# Optical Feedback Sensitivity of a Semiconductor Ring Laser with Tunable Directionality

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Layout of the SRL

#### 2.2. Experimental Setup

_{laser}and the current I

_{SOA1}that controls the strength of the FOF of arm 2. It should be noted that we have obtained similar results when using FOF from feedback arm 1, with the difference being that the roles of the CW and the CCW modes are then reversed. Light emitted in the CW and in the CCW direction was collected outside the laser chip using lensed fibers. Light emitted in the CCW direction was sent through a feedback loop, and was coupled back with a time delay of about 50 ns into the CW directional mode. The COF feedback loop consisted of a circulator, an external SOA, an optical bandpass filter, a 2 × 2 single-mode splitter and a polarization controller. The circulator directed the CCW light from the laser towards the external SOA. The current I

_{SOA2}injected in this external SOA was used to control the COF strength. Next, the amplified light was sent through a tunable bandpass filter with a bandwidth of 0.3 nm of which the center wavelength was tuned to the SRL’s wavelength. This tunable filter was needed to remove the amplified spontaneous emission noise—introduced by the external SOA—from the feedback signal. The polarization controller was used to adjust the polarization of the re-injected light such that it matched the emitted polarization direction. Light was re-injected into the SRL chip using the third port of the circulator. The splitter coupled 50% of the light out of the feedback loop such that we could measure its temporal and spectral properties. The optical spectrum was measured with a scanning spectrum analyzer set at a resolution of 0.02 nm. Time traces of the intensity fluctuations were measured using a 12 GHz photo-detector coupled to a fast oscilloscope of which the input bandwidth was set at 13 GHz in the experiments discussed in Section 3.

#### 2.3. Rate-Equation Model

^{cw}and E

^{ccw}of the CW and CCW directions, respectively. The number of carriers, N, is described by Equation (3). We have limited ourselves to one longitudinal mode (LM). The values of the different parameters are as follows: κ = 200 ns

^{−1}is the field decay rate, α = 3.5 is the linewidth enhancement factor, μ = 1.2 is the normalized injection current, γ = 0.4 ns

^{−1}is the carrier inversion decay rate. The effect of the backscattering is taken into account using the dissipative backscattering parameter k

_{d}= 0.2 ns

^{−1}and the conservative backscattering parameter k

_{c}= 0.88 ns

^{−1}which have been used for both of the two directional modes. The differential gain functions are given by:

^{cw}and E

^{ccw}. ${\tau}_{2}$ is the propagation time in the FOF section which is integrated on the chip and is very small. Therefore, we take τ

_{2}equal to zero in the simulations. Here it is important to mention that the feedback scheme in this study is different from the feedback scheme which has been discussed in [17,18], where self-feedback has been investigated. The last terms in Equations (1) and (2) represent the effect of spontaneous emission noise coupled to the CW/CCW modes [18,19]. D represents the noise strength expressed as D = D

_{0}(N + G

_{0}N

_{0}/κ), where D

_{0}is the spontaneous emission factor, G

_{0}= 10

^{−12}m

^{3}s

^{−1}is the gain parameter, N

_{0}= 1.4 × 10

^{24}m

^{−3}is the transparency carrier density. ξi (t)(i = cw, ccw) are two independent complex Gaussian white noises with zero mean and correlation $\langle {\xi}_{i}\left(t\right){\xi}_{j}^{*}\left(\dot{t}\right)\rangle ={\delta}_{ij}\left(t-\dot{t}\right).$ Time is rescaled to photon lifetime τ

_{ph}= 5 ps.

## 3. Results

#### 3.1. Experimental Results

_{SOA1}injected in the SOA of FOF arm 2 in Figure 1. The laser current I

_{laser}was kept constant, as shown in Figure 4, at a value of 60 mA. For low values of I

_{SOA1}, most power was emitted in the CW direction. But as I

_{SOA1}was ramped up, the power in the CCW direction gradually increased at the expense of the power in the CW direction. This is to be expected from the feedback configuration used in this experiment as the FOF in feedback arm 2 coupled light from the CW direction into the CCW direction. The power distribution over the two directional modes is further detailed at the right-hand side of Figure 4, where we plot the ratio between the power in the CCW direction and the power in the CW direction. This so-called directional mode suppression ratio (DMSR) increased most strongly when I

_{SOA1}increased from 0 to 11 mA, and then continued to increase at a slower pace for still higher values of I

_{SOA1}.

_{SOA1}= 0 mA, as in that case there was no FOF and we measured the feedback sensitivity of the SRL itself. The second bias point, BP2, that we would further investigate corresponds to I

_{SOA1}= 11 mA, as in this case the FOF clearly favored the CCW directional mode. Finally, the third selected bias condition, BP3, corresponds to I

_{SOA1}= 30 mA and in this case the directional mode suppression ratio was greatest. For BP2 and BP3, the SRL emitted a single longitudinal mode whose wavelength of 1551.555 nm was determined by the reflection spectrum of the DBR in feedback arm 2. For BP1, the output of the SRL was also single-mode but the emission wavelength of 1538.405 nm was determined by the gain maximum.

_{SOA2}injected in the external SOA. We first calibrated the amplification of the external amplifier by measuring the power transmitted through the external SOA as a function of its bias current (while keeping the laser current I

_{laser}and the FOF current I

_{SOA1}constant). For small values of I

_{SOA2}, the CW intensity was rather constant with some noise-induced fluctuations around the steady state. This is illustrated in Figure 5 (left) at a setting (I

_{laser}, I

_{SOA1}, I

_{SOA2}) = (60 mA, 11 mA, 500 mA). Increasing I

_{SOA2}eventually led to undamping of the relaxation oscillations as illustrated in Figure 5 (middle) for (I

_{laser}, I

_{SOA1}, I

_{SOA2}) = (60 mA, 11 mA, 600 mA). This marks the onset of the COF-induced dynamics. For larger values of the COF strength, the feedback-induced dynamical fluctuations became stronger and more complex as illustrated in Figure 5 (right) for (I

_{laser}, I

_{SOA1}, I

_{SOA2}) = (60 mA, 11 mA, 700 mA).

_{laser}and the mean value of the detector signal. Calculating this ratio is equivalent to rescaling the time traces such that the average value of the detector signal is equal to one. We performed this rescaling of the STD to make the extracted values independent of the average power coupled to the read-out fiber. The noise of the oscilloscope and the photo-detector are compensated for when extracting the value of σ

_{laser}from the time traces by assuming that the noise of these sources is Gaussian and is independent from the fluctuations in the laser’s intensity. To perform this compensation, we measured a time trace of the detector signal (using the same oscilloscope settings as when measuring the laser’s intensity) without optical input to the detector. From this time-trace, we determined the standard deviation σ

_{det}of the detector and oscilloscope noise (the mean value of the detector and oscilloscope noise was measured to be close to zero). Using the standard deviation σ

_{timetrace}extracted from the intensity time trace, we estimate the standard deviation of the intensity fluctuations σ

_{laser}to be ${\sigma}_{laser}=\sqrt{{\sigma}_{timetrace}^{2}-{\sigma}_{noise}^{2}}$.

_{SOA2}and was obtained by measuring the optical power after the splitter in Figure 2 when the feedback loop was open. For each of the bias conditions, the STD was small for small values of the COF strength, as there were not yet any feedback-induced dynamics in the time traces. When increasing the COF strength, we can see in Figure 6 that the onset of the feedback-induced dynamics was lowest for bias condition BP1, i.e., without FOF to stabilize the laser. When FOF was applied (see measurements for BP2 and BP3 in Figure 6), the onset of the COF dynamics was shifted to larger values of the feedback strength, but this shift was not large for BP2 and BP3: the shift in the onset when comparing BP1 to BP2 was roughly a factor of 2 and was thus rather modest as compared to the suppression of feedback dynamics in strongly unidirectional SRLs [7,8]. Moreover, when increasing the FOF strength from BP2 to BP3, we actually observed a slight drop in the onset of the COF dynamics. The experiments thus show only a limited effectiveness of the proposed FOF scheme to suppress these dynamical fluctuations, and this effectiveness is furthermore dependent on the exact value of the applied FOF strength. The reason behind these observations will be clarified based on numerical simulations of the system in Section 3.2.

#### 3.2. Results from Numerical Simulations

^{–1}(as this is a good setting to show the effect of the FOF on the onset of the laser dynamics). In the red time trace of Figure 7 (left), FOF was not used whereas the FOF strength was set to 2 ns

^{–1}in the blue time trace of Figure 7 (left). Using FOF, the intensity fluctuations in the time trace became smaller as compared to the case without FOF. We also notice that the average intensity in the CCW direction increased due to the FOF, as it enhances the CCW mode (see also Figure 4). As a result, the rescaled STD was smaller for the trace in Figure 7 (left) corresponding to ${\eta}_{2}$ = 2 ns

^{–1}.

_{0}to be 2 × 10

^{−6}ns

^{−1}in order to find the same rescaled STD in the simulations without COF. Similarly to the experiments, we started by calculating the mean value and the STD of the time traces without FOF (${\eta}_{2}$ = 0 ns

^{−1}). We increased the strength of the COF by increasing ${\eta}_{1}$ from 0 to 1.0 ns

^{−1}in steps of 0.05 ns

^{−1}while the rest of the parameters were fixed (${\eta}_{2}$ = 0 ns

^{−1}). Next, we repeated the calculations of the mean value and the STD of the time traces, but this time with FOF by setting ${\eta}_{2}$ to 3 ns

^{−1}, 5 ns

^{−1}and 8 ns

^{−1}, while the rest of parameters were kept unchanged. We plot the rescaled STD from the simulations in Figure 7 (right) as a function of the COF strength ${\eta}_{1}$. At low values of the COF strength, the STD is relatively small and remains approximately constant when changing the COF strength. The onset of COF-induced dynamics is visible in these curves as the point at which the STD starts to rapidly increase with increasing COF strength. Similarly to the experiments, the onset happened first for the laser without FOF around ${\eta}_{1}$ = 0.2 ns

^{−1}. When FOF was applied, the onset first shifted to larger COF strengths, but this shift is albeit rather limited. When further increasing the FOF strength, the onset of the dynamics shifted erratically and we did not observe a continuous increase in the onset. These numerical results thus agree qualitatively with our experimental trends and observations discussed in Section 3.1, and show that the FOF scheme presented in Figure 1 does not really help to reduce the COF-induced dynamics.

_{i}, from the model described in Equations (1)–(3) without noise (setting D = 0). By studying the Lyapunov spectrum, we tried to understand how FOF influences both the stability and complexity of the chaotic dynamics that might have arisen. For the computation of the Lyapunov exponents, we applied the ideas of Farmer [20] to our case. Specifically, we integrated the corresponding delay differential equations with an Euler method. This converts the original delay differential equations in a map. We computed the Lyapunov exponents of this map. Only a finite portion of the infinite set of λ

_{i}can be determined by such a numerical analysis. In Figure 8, we present the five largest Lyapunov exponents vs. the COF strength η

_{1}. Due to the field nature of the equations, one exponent will always be zero. If only the maximal exponent is zero, the SRL will be emitting in a continuous wave. If two exponents are zero, while the others are all negative, the laser output will be periodic. If more exponents are zero, the dynamics can correspond to either periodic or quasi-periodic behavior. Once the maximal Lyapunov exponent becomes positive, the SRL will be operating chaotically. From Figure 7 (right) and Figure 8 (left), in the case of no filtered feedback, the increase of the STD around ${\eta}_{1}$ = 0.1 to 0.4 ns

^{–1}can be attributed to a bifurcation from continuous wave emission to periodic oscillations. It is only later, after a regime of quasi-periodic behavior, that the laser became chaotic (around ${\eta}_{1}$ = 0.8 ns

^{–1}). With FOF (${\eta}_{2}$ = 3.0 ns

^{–1}), in Figure 8 (middle), below ${\eta}_{1}$ = 0.7 ns

^{–1}, the SRL with filtered feedback was continuously lasing except for some very small windows of periodic behavior. While this seems to indicate that the SRL would be more stably lasing, the negative Lyapunov exponents were now much smaller in amplitude. This indicates that the SRL would be much easier to destabilize due to noise, for example. The bifurcation to chaotic behavior hardly moved and still appeared at feedback strengths around ${\eta}_{1}$ = 0.8 ns

^{–1}. However, its accompanying positive Lyapunov exponents were increased significantly, indicating a more complex and less damped dynamical chaotic behavior. For ${\eta}_{2}$ = 8.0 ns

^{–1}(Figure 8 (right)), it is clear that the large region of chaos shifted to lower values of ${\eta}_{1}$(${\eta}_{1}$≈ 0.4 ns

^{–1}). Around ${\eta}_{1}$ = 0.2 ns

^{–1}, the laser was first destabilized as a small window of mildly chaotic behavior appeared (i.e., only one of the Lyapunov exponents was positive). This onset of chaotic oscillations corresponds to the abrupt change in the rescaled STD observed numerically in Figure 7 (right) and experimentally in Figure 6 for I

_{SOA1}= 30 mA. To conclude, with filtered feedback, the dynamical behavior of the SRL was altered considerably. For some values of the filtered feedback this led to a larger but less stable continuous wave regime and chaos which was more complex. Because of the larger continuous wave regime, the feedback sensitivity was somewhat reduced as compared to the device without FOF.

## 4. Discussion

_{d}= k

_{c}= 0) in the numerical simulations. In this case, the SRL without any feedback operates in a unidirectional regime with the full output power concentrated either in the CW or CCW mode. In Figure 9, we show the results from a numerical analysis of the Equations (1)–(3) for k

_{d}= k

_{c}= 0. The left-hand side of Figure 9 shows rescaled STDs obtained from time traces using the procedure described above. For all cases, we find that the STD increases for low COF strengths, which are even lower than in Figure 7. The right-hand side of Figure 9 shows the five largest Lyapunov values describing the noiseless dynamics of the SRL in the case of filtered feedback. Again, at a very low feedback strength (${\eta}_{1}$ > 0.05 ns

^{–1}), the SRL becomes chaotic. It is clear that even in the case of no backscattering, the filtered feedback actually destabilizes the SRL. This indicates that—for the device layout studied here—a feedback signal in the quiescent directional mode is coupled (through the FOF branch) sufficiently strongly to the dominant directional mode in order to invoke delay-induced dynamical fluctuations.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Image of the semiconductor ring laser with filtered feedback, in which the different laser and feedback components are indicated.

**Figure 2.**Schematic of the experimental setup. LF, lensed fiber; SOA2, semiconductor optical amplifier used to tune the coherent optical feedback (COF) strength; TF, tunable optical bandpass filter; Det, fast opto-electronic detector; PC, polarization controller.

**Figure 3.**Output power of the two directional modes versus laser injection current while the current in the filtered optical feedback (FOF) section is equal to 0 mA.

**Figure 4.**Output power of the two directional modes versus current injected in the semiconductor optical amplifier (SOA) in the FOF arm 2 (left) and directional mode suppression ratio as a function of the SOA current in the FOF arm (right) at a laser injection current of 60 mA.

**Figure 5.**Time traces of the laser’s output as measured by the detector in the setup of Figure 2 at a laser injection current of 60 mA and an SOA current in the FOF path of 11 mA for different strengths of the COF by changing the current injected in SOA2 in the COF path: I

_{SOA2}= 500 mA (left), I

_{SOA2}= 600 mA (middle) and I

_{SOA2}= 700 mA (right).

**Figure 6.**Rescaled STD from time traces of the laser’s output as a function of the COF signal strength as measured after the splitter in Figure 2 for different values of the I

_{SOA1}(which controls the FOF strength).

**Figure 7.**Simulated time traces in the CCW direction when the COF strength is η

_{1}= 0.4 ns

^{–1}without FOF in red and with FOF using η

_{2}= 2 ns

^{–1}in blue (left). Standard deviation of the simulated time traces of the laser’s output as a function of the COF strength η

_{1}for different values of the FOF strength η

_{2}(right).

**Figure 8.**The five largest Lyapunov exponents: without FOF (i.e., η

_{2}= 0 ns

^{–1}) (left), with FOF (i.e., η

_{2}= 3 ns

^{–1}) (middle) and with large FOF strength (i.e., η

_{2}= 8 ns

^{–1}) (right).

**Figure 9.**Standard deviation of the simulated time traces of the laser’s output as a function of the COF strength (left) and the five largest Lyapunov exponents (right) with FOF (η

_{2}= 3 ns

^{–1}) when the backscattering is set to zero.

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**MDPI and ACS Style**

Verschaffelt, G.; Khoder, M.; Van der Sande, G.
Optical Feedback Sensitivity of a Semiconductor Ring Laser with Tunable Directionality. *Photonics* **2019**, *6*, 112.
https://doi.org/10.3390/photonics6040112

**AMA Style**

Verschaffelt G, Khoder M, Van der Sande G.
Optical Feedback Sensitivity of a Semiconductor Ring Laser with Tunable Directionality. *Photonics*. 2019; 6(4):112.
https://doi.org/10.3390/photonics6040112

**Chicago/Turabian Style**

Verschaffelt, Guy, Mulham Khoder, and Guy Van der Sande.
2019. "Optical Feedback Sensitivity of a Semiconductor Ring Laser with Tunable Directionality" *Photonics* 6, no. 4: 112.
https://doi.org/10.3390/photonics6040112