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

We model the non-linear gain characteristics of a Fabry-Perot semiconductor optical amplifier using a modified photon density rate equation. Good agreement is found with experimental results, with the simulation accurately reproducing all the major characteristics of the amplifier. To our knowledge, this is the first calculation using only the rate equations that accurately predicts the gain and nonlinear behavior of FPSOAs. ©2003 Optical Society of America OCIS codes: (140.3280) Laser amplifiers; (270.3430) Laser theory; (250.5980) Semiconductor optical amplifiers; (230.4320) Nonlinear optical devices; (190.5970) Semiconductor nonlinear optics including MQW References and Links 1. D. Wiedenmann, B. Moeller, R. Michalzik, and K.J.Ebeling, “Performance characteristics of vertical-cavity semiconductor laser amplifiers,” Electron. Lett. 32, 342-343 (1996) 2. C. Tombing, T. Saitoh and T. Mukai, “Performance prediction for vertical-cavity semiconductor laser amplifiers,” IEEE J. Quantum Electron. 30, 2491-2499 (1994) 3. J. Piprek, S. Bjorlin and E. Bowers, “Design and analysis of vertical-cavity semiconductor optical amplifiers,” IEEE J. Quantum Electron. 37, 127-134 (2001) 4. Daniel T. Cassidy, “ Comparison of rate equation and Fabry-Perot approaches to modeling a diode laser” Appl. Opt. 22, 3321-3326 (1983) 5. Adams, M.J., Collins, J.V., and Henning, I.D., “Analysis of semiconductor laser optical amplifiers”, IEE Proc. J Optoelectron. 132, 58-63, (1985). 6. G. P. Agrawal and N. K. Dutta, Semiconductor lasers, (Kluwer Academic, 1993) 7. M.J. Adams, ‘Time Dependent Analysis of Active and Passive Optical Bistability in Semiconductors”, IEE Proceedings J Optoelectron. 132, 343-348, (1985). 8. P. Wen, M. Sánchez, M. Gross, O. Kibar, S. Esener, “New photon density rate equation for Fabry-Perot semiconductor optical amplifiers (FP SOAs),” in Physics and Simulation of Optoelectronic Devices X, Proc. SPIE 4646, 243-250, (2002) 9. Royo, P; Koda, R; Coldren, L.A., “Vertical cavity semiconductor optical amplifiers: comparison of FabryPerot and rate equation approaches.”, IEEE J. Quantum Electron. 38, 279-84, (2002). 10. M. Sánchez, P. Wen, M. Gross, S. Esener, “Nonlinear gain in vertical-cavity semiconductor optical amplifiers.” IEEE Phot. Tech. Lett. 15, 507-9, (2003). 11. P. Wen, M. Sánchez, M. Gross, S. Esener, “Vertical-cavity optical AND gate”, Opt. Commun. 219, 383387, (2003) 12. T.E. Sale, Vertical Cavity Surface Emitting Lasers, (Research Studies Press, Somerset, England, 1995) 13. Coldren, L, Corzine, S., Diode Lasers and Photonic Integrated Circuits, (Wiley-Interscience, New York, NY, 1995) 14. Shin, J.H., Hwang, J.K, Ha, H, Lee, Y.H., “Anamalous above-threshold spontaneous emission in gainguided vertical-cavity surface-emitting lasers”, Appl. Phys. Lett. 68, 2180-2182, (1996) 15. Ha, KH, Lee, YH, “Determiniation of Cavity Loss in Proton Implanted Vertical-Cavity Surface Emitting Lasers,” Jpn. J. Appl. Phys. 37, L372-L374, (1998)


Introduction
Fabry-Perot semiconductor optical amplifiers (FPSOAs) have been a subject of research for more than two decades.As one type of FPSOAs, vertical-cavity semiconductor optical amplifiers (VCSOAs) have been investigated in recent years [1][2][3].They have at least two advantages over traditional edge-emitting semiconductor optical amplifiers (EESOAs).First, the high coupling loss from the narrow, asymmetric gain region of EESOAs is reduced in VCSOAs by virtue of the relatively large (~10 um), circular input / output structure.Second, the mode-partitioning noise is suppressed, due to single longitudinal mode operation of VCSOAs.Furthermore, 2-D arrays of VCSOAs are easily fabricated, and thus attractive for parallel applications such as optical information processing, optical interconnects and remote imaging systems [2].
A difficulty in the modeling of FPSOAs is that the standard laser rate equations fail to accurately predict the behavior of Fabry-Perot (FP) lasers and amplifiers subjected to external optical injection [4].The Fabry-Perot equations are usually used instead of the rate equations when considering this class of amplifier [3,5].However, an accurate solution based on the rate equations is desirable for a number of reasons.For one, device operation very near and through the lasing threshold can be predicted, whereas the Fabry-Perot equations have a singularity at the lasing threshold.An additional advantage is that the spontaneous emission of the amplifier is included in a self-consistent manner.Finally, the rate equations can be used to model multi-mode operation and noise dynamics [6].
Several authors have suggested hybrid analyses [3,7] or modifications to the rate equations [8,9] in order to more accurately model FPSOAs.Gain saturation has been of particular concern in these reports, both as a general modeling issue and because of the resulting nonlinear behavior.The nonlinearity is a result of the coupling between the carrier concentration and index of refraction in the cavity.This phenomenon is particularly important in resonant amplifier structures, since the resonant wavelength shifts as the input injection is increased and the gain saturates.Under proper conditions, the shift in resonance results in differential gain-like and bistable input/output (I/O) [10].VCSOAs with these characteristics may be useful as nonlinear optical elements, performing simple optical logic or regeneration [11].
The goal of this work is to predict the non-linear gain behavior using the modified rate equations.The rate equations given in [6], with the photon injection term in [8], are used as a basis, with additional modifications introduced to account for dispersive nonlinearity in the device.The equations are numerically solved and the predictions compared to experimental results from a VCSOA.Good agreement is found, with the effects of the nonlinearity clearly reproduced.

Theory
Though the rate equations allow a multi-mode simulation, the VCSOAs measured operate in a single longitudinal and transverse mode near the threshold, and also usually have a dominant polarization state.Thus, the simulation detailed here is limited to the single-mode rate equations, and a single polarization.To begin with, the rate equations are listed in generalized form (as in [6]): These equations represent the time rate of change in the carrier (N e ) and photon (N p ) number in the laser cavity, with several substitutions to simplify the equations.In the carrier rate equation, there are three terms.The first represents the electrical injection of carriers, via the bias current I, differential efficiency η e , and charge of an electron, q.The middle term contains a recombination rate term γ e , which is the inverse of the electron lifetime, τ e : The three coefficients A,B,C represent non-radiative, radiative and Auger recombination, respectively, and n is the carrier density.The last term of the carrier rate equation is the consumption of carriers by stimulated emission, which is directly proportional to the gain rate (G) and the photon number.
Next, consider the photon equation: γ p is the photon loss rate, and R sp is the spontaneous emission rate.N inj is the number of photons externally injected into the cavity, and C p is a coupling coefficient.The gain is approximated using a linear expression, since only a narrow range of bias currents is to be considered: Here, Γ is the lateral confinement factor, a is the differential gain coefficient, n is the carrier density, n 0 the transparency carrier density, µ g is the group refractive index, and c is the speed of light in vacuum.Thus v g is the group velocity of light in the laser cavity.This value is a weak function of the carrier density, but in the first order approximation used here, it can be considered a constant.The photon loss rate is defined as: The mirror loss is α m , and α i is the internal loss, with τ p representing the photon lifetime.
The mirror loss is approximated as a distributed loss over the length of the cavity (L), and is a function of the mirror reflectivites R 1 and R 2 , which are the front and rear mirrors, respectively.The next term of the photon equation in Eq. ( 1) represents the spontaneous emission photons: The spontaneous emission factor is β sp , and V′ is the volume of the gain medium.For edge emitting FPSOAs, the entire laser cavity is the gain medium, but in a quantum well VCSOA, the gain medium is only a fraction of the laser cavity.This fact is accounted for by the use of a longitudinal confinement factor, which is designated Γ l .
To this point, the rate equations listed here follow the standard form in the literature.The modification that allows accurate modeling of FPSOAs is the use of an alternate photon injection coupling term, as proposed by Wen, et al. in [8].The coupling term used is this paper is very similar: The wavenumber is k, the photon roundtrip time is τ RT , and η in is a fitting parameter.The single pass amplitude gain is given by G s and φ represents the single pass phase change.
Equation.( 6) is derived from the summation of the optical field and its multiple reflections inside the laser cavity.This term differs from the expression in [8] by the addition of a fitting parameter η in , and a factor of (1-R 1 ).This factor results from the fact that in [8], the coupling term is expressed as a function of the photon number injected into the cavity.Here, N inj represents the photons delivered to the front face of the amplifier, and so a proportionality factor is required.The fitting parameter is justified as follows: Note that the injection term is made into a rate by dividing by the roundtrip time (τ RT ).Some correction is required to account for the reality that the actual rate at which photons are added to the cavity is somewhat less than the term above, since it takes more than one roundtrip to build up the interference.Such a correction is included in the fitting parameter η in , which also accounts for physical coupling loss in the optical system.
The dependence of the index of refraction on carrier density is included in the single pass phase change.The phase term is derived from the steady state solution of the phase rate equation [6].The single pass gain and phase are given by: ( ) In the phase expression, β c is the linewidth enhancement factor, and φ 0 is the detuning from the resonant frequency.The carrier density without optical injection is given in the parameter n 1 .Note the longitudinal confinement is present in these equations because the gain and index change occur only within the active region.The second term of the phase equation has been included in the calculations of this paper in order to model the nonlinear amplifier behavior.That term couples the optical phase to the carrier density in the amplifier, and Eq. ( 6) is a sensitive function of the phase, φ.Thus, Eqs. ( 6) and ( 7) implicitly form an additional coupling between the photon density and carrier rate equations that is not captured by the traditional rate equations.Now, with substitution of equations 2-7 into equation 1, one can solve the coupled equations to model the behavior of a FPSOA, including non-linear carrier consumption and index of refraction.The calculated photon number is converted to a measurable quantity using well-known Eq. ( 6).An output coupling coefficient η out is included in this conversion, to account for losses in the optical system before the detector.Equations (1-7) define all the expressions necessary to this simulation, but do not have a general analytic solution, so predictions are obtained numerically, using Mathematica (Wolfram Research).These equations are applied to a VCSOA in the next section, and the predictions compared to measured results.

Experiment results
To test the predictions of these equations, a VCSOA operated in reflection mode has been used.A general schematic of an electrically pumped, reflection mode VCSOA structure is shown in Fig. 1, with the transverse and longitudinal optical intensity distributions superimposed on the structural diagram.The VCSOA consists of two multi-layer Bragg reflectors grown vertically on the substrate, with a short cavity and quantum well (QW) gain layer in between.Proton implantation or oxidation confine the electrical current and optical field.The VCSOA can be modeled as a FPSOA if the front and rear mirrors are approximated as a pair of hard mirrors placed at the penetration depth of the longitudinal optical field.The confinement factors, Γ and Γ l are defined by the overlap of the optical mode with the active gain region (QW).The VCSOA used is a proton implanted VCSEL manufactured by Emcore.The aperture size is 8 µm.Further details of the device and the experimental setup can be found in [10].
One difficulty of the rate equations is the introduction of many parameters, most of which are not directly measurable from a completed device.Most of the required parameters have been taken from the literature, and a few could be directly measured.Table 1 lists all the parameters and the values used in the calculation. 1.0 2.5 × 10 -5 Γ l [14,15] 0.1 0.99, 0.9995 η in , η out 0.02, 0.7 A [14,15] 1 × 10 8 B [14] 1 × 10 -10 cm 3 C [14] 5 × 10 -30 cm 6   The parameters listed as "Fixed", are measured or taken from the literature without modification.The "Variable" parameters are fitting parameters.For those values with references listed, the literature has been used to identify an approximate value and reasonable range, then fine-tuned by fitting to measured data.The cavity length and mirror reflectivites are calculated using a SEM picture of the laser die to count the mirror pairs, then applying standard equations for reflectivity and penetration depth [13].All parameters except β c , α i , η in and, η out are fixed by fitting the L-I curve of the laser.The internal loss is determined by fitting to the width of the gain window of the device at low input power (100nW), as in [10], and linewidth enhancement factor is adjusted to fit the measured input/output (I/O) characteristic of a device at a single bias current.Coupling loss can be estimated from characterization of the optical system, but bears some degree of uncertainty.Thus, some variation of η in and η out is also allowed when fitting the I/O characteristic.The value of η in presented here compares favorably (factor of 2) to our estimates based on calculation of the photon buildup time.
The first step is fitting the steady state solution, without optical injection, to the measured L-I curve of a device.The VCSOA used reaches the lasing threshold with about 6.1 mA of pumping current.The fit near threshold is shown in Fig. 2.After achieving a reasonable fit to the L-I curve, the trends of the simulation were investigated, to see if it replicated our experimental observations.One test of the simulation is the calculation of gain vs. detuning.Fig. 3 shows the comparison: The non-linearity introduces an asymmetry to the gain window and the detuning for peak gain shifts with input power.The calculated results look quite similar to the measured data.Additionally, the width and peak shifts of the gain windows are quantitatively equivalent.The gain at small input powers is slightly overestimated by the simulation.At 500 nW input power, the predicted gain is 15% greater than the measured value.The discrepancy in gain at low input powers may be a result of the lack of mixing terms between the amplified spontaneous emission and injected photons.When the injected photon population is of the same order of magnitude as the spontaneously generated photons, competition between the two is expected.
Most importantly, the optical transfer characteristic of the VCSOA can be compared to that of the simulation for a fixed bias current, as shown in Fig. 4. The detunings used in Fig. 4 correspond to -22 to 36 pm of detuning.The experimental data were taken over a detuning range of -20 to 40 pm.The slight offset between calculation and experiment is due to uncertainty in determination of the resonant wavelength of the device.The total span and spacing of wavelengths matches the experimental results within the uncertainty of our measurement.The results of Figs.2-4 suggest that this model is valid.One of the goals of this work is to produce a model that is valid across a range 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 parameters unchanged.The results are plotted against experimental measurement in Fig. 5: The detunings used in Fig. 5 correspond to -27 to 30 pm of detuning.The experimental data were taken over a detuning range of -20 to 35 pm.The range and spacing of detuning once again matches the experimental data with the accuracy of our observation.The offset between the two ranges suggests some error in the estimation of the resonant wavelength.The simulation agrees well with the experimental data, though there is some overshoot in the calculated results.This is likely due to the use of a linear gain expression, where a logarithmic rule would be more accurate.
The agreement obtained in Figs. 4 and 5 is as good as can be achieved using existing models of the nonlinearity in FPSOAs [10], which rely on the Fabry-Perot gain equations.The results demonstrate that this model accurately reproduces all the major features of nonlinear FPSOA operation, and, when properly calibrated, is accurate across a range of bias currents and wavelengths.With additional refinement, such as the use of a logarithmic gain expression, even closer agreement should be achievable.

Conclusion
We have solved a modified form of the rate equations that includes the contribution of the dispersive non-linearity present in Fabry-Perot semiconductor amplifiers.The model is calibrated and compared against measurements of a VCSOA.Good agreement is found and the model reproduces all the major features of the device characteristics.Use of the rate equations enables analyses of FPSOAs that are cumbersome or impossible with FP gain equations, such as noise, multi-mode, and high-speed operation.The rate equations also provide the advantage of automatic inclusion of ASE and the ability to model device operation all the way through the lasing threshold.One drawback of the approach is the large parameter set that must be determined to accurately model a real device.To our knowledge, this is the first calculation using only the rate equations that accurately predicts the gain and nonlinearity of FPSOAs.

Fig. 2 .
Fig. 2. L-I curve.Dots are measured data, solid line is calculation

Fig. 3 .
Fig. 3. Gain vs. detuning for several input powers.Calculated result on the left, measured data on the right.

Table 1 :
Rate equation parameters for VCSOA