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Abstract
Laser cavity-soliton microcombs are robust optical pulsed sources, usually implemented with a microresonator-filtered fibre laser. In such a configuration, a nonlinear microcavity converts the narrowband pulse resulting from bandwidth-limited amplification to a background-free broadband microcomb. Here, we theoretically and experimentally study the soliton conversion efficiency between the narrowband input pulse and the two outputs of a four-port integrated microcavity, namely the ‘Drop’ and ‘Through’ ports. We simultaneously measure on-chip, single-soliton conversion efficiencies of 45% and 25% for the two broadband comb outputs at the ‘Drop’ and ‘Through’ ports of a 48.9 GHz free-spectral range micro-ring resonator, obtaining a total conversion efficiency of 72%.
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1. Introduction
‘Microcombs’ are optical frequency combs based on microresonators, typically operating via the nonlinear Kerr effect [1,2]. This technology has particularly benefitted from the discovery of dissipative temporal cavity-solitons [3,4], which exhibit a broad and smooth optical spectrum suitable for a wide range of practical applications [5–14].
Generally, the cavity-soliton solutions obtained in driven microresonator configurations are well-described by the Lugiato-Lefever equation [15]. Evaluating the performance of such systems has been a central focus, and recent analysis has demonstrated the dependence of the conversion efficiency on the microcavity free-spectral range (FSR) [16–19]. In standard pumping regimes, the optical conversion efficiency drastically falls below a few per cent for resonators with repetition rates within the electronic detection bandwidth (<100 GHz) [19], which are particularly important for metrological applications [1,2]. Soliton crystals and dark solitons appear to exceed the limitations of bright solitons in terms of robustness and conversion efficiency [17,20,21]. Nonetheless, the regular pulsed nature and smooth spectra of bright solitons motivate their further pursuit. Recently, alternate methodologies have been proposed, including microcombs based on pulsed pumping [22,23], self-injection locking configurations [11,24,25], and multiple cavities or modes and nonlinear couplers [26–32].
In such a framework, we recently introduced the concept of laser cavity-soliton microcombs [33–35]. These localised pulses form in a lasing cavity without requiring a saturable absorber, and they are sustained by the Kerr nonlinearity. Laser cavity-solitons are background-free and have a very high modal efficiency. In our configuration, we used a four-port nonlinear microcavity nested in a fibre cavity with optical gain to obtain a broadband microcomb. Unlike most designs, we do not coherently drive the nonlinear microcavity—conversely, the microcomb spontaneously forms as the result of the nonlinear locking properties of the laser itself. Notably, the system’s natural slow nonlinearities, such as thermo-optical and gain saturation effects, allow it to reliably start, maintain and recover the desired soliton state [35].
Here, we present a detailed study of the soliton nonlinear conversion efficiency for microresonator-based laser cavity-solitons. Within the laser system, the microcavity converts a narrowband input into a broadband comb emission. The laser has two important microresonator output ports that provide a large bandwidth comb: the ‘Through’ and ‘Drop’ ports. The former is the typical output port of the system, while the field at the ‘Drop’ port feeds the amplifier input. However, a large part of the ‘Drop’ field can be extracted and used as an additional output. Our system’s net output accounts for both the ‘Through’ and ‘Drop’ ports. We theoretically evaluate and experimentally measure the on-chip conversion efficiencies at these two output ports. These figures allow us to directly compare the performance of the laser to driven configurations [16–19,26,29], especially with pulsed pumped schemes [22,23].
2. Background
2.1 System geometry and experimental setup
In our experiments, similar to those previously reported [22,11], an integrated, CMOS compatible Hydex microcavity with free-spectral range FSR = 48.9 GHz is nested in a polarisation-maintaining Erbium-doped amplifying cavity with FSR = 90 MHz, in the configuration shown in Fig. 1(a).
Fig. 1. (a) Experimental setup. The laser consists of an on-chip, fibre-coupled microring cavity (FSR = 48.9 GHz) and an amplifying fibre-loop. Free-space couplers (FSCs) link a free-space section comprising a bandpass filter (BPF) and a beam splitter (BS) to monitor the ‘Drop’ port. We do not show the setup polarisation controls and the delay line. An isolator (ISO) ensures unidirectional operation. A laser diode (LD) pumps a polarisation maintaining amplifier (EDFA, green). (b) On-chip ‘Through’ and ‘Drop’ transmission profiles for a typical microcavity resonance (around 1545 nm – plotted as circles, exhibiting a linewidth of $\mathrm{\Delta }F$ = 120 MHz) and corresponding Lorentzian fits (solid lines). ${T_D}$ (${T_T}$) is the ‘Drop’ (‘Through’) on-chip power transmission.
The microcavity has four ports and exhibits a linewidth of $\mathrm{\Delta }F$ = 120 MHz, with the typical ‘Through’ and ‘Drop’ transmissions values reported in Fig. 1(b). The Q factor is 1.6 million, and the resonant on-chip transmission recorded at the ‘Through’ and ‘Drop’ ports are ${T_T} \approx 20{\%}$ and ${T_D} \approx 40{\%}$ respectively. The input-output chip coupling losses are ${\approx} {\; }$ 3 dB. The laser cavity has a small free space section comprising a 12 nm (∼ 650 GHz) bandpass filter, a delay line to adjust the cavity length, and a beam splitter providing an additional output to monitor the ‘Drop’ port.
2.2 Theoretical background
We start with the modelling of [33,34,36], and we will indicate quantities related to the microcavity (amplifying cavity) with the ‘a’ (‘b’) letter. Cavity roundtrip times are ${T_{({a,b} )}}$ with $T_a^{ - 1} = $ 50 GHz and $T_b^{ - 1} = $ 90 MHz. We define t and $\tau \in [{0,{T_a}} ]$ as the slow and fast times in [s] and express the optical fields in [$\sqrt {\textrm{W}} $]; the microcavity field is $a({t,\tau } )$ and the laser field is represented by a superposition of 2N+1 supermodes ${b_q}({t,\tau } )$. The system reads:
Here, $v \approx 2\; \times {10^8}{\; \textrm{m}}{\textrm{s}^{ - 1}}$ is the speed of light in fibre, group velocity dispersions are ${\beta _a} ={-} 22\; \textrm{p}{\textrm{s}^2}\textrm{k}{\textrm{m}^{ - 1}}$ and ${\beta _b} ={-} 45\; \textrm{p}{\textrm{s}^2}\textrm{k}{\textrm{m}^{ - 1}}$, γ is the Kerr coefficient in $[{{\textrm{m}^{ - 1}}{\textrm{W}^{ - 1}}} ]$, and $\Delta $ is the phase cavity mismatch, normalised to $T_b^{ - 1}$. We model the dispersive losses with a derivative of order $2r$, with $r = 3$, and the coefficient ${\mathrm{\Omega }_\textrm{G}} = 2\; \pi \times \; 650\; \textrm{GHz}$. The amplification bandwidth is $\mathrm{\Delta \Omega } \approx {\; }2{\mathrm{\Omega }_\textrm{G}}\sqrt[{2r}]{{\ln 2}},$ here corresponding to approximately 24 microcavity modes. The term g, is the actual gain experienced by the field between the ‘Drop’ and the ‘Input’ ports of the microcavity, accounting for both the fibre amplification and the losses of the loop [33]. The variable $\theta = {\theta _i} + {\theta _c}$, including both intrinsic losses ${\theta _i}$ and coupling losses ${\theta _c},$ is the microcavity loss per roundtrip, connected to the microcavity linewidth $\mathrm{\Delta }F = 120$ MHz by the expression $\theta = \pi \mathrm{\Delta }F{T_a}$ and related to the transmissions at resonance (cf. Figure 1(b)). Specifically, ${T_T} = \theta _i^2{\theta ^{ - 2}}$ and ${T_D} = \theta _c^2{\theta ^{ - 2}}$ are the on-chip power transmissions at resonance of the ‘Through’ and ‘Drop’ ports, respectively. Here we use ${\theta _c} = 0.7\; \theta .$
We look for the soliton states with a similar analysis to [23,24]. Soliton states are defined as $a({t,\tau } )= A(\tau )\textrm{exp} [{i\phi t} ]$ and ${b_q}({t,\tau } )= {B_q}(\tau )\textrm{exp} [{i\phi t} ]$, where $\phi $ is the soliton nonlinear phase, and $A(\tau )$ and ${B_q}(\tau )$ are the field envelopes. They have approximately all the energy in the supermode ${B_0}(\tau )$. Hence, for simplicity, we will disregard the other supermodes in the discussion below. We define the average powers of the two stationary fields as ${\; }{E_A} = \mathop \smallint \limits_0^{{T_a}} {|{A(\tau )} |^2}d\tau ,{\; \; }{E_B} = \mathop \smallint \limits_0^{{T_a}} {|{{B_0}(\tau )} |^2}d\tau $. Figure 2(a) reports the energy ${E_A}$ as a function of gain g and phase cavity mismatch $\Delta $ for the complete set of soliton states, along with the instability region of the background state in the ($g,\Delta )\; $ plane, in blue. The region of existence and stability of the cavity-solitons is within $0 < g - {\theta _i}{\theta ^{ - 1}} < 1$ and stable solitons are marked with thick lines. Given the stationary states of Eqs. (1,2), the optical intensities at the ‘Input’ ${I_{IN}}$, ‘Drop’ ${I_D}$ and ‘Through’$\; {I_T}$ ports of the microcavity are straightforwardly evaluated via the quantities
Fig. 2. Single soliton states summary for ${\theta _c} = 0.7\; \theta $, N = 5. (a) Energy of the microcavity field normalised to ${({\gamma v{T_b}} )^{ - 1}}$ for the solitons states as a function of gain and detuning. Blue to yellow lines represent states with increasing gain, as shown in the y-axis, with thick lines representing the stable states. The shadowed blue area represents the instability of the zero state. (b) Spectra of solitons at the ‘Input’ (${\textrm{I}_{\textrm{IN}}}),$ ‘Drop’ (${\textrm{I}_\textrm{D}})$ and ‘Through’ (${\textrm{I}_\textrm{T}})\; $ ports as a function of the gain g for the stable states at the centre of the stability region. The ‘Through’ output ${I_T}$ shows the signature of the interference between the fields in the two cavities, resulting in two pronounced minima around the edges of the gain bandwidth (here, approximately 24 microcavity modes, marked with red arrows). (c) Efficiencies at the ‘Drop’ (${\eta _D},$ circle) and ‘Through’ (${\eta _T},\; $ crosses) ports calculated for the states shown in (b). Underlying data for the figure is available as part of the Supplemental Material (Dataset 1, [40]).
From Eqs. (1) and (2), it directly follows that the ratio between the power stored in the two cavities at a stationary regime is simply ruled by the roundtrip loss of each cavity [34]. This is expressed analytically as:
Here we defined
We use Eq. (4) to calculate our figures-of-merit, namely the microcavity internal conversion efficiency at the ‘Drop’ port, as:
Equations (5) and (6) are directly connected to the transmissions at resonance ${T_D} = \theta _c^2{\theta ^{ - 2}}$ and ${T_T} = \theta _i^2{\theta ^{ - 2}}$ and their dependence on $g\; $ is reported in Fig. 2(c) for the set of stable states in Fig. 2(b). Equation (6) implies that the efficiency at the ‘Through’ port increases with the coupling coefficient ${\theta _c}$. Reference [26], where the authors studied a similar topology but without gain and for a driven configuration, also reported a similar trend.
Conversely, Eq. (5) shows that the ‘Drop’ port has an inverse relationship, with the efficiency increasing for a smaller coupling coefficient. For the broadest soliton in this range of parameters, a microcavity with our experimental coupling coefficient (${\theta _c} = 0.7\; \theta $) provides $\eta _I^D \approx 40\%$ and $\eta _I^T \approx 30\%$. With a similar set as the one in Fig. 2 and no intrinsic losses (${\theta _i} = \; 0$), the broadest soliton would provide $\eta _I^D \approx \; 83\%$ and $\eta _I^T \approx 17\%$. Figure 3 summarises this last case $.$
Fig. 3. Single soliton states summary for ${\theta _c} = \; \theta $, N = 5. (a) Energy of the microcavity field normalised to ${({\gamma v{T_b}} )^{ - 1}}$ for the solitons states as a function of gain and detuning. Blue to yellow lines represent states with increasing gain, as shown in the y-axis, with thick lines representing the stable states. The shadowed blue area represents the instability of the zero state. (b) Spectra of solitons at the ‘Input’ (${\textrm{I}_{\textrm{IN}}}),$ ‘Drop’ (${\textrm{I}_\textrm{D}})$ and ‘Through’ (${\textrm{I}_\textrm{T}})\; $ ports as a function of the gain g for the stable states at the centre of the stability region. The ‘Through’ output ${I_T}$ shows the signature of the interference between the fields in the two cavities. Such interference can result in an almost flat region at the centre of the spectrum. (c) Efficiencies at the ‘Drop’ (${\eta _D},$ circle) and ‘Through’ (${\eta _T},\; $ crosses) ports calculated for the states shown in (b). Underlying data for the figure is available as part of the Supplemental Material (Dataset 1, [40]).
3. Experiments
We verified these findings in our experiments by studying the nonlinear conversion efficiencies of single and two soliton states as a function of the gain, which we controlled by changing the 980 nm pump power of the EDFA as in [35]. Please see also [33,36–38] for further experimental details.
We configured our setup in a condition where the system displayed single or two-soliton states as dominant attractors. For solitary pulses, the microcomb lines were locked in a red-detuned position of the microcavity, here between 40-50 MHz from the centre resonance position. As discussed in [35], the system’s slow nonlinearity can start and maintain the system in a specific state (dominant attractor). Such a state is selected by acting on the global parameters of the laser (EDFA pump power, cavity length and intracavity losses). The cavity operates with approximately 8.4 dB of loss, including the chip’s 3 dB input-output coupling losses. We set the laser cavity length approximately as a multiple of the microcavity length [35]. By varying the EDFA pump power, we observe distinct regions providing single soliton and two-soliton states, which are stable long-term [35]. Figure 4(a) and (b) report the spectra at the ‘Drop’ and ‘Through’ ports, respectively. The drop port was collected with a 10% intracavity beam splitter, while the ‘Through’ port was directly extracted from the chip. Therefore, the available power to detect the ‘Through’ spectrum was larger than the ‘Drop’ one, resulting in a larger SNR at the OSA. For this reason, the spectra in Fig. 4(b) appear broader than those in Fig. 4(a).
Fig. 4. Experimental study of soliton states for 8.4 dB intracavity loss. (a) ‘Drop’ and (b) ‘Through’ optical spectra (logarithmic scale) of the soliton states. The sequential spectra are offset such that they may be better distinguished, where the y-axis in panel (a) represents the EDFA pump power. Colours from blue to yellow are proportional to the EDFA pump power and are used to identify the states also in the panels in Fig. 5. Underlying data for the figure is available as part of the Supplemental Material (Dataset 1, [40]).
A slight dip due to mode-crossing losses in the microcavity is present at around 1545 nm in the spectra at all ports (Fig. 4). Single soliton states show the coexistence of a CW mode around 1538 nm, where the Erbium has a peak in the gain. We verified by laser scanning spectroscopy that it is frequency-shifted and distinct from the soliton state. This feature is strongly diminished in the two-soliton states [35].
This configuration of parameters shows two distinct regions for each type of state. Single soliton states are found for EDFA powers between 330-380 mW and 400-420 mW. On the other hand, two-soliton states are located in the regions 380-400 mW and 420-460 mW. Below 330 mW, the laser provides unstable incoherent pulses, which we show in Fig. 4 for completeness, but do not further discuss in Fig. 5. A signature of the regions with different soliton numbers is visible also in Fig. 5(a) and (b). Here we report the input power to the ring (a) and the output ‘Drop’ and ‘Through’ powers (b, circles and crosses, respectively) for the soliton states. The plots show clear jumps in the EDFA output power whenever the system transitions to a different state (here, one- or two-soliton). As discussed in [35], the system changes its dominant attractor in these transitions.
Fig. 5. (a) Measured power at the microcavity ‘Input’ (off-chip) against EDFA pump power. (b) The off-chip power at ‘Through’ (crosses) and ‘Drop’ (circles) ports against EDFA pump power. Colours from blue to yellow are proportional to the EDFA pump power. They are used to identify the states in all the panels, consistently to Fig. 4. (c) On-chip nonlinear conversion efficiencies for the ‘Through’ and ‘Drop’ ports vs gain for the specific state. Underlying data for the figure is available as part of the Supplemental Material (Dataset 1, [40]).
Figure 5(c) reports the on-chip nonlinear conversion efficiencies obtained directly from the measurements in Fig. 4(a), (b), accounting for the 1.5 dB per-port coupling loss. We plot these values against the experimental gain, evaluated as the ratio between the input and output energies at the EDFA. For EDFA pump powers between 330-460 mW, the intracavity gain increases by about 0.4 dB, producing an optical field at the input of the microring between 80-120 mW (Fig. 5(a)). Erbium-doped amplifiers operate with an efficiency of about 25% [39].
The ‘Through’ and ‘Drop’ efficiencies show a clear linear dependence on the gain, regardless of the type of state or pump power, as predicted by Eqs. (5) and (6). We measure on-chip conversion efficiencies between 25-30% for soliton states at the ‘Through’ port. Such an output is readily available: here, we had off-chip, fibre-coupled powers between 11-16 mW. The ‘Drop’ port shows on-chip conversion efficiencies between 42-46%, which we monitored with a 10% beam splitter. These two properties imply a total conversion efficiency of about 72% in this range. Although we used a setup with relatively low losses, solitons can be achieved with EDFA operating at much higher gain regimes [11], requiring only a few mW of EDFA input power. With proper engineering, it is possible to use most of the on-chip output power, which in this set was between 25 and 40 mW. These results represent a record conversion efficiency in the present state-of-the-art, especially compared with externally driven configurations. For continuous-wave (CW) driven cavities, single port conversion is usually about a few percent. A conversion in the order of 20% would require over-coupled microcavities with FSRs exceeding 1 THz [19]–20 times the value we use here. CW-driven coupled resonators [29] can generate microcombs with higher efficiencies and larger repetition rates (50 GHz repetition rates and conversion in the order of 35%, the authors in this paper also reported 55% for smaller resonators) by finely tuning the two cavities in the proper regime. Engineered dual mode microresonators [31] have also shown efficiencies exceeding 30% [32] in resonators with 1 THz FSR. Similar conversion values for stationary solitons have also been achieved with pulsed driving in larger resonators (22 GHz) requiring Q factors above 10 million [23], almost one order of magnitude larger than the one used here.
4. Conclusions
In conclusion, we report theoretical and experimental investigations of the conversion efficiency properties of laser cavity-solitons in a microcavity-filtered laser. We derive a simple formulation for the nonlinear conversion efficiencies of the stationary states at the ‘Through’ and ‘Drop’ ports, representing the two physical outputs of the system, which have an opposite trend with gain. We show consistent soliton conversion efficiencies between 25-30% and 42-46%, simultaneously available at the microcavity’s ‘Through’ and ‘Drop’ outputs, for a total output conversion efficiency as high as 72%, exceeding the current state-of-the-art for microcombs.
Funding
Engineering and Physical Sciences Research Council (EP/S001018/1); European Research Council (725046, 851758); Defence Science and Technology Laboratory (DSTLX1000142078); Innovate UK (10004412, 77087); Leverhulme Trust (ECF-2020-537); Natural Sciences and Engineering Research Council of Canada; Ministère de l'Économie, de la Science et de l'Innovation - Québec; Strategic Priority Research Program of the Chinese Academy of Sciences (XDB24030300).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data generated and analysed in Figs. 2–5 are provided as part of the Supplemental Material Dataset 1, Ref. [40]. Additional datasets generated during the current study may be obtained from the corresponding author upon reasonable request.
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