Laser diode rate equations
Encyclopedia
The laser diode
rate equations 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 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.
The laser diode rate equations can be formulated with more or less complexity to model different aspects of laser diode behavior with varying accuracy.
. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the optical cavity
modes:
where:
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
region,
is the carrier lifetime, G is the gain coefficient (s−1), 
is the confinement factor, 
is the photon lifetime, 
is the spontaneous emission factor, M is the number of modes modelled, μ is the mode number, and
subscript μ has been added to G, Γ, and β to indicate these properties may vary for the different 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 non-radiative recombination
processes (described by the decay time
) 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 increase 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
and the third term is the contribution of spontaneous emission from carrier non-radiative recombination.
on wavelength as follows:
where:
α is the gain coefficient and ε is the gain compression factor (see below). λμ is the wavelength of the μth mode, δλg is the full width at half maximum (FWHM) of the gain curve, the centre of which is given by
where λ0 is the centre wavelength for N = Nth and k is the spectral shift constant (see below). Nth is the carrier density at threshold and is given by
where Ntr is the carrier density at transparency.
βμ is given by
where
β0 is the spontaneous emission factor, λs is the centre wavelength for spontaneous emission and δλs is the spontaneous emission FWHM. Finally, λμ is the wavelength of the μth mode and is given by
where δλ is the mode spacing.
semiconductor laser diodes. There are several phenomena which cause the gain to
'compress' which are dependent upon optical power. The two main phenomena are
spatial hole burning and spectral hole burning
.
Spatial hole burning occurs as a result of the standing wave nature of the optical
modes. Increased lasing power results in decreased carrier diffusion efficiency which
means that the stimulated recombination time becomes shorter relative to the carrier
diffusion time. Carriers are therefore depleted faster at the crest of the wave causing a
decrease in the modal gain.
Spectral hole burning is related to the gain profile broadening mechanisms such
as short intraband scattering which is related to power density.
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 :
in refractive index in the active region during intensity modulation. It is possible to
evaluate the shift in wavelength by determining the refractive index change of the active
region as a result of carrier injection. A complete analysis of spectral shift during direct
modulation found that the refractive index of the active region varies proportionally to carrier density and hence the wavelength varies proportionally to injected current.
Experimentally, a good fit for the shift in wavelength is given by:
where I0 is the injected current and Ith is the lasing threshold current.
Laser diode
The laser diode is a laser where the active medium is a semiconductor similar to that found in a light-emitting diode. The most common type of laser diode is formed from a p-n junction and powered by injected electric current...
rate equations model the electrical and optical performance of a laser diode. This system of ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s relates the number or density of photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s and charge carrier
Charge carrier
In physics, a charge carrier is a free particle carrying an electric charge, especially the particles that carry electric currents in electrical conductors. Examples are electrons and ions...
s (electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s) 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
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
to obtain a time-domain solution, or used to derive a set of steady state or small signal
Small signal model
Small-signal modeling is a common analysis technique in electrical engineering which is used to approximate the behavior of nonlinear devices with linear equations...
equations to help in further understanding the static and dynamic characteristics of semiconductor lasers.
The laser diode rate equations can be formulated with more or less complexity to model different aspects of laser diode behavior with varying accuracy.
Multimode rate equations
In the multimode formulation, the rate equations model a laser with multiple optical modesNormal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the optical cavity
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...
modes:
where:
N is the carrier density, P is the photon density, I is the applied current, e is the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
, V is the volume of the active
Active laser medium
The active laser medium is the source of optical gain within a laser. The gain results from the stimulated emission of electronic or molecular transitions to a lower energy state from a higher energy state...
region,
is the carrier lifetime, G is the gain coefficient (s−1), is the confinement factor, is the photon lifetime, is the spontaneous emission factor, M is the number of modes modelled, μ is the mode number, andsubscript μ has been added to G, Γ, and β to indicate these properties may vary for the different 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 non-radiative recombination
Non-radiative recombination
Non-radiative recombination is a process in phosphors and semiconductors, whereby charge carriers recombine without releasing photons. A phonon is released instead....
processes (described by the decay time
) and the third term is the carrier depletion due to stimulated recombinationStimulated emission
In optics, stimulated emission is the process by which an atomic electron interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level, transferring its energy to that field. A photon created in this manner has the same phase, frequency, polarization, and...
, 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 increase 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
and the third term is the contribution of spontaneous emission from carrier non-radiative recombination.The modal gain
Gμ, the gain of the μth mode, can be modelled by a parabolic dependence of gainon wavelength as follows:
where:
α is the gain coefficient and ε is the gain compression factor (see below). λμ is the wavelength of the μth mode, δλg is the full width at half maximum (FWHM) of the gain curve, the centre of which is given by
where λ0 is the centre wavelength for N = Nth and k is the spectral shift constant (see below). Nth is the carrier density at threshold and is given by
where Ntr is the carrier density at transparency.
βμ is given by
where
β0 is the spontaneous emission factor, λs is the centre wavelength for spontaneous emission and δλs is the spontaneous emission FWHM. Finally, λμ is the wavelength of the μth mode and is given by
where δλ is the mode spacing.
Gain Compression
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 are
spatial hole burning and spectral hole burning
Spectral hole burning
Spectral hole burning is the frequency selective bleaching of the absorption spectrum of a material, which leads to an increased transmission at the selected frequency....
.
Spatial hole burning occurs as a result of the standing wave nature of the optical
modes. Increased lasing power results in decreased carrier diffusion efficiency which
means that the stimulated recombination time becomes shorter relative to the carrier
diffusion time. Carriers are therefore depleted faster at the crest of the wave causing a
decrease in the modal gain.
Spectral hole burning is related to the gain profile broadening mechanisms such
as short intraband scattering which is related to power density.
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 :
Spectral Shift
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 to
evaluate the shift in wavelength by determining the refractive index change of the active
region as a result of carrier injection. A complete analysis of spectral shift during direct
modulation found that the refractive index of the active region varies proportionally to carrier density and hence the wavelength varies proportionally to injected current.
Experimentally, a good fit for the shift in wavelength is given by:
where I0 is the injected current and Ith is the lasing threshold current.