Introduction. Mode-locked fiber oscillators produce highly stable ultrafast optical pulse trains that drive versatile applications in science and technology, including frequency metrology [1], timing and synchronization in large scale research facilities [2], and high-resolution nonlinear biological imaging [3]. Over the past few decades, extensive research has therefore been directed at the development and improvement of high-performance oscillator setups. In particular, oscillators mode-locked with artificial saturable absorbers (SAs) based on the optical Kerr-effect, e.g., via implementation of nonlinear amplifying/optical loop mirrors (NALM/NOLM) [4–6], are known for their superior environmental stability, versatile mode-locking capabilities, and remarkable noise performance [7,8]. In contrast to NPE oscillators in ring configurations [9], NALM/NOLM oscillators leverage all-polarization-maintaining (PM) fiber segments, significantly enhancing their long-term stability and resilience against environmental perturbations.

In recent years, substantial research has been devoted to an alternative oscillator structure mode-locked via all-PM linear self-stabilized fiber interferometers (LSI) [10–13]. This concept, initially proposed by Fermann et al. in 1994 [14], provides a more streamlined cavity structure compared to NALM/NOLM mode-locked lasers and further offers new ways for adaptations, such as the recently demonstrated implementation of a coherent pulse divider for efficient energy scaling and noise reduction [15].

At first glance, the working principle of the established NALM mode-locked oscillators, such as those in the Figure-of-9 configuration, appears closely related to that of LSI mode-locked oscillators. During each round trip within both cavities, the intra-cavity pulse undergoes a separation into two modes: two counterpropagating pulses in the NALM configuration and co-propagating polarization modes in the LSI configuration’s PM-fiber segment. In both cases, these modes accumulate a differential nonlinear phase shift $\Delta {\varphi _{nl}}$, which is subsequently converted into self-amplitude modulation (SAM) through the interaction with a polarizing element, such as a polarization beam-splitter (PBS).

However, the LSI oscillators introduce a unique element: the temporal overlap of both polarization modes in the fiber enables nonlinear phase distortions via cross-phase modulation (XPM) each round trip, a phenomenon not present in NALM lasers.

This study investigates the impact of XPM on the performance of LSI mode-locked fiber lasers and gives straight-forward guidelines for significant improvements. Numerical simulations reveal that XPM causes significant modulations in the output spectra via distortions of the artificial SA mechanism. We verify this finding experimentally by constructing an LSI mode-locked fiber oscillator with XPM suppression using an intra-cavity birefringent YVO₄ crystal and compare it to a reference oscillator in a standard configuration. In agreement with the numerical results, the XPM suppression enables enhanced laser performance, i.e., a significant reduction of characteristic spectral modulations at the laser output ports, a reduction in mode-locking threshold by more than 45%, and a reduced nonlinear loss in the artificial saturable absorber mechanism.

Experimental setup and working principle. The experimental setup of the enhanced/reference LSI mode-locked fiber oscillator is illustrated in Fig. 1. Its all-PM fiber segment includes a 0.7 m highly ytterbium-doped fiber (YDF, CorActive Yb401-PM) optically pumped with a 1 W laser diode at 976 nm that is coupled to the YDF through a wavelength-division multiplexer (WDM). The free-space arm at the side of collimator C1 contains the cavity end-mirror a 1000 lines/mm transmission grating pair (LightSmyth 1040-Series) for dispersion-management, a polarization beam-splitter (PBS) and the non-reciprocal phase-bias consisting of a Faraday-rotator (FR, 45° single-pass), a half-wave plate (HWP) and an eight-wave plate (EWP) with tunable rotation angles ${\theta _H}$ and ${\theta _E}$, respectively. The quarter-wave plate (QWP) serves as a tunable output coupler to monitor the intra-cavity field at Port T of PBS1. An additional half-wave plate (HWP1) is implemented to align the slow axis of the PM-fiber to the transmission axis of PBS1.

 

Fig. 1. Experimental setup of the enhanced LSI mode-locked fiber oscillator with an intra-cavity YVO₄ crystal and a monitor port for monitoring the separated polarization modes. The reference setup is obtained by removing the YVO₄ crystal. GP, grating pair; QWP, quarter-wave plate; PBS, polarization beam-splitter; EWP, eight-wave plate: HWP, half-wave plate; FR, Faraday-rotator; C, collimator; WDM, wavelength-division multiplexer; YDF, ytterbium-doped fiber; OC, output coupler; ISO, isolator; YDFA, ytterbium-doped fiber amplifier.

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Each round trip in the LSI cavity, the phase-bias rotation angles ${\theta _E}$ and ${\theta _H}$ generate orthogonal polarization modes parallel to fast and slow axes of the PM-fiber with a fixed energy-splitting ratio and a phase-offset. For an asymmetric energy-splitting ratio, both modes accumulate a nonlinear phase difference $\Delta {\varphi _{nl}}$ via the optical Kerr-effect. Simultaneously, the fibers’ birefringence leads to a difference in the linear phase shift, causing a walk-off between both modes by ∼1.25 ps/m (PM980-XP fiber at 1030 nm). To compensate this drift-off, an adapted Faraday-rotator mirror (FRM) is attached at the side of C2, consisting of a FR (45° single-pass) and an 80:20 output coupler. After the double pass through the PM-fiber, the compensation of the linear phase difference ensures the overlap of both polarization modes in time with an accumulated $\Delta {\varphi _{nl}}$ that results in a nonlinear polarization rotation (NPR) of the recombined field. At the PBS, this NPR results in a separation of the intra-cavity field into reflected (Port R) and transmitted (Port T) components, depending on the characteristic sinusoidal transmission function $T({\Delta {\varphi_{nl}}} )$. The shape and offset of $T({\Delta {\varphi_{nl}}} )$ are tunable via ${\theta _E}$ and ${\theta _H}$ and fully characterize the artificial SA [].

The slow walk-off between the polarization modes during the nonlinear propagation in the PM-fiber enables phase distortions via XPM. The setup can take the form of a reference setup (RS) with XPM and an enhanced setup (ES) with XPM suppression. For the ES, a highly birefringent 20 mm long YVO₄ crystal is included between C1 and HWP1 to pre-separate both modes in time for suppressing any nonlinear interaction via XPM in the fiber segment.

Numerical simulations. To investigate the influence of XPM on mode-locked steady-states in LSI oscillators, numerical simulations are conducted. To include the XPM, the nonlinear propagation in the fiber segment is described via coupled nonlinear Schrödinger equations (NLSE) similar to Ref. [16]. The model further includes self-phase modulation (SPM), chromatic dispersion up to the second order [17,18] and gain based on rate-equations which account for the double pass configuration [19]. The artificial SA is described via the Jones formalism to derive the SA transmission $T({\Delta {\varphi_{nl}}} )$ as a function of the differential nonlinear phase shift $\Delta {\varphi _{nl}}$, similar to Ref. [7]. The simulation parameters are matched to the experimental setup in Fig. 1, with ${\theta _H} = 52^\circ $ and ${\theta _E} = 81^\circ $ for optimum starting conditions with a maximum positive slope at $T({\Delta {\varphi_{nl}} = 0} )$. The QWP rotation angle is set to ensure 40% output coupling ratio to Port T. The free-space cavity loss, considering the insertion loss of the GP, the coupling efficiency of C1/2 and the OC is lumped at the sides of C1 and C2 with 70% and 40%, respectively.

For the RS in mode-locked steady-state, Fig. 2(a) shows the simulated evolution of the intensity-dependent $\Delta {\varphi _{nl}}$ influenced by the XPM as a function of the propagation distance in the RS fiber segment. Here, the position marked by 0 m corresponds to the position of the Faraday-rotator in front of the OC in Fig. 1. The simulated Port T/R output spectra of the RS are shown in the inset of Fig. 2(a), demonstrating significant spectral modulations. Similar distortions have been reported over a broad range of LSI oscillator cavity parameters and mode-locking regimes in a variety of experimental studies; e.g., in Refs. [10–13]. The occurrence of these spectral modulations can be explained with two interacting mechanisms: the nonlinear propagation of the drifting orthogonal polarization modes resulting in an XPM-driven creation of a satellite pulse (see S1 in Supplement 1) and the modulation of $\Delta {\varphi _{nl}}$ shown in Fig. 2(b), which is directly proportional to the SA transmission [5,7].

 

Fig. 2. (a) Simulated evolution of $\Delta {\varphi _{nl}}(t )$ in the RS with XPM as function of the fiber segment propagation distance together with the position-dependent polarization mode delay. The inset shows the resulting output spectra at Port T (purple) and Port R (blue). (b) Corresponding total round trip $\Delta {\varphi _{nl}}(t )$. (c) Evolution of $\Delta {\varphi _{nl}}(t )$ in the ES with XPM suppression. Inset: steady-state Port T (purple) and Port R (blue) spectra. (d) Total accumulated round trip $\Delta {\varphi _{nl}}(t )$.

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In contrast, Fig. 2(c) shows the evolution of $\Delta {\varphi _{nl}}$ with identical cavity parameters but in the ES with XPM suppression via the YVO₄ crystal. In this case, an interaction of polarization modes via XPM is prevented by a pre-separation in time as illustrated. The avoidance of XPM prevents modulations in the accumulated $\Delta {\varphi _{nl}}$ (Fig. 2(d)) and the creation of a satellite pulse as shown numerically in S1 of Supplement 1, therefore mitigating both spectral modulations and nonlinear distortions of the SA transmission. As a consequence, smooth output spectra at Ports T and R (inset of Fig. 2(c)) can be obtained.

Experimental results and discussion. To investigate the influence of the XPM suppression experimentally, comparative measurements are conducted on the RS and ES. To ensure comparability, both setups are operated in a stretched-pulse mode-locking regime with ${\sim}{-} 5\ast {10^3}\; f{s^2}$ net dispersion. For both RS and ES, the cw-laser threshold is measured at ∼60 mW, verifying a negligible influence of the YVO₄ crystal on the cavity alignment and linear loss.

To investigate the influence of XPM on the mode-locking threshold, Fig. 3(a) illustrates both oscillators’ slope efficiencies $d{P_{out}}/d{P_{p\; }}$ (${P_p}$, pump power; ${P_{out}}$, Port T output power) together with the corresponding power transfer curve (inset). For this measurement, the pump power is stepwise increased in 25 mW steps up to a maximum of 1.5 W. The transition from cw to the energetically more efficient mode-locked state is clearly visible by a sharp increase in the slope efficiency. Compared to the RS with a measured mode-locking threshold at 1.225 W, the avoidance of XPM in the ES results in a reduced threshold at 0.7 W, corresponding to ∼45% less required pump power. The reduced starting ability of the RS can be explained with XPM-induced distortions and the resulting increase in the nonlinear loss of the artificial SA mechanism that is verified later on.

 

Fig. 3. (a) Measured slope efficiency $d{P_{out}}/d{P_p}$ as a function of the pump power ${P_p}$ for the RS (yellow) and ES (blue). The inset shows the corresponding output versus pump power characteristic ${P_{out}}({{P_p}} )$. (b) AC trace of the polarization modes measured at Port M of the RS. (c) Corresponding AC trace of the ES.

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Once mode-locking is initiated, a stable single-pulse operation can be obtained by reducing the pump power to ∼50 mW for the RS and 55 mW for the ES. Figures 3(b) and 3(c) show AC traces of the separated polarization modes with a maximum delay measured at Port M in a single-pulse regime of the RS and ES, respectively. In the case of the RS, the measured ∼3 ps walk-off is entirely caused by the PM-fibers’ birefringence. For the ES, the YVO₄ crystal with a birefringence of 0.208 at 1030 nm and 20 mm length causes a pre-separation of ∼16 ps. In the experiment, the fast axis of the YVO₄ is aligned parallel to the slow axis of the PM-fiber, resulting in the measured walk-off by ∼13 ps shown in Fig. 3(c).

In LSI mode-locked oscillators, the output power at Port T is directly proportional to the intra-cavity power, while the Port R output power reflects the loss of the artificial SA. A comparison of their power ratio therefore yields information about the relative nonlinear loss introduced by the artificial SA mechanism, independent of the pump power. The measured output power for the RS at Port T and Port R is 0.6 and 2 mW corresponding to pulse energies of 16.7 and 55.6 pJ, respectively. Conversely, the measured output power at the corresponding ES output ports is 1.4 and 0.9 mW with pulse energies of 38.9 and 25 pJ, respectively. Notably, the power ratio between Port T and Port R is inverted between RS and ES. Considering the verified identical linear cavity loss for ES and RS, the suppression of the XPM in the ES therefore seems to improve not only the artificial SAs self-starting capability (Fig. 3(a)) but further its nonlinear transmission [7].

To further investigate the characteristics of this energy transfer difference and also the general quality of the output pulses in both configurations, Figs. 4(a) and 4(b) illustrate the Port T/R output spectra measured for RS and ES, respectively. The low pulse energies at the output ports did not allow for corresponding autocorrelation measurements. Instead, a detailed numerical time-domain analysis is provided in S1/Supplement 1.

 

Fig. 4. (a) Output spectra of the RS in stretched-pulse regime measured at Port T (purple) and Port R (blue). (b) Spectra measured at Port T (purple) and Port R (blue) of the ES with XPM suppression. The insets show RF spectra measured at Port T of the respective configuration.

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To verify the occurrence of periodic spectral perturbations predicted by the simulations with high fidelity, the output spectra are measured with an optical spectrum analyzer (ANDO AQ6315A) at a high spectral resolution of 0.02 nm. The insets in Figs. 4(a) and 4(b) show the broadband RF spectra measured at Port T of RS and ES, respectively, measured with a fast photodetector (ET-3500AF) and a RF spectrum analyzer (Agilent N9030A). The absence of modulations in the amplitude of the higher harmonics is monitored to verify the single-pulse operation.

As shown in Fig. 4(a), the measured RS output spectra at Ports T and R show strong modulations with a full width at half maximum (FWHM) of 13.8 and 18.1 nm respectively. In agreement with the simulations, the spectra at Ports T and R of the ES in Fig. 4(b) have significantly reduced spectral modulations with 13.4 and 12.2 nm FWHM, respectively. Notably, the influence of the XPM suppression on Port R’s spectral shape is larger, possibly due to a phase-bias dependent distribution of XPM-distorted spectral components to the output ports as recently discussed in Ref. [20]. Nevertheless, it is clearly verified that the XPM suppression is a crucial precondition for the high spectral output quality from LSI oscillators comparable to that of e.g., NALM/NOLM and NPE oscillators.

Conclusion. In conclusion, we demonstrated the significant potential of XPM suppression for performance enhancement of all-PM LSI mode-locked fiber oscillators. Numerical simulations are conducted to verify the influence of the XPM as major contribution to distortions of the saturable absorber mechanism an output pulse quality. In the experiment, an oscillator with XPM suppression via the intra-cavity YVO₄ crystal is constructed and compared to a reference laser. In strong agreement with the numerical results, it is shown that the suppression of XPM results in significant improvement of the output pulse quality. Further, a reduction of the mode-locking threshold by more than 45% is verified. This demonstrated performance enhancement will greatly increase the applicability of LSI mode-locked fiber lasers and enable competition with high-performance NALM/NOLM mode-locked oscillators.