# Gain-Switched Short Pulse Generation from 1.55 µm InAs/InP/(113)B Quantum Dot Laser Modeled Using Multi-Population Rate Equations

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{ihom}), the Q-Dot ensemble is divided into 2X + 1 groups, depending on their resonant energies for the interband transition [14,15]. Figure 1 shows the relaxation mechanisms in the xth Q-Dot subgroup. Energies of Exs and Grs of xth Q-Dot are represented as ${E}_{Exs\_x}$ and ${E}_{Grs\_x}$, respectively. As a result, the longitudinal cavity photon modes of up to 2P + 1 are constructed in the cavity [13]. When the index x is equal to X, this situation corresponds to the central Q-Dot group. When index p is equal to P, this case corresponds to the central mode with the transitional energies of ${E}_{Exs0}$ and ${E}_{Grs0}$ for Exs and Grs, respectively. Each Q-Dot group energy width ($\mathsf{\Delta}\mathrm{E}$) and mode energy separation ($\mathsf{\Delta}{E}_{p}$) are assumed to be equal and taken to be as 1 meV [13]. The xth Q-Dot group energy and pth mode energies are indicated by:

- ${\mu}_{Exs,Grs}$ is the degeneracy of the Exs and Grs,
- ${N}_{o}$ is the Q-Dot density and
- ${N}_{Exs\_x,Grs\_x}$ is the carrier density in the Exs and Grs of xth Q-Dot.
- ${G}_{xExs,xGrs}$ is the density rate of xth Q-Dot in the Exs and Grs.

_{inh}= 2.35 σ. In other words, Γ

_{inh}is described as inhomogeneous broadening. The carrier escape time is related to the carrier capture time [17] and given as:

_{B}is the Boltzmann constant, T is the temperature, and E

_{Wly}is the energy of the Wly.

_{g}is the group velocity. $\overline{{\tau}_{Wly-Exs}}$, and $\overline{{\tau}_{Wly-Grs}}$ indicate the average capture times from the Wly to Exs and from the Wly to Grs in Q-Dot ensemble. They are defined as follows:

_{o}, ħ, n

_{r}, and m

_{o}are the speed of light, dielectric constant in free space, Planck constant, refractive index of active medium, and free mass of electron, respectively. ${\left|{P}_{Exs,Grs}^{\sigma}\right|}^{2}$, is the transition matrix element [10] and it is estimated approximately at 2m

_{o}E

_{Exs0,Grs0}for InAs [17]. ${S}_{Exs\_p,Grs\_p}$ is the photon density of the pth mode emitted from Exs and Grs. The homogeneous broadening of the stimulated emission process is assumed to be Lorentzian such that ${L}_{Exs,Grs}\left({E}_{Grs\_p}-{E}_{Grs\_x}\right)$,

_{RT}= 2 L/v

_{g}. This term is equal to the number of photons per second per volume irradiating the Exs level in a single round-trip.

_{i}indicates the applied peak power of the Gaussian pulse to Exs.

## 3. Discussion and Results

_{rf}[21,22].:

_{hom}of 15 meV and Γ

_{ihom}of 45 meV has been used in the following results unless stated otherwise. For these values the gain compression factor, ${\epsilon}_{Exsp,Grsp}$ is calculated as 7.8 × 10

^{−16}cm

^{3}for Exs and Grs. To observe the radiation simultaneously from both Exs and Grs, the I

_{rf}was taken to be 40 mA in the simulations.

_{hom}and Γ

_{ihom}affect the threshold current (I

_{th}), the differential gain and the gain compression factor [10,12], first without EOGB, and the effect of Γ

_{hom}and Γ

_{ihom}on these mentioned parameters were investigated. Subsequently, an EOGB was applied to the Exs to observe how the optical beam illumination affects the gain-switching output pulses.

_{th}was calculated as 30 mA for Exs and 2 mA for Grs for the linear-gain case (${\mathsf{\epsilon}}_{\mathrm{Exs},\mathrm{Grs}}$ = 0) (see Figure 2a); 21 mA for Exs and 2 mA for Grs for the nonlinear-gain case (${\mathsf{\epsilon}}_{\mathrm{Exs},\mathrm{Grs}}$ ≠ 0) were obtained (see Figure 2b). The total threshold current (Grs + Exs) for both cases was calculated as 2 mA. As seen in Figure 2b, deviation from Figure 2a due to ${\mathsf{\epsilon}}_{\mathrm{Exs},\mathrm{Grs}}$ is because of the direct relaxation from the Wly to Grs.

_{ihom}changes between 30 and 80 meV at room temperature [23,24], therefore, we changed it from 30 meV to 80 meV. In this case, Γ

_{hom}is taken as to be 15 meV at room temperature [11]. Similarly, since the range of Γ

_{hom}is between 10 and 30 meV [2], Γ

_{hom}is changed from 10 to 30 meV and Γ

_{ihom}is taken to be 45 meV.

_{ihom}is changed from 30 meV to 80 meV, as the I

_{th}of Exs drops from 27 mA to 10 mA the I

_{th}of Grs increases from 1 mA to 6 mA (see Figure 3). If Γ

_{ihom}is greater than 70 meV, the threshold current increases as the photon density of Grs decreases and finally the threshold currents of Grs and Exs become the same at 11 mA. The effect of Γ

_{ihom}on the differential gain is shown in Figure 4. As seen in the figure, as Γ

_{ihom}increases, the differential gain of Exs and Grs decreases. Similar results were also observed in [12].

_{hom}on I

_{th}and on differential gain is similar to that of Γ

_{ihom}, providing similar differential gain characteristics as in Figure 4 when Γ

_{hom}is increased from 10 meV to 30 meV. For the center subgroup of the Q-Dot, as Γ

_{hom}is increased from 10 meV to 30 meV, I

_{th}of Exs decreases up to 22.5 meV (dropping from 26 mA to 10 mA) and after that point it slightly increases. I

_{th}of Grs increases from 1 mA to 6 mA (see Figure 5). When Γ

_{hom}is greater than 22.5 meV the photon density of Grs decreases, whereas the threshold current increases, yielding a threshold current of 14 mA, which is equal to that of the Exs. Figure 6 indicates the effect of Γ

_{hom}on the gain compression factors of Exs and Grs. As seen in the figure, the gain compression factor decreases with the increasing Γ

_{hom}.

_{hom}and Γ

_{ihom}.

_{rf}of 40 mA. Figure 7 indicates the gain-switched output pulses for an I

_{rf}of 40 mA. As shown in the figures, the Grs pulse width is longer (370 ps), while the Exs pulse width is narrow (43 ps). It can be also observed from the figure that the Exs and Grs together contribute to the output pulses since the applied current magnitude is greater than the threshold currents of both states. Therefore, the generated pulses are due to both Exs and Grs emission. The total (Exs + Grs) pulse width is 255 ps and the peak power is 28 mW. As seen in the results, the pulse width of the gain-switched output pulses are long. We also observed that increasing the injection current leads to both the peak power and pulse width increasing. The reason for the increase in the output pulse width with the current is that, although the photon density of the Grs increases with the current, the Grs photon density decreases slowly after reaching the maximum value, as seen in Figure 7. However, the Exs photon density decreases rapidly compared to that of the Grs, yielding a shorter output pulse. It can be said that the long pulses in the InAs-InP (113)B lasers are emitted from the InP ground state.

_{hom}and Γ

_{ihom}increase the I

_{th}of Exs decreases, whereas I

_{th}of Grs increases (see Figure 3 and Figure 5). Therefore, according to the magnitude of the applied current, even with a smaller value of Γ

_{ihom}, the contribution of Exs to output pulses is possible. In order to show this, 25 mA of I

_{rf}current is applied and the gain-switched output pulses were obtained for Γ

_{ihom}= 30 meV (Γ

_{ihom}< ΔE

_{dif}= E

_{Exs0}-E

_{Grs0}=48 meV) and Γ

_{ihom}= 55 meV (Γ

_{ihom}> ΔE

_{dif}= 48 meV). As seen in Figure 8 and Figure 9, the output pulse with a full-width half-maximum (FWHM) of 386 ps and peak power of 26 mW for Γ

_{ihom}= 30 meV is generated from Grs emission only. However, since I

_{th}of Exs decreased for Γ

_{ihom}= 55 meV, both Grs and Exs contribute the output pulse providing an FWHM of 233 ps and peak power of 10 mW. If we apply a current greater than the peak current of 25 mA, for example 60 mA for Γ

_{ihom}= 30 meV, both Grs and Exs contribute to the lasing process simultaneously, as shown in Figure 10 producing an FWHM of 478 ps and peak power of 53 mW. Briefly, we can say that the contribution of Exs to gain-switched output pulses depend on not only the value of Γ

_{ihom}, which is smaller or greater than the energy difference between Exs and Grs, but also on the magnitude of the applied current. In addition, it can be also observed from the results that the width of pulses are long due to dominant effect of the Grs emission as mentioned before. Wang et al. [12] showed that if Γ

_{ihom}is smaller than the energy difference between Exs and Grs (ΔE

_{dif}= E

_{Exs0}-E

_{Grs0}), lasing occurs only due to Grs if Γ

_{ihom}is greater than the ΔE

_{dif}; both Grs and Exs contribute to the lasing process. However, as seen from our results, Exs lasing depends on the magnitude of current as well as on the value of Γ

_{ihom}.

_{rf}of 12 mA. As seen in the figure, Exs emission is dominant over Grs emission, which means the output pulse is generated due to Exs emission. Therefore, the width of the output pulse is narrow (26 ps) and the peak power is high (82 mW) even though the applied current is low. Additionally, the peak power of EOGB must be increased to further increase the peak power of the output pulse. Figure 13 shows output pulses for an optical beam peak power of 20 mW for I

_{rf}of 12 mA. As seen in the figure, while the peak power of the output increases, the width of the output pulse slightly increases and provides a value of 27 ps. Furthermore, according to the applied current, we can adjust the magnitude of the optical beam to obtain short pulses.

## 4. Conclusions

- (1)
- The differential gain and the gain compression factor decrease with the increasing homogeneous and inhomogeneous broadenings. However, while the threshold current of the ground state increases, that of the excited state decreases as the broadenings are increased.
- (2)
- While gain-switched pulses are produced only due to ground state emission at small currents, the Exs and Grs emissions contribute to the output pulses at greater currents values (greater than the threshold currents of both excited state and ground state). Since the photon density of the ground state decreases gradually after reaching its maximum value, the output pulses originating from ground state emission have long pulse widths. On the other hand, the photon density of the excited state decreases rapidly after reaching its maximum value, yielding a narrower pulse width. Therefore, when an optical Gaussian pulse beam is applied to the quantum dot laser, gain-switching shorter pulses with high peak power are obtained since excite state emissions dominate ground state emissions.
- (3)
- The contribution of excited state to gain-switched output pulses depend on the magnitude of the applied current as well as on the value of the inhomogeneous broadening.
- (4)
- In the absence of the optical beam, the laser output is strongly affected by the change in the laser parameters, whereas in the presence of the optical beam, this effect is negligible.
- (5)
- The behavior of gain-switching characteristics with and without a Gaussian pulse beam are similar for liner-gain and nonlinear-gain cases except that higher peak power and narrower output pulses are obtained for the linear-gain case.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Output power vs. dc current without applying EOGB to the Exs, (

**a**) ${\mathsf{\epsilon}}_{\mathrm{Exs},\mathrm{Grs}}$ = 0 (

**b**) ${\mathsf{\epsilon}}_{\mathrm{Exs},\mathrm{Grs}}$ ≠ 0.

Cavity length, L 0.245 cm Cavity width, w 12 µm Confinement factor, Γ 0.025 Quantum dot density, No 6 × 10 ^{16} cm^{−3}Refractive index, nr 3.27 Cavity internal loss, αint 6 cm ^{−1}Mirror reflectivity, R1, R2 0.95, 0.05 Spontaneous emission ofWly, τwr 500ps Spontaneous emission of Exs, τer 500ps Spontaneous emission of Grs, τp 1.2ns Photon lifetime, τr 8.92 ps Spontaneous coupling factor, β 1 × 10 ^{−4}Emission energy of Wly, Ewly 1.05 eV Emission energy of Exs, Eexs 0.840 eV Emission energy of GS, Egrs 0.792 eV Phonon relaxation of Wly, Awly 1.35 × 10 ^{10} s^{−1}Auger coefficient of Wly, Cwly 5 × 10 ^{−9} cm^{3}s^{−1}Phonon relaxation of Wly, Aexs 1.5 × 10 ^{10} s^{−1}Auger coefficient of Exs, Cexs 9 × 10 ^{−8} cm^{3}s^{−1}Degeneracy of Grs, Exs, Wly, µgrs, exs, wly 2,4,10 Operating frequency, f 1 GHz Wavelength, λ 1.55 µm Homogeneous broadening, Γhom 15 meV Inhomogeneous broadening, Γihom 45 meV |

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**MDPI and ACS Style**

Tunc, H.S.D.; Dogru, N.; Cengiz, E.
Gain-Switched Short Pulse Generation from 1.55 µm InAs/InP/(113)B Quantum Dot Laser Modeled Using Multi-Population Rate Equations. *Mathematics* **2022**, *10*, 4316.
https://doi.org/10.3390/math10224316

**AMA Style**

Tunc HSD, Dogru N, Cengiz E.
Gain-Switched Short Pulse Generation from 1.55 µm InAs/InP/(113)B Quantum Dot Laser Modeled Using Multi-Population Rate Equations. *Mathematics*. 2022; 10(22):4316.
https://doi.org/10.3390/math10224316

**Chicago/Turabian Style**

Tunc, Hilal S. Duranoglu, Nuran Dogru, and Erkan Cengiz.
2022. "Gain-Switched Short Pulse Generation from 1.55 µm InAs/InP/(113)B Quantum Dot Laser Modeled Using Multi-Population Rate Equations" *Mathematics* 10, no. 22: 4316.
https://doi.org/10.3390/math10224316