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Dynamics of the Frequency Shifts in Semiconductor Lasers under the Injection of a Frequency Comb

Dynamics of the Frequency Shifts in Semiconductor Lasers under the Injection of a Frequency Comb hv photonics Article Dynamics of the Frequency Shifts in Semiconductor Lasers under the Injection of a Frequency Comb Najm M. Al-Hosiny Department of Physics, College of Science, Taif University, P.O. Box 1109, Taif 21944, Saudi Arabia; najm@tu.edu.sa Abstract: We have numerically investigated the dynamics of frequency shifts in semiconductor lasers under the injection of a frequency comb. We have studied the effect of comb spacing on the locking bandwidth. Frequency comb spacing was found to play an important role in the boundaries of the locking bandwidth as well as in the frequency shift of the SL peak. Keywords: optical injection; frequency comb; frequency pulling; frequency pushing 1. Introduction Frequency combs have been used in semiconductor lasers since 1992 [1]. In terms of optically injected semiconductor lasers, frequency combs have been studied in different aspects, including comb generation [2–6], selective amplification of the comb [7], producing low-noise microwave signals [8,9], optoelectronic millimeter-wave synthesis [10] and full- duplex coherent optical system [11]. As a key component in communication systems, the investigation of the frequency and its stabilization in semiconductor lasers has been under investigation since 1988 [12,13]. One of the major techniques to stabilize the frequency in semiconductor lasers is optical injection locking [14]. We have previously reported the dynamics of semiconductor lasers under dual optical injection [15,16]. The additional signal was found to enhance the chaos and control the stability map. Arfan et al. [17] have shown Citation: Al-Hosiny, N.M. Dynamics that the injected laser can lock to equipartition points in frequencies between two adjacent of the Frequency Shifts in modes or each individual frequency of the master laser. In 2014, analytical and numerical Semiconductor Lasers under the calculations were performed for a semiconductor laser under the injection of a frequency Injection of a Frequency Comb. Photonics 2022, 9, 886. https:// comb [18]. The study found that the locking depends on the separation of the comb and doi.org/10.3390/photonics9120886 the identification of three major regions of unique amplifications. Later on, another study identified important criteria to maximize the frequency locking range of the semiconductor Received: 24 October 2022 laser under the injection of a frequency comb [19]. It was also found that the slave laser Accepted: 18 November 2022 under the injection of a frequency comb undergoes two different mechanisms, in which Published: 22 November 2022 the output of the frequency comb separation is decreased [20]. The study also found that Publisher’s Note: MDPI stays neutral the relaxation oscillations of the injected laser can lock to the harmonics of the injected with regard to jurisdictional claims in optical comb, generating extra tones in the optical comb around the slave laser ’s frequency. published maps and institutional affil- Recently, a comprehensive theoretical and experimental study investigated the variety iations. of nonlinear dynamics exhibited by a single-frequency semiconductor laser subjected to optical injection from a frequency comb [21]. By varying the comb parameters (number of lines and comb spacing), a rich variety of nonlinear dynamics was identified including wave mixing and irregular chaotic pulsing. In this study, we numerically investigate the Copyright: © 2022 by the author. effect of frequency comb spacing on the locking bandwidth and on the frequency shift of Licensee MDPI, Basel, Switzerland. the slave laser. The locking maps for different comb spacings are drawn and discussed. This article is an open access article The frequency shifts are extracted from the maps and analyzed. distributed under the terms and conditions of the Creative Commons 2. Materials and Methods Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ Our model is based on Lang’s approach [22], where the system is described by three 4.0/). differential equations for electric field amplitude, phase and carrier density. Before we Photonics 2022, 9, 886. https://doi.org/10.3390/photonics9120886 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 9 2. Materials and Methods Our model is based on Lang’s approach [22], where the system is described by three differential equations for electric field amplitude, phase and carrier density. Before we elaborate on the model, let us introduce the frequency comb, which will be injected inside Photonics 2022, 9, 886 2 of 8 the cavity of the slave laser (SL). Our SL is assumed to be a single-mode DFB laser. The frequency comb consists of five peaks as shown in Figure 1. The central peak is f3 and the elaborate on the model, let us introduce the frequency comb, which will be injected inside spacing between the peaks is ∆, which is constant in each case as will be shown later. The the cavity of the slave laser (SL). Our SL is assumed to be a single-mode DFB laser. The frequency comb consists of five peaks as shown in Figure 1. The central peak is f and the intensity of the peaks (and hence the injection level) is equal. spacing between the peaks is D, which is constant in each case as will be shown later. The intensity of the peaks (and hence the injection level) is equal. Figure 1. The frequency comb used in our simulation. Figure 1. The frequency comb used in our simulation. The rate equation for the SL for electric field amplitude, phase and carrier density, can be expressed respectively, as follows: d 1 The rate equation for the SL for electric field amplitude, phase and carrier density, E (t) = G DN(t)E (t) + h E cos(Df ) (1) o N o å m m dt 2 can be expressed respectiv  ely, as follow  s: d 1 E f (t) = aG DN(t) + h sin(Df ) (2) o N å m dt 2 E (t) d N(t) N(t) = J Gd(N(t) N )E1(t) (3) N o dt t s E (t)= G N (t) E (t)+ E cos( ) (1) o N o m m dt 2 where E (t) is the electric field of the SL, G is the material gain coefficient, DN(t) is o N the population inversion (N–N ) where N is the carrier density and N is its value at th th threshold, h is the coupling coefficient, m represents the order of the frequency in the comb (m = 1, 2, 3, 4, 5), Df = Dw t f (t), where Dw = w w (the angular frequency m m o m m o   d 1 E detuning between the free-running SL laser and the master signals ML ), f (t) is the SL m o m  (t)=  G N (t) +  sin( ) (2)   o N m phase, a is the linewidth enhancement factor and N is transparent carrier density. J is the dt 2 E (t)   injected current density and t is the lifetime for spontaneous emission and non-radiative  recombination. The injection strength K , can be defined as the ratio of the injected field (E ) to the free-running SL field (E ), which is given by E = t (J N /t ), where m os os p s th t is the photon lifetime. Note that throughout our simulation the injection level of the d N (t) N (t)= J − − G (N (t)− N ) E (t) comb peaks is kept equal. In terms of the locking map, the map is drawn considering N o o (3) dt  the detuning and injection level of the central peak (i.e., f ). We numerically perform full 3 s integration for the rate Equations (1)–(3) using Runge–Kutta method. The theoretical power spectra were obtained through fast Fourier transform (FFT). The dominant peak is then recorded to determine the dynamic of the system (locking, pulling, pushing, etc.). The where Eo (t) is the electric field of the SL, GN is the material gain coefficient, ΔN(t) is the parameters used in our simulation are obtained through experimental characterization of the SL [22], and are shown in Table 1. population inversion (N–Nth) where N is the carrier density and Nth is its value at thresh- old, η is the coupling coefficient, m represents the order of the frequency in the comb (m = 1, 2, 3, 4, 5), , where (the angular frequency detuning  = t− (t)  = − m m o m m o between the free-running SL laser and the master signals MLm), ϕ (t) is the SL phase, α is the linewidth enhancement factor and No is transparent carrier density. J is the injected current density and τs is the lifetime for spontaneous emission and non-radiative recom- bination. The injection strength Km, can be defined as the ratio of the injected field (Em) to the free-running SL field (Eos), which is given by , where τp is the pho- E =  (J − N / ) os p th s ton lifetime. Note that throughout our simulation the injection level of the comb peaks is kept equal. In terms of the locking map, the map is drawn considering the detuning and injection level of the central peak (i.e., f3). We numerically perform full integration for the rate Equations (1)–(3) using Runge–Kutta method. The theoretical power spectra were ob- tained through fast Fourier transform (FFT). The dominant peak is then recorded to de- termine the dynamic of the system (locking, pulling, pushing, etc.). The parameters used in our simulation are obtained through experimental characterization of the SL [22], and are shown in Table 1. Photonics 2022, 9, x FOR PEER REVIEW 3 of 9 Photonics 2022, 9, 886 3 of 8 Table 1. Parameters used in our simulation. Parameter Symbol Value Table 1. Parameters used in our simulation. Wavelength λ 1556.6 nm Parameter Symbol V −12 alue 3 −1 Differential Gain GN 1.4 × 10 m s Wavelength  1556.6 nm Carrier lifetime τs 0.43 ns 12 3 1 Differential Gain G 1.4  10 m s Photon lifetime τp 1.8 ps Carrier lifetime t 0.43 ns 10 −1 Coupling rate η 9 × 10 s Photon lifetime t 1.8 ps 24 −3 Transparency carrier density No 1.1 × 10 m 10 1 Coupling rate h 9  10 s 24 −3 24 3 Threshold carrier density Nth 1.5 × 10 m Transparency carrier density N 1.1  10 m 24 3 Threshold carrier density N Normalized injection current I/Ith 1.5  2 10 m th Normalized injection current I/I 2 th 3. Results and Discussion 3. Results and Discussion In order to study the dynamics of the frequency shift, we first draw the locking map In order to study the dynamics of the frequency shift, we first draw the locking map of the system. The master frequency comb MLm (with the spacing ∆ = 0.2 GHz) is injected of the system. The master frequency comb ML (with the spacing D = 0.2 GHz) is injected inside the cavity of the SL. Its central peak (f3) is then swapped from −20 GHz to +20 GHz inside the cavity of the SL. Its central peak (f ) is then swapped from 20 GHz to +20 GHz at each injection level (from 0 to 0.7) to draw the map shown in Figure 2. We record the at each injection level (from 0 to 0.7) to draw the map shown in Figure 2. We record the SL peak position at each point so that the map shows the frequency shift all over the map. SL peak position at each point so that the map shows the frequency shift all over the map. The inset on the right shows the general characteristics of the locking map. The map looks The inset on the right shows the general characteristics of the locking map. The map looks like a typical injection locking map but it appears wider than the case in a single injection like a typical injection locking map but it appears wider than the case in a single injection and that is obviously because the injected signal consists of five peaks. Inside the locking and that is obviously because the injected signal consists of five peaks. Inside the locking region (marked locked in the inset), the SL is unstably locked to the comb signals. The region (marked locked in the inset), the SL is unstably locked to the comb signals. The stability condition is that the side peak, including the relaxation oscillation frequency stability condition is that the side peak, including the relaxation oscillation frequency (ROF), (ROF), should be lower than −20 dB [23]. This condition is not fulfilled in this case and the should be lower than 20 dB [23]. This condition is not fulfilled in this case and the map map showed only the unstable locking region. Therefore, the frequency shifts of the SL showed only the unstable locking region. Therefore, the frequency shifts of the SL (inside (inside the locking region) have the same frequency detuning value, meaning that the SL the locking region) have the same frequency detuning value, meaning that the SL is locked is locked to the ML. to the ML. Figure 2. The locking map of the SL under the injection of a frequency comb with ∆ = 0.2 GHz. The Figure 2. The locking map of the SL under the injection of a frequency comb with D = 0.2 GHz. The inset shows the general characteristics of the map. The labels (a-d) indicate the operation points at inset shows the general characteristics of the map. The labels (a–d) indicate the operation points at which the stability maps in Figure 4 are taken. which the stability maps in Figure 4 are taken. It can also be seen that the lower boundary of the locking bandwidth has a clear in- It can also be seen that the lower boundary of the locking bandwidth has a clear termittency in the locking shown by the small locking islands outside the band. This re- intermittency in the locking shown by the small locking islands outside the band. This gion was previously reported as having chaotic behavior in the case of a single optical region was previously reported as having chaotic behavior in the case of a single optical injection [15]. In terms of the frequency shifts, we observe frequency pushing outside the injection [15]. In terms of the frequency shifts, we observe frequency pushing outside the locking bandwidth. That is to say, the SL shift is positive when the comb is injected in the locking bandwidth. That is to say, the SL shift is positive when the comb is injected in the negative negative detuning detuning side side (outside (outsidethe the locking locking bandwidth) bandwidth) and andvice viceversa. versa. In In this this rregion, egion, we we have previously reported the frequency-pulling effect that leads to the secondary locking region (SLR) in semiconductor lasers under single and dual optical injections [24]. This Photonics 2022, 9, x FOR PEER REVIEW 4 of 9 Photonics 2022, 9, 886 4 of 8 have previously reported the frequency-pulling effect that leads to the secondary locking region (SLR) in semiconductor lasers under single and dual optical injections [24]. This behavior was attributed to the carrier density dynamics [22]. Frequency pushing was also behavior was attributed to the carrier density dynamics [22]. Frequency pushing was also reported before for single [25] and dual [26] optical injections. This pushing was found in reported before for single [25] and dual [26] optical injections. This pushing was found in the low injection region and was also attributed to the variation in carriers’ density. We the low injection region and was also attributed to the variation in carriers’ density. We shall elaborate on this pushing later on. To show the locking and pushing described be- shall elaborate on this pushing later on. To show the locking and pushing described before, fore, we recorded the power spectra of the SL at four different points (a, b, c, and d) shown we recorded the power spectra of the SL at four different points (a, b, c, and d) shown in in Figure 2. The corresponding spectra are shown in Figure 3. Figure 2. The corresponding spectra are shown in Figure 3. Figure 3. Power spectra of the system at the points marked in Figure 2. The insets are magnifica- Figure 3. Power spectra of the system at the points marked in Figure 2. The insets are magnification tion part of the spectra as indicated by the arrows. part of the spectra as indicated by the arrows. In Figure 3a, the comb is injected outside the locking region (at label a in Figure 2, ∆f In Figure 3a, the comb is injected outside the locking region (at label a in Figure 2, = −13 GHz and K = 0.1). The SL is not locked to the comb but rather pushed to the positive Df = 13 GHz and K = 0.1). The SL is not locked to the comb but rather pushed to the detuning side as shown in the inset. However, due to the gain inside the cavity, the comb positive detuning side as shown in the inset. However, due to the gain inside the cavity, number and power largely altered according to their position relative to the SL peak. Fig- the comb number and power largely altered according to their position relative to the SL ure 3b shows the spectra when the comb is injected inside the locking bandwidth (at label peak. Figure 3b shows the spectra when the comb is injected inside the locking bandwidth b in Figure 2, ∆f = −6 GHz and K = 0.5). The SL is locked to the comb with the generation (at label b in Figure 2, Df = 6 GHz and K = 0.5). The SL is locked to the comb with of many small combs (four-wave mixing FWM) around the SL peaks with the same spac- the generation of many small combs (four-wave mixing FWM) around the SL peaks with ing value. The same behavior is observed when the comb is injected in the positive detun- the same spacing value. The same behavior is observed when the comb is injected in the ing side (at label c in Figure 2, ∆f = 10 GHz and K = 0.7). The spectra of such behavior are positive detuning side (at label c in Figure 2, Df = 10 GHz and K = 0.7). The spectra of such shown in Figure 3c. Finally, when the comb is injected outside the locking region in the behavior are shown in Figure 3c. Finally, when the comb is injected outside the locking positive detuning side (at label d in Figure 2, ∆f = 15 GHz and K = 0.1), the SL is pushed region in the positive detuning side (at label d in Figure 2, Df = 15 GHz and K = 0.1), the SL is towards the negative detuning side as shown in Figure 3d. Note here that the cavity mode pushed towards the negative detuning side as shown in Figure 3d. Note here that the cavity spaci mode ng spacing in our mo in our del is model aboutis 0.3 about nm acc 0.3ording nm accor to o ding ur char to act our eri characterization zation [22]. Since [22 our ]. Since laser is our ass laser umed is tassumed o be a sing to le be -ma ode single-mode DFB laser DFB and is laser driven and far is driven above tfar he tabove hresho the ld (se thr e eshold Table (see Table 1), the effect of mode spacing can be neglected. This spacing is believed to play 1), the effect of mode spacing can be neglected. This spacing is believed to play a major role a major in the role dyin namics the dynamics if the SL is if driven the SL n is ea driven r the thr near eshold the wher threshold e the Amp wherlified e the Amplified Spontane- ous Spontaneous Emission Emission can be not can iced. be noticed. To see the effect of the comb on the locking map, we generate the locking map for To see the effect of the comb on the locking map, we generate the locking map for different comb spacing (D = 0.2, 0.4, 0.6 and 0.8 GHz), as shown in Figure 4. As can different comb spacing (∆ = 0.2, 0.4, 0.6 and 0.8 GHz), as shown in Figure 4. As can be seen be seen in the figure, the main effect of the comb spacing on the map appears on the in the figure, the main effect of the comb spacing on the map appears on the boundaries boundaries of the map, especially on the negative detuning side. As the spacing increases, of the map, especially on the negative detuning side. As the spacing increases, the inter- the intermittency at the boundary of the map is enhanced. At higher spacing (0.6 and mittency at the boundary of the map is enhanced. At higher spacing (0.6 and 0.8 GHz), 0.8 GHz), this intermittency appears inside the locking bandwidth in the negative detuning this intermittency appears inside the locking bandwidth in the negative detuning side and side and at high-frequency detuning, as shown in Figure 4c,d. There are many spots at at high-frequency detuning, as shown in Figure 4c,d. There are many spots at which the which the SL is not locked to the comb but pushed (the green spot inside the locking SL is not locked to the comb but pushed (the green spot inside the locking bandwidth). bandwidth). This clearly indicates that the comb spacing has a crucial effect on the locking map. This can be attributed to the fact that higher spacing provides a better opportunity Photonics 2022, 9, x FOR PEER REVIEW 5 of 9 Photonics 2022, 9, 886 5 of 8 This clearly indicates that the comb spacing has a crucial effect on the locking map. This can be attributed to the fact that higher spacing provides a better opportunity for the FWM and other nonlinear dynamics to occur in the negative detuning side, where chaos is re- for the FWM and other nonlinear dynamics to occur in the negative detuning side, where ported in the single optical injection case [15]. chaos is reported in the single optical injection case [15]. Figure 4. Locking map of the SL under the injection of the comb with different comb spacings: (a) ∆ Figure 4. Locking map of the SL under the injection of the comb with different comb spacings: = 0.2 GHz, (b) ∆ = 0.4 GHz, (c) ∆ = 0.6 GHz and (d) ∆ = 0.8 GHz. The vertical and horizontal white (a) D = 0.2 GHz, (b) D = 0.4 GHz, (c) D = 0.6 GHz and (d) D = 0.8 GHz. The vertical and horizontal lines in (a) indicate the lines at which the frequency shift in Figures 5 and 6 are extracted. white lines in (a) indicate the lines at which the frequency shift in Figures 5 and 6 are extracted. Now, we investigate the frequency shifts extracted from the maps above. We first scan Now, we investigate the frequency shifts extracted from the maps above. We first the shift as a function of frequency detuning at constant injection levels (0.1, 0.3, 0.5 and 0.7), scan the shift as a function of frequency detuning at constant injection levels (0.1, 0.3, 0.5 as shown by the vertical white lines in Figure 4a. The extracted frequency shifts are shown and 0.7), as shown by the vertical white lines in Figure 4a. The extracted frequency shifts in Figure 5. As expected, the frequency shift is more evident with a low injection level (at are shown in Figure 5. As expected, the frequency shift is more evident with a low injec- 0.1, i.e., Figure 5a). In this case, the SL peak experiences first frequency pushing when the tion level (at 0.1, i.e., Figure 5a). In this case, the SL peak experiences first frequency push- comb is injected far away from the free-running SL. As the injected comb approaches the ing when the comb is injected far away from the free-running SL. As the injected comb locking bandwidth, the SL peak is pulled toward the injected signals. These behaviors are attributed approache tos the the dynamics locking of bandwi carrierdt density h, the inside SL pe the akcavity is pul of lethe d to SL. ward the injected signals. As the injection level increases (to 0.3, i.e., Figure 5b), the frequency shift is clearly These behaviors are attributed to the dynamics of carrier density inside the cavity of the damped due to the carriers’ suppression. For higher injection levels (0.5 and 0.7, i.e., SL. Figure 5c,d), the maps are covered by the locking region so that the frequency shift is always equal to the frequency detuning except for a few points here and there, especially at higher comb spacings. We have also scanned the frequency shift horizontally in the maps as a function of the injection levels at constant frequency detunings (the horizontal white lines in Figure 4a, at Df = 15, 7, 0, 7, 15 GHz). The extracted shifts are shown in Figure 6. At Df = 15 GHz, both frequency pushing and pulling are observed, as shown in Figure 6a. However, at Df = +15 GHz, we only observed frequency pushing, as shown in Figure 6b. This is probably due to the asymmetric characteristics originating from the value of the linewidth enhancement factor (LEF) as reported before in the case of a single optical injection [27]. In the case of Df = 7 and 7 GHz shown in Figure 6c,d, we only observe frequency pulling as the locking bandwidth is very close. Finally, when the comb is injected at the same frequency as the free-running SL (Df = 0 GHz, i.e., Figure 6e), we observe that the SL is slightly shifted towards the positive detuning side at a low injection level (<0.2), and for higher than that, the SL peak is shifted towards the negative detuning side. This is Photonics 2022, 9, 886 6 of 8 Photonics 2022, 9, x FOR PEER REVIEW 6 of 9 Photonics 2022, 9, x FOR PEER REVIEW 6 of 9 again due to the change in the refractive index caused by the change in carriers’ density and described by the LEF. Figure 5. Frequency shifts vs. frequency detuning at different injection levels for the four cases of the frequency comb: (a) K = 0.1, (b) K = 0.3, (c) K = 0.5 and (d) K = 0.7. As the injection level increases (to 0.3, i.e., Figure 5b), the frequency shift is clearly damped due to the carriers’ suppression. For higher injection levels (0.5 and 0.7, i.e., Fig- ure 5c,d), the maps are covered by the locking region so that the frequency shift is always equal to the frequency detuning except for a few points here and there, especially at higher Figure Figure 5. Fr 5.equency Frequen shifts cy vs. shif frts equency vs. frequ detuning ency atde dif tfer un ent ing injection at different levels for injec theti four on cases levelof s f the or the four cases of comb spacings. frequency comb: (a) K = 0.1, (b) K = 0.3, (c) K = 0.5 and (d) K = 0.7. the frequency comb: (a) K = 0.1, (b) K = 0.3, (c) K = 0.5 and (d) K = 0.7. As the injection level increases (to 0.3, i.e., Figure 5b), the frequency shift is clearly damped due to the carriers’ suppression. For higher injection levels (0.5 and 0.7, i.e., Fig- ure 5c,d), the maps are covered by the locking region so that the frequency shift is always equal to the frequency detuning except for a few points here and there, especially at higher comb spacings. Figure 6. Frequency shifts vs. injection level at different frequency detunings for the four cases of Figure 6. Frequency shifts vs. injection level at different frequency detunings for the four cases of the the frequency comb spacing: (a) ∆f = −15 GHz, (b) ∆f = 15 GHz, (c) ∆f = −7 GHz, (d) ∆f = 7 GHz and frequency comb spacing: (a) Df = 15 GHz, (b) Df = 15 GHz, (c) Df = 7 GHz, (d) Df = 7 GHz and (e) ∆f = 0 GHz. (e) Df = 0 GHz. Figure 6. Frequency shifts vs. injection level at different frequency detunings for the four cases of the frequency comb spacing: (a) ∆f = −15 GHz, (b) ∆f = 15 GHz, (c) ∆f = −7 GHz, (d) ∆f = 7 GHz and (e) ∆f = 0 GHz. Photonics 2022, 9, x FOR PEER REVIEW 7 of 9 We have also scanned the frequency shift horizontally in the maps as a function of the injection levels at constant frequency detunings (the horizontal white lines in Figure 4a, at ∆f = 15, 7, 0, −7, −15 GHz). The extracted shifts are shown in Figure 6. At ∆f = −15 GHz, both frequency pushing and pulling are observed, as shown in Figure 6a. However, at ∆f = +15 GHz, we only observed frequency pushing, as shown in Figure 6b. This is probably due to the asymmetric characteristics originating from the value of the linewidth enhancement factor (LEF) as reported before in the case of a single optical injection [27]. In the case of ∆f = −7 and 7 GHz shown in Figure 6c,d, we only observe frequency pulling as the locking bandwidth is very close. Finally, when the comb is injected at the same frequency as the free-running SL (∆f = 0 GHz, i.e., Figure 6e), we observe that the SL is slightly shifted towards the positive detuning side at a low injection level (<0.2), and for higher than that, the SL peak is shifted towards the negative detuning side. This is again Photonics 2022, 9, 886 7 of 8 due to the change in the refractive index caused by the change in carriers’ density and described by the LEF. Finally, we concentrate on the frequency pushing occurring at a low injection level Finally, we concentrate on the frequency pushing occurring at a low injection level (K = 0.1) when the comb is injected very far from the free-running SL (i.e., at ∆f > 10 GHz (K = 0.1) when the comb is injected very far from the free-running SL (i.e., at Df > 10 GHz and ∆f > −10 GHz), as shown in Figure 7. In other words, this figure is a magnification of and Df > 10 GHz), as shown in Figure 7. In other words, this figure is a magnification of parts of Figure 5a. It can be seen that the frequency pushing in general is enhanced as the parts of Figure 5a. It can be seen that the frequency pushing in general is enhanced as the spacing of the comb increases. The spread of the data is due to the fact that the dominant spacing of the comb increases. The spread of the data is due to the fact that the dominant peak in the comb is not always the same and changes as the comb is detuned closer or peak in the comb is not always the same and changes as the comb is detuned closer or further from the free-running SL. This random FWM can also cause the frequency pulling further from the free-running SL. This random FWM can also cause the frequency pulling shown in some points in the figure (red and green). shown in some points in the figure (red and green). Figure 7. Frequency shifts vs. frequency detuning at K = 0.1 for the four cases of the frequency comb. Figure 7. Frequency shifts vs. frequency detuning at K = 0.1 for the four cases of the frequency comb. 4. Conclusions 4. Conclusions We have shown theoretically that the frequency comb spacing has a crucial role in the We have shown theoretically that the frequency comb spacing has a crucial role in locking map of a semiconductor laser under the injection of the frequency comb. As the the locking map of a semiconductor laser under the injection of the frequency comb. As comb spacing increases, the boundaries of the locking area become intermittent, especially the comb spacing increases, the boundaries of the locking area become intermittent, espe- the lower boundary at the negative detuning side. This spacing of the comb has been found cially the lower boundary at the negative detuning side. This spacing of the comb has been to affect the frequency shift of the free-running SL outside the locking bandwidth. These found to affect the frequency shift of the free-running SL outside the locking bandwidth. results are believed to contribute to a better understanding of the frequency comb injection These results are believed to contribute to a better understanding of the frequency comb as they can be utilized in many modern communication applications. Future work should injection as they can be utilized in many modern communication applications. Future include the effect of changing the number of the comb lines, varying the power of the lines, work should include the effect of changing the number of the comb lines, varying the the dynamics of carriers’ density and the experimental validation of these results. power of the lines, the dynamics of carriers’ density and the experimental validation of these results. Funding: This research was funded by Taif University, Researchers Supporting Project number (TURSP-2020/25), Taif, Saudi Arabia. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: This work was supported by Taif University, Researchers Supporting Project number (TURSP-2020/25), Taif, Saudi Arabia. Conflicts of Interest: The author declares no conflict of interest. References 1. Ohtsu, M.; Nakagawa, K.; Kourogi, M.; Wang, W. Frequency control of semiconductor lasers. J. Appl. Phys. 1993, 73, R1–R17. [CrossRef] 2. Sooudi, E.; Sygletos, S.; Ellis, A.D.; Huyet, G.; McInerney, J.G.; Lelarge, F.; Merghem, K.; Rosales, R.; Martinez, A.; Ramdane, A.; et al. Optical Frequency Comb Generation Using Dual-Mode Injection-Locking of Quantum-Dash Mode-Locked Lasers: Properties and Applications. IEEE J. Quantum Electron. 2012, 48, 1327–1338. [CrossRef] Photonics 2022, 9, 886 8 of 8 3. Rosado, A.; Pérez-Serrano, A.; Tijero JM, G.; Valle, Á.; Pesquera, L.; Esquivias, I. Experimental study of optical frequency comb generation in gain-switched semiconductor lasers. Opt. Laser Technol. 2018, 108, 542–550. [CrossRef] 4. Rosado, A.; Perez-Serrano, A.; Tijero, J.M.G.; Gutierrez, A.V.; Pesquera, L.; Esquivias, I. Numerical and Experimental Analysis of Optical Frequency Comb Generation in Gain-Switched Semiconductor Lasers. IEEE J. Quantum Electron. 2019, 55, 2001012. [CrossRef] 5. Zhao, B.-B.; Kovanis, V.; Wang, C. Tunable Frequency Comb Generation Using Quantum Cascade Lasers Subject to Optical Injection. IEEE J. Sel. Top. Quantum Electron. 2019, 25, 1900207. [CrossRef] 6. Ren, H.; Fan, L.; Liu, N.; Wu, Z.; Xia, G. Generation of Broadband Optical Frequency Comb Based on a Gain-Switching 1550 nm Vertical-Cavity Surface-Emitting Laser under Optical Injection. Photonics 2020, 7, 95. [CrossRef] 7. Moon, H.S.; Kim, E.B.; Park, S.E.; Park, C.Y. Selection and amplification of modes of an optical frequency comb using a femtosecond laser injection-locking technique. Appl. Phys. Lett. 2006, 89, 181110. [CrossRef] 8. Ramond, T.M.; Hollberg, L.; Juodawlkis, P.W.; Calawa, S.D. Low-noise optical injection locking of a resonant tunneling diode to a stable optical frequency comb. Appl. Phys. Lett. 2007, 90, 171124. [CrossRef] 9. Chan, S.-C.; Xia, G.-Q.; Liu, J.-M. Optical generation of a precise microwave frequency comb by harmonic frequency locking. Opt. Lett. 2007, 32, 1917. [CrossRef] 10. Fukushima, S.; Silva, C.F.C.; Muramoto, Y.; Seeds, A.J. Optoelectronic millimeter-wave synthesis using an optical frequency comb generator, optically injection locked lasers and a unitraveling-carrier photodiode. J. Light. Technol. 2003, 21, 3043–3051. [CrossRef] 11. Zhang, H.; Xu, M.; Zhang, J.; Jia, Z.; Campos, L.A.; Knittle, C. Highly Efficient Full-Duplex Coherent Optical System Enabled by Combined Use of Optical Injection Locking and Frequency Comb. J. Light. Technol. 2021, 39, 1271–1277. [CrossRef] 12. Ohtsu, M. Frequency stabilization in semiconductor lasers. Opt. Quantum Electron. 1988, 20, 283–300. [CrossRef] 13. Manamanni, K.; Steshchenko, T.; Wiotte, F.; Ramdane, A.C.; Sahni, M.-O.; Roncin, V.; Du-Burck, F. Frequency Stability Transfer in Passive Mode-Locked Quantum-Dash Laser Diode Using Optical Injection Locking. IEEE J. Quantum Electron. 2022, 58, 1300409. [CrossRef] 14. Liu, Z.; Slavik, R. Optical Injection Locking: From Principle to Applications. J. Light. Technol. 2020, 38, 43–59. [CrossRef] 15. Al-Hosiny, N.M.; Henning, I.D.; Adams, M.J. Tailoring enhanced chaos in optically injected semiconductor lasers. Opt. Commun. 2007, 269, 166–173. [CrossRef] 16. Al-Hosiny, N.M. 3D Injection-locking maps of semiconductor laser under multiple optical injections. J. Opt. 2021, 50, 629–636. [CrossRef] 17. Tistomo, A.S.; Gee, S. Laser frequency fixation by multimode optical injection locking. Opt. Express 2011, 19, 1081. [CrossRef] 18. Gavrielides, A. Comb Injection and Sidebands Suppression. IEEE J. Quantum Electron. 2014, 50, 364–371. [CrossRef] 19. Ó Duill, S.P.; Anandarajah, P.M.; Smyth, F.; Barry, L.P. Injection-locking criteria for simultaneously locking single-mode lasers to optical frequency combs from gain-switched lasers. In Physics and Simulation of Optoelectronic Devices XXV; Witzigmann, B., Osinski, ´ M., Arakawa, Y., Eds.; SPIE: Bellingham, WA, USA, 2017; Volume 10098, pp. 51–59. [CrossRef] 20. Shortiss, K.; Lingnau, B.; Dubois, F.; Kelleher, B.; Peters, F.H. Harmonic frequency locking and tuning of comb frequency spacing through optical injection. Opt. Express 2019, 27, 36976. [CrossRef] 21. Doumbia, Y.; Malica, T.; Wolfersberger, D.; Panajotov, K.; Sciamanna, M. Nonlinear dynamics of a laser diode with an injection of an optical frequency comb. Optics Express 2020, 28, 30379. [CrossRef] 22. Al-Hosiny, N.M.; Henning, I.D.; Adams, M.J. Correlation of Electron Density Changes with Optical Frequency Shifts in Optically Injected Semiconductor Lasers. IEEE J. Quantum Electron. 2006, 42, 570–580. [CrossRef] 23. Hui, R.; D’Ottavi, A.; Mecozzi, A.; Spano, P. Injection locking in distributed feedback semiconductor lasers. IEEE J. Quantum Electron. 1991, 27, 1688–1695. [CrossRef] 24. Al-Hosiny, N.; Henning, I.D.; Adams, M.J. Secondary locking regions in laser diode subject to optical injection from two lasers. Electron. Lett. 2006, 42, 759. [CrossRef] 25. Simpson, T.B.; Liu, J.M.; Huang, K.F.; Tai, K. Nonlinear dynamics induced by external optical injection in semiconductor lasers. Quantum Semiclassical Opt. J. Eur. Opt. Soc. Part B 1997, 9, 765–784. [CrossRef] 26. Liao, Y.-H.; Liu, J.-M.; Lin, F.-Y. Dynamical Characteristics of a Dual-Beam Optically Injected Semiconductor Laser. IEEE J. Sel. Top. Quantum Electron. 2013, 19, 1500606. [CrossRef] 27. Goldberg, L.; Taylor, H.F.; Weller, J.F. Locking bandwidth asymmetry in injection-locked GaAlAs lasers. Electron. Lett. 1982, 18, 986. [CrossRef] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute

Dynamics of the Frequency Shifts in Semiconductor Lasers under the Injection of a Frequency Comb

Al-Hosiny, Najm M.
Photonics , Volume 9 (12) – Nov 22, 2022
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Abstract

hv photonics Article Dynamics of the Frequency Shifts in Semiconductor Lasers under the Injection of a Frequency Comb Najm M. Al-Hosiny Department of Physics, College of Science, Taif University, P.O. Box 1109, Taif 21944, Saudi Arabia; najm@tu.edu.sa Abstract: We have numerically investigated the dynamics of frequency shifts in semiconductor lasers under the injection of a frequency comb. We have studied the effect of comb spacing on the locking bandwidth. Frequency comb spacing was found to play an important role in the boundaries of the locking bandwidth as well as in the frequency shift of the SL peak. Keywords: optical injection; frequency comb; frequency pulling; frequency pushing 1. Introduction Frequency combs have been used in semiconductor lasers since 1992 [1]. In terms of optically injected semiconductor lasers, frequency combs have been studied in different aspects, including comb generation [2–6], selective amplification of the comb [7], producing low-noise microwave signals [8,9], optoelectronic millimeter-wave synthesis [10] and full- duplex coherent optical system [11]. As a key component in communication systems, the investigation of the frequency and its stabilization in semiconductor lasers has been under investigation since 1988 [12,13]. One of the major techniques to stabilize the frequency in semiconductor lasers is optical injection locking [14]. We have previously reported the dynamics of semiconductor lasers under dual optical injection [15,16]. The additional signal was found to enhance the chaos and control the stability map. Arfan et al. [17] have shown Citation: Al-Hosiny, N.M. Dynamics that the injected laser can lock to equipartition points in frequencies between two adjacent of the Frequency Shifts in modes or each individual frequency of the master laser. In 2014, analytical and numerical Semiconductor Lasers under the calculations were performed for a semiconductor laser under the injection of a frequency Injection of a Frequency Comb. Photonics 2022, 9, 886. https:// comb [18]. The study found that the locking depends on the separation of the comb and doi.org/10.3390/photonics9120886 the identification of three major regions of unique amplifications. Later on, another study identified important criteria to maximize the frequency locking range of the semiconductor Received: 24 October 2022 laser under the injection of a frequency comb [19]. It was also found that the slave laser Accepted: 18 November 2022 under the injection of a frequency comb undergoes two different mechanisms, in which Published: 22 November 2022 the output of the frequency comb separation is decreased [20]. The study also found that Publisher’s Note: MDPI stays neutral the relaxation oscillations of the injected laser can lock to the harmonics of the injected with regard to jurisdictional claims in optical comb, generating extra tones in the optical comb around the slave laser ’s frequency. published maps and institutional affil- Recently, a comprehensive theoretical and experimental study investigated the variety iations. of nonlinear dynamics exhibited by a single-frequency semiconductor laser subjected to optical injection from a frequency comb [21]. By varying the comb parameters (number of lines and comb spacing), a rich variety of nonlinear dynamics was identified including wave mixing and irregular chaotic pulsing. In this study, we numerically investigate the Copyright: © 2022 by the author. effect of frequency comb spacing on the locking bandwidth and on the frequency shift of Licensee MDPI, Basel, Switzerland. the slave laser. The locking maps for different comb spacings are drawn and discussed. This article is an open access article The frequency shifts are extracted from the maps and analyzed. distributed under the terms and conditions of the Creative Commons 2. Materials and Methods Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ Our model is based on Lang’s approach [22], where the system is described by three 4.0/). differential equations for electric field amplitude, phase and carrier density. Before we Photonics 2022, 9, 886. https://doi.org/10.3390/photonics9120886 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 9 2. Materials and Methods Our model is based on Lang’s approach [22], where the system is described by three differential equations for electric field amplitude, phase and carrier density. Before we elaborate on the model, let us introduce the frequency comb, which will be injected inside Photonics 2022, 9, 886 2 of 8 the cavity of the slave laser (SL). Our SL is assumed to be a single-mode DFB laser. The frequency comb consists of five peaks as shown in Figure 1. The central peak is f3 and the elaborate on the model, let us introduce the frequency comb, which will be injected inside spacing between the peaks is ∆, which is constant in each case as will be shown later. The the cavity of the slave laser (SL). Our SL is assumed to be a single-mode DFB laser. The frequency comb consists of five peaks as shown in Figure 1. The central peak is f and the intensity of the peaks (and hence the injection level) is equal. spacing between the peaks is D, which is constant in each case as will be shown later. The intensity of the peaks (and hence the injection level) is equal. Figure 1. The frequency comb used in our simulation. Figure 1. The frequency comb used in our simulation. The rate equation for the SL for electric field amplitude, phase and carrier density, can be expressed respectively, as follows: d 1 The rate equation for the SL for electric field amplitude, phase and carrier density, E (t) = G DN(t)E (t) + h E cos(Df ) (1) o N o å m m dt 2 can be expressed respectiv  ely, as follow  s: d 1 E f (t) = aG DN(t) + h sin(Df ) (2) o N å m dt 2 E (t) d N(t) N(t) = J Gd(N(t) N )E1(t) (3) N o dt t s E (t)= G N (t) E (t)+ E cos( ) (1) o N o m m dt 2 where E (t) is the electric field of the SL, G is the material gain coefficient, DN(t) is o N the population inversion (N–N ) where N is the carrier density and N is its value at th th threshold, h is the coupling coefficient, m represents the order of the frequency in the comb (m = 1, 2, 3, 4, 5), Df = Dw t f (t), where Dw = w w (the angular frequency m m o m m o   d 1 E detuning between the free-running SL laser and the master signals ML ), f (t) is the SL m o m  (t)=  G N (t) +  sin( ) (2)   o N m phase, a is the linewidth enhancement factor and N is transparent carrier density. J is the dt 2 E (t)   injected current density and t is the lifetime for spontaneous emission and non-radiative  recombination. The injection strength K , can be defined as the ratio of the injected field (E ) to the free-running SL field (E ), which is given by E = t (J N /t ), where m os os p s th t is the photon lifetime. Note that throughout our simulation the injection level of the d N (t) N (t)= J − − G (N (t)− N ) E (t) comb peaks is kept equal. In terms of the locking map, the map is drawn considering N o o (3) dt  the detuning and injection level of the central peak (i.e., f ). We numerically perform full 3 s integration for the rate Equations (1)–(3) using Runge–Kutta method. The theoretical power spectra were obtained through fast Fourier transform (FFT). The dominant peak is then recorded to determine the dynamic of the system (locking, pulling, pushing, etc.). The where Eo (t) is the electric field of the SL, GN is the material gain coefficient, ΔN(t) is the parameters used in our simulation are obtained through experimental characterization of the SL [22], and are shown in Table 1. population inversion (N–Nth) where N is the carrier density and Nth is its value at thresh- old, η is the coupling coefficient, m represents the order of the frequency in the comb (m = 1, 2, 3, 4, 5), , where (the angular frequency detuning  = t− (t)  = − m m o m m o between the free-running SL laser and the master signals MLm), ϕ (t) is the SL phase, α is the linewidth enhancement factor and No is transparent carrier density. J is the injected current density and τs is the lifetime for spontaneous emission and non-radiative recom- bination. The injection strength Km, can be defined as the ratio of the injected field (Em) to the free-running SL field (Eos), which is given by , where τp is the pho- E =  (J − N / ) os p th s ton lifetime. Note that throughout our simulation the injection level of the comb peaks is kept equal. In terms of the locking map, the map is drawn considering the detuning and injection level of the central peak (i.e., f3). We numerically perform full integration for the rate Equations (1)–(3) using Runge–Kutta method. The theoretical power spectra were ob- tained through fast Fourier transform (FFT). The dominant peak is then recorded to de- termine the dynamic of the system (locking, pulling, pushing, etc.). The parameters used in our simulation are obtained through experimental characterization of the SL [22], and are shown in Table 1. Photonics 2022, 9, x FOR PEER REVIEW 3 of 9 Photonics 2022, 9, 886 3 of 8 Table 1. Parameters used in our simulation. Parameter Symbol Value Table 1. Parameters used in our simulation. Wavelength λ 1556.6 nm Parameter Symbol V −12 alue 3 −1 Differential Gain GN 1.4 × 10 m s Wavelength  1556.6 nm Carrier lifetime τs 0.43 ns 12 3 1 Differential Gain G 1.4  10 m s Photon lifetime τp 1.8 ps Carrier lifetime t 0.43 ns 10 −1 Coupling rate η 9 × 10 s Photon lifetime t 1.8 ps 24 −3 Transparency carrier density No 1.1 × 10 m 10 1 Coupling rate h 9  10 s 24 −3 24 3 Threshold carrier density Nth 1.5 × 10 m Transparency carrier density N 1.1  10 m 24 3 Threshold carrier density N Normalized injection current I/Ith 1.5  2 10 m th Normalized injection current I/I 2 th 3. Results and Discussion 3. Results and Discussion In order to study the dynamics of the frequency shift, we first draw the locking map In order to study the dynamics of the frequency shift, we first draw the locking map of the system. The master frequency comb MLm (with the spacing ∆ = 0.2 GHz) is injected of the system. The master frequency comb ML (with the spacing D = 0.2 GHz) is injected inside the cavity of the SL. Its central peak (f3) is then swapped from −20 GHz to +20 GHz inside the cavity of the SL. Its central peak (f ) is then swapped from 20 GHz to +20 GHz at each injection level (from 0 to 0.7) to draw the map shown in Figure 2. We record the at each injection level (from 0 to 0.7) to draw the map shown in Figure 2. We record the SL peak position at each point so that the map shows the frequency shift all over the map. SL peak position at each point so that the map shows the frequency shift all over the map. The inset on the right shows the general characteristics of the locking map. The map looks The inset on the right shows the general characteristics of the locking map. The map looks like a typical injection locking map but it appears wider than the case in a single injection like a typical injection locking map but it appears wider than the case in a single injection and that is obviously because the injected signal consists of five peaks. Inside the locking and that is obviously because the injected signal consists of five peaks. Inside the locking region (marked locked in the inset), the SL is unstably locked to the comb signals. The region (marked locked in the inset), the SL is unstably locked to the comb signals. The stability condition is that the side peak, including the relaxation oscillation frequency stability condition is that the side peak, including the relaxation oscillation frequency (ROF), (ROF), should be lower than −20 dB [23]. This condition is not fulfilled in this case and the should be lower than 20 dB [23]. This condition is not fulfilled in this case and the map map showed only the unstable locking region. Therefore, the frequency shifts of the SL showed only the unstable locking region. Therefore, the frequency shifts of the SL (inside (inside the locking region) have the same frequency detuning value, meaning that the SL the locking region) have the same frequency detuning value, meaning that the SL is locked is locked to the ML. to the ML. Figure 2. The locking map of the SL under the injection of a frequency comb with ∆ = 0.2 GHz. The Figure 2. The locking map of the SL under the injection of a frequency comb with D = 0.2 GHz. The inset shows the general characteristics of the map. The labels (a-d) indicate the operation points at inset shows the general characteristics of the map. The labels (a–d) indicate the operation points at which the stability maps in Figure 4 are taken. which the stability maps in Figure 4 are taken. It can also be seen that the lower boundary of the locking bandwidth has a clear in- It can also be seen that the lower boundary of the locking bandwidth has a clear termittency in the locking shown by the small locking islands outside the band. This re- intermittency in the locking shown by the small locking islands outside the band. This gion was previously reported as having chaotic behavior in the case of a single optical region was previously reported as having chaotic behavior in the case of a single optical injection [15]. In terms of the frequency shifts, we observe frequency pushing outside the injection [15]. In terms of the frequency shifts, we observe frequency pushing outside the locking bandwidth. That is to say, the SL shift is positive when the comb is injected in the locking bandwidth. That is to say, the SL shift is positive when the comb is injected in the negative negative detuning detuning side side (outside (outsidethe the locking locking bandwidth) bandwidth) and andvice viceversa. versa. In In this this rregion, egion, we we have previously reported the frequency-pulling effect that leads to the secondary locking region (SLR) in semiconductor lasers under single and dual optical injections [24]. This Photonics 2022, 9, x FOR PEER REVIEW 4 of 9 Photonics 2022, 9, 886 4 of 8 have previously reported the frequency-pulling effect that leads to the secondary locking region (SLR) in semiconductor lasers under single and dual optical injections [24]. This behavior was attributed to the carrier density dynamics [22]. Frequency pushing was also behavior was attributed to the carrier density dynamics [22]. Frequency pushing was also reported before for single [25] and dual [26] optical injections. This pushing was found in reported before for single [25] and dual [26] optical injections. This pushing was found in the low injection region and was also attributed to the variation in carriers’ density. We the low injection region and was also attributed to the variation in carriers’ density. We shall elaborate on this pushing later on. To show the locking and pushing described be- shall elaborate on this pushing later on. To show the locking and pushing described before, fore, we recorded the power spectra of the SL at four different points (a, b, c, and d) shown we recorded the power spectra of the SL at four different points (a, b, c, and d) shown in in Figure 2. The corresponding spectra are shown in Figure 3. Figure 2. The corresponding spectra are shown in Figure 3. Figure 3. Power spectra of the system at the points marked in Figure 2. The insets are magnifica- Figure 3. Power spectra of the system at the points marked in Figure 2. The insets are magnification tion part of the spectra as indicated by the arrows. part of the spectra as indicated by the arrows. In Figure 3a, the comb is injected outside the locking region (at label a in Figure 2, ∆f In Figure 3a, the comb is injected outside the locking region (at label a in Figure 2, = −13 GHz and K = 0.1). The SL is not locked to the comb but rather pushed to the positive Df = 13 GHz and K = 0.1). The SL is not locked to the comb but rather pushed to the detuning side as shown in the inset. However, due to the gain inside the cavity, the comb positive detuning side as shown in the inset. However, due to the gain inside the cavity, number and power largely altered according to their position relative to the SL peak. Fig- the comb number and power largely altered according to their position relative to the SL ure 3b shows the spectra when the comb is injected inside the locking bandwidth (at label peak. Figure 3b shows the spectra when the comb is injected inside the locking bandwidth b in Figure 2, ∆f = −6 GHz and K = 0.5). The SL is locked to the comb with the generation (at label b in Figure 2, Df = 6 GHz and K = 0.5). The SL is locked to the comb with of many small combs (four-wave mixing FWM) around the SL peaks with the same spac- the generation of many small combs (four-wave mixing FWM) around the SL peaks with ing value. The same behavior is observed when the comb is injected in the positive detun- the same spacing value. The same behavior is observed when the comb is injected in the ing side (at label c in Figure 2, ∆f = 10 GHz and K = 0.7). The spectra of such behavior are positive detuning side (at label c in Figure 2, Df = 10 GHz and K = 0.7). The spectra of such shown in Figure 3c. Finally, when the comb is injected outside the locking region in the behavior are shown in Figure 3c. Finally, when the comb is injected outside the locking positive detuning side (at label d in Figure 2, ∆f = 15 GHz and K = 0.1), the SL is pushed region in the positive detuning side (at label d in Figure 2, Df = 15 GHz and K = 0.1), the SL is towards the negative detuning side as shown in Figure 3d. Note here that the cavity mode pushed towards the negative detuning side as shown in Figure 3d. Note here that the cavity spaci mode ng spacing in our mo in our del is model aboutis 0.3 about nm acc 0.3ording nm accor to o ding ur char to act our eri characterization zation [22]. Since [22 our ]. Since laser is our ass laser umed is tassumed o be a sing to le be -ma ode single-mode DFB laser DFB and is laser driven and far is driven above tfar he tabove hresho the ld (se thr e eshold Table (see Table 1), the effect of mode spacing can be neglected. This spacing is believed to play 1), the effect of mode spacing can be neglected. This spacing is believed to play a major role a major in the role dyin namics the dynamics if the SL is if driven the SL n is ea driven r the thr near eshold the wher threshold e the Amp wherlified e the Amplified Spontane- ous Spontaneous Emission Emission can be not can iced. be noticed. To see the effect of the comb on the locking map, we generate the locking map for To see the effect of the comb on the locking map, we generate the locking map for different comb spacing (D = 0.2, 0.4, 0.6 and 0.8 GHz), as shown in Figure 4. As can different comb spacing (∆ = 0.2, 0.4, 0.6 and 0.8 GHz), as shown in Figure 4. As can be seen be seen in the figure, the main effect of the comb spacing on the map appears on the in the figure, the main effect of the comb spacing on the map appears on the boundaries boundaries of the map, especially on the negative detuning side. As the spacing increases, of the map, especially on the negative detuning side. As the spacing increases, the inter- the intermittency at the boundary of the map is enhanced. At higher spacing (0.6 and mittency at the boundary of the map is enhanced. At higher spacing (0.6 and 0.8 GHz), 0.8 GHz), this intermittency appears inside the locking bandwidth in the negative detuning this intermittency appears inside the locking bandwidth in the negative detuning side and side and at high-frequency detuning, as shown in Figure 4c,d. There are many spots at at high-frequency detuning, as shown in Figure 4c,d. There are many spots at which the which the SL is not locked to the comb but pushed (the green spot inside the locking SL is not locked to the comb but pushed (the green spot inside the locking bandwidth). bandwidth). This clearly indicates that the comb spacing has a crucial effect on the locking map. This can be attributed to the fact that higher spacing provides a better opportunity Photonics 2022, 9, x FOR PEER REVIEW 5 of 9 Photonics 2022, 9, 886 5 of 8 This clearly indicates that the comb spacing has a crucial effect on the locking map. This can be attributed to the fact that higher spacing provides a better opportunity for the FWM and other nonlinear dynamics to occur in the negative detuning side, where chaos is re- for the FWM and other nonlinear dynamics to occur in the negative detuning side, where ported in the single optical injection case [15]. chaos is reported in the single optical injection case [15]. Figure 4. Locking map of the SL under the injection of the comb with different comb spacings: (a) ∆ Figure 4. Locking map of the SL under the injection of the comb with different comb spacings: = 0.2 GHz, (b) ∆ = 0.4 GHz, (c) ∆ = 0.6 GHz and (d) ∆ = 0.8 GHz. The vertical and horizontal white (a) D = 0.2 GHz, (b) D = 0.4 GHz, (c) D = 0.6 GHz and (d) D = 0.8 GHz. The vertical and horizontal lines in (a) indicate the lines at which the frequency shift in Figures 5 and 6 are extracted. white lines in (a) indicate the lines at which the frequency shift in Figures 5 and 6 are extracted. Now, we investigate the frequency shifts extracted from the maps above. We first scan Now, we investigate the frequency shifts extracted from the maps above. We first the shift as a function of frequency detuning at constant injection levels (0.1, 0.3, 0.5 and 0.7), scan the shift as a function of frequency detuning at constant injection levels (0.1, 0.3, 0.5 as shown by the vertical white lines in Figure 4a. The extracted frequency shifts are shown and 0.7), as shown by the vertical white lines in Figure 4a. The extracted frequency shifts in Figure 5. As expected, the frequency shift is more evident with a low injection level (at are shown in Figure 5. As expected, the frequency shift is more evident with a low injec- 0.1, i.e., Figure 5a). In this case, the SL peak experiences first frequency pushing when the tion level (at 0.1, i.e., Figure 5a). In this case, the SL peak experiences first frequency push- comb is injected far away from the free-running SL. As the injected comb approaches the ing when the comb is injected far away from the free-running SL. As the injected comb locking bandwidth, the SL peak is pulled toward the injected signals. These behaviors are attributed approache tos the the dynamics locking of bandwi carrierdt density h, the inside SL pe the akcavity is pul of lethe d to SL. ward the injected signals. As the injection level increases (to 0.3, i.e., Figure 5b), the frequency shift is clearly These behaviors are attributed to the dynamics of carrier density inside the cavity of the damped due to the carriers’ suppression. For higher injection levels (0.5 and 0.7, i.e., SL. Figure 5c,d), the maps are covered by the locking region so that the frequency shift is always equal to the frequency detuning except for a few points here and there, especially at higher comb spacings. We have also scanned the frequency shift horizontally in the maps as a function of the injection levels at constant frequency detunings (the horizontal white lines in Figure 4a, at Df = 15, 7, 0, 7, 15 GHz). The extracted shifts are shown in Figure 6. At Df = 15 GHz, both frequency pushing and pulling are observed, as shown in Figure 6a. However, at Df = +15 GHz, we only observed frequency pushing, as shown in Figure 6b. This is probably due to the asymmetric characteristics originating from the value of the linewidth enhancement factor (LEF) as reported before in the case of a single optical injection [27]. In the case of Df = 7 and 7 GHz shown in Figure 6c,d, we only observe frequency pulling as the locking bandwidth is very close. Finally, when the comb is injected at the same frequency as the free-running SL (Df = 0 GHz, i.e., Figure 6e), we observe that the SL is slightly shifted towards the positive detuning side at a low injection level (<0.2), and for higher than that, the SL peak is shifted towards the negative detuning side. This is Photonics 2022, 9, 886 6 of 8 Photonics 2022, 9, x FOR PEER REVIEW 6 of 9 Photonics 2022, 9, x FOR PEER REVIEW 6 of 9 again due to the change in the refractive index caused by the change in carriers’ density and described by the LEF. Figure 5. Frequency shifts vs. frequency detuning at different injection levels for the four cases of the frequency comb: (a) K = 0.1, (b) K = 0.3, (c) K = 0.5 and (d) K = 0.7. As the injection level increases (to 0.3, i.e., Figure 5b), the frequency shift is clearly damped due to the carriers’ suppression. For higher injection levels (0.5 and 0.7, i.e., Fig- ure 5c,d), the maps are covered by the locking region so that the frequency shift is always equal to the frequency detuning except for a few points here and there, especially at higher Figure Figure 5. Fr 5.equency Frequen shifts cy vs. shif frts equency vs. frequ detuning ency atde dif tfer un ent ing injection at different levels for injec theti four on cases levelof s f the or the four cases of comb spacings. frequency comb: (a) K = 0.1, (b) K = 0.3, (c) K = 0.5 and (d) K = 0.7. the frequency comb: (a) K = 0.1, (b) K = 0.3, (c) K = 0.5 and (d) K = 0.7. As the injection level increases (to 0.3, i.e., Figure 5b), the frequency shift is clearly damped due to the carriers’ suppression. For higher injection levels (0.5 and 0.7, i.e., Fig- ure 5c,d), the maps are covered by the locking region so that the frequency shift is always equal to the frequency detuning except for a few points here and there, especially at higher comb spacings. Figure 6. Frequency shifts vs. injection level at different frequency detunings for the four cases of Figure 6. Frequency shifts vs. injection level at different frequency detunings for the four cases of the the frequency comb spacing: (a) ∆f = −15 GHz, (b) ∆f = 15 GHz, (c) ∆f = −7 GHz, (d) ∆f = 7 GHz and frequency comb spacing: (a) Df = 15 GHz, (b) Df = 15 GHz, (c) Df = 7 GHz, (d) Df = 7 GHz and (e) ∆f = 0 GHz. (e) Df = 0 GHz. Figure 6. Frequency shifts vs. injection level at different frequency detunings for the four cases of the frequency comb spacing: (a) ∆f = −15 GHz, (b) ∆f = 15 GHz, (c) ∆f = −7 GHz, (d) ∆f = 7 GHz and (e) ∆f = 0 GHz. Photonics 2022, 9, x FOR PEER REVIEW 7 of 9 We have also scanned the frequency shift horizontally in the maps as a function of the injection levels at constant frequency detunings (the horizontal white lines in Figure 4a, at ∆f = 15, 7, 0, −7, −15 GHz). The extracted shifts are shown in Figure 6. At ∆f = −15 GHz, both frequency pushing and pulling are observed, as shown in Figure 6a. However, at ∆f = +15 GHz, we only observed frequency pushing, as shown in Figure 6b. This is probably due to the asymmetric characteristics originating from the value of the linewidth enhancement factor (LEF) as reported before in the case of a single optical injection [27]. In the case of ∆f = −7 and 7 GHz shown in Figure 6c,d, we only observe frequency pulling as the locking bandwidth is very close. Finally, when the comb is injected at the same frequency as the free-running SL (∆f = 0 GHz, i.e., Figure 6e), we observe that the SL is slightly shifted towards the positive detuning side at a low injection level (<0.2), and for higher than that, the SL peak is shifted towards the negative detuning side. This is again Photonics 2022, 9, 886 7 of 8 due to the change in the refractive index caused by the change in carriers’ density and described by the LEF. Finally, we concentrate on the frequency pushing occurring at a low injection level Finally, we concentrate on the frequency pushing occurring at a low injection level (K = 0.1) when the comb is injected very far from the free-running SL (i.e., at ∆f > 10 GHz (K = 0.1) when the comb is injected very far from the free-running SL (i.e., at Df > 10 GHz and ∆f > −10 GHz), as shown in Figure 7. In other words, this figure is a magnification of and Df > 10 GHz), as shown in Figure 7. In other words, this figure is a magnification of parts of Figure 5a. It can be seen that the frequency pushing in general is enhanced as the parts of Figure 5a. It can be seen that the frequency pushing in general is enhanced as the spacing of the comb increases. The spread of the data is due to the fact that the dominant spacing of the comb increases. The spread of the data is due to the fact that the dominant peak in the comb is not always the same and changes as the comb is detuned closer or peak in the comb is not always the same and changes as the comb is detuned closer or further from the free-running SL. This random FWM can also cause the frequency pulling further from the free-running SL. This random FWM can also cause the frequency pulling shown in some points in the figure (red and green). shown in some points in the figure (red and green). Figure 7. Frequency shifts vs. frequency detuning at K = 0.1 for the four cases of the frequency comb. Figure 7. Frequency shifts vs. frequency detuning at K = 0.1 for the four cases of the frequency comb. 4. Conclusions 4. Conclusions We have shown theoretically that the frequency comb spacing has a crucial role in the We have shown theoretically that the frequency comb spacing has a crucial role in locking map of a semiconductor laser under the injection of the frequency comb. As the the locking map of a semiconductor laser under the injection of the frequency comb. As comb spacing increases, the boundaries of the locking area become intermittent, especially the comb spacing increases, the boundaries of the locking area become intermittent, espe- the lower boundary at the negative detuning side. This spacing of the comb has been found cially the lower boundary at the negative detuning side. This spacing of the comb has been to affect the frequency shift of the free-running SL outside the locking bandwidth. These found to affect the frequency shift of the free-running SL outside the locking bandwidth. results are believed to contribute to a better understanding of the frequency comb injection These results are believed to contribute to a better understanding of the frequency comb as they can be utilized in many modern communication applications. Future work should injection as they can be utilized in many modern communication applications. Future include the effect of changing the number of the comb lines, varying the power of the lines, work should include the effect of changing the number of the comb lines, varying the the dynamics of carriers’ density and the experimental validation of these results. power of the lines, the dynamics of carriers’ density and the experimental validation of these results. Funding: This research was funded by Taif University, Researchers Supporting Project number (TURSP-2020/25), Taif, Saudi Arabia. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: This work was supported by Taif University, Researchers Supporting Project number (TURSP-2020/25), Taif, Saudi Arabia. Conflicts of Interest: The author declares no conflict of interest. References 1. Ohtsu, M.; Nakagawa, K.; Kourogi, M.; Wang, W. Frequency control of semiconductor lasers. J. Appl. Phys. 1993, 73, R1–R17. [CrossRef] 2. Sooudi, E.; Sygletos, S.; Ellis, A.D.; Huyet, G.; McInerney, J.G.; Lelarge, F.; Merghem, K.; Rosales, R.; Martinez, A.; Ramdane, A.; et al. Optical Frequency Comb Generation Using Dual-Mode Injection-Locking of Quantum-Dash Mode-Locked Lasers: Properties and Applications. IEEE J. Quantum Electron. 2012, 48, 1327–1338. [CrossRef] Photonics 2022, 9, 886 8 of 8 3. Rosado, A.; Pérez-Serrano, A.; Tijero JM, G.; Valle, Á.; Pesquera, L.; Esquivias, I. Experimental study of optical frequency comb generation in gain-switched semiconductor lasers. Opt. Laser Technol. 2018, 108, 542–550. [CrossRef] 4. Rosado, A.; Perez-Serrano, A.; Tijero, J.M.G.; Gutierrez, A.V.; Pesquera, L.; Esquivias, I. Numerical and Experimental Analysis of Optical Frequency Comb Generation in Gain-Switched Semiconductor Lasers. IEEE J. Quantum Electron. 2019, 55, 2001012. [CrossRef] 5. Zhao, B.-B.; Kovanis, V.; Wang, C. Tunable Frequency Comb Generation Using Quantum Cascade Lasers Subject to Optical Injection. IEEE J. Sel. Top. Quantum Electron. 2019, 25, 1900207. [CrossRef] 6. Ren, H.; Fan, L.; Liu, N.; Wu, Z.; Xia, G. Generation of Broadband Optical Frequency Comb Based on a Gain-Switching 1550 nm Vertical-Cavity Surface-Emitting Laser under Optical Injection. Photonics 2020, 7, 95. [CrossRef] 7. Moon, H.S.; Kim, E.B.; Park, S.E.; Park, C.Y. Selection and amplification of modes of an optical frequency comb using a femtosecond laser injection-locking technique. Appl. Phys. Lett. 2006, 89, 181110. [CrossRef] 8. Ramond, T.M.; Hollberg, L.; Juodawlkis, P.W.; Calawa, S.D. Low-noise optical injection locking of a resonant tunneling diode to a stable optical frequency comb. Appl. Phys. Lett. 2007, 90, 171124. [CrossRef] 9. Chan, S.-C.; Xia, G.-Q.; Liu, J.-M. Optical generation of a precise microwave frequency comb by harmonic frequency locking. Opt. Lett. 2007, 32, 1917. [CrossRef] 10. Fukushima, S.; Silva, C.F.C.; Muramoto, Y.; Seeds, A.J. Optoelectronic millimeter-wave synthesis using an optical frequency comb generator, optically injection locked lasers and a unitraveling-carrier photodiode. J. Light. Technol. 2003, 21, 3043–3051. [CrossRef] 11. Zhang, H.; Xu, M.; Zhang, J.; Jia, Z.; Campos, L.A.; Knittle, C. Highly Efficient Full-Duplex Coherent Optical System Enabled by Combined Use of Optical Injection Locking and Frequency Comb. J. Light. Technol. 2021, 39, 1271–1277. [CrossRef] 12. Ohtsu, M. Frequency stabilization in semiconductor lasers. Opt. Quantum Electron. 1988, 20, 283–300. [CrossRef] 13. Manamanni, K.; Steshchenko, T.; Wiotte, F.; Ramdane, A.C.; Sahni, M.-O.; Roncin, V.; Du-Burck, F. Frequency Stability Transfer in Passive Mode-Locked Quantum-Dash Laser Diode Using Optical Injection Locking. IEEE J. Quantum Electron. 2022, 58, 1300409. [CrossRef] 14. Liu, Z.; Slavik, R. Optical Injection Locking: From Principle to Applications. J. Light. Technol. 2020, 38, 43–59. [CrossRef] 15. Al-Hosiny, N.M.; Henning, I.D.; Adams, M.J. Tailoring enhanced chaos in optically injected semiconductor lasers. Opt. Commun. 2007, 269, 166–173. [CrossRef] 16. Al-Hosiny, N.M. 3D Injection-locking maps of semiconductor laser under multiple optical injections. J. Opt. 2021, 50, 629–636. [CrossRef] 17. Tistomo, A.S.; Gee, S. Laser frequency fixation by multimode optical injection locking. Opt. Express 2011, 19, 1081. [CrossRef] 18. Gavrielides, A. Comb Injection and Sidebands Suppression. IEEE J. Quantum Electron. 2014, 50, 364–371. [CrossRef] 19. Ó Duill, S.P.; Anandarajah, P.M.; Smyth, F.; Barry, L.P. Injection-locking criteria for simultaneously locking single-mode lasers to optical frequency combs from gain-switched lasers. In Physics and Simulation of Optoelectronic Devices XXV; Witzigmann, B., Osinski, ´ M., Arakawa, Y., Eds.; SPIE: Bellingham, WA, USA, 2017; Volume 10098, pp. 51–59. [CrossRef] 20. Shortiss, K.; Lingnau, B.; Dubois, F.; Kelleher, B.; Peters, F.H. Harmonic frequency locking and tuning of comb frequency spacing through optical injection. Opt. Express 2019, 27, 36976. [CrossRef] 21. Doumbia, Y.; Malica, T.; Wolfersberger, D.; Panajotov, K.; Sciamanna, M. Nonlinear dynamics of a laser diode with an injection of an optical frequency comb. Optics Express 2020, 28, 30379. [CrossRef] 22. Al-Hosiny, N.M.; Henning, I.D.; Adams, M.J. Correlation of Electron Density Changes with Optical Frequency Shifts in Optically Injected Semiconductor Lasers. IEEE J. Quantum Electron. 2006, 42, 570–580. [CrossRef] 23. Hui, R.; D’Ottavi, A.; Mecozzi, A.; Spano, P. Injection locking in distributed feedback semiconductor lasers. IEEE J. Quantum Electron. 1991, 27, 1688–1695. [CrossRef] 24. Al-Hosiny, N.; Henning, I.D.; Adams, M.J. Secondary locking regions in laser diode subject to optical injection from two lasers. Electron. Lett. 2006, 42, 759. [CrossRef] 25. Simpson, T.B.; Liu, J.M.; Huang, K.F.; Tai, K. Nonlinear dynamics induced by external optical injection in semiconductor lasers. Quantum Semiclassical Opt. J. Eur. Opt. Soc. Part B 1997, 9, 765–784. [CrossRef] 26. Liao, Y.-H.; Liu, J.-M.; Lin, F.-Y. Dynamical Characteristics of a Dual-Beam Optically Injected Semiconductor Laser. IEEE J. Sel. Top. Quantum Electron. 2013, 19, 1500606. [CrossRef] 27. Goldberg, L.; Taylor, H.F.; Weller, J.F. Locking bandwidth asymmetry in injection-locked GaAlAs lasers. Electron. Lett. 1982, 18, 986. [CrossRef]

Journal

Photonics – Multidisciplinary Digital Publishing Institute

Published: Nov 22, 2022

Keywords: optical injection; frequency comb; frequency pulling; frequency pushing

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