Ultrafast pump-probe measurement has played a pivotal role in the study of femtosecond light–matter interaction and carrier dynamics as well as the characterization of chemical, solid-state, and biological materials [1]. Among all the variants, the dual-comb pump-probe technique offers a unique nonmechanical delay scanning mechanism that solves the measurement speed bottleneck and enables high-speed applications including terahertz time-domain spectroscopy [2–4], ultrafast photoacoustic characterization [5,6], high-sensitivity photothermal spectroscopy [7], multidimensional coherent spectroscopy [8,9], and coherent Raman spectroscopy [10–13]. In the dual-comb pump-probe technique, the delay scanning is controlled by setting the repetition rate difference (Δfrep) between the two combs such that the two pulse trains will temporally walk off each other by steps of Δfrep/frep2, where frep is the average repetition rate. The total delay scan is 1/frep, and the measurement speed is limited by 1/Δfrep.

The single-cavity dual-comb laser (SCDCL) is an emerging dual-comb architecture that has been studied over the past few years in various platforms including on-chip microresonators [14], Kerr-lens mode-locked lasers [15], semiconductor disk lasers [16], and fiber lasers [17–22]. Therein, fiber SCDCLs are a particularly attractive architecture owing to their compactness and environmental robustness. Fiber SCDCL opens up the possibility for low-complexity dual-comb systems and can potentially lead to a paradigm shift in dual-comb light-source development [23,24]. In particular, we recently demonstrated the counterpropagating all-normal dispersion (CANDi) fiber laser that pushed the energy limit in fiber SCDCLs by 2 orders of magnitude to 8-nJ pulse energy in each comb [25,26]. 190-fs compressed pulse duration and 39-fs integrated relative timing jitter (1 kHz, 20 MHz) have also been demonstrated and characterized. All these features render CANDi fiber laser an ideal source candidate for ultrafast dual-comb pump-probe measurements.

Another obstacle to the widespread adoption of the dual-comb pump-probe technique is the fundamental trade-off between the time resolution and measurement speed, due to their opposite scaling laws with respect to the Δfrep [27]. In addition, for applications such as coherent Raman spectroscopy, where the vibrational coherence lifetime of most molecules is only a few picoseconds, the dual-comb pump-probe technique’s typical >1-ns total delay scan is much longer than necessary, resulting in a <1% measurement duty cycle and a significant waste of dual-comb power that effectively lowers the measurement sensitivity [12]. To break the fundamental trade-off and enhance the time and energy efficiency, a few elegant ways have been devised recently by nonmechanical cavity length control of one of the dual combs [11,13,28]. Impressive results such as orders-of-magnitude measurement speed enhancement without sacrificing resolution and sensitivity and near 100% duty cycle coherent Raman spectroscopy have been demonstrated [11,13]. However, none of these methods are compatible with the SCDCL architecture, let alone fiber SCDCL.

In this paper, we devise an all-optical dynamic Δfrep modulation technique that is applicable to SCDCL to overcome the fundamental trade-off between time resolution and measurement speed. We demonstrate a dynamic CANDi (DCANDi) fiber laser architecture and illustrate a dynamic Δfrep switching between >+10Hz and <−10Hz at kilohertz-level modulation frequency. We characterize the relative timing jitter of DCANDi fiber laser to be the same as that of CANDi fiber laser, confirming that our all-optical dynamic Δfrep modulation technique does not introduce excessive noise. Furthermore, the dynamic modulation mechanism is studied and validated both theoretically and experimentally. As a proof-of-principle experiment, we apply DCANDi fiber laser to measure the relaxation time of a semiconductor saturable absorber mirror (SESAM), achieving a measurement speed and duty cycle enhancement of 143 times compared to traditional dual-comb pump-probe measurement. DCANDi fiber laser is a promising light source for low-complexity, high-speed, high-sensitivity ultrafast dual-comb pump-probe measurements.

The simplified DCANDi laser configuration is shown in Fig. 1(a). The cavity of DCANDi laser is constructed similar to that in Ref. [25]. However, in order to bring Δfrep closer to zero, the lengths of the passive fiber on the two sides of the ytterbium-doped fiber (YDF) are deliberately kept the same, which makes the cavity symmetric for both directions. The fiber section consists of ∼2m long 6-μm-core YDF (Thorlabs YB1200-6/125DC) and ∼1m of HI1060 passive fiber on each side of the YDF. The length of gain fiber is chosen to be long enough to achieve the gain asymmetry for dynamic modulation (mechanism discussed in Section 3 and Appendix A) but short enough to avoid reabsorption loss for efficient lasing. The mode locking is achieved using nonlinear polarization rotation. Moreover, the YDF is pumped bidirectionally by two 980-nm laser diodes with same power level to make the gain distribution inside the YDF symmetric for both directions. In this way, Δfrep can be easily tuned to close to zero by fine-tuning the intracavity wave plates.

Under this condition, by modulating the current of either of the pumps (pump2 is modulated here) with a square waveform, the sign of Δfrep can be dynamically flipped, as schematically shown in Fig. 1(b). In order to make sure the absolute value of Δfrep is the same before and after flipping, a slow phase-locked loop (PLL) with 100 Hz control bandwidth is applied to pump1 to lock the average Δfrep to zero. More details on the PLL can be found in Appendix E. Since the control bandwidth is much smaller than the modulation frequency (kilohertz), the PLL does not counteract the modulation. When the average value of Δfrep is successfully locked to zero, the pulses from the two combs will temporally walk off each other alternatively as the sign of Δfrep flips [Fig. 1(b)] without needing to walk off for a whole round trip. As a result, the temporal interferogram frequency (i.e., the DC detection frequency) will be twice the modulation frequency, instead of Δfrep, as in a conventional DC system. In this way, the DCANDi system can dramatically increase its DC detection speed without sacrificing its temporal sampling resolution and pulse energy, thus providing an ideal tool for high-speed nonlinear DC applications.

The mechanism of pump power change-induced dynamic Δfrep modulation is mainly attributed to the nonidentical center frequency change coupled with intracavity group delay dispersion (β2), which is identified here through numerical simulation. The dependence of frep on pump power has been extensively studied in Refs. [25,26]. The change in frep due to change in pump power can be due to several factors, including spectral shift (ωΔ) coupled with β2, change in root mean square spectral width coupled with third-order dispersion, gain, and self-steepening terms. In Ref. [26], we identified the ωΔ term coupled with β2 to be the main coupling mechanism, resulting in pump-power-induced frep and Δfrep change in the CANDi laser. Considering only this term, the total pump-power-(P)-induced changes in frep and Δfrep are governed by the following equations, as explained in Ref. [26]: dfrepdP=−frep2×(β2dωΔdP),(1)dΔfrepdP=−frep2×(β2dωΔCWdP−β2dωΔCCWdP)=−frep2×(β2dδωΔdP).(2)

In the above equation, ωΔCW and ωΔCCW are the spectral shift in clockwise (CW) and counterclockwise (CCW) directions, while δωΔ is the relative spectral shift between the two directions. Since both combs in the DCANDi laser share the same pump, the mechanism for pump-power-induced change on Δfrep can be attributed to the asymmetric response of the two directions to pump-power change. When only one of the pumps is modulated, the gain distribution along the length of the YDF becomes asymmetric for the two directions. This results in unequal gain center shift, resulting in different spectral shift for the two directions for a given pump-power change. This causes unequal change in frep in two directions and hence a change in Δfrep.

To confirm this hypothesis, we developed a simulation routine to model DCANDi based on solving the generalized complex Ginzburg–Landau equation coupled with rate equations. Since the gain inside the YDF dynamically changes as the pump power is modulated, a key requirement of the simulation is to calculate the true gain by solving the rate equations and generalized complex Ginzburg–Landau equation simultaneously. Note that a steady state condition of upper state population and a perfect symmetric cavity is assumed in the simulation. The results of simulation are shown in the Figs. 2(a) and 2(b). Figure 2(a) illustrates the δωΔ as a function of round trip for a particular mode-locked state as the pump power is modulated. Note that the simulation is displayed for 600 round trips, with the pump power modulated every 100 round trips. Figure 2(b) shows the comparison of the simulated Δfrep deduced from the difference in pulse timing against Δfrep calculated from the values in Fig. 2(a) using Eq. (2). The simulated Δfrep matches well with the Δfrep calculated using β2 coupled with the δωΔ. This confirms that unequal gain center shift, resulting in different ωΔ for the two directions for a given pump-power change is the main reason behind dynamic modulation of Δfrep. More insights on the importance of longitudinal gain profile in causing pump-power-dependent δωΔ can be found in the Appendix section. Later, in Section 4, we experimentally confirm the findings of the simulation to cement our understanding on the mechanism. More details on the simulation and simulation results are included in Appendix A.

These simulation results help us understand the mechanism of Δfrep modulation in DCANDi laser and pave the way for custom designing DCANDis in the future. Beyond unraveling the mechanism of dynamic modulation in DCANDi, the simulation routine developed in this work contributes significantly to the broader field of SCDCL. The simulation can model complex interplay of parameters governing bidirectional SCDCL performance and help achieve precision controlled bidirectional SCDCL. The theoretical foundation discussed in this work can not only help optimize existing SCDCL designs but also lead to novel designs and applications.

Figure 3 shows the basic performance of the DCANDi laser. Under static state (no current modulation), the optical spectra of the two combs are shown in Fig. 3(a). The two spectra show high spectral overlapping, and both represent a “batman” spectrum typical for ANDi lasers. The DCANDi laser has a repetition rate of 48 MHz and pulse energies of about 2 nJ for both directions. Higher pulse energies can be obtained by replacing the current 6-μm-core fibers with 10-μm-core large-mode area fibers [26]. A single-pulse compressor made of transmission grating pairs is used to compress pulses from both directions to around 180 fs, with total transmission of over 85% (see Appendix B for more details). Therefore, the peak power for compressed pulses is about 11 kW, which directly enables many nonlinear DC applications. To intuitively show the operation of dynamic Δfrep modulation, we first modulate the current of pump2 with about 1 Hz frequency and simultaneously monitor the repetition rate of both combs using two frequency counters referenced to the same clock. The corresponding current modulation depth is around 3%, which does not destroy the mode-locking state, thanks to the wide mode-locking range of ANDi lasers. The resulting Δfrep is shown in Fig. 3(b), which periodically and deterministically switches from ∼+11 to ∼−11Hz. The local fluctuation results from the limited resolution of frequency counter under 0.03 s arm time. To demonstrate the real potential of dynamic operation of the DCANDi laser, the modulation frequency is then increased to 1 kHz and the PLL is turned on. The two combs are then combined through a 50/50 coupler and launched together into an amplified photodetector. Since the pulses from the two combs do not walk off monotonically when the average Δfrep is locked to zero, a variable optical delay line (ODL) is deployed on the path of one comb to make sure the pulses can temporally overlap each other at the PD. The output of the PD is low-pass filtered with 28 MHz; the corresponding temporal waveform is shown in Fig. 3(c). As expected, the temporal interferogram between the two combs shows a repeating frequency of 1 kHz, while two interferograms appear in each period, which results from forward and backward (shadowed with red) scanning. The time separations between a forward-scanning interferogram and two neighboring backward-scanning interferograms can be tuned to be identical by adjusting the relative delay between the two combs, leading to an effective DC detection speed of 2 kHz. The zoomed-in temporal waveforms of a forward-scanning and a backward-scanning interferograms are shown in Figs. 3(d) and 3(f), respectively, and their corresponding spectra are obtained by performing fast Fourier transform (FFT), as shown in Figs. 3(e) and 3(g). As observed, the two spectra are located at different center RF frequencies. This is due to the change of carrier-envelope-offset frequency (Δfceo) in response to the modulation of pump power. Nevertheless, both Figs. 3(e) and 3(g) show an RF spectrum that matches well with the overlapping optical spectrum in Fig. 3(a), which indicates the feasibility of fast and accurate DC metrology using the DCANDi laser.

In the previous section, using simulation, we identified the unequal ωΔ in the two directions coupled with β2 as the main reason behind the dependence of Δfrep on pump power change. Here, we experimentally confirm this by measuring the evolution of the pulse spectrum in both directions during the pump-power modulation using the dispersive Fourier transform (DFT) technique. More information on measurement setup can be found in Appendix F. The modulation frequency was set at 5 kHz. The measured spectral evolution is shown in the appendix, while the δωΔ calculated from the DFT measurements are shown in Fig. 4(a). Figure 4(b) shows the expected Δfrep calculated based on Eq. (2) and data from Fig. 4(a). The β2 coupled with δωΔ results in Δfrep change from ∼−11 to ∼+11Hz. This matches with magnitude of dynamic modulation in Δfrep achieved experimentally, as shown in the frequency counter measurement in Fig. 3(b). This validates the results from simulation and confirms that the unequal ωΔ change due to pump-power change contributes the most in dynamic modulation of Δfrep.

It should be noted that the difference in the modulation depth of Δfrep in simulation and experiments is due to the difference in the mode-locking state in the experiment and simulation and also due to the fact that the simulation assumes a perfectly symmetric cavity, while it could be slightly asymmetric in the real experiments due to human error. However, the fact that dynamic modulation can be achieved even in a perfectly symmetric cavity, as shown through simulation results, rules out the necessity of cavity asymmetry (asymmetry in passive fiber length) for creating unequal response of the two directions to pump power and strengthens our understanding of spectral shifting playing the major role in this process. Since the asymmetric gain distribution along the length of the YDF in the two directions is the primary cause for realizing the dynamic tunable DC presented in this study, such dynamic modulation can be achieved in any symmetric bidirectional SCDCL systems where such an imbalanced longitudinal gain in the gain medium can be achieved. This is another advantage of bidirectional geometry compared to dual-wavelength and dual-polarization SCDCL systems, along with lesser cross talk and overlapping spectra. Hence, this technique is widely applicable in bidirectional SCDCL geometries, such as soliton fiber DCs and free-space DCs, and is not limited to CANDi lasers.

To ensure the enhanced frame rate in DCANDi does not come at the expense of CANDi’s timing stability, we use DFT-based spectral interferometry to measure the relative timing jitter during dynamic modulation with femtosecond resolution. Experimental details on the measurement technique can be found in the appendix and in Ref. [26]. The spectral interferogram formed by the two separation-evolving comb pulses is shown in Fig. 5(a). The two positions around 140 and 290 μs, where the fringe density drops to zero, correspond to the times when the two comb pulses meet each other during the forward and backward scanning. By performing an FFT on the spectral fringes in each round trip, the evolution of pulse separation is reconstructed, as shown in Fig. 5(b). Furthermore, the corresponding Δfrep can be obtained by taking the derivative of separation evolution in neighboring round trips, as shown in Fig. 5(c). The local fluctuation at 140 and 290 μs was caused by the separation-retrieving error when the two pulses overlap with each other. The near-zero spectral fringe density makes the separation calculation inaccurate. Nevertheless, Figs. 5(b) and 5(c) clearly show the separation and Δfrep evolution during the switching event. As observed, the two pulses first walk off with each other with constant speed before 195 μs. The corresponding Δfrep was about 11 Hz. Then, the pump power dropped, which drove the Δfrep to the negative value. The transition process took around 90 μs, after which the Δfrep appeared stable at −11Hz. The duration of transition process is attributed to the laser response bandwidth, which is around 10 kHz. This also sets the upper speed limit for the dynamic Δfrep modulation. To verify the laser stability during dynamic operation, we calculate the relative timing jitter during 1000 consecutive round trips (48 μs) before and after the transition stage [marked by the red and blue dashed areas in Fig. 5(b)]. The corresponding power spectral densities (PSDs) and the corresponding integrated jitters are shown in Figs. 5(d) and 5(e), respectively. As observed, the PSDs for both time spans appear almost identical, and the integrated jitter (from 24 MHz to 20 kHz) is only around 5 fs, which is as stable as a conventional CANDi laser without modulation [26]. This result confidently proves that the dynamic modulation does not sacrifice the relative timing stability, thus providing a powerful technique for significantly enhancing the DC detection speed using SCDCL.

To demonstrate the application of high frame rate and temporal resolution of the proposed dynamic Δfrep modulation technique, we characterize the relaxation time constant of an off-the-shelf SESAM sample (from BATOP) using the DCANDi laser. The pump fluence incident on the SESAM is ∼58μJcm−2, while the probe fluence is ∼0.5μJcm−2. The pump-probe experiment was done in a reflection geometry, as shown in Fig. 6(a). The reflected probe was collected using a Si-switchable gain detector (Thorlabs PDA100A). The relaxation time constant is characterized in both conventional DC pump-probe and the dynamic Δfrep modulation DC pump-probe methods. The results are shown in Figs. 6(b)–6(e). In the conventional DC method, where the Δfrep was fixed at 14 Hz (temporal sampling time step is ∼6fs), Fig. 6(b) shows the collected pump-probe signal for 600 ms (nine pump-probe curves) and Fig. 6(c) shows the averaged SESAM response, which reveals a time constant of 5.18±0.18ps. The same experiment was repeated using the dynamic modulation of Δfrep at 1 kHz. Figure 6(d) shows the measured pump-probe signal using dynamic Δfrep modulation technique for 5 ms. The temporal sampling time step is maintained at ∼6fs. Figure 6(e) shows the average SESAM response, revealing a fitted time constant of 5.19±0.22ps. The number of pump-probe curves used for calculating the average response is kept at nine for both the techniques to ensure the same SNR.

Figure 6 confirms a frame rate enhancement of ∼143 (2000 Hz/14 Hz) without sacrificing resolution, sensitivity, and accuracy. Of note, while similar enhancement has been demonstrated in two-laser dual-comb systems [11,13], this is the first dynamic Δfrep modulation demonstration using an SCDCL. The inherent common-mode-noise cancellation of SCDCLs helps in passively reducing relative timing jitter, which opens up the possibility for low-complexity dual-comb pump-probe measurements. Apart from frame rate enhancement, as a byproduct, assuming the noise is uncorrelated, one can also expect to obtain a higher SNR in the proposed dynamic Δfrep modulation technique for same averaging time. From the standard deviation of the fitted response in both cases, it is also clear that dynamic modulation measurement has a similar error (or SNR) compared to the conventional case. Hence, the suggested technique can be used for employing higher frame rate or higher SNR depending on application.

In summary, we demonstrate an all-optical technique to break the measurement speed-time resolution trade-off in SCDCL dual-comb pump-probe measurements. We achieve the all-optical dynamic Δfrep modulation in a dual-pumped CANDi fiber laser by applying pump-power modulation to one of the pumps and showcase its basic operation. We present a simulation routine to accurately model the complex interplay of parameters governing bidirectional SCDCLs and use it to identify the unequal change in center frequency due to the asymmetric longitudinal gain distribution in the YDF as the mechanism of dynamic Δfrep modulation of the DCANDi fiber laser. In addition, we confirm that the demonstrated all-optical dynamic Δfrep modulation technique does not introduce excessive relative timing jitter and identify that the measurement speed is limited by the laser response bandwidth to 10 kHz. As a proof-of-principle experiment, we apply the DCANDi fiber laser to measure the relaxation time of an off-the-shelf SESAM and achieve a 143-fold enhancement in measurement speed and duty cycle. The DCANDi fiber laser is a promising light source for low-complexity, high-speed, high-sensitivity ultrafast DC pump-probe measurements. The working principle can be generalized to other SCDCL platforms where gain asymmetry in the two directions dominates the Δfrep, thus representing a universal solution for breaking the measurement speed-time resolution trade-off in bidirectional SCDCLs. In particular, gain media with a shorter upper-state lifetime, such as highly doped Yb multicomponent glass [29], phosphate glass fiber [30], and Ti:sapphire crystal [31] can be utilized to further enhance the dynamic modulation speed above 10 kHz.

Acknowledgment. The authors acknowledge the Optics and Photonics Research Group at the University of Colorado Boulder supervised by Dr. Juliet Gopinath for providing the SESAM used in this work.