Quantum Dot Semiconductor Optical Amplifier: Simulation Results
Overview: In the following, we discuss simulation results related to a
quantum dot semiconductor amplifier (QDSOA) during steady-state and during
excitation by an optical pulse of picosecond duration. The effect of gain
saturation of homogeneously and inhomogeneously broadened QD gain media is
visualised. The numerical results presented here have been obtained with the simulator
introduced here.
Contents
1 Quantum Dot Optical Amplifier
1.1 QDSOA at Steady State
1.2 QDSOA During Injection of an Optical Pulse
1.1 QDSOA at Steady State
1.2 QDSOA During Injection of an Optical Pulse
Quantum Dot Optical Amplifier
In the following, we discuss numerical results obtained with the simulator
introduced here.
We assume an edge emitting
quantum dot semiconductor amplifier (QDSOA) with a cavity length of
1000μm and ridge injection stripe with a width of 5μm.
The geometry of the device and the active area
are shown in Fig. 1.1.
In order to simulate a QDSOA with anti-reflective coated facets, the front and rear facet reflectivity of the optical cavity have been set to a low value (10-6). The low facet reflectivity limits back-reflection of injected optical pulses and prevents a build-up of the optical field intensity inside the QDSOA cavity.
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.
In order to simulate a QDSOA with anti-reflective coated facets, the front and rear facet reflectivity of the optical cavity have been set to a low value (10-6). The low facet reflectivity limits back-reflection of injected optical pulses and prevents a build-up of the optical field intensity inside the QDSOA cavity.
1.1 QDSOA At Steady State
As initial starting condition for the differential equations describing the QD carrier occupation
probability (1.2 and 1.3), we assume a
carrier occupation probability of 0.0 for the bound QD states (empty QDs).
The initial 2-D charge density required by differential equations 1.8 and 1.9 is calculated according to the equations described here using the applied bias voltage (a model parameter that in this case is set to 1.19V).
Note: The boundary conditions mentioned above are non-critical, i.e. they are to some extent arbitrary (keeping in mind that the occupation probability can only take values between 0 and 1). Nevertheless, chosing the initial 2-D charge density close to the steady state value ensures that the total system quickly reaches steady state.
After starting the simulation, scattering processes transport carriers from the 2-D layers to the QDs. This leads to a temporary reduction of the 2-D charge density in the quantum well (QWell) surrounding the QDs. The steady-state value of the 2-D charge density (corresponding to the set bias voltage) is restored within approx. 20ps (compare with Fig. 1.2).
At steady-state carrier capture into the QDs and carrier escape from the QDs balance. Fig. 1.4 shows the carrier occupation probability of QD electron and hole states, respectively. For this simulation, we have assumed empty QDs at the initial time-point.
Fig 1.4 shows that the steady-state carrier occupation of the QD levels is reached within a time-frame of approx. 20ps. The actual value of the carrier occupation probability of individual QD states depends on the 2-D charge density and the energetic separation between the bound QD states and between 2-D states and 0-D QD states, respectively. This is due to the fact that phonon induced carrier scattering depends strongly on the energy difference between the initial and the final state carrier state (see section on 'Carrier Scattering and Relaxation Processes').
The initial 2-D charge density required by differential equations 1.8 and 1.9 is calculated according to the equations described here using the applied bias voltage (a model parameter that in this case is set to 1.19V).
Note: The boundary conditions mentioned above are non-critical, i.e. they are to some extent arbitrary (keeping in mind that the occupation probability can only take values between 0 and 1). Nevertheless, chosing the initial 2-D charge density close to the steady state value ensures that the total system quickly reaches steady state.
After starting the simulation, scattering processes transport carriers from the 2-D layers to the QDs. This leads to a temporary reduction of the 2-D charge density in the quantum well (QWell) surrounding the QDs. The steady-state value of the 2-D charge density (corresponding to the set bias voltage) is restored within approx. 20ps (compare with Fig. 1.2).
Fig. 1.2: 2-D charge density in the quantum well surrounding the quantum dots at the start of the simulation. Note: The charge density is stated in semiconductor units using picoseconds 'ps' as the unit of time and nanometers 'nm' as the unit of length.
The drop of the 2-D charge density causes a temporary increase of the injection current density (see Fig. 1.3). Charge carriers that are captured from 2-D states to 0-D auantum dot states are replaced by carriers injected into the quantum well. The injection current density flowing accross the device at steady-state is due to the leakage current caused by loss processes (see section 1.3 ). Fig. 1.3: Injection current density. Note: We simulate direct injection into the 2-D layer neglecting carrier scattering between the bulk medium and the 2-D semiconductor structure.
At steady-state carrier capture into the QDs and carrier escape from the QDs balance. Fig. 1.4 shows the carrier occupation probability of QD electron and hole states, respectively. For this simulation, we have assumed empty QDs at the initial time-point.
Fig 1.4 shows that the steady-state carrier occupation of the QD levels is reached within a time-frame of approx. 20ps. The actual value of the carrier occupation probability of individual QD states depends on the 2-D charge density and the energetic separation between the bound QD states and between 2-D states and 0-D QD states, respectively. This is due to the fact that phonon induced carrier scattering depends strongly on the energy difference between the initial and the final state carrier state (see section on 'Carrier Scattering and Relaxation Processes').
Fig. 1.4: QD carrier occupation probability at steady-state (applied bias voltage 1.19V). Note: For this simulation, we have assumed 'empty' QDs at time-point 0ps.
1.2 QDSOA During the Injection of an Optical Pulse
After the QDSOA has reached a steady-state (as shown in section 1.1),
we simulate the injection of an optical pulse with a duration of 0.5ps, an energy of 0.5pJ, and
a central frequency of 1671ps-1.
The numerical term simulating the injected pulse is discussed
here.
Fig. 1.5 refers to a homogeneously broadenend QDSOA (broadening 30meV) with a cavity length of 1000μm and an injection stripe of 5μm. A homogeneously broadened QDSOA is an idealised QDSOA containing identical QDs.
The left plot in Fig. 1.5 shows the amplification of the injected optical pulse during the passage through the QDSOA. The gain of the pumped active medium leads to an exponential increase of the pulse intensity while the transverse variation of the refractive index causes a reshaping of the pulse.
The optical pulse is amplified via induced electron-hole recombination of carriers in bound QD states. The higher the energy of the pulse, the more electron-hole recombination processes are induced. This leads to a selective depleation of carriers in those QD states that have a optical transition that is close to the central frequency of the injected light pulse. This process is called spectral hole-burning.
The local injected current density is shown in the plot on the right side of Fig. 1.5. We notice a region of increased current density following the optical pulse with a delay in the order of picoseconds. The delay indicates the time required by 2-D carriers to refill the QD states depleated by the interaction with the injected optical pulse.
The simulation results presented in Fig. 1.5 refer to an idealised QDSOA containing an ensemble of identical QDs. Fig. 1.6 refers to more realistic inhomogeneously broadened QDSOA.
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. 1.6, 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 and by the efficency of carrier capture processes. (Carrier capture times depends on the transition energy and may be higher for QDs with confinement energies with respect to 2-D states exceeding the energy of LO-phonons.)
As can be seen in the left animation in Fig. 1.6, the non-uniform gain medium leads to a reshaping of the optical pulse generating a modulation in longitudinal direction. Compared to the homogeneous QDSOA (see Fig. 1.5) the peak intensity of the optical pulse amplified in the inhomogeneously broadened QDSOA is lower.
Fig. 1.5 refers to a homogeneously broadenend QDSOA (broadening 30meV) with a cavity length of 1000μm and an injection stripe of 5μm. A homogeneously broadened QDSOA is an idealised QDSOA containing identical QDs.
The left plot in Fig. 1.5 shows the amplification of the injected optical pulse during the passage through the QDSOA. The gain of the pumped active medium leads to an exponential increase of the pulse intensity while the transverse variation of the refractive index causes a reshaping of the pulse.
The optical pulse is amplified via induced electron-hole recombination of carriers in bound QD states. The higher the energy of the pulse, the more electron-hole recombination processes are induced. This leads to a selective depleation of carriers in those QD states that have a optical transition that is close to the central frequency of the injected light pulse. This process is called spectral hole-burning.
Fig. 1.5: Animated plot showing the field intensity of an injected optical pulse (with a duration of 0.5ps) propagating through a homogeneously broadened QDSOA (left). The plot in the center shows the local optical gain (at a frequency of 1671ps-1) induced by charge carriers captured into QDs. The plot on the right shows the local injected current density.
The plot in the center of Fig. 1.5 shows the influence of the optial pulse on the local material gain of the QDSOA. Dark areas correspond to absorptive regions (negative gain), a red shade indicates a transparent medium, while a yellow shade indicates an inverted active medium with positive optical gain. At steady-state (before the injection of the pulse), the region along the injection stripe is inverted providing positive gain. The energy of the injected optical pulse is sufficiently high to deplete the gain medium locally, creating a region of reduced gain. This process is called spatial hole-burning. After the passage of the pulse the gain recovers within a time-frame of 20ps.The local injected current density is shown in the plot on the right side of Fig. 1.5. We notice a region of increased current density following the optical pulse with a delay in the order of picoseconds. The delay indicates the time required by 2-D carriers to refill the QD states depleated by the interaction with the injected optical pulse.
The simulation results presented in Fig. 1.5 refer to an idealised QDSOA containing an ensemble of identical QDs. Fig. 1.6 refers to more realistic inhomogeneously broadened QDSOA.
Fig. 1.6: Animated plot showing the field intensity of an injected optical pulse (with a duration of 0.5ps) propagating through an inhomogeneously broadened QDSOA (left). The plot in the center shows the local optical gain (at a frequency of 1671ps-1) induced by charge carriers captured into QDs. The plot on the right shows the local injected current density.
For the QD ground state transition, we have assumed an inhomogeneous broadening of 30meV. For transitions involving excited QD states, we have assumed an inhomogeneous broadening of 50meV. Generating a set of inhomogenously broadened QDs is discussed here.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. 1.6, 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 and by the efficency of carrier capture processes. (Carrier capture times depends on the transition energy and may be higher for QDs with confinement energies with respect to 2-D states exceeding the energy of LO-phonons.)
As can be seen in the left animation in Fig. 1.6, the non-uniform gain medium leads to a reshaping of the optical pulse generating a modulation in longitudinal direction. Compared to the homogeneous QDSOA (see Fig. 1.5) the peak intensity of the optical pulse amplified in the inhomogeneously broadened QDSOA is lower.