# Optimal Conditions for a Multimode Laser Diode with Delayed Optical Feedback in Terahertz Time-Domain Spectroscopy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Rate Equations for a Multimode Laser Diode with Delayed Optical Feedback

_{m}, E

_{m}(t−τ

_{fb}), S

_{m}, and g

_{m}are the complex optical electric field, the complex delayed optical feedback field, the photon density, and the modal gain for the mth mode, respectively; N is the carrier density; α and R

_{spn}are the linewidth enhancement factor and the carrier density-dependent spontaneous emission coefficient, respectively; δf is the longitudinal mode spacing; τ

_{p}and T

_{1}are the photon lifetime and the carrier density-dependent carrier lifetime, respectively; f

_{c}is the coupling coefficient of the optical feedback fields; T

_{rt}and τ

_{fb}are the round-trip time of the LD cavity and the delay time of the optical feedback fields, respectively; I, e, and V are the injection current, the elementary electric charge, and the laser cavity volume, respectively; ∆τ

_{c}is the coherence time of the amplified spontaneous emission; and ξ

_{Sm}and ξ

_{N}are zero-mean and unit-variance Gaussian distributions, respectively, whose amplitudes are varied every ∆τ

_{c}as step functions [29,30].

_{m}, for the mth mode is expressed below as a function of the differential gain coefficient, G

_{0m}, the carrier density at transparency, N

_{0m}, and the intrinsic gain saturation coefficient, ε

_{Nm}.

_{S}is the gain compression factor and G

_{0m}, N

_{0m}, and ε

_{Nm}for the mth mode are represented by polynomial equations of mode number m [28,29,30]. The condition m = 0 stands for the central mode, which was set to 375 THz (800 nm) by taking into account wavelength matching with a GaAs photoconductive antenna. A negative mode number indicated that the oscillation frequency for the mode of interest was lower than the central frequency, while a positive mode number indicated that it was higher. The m-value was varied over a range of −30 to +30, allowing for the simulation of multimode oscillation involving a total of 61 modes.

_{1}, and the carrier density-dependent spontaneous emission coefficient, R

_{spn}, are expressed as follows:

_{1}, C

_{2}, and C

_{3}are the nonradiative recombination rate, the radiative recombination coefficient, and the Auger recombination coefficient, respectively, and β is the spontaneous emission factor.

_{m}, and the photon density, S

_{m}, for the mth mode is given by:

_{r}, ε

_{0}, h, and f

_{m}represent the refractive index of the active layer of the LD, the dielectric constant for the vacuum, the Planck’s constant, and the central oscillation frequency of the mth mode, respectively. The notations and values for these parameters are listed in Table 1.

_{c}. This formulation was based on the Lang–Kobayashi equation for single-mode oscillation [31]. Since the MMLD output released approximately 90% of the optical energy from the cavity to the outside during the round-trip time, up to 81% of the optical energy was fed back into the MMLD cavity through the delayed optical feedback. As a result, the maximum value of f

_{c}was 0.9 ($=\sqrt{0.81}$). To simplify the interaction between the complex delayed optical feedback fields and the complex optical electric fields in the LD cavity, we assumed a coupling phase between them of zero, and therefore, f

_{c}was treated as a real number. This choice reflected the positioning of the external mirror at an integer multiple of the optical path length of the LD cavity.

_{m}(t) (61 modes) and N(t) by numerically integrating the aforementioned multimode rate equations using the fourth-order Runge–Kutta method. The time interval for the numerical integration was set to approximately 0.03 ps by dividing 1 ns into 2

^{15}(=32,768) time steps. This resulted in a frequency bandwidth of 32.768 THz that could be handled in the calculation. After calculating the initial 5 ns of the transient temporal waveforms for the variables, we recorded time-series data every 1 ns. This process was repeated 1000 times, resulting in a total time-series data collection of 1 μs. The 1 ns time series data for the 61-mode LD output, denoted as E

_{m}(t), were summed together with the random phases. Subsequently, a fast Fourier-transform was applied to obtain the complex optical spectrum data, ${\stackrel{~}{E}}_{sum}\left(\omega \right)$, and their absolute values were squared to record power spectrum data, ${\left|{\stackrel{~}{E}}_{sum}\left(\omega \right)\right|}^{2}$, every 1 ns. This procedure was repeated 1000 times, resulting in a dataset comprising 1000 power spectra.

#### 2.2. Simulation Model for THz Time-Domain Spectroscopy

_{m}represents an arbitrary phase for the mth mode. The photo-carriers induced a transient current, J(t), when a bias voltage was applied to the gap. Consequently, electromagnetic waves were emitted into free space, with their amplitudes being proportional to the time derivative of the transient current, dJ(t)/dt. In cases where an antenna exhibits wideband characteristics due to the ultrafast relaxation times of the photo-carriers, the THz wave, E

_{THz}(t), can be represented as follows:

## 3. Results and Discussion

#### 3.1. Classification of the LD Oscillation State by the Coupling Coefficient of the Optical Feedback

_{c}: 0 (a1–a3), 0.1 (b1–b3), 0.4 (c1–c3), and 0.6 (d1–d3), where the pumping rate r and the delay time of the optical feedback τ

_{fb}were set to 1.5 and 1 ns, respectively. The power spectra of the MMLD outputs (a2–d2) and the THz-TDS outputs (a3–d3) were obtained by averaging 1000 spectral data points, each acquired at 1 ns intervals. When f

_{c}was set to zero (no optical feedback), the MMLD exhibited steady-state oscillation (a1), albeit with a relatively significant amount of noise resulting from the Langevin noise. The side-mode suppression ratio attained 26.4 dB (a2), indicating that the MMLD operated in single-mode oscillation at the negative first mode. As a result, the generation of optical beats between the multimode, based on a 139 GHz longitudinal mode spacing, was effectively suppressed, leading to a low THz-TDS output (a3). When f

_{c}was set to 0.1, the temporal waveform exhibited chaotic behavior (b1), and the spectrum demonstrated multimode oscillation, primarily operating in four modes ranging from the negative fourth to the negative first modes (b2). This behavior arose from the periodic optical feedback, which acted as a seed light for each mode, preventing the concentration of gain in a particular mode. The presence of optical feedback disrupted the MMLD’s ability to settle into steady-state oscillation, resulting in a continuous transient state. Consequently, the THz-TDS output spectrum (b3) showed an increase in the 139, 278, and 417 GHz spectral components, corresponding to one, two, and three times the longitudinal mode spacing of the MMLD, respectively. Subsequently, when f

_{c}was raised to 0.4, the temporal waveform exhibited intermittent chaotic oscillation, characterized by intense peaks at the following specific time points: 6, 116, 134, 189, 284, 420, 508, 579, 648, 736, 875, and 959 ns (c1). Concurrently, the multimode spectrum of the MMLD output broadened asymmetrically (c2), leading to a corresponding broadening of the THz-TDS output spectrum to approximately 2 THz (c3). Moreover, the widening of each linewidth observed in both the LD output spectrum (c2) and the corresponding THz-TDS output spectrum (c3) indicated that the intermittent chaotic oscillation was accompanied by the presence of short optical pulses. However, when f

_{c}was further increased to 0.6, the MMLD returned to steady-state oscillation (d1) with two (the negative fifth and negative sixth) modes (d2). The resulting THz-TDS output spectrum displayed only a 139 GHz spectral component (d3). This result could be attributed to the delicate equilibrium between the optical feedback and the optical gain, with the presence of relatively strong optical feedback leading to the phenomenon of optical injection locking in the MMLD. By performing area integration on the spectra presented in Figure 2(a2–d2), we observed moderate increases in the MMLD outputs by factors of 1.06 (f

_{c}= 0.1), 1.46 (f

_{c}= 0.4), and 1.91 (f

_{c}= 0.6) compared to the case without optical feedback (a2). These increases were attributed to the presence of optical feedback. In contrast, upon analyzing the corresponding THz-TDS outputs from the area integration of the spectra in Figure 2(a3–d3), a significant surge was estimated. The THz-TDS outputs experienced increases by factors of 22 (f

_{c}= 0.1), 676 (f

_{c}= 0.4), and 466 (f

_{c}= 0.6) due to the effect of optical feedback. These findings demonstrated that an appropriate amount of optical feedback was necessary to achieve broadband and high-intensity THz-TDS outputs using an MMLD with delayed optical feedback. In this numerical calculation, while environmental and equipment noise were not included, a noise floor was generated based on Langevin noise. As a result, the signal-to-noise ratios of the THz-TDS outputs shown in (a3–d3) were estimated to be 5 to 9 dB lower than those obtained without Langevin noise.

#### 3.2. Characteristics of the Intermittent Chaotic Oscillations

_{c}= 0.4. Figure 3a depicts the temporal waveform of the carrier density when f

_{c}was set to 0.4, revealing significant accumulations of carrier densities at the following specific time points: 0, 109, 130, 179, 276, 414, 502, 576, 644, 728, 866, and 956 ns. It is worth noting that the intense peak intensities observed in the MMLD output shown in Figure 2(c1) occurred between 3 and 10 ns after the carrier density accumulation timings, indicating that the MMLD operated similar to a Q-switched laser [32] under the given optical feedback condition. Figure 3b,c illustrates the temporal variations in the MMLD output energy (b) and the THz-TDS output energy (c), respectively, integrated every 1 ns. The distinct energy drops observed in Figure 3b precisely corresponded to the significant carrier density accumulations shown in Figure 3a. These rapid energy drops shown in Figure 3b could be observed when the temporal waveform, depicted in Figure 2(c1), was measured with a slightly slower optical detector, commonly referred to as “low-frequency fluctuation (LFF)” in the field of laser chaos [33]. In contrast, the distinct energy increases observed in Figure 3c aligned precisely with the strong peak intensities of the MMLD output shown in Figure 2(c1).

_{c}= 0.1. In contrast, the spectrum in Figure 6a was approximately three times wider than that shown in Figure 6b. Furthermore, the spectral peak shown in Figure 6a had shifted toward shorter wavelengths by approximately 1 nm in comparison to that shown in Figure 6b. This shift reflected the band-filling effect resulting from the abrupt accumulations of carrier densities just prior to the generation of the picosecond optical pulses, as seen in Figure 3a. As shown in Figure 4b, the temporal waveform exhibited a mixture of picosecond pulses and chaotic oscillations with relatively low intensities, with both elements present in nearly equal proportions. As a result, the corresponding power spectrum shown in Figure 5a clearly resembled the composite shape of the power spectra shown in Figure 6a,b, with significant asymmetry in its spectral profile. The power spectrum averaged over 1 μs, as shown in Figure 2(c2), and it also exhibited a similar asymmetric shape, which was evidence that the MMLD output exhibited intermittent chaotic oscillations.

_{fb}= 2 ns) was used. The observed asymmetric spectral shape depicted in Figure 6c was consistent with the calculation shown in Figure 2(c2), indicating that the MMLD with delayed optical feedback used in the experiment exhibited intermittent chaotic oscillations, including picosecond optical pulse generation. In this sense, the COLD (chaotic oscillating laser diode) used in the THz-TDS experiment could be specifically referred to as ICOLD (intermittent chaotic oscillating laser diode).

#### 3.3. Time Convergence of the Output Spectral Shapes

_{LD}(t), between the adjacent MMLD output spectra, which were averaged every 1 ns using the rate equations mentioned above. This intensity difference was calculated using the following equation:

_{0}and t

_{1}, respectively. The time difference between t

_{0}and t

_{1}was set to 1 ns, and this calculation continued until t

_{1}reached 1 μs. The intensity difference demonstrated a convergence trend, approximately reaching a reciprocal time. Figure 7b–d depict the power spectra of the MMLD outputs averaged over 10 ns, 100 ns, and 500 ns, respectively. Taking into account the convergence of the intensity difference displayed in Figure 7a, the power spectrum shape in Figure 7d (averaged over 500 ns) closely resembled that shown in Figure 2(c2) (averaged over 1000 ns). Hence, despite the chaotic changes in the temporal waveform of the MMLD output with delayed optical feedback from moment to moment, the averaged power spectrum consistently converged to a constant shape, reflecting the attractor characteristics of the MMLD system with delayed optical feedback. These findings were consistent with the previously reported experimental result that the averaged power spectrum was stably observed with a constant shape [26]. The results presented in Figure 7 demonstrate that the power spectrum of the MMLD with delayed optical feedback stabilized to a constant shape through a time average in the order of microseconds.

_{THz}(t), between the adjacent THz-TDS output spectra averaged every 1 ns. This intensity difference was calculated using the following equation:

_{0}and t

_{1}, respectively. Similar to Figure 7, the time difference between t

_{0}and t

_{1}was set to 1 ns, and this calculation continued until t

_{1}reached 1 μs. Additionally, Figure 8b–d presents the power spectra of the THz-TDS outputs averaged over 10 ns, 100 ns, and 500 ns, respectively. The observed fluctuations shown in Figure 8a could be attributed to the time derivative of the chaotic temporal waveform of the photo-carriers during the THz wave generation described in Equation (8).

#### 3.4. Expansion of the Intermittent Chaotic Oscillation Region

_{c}for the time period 900–1000 ns, with r = 1.5 and τ

_{fb}= 1 ns (the same conditions shown in Figure 2). Notably, significant peak fluctuations with intensities exceeding 100 (arb. unit), indicative of intermittent chaos, were observed only when f

_{c}was set to 0.27, 0.32, and 0.40. These values corresponded to situations where even slight variations in f

_{c}caused sharp decreases in the THz-TDS outputs. However, previous experiments reported that the THz-TDS outputs exhibited only minor fluctuations in response to slight variations in f

_{c}[26]. To reconcile these calculation results with the experimental results, the controllable parameter values (the pumping rate, r, and the delay time of the optical feedback, τ

_{fb}) were systematically varied, and the corresponding bifurcation diagrams were plotted, as shown in Figure 9(b1–c2). Specifically, Figure 9(b1,b2) demonstrates that under higher r values, the region of intermittent chaos expanded towards the higher f

_{c}side. This expansion was attributed to the increase in the relaxation oscillation frequency of the MMLD resulting from an increase in the pumping rate, r. The relaxation oscillation frequency of the MMLD could be expressed as Equation (13), introducing mode dependence in the form of the relaxation oscillation frequency of the single-mode LD [36].

_{Rm}, I

_{thm}, and I

_{0m}represent the relaxation oscillation frequency, the threshold current, and the current level corresponding to the carrier density at transparency, respectively, for the mth mode. Thus, at high excitation, the relaxation oscillation frequency was higher and fast intermittent chaotic oscillations with picosecond pulse oscillations were more likely to occur. Figure 9(c1) demonstrates that when τ

_{fb}was set to a short time (e.g., 0.5 ns), the region of intermittent chaos completely disappeared. This phenomenon could be attributed to the high frequency of the optical modulation caused by the optical feedback (1/τ

_{fb}~2 GHz), which approached the relaxation oscillation frequency of the MMLD driven at r = 1.5 (~4 GHz). In contrast, as shown in Figure 9(c2), when τ

_{fb}was set to 2 ns, the frequency of the optical modulation (1/τ

_{fb}~0.5 GHz) became significantly lower than the relaxation oscillation frequency, resulting in an expansion of the intermittent chaos region. Thus, increasing the injection current or delay time expanded the intermittent chaotic area of the MMLD with delayed optical feedback.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Temporal waveforms of the MMLD outputs (

**a1**–

**d1**), along with the corresponding power spectra (

**a2**–

**d2**) and THz-TDS outputs (

**a3**–

**d3**) for the different values of f

_{c}(0 (

**a1**–

**a3**), 0.1 (

**b1**–

**b3**), 0.4 (

**c1**–

**c3**), and 0.6 (

**d1**–

**d3**)). The parameters r and τ

_{fb}were set to 1.5 and 1 ns, respectively.

**Figure 3.**(

**a**) Temporal waveform of the carrier density when f

_{c}was set to 0.4. The temporal variations in the (

**b**) MMLD output energy and (

**c**) THz-TDS output energy integrated every 1 ns.

**Figure 4.**(

**a**) A magnified view of a section from Figure 3c. The temporal waveforms of the MMLD outputs are shown during the time periods (

**b**) 644–645 ns, (

**c**) 648–649 ns, and (

**d**) 690–691 ns, respectively.

**Figure 5.**(

**a**) Power spectrum of the MMLD output corresponding to the temporal waveform in Figure 4b. (

**b**) Temporal waveforms of the MMLD output for the respective (the negative second to the fifth) modes.

**Figure 7.**(

**a**) Temporal variation in the intensity difference between the adjacent averaged MMLD output spectra. Examples of the MMLD output spectra averaged over (

**b**) 10 ns, (

**c**) 100 ns, and (

**d**) 500 ns, respectively.

**Figure 8.**(

**a**) Temporal variation in the intensity difference between the adjacent averaged THz-TDS output spectra. Examples of the THz-TDS output spectra averaged over (

**b**) 10 ns, (

**c**) 100 ns, and (

**d**) 500 ns, respectively.

**Figure 9.**Peak plots as functions of f

_{c}when the values of (r, τ

_{fb}) were set to (

**a**) (1.5, 1 ns), (

**b1**) (2.0, 1 ns), (

**b2**) (2.5, 1 ns), (

**c1**) (1.5, 0.5 ns), and (

**c2**) (1.5, 2 ns), respectively.

**Table 1.**Notations and values for the parameters [30].

Notation | Parameter | Value | Unit |
---|---|---|---|

m | Mode number | −30~+30 | |

τ_{p} | Photon lifetime | 2.0 | ps |

T_{rt} | Round-trip time of the LD cavity | 7.1 | ps |

τ_{fb} | Delay time of the optical feedback fields | 1 | ns |

C_{1} | Nonradiative recombination rate | 2.0 × 10^{8} | s^{−1} |

C_{2} | Radiative recombination coefficient | 2.0 × 10^{−16} | m^{3} s^{−1} |

C_{3} | Auger recombination coefficient | 0 | m^{6} s^{−1} |

f_{c} | Coupling coefficient of the optical feedback fields | 0~0.6 | |

α | Linewidth enhancement factor | 3.0 | |

β | Spontaneous emission factor | 1.0 × 10^{−6} | |

ε_{S} | Gain compression factor | 0.05 × 10^{−23} | m^{3} |

δf | Longitudinal mode spacing | 0.139 | THz |

∆τ_{c} | Coherence time of amplified spontaneous emission | 7.1 | ps |

e | Elementary electric charge | 1.60 × 10^{−19} | C |

L | Laser cavity length | 300 | μm |

V | Laser cavity volume | 180 | μm^{3} |

n_{r} | Refractive index of the active layer | 3.6 | |

ε_{0} | Dielectric constant for the vacuum | 8.85 × 10^{−12} | F/m |

H | Planck’s constant | 6.63 × 10^{−34} | Js |

f_{m} | Central oscillation frequency of the mth mode | 375 + 0.139 m | THz |

I | Injection current | r × I_{th}_{0} | mA |

r | Pumping rate | 1.5~2.5 | |

I_{th}_{0} | Threshold current for the central mode | 24 | mA |

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## Share and Cite

**MDPI and ACS Style**

Wada, K.; Kitagawa, T.; Matsuyama, T.; Okamoto, K.; Kuwashima, F.
Optimal Conditions for a Multimode Laser Diode with Delayed Optical Feedback in Terahertz Time-Domain Spectroscopy. *Spectrosc. J.* **2023**, *1*, 137-151.
https://doi.org/10.3390/spectroscj1030012

**AMA Style**

Wada K, Kitagawa T, Matsuyama T, Okamoto K, Kuwashima F.
Optimal Conditions for a Multimode Laser Diode with Delayed Optical Feedback in Terahertz Time-Domain Spectroscopy. *Spectroscopy Journal*. 2023; 1(3):137-151.
https://doi.org/10.3390/spectroscj1030012

**Chicago/Turabian Style**

Wada, Kenji, Tokihiro Kitagawa, Tetsuya Matsuyama, Koichi Okamoto, and Fumiyoshi Kuwashima.
2023. "Optimal Conditions for a Multimode Laser Diode with Delayed Optical Feedback in Terahertz Time-Domain Spectroscopy" *Spectroscopy Journal* 1, no. 3: 137-151.
https://doi.org/10.3390/spectroscj1030012