Super-mode noise suppression of a silicon hybrid mode-locked ring laser using a symmetric
Fabry–Perot filter
Mohammad Shekarpour
ANDMohammad Hasan Yavari*
Faculty of Engineering, Shahed University, Tehran, Iran
*Corresponding author: [email protected]
Received 11 October 2021; revised 21 December 2021; accepted 22 December 2021; posted 22 December 2021;
published 28 January 2022
In this research, we propose a harmonically mode-locked hybrid silicon laser by introducing intracavity reflectors (ICRs) to suppress super-mode noise. Using a delay differential equation model, the dynamics, phase noise, and timing jitter of the proposed structure (PS) and the without-ICR structure (WOICRS) are investigated. The simula- tion results show a significant increase of 20 GHz harmonic regime of PS compared to WOICRS. Additionally, this regime of PS can be achieved at a lower current than WOICRS. The phase noise of the PS is improved to about 8 dB.
Given the better selectivity of optical modes in PS, the phase noise is reduced compared to that of with-intracavity structure. Furthermore, numerical results show that timing jitter reduction has periodic behavior by changing the harmonic inter-spike interval time. The analyses demonstrate that harmonic mode-locking of a long cavity laser is a valid strategy for reducing phase noise and timing jitter. The simulation results of WOICRS are in a good agreement with experimental results. © 2022 Optica Publishing Group
https://doi.org/10.1364/JOSAB.445545
1. INTRODUCTION
Research on passively mode-locked semiconductor lasers have a long history [1–8]. In the past several decades, these types of optical pulse sources have played an important role in high- bit-rate communication systems [5], optical clocking [9], optically sampled analog to digital converters [10], frequency measurement [11], and spectroscopy [12]. Large noise figures are, however, restricting applications of semiconductor mode- locked lasers (MLLs) such as analog-to-digital converters and optical clocking in reaching a higher sample rate and repetition frequency, respectively [13]. Low-noise operation of MLLs can be achieved using high-quality-factor long cavities [14]. In long cavity lasers, to obtain desired a high frequency repetition rate, harmonic mode-locked operation is inevitable [15]. However, the super-mode noise caused by independent propagation of multiple pulses circulating in the ring cavity is a major drawback of harmonic mode-locking (HML) [15]. Use of an intracavity Fabry–Perot (FP) etalon, as an effective method, can suppress the super-mode noise [16].
In silicon photonic integrated circuits (PICs), passive MLLs have attracted increasing attention in recent years [17–21].
Silicon photonic passive waveguides make long on-chip cavi- ties possible while also reducing cavity loss, which is crucial in designing long cavity lasers [17–20]. In [17], a RF linewidth of a harmonically long cavity silicon MLL by employing an intra- cavity filter was reduced to 52 KHz in 20 GHz repetition rate by
a factor of 30 compared to a fundamental 20 GHz ring cavity.
Indeed, using an intracavity filter in long cavity structures, the narrow RF linewidth can be maintained at high repetition frequency. Unlike hybrid mode-locking (ML) [22] and optical injection [23] methods, which are complicated and highly expensive due to requiring external modulation and narrow bandwidth continuous wave (CW) lasers, long cavity structures do not need external sources. Integration of such MLLs on a silicon photonics platform is also promising for solving the difficulties in implementation of high-speed PICs [24].
In this work, we propose a modification of with-intracavity structure [17] [Fig. 1(a)] using intracavity reflectors (ICRs) instead of intracavity filters [Fig. 1(b)]. So, the influence of ICRs is investigated on the dynamic behavior of a passively mode-locked hybrid silicon laser. Finally, the timing jitter and phase noise of the proposed structure (PS) are simulated and compared with those of without-ICR structure (WOICRS).
2. PROPOSED STRUCTURE
Figure 1(b) shows the harmonically mode-locked hybrid sil- icon laser. This structure has two gain sections separated by a saturable absorber (SA). A 10:90 directional coupler couples light from the ring laser to the bus waveguide. The 20 GHz spectral filter by symmetrical ICRs is used inside the laser cavity.
The propagation loss of the waveguide is about 1 dB/cm. The
0740-3224/22/020527-08 Journal © 2022 Optica Publishing Group
Fig. 1. Schematic of harmonically mode-locked hybrid silicon lasers using (a) intracavity filter [17] and (b) ICRs.
low-propagation-loss silicon waveguide is an effective method for achieving a high quality factor and consequently better side-mode suppression.
One of the techniques to achieve the harmonics of a funda- mental round trip is employing compound-cavity ML formed by intermediate reflectors [25] in a linear cavity and equivalently in a ring cavity by intra- or extra-cavity [17,26]. Figures2(a)–
2(j) indicate the different configurations of the fundamental and harmonic linear lasers as well as the equivalent ring struc- tures. In the linear structure, by one ICR [Fig.2(e)], the cavity is divided into sub-cavities with length ratio L1:L2=m:n (M=m+n), wherem andn are integers without common factors [25], and equivalently in the ring structure [Fig.2(f )], by an intra- or extra-ring cavity with length ratioL1:L2=m:n [17,26]. In Figs.2(e) and2(f ) theMth harmonic has minimum loss and thus mode locking is predicted at HM. To achieve a uniform pulse train, for a higher harmonic number ofM, the single reflector cannot provide sufficient spectral selectivity [25].
Use of two ICRs, positioned at different fractions of sub-cavity length [L1:(L2+L3)=m:nandL2:L3=1:k, andm,n, andkare integers without common factors] leads to generating Mth (M=m+n) harmonic [Fig.2(i)], and thus the interme- diate modes are sufficiently suppressed [25]. The equivalent ring structure of two ICRs is shown in Fig.2(j). Indeed, each of the ICRs [Fig.2(i)] or equivalent rings [Fig.2(j)] is responsible for suppressing a different subset of nonharmonic modes [25].
Figures2(g) and2(h) depict the combination of colliding-pulse ML (CPM) and compound-cavity ML techniques to achieve higher harmonics [26].
The PS depicted in Fig.2(k) consists of a combination of linear and ring configurations due to the existence of ICRs, as described in [27,28]. Indeed, symmetric ICRs can be considered as a FP cavity. Because of implementing two intermediate reflec- tors positioned on both sides of SA symmetrically, this design is symmetrical; thus, the pulse train can be generated in clockwise and counterclockwise directions. In with-intracavity structure [17], as reported previously, to have pulses in both directions,
Fig. 2. Configurations of mode-locked linear and equivalent ring lasers (fundamental and harmonic ML). (a) FML [2]. (b) FML [2,17,19]. (c) CPM [18,26]. (d) CPM with two SAs [26]. (e) HML with one ICR [25,26]. (f ) HML with compound cavity [17,26].
(g) Combination of CPM and CCM [26]. (h) Combination of CPM and CCM. (i) HML with two ICRs [25]. (j) HML with two ring cavities. (k) proposed structure [27,28].
two 50:50 multi-mode interference couplers should be imple- mented to establish symmetry. However, in the PS, ICRs are inherently symmetric, and their reflectivity can be optimized to further reduce phase noise and timing jitter. Also, using ICRs,
Fig. 3. Ring resonator and equivalent linear cavity for (a), (b) without- and (c), (d) with-FP cavity.
Fig. 4. Magnitude of transmission coefficient (|Sciaj|2= |ci/aj|2) for length ratio (LFP:Lring) of (a) 2/9 and (b) 2/10.
we can take advantage of the colliding-pulse effect to achieve HML without using two SAs. Note that in conventional ring MLL [26], two SAs are needed to obtain higher harmonics by the colliding-pulse effect [Fig.2(d)].
To suppress effectively intermediate modes of PS, the length ratios of FP and the ring cavity must be properly designed. The multiple circulations of optical field inside the ring resonator are physically identical to the multiple reflections inside a linear cavity [29].
Therefore, an equivalent optical circuit is used to investi- gate the influence of the FP cavity on the filtering behavior of PS. Figure 3 shows the ring resonator with and without the FP cavity as well as the equivalent linear cavity. The round trip time between ICRs is designed to be half of the desired repetition rate (20 GHz). The length ratio of FP and the ring cavity isLFP:Lring=m:n. Ifmandnare considered as integer values without common factors, the intermediate modes of the ring cavity can be better suppressed by the FP filter [25].
So, the length ratio can be selected as 2/9 or 2/11. Figures4(a) and 4(b) show the magnitude of transmission coefficient (|Sciaj|2= |ci/aj|2) versus 2βLeq(whereβis propagation con- stant) for LFP/Lring=2/9 and 2/10, respectively. As can be seen, for LFP/Lring=2/10, the fifth mode of the ring is not suppressed due to common factors of two and 10. In the ML regime, LFP/Lring=2/9 in PS leads to generation of a pulse train with 20 GHz, which isH9of the fundamental frequency.
Also, the FP cavity constructed by ICRs operates as two spectral filters with a length ratio of 1/9, due to symmetric colliding- pulse effects in SA. So the suppression of the mode in PS is
higher compared to the structure introduced in [17], which has one intracavity filter with a length ratio of 1/10.
3. DDE MODEL OF PS AND WOICRS
Based on the model proposed in [30], in this paper, we extend the delay differential equation (DDE) model to study HML of long cavity lasers as shown in Fig.1(b). Equations (1) and (2) describe the slowly varying field amplitudeE in WOICRS and PS, respectively. The final set of three coupled DDEs for the slowly varying field amplitude E, saturable gainG, and saturable lossQis
γ−1E˙(t)= −E(t)+R0(t−T0)e−i1T0E(t−T0)+Dξ(t), (1) γ−1E˙(t)= −E(t)+R1(t−T1)e−i1T1E(t−T1)
+
∞
X
l=1
e−il c−i1(T1+lτ)R2(t−T1−lτ)
×E(t−T1−lτ)+
∞
X
l=1
e−i2l c−i1(T1+2lτ)
×R3(t−T1−l(2τ))E(t−T1−l(2τ))+Dξ(t), (2) G(t˙ )=Jg−γgG(t)−e−Q(t)(eG(t)−1)|E(t)|2, (3)
Q(t)˙ =Jq−γqQ(t)−rse−Q(t)(eQ(t)−1)|E(t)|2, (4)
Ri(t)=p
kie1/2(1−iαg)G(t)−1/2(1−iαq)Q(t), (5) where Eqs. (1), (3), and (4) model the WOICRS, while Eqs. (2), (3), and (4) model the PS.
Due to the pulse colliding, the transient grating (TG) is generated in a SA [Fig. 1(b)]. By considering the coupling coefficient in the traveling wave equation, the effects of CPM in a linear configuration have been modeled [31]. However, in DDEs, it is assumed that the lasing is unidirectional [30] so that for modeling TG of CPM, we define an effective mirror model [see Fig.1(b)]. Note that in WOICRS, the clockwise and counterclockwise pulses also collide within the SA, and thus TG is produced. However, because of the unidirectionality of DDEs in modeling these lasers, the part of the optical field reflected by TG cannot be taken into account. On the other hand, in addition to the TG, if one or more reflectors exist, the direction of the part of the optical field reflected by TG can be reversed and thus coincide with simulated direction.
Accordingly, for the proposed configuration, in HML analysis, the definition of the effective mirror is vital due to the existence of ICRs [Fig.1(b)]. As shown in Fig.1(b), the effective length (Leff) of TG is Leff =tanh(κLTG)[32], Whereκ is the cou- pling coefficient. For the small index difference of TG (tanh κLTG→κLTG), the effective mirror is at the center of the TG.
The absorption of TG is modeled byQin DDEs.
Jg is unsaturated gain and Jq is unsaturated absorption.
The carrier lifetimes in the gain and absorber sections are
Fig. 5. Corresponding paths for time delays used for the DDE model in (a) WOICRS and (b) PS.
given by 1/γg and 1/γq, respectively. The factorrs is propor- tional to the ratio of the saturation energies in the gain and absorber sections. The bandwidth of the laser is taken into account by a Lorentzian-shaped filter function with full-width at half maximum (FWHM) γ. Here, l indicates the num- ber of round trips in the FP filter, andC denotes the phase of light due to one round trip in the FP filter.1accounts for a possible detuning between the frequency of the maxi- mum of the gain spectrum and the frequency of the nearest cavity mode. The linewidth enhancement factors in the gain and absorber sections are denoted byαg andαq, respectively.
Spontaneous emission is modeled in Eqs. (1) and (2) by a complex Gaussian white-noise term ξ(t) with strength D.
In Eq. (5),Ri(t)describes the amplification and losses of the electric field during one round trip in the ring cavity. The cavity intensity loss for WOICS is kk0=Cf +(LLw)+Lc+Sc, whereCf is the correction factor, andLis cavity length of the ring laser. LW and Lc are waveguide loss (dB/cm) and cou- pler loss, respectively.Sc is the splitting ratio of the coupler to maintain light in the ring laser. The cavity intensity losses for PS (l=1) are kk1=Cf +(0.9LLw)+Lc+Sc+RICR, kk2=Cf +(LLw)+Lc+Sc+Reff and kk3=Cf+ (1.1LLw)+Lc+Sc+RICR. The relation betweenkkiandki
iskki=10 logki. As shown in Fig.5, the cold cavity round trip
Table 1. Parameters Used in Numerical Simulations [4,5,17,33]
Symbol Quantity Value
γ Full width at half maximum 0.125 ps−1
γg Carrier relaxation rate in gain section 1 ns−1 γq Carrier relaxation rate in absorber section 75 ns−1
rs Ratio of saturation energies 25
Jg Pump rate in gain section 1–160
Jq Pump rate in absorber section 50–400
Lw Propagation loss of waveguide 1 dB/cm
Lc Propagation loss of coupler 0.5 dB
Sc Splitting ratio of couplers 10 dB
L Cavity length of ring laser 3.967 cm
T0 Cold-cavity round trip time 500 ps
TISI,H1 Inter-spike interval time in WOICRS 1.005T0
Cf Correction factor 4.7 dB
τ Intracavity round trip time 0.1T0
RICR Intensity reflection of intracavity reflector 10 dB D Strength of spontaneous emission noise 0.5 e6
time for the WOICRS isT0, while for PS, the DDE is based on three delay times:T1=0.9T0,T1+τ, andT1+2τ. The round trip time between ICR and the center of TG (τ) determines the repetition frequency of HML (20 GHz); consequently, the value ofτis 0.1T0. Note that in PS, a repetition frequency of 20 GHz is the ninth harmonic (H9) of the fundamental frequency (∼2.22 GHz). Other parameters for simulations are reported in Table1.
4. RESULTS AND DISCUSSION A. Dynamics of WOICRS and PS
As depicted in Fig.6, in the ML regime of WOICRS, the opti- cal period of pulses is divided into a slow stage (|E|2 ∼ 0), whose gain and absorption media are recovered, and a fast stage, whose electric field intensity is large [30]. The recov- ery time of absorbing media is lower than that of the gain media, which is also the necessary condition for stable ML [1]. Figure 6 also reveals time traces of the amplitude |E|, saturable gain G, total loss Qt=Q+ |lnk0|, and net gain Gt=G(t)−Qt for H1 and H10 of WOICRS. The net gain at the beginning and end of the slow stage is negative, which is the stability criterion for ML pulses [30]. In WOICRS, the inter-spike interval time of the fundamental frequency (TISI,H1 ≈T0+eγ−1=T0+O(γ−1)) is longer than the cold-cavity round trip time T0. The perturbations caused by noise appear in the leading edge of a pulse when e <1 and trailing edge when e > 1 [30]. For the nth harmonic, TISI,Hn ≈ (T0+O(γ−1))/n.
Fig. 6. Time traces of the absolute value of the normalized ampli- tude|E|, gainG, total lossQtand net gainGtfor WOICRS (a)H1
and (b)H10.
Fig. 7. Dynamic results of WOICRS and PS.
Figure7helps to identify the regimes of CW,Q-switching (QS), fundamental ML (FML), and HML. In WOICRS, various harmonic regimes of ML can be obtained by changing Jg and Jq [Fig.7(a)]. As can be seen, allH1toH10are gener- ated with respect to the fundamental frequency, i.e., 2 GHz.
By increasing the unsaturated absorption (Jq), which corre- sponds to reverse voltage, the recovery time of the absorber decreases. On the other hand, rapid absorption recovery causes the unsaturated gain (Jg) required for the desired repetition frequency to rise [1]. This behavior is in good agreement with harmonic regimes of repetition frequency as shown in Fig.7(a).
Meanwhile, HML regimes of the PS exhibit only H9 plus a significant increase in the 20 GHz harmonic regime compared to WOICRS. To have a ML regime, as PS designed for the 20 GHz harmonic regime, the requiredJgandJqare increased compared to WOICRS. So, the regime of CW operation increases in PS [see Fig. 7(b)]. There is a trade-off between harmonic number and required unsaturated gain (Jg) as well as absorption (Jq).
The numerical bifurcation analysis of Eqs. (1), (3), and (4) for WOICRS is performed using the DDEBIFTOLL package [34]. The Andronov–Hopf bifurcation curve H1 corresponds to a FML regime with the pulse repetition period close to the cavity round trip time T0. Curves Hn with n=2,· · · ,10 correspond tonth harmonic ML regimes with repetition peri- odsTHn ∼= T0/n. There is a good match between bifurcation analysis (Fig.9) and dynamic range of the repetition frequency [Fig.7(a)], where a gradual transition from a FML toH10was observed with the rise ofJgandJq.
Fig. 8. Peak of pulses and pulse width of WOICRS and PS as a func- tion of (a)J q(J g=100) and (b)J g(J q=400).
Indeed, the FP filter suppresses intermediate harmonics.
We investigate this influence of the FP filter by applying dis- crete Fourier transform (DFT) to field power, for WOICRS and PS in the 20 GHz harmonic regime (see Fig.10). From the power spectrum in Fig.10(a), only the 20 GHz harmonic component can be seen in PS, but in another configuration, all harmonic components appear [see Fig.10(b)], causing ampli- tude distortions in the time domain. This is in agreement with experimental results [17].
In WOICRS, by increasing unsaturated absorption (Jq), the pulse width decreases, and thus the peak of the pulse train is increased [see Figs.7(c) and7(e)]. Also, the ML operation tends to shift from a higher harmonic to lower harmonic frequency.
This behavior is in good agreement with experimental results [1]. In contrast to WOICRS, the changes ofJqdo not lead to a significant effect on widths and peaks of pulses [see Figs.7(d) and7(f )]. Figure8(a) shows this comparison for pulse width and peak of pulse trains of WOICS and PS as a function ofJq. Since PS operates in the 20 GHz harmonic regime, WOICRS oper- ates in different harmonic regimes; thus, the dynamics of pulse widths and peaks of pulses in WOICS and PS have different behavior [Fig.8(b)]. Also, Figs.8(a) and8(b) reveal the 20 GHz harmonic frequency produced in WOICRS, which has a lower
Fig. 9. Andronov–Hopf bifurcations of the CW solution of WOICRS. Curve H1 corresponds to a FML, and curves Hn with n=2,· · · ,10 correspond tonth harmonic ML regimes with the repetition periodsTHn∼=T0/n.
Fig. 10. Electrical spectrum of 20 GHz harmonic regime for (a) PS and (b) WOICRS.
pulse width and higher peak of output pulses compared to the 20 GHz harmonic frequency produced in PS.
B. Phase Noise and Timing Jitter of Dynamics of WOICRS and PS
For the dynamics of the ring MLL presented in previous section, spontaneous emission was not taken into account (D=0).
By considering spontaneous emission in modeling as a noise strength of D, fluctuations in arrival times of pulses appear.
These fluctuations can be quantified by the timing jitter.
Using the procedure proposed in [35], we determine the root-mean-square (rms) timing jitter.
The phase noise spectrum is obtained by averaging over M=30 noise realizations. Subsequently, rms timing jitter is calculated by integrating the phase noise spectrum over the frequency range fromνlow=0.5 MHz toνhigh=5 GHz. The single sideband phase noise of with-intracavity structure, PS, and WOICRS is plotted in Fig.11(a). The use of a selectivity filter in PS and with-intracvity structure leads to a reduction of phase noise by about 8 dB. Also, considering two sub-cavities in PS, the selectivity and therefore the suppression of super-mode noise in PS can be better compared to with-intracavity structure introduced in [8,17]. In Fig.11(b), by increasing the strength of spontaneous emission noise, the difference between phase noise of PS and with-intracavity structure is demonstrated. The
Fig. 11. Single sideband phase noise of WOICRS, PS, and with-intracavity structure for constant J g=130 and J q=400;
(a)D=0.5e6 and (b)D=10×0.5e6.
rms timing jitter of PS normalized to the rms timing jitter of WOICRS as a function ofJq and Jg is depicted in Fig.12(a).
As can be seen, the rms timing jitter of PS is decreased com- pared to WOICRS. Furthermore, a comparison of dynamic results shown in Figs.7(a) and7(b) indicates that the 20 GHz frequency repetition rate of PS can be achieved at lower current than WOICRS. So, the spontaneous emission noise decreases, and further reduction of phase noise and rms timing jitter can be achieved.
Figure 12(a) illustrates a nearly periodic dependence of timing jitter on Jq and Jg. Also, by limiting the range dis- played in Fig.12(b), we observe that the variation of frequency repetition rate in the 20 GHz harmonic regime is similar to the pattern of Fig. 7(a). By considering this variation, we
Fig. 12. (a) rms timing jitter of PS normalized to the rms timing jit- ter of WOICRS and (b) repetition frequency of PS inH9regime.
Fig. 13. Normalized rms timing jitter as a function of distance between ICRs for constantJg=60 andJq=400.
find that when TISI,H9 falls within the resonance regions, the stability of pulses is increased, i.e., greater timing jitter reduction occurs. The dependence of TISI,Hn (repetition rate frequency=1/TISI,Hn) onJqandJgcan be expressed as
TISI,Hn ∼=
T1+O(γ−1) n
c(Jg,Jq). (6)
We define c(Jg, Jq) as a correction coefficient to take into account the variation of repetition frequency, which depends on
JgandJq.
To clarify the dependence of rms timing jitter on TISI,H9, the influence of distance between ICRs as an important factor in determiningTISI,H9is investigated (Fig.13). In this regard, the normalized rms timing jitter as a function of normalized deviation from the initial distances between ICRs (L0,ICR) is calculated. Noted thatL0,ICRis assumed to be equal to 20 GHz.
As can be seen, for constant Jg =60 and Jq=400, the nor- malized rms timing jitter such as the pattern in Fig.12(a) has periodic behavior. Thus, changes inTISI,H9, which are caused by deviation fromL0,ICR, lead to resonance and nonresonance regions affecting the stability of pulses as well as rms timing jitter.
5. CONCLUSION
As an important contribution, we have proposed harmonic MLL using ICRs instead of intracavity filters. We have extended the DDE model to study the influence of ICRs on dynamics, phase noise, and timing jitter of WOICRS and PS. A 20 GHz harmonic regime can be achieved for WOICRS and PS inH10
and H9, respectively. Dynamic analysis demonstrated a signifi- cant increase in the 20 GHz harmonic regime of PS compared to WOICRS. The results revealed that a FP filter effectively sup- presses intermediate harmonics in PS. Numerical results showed the reduction of phase noise and rms timing jitter in PS com- pared to WOICRS. Also, the better selectivity of optical modes in PS leads to more suppression of super-mode noise compared to with-intracavity structure. Regarding dynamic behavior, a 20 GHz frequency repetition rate of the PS can be achieved at a much lower pumping current compared to WOICRS. Thus, the spontaneous emission noise decreases, and further reduction
of phase noise and rms timing jitter is achieved. The numeri- cal results show a nearly periodic dependence of timing jitter, while a greater reduction of the rms timing jitter occurred in the resonance region.
Disclosures. The authors declare no conflicts of interest.
Data Availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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