Overview: In the following, we discuss simulation results related to a
quantum dot semiconductor laser (QDL) in non-equilibrium conditions immediately
after the 'switch-on'.
The numerical results presented here have been obtained with the simulator
introduced here.
Contents
Quantum Dot Laser
In the following, we discuss numerical results obtained with the simulator
introduced here.
We assume an edge emitting
quantum dot semiconductor laser (QDSOA) with a cavity length of
400μm and ridge injection stripe with a width of 8μm.
The geometry of the device and the active area
are shown in Fig. 1.1.
Note: We have assumed a QD surface density of 8.0 1010 cm-2. The increase the gain of the active area, we assume that 15 layers of QDs have been stacked.
Fig. 1.1: Edge emitting quantum dot semiconductor amplifier with ridge-geometry. The inset shows the active area including stacked layers of 'dots-in-a-well' structures, overgrown quantum dots in a quantum well.
Note: We have assumed a QD surface density of 8.0 1010 cm-2. The increase the gain of the active area, we assume that 15 layers of QDs have been stacked.
1.1 Quantum Dot Laser Dynamics After Switch-On
To simulate a QDL, the light generated inside the device
has to be (partially) reflected at the facets.
This leads to the build-up of coherent optical fields with
high intensity in the optical resonator. Cleaved semiconductor material has a
reflectivity of around 0.3-0.4. To reach higher reflectivities, the material has to
coated. Reflectivities close to 1 can be achieved by using
distributed Bragg reflectors (DBRs).
For the simulation results presented here, we have set the reflectivity of the front facet to 0.6 and that of the rear
(output) facet to 0.3.
At the beginning of the simulation, the QD carrier occupation probability has been set to 0.49. After switching on the bias voltage (in this case 1.19V) at the time-point 0ps, the 2-D charge density increases and leads to an increased relaxation of carriers from 2-D states to 0-D quantum dot states.
The animation in the center of Fig. 1.2 shows the local gain of the active laser material in a time-frame of 10ps to 25ps after switch-on. The first snap-shot, at 10ps, shows that the optical gain (at the central frequency of the light field) along the injection stripe is positive, exceeding 6.0x10-6nm-1. At this time-point the optical field intensity (left animation) and the injected current density (right animation) are still low.
Spontaneous emission, modelled using a Langevin noise term, generates a non-coherent optical field inside the resonator cavity. The light is amplified via stimulated electron-hole recombination and partially reflected at the laser facets. After a few round-trips, the light intensity increases (see frames at 11ps - 15ps).
Eventually, the light intensity starts to saturate the gain medium (frames 16ps-25ps). The gain reduction is visualised as regions of red shade (see animation in the center of Fig. 1.2 ). At the same time the injection current density increases to replace charge carriers lost via stimulate recombination and maintain the 2-D carrier density at the value corresponding to the applied bias voltage.
At the beginning of the simulation, the QD carrier occupation probability has been set to 0.49. After switching on the bias voltage (in this case 1.19V) at the time-point 0ps, the 2-D charge density increases and leads to an increased relaxation of carriers from 2-D states to 0-D quantum dot states.
The animation in the center of Fig. 1.2 shows the local gain of the active laser material in a time-frame of 10ps to 25ps after switch-on. The first snap-shot, at 10ps, shows that the optical gain (at the central frequency of the light field) along the injection stripe is positive, exceeding 6.0x10-6nm-1. At this time-point the optical field intensity (left animation) and the injected current density (right animation) are still low.
Spontaneous emission, modelled using a Langevin noise term, generates a non-coherent optical field inside the resonator cavity. The light is amplified via stimulated electron-hole recombination and partially reflected at the laser facets. After a few round-trips, the light intensity increases (see frames at 11ps - 15ps).
Eventually, the light intensity starts to saturate the gain medium (frames 16ps-25ps). The gain reduction is visualised as regions of red shade (see animation in the center of Fig. 1.2 ). At the same time the injection current density increases to replace charge carriers lost via stimulate recombination and maintain the 2-D carrier density at the value corresponding to the applied bias voltage.
Fig. 1.2: Animated plot showing the field intensity in the resonator cavity of a quantum dot laser with an injection stripe width of 8μm and a cavity length of 400μm (left plot). The plot in the center shows the local material gain. The plot on the right shows the local injected current density. The QDL is homogeneously broadened (broadening 25meV). Note: Snap-shots have been taken at a time interval of 1ps.
The simulation results presented in Fig. 1.2 refer to an idealised QDL containing an ensemble of identical QDs. Fig. 1.3 refers to more realistic inhomogeneously broadened QDL. For details related to the broadening parameters see the captions of Fig. 1.2 and Fig. 1.3. The inhomogenously broadened QD ensemble in the active area generates a highly non-uniform gain medium. This is best visualised in the central plot in Fig. 1.3, showing the local material gain at the frequency of the injected optical pulse. The contribution of each QD to the gain of the medium is determined by the detuning between the frequency of the optical pulse and the frequency of the relevant QD transition. Fig. 1.3: Animated plot showing the field intensity in the resonator cavity of a quantum dot laser (QDL) with an injection stripe width of 8μm and a cavity length of 400μm (left plot). The plot in the center shows the local material gain. The plot on the right shows the local injected current density. Inhomogeneous broadening QD ground state: 25meV. Inhomogeneous broadening QD excited states: 30meV. Note: Snap-shots have been taken at a time interval of 1ps.
The optical fields inside the laser cavity (see left animation in Fig. 1.2 and Fig. 1.3) propagate in longitudinal and transverse direction. The complex pattern is caused by light diffraction, wave-guiding along the injection stripe, and the interaction of the light with the active gain medium (via carrier induced changes of the optical gain and the refractive index). The light field dynamics leads to pulsations of the light intensity at the output facet. Depending on the operating conditions and the geometry of the wave-guide the laser may relax to a steady-state continuous wave emission mode in a time-frame of hundreds of pico-seconds to nano-seconds.1.2 Quantum Dot Laser: Emission and Gain Spectra
The optical properties of active media containing QDs depends on the optical transitions involving QD states and the
related transition matrix elements. The gain spectrum also depends on the occupation probability of the
QD states with charge carriers. Empty QDs can absorb light via electron-hole pair generation, while QDs with
charge carriers can generate light via spontaneous or stimulated electron-hole recombination.
Fig. 1.4 shows the occupation probability of QD electron and hole states during the first 50ps after the start of the laser. During the first 10ps the carrier occupation probability of the QD electron ground state and hole ground state increases leading to a high material gain. This leads to the amplification of spontaneous emission inside the laser cavity (see Fig. 1.2). At approximately 10ps the intensity of the light field is sufficiently high to depleat the QD ground state. (The electron-hole QD ground state transition is in resonance with the optical field, i.e. the transition frequency is close to the optical frequency. Fluctuations of the QD ground state carrier occupation probability are related to the non-equilibrium state of the laser. The internal light fields also show strong fluctuations (see Fig. 1.2).
As the laser dynamics approaches a quasi-steady state, the QD carrier occupation probability is 'pinned' at a value that leads to just sufficient optical gain to overcome laser facet losses and internal light absorbtion. The optical loss due the partial reflectivity of the facet mirrors is given by [1]:
where Γ is the confinement factor,
L the cavity length and R1 and R2 is the front and rear
reflectivity, respectively. For the waveguide considered here,
we have determined an optical confinement factor of: Γ = 0.618.
The optical mirror loss is therefore:
αloss = 34.6 x 10-6nm−1. The actual steady-state
gain is higher due to internal light absorption and diffraction, respectively.
Fig. 1.5 shows the gain spectrum of an inhomogeneously broadened QDL during the initial 35ps after startup. At 10ps the maximum of the gain reaches 1.3x10-5nm-1. At this time-point, the QD ground state population approaches its maximum value (see Fig. 1.4 ) before saturation by the lasing field. The gain then gradually decreases, approaching the steady-state threshold gain. The animation in Fig. 1.5 shows that the gain at 20ps-35ps still fluctuates indicating a non-equilibrium state.
The emission spectrum of a QDL is strongly influenced by the gain spectrum. Fig. 1.6 shows the transient gain spectrum and the emission spectrum of a QDL during startup. At 0ps the gain is negative corresponding to an absorptive medium. After 3ps the active medium starts to become inverted (positive optical gain). It leads to a selective amplification of light with frequencies corresponding to positive region of the gain spectrum.
Another factor influencing the emission spectrum of a QDL is the geometry of the optical resonator. The optical feedback provided by the facet mirrors leads to the build-up of the laser field but also to a selection of longitudinal modes supported by the laser cavity. These modes are reflected in the emission spectrum as a series of equidistant spectral lines. The frequency difference between consecutive longitudinal modes is inversely proportional to the length of the resonator and is given by: Δν=c/(2nL), where c is the velcity of light in vacuum, L is the length of the resonator and n is the refractive index.
Fig. 1.4: Carrier occupation probability of QD states during the initial 50ps after the start of the QDL.
Fig. 1.4 shows the occupation probability of QD electron and hole states during the first 50ps after the start of the laser. During the first 10ps the carrier occupation probability of the QD electron ground state and hole ground state increases leading to a high material gain. This leads to the amplification of spontaneous emission inside the laser cavity (see Fig. 1.2). At approximately 10ps the intensity of the light field is sufficiently high to depleat the QD ground state. (The electron-hole QD ground state transition is in resonance with the optical field, i.e. the transition frequency is close to the optical frequency. Fluctuations of the QD ground state carrier occupation probability are related to the non-equilibrium state of the laser. The internal light fields also show strong fluctuations (see Fig. 1.2).
As the laser dynamics approaches a quasi-steady state, the QD carrier occupation probability is 'pinned' at a value that leads to just sufficient optical gain to overcome laser facet losses and internal light absorbtion. The optical loss due the partial reflectivity of the facet mirrors is given by [1]:
| (1.1) |
Fig. 1.5 shows the gain spectrum of an inhomogeneously broadened QDL during the initial 35ps after startup. At 10ps the maximum of the gain reaches 1.3x10-5nm-1. At this time-point, the QD ground state population approaches its maximum value (see Fig. 1.4 ) before saturation by the lasing field. The gain then gradually decreases, approaching the steady-state threshold gain. The animation in Fig. 1.5 shows that the gain at 20ps-35ps still fluctuates indicating a non-equilibrium state.
Fig. 1.5: Gain spectrum of an inhomogeneously broadened QDL during the first 30ps after starting the device. Inhomogenous broadening QD ground state: 25meV. Inhomogeneous broadening QD excited state: 30meV. Note: During the initial frames the gain maximum is beyond the range of the vertical axis.
The emission spectrum of a QDL is strongly influenced by the gain spectrum. Fig. 1.6 shows the transient gain spectrum and the emission spectrum of a QDL during startup. At 0ps the gain is negative corresponding to an absorptive medium. After 3ps the active medium starts to become inverted (positive optical gain). It leads to a selective amplification of light with frequencies corresponding to positive region of the gain spectrum.
Fig. 1.6: Transient gain spectrum (light blue curve) and emission spectrum (orange curve) of a QDL during startup. The gain spectrum is calculated at time intervals of 1.5ps. The emission spectrum corresponds to the Fourier transform of the complex optical fields (sampled over a period of 15ps).
Another factor influencing the emission spectrum of a QDL is the geometry of the optical resonator. The optical feedback provided by the facet mirrors leads to the build-up of the laser field but also to a selection of longitudinal modes supported by the laser cavity. These modes are reflected in the emission spectrum as a series of equidistant spectral lines. The frequency difference between consecutive longitudinal modes is inversely proportional to the length of the resonator and is given by: Δν=c/(2nL), where c is the velcity of light in vacuum, L is the length of the resonator and n is the refractive index.
Fig. 1.7: Longitudinal modes visible in the emission of QDL lasers. The spacing between consecutive longitudinal mode lines depends on the length of the optical resonator.
Bibliography
[1]
Y. Suematsu and A.R. Adams. Handbook of Semiconductor Lasers
and Photonic Integrated
Circuits.
Chapman and Hall,London, 1994.