Modeling and analysis of distributed feedback quantum dot passively mode-locked lasers

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Modeling and analysis of distributed feedback quantum dot passively mode-locked lasers

JAVAD RAHIMI,¹ VAHID AHMADI,^1,* AND MOHAMMAD HASAN YAVARI²

1Faculty of Electrical and Computer Engineering, Simulation of Optoelectronic Device and Systems Lab, Tarbiat Modares University, Tehran, Iran

2School of Electrical Engineering, Shahed University, Tehran, Iran

*Corresponding author: [email protected]

Received 15 March 2016; revised 18 May 2016; accepted 19 May 2016; posted 20 May 2016 (Doc. ID 261060); published 23 June 2016

In this paper, we investigate numerically two proposed monolithic distributed feedback quantum dot passively mode-locked lasers (DFB-QDMLLs) with and without gratings in the saturable absorber (SA) section in order to enhance two important performances of QDMLLs for ultrahigh-bit-rate and single-mode applications.

We find out that depending on the length of the grating, optical pulses with durations of about 3–8 ps at approximately 2nd and 4th harmonics of cavity round-trip frequencies can be generated by the proposed structures.

We also compare the temporal and spectral behaviors of these structures under specified bias conditions and SA lengths. It is shown that DFB-QDMLLs have the ability to generate optical pulses with more peak power than grating-embedded saturable absorber (GESA-DFB-QDMLL) structures which generate shorter pulses with narrower spectral bandwidths. We also show that DFB-QDMLLs operate in a larger range of absorber voltages while the other structure is very sensitive to absorber voltage and operates well for middle ranges of this parameter. © 2016 Optical Society of America

OCIS codes:(140.3490) Lasers, distributed-feedback; (140.4050) Mode-locked lasers; (250.0250) Optoelectronics.

http://dx.doi.org/10.1364/AO.55.005102

1. INTRODUCTION

Optical short pulses with multigigahertz repetition rates generated by monolithic semiconductor mode-locked lasers (MLLs) attract considerable attention in optoelectronic applications, such as ultrahigh-bit-rate optical communications, ultrafast data processing, electro-optic sampling, soliton transition systems, and optical A–D conversion for examining biological or chemical processes [1–8]. Semiconductor lasers based on self- assembled quantum dots (QDs) are particularly interesting for the mode-locked (ML) regime, including the features of low threshold current densities, insensitivity to temperature, broad optical gain spectra due to the inhomogeneous broadening, ultrafast carrier dynamics, and low linewidth enhancement factors [8–10]. So far, there have been many advances in improving the time-domain performance of short optical pulses generated by semiconductor lasers such as peak power, pulse width [11,12], temporal stability, and time jitter reduction [13–15]. On the other hand, for single-mode applications, such as wavelength-division-multiplexed optical communications, a narrow spectrum is needed [7]. In order to achieve this goal, manipulation of multisection mode-locked distributed Bragg reflector (DBR) semiconductor lasers and mode-locked distributed feedback (DFB) lasers has been studied both in experiments and simulations. According to the literature, by

using these structures, optical short pulses not only with narrower spectra but also with higher repetition rates can be generated [13,16–20].

Numerical analysis of MLL is discussed in many papers [13,21,22]. Among them there are two models which are widely used by researchers. Models based on finite difference traveling waves (FDTWs) and delay differential equations (DDEs) demonstrate powerful methods to study and analyze ML regimes [23–26]. With these two methods, one can consider spatial distributions of carriers and electric field amplitudes in order to study MLL. The DDE model assumes unidirectional operation like in a ring laser [21,27], so it re- duces the computational cost in comparison to the FDTW model which is based on the direct solution via the finite difference method for solving counterpropagating optical fields circulating in the laser cavity. However, the FDTW approach is more appropriate because of the high accuracy and the pos- sibility of considering real effects like spontaneous emission noise [28,29].

In this paper, we unveil how to exploit the monolithic DFB configuration in quantum dot mode-locked lasers (QDMLLs).

Our aim is to design MLLs capable of generating optical short pulses with higher repetition rates and improved optical spectra in terms of having narrow spectral widths and larger free

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spectral ranges (FSRs). In this regard, we use the FDTW model to study a monolithic two-section DFB-QD-MLL with and without gratings in saturable absorber (SA). In order to do this, we consider two sets of rate equations for gain and absorber sections with appropriate terms of the electric field circulating in the laser cavity. The structure of the paper is as follows. We present the proposed structures in Section2, and then the theoretical model is discussed in Section3. In Section4, we discuss the results of the simulations, and our conclusion is given in Section 5.

2. STRUCTURE

The proposed structures of the two-section DFB-QD-MLLs are illustrated in Fig.1, which consist of a reverse-biased SA and a forward-biased gain section. We consider a two-section DFB-QD-MLL in which the absorber section has no grating [Fig. 1(a)] and the other similar structure with a grating- embedded saturable absorber (GESA) section [Fig.1(b)]. The active region of both structures is composed of fifteen stacks of self-assembled QD InAs layers, and the details of the epitaxial layers can be found in [28].

3. THEORETICAL MODEL

In this section, the one-dimensional first-order wave equations governing the optical field evolution inside the laser cavity as well as the carrier rate equations are presented.

A. Field Equations

With some proper approximations, the propagation of the electromagnetic field inside the laser cavity can be described by a slowly varying envelop (SWE) of a counterpropagating

electric field with complex amplitudes,Ez; tandE⁻z; t, which satisfy the traveling wave (TW) equations [6,29]

1 vg

∂

∂t ∂

∂z

E−iβE−iκ_E⁻_EE⁻F_sp; (1a)

1 vg

∂

∂t−

∂

∂z

E⁻ −iβE⁻−iκ_EE⁻EF⁻sp; (1b)

wheret indicates time, andZε0; Lis the longitudinal direction along the laser cavity. To solve these equations, we use reflection boundary conditions atZ 0and Z L for two laser facets as follow [28]:

E0; t r₀E⁻0; t; E⁻L; t r_LEL; t; (2) whereν_gc0∕ngris the group velocity (c0is the speed of light in vacuum, and ngr is the group refractive index), r0 and rL

describe reflection coefficient of the laser facets, κ_E−E and κ_EE⁻ (κ_E−E κ_EE⁻κ_i,i∈a; b) are the coupling coef- ficients between forward and backward waves due to the grating, and Fsp and F⁻sp denote spontaneous emission noise [29,30]. By considering only ground-state (GS) lasing, the propagation factor entering Eq. (1) can be described as [28]

ββnGSz; t; z δ−iα

2i−α_H

2 g⁰2nGS−1; (3) where α, α_H, g⁰, and δ denote the internal absorption, the linewidth enhancement factor, the effective differential gain (including the transverse confinement factor), and the static detuning factor, respectively. Moreover,nGS is the ground-state carrier probability that will be described later. In this study, we neglect the inhomogeneous broadening due to the nonuni- form distribution of quantum dots and we use a single- Lorentzian gain spectrum profile in simulations [28,31–33].

B. Carrier Rate Equations

In this section, we describe two sets of rate equations in order to properly take into account the carrier dynamics between the carrier reservoir (CR), exited state (ES), and ground state (GS) of the quantum dots in a reverse-biased SA and a forward- biased gain section. In this regard, the carrier dynamics in the gain section [as in Fig. 2(a)] are defined by following coupled rate equations [28,34]:

d

d tnCRz;t Iz eθI −

n_CR τ_CR−

n_CR1−n_ES

τ_CR→ES 4 n_ES

τ_ES→CR; (4a)

Fig. 1. Proposed structures of DFB-QD-MLLs. (a) DFB-QD- MLL without a grating in the absorber section. (b) DFB-QD-MLL with a grating-embedded saturable absorber section (GESA-DFB- QD-MLL).

Fig. 2. Energy band structure of QD. (a) Gain section.

(b) Saturable absorber section.

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d

d tnESz; t − n_ES τ_ES−

n_ES1−n_GS

τ_ES→GS n_GS1−n_ES 2τ_GS→ES nCR1−nES

4τ_CR→ES − nES

τ_ES→CR; (4b)

d

d tnGSz; t − nGS

τ_GS2nES1−nGS τ_ES→GS −

nGS1−nES τ_GS→ES

−RstnGS; E; (4c) whereR_stis a term of stimulated recombination given by [28]

RstnGS; E 1 θ_E

X

v

E^v g⁰2nGS−1E^v (5)

for the reverse-biased absorber section [see Fig.2(b)]. We use a simplified carrier rate equation model according to [28,35]

d

d tnGSz; t − nGS

τ_GS2nES1−nGS τ_ES→GS −

nGS1−nES τ_GS→ES

−R_stn_GS; E; (6a)

d

d tnESz; t − nES

τ_ES→CR−

nES1−nGS

τ_ES→GS nGS1−nES 2τ_GS→ES ;

(6b) where e is the electron charge,θ_I and θ_E are scaling factors, and Iz is the injection current which is equal to zero at the absorber section. In rate Eqs. (45)–(6), τ⁻¹_n and τ⁻¹_n→m, n; m∈fGS;ES;CRg, represent spontaneous relaxation and transition rates between GS, ES, and CR, respectively, and jEj² is equal tojEj² jE⁻j²which is the optical normalized intensity that denotes optical output power. Other parameters used in the simulation are given in Table1[28].

4. SIMULATIONS AND RESULTS

A rigorous way to analyze the monolithic photonic integrated circuits (PICs) is by using finite difference methods in order to solve the counterpropagating traveling waves with coupled rate equations. For this, first we consider an appropriate discretiza- tion of the time axis with a step unit ofΔt and then take the step lengthΔz in the longitudinal direction (z-axis) to be related byΔt·vg Δz. The field and rate equations of quantum dots will then be solved in each time step for all of the longitudinal subsections using FDTW and Runge–Kutta methods, respectively [36]. In the following, we will discuss the results of our simulations with the mentioned approach for the structures given in Fig.1.

A. Temporal Behaviors of the Proposed Structures Figure3 shows the output power of the DFB-QDMLL as a function of time. The gain injection current is 120 mA, and the SA with a length of 100μm is biased at various voltages.

The coupling coefficient between forward and backward waves in the first structure (κ_a) is equal to15 cm⁻¹. For very small and large values of theκ_aLfactor in this structure, we have no pulse in the output and the laser usually operates in the continuous wave (CW) regime. Within the coupling coefficient ranges of 15 cm⁻¹<κ_a<20 cm⁻¹ our simulations show a stable ML regime. As can be seen in this figure, the output of the laser is harmonically mode locked and we have stable pulses at a repetition rate of about 70 GHz with a duration of 6–8 ps, depending on the absorber voltage. According to this figure, the laser reaches the stable ML regime faster as the absorber voltage increases. However, the turn-on delay time of the laser is almost invariant with the variation of the absorber reverse voltage.

In order to see the behavior of the device in terms of negative voltage applied to the absorber, Fig. 4 shows the steady- state pulses for different absorber voltages. According to this figure, it is seen that the output of the laser is a CW when the absorber is grounded. By increasing the reverse-bias voltage fromU 0 VtoU −7 V, which leads to an increase in the Table 1. Parameters Used in the Simulation [28]

Symbol Quantity Value

λ₀ Central wavelength 1.3μm

ngr Group refractive index 3.75

L Total length 1 mm

δ Static detuning 0 cm⁻¹

α Internal absorption 5 cm⁻¹

α_H Henry factor 2

g⁰ Effective differential gain/absorption 40∕200 cm⁻¹

τ_GS GS relaxation rate 1 ns

τ_ES ES relaxation rate 1 ns

τ_CR CR relaxation rate 1 ns

τ_ES_→_GS ES to GS transition time 2 ps τ_GS_→_ES GS to ES transition time 5 ps τ⁻¹_CR

→ES CR to ES transition rate (SA) 0 τ_CR_→_ES CR to ES transition rate (G) 5 ps τ_ES→CR ES to CR transition rate (SA) 18e^U^∕2V ps τ_ES→CR ES to CR transition rate (G) 80 ps θ_I Injection scaling factor 9.36·10⁶

θ_E Field scaling factor 239 ps

Γ Transversal confinement factor 0.075 r₀ SA section facet reflectivity ffiffiffiffiffiffiffiffiffi

p0.95 r₀ GESA section facet reflectivity 0 r_L Gain section facet reflectivity 0

Fig. 3. Output power of the DFB-QD-MLL for ML pulsation parameters I120 mA and (a) U 0 V, (b) U−2.5 V, (c)U−5 V, and (d)U −7 V.

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saturation absorption in SA, the full width of the pulse at its half-maximum (FWHM) decreases and the shape of the optical pulse becomes more asymmetric due to the manifestation of an additional extreme.

As can be observed in Fig. 5, the output powers of the GESA-DFB-QDMLL [Fig. 1(b)] for the injection current of 120 mA and different absorber voltages are plotted. The value of the coupling coefficient for this structure (κ_b) is20 cm⁻¹, where in comparison to the first structure the range of the coupling coefficient having stable ML pulses is even less (19 cm⁻¹<κ_b<21 cm⁻¹). This structure is very sensitive to laser parameters such as absorber voltage, gain current, and the coupling coefficient. In the absorber length of 100 μm the device shows a stable ML regime. In contrast to the DFB- QDMLL, this structure operates in smaller range of reverse voltage, as shown in Fig. 5. As we can see, there is no pulse in the output of the laser for large and small values of reverse voltages [see Figs.5(a),5(c), and 5(d)]. However, for middle ranges of the absorber voltages, we have stable harmonic mode locking.

In Fig.6, we show stable harmonic ML pulses at a repetition rate of about 88 GHz with an estimated pulse width of 5 ps for

I 120 mA and U −3 V. By analyzing the results of Figs. 5 and 6, we see that the effect of the variation of the absorber reverse voltage on the output of the laser is totally different from that in the DFB-QDMLL. One can explain this behavior of the GESA-DFB-QDMLL by the spectrum-filtering property of the grating and also the AR facets used in this structure.

So far, we have only observed approximately the second harmonic of ML pulses with two different structures presented in Fig.1, and we have not observed the fundamental ML regime because of the length of the grating used in these structures.

As discussed before, the DFB-QDMLL is well behaved against the variation of the parameterκ_a and it operates in the ML regime for middle values of κ_a, that κ_aL factor is sufficient.

However, the GESA-DFB-QDMLL behaves in a way that is only related to the spectral behavior of the grating structure and its selectivity of longitudinal modes. This structure shows harmonic mode locking other than in the middle range ofκ_bL.

To elaborate more on issues discussed so far, we plot the output optical power of the GESA-DFB-QDMLL for the various absorber section lengths in Fig. 7. This figure depicts the dependency relationship between the absorber length and device operation regime atκ_bL6, where the total fixed length of the laser is 1 mm. When the absorber length is too short, mode locking does not occur due to insufficient saturable absorption. By increasing the SA length, optical pulses with strong backgrounds are produced. With a further increase in Fig. 4. Output power of the DFB-QDMLL for ML pulsation

parameters I120 mA and different values of saturable absorber voltage in the steady state.

Power (mW)Power (mW) Power (mW)Power (mW)

Fig. 5. Output power of the DFB-QDMLL for ML pulsation parameters I120 mA and (a) U0 V, (b) U−2.5 V, (c)U−5 V, and (d)U −7 V.

Power (mV) Power (mV)

Fig. 6. Output power of the GESA-DFB-QDMLL for ML pulsation parametersI120 mAandU−3 V.

Fig. 7. Peak power of the GESA-DFB-QDMLL for different absorber lengths in κ60 cm⁻¹. The ML pulsation parameters areI120 mAandU −5 V.

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absorber length (120≤lSA ≤180), stable harmonic mode locking is observed at the output of the device. WhenlSA becomes longer than 180μm, the output pulses again experience some background due to the increase of the absorber loss.

Finally, for very larger SA lengths the lasing is not achieved due to the strong loss of the SA section and insufficient gain.

Figure 8 shows the output power of the GESA-DFB- QDMLL atI 120 mA,U −5V, andlSA 160μmfor κ_b60 cm⁻¹which is harmonically mode locked at a repetition rate of about 170 GHz, corresponding to the almost 4th harmonic of the cavity round-trip frequency. The pulse width is estimated at about 3 ps, and the peak power is around 4 mW.

In Fig.9, the output optical power of the DFB-QDMLL is plotted against the gain injection current forU −3 V, and κ_a15 cm⁻¹. This figure also depicts the typical sequence of operation regimes with increasing injection current. As shown in this figure, the laser is off for low values of gain currents (I ≤I_th≈16 mA), which is shown by zero intensity in the figure. By increasing the injection current (Ith< I ≤68 mA), the laser operates in a Q-switching regime with relatively high output power. After this region, we have an incomplete ML with almost low power for a large range of injection currents (68 mA< I≤86 mA). At higher injection currents (86 mA< I ≤128 mA), a stable harmonic ML is observed at a repetition rate of 70 GHz. It is seen that with increasing current in this region, the pulse power increases in contrast to the results represented in [28]. The occurrence of the observed behavior can be explained by the selectivity property of the DFB structure which can reduce the effect of the other groups of locked modes. This means that with increasing gain current, the power is not consumed by the other groups of locked phase because they are suppressed by the spectral filtering of the DFB structure. Last, with more increase in the gain current, the output will be a CW, as shown in the last part of Fig.9.

The typical sequence of the operation regime with increasing injection current for a GESA structure is also depicted in Fig.10. This laser behaves approximately similar to the former one, except for few differences. First, we see the reduction of threshold current down to about 10 mA that can be explained by increasing theκ_bLfactor in the second proposed structure.

Another difference between the two figures is observed in the

second region where we have no Q-switching regime in the GESA structure and the output is a CW with low power.

The behavior of the output power in other ranges of the injection current is almost the same.

B. Spectral Behaviors of the Proposed Structures The data presented and discussed thus far reveal the temporal behavior of the mentioned devices. Figure11shows the optical spectrum of fundamental mode-locked pulses of a conventional QD-MLL with I 120 mA and U −3 V presented in [28]. It can be seen that the longitudinal modes separated by 40 GHz correspond to the laser repetition rate given in [28]. Figure12shows the optical spectrum of the 8 ps pulses shown in Fig.3(b). As we can see, the free spectral range of the laser is about 70 GHz, which is the repetition rate of the DFB-QDMLL shown in Fig. 1(a) under the bias condition mentioned in Fig. 3(b).

In Figs. 13 and 14, the corresponding optical spectra of Figs.6and8are plotted, respectively. These two spectra show the spectral behavior of the GESA-DFB-QDMLL shown in Fig.1(b) under the specified bias condition and laser parameters. As can observed in Fig.13, the free spectral range of the laser modes is about 88 GHz which is almost the 2nd harmonic Fig. 8. Output power of the GESA-DFB-QDMLL for ML pulsa-

tion parameters I120 mA and different values of saturable absorber voltage withκ_b60 cm⁻¹.

0 20 40 60 80 100 120 140 160

0 5 10 15

Injection Current (mA) Incomplete

ML ML CW

off Q switching

Power (mW)

Fig. 9. Typical sequence of the operation regime versus injection current in the DFB-QDMLL forU−3 V.

Fig. 10. Typical sequence of the operation regime versus injection current in the GESA-DFB-QDMLL forU −3 V.

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of the cavity round-trip frequency. Because of the length of the grating and also using AR facets in the GESA-DFB-QDMLL, the free spectral range of the longitudinal modes is increased in comparison to the first structure. It can also be seen from Fig.14 that the 4th harmonic of the ML pulses is generated and the spectrum has good purity so that only longitudinal modes separated by 170 GHz are enhanced and the other modes are strongly suppressed.

As can be seen in Figs.13 and 14, both spectra have the central frequency in their spectral response. However, in conventional uniform DFBs with AR facets, the central frequency is stopped due to the spatial-hole burning and thus two side modes will oscillate instead. In our case, the scenario is totally different; however, both facets are AR and the structure has uniform grating. One can explain the complex spectral behavior of the GESA-QD-MLL by Fig.15and Eq. (3). In Fig.15, the ground-state carrier probability is plotted both in gain [nGSG] and absorber [nGSSA] sections [37]. It is obvious that the value of GS occupation probabilitynGSin the absorber section is always less than a specific threshold value (nGS0.5) which determines the absorption or amplification performance Fig. 11. Optical spectrum of ML pulses for a conventional QD-

MLL forI120 mAandU−3 Vpresented in [19].

Fig. 12. Optical spectrum of 8 ps pulses shown in Fig.3(b).

Fig. 13. Ground-state carrier probability in the GESA-QD-MLL forI120 mA,U−3 V, andκ20 cm⁻¹.

Fig. 14. Optical spectrum of the 3 ps pulses shown in Fig.8.

Fig. 15. Ground-state carrier probability in the GESA-QD-MLL forI120 mA,U−3 V, andκ_b20 cm⁻¹.

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of the MLL sections. Indeed, this behavior of the carrier density in the absorber is a prerequisite for a reverse-biased absorber to act as a SA. The opposite behavior happens in the gain section, which causes the GS occupation probability to be above the threshold value. This difference between the pattern of the carrier probability in the amplification and absorption sections affects the propagation factor by applying the mentioned mechanism in the last term of Eq. (3), which consequently injects different phases to the traveling waves circulating in both sections and removes grating uniformity.

5. CONCLUSION

We have investigated a DFB-QD-MLL with and without a grating in the absorber section. Our simulations reveal that the SA section in the two discussed structures plays an important role.

In the first structure in this study (without a grating in the SA), there is no mandatory situation where the device operates in a single longitudinal mode. However, for the second structure, again we observe several longitudinal modes because in this structure the SA acts as a passive section for the grating and itself induces additional modes, thus removing grating uniformity. As a result, we cannot assume these structures like a conventional DFB construction that favors single-mode operation. It has also been shown that based on the grating length and laser parameters, we only have approximately 2nd and 4th harmonics of the cavity round-trip frequencies in our model and we have not seen the fundamental ML. Simulations show that the output pulses are very sensitive to device parameters such as gain injection current, SA reverse voltage, and the coupling coefficient. Furthermore, we have presented optical spectra of output pulses and discuss the differences between the proposed structures with conventional mode-locked and DFB lasers. We have studied the performances of the proposed structures in improving the spectral response and increasing the free spectral range of the optical spectrum from 40 (conventional QD-MLL) to 170 GHz.

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