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| The [[laser diode]] [[rate equation]]s model the electrical and optical performance of a laser diode. This system of [[ordinary differential equation]]s relates the number or density of [[photon]]s and [[charge carrier]]s ([[electron]]s) in the device to the injection [[current (electricity)|current]] and to device and material parameters such as [[carrier lifetime]], photon lifetime, and the optical gain.
| | Translator Fiorita from Inuvik, has hobbies and interests which includes geocaching, sleep apnea and netball. Has travelled since childhood and has visited numerous locales, such as Kolomenskoye.<br><br>Look into my web blog: [http://vineveraskincar.livejournal.com/584.html vine vera] |
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| 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 model|small signal]] equations to help in further understanding the static and dynamic characteristics of [[semiconductor lasers]].
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| The laser diode rate equations can be formulated with more or less complexity to model different aspects of laser diode behavior with varying accuracy.
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| ==Multimode rate equations==
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| In the multimode formulation, the rate equations<ref>G. P. Agrawal, "Fiber-Optic Communication Systems", Wiley Interscience, Chap. 3</ref> model a laser with multiple optical [[normal mode|modes]]. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the [[optical cavity]] modes:
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| :<math>\frac{dN}{dt} = \frac{I}{eV} - \frac{N}{\tau_n} - \sum_{\mu=1}^{\mu=M}G_\mu P_\mu</math>
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| :<math>\frac{dP_\mu}{dt} = \Gamma_\mu(G_\mu - \frac{1}{\tau_p})P_\mu + \beta_\mu \frac{N}{\tau_n}</math>
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| where:
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| N is the carrier density, P is the photon density, I is the applied current, e is the [[elementary charge]], V is the volume of the [[Active laser medium|active]] region, <math>{\tau_n}</math> is the carrier lifetime, G is the gain coefficient (s<sup>−1</sup>), <math>\Gamma</math> is the confinement factor, <math>{\tau_p}</math> is the photon lifetime, <math>{\beta}</math> is the spontaneous emission factor, M is the number of modes modelled, μ is the mode number, and
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| subscript μ has been added to G, Γ, and β to indicate these properties may vary for the different modes.
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| 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 [[non-radiative recombination]] processes (described by the decay time <math>{\tau_n}</math>) and the third term is the carrier depletion due to [[Stimulated emission|stimulated recombination]], which is proportional to the photon density and medium gain.
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| 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 <math>{\tau_p}</math> and the third term is the contribution of spontaneous emission from carrier non-radiative recombination.
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| ==The modal gain==
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| G<sub>μ</sub>, the gain of the μ<sup>th</sup> mode, can be modelled by a parabolic dependence of gain
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| on wavelength as follows:
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| :<math> G_\mu = \frac{\alpha N [1-(2\frac{\lambda(t)-\lambda_\mu}{\delta\lambda_g})^2] - \alpha N_0}{1 + \epsilon \sum_{\mu=1}^{\mu=M}P_\mu}</math>
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| where:
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| α is the gain coefficient and ε is the gain compression factor (see below). λ<sub>μ</sub> is the wavelength of the μ<sup>th</sup> mode, δλ<sub>g</sub> is the full width at half maximum (FWHM) of the gain curve, the centre of which is given by
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| :<math>\lambda(t)=\lambda_0 + \frac{k(N_{th} - N(t))}{N_{th}}</math>
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| where λ<sub>0</sub> is the centre wavelength for N = N<sub>th</sub> and k is the spectral shift constant (see below). N<sub>th</sub> is the carrier density at threshold and is given by
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| :<math>N_{th}=N_{tr} + \frac{1}{\alpha\tau_p\Gamma}</math>
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| where N<sub>tr</sub> is the carrier density at transparency.
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| β<sub>μ</sub> is given by
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| :<math>\beta_\mu=\frac{\beta_0}{1+(2(\lambda_s-\lambda_\mu)/\delta\lambda_s)^2}</math>
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| where
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| β<sub>0</sub> is the spontaneous emission factor, λ<sub>s</sub> is the centre wavelength for spontaneous emission and δλ<sub>s</sub> is the spontaneous emission FWHM. Finally, λ<sub>μ</sub> is the wavelength of the μ<sup>th</sup> mode and is given by
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| :<math>\lambda_\mu=\lambda_0 - \mu\delta\lambda + \frac{(n-1)\delta\lambda}{2}</math> | |
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| where δλ is the mode spacing.
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| ==Gain Compression==
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| The gain term, G, cannot be independent of the high power densities found in
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| semiconductor laser diodes. There are several phenomena which cause the gain to
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| 'compress' which are dependent upon optical power. The two main phenomena are
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| [[spatial hole burning]] and [[spectral hole burning]].
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| Spatial hole burning occurs as a result of the standing wave nature of the optical
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| modes. Increased lasing power results in decreased carrier diffusion efficiency which
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| means that the stimulated recombination time becomes shorter relative to the carrier
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| diffusion time. Carriers are therefore depleted faster at the crest of the wave causing a
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| decrease in the modal gain.
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| Spectral hole burning is related to the gain profile broadening mechanisms such
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| as short intraband scattering which is related to power density.
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| 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 :
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| :<math>1 + \epsilon \sum_{\mu=1}^{\mu=M}P_\mu</math>
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| ==Spectral Shift==
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| Dynamic wavelength shift in semiconductor lasers occurs as a result of the change
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| in refractive index in the active region during intensity modulation. It is possible to
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| evaluate the shift in wavelength by determining the refractive index change of the active
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| region as a result of carrier injection. A complete analysis of spectral shift during direct
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| modulation found that the refractive index of the active region varies proportionally to carrier density and hence the wavelength varies proportionally to injected current.
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| Experimentally, a good fit for the shift in wavelength is given by:
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| :<math>\delta\lambda=k\left(\sqrt{\frac{I_0}{I_{th}}}-1\right)</math>
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| where I<sub>0</sub> is the injected current and I<sub>th</sub> is the lasing threshold current.
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| ==References==
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| {{Reflist}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Semiconductor lasers]]
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Translator Fiorita from Inuvik, has hobbies and interests which includes geocaching, sleep apnea and netball. Has travelled since childhood and has visited numerous locales, such as Kolomenskoye.
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