Nonlinear pulse propagation in a quantum dot laser

O. Karni; A. Capua; G. Eisenstein; D. Franke; J. Kreissl; H. Kuenzel; D. Arsenijević; H. Schmeckebier; M. Stubenrauch; M. Kleinert; D. Bimberg; C. Gilfert; J. P Reithmaier

doi:10.1364/OE.21.005715

1. Introduction

Ultra-fast nonlinear dynamics in quantum dot (QD) edge emitting lasers depend on a number of phenomena which are often hard to distinguish and identify experimentally. The dynamics are usually studied using time [1–3] or frequency [4,5] domain techniques to probe semiconductor optical amplifiers (SOAs), rather than lasers operating above threshold. However, since the rate of steady state stimulated emission is much higher in lasers compared to SOAs, there are important differences in the carrier and gain dynamics of the two structures. One way to examine directly the ultra-fast dynamics of a laser is by mapping the nonlinear propagation of an ultra-short pulse along the waveguide, while operating above threshold [6,7]. The nonlinear interaction between the pulse and the two counter-propagating cw fields of the laser imprints a distinct signature on the amplitude and spectral content of the pulse. The nature of this signature is determined by a variety of nonlinear processes, the most important ones being four-wave mixing (FWM) [8–11] and self-phase modulation (SPM) [12].

This paper describes results of modeling and experimental characterization of these nonlinear interactions for InAs/InP QD lasers. We observe the creation of a hole in the spectrum of the pulse resulting from FWM among all the fields propagating within the cavity. The spectral hole is located in the vicinity of the cw frequency and its width is determined by an effective relaxation time of the carriers. The time-domain manifestation of this signature is an amplitude tail. In addition, second order FWM processes are predicted to take place within the cavity generating two counter-propagating idlers. One of these secondary idlers co-propagates with the original pulse and contributes to the modification of the trailing edge of the output signal. Similar results were obtained for quantum dash lasers (such as the ones described in [11]) but here we present only QD laser results.

The nonlinear interactions are analyzed using two approaches. First, a semi-analytical model is presented which considers only FWM interaction among six waves which propagate in the cavity. This model isolates the wave mixing process neglecting saturation and SPM. Second, the complete interaction is analyzed using a numerical model based on a finite-difference time-domain (FDTD) simulation of the Maxwell-Schrödinger equations. The QD gain medium is described in the simulation as a sequence of two-level systems, which are fed, under an electrical bias, from a high energy carrier reservoir.

Experimental verification of the model requires the pulse to be characterized using ultra-fast coherent techniques. We have used cross-frequency-resolved optical gating (X-FROG) measurements [13,14] which reveal all the predicted modifications induced on the complex field of the pulse, such as: a distinct hole in the pulse spectrum (in the vicinity of the cw oscillation wavelength) and deformation of the pulse trailing edge. Detection of such complex ultra-fast effects requires separating the pulse response from any steady state background signal using FROG-like sampling techniques [15].

The paper is organized as follows. Section 2 describes two theoretical models. The experimental results are presented in Section 3 together with an analysis and a comparison to the predictions, while Section 4 is devoted to conclusions.

2. Theory

The interaction between a propagating pulse and the two counter-propagating cw oscillating fields is analyzed in this section. A semi-analytical FWM model is described in sub-section 2.1 and a comprehensive numerical simulation in 2.2.

2.1. Semi-analytical FWM model

The propagation of a short pulse along a laser diode driven above threshold initiates a complicated FWM process between the pulse and the two counter-propagating cw fields. The formalism we present for describing this interaction is an extension of the models by Agrawal for counter-propagating cw signals [8] and for pulse amplification in a saturated medium [12] as well as of Shtaif for FWM between two co-propagating pulses [9].

Assuming propagation along the waveguide of the laser, parallel to the z-axis, the wave equation for the electric field is:

\nabla^{2} \vec{E} - \frac{n_{b}^{2}}{c^{2}} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = \frac{1}{ε_{0} c^{2}} \frac{\partial^{2} \vec{P}}{\partial t^{2}}

where

\vec{E}

is the electric field,

ε_{0}

is the vacuum permittivity, c is the speed of light in vacuum, and

n_{b}

is the refractive index of the waveguide which varies with the transversal coordinates, x and y.

\vec{P}

is the induced polarization,

\vec{P} = ε_{0} χ \vec{E}

, where

χ

is the complex susceptibility. Thus, Eq. (1) becomes:

\nabla^{2} \vec{E} - \frac{n_{b}^{2}}{c^{2}} \frac{\partial^{2} \vec{E}}{\partial t^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} (χ \vec{E})}{\partial t^{2}} = 0

The last term in Eq. (2) introduces the dynamic response of the susceptibility. The temporal evolution of

χ

in the saturated regime is⁹

\frac{\partial χ}{\partial t} = \frac{χ_{s s} - χ}{τ} - \frac{χ}{τ} \frac{{| E |}^{2}}{I_{s}}

Where

χ_{s s}

is the unsaturated susceptibility,

I_{s}

is the saturation intensity, and

τ

is an effective relaxation time of the susceptibility. The propagating pulse experiences a variety of nonlinear processes but the present model considers only the FWM interaction.

In order to properly account for all the possible FWM products, one needs to consider six field components (as illustrated in Fig. 1): Two counter-propagating cw fields, representing the oscillating mode of the cavity, and serving as the pumps of the FWM interaction; The injected pulse, whose carrier wavelength is defined as the probe wavelength; Two counter-propagating idler signals, carried by a conjugate wavelength; Another signal having the probe wavelength, propagating in the opposite direction relative to the input pulse.

Fig. 1 Illustration of the various signals accounted for in the analytical model. Their directions of propagation are marked by the colored arrows, where their central angular frequencies are noted as well.

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The fields are assumed to propagate adiabatically, with negligible modifications to their (single) lateral mode profile and to their TE polarization. For simplicity, we model the laser fields as two external cw signals, which are injected from both sides of the waveguide with equal frequencies, amplitudes and phases, therefore:

\begin{array}{l} E = F (x, y) \cdot \frac{1}{2} {E_{f} (z) e^{i (ω_{0} t - k_{0} z)} + E_{b} (z) e^{i (ω_{0} t + k_{0} z)} + c . c . \\ + A_{p f} (z, t) e^{i (ω_{p} t - k_{p} z)} + A_{p b} (z, t) e^{i (ω_{p} t + k_{p} z)} + c . c . \\ + A_{i f} (z, t) e^{i (ω_{i} t - k_{i} z)} + A_{i b} (z, t) e^{i (ω_{i} t + k_{i} z)} + c . c .} \end{array}

Where

F (x, y)

represents the lateral mode profile.

E_{f}, E_{b}, A_{p f}, A_{p b}, A_{i f}, A_{i b}

are respectively the field amplitudes of the right (forward) and left (backward) propagating cw fields, the pulses and the idlers. Their corresponding angular frequencies and wave numbers are

ω_{0}, ω_{p}, ω_{i}

and

k_{0}, k_{p}, k_{i}

. The angular frequencies are related by

ω_{i} = 2 ω_{0} - ω_{p} ≜ ω_{0} - Ω

and the wave numbers by

k_{i} = | 2 k_{0} - k_{p} |

(assuming negligible chromatic dispersion). The pulses and idlers are assumed to be much weaker than the cw signals so they do not affect the cw amplitudes. This small-signal approximation is common [8] since it allows for semi-analytical solutions. Furthermore, the spectral separation between the pulse and the cw fields is taken to be wider than the spectral width of the pulse, enabling the decomposition of the wave equation into separate frequency components. Finally, all the amplitudes are treated as slowly varying envelopes.

Substitution of the total electric field Eq. (4) into Eq. (3), and applying the small-signal approximation, yields the evolution of several time and space dependent components of the susceptibility:

\begin{array}{l} χ = χ_{0} + (χ_{l o n g_f} (z, t) e^{i (Ω t - (k_{0} - k_{p}) z)} + {\tilde{χ}}_{l o n g_f} (z, t) e^{- i (Ω t - (k_{0} - k_{p}) z)}) \\ + (χ_{l o n g_b} (z, t) e^{i (Ω t + (k_{0} - k_{p}) z)} + {\tilde{χ}}_{l o n g_b} (z, t) e^{- i (Ω t + (k_{0} - k_{p}) z)}) \\ + (χ_{s h o r t_f p} (z, t) e^{i (Ω t - (k_{0} + k_{p}) z)} + {\tilde{χ}}_{s h o r t_f p} (z, t) e^{- i (Ω t - (k_{0} + k_{p}) z)}) \\ + (χ_{s h o r t_b p} (z, t) e^{i (Ω t + (k_{0} + k_{p}) z)} + {\tilde{χ}}_{s h o r t_b p} (z, t) e^{- i (Ω t + (k_{0} + k_{p}) z)}) \\ + (χ_{s h o r t_f i} (z, t) e^{i (Ω t - (k_{0} + k_{i}) z)} + {\tilde{χ}}_{s h o r t_f i} (z, t) e^{- i (Ω t - (k_{0} + k_{i}) z)}) \\ + (χ_{s h o r t_b i} (z, t) e^{i (Ω t + (k_{0} + k_{i}) z)} + {\tilde{χ}}_{s h o r t_b i} (z, t) e^{- i (Ω t + (k_{0} + k_{i}) z)}) \end{array}

where

χ_{0}

is the average susceptibility. The notation in Eq. (5) serves to discriminate short period components from long period ones, as well as right (forward) from left (backward) propagating components. Also, the wavenumbers separate components originating from the probe wave and the conjugate wave interactions. The components oscillating at frequency

Ω

are responsible for coupling the cw fields with the pulses, while those oscillating at

- Ω

couple the cw fields with the idlers; they are not the complex conjugates of their up-converting counterparts, and therefore are marked by a tilde.

Using Eq. (5) we can separate Eq. (3) into individual evolution equations:

χ_{0} = \frac{χ_{s s}}{1 + I_{0}}

\begin{array}{l} \frac{\partial χ_{l o n g_f}}{\partial t} = - χ_{l o n g_f} \frac{1 + I_{0} + i Ω τ}{τ} - \frac{χ_{0}}{4 τ} (\frac{E_{b} A_{i b}^{*}}{I_{s}} + \frac{A_{p b} E_{b}^{*}}{I_{s}}) \\ - \frac{1}{4 τ} (\frac{E_{f} E_{b}^{*}}{I_{s}} χ_{s h o r t_b p} + \frac{E_{f}^{*} E_{b}}{I_{s}} χ_{s h o r t_f i}^{*}) \end{array}

\begin{array}{l} \frac{\partial χ_{l o n g_b}}{\partial t} = - χ_{l o n g_b} \frac{1 + I_{0} + i Ω τ}{τ} - \frac{χ_{0}}{4 τ} (\frac{E_{f} A_{i f}^{*}}{I_{s}} + \frac{A_{p f} E_{f}^{*}}{I_{s}}) \\ - \frac{1}{4 τ} (\frac{E_{f} E_{b}^{*}}{I_{s}} χ_{s h o r t_b i} + \frac{E_{f}^{*} E_{b}}{I_{s}} χ_{s h o r t_f p}^{*}) \end{array}

\frac{\partial χ_{s h o r t_f p}}{\partial t} = - χ_{s h o r t_f p} \frac{1 + I_{0} + i Ω τ}{τ} - \frac{χ_{0}}{4 τ} \frac{A_{p f} E_{b}^{*}}{I_{s}} - \frac{χ_{l o n g_b}}{4 τ} \frac{E_{f} E_{b}^{*}}{I_{s}}

\frac{\partial χ_{s h o r t_b p}}{\partial t} = - χ_{s h o r t_b p} \frac{1 + I_{0} + i Ω τ}{τ} - \frac{χ_{0}}{4 τ} \frac{A_{p b} E_{f}^{*}}{I_{s}} - \frac{χ_{l o n g_f}}{4 τ} \frac{E_{b} E_{f}^{*}}{I_{s}}

\frac{\partial χ_{s h o r t_f i}}{\partial t} = - χ_{s h o r t_f i} \frac{1 + I_{0} + i Ω τ}{τ} - \frac{χ_{0}}{4 τ} \frac{A_{i b}^{*} E_{f}}{I_{s}} - \frac{χ_{l o n g_f}}{4 τ} \frac{E_{f} E_{b}^{*}}{I_{s}}

\frac{\partial χ_{s h o r t_b i}}{\partial t} = - χ_{s h o r t_b i} \frac{1 + I_{0} + i Ω τ}{τ} - \frac{χ_{0}}{4 τ} \frac{A_{i f}^{*} E_{b}}{I_{s}} - \frac{χ_{l o n g_b}}{4 τ} \frac{E_{b} E_{f}^{*}}{I_{s}}

where * indicates complex conjugate,

I_{0} = \frac{{| E_{f} |}^{2} + {| E_{b} |}^{2}}{2 I_{s}}

and

Δ k ≜ k_{i} - k_{p}

. A separate set of evolution equations for the

- Ω

oscillating components is obtained similarly, by taking a complex conjugate of the coefficients of the above equations, and replace the

Ω

oscillating susceptibility components with their tilde counterparts (with

χ_{0} = {\tilde{χ}}_{0}

).

Eliminating the lateral profile of the fields in the wave Eq. (4), using

\nabla_{⊥}^{2} F - \frac{ω_{0}^{2}}{c^{2}} ({\bar{n}}^{2} - n_{b}^{2}) F = 0

(with

\bar{n}

being the effective index of the guided mode), and integration over the lateral coordinates, leads to the following set of propagation equations:

\frac{\partial E_{f}}{\partial z} = - \frac{i}{2} Γ \frac{ω_{0}}{\bar{n} c} χ_{0} E_{f}

\frac{\partial E_{b}}{\partial z} = \frac{i}{2} Γ \frac{ω_{0}}{\bar{n} c} χ_{0} E_{b}

\begin{array}{l} \frac{\partial A_{p f}}{\partial z} + \frac{1}{v_{g}} \frac{\partial A_{p f}}{\partial t} = i \frac{ω_{0} Ω}{ω_{p}} (\frac{\bar{n}}{c} - \frac{1}{v_{g}}) A_{p f} - \frac{i}{2} Γ \frac{ω_{p}}{\bar{n} c} χ_{0} A_{p f} \\ - \frac{i}{2} Γ \frac{ω_{p}}{\bar{n} c} [(χ_{l o n g_b} + χ_{l o n g_f} e^{- i Δ k z}) E_{f} + (χ_{s h o r t_f p} + χ_{s h o r t_f i} e^{- i Δ k z}) E_{b}] \\ - \frac{Γ}{\bar{n} c} [(\frac{\partial χ_{l o n g_b}}{\partial t} + \frac{\partial χ_{l o n g_f}}{\partial t} e^{- i Δ k z}) E_{f} + (\frac{\partial χ_{s h o r t_f p}}{\partial t} + \frac{\partial χ_{s h o r t_f i}}{\partial t} e^{- i Δ k z}) E_{b}] \end{array}

\begin{array}{l} \frac{\partial A_{p b}}{\partial z} - \frac{1}{v_{g}} \frac{\partial A_{p b}}{\partial t} = - i \frac{ω_{0} Ω}{ω_{p}} (\frac{\bar{n}}{c} - \frac{1}{v_{g}}) A_{p b} + \frac{i}{2} Γ \frac{ω_{p}}{\bar{n} c} χ_{0} A_{p b} \\ + \frac{i}{2} Γ \frac{ω_{p}}{\bar{n} c} [(χ_{s h o r t_b p} + χ_{s h o r t_b i} e^{i Δ k z}) E_{f} + (χ_{l o n g_f} + χ_{l o n g_b} e^{i Δ k z}) E_{b}] \\ + \frac{i}{2} \frac{Γ}{\bar{n} c} [(\frac{\partial χ_{s h o r t_b p}}{\partial t} + \frac{\partial χ_{s h o r t_b i}}{\partial t} e^{i Δ k z}) E_{f} + (\frac{\partial χ_{l o n g_f}}{\partial t} + \frac{\partial χ_{l o n g_b}}{\partial t} e^{i Δ k z}) E_{b}] \end{array}

\begin{array}{l} \frac{\partial A_{i f}}{\partial z} + \frac{1}{v_{g}} \frac{\partial A_{i f}}{\partial t} = - i \frac{ω_{0} Ω}{ω_{i}} (\frac{\bar{n}}{c} - \frac{1}{v_{g}}) A_{i f} - \frac{i}{2} Γ \frac{ω_{i}}{\bar{n} c} χ_{0} A_{i f} \\ - \frac{i}{2} Γ \frac{ω_{i}}{\bar{n} c} [({\tilde{χ}}_{l o n g_b} + {\tilde{χ}}_{l o n g_f} e^{i Δ k z}) E_{f} + ({\tilde{χ}}_{s h o r t_b i} + {\tilde{χ}}_{s h o r t_b p} e^{i Δ k z}) E_{b}] \\ - \frac{Γ}{\bar{n} c} [(\frac{\partial {\tilde{χ}}_{l o n g_b}}{\partial t} + \frac{\partial {\tilde{χ}}_{l o n g_f}}{\partial t} e^{i Δ k z}) E_{f} + (\frac{\partial {\tilde{χ}}_{s h o r t_b i}}{\partial t} + \frac{\partial {\tilde{χ}}_{s h o r t_b p}}{\partial t} e^{i Δ k z}) E_{b}] \end{array}

\begin{array}{l} \frac{\partial A_{i b}}{\partial z} - \frac{1}{v_{g}} \frac{\partial A_{i b}}{\partial t} = i \frac{ω_{0} Ω}{ω_{i}} (\frac{\bar{n}}{c} - \frac{1}{v_{g}}) A_{i b} + \frac{i}{2} Γ \frac{ω_{i}}{\bar{n} c} χ_{0} A_{i b} \\ + \frac{i}{2} Γ \frac{ω_{i}}{\bar{n} c} [({\tilde{χ}}_{s h o r t_f i} + {\tilde{χ}}_{s h o r t_f p} e^{- i Δ k z}) E_{f} + ({\tilde{χ}}_{l o n g_f} + {\tilde{χ}}_{l o n g_b} e^{- i Δ k z}) E_{b}] \\ - \frac{Γ}{\bar{n} c} [(\frac{\partial {\tilde{χ}}_{s h o r t_f i}}{\partial t} + \frac{\partial {\tilde{χ}}_{s h o r t_f p}}{\partial t} e^{- i Δ k z}) E_{f} + (\frac{\partial {\tilde{χ}}_{l o n g_f}}{\partial t} + \frac{\partial {\tilde{χ}}_{l o n g_b}}{\partial t} e^{- i Δ k z}) E_{b}] \end{array}

where

v_{g}

is the group velocity of the pulses, and

Γ

is the confinement factor of the waveguide.

The susceptibility is related to the gain and refractive index by:

χ (z, t) = - \frac{c \bar{n}}{ω_{0}} (α - i) g (z, t)

Where

α

is the linewidth enhancement factor [16], and g is the gain profile.

Equations (6)-(12), together with the matching equations for the down-converting susceptibility components, and Eqs. (14)-(19), are solved numerically in time and length of propagation. As a first stage, the steady-state susceptibility is calculated together with the cw fields. A transform limited Gaussian pulse is introduced after laser oscillations have stabilized. At each time step the pulse and the other wave mixing products are advanced in space and the susceptibility components are evaluated at each point along the waveguide.

The parameters used in the calculations are listed in Table 1. They are set so that the net average modal gain under steady-state conditions (which represents the threshold modal gain of the laser) is about 11 dB, while matching the basic assumptions of the theory.

Table 1. List of parameters used for the semi-analytical FWM evaluation.

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The evolution of the signals along the waveguide is presented in Fig. 2(a), 2(b) and 2(c). Shown in logarithmic scale are the amplitudes of the injected pulse and the wave mixing products (relative to the pulse peak), calculated when the pulse is located near the input facet, in the middle, and close to the output facet, respectively. The injected pulse (shown in blue) creates, via FWM with the two cw fields, both counter and co-propagating idlers (shown in red and green, respectively). Upon propagation, the idlers interact with the cw fields in a cascaded FWM process which produces additional idlers. One of these (shown in black) propagates towards the left (input) facet, and is marked in the figure as the left propagating pulse. The other secondary idler co-propagates with the injected pulse and contributes to the modification of the trailing edge of the output signal in addition to the pulse deformation originating from the first order FWM process.

Fig. 2 Calculated propagation of the pulse and the generation of the FWM products along the waveguide. (a) near the input facet, (b) at the center and (c) near the output facet. The traces show in logarithmic scale the features at low intensity relative to the pulse peak. The color code of the plot matches the one used in Fig. 1.

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Figure 3 presents the total output signal and the various wave mixing products, as obtained at both facets of the cavity, as function of propagation time. The output signal emerges about 12 ps after the original pulse was injected to the cavity. The low level tail is observed once more. The peak of the left propagating idler (shown in red) is calculated at the input (left) facet after it has been amplified along the entire length of the cavity. After it reaches its peak, this idler is not immediately suppressed because of the secondary FWM processes, which mediate continuous transfer of energy between the idler and the pulse signals (in both propagation directions). This is also inferred by the inset of Fig. 3(a), which shows the trailing edge of the total output signal at the right facet, as well as the pulse and idler signals of which it is comprised. In this image, the signals show a cyclic transfer of energy between pulse and idler after the main part of the pulse has passed. Finally, the existence of the secondary FWM process is also implied by the presence of the pulse signal that propagates towards the left facet, and plotted in black in Fig. 3(b). This signal could only be created by FWM between the idler signals and the cw fields, which is a cascaded process.

Fig. 3 Calculated time-dependent amplitudes of (a) the output signal at the right output facet, (b) the signals exiting the left input facet. The inset shows a magnification of the trailing edge of the output signal, as composed of the right propagating pulse and idler.

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Figure 4 shows the time-dependent phase of the total output signal. The figure actually shows an inverted phase profile (with respect to the semi-analytical formalism) in order to match the notations used in the FDTD simulation in sub-section 2.2, and in the measurements described in section 3. The phase profile shows a distinct pattern on the trailing edge, followed by an almost linear decrease (folded to fit the $[- π, π]$ domain), indicating a frequency shift of the FWM products relative to the pulse carrier frequency.

Fig. 4 Calculated time-dependent amplitude and inverted phase of the output signal.

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The spectral modifications of the output signal are shown in Fig. 5. The output spectra, normalized to an arbitrary background level, are shown for effective gain recovery times of 1 ps, 2 ps and 5 ps. The spectral hole due to FWM mediated wave mixing is consistent with previous results [5,8,17] and is clearly visible at about 1500 nm. It is asymmetric relative to the cw frequency due to the well-known Bogatov effect [18]. The hole width is approximately proportional to the inverse of the gain recovery time, as expected [17,19]. This signature is consistent with the time-domain phase profile, since it represents a relatively narrow-banded signal, which is also shifted in wavelength from the pulse carrier wavelength. The inset shows a spectrum for the case of 5 ps gain recovery time indicating that the spectral hole is created on the short wavelength side of the pulse peak. At the wavelength of the cw signal, the pulse amplitude is reduced from its value at its peak wavelength by about 5dB. The assumption of no spectral overlap between the pulse spectrum and the cw field is not fully satisfied, but since the spectral width of the wave mixing signature is much smaller than 10 nm, the approximation is reasonably accurate.

Fig. 5 Calculated FWM signatures in the spectra of the output signal for gain recovery times of 1 ps, 2 ps and 5 ps. The inset presents an overview of the spectrum for the case of 5 ps.

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A calculated X-FROG spectrogram of the output signal (gated by the input pulse) is shown in Fig. 6 (enhanced by power of 1/5 for improved visualization). The long pattern on the right is a trace of the temporal tail, and the distortion at a delay of 400 fs represents the spectral hole due to wave mixing.

Fig. 6 Calculated sum-frequency spectrogram of the output pulse gated by the input pulse. The trace is magnified by power of 1/5 for visualization of details. Clear dips are visible near the trailing edge, followed by the spectrally-narrow wave mixing tail (circled in red).

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2.2. Numerical Model

In order to widen the scope of the modeling and to include additional effects, we developed a comprehensive numerical model, which solves the Maxwell-Schrödinger equations. It uses the density matrix formalism under the dipole approximation together with the rate equations, all of which are solved self-consistently using the FDTD technique [20,21].

Our model calculates the light-matter interaction along the propagation (z) axis and therefore we evaluate the x-component of the electrical field, E_x, and the y-component of the magnetic field, H_y. Each z-interval contains one QD with a single excited state for the electrons, which is coupled to a carrier reservoir and is biased electrically. We neglect the fact that the field addresses actually more than one QD simultaneously. The carrier reservoir supplies both electrons and holes. The valance band contains several closely placed states which are highly occupied, and from which holes relax to the ground state at a high rate. For describing the dynamical behavior, it is sufficient therefore to consider for holes only the ground state. Spontaneous electron-hole recombination processes are assumed to exist for electrons in the reservoir, in the excited state and in the ground state of the QD. A schematic diagram of the energetic arrangement of the QD is shown in Fig. 7.

Fig. 7 A schematic description of the energy levels and interactions considered in the numerical model. The electron and hole reservoir is fed by the electrical bias.

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The interaction is expressed by calculating the co-evolution of the coupled electromagnetic field and the two-level system as given by the Maxwell-Schrödinger equations; the two-level system interacts with the field through the dipole-moment driven polarization, while the field affects the two-level system through the potential term in the Schrödinger’s equations.

The evaluation of the ground state electron and hole level occupation probability-amplitudes, $ρ_{11} (t, z)$ and $ρ_{22} (t, z)$ , respectively, and the off-diagonal elements, $ρ_{12} (t, z)$ and $ρ_{21} (t, z)$ , is calculated using Eqs. (21)–(23), which are written here for simplicity without the time and space notations.

\begin{array}{l} \frac{\partial ρ_{11}}{\partial t} = \frac{N_{e s}}{2 N_{D} τ_{e s \to 11}} (1 - ρ_{11}) - \frac{ρ_{11}}{τ_{11 \to e s}} (1 - \frac{N_{e s}}{2 N_{D}}) \\ - γ_{c} ρ_{11} - i \frac{μ_{x}}{ℏ} (ρ_{12} - ρ_{21}) E_{x} \end{array}

\begin{array}{l} \frac{\partial ρ_{22}}{\partial t} = \frac{1}{τ_{e s c}^{h}} (1 - ρ_{22}) (1 - \frac{P_{r e s}}{D_{r e s}}) - \frac{1}{τ_{c a p}^{h}} \frac{P_{r e s}}{2 N_{D}} ρ_{22} \\ - γ_{v} ρ_{22} + i \frac{μ_{x}}{ℏ} (ρ_{12} - ρ_{21}) E_{x} \end{array}

\frac{\partial ρ_{12}}{\partial t} = - (i ω + γ_{i n}) ρ_{12} - i \frac{μ_{x}}{ℏ} (ρ_{11} - ρ_{22}) E_{x}

with

ρ_{12} (z, t) = ρ_{21}^{*} (z, t)

.

P_res and N_es are the hole reservoir and the electron excited state carrier densities, respectively. N_D is the spatial dot density and D_res is the electron reservoir density of states. The total QD density of electrons in the ground state, which is two-fold degenerate, equals $2 N_{D} ρ_{11}$ .

$μ_{x}$ is the x-component of the QD dipole moment and ω is the resonance angular frequency. $γ_{c}$ , $γ_{v}$ are the upper and lower level decay rates, while $γ_{i n}$ is the off-diagonal element decay rate which determines the linewidth of the resonant transition.

The additional terms in Eq. (21) are responsible for the interaction with the excited-state and for the coupling to the host reservoir. $τ_{e s \to 11}$ and $τ_{11 \to e s}$ , are the relaxation and emission time constants between the excited state and the ground state of the electrons. Similarly, $τ_{e s c}^{h}$ and $τ_{c a p}^{h}$ are the escape and capture time constants to and from the hole reservoir.

The dynamical behavior of the QD and the reservoir are evaluated using a set of rate equations:

\frac{\partial N_{r e s}}{\partial t} = η_{i} \frac{I}{q V} - \frac{N_{r e s}}{τ_{r e s}} - \frac{N_{r e s}}{τ_{c a p}} (1 - \frac{N_{e s}}{2 N_{D}}) + \frac{N_{e s}}{τ_{e s c}} (1 - \frac{N_{r e s}}{D_{r e s}})

\begin{array}{l} \frac{\partial N_{e s}}{\partial t} = \frac{N_{r e s}}{τ_{c a p}} (1 - \frac{N_{e s}}{2 N_{D}}) - \frac{N_{e s}}{τ_{e s c}} (1 - \frac{N_{r e s}}{D_{r e s}}) - \frac{N_{e s}}{τ_{e s \to 11}} (1 - ρ_{11}) \\ + \frac{2 N_{D} ρ_{11}}{τ_{11 \to e s}} (1 - \frac{N_{e s}}{2 N_{D}}) - \frac{N_{e s}}{τ_{e s}} \end{array}

\begin{array}{l} \frac{\partial P_{r e s}}{\partial t} = η_{i} \frac{I}{q V} - \frac{N_{r e s}}{τ_{r e s}} - \frac{N_{e s}}{τ_{e s}} - γ_{c} 2 N_{D} ρ_{11} - \frac{P_{r e s}}{τ_{c a p}^{h}} ρ_{22} \\ + \frac{1}{τ_{e s c}^{h}} 2 N_{D} (1 - ρ_{22}) (1 - \frac{P_{r e s}}{D_{r e s}}) - γ_{v} 2 N_{D} ρ_{22} \end{array}

with N_res being the reservoir carrier density. I, η_i and V are the bias current, injection efficiency and the active layer volume, respectively.

τ_{r e s}

, is the reservoir lifetime.

τ_{c a p}

and

τ_{e s c}

are the reservoir to dot capture and escape lifetimes which are related, for electrons and holes separately, according to the principle of detailed balance [19].

Applying Maxwell's equations to calculate E_x(t,z) and H_y(t,z), completes the description of the light-matter interaction.

\frac{\partial E_{x}}{\partial z} = - μ_{0} \frac{\partial H_{y}}{\partial t}

- \frac{\partial H_{y}}{\partial z} = σ E_{x} + \frac{\partial D_{x}}{\partial t}

D_{x} = ε_{0} E_{x} + P_{x}

where:

P_{x} = P_{m a t} + P_{d o t}

P_{m a t} = ε_{0} χ_{e} E_{x}

P_{d o t} = 2 N_{D} μ_{x} (ρ_{12} + ρ_{12}^{*}) Γ

We distinguish between the polarization which originates from the background material refractive index and the dot polarization as expressed in Eqs. (30)–(32). The background electrical susceptibility is χ_e. The confinement factor Г accounts for the difference between the volumes of the optical mode and the gain medium. The internal losses are included through the conductance, σ, which is assumed to be uniform throughout the structure.

The linewidth enhancement factor α is accounted for by formulating a parametric dependence of the refractive index on carrier population:

n^{2} = ε_{r} = ε_{r_{0}} - ε (N_{r e s}, 2 N_{D} ρ_{11}) = ε_{r_{0}} - C_{r e s} N_{r e s} - C_{Q D} 2 N_{D} ρ_{11}

Where C_res and C_QD are chosen so that the index depends only on the reservoir population [22].

The numerical calculation space is arranged in a way that the QD laser cavity is placed in the center of the z-axis computational space and is surrounded by free-space where the pulse excitation occurs. At the end points of the calculation space (which is longer than the laser cavity), absorbing boundary conditions are artificially applied in order to prevent the outgoing E_x and H_y fields from being reflected back into the calculation space. In order to achieve both high precision and rapid calculations, the Schrödinger and Maxwell equations were calculated simultaneously using a central-difference method while the rate equations, which are sampled at a significantly higher rate than actually needed, were evaluated using a forward-difference scheme.

The model does not account for the inhomogeneous broadening of the self-assembled QDs although additional energy levels may be added easily at the expense of increasing the computational effort. In order to accommodate the entire spectrum of the short, 150 fs pulses, we increase the linewidth of the resonant transition to 16.5 meV (30 nm).

Dielectric coatings were added to the laser facets yielding reflectivities of ~1%. The increased mirror losses yield a large threshold gain as in the experiment, which is presented in Section 3.

The simulation proceeded in two stages. Steady-state laser oscillations were first built-up from a random electrical field noise. Following the stabilization of the various transient effects, a short Gaussian pulse was applied which experienced the various nonlinear effects upon propagation; its complex properties were analyzed at the output. The parameters used for the calculation are listed in Table 2.

Table 2. List of parameters used for the FDTD simulation.

View Table | View all tables in this article

In the second stage we propagated a 150 fs wide unchirped Gaussian pulse with an energy of 14 pJ through the oscillating medium. Figure 8 shows the calculated multimode laser oscillations (in red, and enlarged in the inset), and the pulse spectrum before (green) and after propagation (blue). The figure shows a shift of the pulse peak wavelength by approximately 2 nm; such a shift was indeed observed in previous experiments [14]. As in the analytical model of sub-section 2.1, the output pulse spectrum exhibits a spectral hole near the wavelength of the oscillating field. The width of the hole is estimated to be about 10 nm which corresponds to an effective response time of about 1 ps, consistent with the QD capture time used in the simulation. Nevertheless, since other nonlinear effects such as SPM take place simultaneously, the width of the hole is not determined solely by the capture rate.

Fig. 8 Calculated spectrum of the output pulse (blue), with respect to the spectrum of the initial pulse (green). The spectrum of the cw multi-mode oscillations is plotted in red as a frequency reference. The FWM induced spectral hole is visible on the left of the cw trace. The inset zooms into the cw spectrum, noting its multi-mode nature. The traces are shifted in intensity for clarity.

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The complex temporal field envelopes (phase and amplitude) of the pulse before and after propagation are shown in Fig. 9. Both the phase and amplitude are presented without the fast oscillating carrier frequency and have been extracted according to the relation:

E_{x} (t) = \frac{1}{2} A (t) (e^{i ω_{0} t - i ϕ (t)} + e^{- i ω_{0} t + i ϕ (t)})

with A(t) and φ(t) being the slowly varying temporal amplitude and phase, respectively. The figure shows the complex field of both the unchirped input reference and the output pulse. The phase is not well defined when the amplitude is zero which explains the corrugations in the phase at t = 1.1 ps. The functional form of the output amplitude and phase are almost identical to those obtained with the semi-analytical model (Fig. 4) showing the trailing excitation at around t = 0.25 ps as well as its linear phase evolution. Nevertheless, the temporal trace obtained with the numerical model differs from the analytical model since it contains the chirp that is accumulated due to SPM.

Fig. 9 Calculated time-dependent amplitude and phase of the output pulse. The phase on the tail of the pulse decreases linearly due to the spectral shift of the FWM generated tail from the pulse carrier frequency.

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Another feature of the interaction is that the spectral hole vanishes for high peak pulse peak powers, due to an increased spectral modification, caused by SPM which masks the effects of FWM, as seen in Fig. 10. This behavior is also observed in the experiments described in Section 3.

Fig. 10 Calculated spectra of the output pulse for different input pulse energies. The spectral hole is less distinct as the input energy increases.

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Finally, in order to be able to compare the numerical simulation to the experiments and the semi-analytical calculation, we plot in Fig. 11 the calculated spectrogram (X-FROG trace) of the emitted pulse, where the trace is enhanced by power of 1/5 to improve the visualization of the low power features. The trace shows a somewhat symmetric shift at delays between −0.2 ps to 0.2 ps which is a well-known signature of SPM [15]. In addition, the trace shows the trailing excitation at delays longer than 0.4 ps which originate from the wave mixing process.

Fig. 11 Calculated sum-frequency spectrogram of the FDTD simulation output pulse gated by the input pulse. The trace is enhanced by a power of 1/5 for clarity. A clear dip is visible near the trailing edge, followed by the spectrally-narrow wave mixing tail (circled in red). The other holes are SPM footprints.

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3. Experimental results

Time dependent amplitudes and phases, as well as spectra of a 150 fs FWHM pulse, generated by a tunable optical parametric oscillator propagating through a QD laser, were measured by a X-FROG system under various operating conditions. The X-FROG measurement generates a spectrogram (X-FROG trace) of the sum-frequency product (in a thin LiNbO₃ crystal) of the sampled pulse with a pre-characterized pulse, at different temporal delays between them. The bandwidth of the nonlinear sampling and detection is sufficiently large to easily support the 150 fs pulse namely, it is about 80 nm at the doubled frequency. A phase retrieval technique enables to extract the precise temporal and spectral characteristics of the sampled pulse, which otherwise could not be separated from the background amplified spontaneous emission (ASE).

The laser we tested was a 1.5-mm-long, buried-channel, InAs/InP QD laser with reduced facet reflectivity yielding a high threshold gain. Its bias dependent ASE spectra are depicted in Fig. 12. The wide gain spectrum below threshold results from the inhomogeneously broadened gain. The oscillation threshold is reached at around 75 mA. Short (150 fs) pulses were launched into the laser and propagated along its waveguide in the presence of the cw oscillating fields.

Fig. 12 Measured bias dependent ASE spectra. The gain peak experiences a blue shift with increasing bias. The oscillation threshold is around 75 mA.

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Figure 13(a) and 13(b) show a typical sum-frequency X-FROG trace, and the corresponding X-FROG trace after retrieval, respectively, obtained for a 0.25 pJ input pulse at 1510 nm (which is longer than the cw oscillation wavelength) and a bias of 100 mA. Both traces are enhanced by power of 1/3 for improved visualization of details. The retrieved trace, which is clearer than the measured one, helps in matching the experimental results to the theory. The traces are similar to those predicted by both the semi-analytical calculation and the FDTD simulation (Fig. 6 and 11). The tail and dip in its base, which originate from the FWM process, are circled in red. The weak, trailing signal and the dip are observable at wavelengths longer than the marked tail, and are consistent with the spectrogram obtained by the FDTD simulation. The differences between the measured (retrieved) and the predicted spectrograms originate from the fact that the simulations assume a perfect input pulse while the experiment used an imperfect input pulse which was not purely Gaussian and also contained some chirp. Using the real pulse as an input to the simulation can improve the accuracy of the model. Similar results were also obtained for an input pulse centered at 1470 nm.

Fig. 13 (a) Measured X-FROG trace (spectrogram). (b) Retrieved X-FROG trace. The images are enhanced by a power of 1/3 for clarity. The wave mixing-generated tail and dip are circled in red.

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The temporal profiles of the retrieved pulses, at various bias levels are presented in Fig. 14(a) and 14(b), for pulses centered at 1470 nm and 1510 nm, respectively. A pulse asymmetry rises with bias level, as the medium gets saturated more deeply. The saturation evolution, with respect to the pulse wavelength, follows the spectral shape of the ASE. Furthermore, for above-threshold bias levels, the predicted tail is visible (magnified in the right inset of Fig. 14(b)), and grows with bias in proportion to the amount of ASE in the spectral domain of the pulse. The time-dependent phase of the pulses is plotted in the insets, on the left of Fig. 14(a) and 14(b). Once more, the predictions of the simulations are confirmed, as the phase rises or falls linearly according to the shift of the oscillating mode frequency relative to the pulse carrier frequency. Thus, we conclude that FWM originated wave mixing is indeed dominant, together with some SPM, in the nonlinear interactions of these experiments.

Fig. 14 Retrieved time-dependent amplitudes and spectra of the amplified pulse at different bias levels. (a) and (b) show time-dependent pulses initially at 1470 nm and 1510 nm, respectively. The inset in (a) and the left inset in (b) show the instantaneous phase of the amplified pulse at 100 mA bias. The right inset in (b) is a magnification of the trailing edge of the amplified pulse at 1510 nm. (c) and (d) are the spectra of the 1470 nm and 1510 nm pulses, respectively. The wave mixing hole is visible in the 1485 nm region in (c), and near 1495 nm in (d).

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The retrieved output pulse spectra for a pulse at 1470 nm are shown in Fig. 14(c). For bias levels above threshold, the spectra exhibit the predicted hole in the vicinity of the cw oscillation wavelength. As the pulse is tuned to 1510 nm, the spectral location of the hole remains approximately unchanged (Fig. 14(d)) proving, unequivocally, that it originates from the FWM process with the oscillating cw fields. The width of the spectral hole is 3-4 nm, (~500 GHz), consistent with the reciprocal time constant for carrier relaxation in QD gain media [19]. The nature of the X-FROG system is such that the cw spectral line is not observed; its wavelength is determined separately by an optical spectrum analyzer.

The spectral hole is most significant for a pulse energy of 0.25 pJ, while at larger energies other effects, such as SPM, mask it. This is demonstrated in Fig. 15, which shows the retrieved spectra of 1510 nm pulses, with 50 pJ energies, at various bias levels.

Fig. 15 Retrieved spectra of 1510 nm, 50 pJ pulses for different bias levels. The wave mixing signature is masked by other effects.

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The wave mixing spectral dip is seen; however, it is either shifted or masked, and in general it is significantly less clear compared to the case of the 0.25 pJ pulses. This observation is consistent with the numerical model, Fig. 10.

4. Conclusion

To conclude, we analyzed theoretically the properties of an ultra-short pulse, which propagates through a QD laser operating above threshold, and interacts with the oscillating modes. A semi-analytic calculation of the FWM process between the pulse and the oscillating field predicts a dip in the spectrum of the output signal close the cw wavelength. The width of the spectral hole is determined by an effective gain recovery time. It also predicts a tail in the time dependent amplitude which is spectrally narrow and is centered near the oscillating mode frequency; These patterns are predicted as well, and in more detail, by a FDTD simulation, which considers charge carrier dynamics as well as two-level interactions with the electromagnetic fields and includes additional nonlinear effects other than FWM such as saturation induced SPM. However, the FWM interactions are dominant and are easily observed in both the time and frequency domains, as well as in the calculated spectrograms.

Finally, a X-FROG characterization was used to examine experimentally the properties of a 150 fs pulse which propagated along an InAs/InP QD laser. The experimental results are consistent with all the numerical predictions. An effective gain recovery time of about 2 ps was measured, in agreement with known values for such lasers. Many fine details, including small trailing pulses as well as spectral modifications predicted by the models, were confirmed by the X-FROG experiments, leading to the conclusion that FWM, accompanied by SPM, is the dominant nonlinear interaction in these experiments.

Acknowledgments

This work was partially supported by the Israeli Science Foundation and by the European Commission through the project GOSPEL. The work of the authors from HHI and TUB was funded by DFG in the framework of SFB 787. G.E. would like to thank A. von Humboldt Foundation and DFG/SFB 787 for generous support. O.K acknowledges support of RBNI and A.C thanks the Clore Foundation.

References and links

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2. P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Ultrafast gain dynamics in InAs-InGaAs quantum-dot amplifiers,” IEEE Photon. Technol. Lett. 12(6), 594–596 (2000). [CrossRef]

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5. A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud, and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of a saturated InAs/InP quantum dash amplifier operating near 1550 nm,” Opt. Express 16(3), 2141–2146 (2008). [CrossRef] [PubMed]

6. J. Zimmermann, S. T. Cundiff, G. von Plessen, J. Feldmann, M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, “Dark pulse formation in a quantum-dot laser,” Appl. Phys. Lett. 79(1), 18–20 (2001). [CrossRef]

7. C. Sun, B. Golubovic, H. Choi, C. A. Wang, and J. G. Fujimoto, “Femtosecond investigations of spectral hole burning in semiconductor lasers,” Appl. Phys. Lett. 66(13), 1650–1652 (1995). [CrossRef]

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9. M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett. 66(12), 1458–1460 (1995). [CrossRef]

10. A. Mecozzi and J. Mork, “Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3(5), 1190–1207 (1997). [CrossRef]

11. A. Bilenca, R. Alizon, V. Mikhelashvili, D. Dahan, G. Eisenstein, R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Broad-band wavelength conversion based on cross-gain modulation and four-wave mixing in InAs-InP quantum-dash semiconductor optical amplifiers operating at 1550 nm,” IEEE Photon. Technol. Lett. 15(4), 563–565 (2003).

12. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989). [CrossRef]

13. S. Linden, H. Giessen, and J. Kuhl, “XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi, B Basic Res. 206(1), 119–124 (1998). [CrossRef]

14. A. Capua, A. Saal, O. Karni, G. Eisenstein, J. P. Reithmaier, and K. Yvind, “Complex characterization of short-pulse propagation through InAs/InP quantum-dash optical amplifiers: from the quasi-linear to the two-photon-dominated regime,” Opt. Express 20(1), 347–353 (2012). [CrossRef] [PubMed]

15. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Norwell, 2002).

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Parameter	Value	Units
Cavity length (L)	900	µm
CW wavelength ( $λ_{0}$ )	1500	nm
Pulse carrier wavelength ( $λ_{1}$ )	1510	nm
Gain recovery time (τ)	2	ps
Optical confinement factor (Г)	0.25	^%
Linewidth enhancement factor ( $α$ )	5
Small-signal gain	14,000	cm⁻¹
Modal refractive index ( $\bar{n}$ )	3.5
Group index ( $n_{g}$ )	4
Saturation power ( $\propto I_{s}$ )	13	dBm
CW power ( $\propto {\| E_{f} \|}^{2}$ )	0	dBm
Pulse peak power ( $\propto {\| A_{p} \|}^{2}_{\max}$ )	−10	dBm
Pulse width – FWHM	300	fs

Parameter	Value	Units
Cavity length (L)	250	µm
Electrical bias (I)	600	mA
Injection efficiency (η_i)	50	%
QD density of states (N_D)	8 × 10²³	m⁻³
Carrier reservoir density of states (D_res)	2 × 10²⁵	m⁻³
Upper level decay rate (γ_c)	2.5 × 10⁹	s⁻¹
Lower level decay rate (γ_v)	2.5 × 10⁹	s⁻¹
Off-diagonal element decay rate (γ_in)	3.6 × 10¹²	s⁻¹
Carrier reservoir lifetime (τ_res)	0.4	ns
Excited state lifetime (τ_es)	0.4	ns
Dipole moment (μ_x)	1.5 × 10⁻²⁸	C × m
Optical confinement factor (Г)	0.0025
Active layer depth (five QD layers)	50	nm
Waveguide width	4	µm
Reservoir-QD electron capture time (τ_cap)	1	ps
Excited state relaxation time (τ_es→11)	100	fs
Hole capture time (τ^h_cap)	100	fs
Hole escape time (τ^h_esc)	100	fs
Background dielectric constant (ε_r0)	12
Carrier reservoir index coefficient (C_res)	0.9 × 10⁻²⁵	m³
QD index coefficient (C_QD)	0.2 × 10⁻²⁵	m³
Conductivity (σ)	13	(Ω × m)⁻¹

Parameter	Value	Units
Cavity length (L)	900	µm
CW wavelength ( $λ_{0}$ )	1500	nm
Pulse carrier wavelength ( $λ_{1}$ )	1510	nm
Gain recovery time (τ)	2	ps
Optical confinement factor (Г)	0.25	^%
Linewidth enhancement factor ( $α$ )	5
Small-signal gain	14,000	cm⁻¹
Modal refractive index ( $\bar{n}$ )	3.5
Group index ( $n_{g}$ )	4
Saturation power ( $\propto I_{s}$ )	13	dBm
CW power ( $\propto {\| E_{f} \|}^{2}$ )	0	dBm
Pulse peak power ( $\propto {\| A_{p} \|}^{2}_{\max}$ )	−10	dBm
Pulse width – FWHM	300	fs

Parameter	Value	Units
Cavity length (L)	250	µm
Electrical bias (I)	600	mA
Injection efficiency (η_i)	50	%
QD density of states (N_D)	8 × 10²³	m⁻³
Carrier reservoir density of states (D_res)	2 × 10²⁵	m⁻³
Upper level decay rate (γ_c)	2.5 × 10⁹	s⁻¹
Lower level decay rate (γ_v)	2.5 × 10⁹	s⁻¹
Off-diagonal element decay rate (γ_in)	3.6 × 10¹²	s⁻¹
Carrier reservoir lifetime (τ_res)	0.4	ns
Excited state lifetime (τ_es)	0.4	ns
Dipole moment (μ_x)	1.5 × 10⁻²⁸	C × m
Optical confinement factor (Г)	0.0025
Active layer depth (five QD layers)	50	nm
Waveguide width	4	µm
Reservoir-QD electron capture time (τ_cap)	1	ps
Excited state relaxation time (τ_es→11)	100	fs
Hole capture time (τ^h_cap)	100	fs
Hole escape time (τ^h_esc)	100	fs
Background dielectric constant (ε_r0)	12
Carrier reservoir index coefficient (C_res)	0.9 × 10⁻²⁵	m³
QD index coefficient (C_QD)	0.2 × 10⁻²⁵	m³
Conductivity (σ)	13	(Ω × m)⁻¹

Nonlinear pulse propagation in a quantum dot laser

Abstract

1. Introduction

2. Theory

2.1. Semi-analytical FWM model

2.2. Numerical Model

3. Experimental results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (15)

Tables (2)

Equations (34)

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