin for several inut owers. Calculated result on the left, measured data on the riht. The non-linearity introduces an asymmetry to the ain window and the detunin for eak ain shifts with inut ower. The calculated results look quite similar to the measured data. Additionally, the width and eak shifts of the ain windows are quantitatively equivalent. The ain at small inut owers is slihtly overestimated by the simulation. At 500 nw inut ower, the redicted ain is 15% reater than the measured value. The discreancy in ain at low inut owers may be a result of the lack of mixin terms between the amlified sontaneous emission and injected hotons. When the injected hoton oulation is of the same order of manitude as the sontaneously enerated hotons, cometition between the two is exected. Most imortantly, the otical transfer characteristic of the VCSOA can be comared to that of the simulation for a fixed bias current, as shown in Fi. 4. The detunins used in Fi. 4 corresond to to 36 m of detunin. The exerimental data were taken over a detunin rane of -0 to 40 m. The sliht offset between calculation and exeriment is due to uncertainty in determination of the resonant wavelenth of the device. The total san and sacin of wavelenths matches the exerimental results within the uncertainty of our measurement. The results of Fis. -4 suest that this model is valid. (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 694

7 50 d Pout (µw) c a b P in (µw) Fi. 4. Outut vs. inut ower for several detunins at 5.8mA bias. Symbols are measured data, solid lines are calculated. Detunins are (mrad): a) 0.5, b) 0.05, c) 0.4, d) 0.78 One of the oals of this work is to roduce a model that is valid across a rane of bias currents. Thus, as an additional test, the I/O for this device is calculated at a different bias current (5.6mA), with all other arameters unchaned. The results are lotted aainst exerimental measurement in Fi. 5: 00 d out (µw) c b a P in (µw) Fi. 5. Outut vs. inut ower for several detunins at 5.6mA bias. Symbols are measured data, solid lines are calculated. Detunins are (mrad): a) 0.6, b) 0.1, c) 0.35, d) The detunins used in Fi. 5 corresond to 7 to 30 m of detunin. The exerimental data were taken over a detunin rane of -0 to 35 m. The rane and sacin of detunin once aain matches the exerimental data with the accuracy of our observation. The offset between the two ranes suests some error in the estimation of the resonant wavelenth. The simulation arees well with the exerimental data, thouh there is some overshoot in the calculated results. This is likely due to the use of a linear ain exression, where a loarithmic rule would be more accurate. The areement obtained in Fis. 4 and 5 is as ood as can be achieved usin existin models of the nonlinearity in FPSOAs [10], which rely on the Fabry-Perot ain equations. The results demonstrate that this model accurately reroduces all the major features of nonlinear FPSOA oeration, and, when roerly calibrated, is accurate across a rane of bias (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 695

8 currents and wavelenths. With additional refinement, such as the use of a loarithmic ain exression, even closer areement should be achievable. Conclusion We have solved a modified form of the rate equations that includes the contribution of the disersive non-linearity resent in Fabry-Perot semiconductor amlifiers. The model is calibrated and comared aainst measurements of a VCSOA. Good areement is found and the model reroduces all the major features of the device characteristics. Use of the rate equations enables analyses of FPSOAs that are cumbersome or imossible with FP ain equations, such as noise, multi-mode, and hih-seed oeration. The rate equations also rovide the advantae of automatic inclusion of ASE and the ability to model device oeration all the way throuh the lasin threshold. One drawback of the aroach is the lare arameter set that must be determined to accurately model a real device. To our knowlede, this is the first calculation usin only the rate equations that accurately redicts the ain and nonlinearity of FPSOAs. (C) 003 OSA 0 October 003 / Vol. 11, No. 1 / OPTICS EXPRESS 696