During the past few decades, ultrafast lasers have been well-developed and widely employed in various fields, such as fundamental scientific research^[1], ultrafast biophotonics^[2], and high-precision laser processing^[3]. Generally, parameters such as the average power, pulse energy, pulse duration, and pulse repetition rate are first noticed. Another important parameter, the temporal quality or contrast of pulses, is also characterized when high-field physics is referred to^[4], for which the involved lasers usually operate at very low repetition rates of far less than 10 Hz. As far as the high-repetition-rate ($>100\mathrm{kHz}$) high-average-power ultrafast lasers are concerned, temporal pulse quality is not always mentioned, which, however, is becoming a new issue when laser processing quality is cared about^[5-7] because of the appearance of the pedestals.

For high-power ultrafast lasers with repetition rates of higher than 100 kHz, both all-fiber and hybrid “PM-fiber + bulk crystal” configurations are effective and promising technological approaches^[8,9]. However, we recently noticed such a problem of dense spectral modulations in common all-PM-fiber^[10-12] and hybrid “PM-fiber + bulk crystal”^[13-18] ultrafast lasers. For the worst conditions, the modulated spectral valley could be down to the bottom, and the modulation periods are even less than 0.5 nm (roughly calculated from the spectra given in Refs. [10,15,18]). Here, one thing to mention is that the spectral modulations can only be clearly resolved when the optical spectrum analyzer (OSA) is in high-resolution mode; otherwise, the modulation details can be smoothed. Therefore, the real modulation periods could be even denser because of the unknown resolution choices in reported literatures, as well as the limitations due to the highest resolution of available OSAs.

To unveil the mechanism that is responsible for severe spectral modulations in PM-fiber-based ultrafast lasers, some explanations have been proposed, and without exception, nearly all were ascribed to the self-phase modulation (SPM) effect^{[10,12,14,19,20]}. Affirmatively, the SPM resulting from the optical Kerr effect plays a prominent role in high-power and high-energy lasers and manifests itself by the broadening of the spectral bandwidth, the extent to which depends on how large the B-integral is^[21]. Notably, in Refs. [10,11,18], all the B-integrals were below $5\pi$, but the number of modulated spectral peaks was much more than the SPM effect predicted one (roughly equal to the ratio between the B-integral and $\pi$^[21]. Therefore, there must be another underlying mechanism responsible for this phenomenon.

Apart from the SPM, other explanations are also thought to be the reasons for the dense spectral modulations. The chirped pulse fiber Bragg grating (CFBG) was considered to be a source of the original spectral ripple for the chirped pulse amplification systems^[11]. However, it is inapplicable to non-chirped pulse amplifiers^[22]. Additionally, the surface reflection of optical elements is believed to be another potential source of primary spectral modulations^[18], whereas significant spectral modulations have been rarely reported in all-solid-state crystal-based ultrafast lasers^[23,24]. Additionally, the sub-pulses introduced by the surface reflection of fiber-coupled optical elements should follow behind the main pulse with a fixed temporal delay along the PM fibers, thus introducing the spectral modulation with an unchanged modulated period. Theoretically, the two pulses can be distinguished using autocorrelators for this condition, but it was hardly ever mentioned in those reports with the significant spectral modulations and was not observed during our experiments either. Therefore, neither chirped devices nor surface reflection of optical elements can be recognized as the major inducers of spectral modulations.

Therefore, although many researchers have attempted to explain this phenomenon, a unified and widely accepted explanation has not yet been formed. It is common sense that spectral modulations typically correspond to the presence of pulse pairs with a short time delay in the temporal domain. Polarization mode dispersion (PMD), arising from the fiber birefringence, is defined as the time delay per unit length experienced by different polarization components propagating in fibers. Although the PMD coefficients of common non-PM fibers are small (generally at the level of $\sim 1.0\times {10}^{-4}\mathrm{ps}·{\mathrm{m}}^{-1}$), it remains a key limiting factor for high-data-rate and long-distance fiber-optic communications^[25]. In contrast, the PMD coefficients of PM fibers are nearly 4 orders of magnitude higher than those of non-PM fibers, as summarized in Table 1. This substantial difference results in significant temporal delays when different polarization components travel in PM fibers. Besides, the PM fibers feature very high but limited PM abilities; therefore, the main pulse and depolarized sub-pulse can transmit along slow and fast axes at the same time. We also noticed that the PMD has been pointed out as the main limiting factor for the realization of mode-locked fiber lasers based on the well-known nonlinear polarization rotation (NPR) effect. To compensate for the PMD, one can employ the cross-splicing method, as demonstrated in Ref. [26]. Therefore, we naturally suppose that maybe PMD is the right reason responsible for dense spectral modulations.

Based on the relevant reports and our experimental results, we believe there are two main reasons that account for the current explanation of the dense spectral modulation in PM-fiber-based ultrafast laser systems: (I)In the frequency domain, practical PM fiber amplifiers typically exhibit high signal power and energy, which are consequently accompanied by significant nonlinear effects. The spectral modulations are the joint results of PMD and SPM, which makes it very hard to distinguish between the PMD and SPM contributions directly from the modulated spectra.(II)In the time domain, it is common sense that the main pulse and sub-pulse can be measured simultaneously using an autocorrelator once the spectral modulation is induced by the interference of two laser pulses with a temporal delay. However, commercial autocorrelators commonly used in laboratories have the polarization-dependent property, such as pulseCheck autocorrelators from APE company. Consequently, two pulses with the same polarization can be measured by autocorrelators simultaneously, but only one pulse can be recognized for two cross-polarized pulses. So, the sub-pulses that arise from the surface reflection of optical elements are easily resolved by autocorrelators. We believe this is also a key reason that people naturally attribute the spectral modulation to the SPM effect and the surface reflection of optical elements but completely ignore the effect of PMD.

In this Letter, based on the above analysis and the existing research results, we propose that PMD is a primary mechanism for the dense spectral modulation in PM-fiber-based ultrafast amplifiers, which is extensively investigated by both numerical calculations and practical experiments.

In an ideal PM fiber, a linearly polarized laser pulse can maintain its single polarized state along the slow axis throughout its propagation, as depicted in Fig. 1(a). However, for a real PM fiber, the limited polarization extinction ratio (PER) makes it behave akin to a combination of an ideal PM fiber and an angled half-waveplate (HWP), as illustrated in Figs. 1(b) and 1(c). The angle of the HWP is defined as the relative axis difference between the HWP and the slow axis of PM fiber. Meanwhile, the PMD introduces a temporal delay between the main pulse and the depolarized sub-pulse. Furthermore, the two modes can be amplified to the higher power levels simultaneously, thus leading to more severe mode interference, as shown in Fig. 1(d).

The numerical simulations were first conducted to analyze the mechanism of spectral modulations in both the frequency and temporal domains. The envelope of spectral modulations between the main pulse and sub-pulse can be expressed as^[27]$$I_M(\omega )=A^2+1+2A·\cos (B·\omega +\phi ),$$where $A$ represents the modulation depth, which is associated with the powers of the main pulse and sub-pulse; $B$ is the modulation frequency, which is associated with the pulse delay between the main pulse and sub-pulse; and $\phi$ is the phase shift.

The numerical simulations were mainly conducted to verify the relationship between modulated periods and the PMD effect, as well as between the modulated amplitude and PER of PM fibers. Therefore, passive PM fibers rather than gain fibers were adopted for calculation to avoid the effect of wavelength-dependent signal gain, which can influence the modulated amplitude and make it difficult to distinguish the real effect of the PER on spectral modulation. Additionally, to simplify the calculation and clearly show the effect of PMD on the spectral modulation, the ideal passive PM fibers (without considering the nonlinear effects, dispersion, and depolarization) as shown in Fig. 1(a) were adopted for the simulation. Since mode interference primarily depends on the relative intensity and temporal delay between the main pulse and sub-pulse, rather than on real-time pulse durations, neglecting the pulse stretching effect that arises from the second-order dispersion of PM fibers in simulation was considered a reasonable simplification.

In addition, an 80 fs ultrashort pulse, corresponding to a Fourier-transform-limited Gaussian spectrum with a spectral bandwidth (FWHM, full width at half-maximum) of 20 nm and a central wavelength of 1030 nm, was adopted as the initial signal, as shown in Figs. 2(a) and 2(b). Furthermore, 4096 points were used for the simulation, and no optical filters were introduced into the calculation to avoid its influence on the calculated spectra and pulses.

First, spectral evolution versus parameter B was simulated with a fixed parameter A of 0.1, as shown in Figs. 2(c) and 2(d). The sub-pulses were gradually shifted from $\sim 650$ to $\sim 3250\mathrm{fs}$ with parameter B being varied from 5 to 25, companied by a corresponding decrease in modulation periods from about 5.2 to 1.0 nm, as shown in the zoomed-in images of Fig. 2(d). Second, the spectral evolution versus parameter A was analyzed with a fixed parameter B of 15, as depicted in Figs. 2(e) and 2(f). As parameter A increased from 0.1 to 0.5, the modulation periods remained unchanged but with a deepened spectral valley, concurrently accompanied by a decrease in the pulse contrast ratio.

The simulated results demonstrate that the spectral modulation periods are primarily determined by the temporal delay between the main pulse and sub-pulse, which closely correspond to the delays induced by PMD on the same order of magnitude. Meanwhile, the depths of the spectral modulations are strongly dependent on the relative intensities of the main pulse and sub-pulse, characterized by the pulse contrast ratio.

To measure the main pulse and sub-pulse simultaneously, an experimental setup based on a Wollaston prism (WP) was arranged to measure the cross-polarized pulses, as shown in Fig. 3. First, the main pulse and depolarized sub-pulse, which correspond to the slow and fast axis of PM fiber, respectively, are divided into two beams by the WP. Then, one beam with an S-polarization state was directly injected into the autocorrelator, while the polarization of another beam was translated from S-polarization to P-polarization by an HWP. Additionally, the time synchronization between two beams was realized with a translation stage.

The seed pulse was first injected into the system for time synchronization. As shown in Fig. 4, the sub-pulse gradually shifted closer to the main pulse with the decrease of optical path difference ($L$) between the two beams. Finally, when $L$ was zero, the two pulses were fully overlapped with the narrowest pulse duration. In this way, the extra time delay that accumulated in the system was effectively eliminated, and the depolarized sub-pulse and main pulse were successfully retrieved at the same time in this work.

Next, a multi-stage PM fiber amplifier was arranged to verify the plausibility of simulation results. As shown in Fig. 5, the seed was a passively mode-locked fiber oscillator with a central wavelength of 1034 nm, a spectral bandwidth of 15 nm, and an average power of 10 mW. An obvious Kelly sideband appears at $\sim 1045\mathrm{nm}$ because of the soliton mode-locking state of the oscillator. The PM fiber amplifiers included a single-mode (PM-YDF-6/125) and two multi-mode (PLMA-YDF-10/125, PLMA-YDF-25/250) fiber amplifiers, with fiber coil diameters of about 8, 10, and 15 cm, respectively. Accordingly, the total fiber lengths were around 400, 440, and 550 cm. An HWP combined with two fiber collimators was inserted between each stage of fiber amplifiers to adjust the polarization state of the signal. The zero-degree HWP corresponds to the slow-axis working state for PM fibers. An OSA (AQ6370D, YOKOGAWA) with a maximum spectral resolution of 0.02 nm was adopted, ensuring the capture of modulated details.

Before the amplification, the seed laser was first injected into PM-980 fibers to analyze the spectral properties. In this test, the angle of the HWP and the fiber length correspond to parameter A and B of the simulations, respectively. As shown in Figs. 6(a) and 6(b), holding $\theta$ constant at 15°, the modulation periods were gradually reduced with the increase of fiber length, and the pulse delays were shifted from about 9.6 to 19.6 ps accordingly. Notably, the pulse delay in real PM fibers is accompanied by a pulse stretching effect due to second-order dispersion. For a fixed fiber length, the pulse contrast ratio was decreased as θ changed, accompanied by an increase in modulation depth, as depicted in Figs. 6(c) and 6(d). These results aligned well with the numerical simulations.

In real PM fiber amplifiers, the sub-pulse primarily originates from the depolarization, rather than the tuning of waveplates. Figure 7 illustrates the evolution of the signal power and PER with respect to the pump power. The PER is defined as the ratio of power in the principal polarization mode to that in the orthogonal one, expressed as $\mathrm{PER}=10·\log (P_{\text{principal}}/P_{\text{orthogonal}})$. The PM-YDF-6/125 fiber exhibits the best PM capability with a PER of 23 dB, as well as the weakest spectral modulation among the three stages of fiber amplifiers, as shown in Fig. 8. However, the PER rapidly declines from 21.9 dB for PLMA-YDF-10/125 fiber to 12.2 dB for PLMA-YDF-25/250 fiber amplifiers.

To minimize the impact of SPM, the spectral properties of fiber amplifiers were investigated at a low power within 400 mW. The accumulated B-integral reached approximately $0.9\pi$ and $1.1\pi$ for PLMA-YDF-10/125 and PLMA-YDF-25/250 fiber amplifiers, respectively. The larger mode area of PLMA-YDF-25/250 fiber contributes to the reduction of the nonlinear effect but with a worse PM ability. Consequently, the PLMA-YDF-25/250 fiber always exhibited a stronger modulation intensity than PLMA-YDF-10/125 fiber amplifiers at the same power levels. As depicted in Fig. 9, both the PLMA-YDF-10/125 and PLMA-YDF-25/250 fiber amplifiers show the same evolutions with the polarization modes in PM-980 fibers.

Here it needs to be pointed out that the smaller modulation periods of the PLMA-YDF-25/250 fiber amplifier compared with the PLMA-YDF-10/125 fiber amplifier were attributed to the longer fiber length with reduced modulation periods. Therefore, PMD combined with the PER provides a more plausible explanation than SPM for severe spectral modulations in PM-fiber-based ultrafast lasers.

Because of such effects as the SPM, cross-phase modulation (XPM), and signal amplification being neglected in the numerical simulation and experiments based on the passive PM-980 fibers, the spectral modulations were solely determined by the PMD effect. Therefore, the spectral modulation period could be clearly resolved in the spectra. However, the spectral envelopes and peaks varied slightly during amplification in multi-mode fiber amplifiers (PLMA-YDF-10/125, PLMA-YDF-25/250) due to the minor nonlinear effects and the difference of wavelength-dependent signal gain. In these conditions, the spectral modulation period induced by the PMD could not be fully distinguished from other effects. The measured spectral modulation period may not fully agree with the real pulse intervals induced by PMD. Therefore, here we did not add the zoomed-in images of the detailed spectral modulation periods into Fig. 9.

In conclusion, the mechanism of spectral modulations in PM-fiber-based ultrafast amplifiers is thoroughly investigated by both numerical simulations and experiments in this work. To the best of our knowledge, the PMD is first proved to be a crucial factor of spectral modulations. Actually, all effects may need to be taken into consideration in practical applications, particularly for high-power and high-energy laser systems, and PMD should be prioritized when understanding initial spectral ripples, as schematically illustrated in Fig. 10. Notably, this model theoretically applies to chirped and unchirped pulse amplification systems.

In theory, a better PM ability of fibers can result in milder spectral modulations. Nevertheless, commercially available PM fibers are still limited in their PER. As a practical solution, the cross-splicing of PM fibers could be a viable method to compensate for the pulse delay between the main pulse and sub-pulse, thereby mitigating the impact of spectral modulations.