Semiconductor Laser Rate Equation Pdf
Haldis Momeni
The laser diode rate equations model the electrical and optical performance of a laser diode. This system of ordinary differential equations relates the number or density of photons and charge carriers (electrons) in the device to the injection current and to device and material parameters such as carrier lifetime, photon lifetime, and the optical gain.
The rate equations may be solved by numerical integration to obtain a time-domain solution, or used to derive a set of steady state or small signal equations to help in further understanding the static and dynamic characteristics of semiconductor lasers.
In the multimode formulation, the rate equations[1] model a laser with multiple optical modes. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the optical cavity modes:
The first term on the right side of the carrier rate equation is the injected electrons rate (I/eV), the second term is the carrier depletion rate due to all recombination processes (described by the decay time τ n \displaystyle \tau _n ) and the third term is the carrier depletion due to stimulated recombination, which is proportional to the photon density and medium gain.
In the photon density rate equation, the first term ΓGP is the rate at which photon density increases due to stimulated emission (the same term in carrier rate equation, with positive sign and multiplied for the confinement factor Γ), the second term is the rate at which photons leave the cavity, for internal absorption or exiting the mirrors, expressed via the decay time constant τ p \displaystyle \tau _p and the third term is the contribution of spontaneous emission from the carrier radiative recombination into the laser mode.
The gain term, G, cannot be independent of the high power densities found insemiconductor laser diodes. There are several phenomena which cause the gain to'compress' which are dependent upon optical power. The two main phenomena arespatial hole burning and spectral hole burning.
To account for gain compression due to the high power densities in semiconductor lasers, the gain equation is modified such that it becomes related to the inverse of the optical power. Hence, the following term in the denominator of the gain equation :
Dynamic wavelength shift in semiconductor lasers occurs as a result of the changein refractive index in the active region during intensity modulation. It is possible toevaluate the shift in wavelength by determining the refractive index change of the activeregion as a result of carrier injection. A complete analysis of spectral shift during directmodulation found that the refractive index of the active region varies proportionally to carrier density and hence the wavelength varies proportionally to injected current.
In this paper, we introduce the concept of the cascaded optically injection-locked (OIL) semiconductor laser and present its novel rate equations. Then, the new locking range for this configuration has been obtained by mathematical demonstration. Subsequently, we modified a new adjustment for the detuning frequencies (Δ f i n j ) of the cascaded OIL system as well as the linewidth enhancement factors (α) values. Utilizing these tunings, improvements in the steady-state photon number and the phase modulation (PM) range become possible. Afterward, we define the generation of the complex optical signal area and extract the transfer function for investigating the frequency response of the cascaded system. The simulations have been performed once with identical α values and once with the various α values in the slave laser (SL) stages. We conclude that these novel proposed adjustments, combined with a strong injection ratio (Rinj) of 15 dB and a high bias current, can significantly broaden the bandwidth near 700 GHz while maintaining the fair gain available up to 180 GHz. Further, the generation of complex optical signal areas has been boosted for high-quality complex modulation applications. Eventually, we exhibit a novel approach for generating different α values in the SL stages by applying managed temperature variations in the experimental setup of the cascaded system, regardless of employing similar SLs.
Rate equations for micro- and nanocavity lasers are formulated which take account of the finite number of emitters, Purcell effects as well as stochastic effects of spontaneous emission quantum noise. Analytical results are derived for the intensity noise and intensity correlation properties, g(2), using a Langevin approach and are compared with simulations using a stochastic approach avoiding the mean-field approximation of the rate equations. Good agreement between the two approaches is found even for large values of the spontaneous emission beta-factor, i.e., for threshold-less lasers, as long as more than about ten emitters contribute to lasing. A large value of the beta-factor improves the noise properties.
In this paper, we generalize the standard semiconductor laser rate equations4 to cover the case of ultra-small and high-β lasers containing only a few discrete emitters. Analytical expressions for the steady-state intensity noise and second-order intensity correlation, g(2)(0), are derived using a small-signal Langevin approach and compared with stochastic simulations taking into account the discrete nature of photons and electrons. Very good agreement between the two approaches is found in general, with small deviations occurring around threshold when less than 10 emitters contribute to lasing.
Figure 1 shows a schematic defining important variables describing the nanocavity laser, i.e., number n0 of emitters (dipoles), number np of photons in the cavity mode, decay rate γc of the cavity population, and the coupling rate γr between photons in the cavity mode and a single emitter. If the polarization of the medium decays on a timescale that is short compared to the other characteristic time constants of the laser, due to various dephasing processes, the polarization can be eliminated adiabatically. These de-coherence processes also allow one to neglect quantum mechanical correlations,16 and the laser dynamics can be described as rate equations for the (classical) photon and carrier number
Here, Ppu is the pump rate into the upper laser level, taking into account Pauli blocking,16,Rst and Rsp are the net rates of stimulated and spontaneous emission into the cavity mode, Rbg is the (background) rate of spontaneous emission into all modes, but the cavity mode, as well as non-radiative emission, and Rc is the photon escape rate from the cavity. Furthermore, Fn(t) and Fp(t) are stochastic Langevin forces. The open nature of the cavity may be given firm ground by the use of a quasi-bound state basis, allowing consideration of complex cavities.21,22
In laser cavities realized using photonic crystals, e.g. line-defect cavities,5,7,30,31 the group velocity is reduced due to strong dispersion. It has been shown that slow light effects give rise to an enhancement of the gain per unit length.32,33 However, the temporal gain coefficient entering laser rate equations for cavity quantities, such as the total photon number, is unaffected since the laser roundtrip time is also increased in proportion to the group refractive index.31 The group velocity entering (8) is thus the background value not taking into account the longitudinal perturbation of the refractive index.31,34
For nanolasers with a small photon number, it is not obvious that a small-signal analysis correctly describes the noise properties. Close to laser threshold and below, spontaneous emission of a single photon may thus constitute a large perturbation of the state of the laser, which is not well described by linearized equations. Furthermore, for high-β lasers, the threshold regime extends over a large range of pump rates. In order to establish the accuracy of the analytical results, we compare these with simulations without the assumptions of the Langevin analysis.
In conclusion, we analyzed a rate equation model valid for nanolasers with discrete emitters. We considered the good-cavity limit, valid for most semiconductor lasers, where the emitter broadening is larger than the cavity linewidth. Several conclusions are drawn: (1) The quantum noise properties, including the second-order intensity correlation, g(2)(0), are well accounted for by analytical expressions, derived using the Langevin approach, in the entire region from below to above threshold, at least for more than 10 emitters; (2) The noise properties are improved by increasing the spontaneous emission β-factor; (3) Purcell enhancement effects are already included in standard semiconductor laser rate equations when excluding the bad-cavity limit; and (4) for a given laser cavity, there is an optimum number of emitters.
A set of rate equations is derived describing the deterministic multi-mode dynamics of a semiconductor laser. Mutual interactions among the lasing modes, induced by high-frequency modulations of the carrier distribution, are described by carrier-inversion moments and lead to special spectral content of each spatial mode. The diffusion of carriers is shown to play an important role in determining the spectral properties of the field. The Bogatov effect of asymmetric gain suppression in semiconductor lasers will be derived. We will explicitly discuss the nontrivial relationship between the modes of the nonlinear cavity and the optical spectrum of the laser output and illustrate this for a two and three-mode laser.
This contribution attempts to review certain resonance effects which occur in the dynamic behaviour of semiconductor lasers. To study these effects theoretically, a rate equation approach is used for single-mode operation in the region of lasing threshold.
Semiconductor laser is one of the core light sources of optical fiber communication systems. Better semiconductor laser performance helps to increase the capacity and quality of the entire fiber optic communication system. In order to design better semiconductor lasers, it is often necessary to simulate various performances through rate equations. However, some parameters need to be obtained through experimental results, so it is necessary to fabricate semiconductor laser chip to extract the parameters. Through multiple rounds of iterative experiments and theoretical calculation, chip optimization is continuously carried out to achieve the final goal. In the process of optimization analysis, in order to better analyze the performance of laser chips, laser parameters are usually divided into eigen parameters and parasitic parameters. Parasitic parameters mainly include capacitance, resistance and other properties. They can be obtained by means of a fitting method by combining a reflection curve of the small-signal frequency response (S11) with an equivalent circuit model. These parameters are basically not affected by the laser-current change. However, the extraction of the eigenmetric parameters is related to the selected laser-current range, and the parameter extraction results themselves have a variety of possible answers. How to obtain more accurate and suitable for a wide operating current range of eigen parameters is the focus of this paper. In recent years, with the development of emerging applications such as 5G optical communication and Radio On Fiber (ROF) analog communication, semiconductor lasers are required to operate at larger drive currents to improve high-speed performance and are required to have better linearity. In such ROF analog communication systems, semiconductor lasers need to work at a lager bias laser-current and in the linear region to obtain excellent microwave performance. Besides, it is need to test and evaluate many microwave performance indicators, such as third-order intermodulation, second harmonic, relative intensity noise, etc. The acquisition of these indicators requires complex test systems, expensive equipment and difficult testing. Therefore, how to accurately and conveniently obtain the microwave performance of the laser at a large drive current is particularly important. By testing some basic performance of the laser, the eigenvalue and parasitic parameters of the semiconductor laser can be extracted. These extracted parameters, combined with system characteristics, can be used to quickly calculate and evaluate most of the microwave performance of the system. This method greatly reduces the test requirements and the costs of test equipment, greatly speeding up the performance evaluation of semiconductor lasers and the entire design process. The traditional parameter extraction method can obtain the parameter values of semiconductor lasers operating at low bias currents (1040 mA). But the parameter extraction at large currents is not very accurate, and more complicated tests such as measuring chirps are required. This paper re-evaluates the applicability of approximate conditions in common semiconductor laser parameter extraction methods and finds that approximate conditions fail at large currents. To this end, it is proposed to use a more general formula for the parameter extraction of the semiconductor laser rate equation. Taking the DFB laser chip as an example, the small-signal frequency response curve (S21) is used to accurately extract the resonant frequency fr and the damping factor γ of the semiconductor laser, and combined with the laser power-current (P-I) response curve of the laser, the various parameters of the semiconductor laser rate equation can be calculated. Compared with the previous parameter extraction approximation calculation method, the present method has three advantages. First, the range of laser drive currents that can be analyzed is wider. Second, there are fewer items to test. The third is that the extraction of parameters such as the resonant frequency fr at large currents is more accurate. This method is an important reference for the optimization and improvement of semiconductor laser with wide operating currents such as analog laser.
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