Multimode description of self-mode locking in a single-section quantum-dot laser

Weng W. Chow; Songtao Liu; Zeyu Zhang; John E. Bowers; Murray Sargent

doi:10.1364/OE.382821

1. Introduction

Laser mode locking has a rich history [1]. The high peak intensities find uses ranging from nonlinear optics to laser machining and surgery. Equally important, the ultrashort pulses are valuable for studying kinetics in chemistry and biology, as well as electron dynamics in condensed matter. Radio frequency combs generated by the beating of mode-locked lasing modes have significantly improved frequency metrology. Recent advances made in mode locking semiconductor quantum-dot (QD) lasers have potential for commercial applications, e.g. in wavelength division multiplexing for telecommunications [2,3,4].

Mode locking is achieved with both single- and multi-section diode lasers [5,6]. In the case of single-section experiments, there is some debate on whether one is seeing self-mode locking or unaccounted saturable absorption from, e.g., uneven current injection. With self-mode locking, the gain medium alone produces a) multimode lasing resulting in broad emission bandwidth and b) four-wave mixing causing the locking of beat frequencies among lasing modes. Both mechanisms arise from nonlinearities in light-matter interactions within an inhomogeneously-broadened distribution of QDs providing gain to a large number of lasing modes.

Mode locking may be investigated using time- or frequency-domain approaches [7,8,9,10,11]. Time-domain or traveling-wave approaches have the advantage of giving a physical picture of pulse train formation, and therefore, are frequently used in device models. Also, they have the advantage of being able to treat resonator outcoupling. However, the effects of group velocity dispersion and self-phase modulation, which are needed to model mode locking in single-section lasers, cannot be readily incorporated [12]. The former complicates the numerical analysis, while the latter leads to an additional input parameter that is not independent of the gain compression factor used to account for gain saturation. With a frequency-domain or multimode semiclassical laser approach, group velocity dispersion is directly input into the specification of the passive cavity modes. Moreover, mode-locking mechanisms, such as self-phase modulation, are treated at a microscopic (electron, hole, radiation field) level, so that mode-locking strength may be directly connected to electronic structure. With the locking mechanism, gain saturation, mode competition and carrier—induced refractive index change treated on equal footing, one has a consistent description of the active medium nonlinearities. However, there is a compromise. The frequency domain approach has a weakness in treating lossy cavities. One may describe an open cavity by the modes of the universe, [13,14] but doing increases computational demand.

This paper describes a frequency-domain-based investigation of self-mode locking in a single-section QD laser. One motivation is to address the question of whether self-mode locking is indeed possible with only an intrinsic QD gain medium producing all the necessary optical nonlinearities to overcome dispersion in the optical cavity. A second motivation is to have an analytical tool for band-structure engineering to optimize device design. We note that to accomplish the second goal, further refinement of the present model is necessary. For example, to facilitate numerical analysis so that we can focus on the first goal, we assumed quasi-equilibrium among quantum dots within an inhomogeneous distribution and approximated carrier transport from quantum well to quantum dot states with a carrier injection efficiency. These assumptions will lead to inaccurate predictions of mode-locking performance, e.g., on the fundamental limitations to RF linewidth and pulse duration. The present paper does not make those predictions.

In Section 2, we describe the derivation of a multimode semiclassical QD laser theory that treats, on equal footing, the nonlinearities contributing to mode saturation and competition, gain-induced dispersion and mode locking. Section 3 discusses numerical simulations using the model. We are able to reproduce the basic features of progression to mode-locked pulses and frequency combs with increasing excitation. The section also illustrates the effects of the linewidth enhancement factor, which is contained within semiclassical laser theory via the frequency pulling and pushing contributions. Partial mode locking and frequency comb jitter are also discussed in Section 3. Section 3 ends with a discussion of comparison between model and experiment.

2. Theory

This section describes a theory for a QD laser consisting of an optical cavity supporting multiple modes and an active medium containing an inhomogeneously-broadened distribution of QDs. Assuming that transverse field effects may be approximated by a mode confinement factor, the laser field equation may be written as

(1)$$E({z,t} )= \frac{1}{2}{e^{ - i{\Omega _0}t}}\mathop \sum \limits_n {E_n}(t ){e^{ - i{\phi _n}(t )}}{u_n}(z )+ c.c.,$$

where t is the time, z is the position along the laser axis and ${u_n}(z )$ is the ${n^{th}}$ passive resonator eigenfunction. For that mode, ${E_n}$ and ${{\Omega }_0} + {\dot{\phi }_n}$ are the electric field amplitude and frequency, where ${{\Omega }_0}$ is the center of the multimode lasing spectrum.

Starting from Maxwell’s equations and assuming slowly varying laser field amplitude and phase compared to optical frequency, we obtain [15]

(2)$$\frac{d}{{dt}}{E_n} ={-} \frac{{{{\Omega }_0}}}{{2Q}}{E_n} - \frac{{{{\Omega }_0}}}{{2{\varepsilon _B}}}Im({{P_n}} )\; \; ,\; $$

(3)$$\frac{d}{{dt}}{\phi _n} = {{\Omega }_n} - {{\Omega }_0} - \frac{{{{\Omega }_0}}}{{2{\varepsilon _B}}}\frac{1}{{{E_n}}}Re({{P_n}} )\; \; ,$$

where $Q$ is the cavity quality factor, $\varepsilon_{B}$ is the background permittivity and $\Omega_{n}$ is the n^th mode passive cavity frequency. In the above equations, the QD gain medium appears as a macroscopic polarization

(4)$$P({z,t} )= \frac{1}{2}{e^{ - i{{\Omega }_0}t}}\mathop \sum \limits_n {P_n}(t ){e^{ - i{\phi _n}(t )}}{u_n}(z )+ c.c.\; \; ,$$

where ${P_n}(t )$ is the complex polarization amplitude that connects to a microscopic (quantum mechanical) polarization ${p_q}(t )= \langle{b_q}{c_q}\rangle\textrm{exp}[{({i{\omega_q} + \gamma } )t} ]\; $from an electron-hole pair via

(5)$${P_n}(t )= \frac{{2{\Gamma }\wp }}{V}\mathop \sum \limits_q {e^{i{{\Omega }_0}t + i{\phi _n}(t )}}\frac{2}{L}\mathop \smallint \nolimits_0^L dz\,{u_n}(z ){p_q}(t )\; \; ,$$

${\Gamma }$ is the mode confinement factor, ${\omega _q}$ is the transition energy for the QD state q, $\gamma $ is the dephasing rate, and $\wp $ is the dipole matrix element for the interaction between an electron-hole pair and the laser field.

The equations of motion necessary for determining ${p_q}(t )$ are derived using the Hamiltonian [16]

(6)$$H = \mathop \sum \limits_q {\varepsilon _{e,q}}c_q^\dagger {c_q} + \mathop \sum \limits_q {\varepsilon _{h,q}}b_q^\dagger {b_q} - \wp \mathop \sum \limits_q ({c_q^\dagger b_q^\dagger + {b_q}{c_q}} )E({z,t} )\; \; ,\; $$

where ${c_q}$ and $c_q^\dagger $ are electron annihilation and creation operators, ${b_q}$ and $b_q^\dagger $ are the corresponding operators for holes, and ${\varepsilon _{\sigma ,q}}$ with $\sigma = e$ or h is the electron or hole energy in QD state q. Based on experiments, we consider only ground-state lasing, in which case, q labels a group of QDs with the same electronic structure. In the numerical simulations, the QDs are grouped into bins, each with width given by the homogeneous broadening $\hbar \gamma $ and average transition frequency ${\omega _q}$.

Working in either the Schrödinger or Heisenberg Picture of quantum mechanics and using perturbation theory, we obtained to 3^rd order in electron-light interaction,

(7)$${p_q}(t )= p_q^{(1 )}(t )+ p_q^{(3 )}(t )\; \; ,$$

where the 1^st and 3^rd order contributions are

(8)$$p_q^{(1 )}(t )={-} i({{n_{e,q}} + \; {n_{h,q}} - 1} )\mathop \sum \limits_n {u_n}(z ){{\cal E}_n}(t ){e^{ - i{\phi _n}(t )}}{D_\gamma }({{\omega_q} - {{\Omega }_n}} )\; \; ,\; $$

(9)$$\begin{aligned}p_q^{(3 )}(t )& =2i({{n_{e,q}} + \; {n_{h,q}} - 1} )\mathop \sum \limits_{n^{\prime},m,m^{\prime}} {u_{n^{\prime}}}(z ){u_m}(z ){u_m}(z ){{\cal E}_{n^{\prime}}}(t ){{\cal E}_m}(t ){{\cal E}_{m^{\prime}}}(t )\; \\ & \times {e^{ - i[{{\phi_{n^{\prime}}}(t )- {\phi_m}(t )+ {\phi_{m^{\prime}}}(t )} ]}}{e^{i({{\omega_q} - {{\Omega }_{n^{\prime}}} + {{\Omega }_m} - {{\Omega }_{m^{\prime}}}} )t + \gamma t}}{D_\gamma }({{{\Omega }_m} - {{\Omega }_{m^{\prime}}}} )\; \\ & \times \frac{{{\gamma _{ab}}}}{\gamma }{D_\gamma }({{\omega_q} - {{\Omega }_{n^{\prime}}} + {{\Omega }_m} - {{\Omega }_{m^{\prime}}}} )[{{D_\gamma }({{\omega_q} - {{\Omega }_m}} )+ {D_\gamma }({{\omega_q} - {{\Omega }_{m^{\prime}}}} )} ]\; \; ,\; \end{aligned}$$

${n_{e,q}} = \langle c_q^{\dagger} {c_q}\rangle$, ${n_{h,q}} = \langle b_q^{\dagger} {b_q}\rangle$, ${\gamma _{ab}}$ is the carrier population relaxation rate between QDs, ${D_\gamma }(x )= {({1 + ix/\gamma } )^{ - 1}}$ and we define a dimensionless laser electric field amplitude ${{\cal E}_n} = \wp {E_n}/({2\hbar \gamma } )$. The 3^rd order contribution gives rise to 4-wave mixing.

Substituting Eqs. (8) and (9) into Eq. (5) to obtain the macroscopic polarization, and then using that polarization in Eqs. (2) and (3), give the slowly varying amplitude and phase equations for each cavity mode. They may be expressed in the form:

(10)$$\begin{aligned}{{{\dot{{\cal E}}}}_n}(t )& =\left[ {g_n^{sat}({{N^{({2d} )}}} )- \frac{{{{\Omega }_0}}}{{2Q}}} \right]{{\cal E}_n}(t )\\ \; &\quad\quad - \mathop \sum \limits_{\{{n^{\prime},m,m^{\prime}} \}} \textrm{Re}[{{\vartheta_{nn^{\prime}mm^{\prime}}}(N^{(2d)} ){e^{ - i{\psi_{nn^{\prime}mm^{\prime}}}}}} ]{{\cal E}_{n^{\prime}}}(t ){{\cal E}_m}(t ){{\cal E}_{m^{\prime}}}(t )\; \; ,\end{aligned}$$

(11)$$\begin{aligned}{{\dot{\phi }}_n}(t )& =[{{\Omega }_n^{sat}(N^{(2d)} )- {{\Omega }_0}} ]\; \\ & - \mathop \sum \limits_{\{{n^{\prime},m,m^{\prime}} \}} \textrm{Im}[{{\vartheta_{nn^{\prime}mm^{\prime}}}{(N^{(2d)})e^{ - i{\psi_{nn^{\prime}mm^{\prime}}}}}} ]{{\cal E}_{n^{\prime}}}(t ){{\cal E}_m}(t ){{\cal E}_{m^{\prime}}}(t ){\cal E}_n^{ - 1}(t )\; \; .\; \end{aligned}$$

In Eq. (10),

(12)$$g_n^{sat} = \frac{{{g_n}}}{{1 + {A_n}{\cal E}_n^2 + \mathop \sum \nolimits_{m \ne n} {B_{n,m}}{\cal E}_m^2}}\; $$

is the saturated gain, where the denominator contains the self- and cross-mode saturation contributions, ${A_n}({{N^{({2d} )}}} ){\cal E}_n^2$ and $\mathop \sum \nolimits_{m \ne n} {B_{n,m}}({{N^{({2d} )}}} ){\cal E}_m^2$, respectively, which deplete the linear gain ${g_n}$. The saturation coefficients are evaluated at the saturated total carrier density, ${N^{({2d} )}}$. The gain medium also modifies the passive cavity, changing ${\Omega _n}$ to

(13)$${\Omega }_n^{sat} = {\varOmega _n} + {\sigma _n}({{N^{({2d} )}}} )- {\tau _n}({{N^{({2d} )}}} ){\cal E}_n^2 - \mathop \sum \limits_{m \ne n} {\tau _{n,m}}({{N^{({2d} )}}} ){\cal E}_m^2{\; }$$

via the 1^st order frequency pulling term ${\sigma _n}$, which in turn is modified by the 3^rd order self- and cross-frequency pushing contributions ${\tau _n}({{N^{({2d} )}}} ){\cal E}_n^2$ and $\mathop \sum \nolimits_{m \ne n} {\tau _{n,m}}({{N^{({2d} )}}} ){\cal E}_m^2$, respectively [15]. Lastly, in Eqs. (10) and (11) are the relative phase angle terms, giving rise to mode locking [10]. For the summations, the bracket $\{{} \}$ indicates summing over only the combinations of $n^{\prime},m,m^{\prime}$ not include in Eqs. (12) and (13). Table 1 shows the equations for the various active medium contributions.

Table 1. Formulas for the 1^st and 3^rd order active medium coefficients.

View Table

When numerically evaluating the coefficients, we make the replacement:

(14)$$\frac{1}{A}\mathop \sum \limits_q \to \mathop \smallint \nolimits_{ - \infty }^\infty d{\varepsilon _q}\; D({{\varepsilon_q}} )\; \; ,\; $$

where the electronic density of state is

(15)$$D({{\varepsilon_q}} )= 2\frac{{N_{QD}^{({2d} )}}}{{\sqrt {2\pi } {{\Delta }_{inh}}}}\textrm{exp}\left[ { - {{\left( {\frac{{{\varepsilon_q}}}{{\sqrt 2 {{\Delta }_{inh}}}}} \right)}^2}\; } \right]\; \; ,\; $$

$A$ is the active region area, $N_{QD}^{({2d} )}$ is the QD density and ${{\Delta }_{inh}}$ is the inhomogeneous width. We assume that carrier scattering is sufficiently fast to maintain quasi-equilibrium carrier distributions. Then,

(16)$${n_{\sigma ,q}} = f({\varepsilon_q^\sigma ,{\mu_\sigma }} )= \frac{1}{{exp[{({\varepsilon_q^\sigma - {\mu_\sigma }} )/({{k_B}T} )} ]+ 1}}\; ,\; $$

where $\sigma = e\; (h )$ for electrons (holes) and T is the active medium temperature. The chemical potential ${\mu _\sigma }$ is determined by the total saturated carrier density:

(17)$${N^{({2d} )}} = \frac{1}{A}\mathop \sum \limits_q {n_{\sigma ,q}} = \mathop \smallint \nolimits_{ - \infty }^\infty d{\varepsilon _q}\; D({{\varepsilon_q}} )f({\varepsilon_q^\sigma ,{\mu_\sigma }} )\; .\; $$

With the quasi-equilibrium assumption, ${N^{({2d} )}}$ is obtained from the second order in electron-light interaction:

(18)$$\frac{d}{{dt}}{n_{\sigma ,q}} ={-} 2i{e^{ - \gamma t}}\mathop \sum \limits_n {u_n}(z ){{\cal E}_n}Im[{{e^{i({{{\Omega }_n} - {\omega_q}} )t + i{\phi_n}}}p_q^{(1 )}} ]\; \; \; .\; $$

Summing Eq. (18) for either $\sigma = e$ or h, we get

(19)$$\frac{{d{N^{({2d} )}}}}{{dt}} = \frac{{\eta J}}{e}\left( {1 - \frac{{{N^{({2d} )}}}}{{2N_{QD}^{({2d} )}}}} \right) - {\gamma _{nr}}{N^{({2d} )}} - \frac{{4{\varepsilon _0}\hbar d}}{{{{\Omega }_0}}}{\left( {\frac{{\gamma {n_B}}}{\wp }} \right)^2}\mathop \sum \limits_n g_n^{sat}({{N^{({2d} )}}} ){\cal E}_n^2\; ,$$

where the pump term consists the current density J, injection efficiency from electrodes to QD states $\eta $, and carrier blocking due to the Exclusion Principle $({1 - {N^{({2d} )}}/2N_{QD}^{({2d} )}} )$. This approach to treating intraband relaxation lies between a full quantum kinetic theory [17] and a simple (but inadequate for our purpose) total carrier density, rate equation treatment [18]. It reasonably mimics microscopic features of carrier scattering, while allowing inclusion of engineering details and reduction of computational demand [19,20].

3. Results and discussion

This section describes results from solving Eqs. (10), (11) and (19) until steady state. A goal is to illustrate the capability of the model to reproduce behaviors observed in self-mode-locking experiments.

3.1 Laser bandwidth and mode locking

The simulations are performed for laser operating at $T = 300K$ with a $1.4\; mm \times 6\; \; \mu m \times 0.5\; \; \mu m$ GaAs cavity and an active region consisting of 5 $I{n_{0.15}}G{a_{0.85}}As$ QWs, each $7\; nm$ thick and embedding a density of $2 \times {10^{10}}\; c{m^{ - 2}}$ InAs QDs [6]. For this active medium, we use background refractive index ${n_b} = 3.66$, dipole matrix element $\wp = e \times 0.6nm$ and central transition energy ${\omega _0} = 0.943eV$. The simulations use dephasing and inter-QD population relaxation rates $\gamma = 2 \times {10^{12}}{s^{ - 1}}\; \gamma $ and ${\gamma _{ab}} = {10^{11}}{s^{ - 1}}$, nonradiative decay rate ${\gamma _{nr}} = {10^9}{s^{ - 1}}$ and carrier injection efficiency $\eta = 0.35$ . Assuming that the dispersion to be compensated by mode locking is only from GaAs, we write the passive cavity frequency in Eq. (13) as ${{\Omega }_m} = mc/({2L{n_{GaAs}}} )$ and use the Sellmeier-type function with coefficients extracted from GaAs experiments [21] for the refractive index ${n_{GaAs}}(\lambda )$, where $\lambda = 2L/m$, c is the speed of light in vacuum, m is the mode number ranging from $1900$ to $2100$ for lasing around $1.315\; um$ in a cavity of length $1.33\; mm$.

The grey curve in Fig. 1(a) is the QD inhomogeneous distribution given in Eq. (15) with ${{\Delta }_{inh}} = 10\; meV$. We chose a small inhomogeneous broadening of ${{\Delta }_{inh}} = 10\; meV$ and consider only the intrinsic dispersion from GaAs, so as to illustrate the capability of the model to describe the entire range of mode-locking behavior: unlocked, partially locked and fully locked. More typical values for ${{\Delta }_{inh}}$ range from $15\; meV$ to $25\; meV$ [22,23], and other contributions to cavity mode dispersion are expected. Both, number of lasing modes and power in each mode increase with increasing injection current. The number of modes considered in each stimulation depended on the number of lasing modes. We made sure that there are at least 10 non-lasing modes on both low and high frequency sides of the lasing modes. The 3 dotted curves (with each dot representing a cavity mode) are for 1.2, 6.5 and 8 times threshold current. The steady-state solutions give the electric field amplitude in each mode ${E_n}$, from which we obtain the modal power (right vertical axis, Fig. 1(a)),

(20)$${P_n} = \frac{1}{4}{\varepsilon _0}cn_B^2({1 - {R_0}} )wdE_n^2\; $$

where ${R_0} = 0.6$ is the output mirror reflectivity, $w \times d$ is the laser beam cross section area. (The back-mirror reflectivity is $0.9$) There are 3 lasing modes at 1.2 times threshold, which increases to 11 lasing modes at 6.5 times threshold and 13 lasing modes at 8 times threshold. The time averaged laser power is given by summing over all modes: $P = \mathop \sum \nolimits_n {P_n}$. Figure 1(b) shows its dependence on excitation. In Fig. 1(a), note that the peak of the grey curve is the peak of the quantum-dot density. The mode with the maximum power is located at the gain peak. The shift of gain peak from quantum-dot density peak comes from the lower energy quantum dots having higher population inversion, when all quantum dots have a common chemical potential.

Fig. 1. (a) Density distribution of inhomogeneously-broadened QDs (grey curve and left vertical axis) and intensity spectra of lasing modes for 1.2, 6.5 and 8 times threshold current (dots and right vertical axis). (b) Average output power versus injection current. (c) Number of lasing modes (left vertical axis) and laser bandwidth (right vertical axis) versus injection current. In both (b) and (c) the excitation relative to threshold are given at the top horizontal axis.

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A requirement for generating short laser pulses or a broad frequency comb is a large number of lasing modes covering a wide frequency range. Figure 1(c) gives the dependence of the number of lasing modes on excitation. On the right axis is the corresponding emission bandwidth (full width at half maximum), the reciprocal of which gives an estimation of the shortest possible duration of each mode-locked pulse.

Whether a particular mode operates above the lasing threshold depends on its saturated gain exceeding the cavity loss. As given by Eq. (12), the saturated gain depends on the modal linear gain together with the self and cross saturation terms in Table 1. The importance of carrying out the derivation to 3^rd order is that the cross-saturation contribution contains both population hole burning and population pulsation. Then, strong signal effects are approximated by resuming the series as indicated in Eq. (12). Rate equation approaches neglect the population pulsation, and consequently overestimate mode competition [15].

A second requirement for producing mode lock pulses is the locking of beat frequencies. One indication of mode-locking is from the time traces of output power computed by evaluating

(21)$$P(t )= \frac{1}{4}{\varepsilon _0}cn_B^2({1 - {R_0}} )wd{\left|{\mathop \sum \limits_n {E_n(t)}{e^{ - i{\phi_n}(t )}}} \right|^2}\; ,\; $$

with the modal field amplitudes and phases taken at times sufficiently long so that steady-state is reached in the modal field amplitude and frequency, ${E_n}$ and ${\dot{\phi }_n}$. Figure 2(a) shows laser output at injection currents chosen to illustrate the three regimes of mode-locking operation. The top plot shows regularly occurring pulses from multimode operation even at the low excitation of $1.2\; {I_{th}}$. At lower excitations, simulations show simply multimode mode operation, with intensity variations but no regular pulses and with periodicity depending on number of modes. Semiclassical laser theory, which neglects laser field quantization effects such as spontaneous emission, is expected to be accurate at $1.2$ times threshold [15]. We have chosen input parameters that accentuate the self-mode locking effect so that there is already partial mode locking as depicted by the RF spectrum. This leads to onset of pulse train formation. However, pulse duration exceeds $10\; ps$ because of the limited laser bandwidth, and a low peak-to-floor contrast. At $6.5\; {I_{th}}$, the broad (308 GHz) emission bandwidth with almost complete mode locking combine to give 2.8 ps pulses with a high 10³ peak-to-floor contrast (Fig. 2(b)). Figure 2(c) illustrates the effects of overexcitation, when the dispersion with increasing number of lasing modes becomes too large for the relative phase angle terms to overcome. The degradation in mode locking at $8\; {I_{th}}$ appears in the form of satellite pulses immediately following the main one (Fig. 2(c)). A time domain approach gives a more physical explanation by associating the satellite pulses to the insufficient recovery of saturable absorption.

Fig. 2. Laser pulse trains (left column) and RF spectra (right column) for excitations 1.2, 6.5, and 8 times above threshold current.

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Another gauge of mode-locking effectiveness is the beat-note spectrum,

(22)$$\begin{aligned}S({{{\Omega }_{RF}}} )& =\mathop \smallint \nolimits_0^\infty dtP(t )\textrm{cos}({{{\Omega }_{RF}}t} ){e^{ - {\gamma _{RF}}t}}\\ & =\frac{{{\varepsilon _0}cn_B^2wd}}{{4{\gamma _{RF}}}}({1 - {R_0}} )\mathop \sum \limits_n {E_n}{E_{n - 1}}{L_{{\gamma _{RF}}}}({{{\Omega }_{RF}} - {{\Omega }_n} + {{\Omega }_{n - 1}}} )\; ,\; \end{aligned}$$

where ${L_a}(x )= {[{1 + {{({x/a} )}^2}} ]^{ - 1}}$ is the Lorzentian lineshape function and ${\gamma _{RF}}$ is the broadening due to, e.g., spectrometer resolution. $S({{{\Omega }_{RF}}} )$ is also referred to as the radio frequency (RF) spectrum, since the beat frequencies are in the tens of GHz range. For excitations below 1.1 ${I_{th}}$, simulations show absence of mode locking, with a RF spectrum that consists a series of peaks. Higher excitation gives rise to partial mode locking. Figure 2(d) shows the RF spectrum for $1.2\; {I_{th}}$, with two closely spaced resonances indicating some condensing or partial locking of two of the four beat frequencies. At $6.5\; {I_{th}}$, there is essentially perfect mode locking, with the next strongest beat frequency two orders lower in magnitude (see Fig. 2(e)). Figure 2(f) shows sizable sidebands in the RF spectrum with further excitation to $8\; {I_{th}}$.

Figure 3 gives a more detail picture of behaviors occurring with increasing injection current. From Fig. 3(a), one initially sees a decreasing pulse width because of increasing emission bandwidth. Then, above 7 times threshold, there is a slight increase in pulse width, coinciding with the appearance of satellite pulses in the cavity. Figure 3(b) plots the time averaged RF spectral width (full width at half maximum) $\langle{\sigma _{RF}}\rangle$. For most excitations, the RF spectra exhibit fluctuations in spectral position and width. At low injection current, the variation in the average linewidth with excitation is from mode hopping. The locked region that follows shows relatively stable behavior, with most of the beat frequencies locked to a common value. Above $7\; {I_{th}}$, the average RF linewidth increases when the beat frequencies begin to unlock as depicted in Fig. 2(f).

Fig. 3. (a) Pulse width and (b) time average RF linewidth versus relative excitation.

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The effect of only partial mode locking also appear in the peak power of the pulses. In Fig. 4, the solid curve is a plot of the peak power in the output pulse train versus excitation. The abrupt drop in peak power corresponds to the loss of locking and the appearance of satellite peaks. For comparison, the dashed curve shows the transform limit:

(23)$$P_{pk}^{xform} = \frac{1}{4}{\varepsilon _0}cn_B^2({1 - {R_0}} ){A_b}{\left|{\mathop \sum \limits_n {E_n}} \right|^2}$$

Fig. 4. Peak power of pulses versus excitation relative to threshold. The dashed curve shows the transform limit, with complete locking of all beat notes.

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3.2 Influence of linewidth enhancement factor

Earlier studies have indicated sensitivity of laser dynamics to the linewidth enhancement factor $\alpha $ [24,25,26]. The contributions to $\alpha $, i.e. the real and imaginary parts of the active medium susceptibility, come out of the perturbation analysis in the form of the frequency pulling and linear gain, ${\sigma _n}$ and ${g_n}$, respectively. Figure 5(a) shows the two quantities for operation at 6.5 times threshold.

Fig. 5. (a) Linear gain and frequency pulling versus frequency for I/I_th = 6.5. (b) Linewidth enhancement factor versus frequency according to g_n and σ_n in 6(a). The dashed lines indicate α at gain peak

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In terms of gain and frequency pulling, the linewidth enhancement factor is

(24)$$\alpha ({\Omega } )= \frac{{d\sigma ({\Omega } )}}{{dN}}{\left( {\frac{{dg({\Omega } )}}{{dN}}} \right)^{ - 1}}\; ,\; $$

which is plotted in Fig. 5(b) for the frequency range where gain is present. Table 1 shows ${\sigma _n}$ and ${g_n}$ to be convolutions of the Gaussian QD distribution and the complex dispersion function D. This gives $\alpha $ its dependence on the inhomogeneous broadening. At 6.5 times threshold, a linewidth enhancement factor of 0.17 is extracted at gain peak, which is located away from the center of the QD distribution. The curve predicts $\alpha = 0$ when operating at the center of the inhomogeneous QD distribution. However, the gain peak does not coincide with the center of the QD distribution until excitation is so high as to be close to gain rollover. Lastly, it is important to note that the present results neglect many-body effects. There will be quantitative modifications as reported in the literature [27]

The linewidth enhancement factor is generally regarded as detrimental to mode locking [28]. We find that to not necessarily be the case. In a GaAs based laser, the intrinsic dispersion from bulk GaAs, where $d({\delta n} )/d\lambda $ is negative, may be mitigated by the carrier-induced dispersion. Figure 6 illustrates this by showing the narrowing of the RF spectrum by frequency pulling (compare Figs. 6(a) and 6(b)). The resulting spectral width is still broad. The drastic narrowing comes from turning on the relative phase angle terms, as shown in Fig. 6(c). To describe the transition from Figs. 6(a) to 6(b), the carrier densities and gain medium properties have to be spectrally resolved, which is not the case in typical time-domain models.

Fig. 6. RF spectra at 6.5 times threshold current: (a) with only intrinsic dispersion from bulk GaAs, (b) with intrinsic GaAs dispersion and frequency pulling, (c) with relative phase angle contribution added. The full width at half maximum for each case is indicated.

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3.3 Partial mode locking and frequency comb jitter

In this paper, we have chosen to strictly follow semiclassical laser theory and have not used a phenomenological noise term to mimic spontaneous emission noise. Nevertheless, simulations show fluctuations in the laser field, except for the fully mode-locked region, $6.5\;<\;I/{I_{th}}\;<\;7.5$, where the laser produces a frequency comb that is stable in absolute spectral width and position. Otherwise, outside of the locked band, jitter in both quantities exist. Figure 7(a) is a plot of the variation in RF spectrum (FWHM) relative to the average RF width (see Fig. 3(b)), where

(25)$${\sigma _{RF}} = \sqrt {\langle{{({{{\Delta }_{RF}} - \langle{{\Delta }_{RF}}\rangle} )}^2}\rangle}$$

is the standard deviation in the FWHM obtained from integrating over time. The plot shows a relatively constant 10% fluctuation compared to the average RF FWHM $ \langle{{\Delta }_{RF}}\rangle$ . One should note that $\langle{{\Delta }_{RF}}\rangle$ increases by about 2 orders of magnitude once outside the fully mode-locked region. Figure 7(b) indicates a blue shift in the frequency comb with increasing excitation. With partial mode locking, there is jitter in the central RF frequency, as shown in Fig. 7(c). This jitter in the frequency comb varies between 1 to 20 MHz.

Fig. 7. Excitation dependences of a) fluctuation in RF spectral FWHM, (b) central frequency and (c) fluctuation in central frequency.

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In our theory, the frequency comb fluctuations come entirely from Eqs. (10), (11) and (19). They result from partial mode locking, when the relative phase angle terms are unable to fully compensate the GaAs intracavity dispersion. In a time-domain model, the jitter has to come from introducing a noise term in the field equations [8].

3.4 Comparison with experiment

We end this paper with a comparison with experiment. The single-section laser is basically that modeled in the earlier sections, except we use a QD inhomogeneous broadening of ${{\Delta }_{inh}} = 15\; meV$, which is extracted from characterization of similar laser samples [22]. Also, we continue to assume that dispersion is entirely from the GaAs waveguide.

In Fig. 8, the red and blue curves are from experiment [6]. and the black curve is from the model. An excitation of 10 times above threshold is picked to give the narrowest RF linewidth of $143\; kHz$, compared to the measured value of $100\; kHz$ at 14 times threshold (Fig. 8(a)). Figure 8(b) shows good agreement with the measured pulse shape. The experimental pulse is measured by second harmonic generation autocorrelation. Fit to a sech pulse shape gives approximately $500\; fs$ pulse duration.

Fig. 8. (a) RF spectrum and (b) mode-locked pulse from theory (black curve) and experiment (blue and red curves). The frequency and time are referenced to the peak values.

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Simulations up to 15 times threshold show only partial mode locking, with peak pulse intensities around 20% that of the transform limit. The relative phase angle terms leading to mode locking cannot totally overcome the combined dispersion of from the GaAs waveguide and the carrier-induced refractive index from the quantum dots (blue curve, Fig. 9). The result is that only some of the beat notes from the lasing modes are locked to a common value (flat portion of black curve, Fig. 9).

Fig. 9. The dotted curve is the mode intensities, the blue curve plots the combined dispersion from the GaAs waveguide and InAs QDs, the black curve shows the effectiveness of the self-mode locking.

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4. Conclusion

This paper treats self-mode locking and frequency comb generation in a single-section semiconductor quantum-dot laser. The approach describes the progression from unlocked to mode-locked operation as observed in experiments and time-domain modeling. In addition, the simulations indicate an extreme sensitivity of mode-locked behavior to injection current. With excessive excitation, mode locking degrades when secondary pulses appear within the laser cavity. Concurrently, the unlocking of beat notes in the frequency comb is evident from the appearance of multiple resonances in the radio frequency (RF) spectrum. Analysis of the modeling results finds a critical balance between lasing bandwidth and mode locking that strongly depends on QD inhomogeneous broadening. We extract an inhomogeneous broadening (${\Delta}_{inh}$ in Eq. (15)) in the 10 meV to 15 meV range as optimum for quantum-dot optical nonlinearities to compensate for dispersion by GaAs in the optical cavity.

A frequency-domain approach is used in the investigation because it unambiguously identifies contributions responsible for gain saturation, mode competition, linewidth enhancement and self-mode locking. Equally important is the direct connection of mode locking performance to the active medium electronic structure. These advantages come from treating the active medium microscopically, with all effects from active medium optical nonlinearities coming from the quantum mechanical electron-hole polarization created by light-carrier interaction. Lastly, the present formulation provides the basis for a more detailed model. Work is in progress to include the quantum-well wetting layer and bulk cladding regions, in order to account for carrier transport and heating effects. Equally important is development of a capability to treat non-equilibrium electron or hole distributions, as well as many-body carrier effects.

Funding

Advanced Research Projects Agency - Energy (DE-AR000067); U.S. Department of Energy (DE-NA0003525); American Institute for Manufacturing Integrated Photonics.

Acknowledgments

We thank Frederic Grillot for discussions. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science.

Disclosures

The authors declare no conflicts of interest.

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Coefficients	Physical context
$g_{n} + i σ_{n} = F_{1} \sum_{q} n_{q}^{i n v} D_{γ} (Ω_{n} - ω_{q})$	Linear gain and frequency pulling
$n_{q}^{i n v} = f (ε_{q}^{e}, μ_{e}) + f (ε_{q}^{h}, μ_{h}) - 1$	Inversion
$D_{γ} (Δ) = 1 / (1 + i Δ / γ)$	Complex Lorentzian
$F_{1} = Γ Ω_{0} ℘^{2} / (2 ε_{B} ℏ γ d)$
$\begin{aligned} θ_{n n^{'} m m^{'}} = & F_{1} \frac{γ_{a b}}{γ} D (Ω_{m} - Ω_{m^{'}}) \sum_{q} n_{q}^{i n v} η_{n^{'} m m^{'}} \\ \times D (ω_{q} - Ω_{n^{'}} + Ω_{m} - Ω_{m^{'}}) \\ \times [D (Ω_{m} - ω_{q}) + D (ω_{q} - Ω_{m^{'}})] \end{aligned}$	General 3^rd order coefficient
$\begin{aligned} η_{n^{'} m m^{'}} = & \frac{1}{8} {1 + \frac{2}{L} \int_{0}^{L} d z \\ \times [cos (\frac{2 π (m - m^{'}) z}{L}) + cos (\frac{2 π (m - m^{'}) z}{L})]} \end{aligned}$	Spatial hole burning
$g_{n} A_{n} + i τ_{n} = θ_{n n n n}$	Self saturation and frequency pushing
$g_{n} B_{n, m} + i τ_{n, m} = θ_{n n m m} + θ_{n m m n}$	Cross saturation and frequency pushing
$ψ_{n n^{'} m m^{'}} = (ν_{n} - ν_{n^{'}} + ν_{m} - ν_{m^{'}}) t + ϕ_{n} - ϕ_{n^{'}} + ϕ_{m} - ϕ_{m^{'}}$	Relative phase angle

Multimode description of self-mode locking in a single-section quantum-dot laser

Abstract

1. Introduction

2. Theory

3. Results and discussion

3.1 Laser bandwidth and mode locking

3.2 Influence of linewidth enhancement factor

3.3 Partial mode locking and frequency comb jitter

3.4 Comparison with experiment

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (9)

Tables (1)

Equations (25)

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