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Abstract
What we believe to be a first theoretical study of transverse mode instability (TMI) in oscillators based on a stimulated thermal Rayleigh scattering (STRS) model is conducted. Higher order mode (HOM) lasing is found to happen at high powers. Further fundamental mode (FM) growth is limited once HOM lasing takes place, with further increase of pump power mostly going to HOM growth, a fundamentally different phenomenon from that in fiber amplifiers. TMI thresholds defined as when the HOM lasing condition is met is studied. The results are consistent with the measured TMI thresholds and their dependence on pumping configurations and pump wavelengths.
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1. Introduction
High power diffraction-limited fiber lasers have a wide range of applications. It is now well understood that transverse mode instability (TMI) is the major limit to further average power scaling [1–12]. Recently, an accurate TMI model has been developed based on combining a stimulated thermal Rayleigh scattering (STRS) model and a quasi-three-dimensional (3D) fiber amplifier model [10]. The STRS approach has been well established in several theoretical studies [3,4,10,11,12]. It is also supported by direct experimental observations [13]. Recently, the standard STRS model has been expanded, by including four wave mixing (FWM) effects, to fully describe dynamic TMI effects in high power amplifiers [14].
The addition of the quasi-3D fiber amplifier model enables the evaluation of gain saturation both across the core and along the fiber to accurately assess its impact on TMI [10]. The model is bench-marked to the full 3D model in Ref. [5]. It is very efficient and, more importantly, agrees well with experimentally measured TMI thresholds in fiber amplifiers [10], giving significant credence to its accuracy. Strong TMI suppression under gain saturation was found to be a dominating factor in TMI and can explain many of the experimentally observed behaviors.
While there have been many theoretical studies of TMI for fiber amplifiers [1–12,14], there has been very few for oscillators. There is a theoretical study of a double-pass amplifier which predicts static instability [15], in contrast to the traveling-wave instability in STRS. There is, however, no evidence for this so far in the numerous experimental studies of TMI in fiber oscillators. There are two recent studies of TMI for oscillators based on stability analysis of a dynamic model [16,17], along the lines of stability analysis applied for the first time in high power fiber amplifiers in Ref. [8]. As in the case of amplifiers, the oscillator TMI is attributed to FWM effects [17]. However, these models are based on the constant-field cavity approximation, strictly applicable to oscillators with very strong output coupler grating reflectivity (close to 1), which is a significant departure from practical fiber oscillators. The underlying physics is less clear in these mathematics-based models despite the simplified analytical analysis.
A double-pass amplifier was experimentally studied in Ref. [18], finding that it has a lower TMI threshold of ∼150W than the ∼230W in a single-pass amplifier. This may just simply be due to that high-order mode (HOM) was fed back for a second pass through the same amplifier, producing an enhanced TMI on the second pass. An oscillator is, however, significantly different from an amplifier in the context of TMI. It generally operates with much less gain and much higher laser power in the fiber due to the enhanced intracavity power traveling in both directions. An oscillator therefore operates in much deeper gain saturation which is now well understood to significantly suppress TMI. In an oscillator, in addition to the FM, HOMs can, however, circulate and resonate in a parasitic competing cavity thereby enhancing modal competition and TMI effects [19].
An experimental comparison of an oscillator and an amplifier based on the same fiber was conducted in Ref. [20], finding that bidirectionally pumped oscillator and amplifier have higher TMI thresholds than that of a counter-pumped amplifier. Pumping at ∼915 nm was found to increase TMI threshold beyond those pumped at ∼976 nm in an oscillator, with near diffraction-limited ∼2 kW achieved using a combination of ∼915 nm and ∼976 nm pumps in a co-pumped oscillator in 2016 [21]. The same Chinese group demonstrated that bidirectional pumping is better at suppressing TMI with diffraction-limited ∼2.5 kW achieved using bidirectional pumping at ∼976 nm in 2016 [22]. Diffraction-limited ∼3 kW was achieved by the same group in a slightly improved bidirectional pumping scheme at ∼976 nm in 2017-2018 [23,24]. The same group finally achieved near diffraction-limited ∼5 kW using a bidirectional pumping scheme at 915 nm in 2019 [25]. The first diffraction-limited ∼5 kW oscillator was demonstrated by bidirectional pumping at ∼970 nm in 2018 by a Japanese group [26]. Similar results to these in Refs. [21–24] were achieved in Ref. [27].
Spurious external reflection was found to lower stimulated Raman scattering (SRS) threshold in an oscillator, leading to a lower TMI threshold in Ref. [25]. Similar effect was also observed in amplifiers [28,29]. Instability can lead to the onset of SRS due to its relatively higher peak powers. Since SRS is initiated by noise and is expected to be very noisy, it can lead to a significantly increased HOM seed in the active fiber (our analysis in Ref. [10] indicates HOM input is seeded mostly by the RIN of the amplifier seed), leading to a significantly lower TMI threshold.
In this work, we conducted the first theoretical study of TMI in fiber oscillators, lasing in the FM cavity, based on a STRS model. The STRS can provide gain to HOMs, in addition to inversion, and can lead to HOM lasing in a secondary competing cavity. The dominant HOM lasing takes place at a frequency slightly lower than that of the corresponding fundamental mode (FM). HOM can also be generated at a frequency slightly higher than that of the corresponding FM. In these two cases, the round-trip gains are at similar orders of magnitude. Other higher order effects can also happen, generating HOM at frequencies of harmonics and mixing frequencies, but they are much weaker.
The TMI threshold in an oscillator is characterized by the lasing action at the dominant HOM frequency. Once the threshold of the HOM secondary cavity is reached, the FM cavity output competes strongly with the HOM cavity output with any further increase of pump power mostly fueling the HOM growth, a behavior markedly different from that in an amplifier and observed experimentally [21–24,27].
Using the numerical model based on STRS and quasi-3D fiber amplifier, we numerically studied this new TMI phenomena in oscillators and find that our model is quantitatively consistent with the measured TMI thresholds in oscillators using HOM losses consistent with the fibers and coiling arrangements used in the experiments. Our model can also predict the observed TMI behaviors with regard to bidirectional pumping and pump wavelength, and also shows that the benefits of bidirectional and off-resonant pumping are much more pronounced for oscillators than amplifiers.
2. Theory
A typical oscillator is illustrated in Fig. 1(a). The FM primary cavity is formed by high reflector (HR) with reflectivity $R_1^{FM}$ and an output coupler with reflectivity $R_2^{FM}$. The corresponding reflectors forming the secondary HOM cavity are $R_1^{HOM}$ and $R_2^{HOM}$ respectively. The respective reflection for each mode is determined by its spatial overlap with the grating and can be easily evaluated once all the relevant spatial distributions are known. Usually, $R_1^{HOM} \approx R_1^{FM}$ and $R_2^{HOM} < R_2^{FM}$, resulting in HOM cavity having a lower quality factor (Q). Usually, the HOM suffers much larger differential round-trip losses, reducing the effective HOM cavity Q-factor even further.
Fig. 1. (a) An illustration of an oscillator operating in FM at frequency f0 with two corresponding HOM oscillators due to TMI operating at frequencies f0−Ω1 and f0 + Ω2. The reflectivity and two counter-propagating traveling thermal waves are also illustrated as χ+ and χ− respectively. (b) with additional illustrated paths for HOM to FM coupling at the oscillator output at frequencies f0−(Ω1−Ω2). Only the dominant frequencies are considered.
The dominant fundamental mode (FM) operates at frequency f0, consisting of many longitudinal modes, shown at the bottom of Fig. 1(a). The forward-propagating FM through STRS amplifies the forward-propagating HOM at f0−Ω1 and creates the forward-propagating thermal wave shown as χ+ just above the fiber in Fig. 1(a). The HOM operates at its own longitudinal modes with a slightly larger mode spacing from that of the FM modes, see bottom of Fig. 1(a). Since the STRS gain peaks at a frequency slightly below the driving FM mode (see the STRS gain spectrum illustrated in red at the bottom Fig. 1, marked by χ), Ω1 corresponds to the FM longitudinal mode where Ω1 is the closest to the STRS gain peak. Steady state is assumed, and all transient effects are ignored in this study.
It is important to point out that the propagating thermal wave is not from thermal diffusion along the fiber, but is generated by the traveling mode interference pattern in an amplifier. The period of the interference pattern is typically much larger than fiber diameter and thermal diffusion dominates in the radial direction. The traveling thermal wave is effectively damped at the rate of thermal diffusion in the radial direction. It is also worth noting that thermal waves generated by each FM longitudinal mode interacting with its corresponding HOM longitudinal mode are in phase and behave like a single wave in this case.
Correspondingly, the counter-propagating FM amplifies the counter-propagating HOM at f0−Ω1 and creates the counter-propagating thermal wave shown as χ− just below the fiber in Fig. 1(a). Interacting with χ−, the forward-propagating FM can also amplify the forward-propagating HOM at f0 + Ω2 and alternatively, interacting with χ+, counter-propagating FM can also amplify the counter-propagating HOM at f0 + Ω2. Again, the HOM also operates at its own longitudinal modes and Ω2 corresponds to the FM longitudinal mode where Ω2 is the closest to the STRS gain peak, see the bottom of Fig. 1 (a) for the illustration of STRS gain spectrum in red and Ω2, noting that HOM at frequencies f0−Ω1 and f0 + Ω2 is driven by different longitudinal mode of FM, and, Ω1 and Ω2 are most likely not the same due to the constraints of FM and HOM longitudinal modes and the STRS gain spectrum. Only the dominant frequencies are considered here. There are also higher-order effects at harmonic and mixing frequencies, but they are generally much weaker.
Additionally, conversion of HOM back to FM at the laser output at frequencies of f0−(Ω1−Ω2) are also possible through multiple paths as illustrated in Fig. 1(b). This can generate significant components at frequency of f0−(Ω1−Ω2) and related harmonics at the FM laser output once TMI threshold is reached. It is worth noting that the FM components at frequency of f0−(Ω1−Ω2) is likely not at the FM longitudinal mode and will not oscillate, unless Ω1 is equal to Ω2.
Clear amplitude noise at frequencies of ∼150 Hz, ∼300 Hz and ∼450 Hz above TMI threshold in an oscillator was observed in a co-pumping configuration in Ref. [21] and at frequencies up to 250 Hz was observed in a co-pumping configuration in Ref. [22]. Amplitude noise at frequencies of ∼220 Hz and ∼410 Hz above TMI threshold in an oscillator was observed in a co-pumping configuration and at frequencies of ∼2kHz and ∼4kHz in a counter-pumping configuration in Ref. [23,24]. Clear amplitude noise at frequency of ∼280 Hz above TMI threshold in an oscillator was also observed in a co-pumping configuration in Ref. [27]. The amplitude noise frequencies from the interference of FM and HOM at the peak STRS gain is expected to be 4 − 6kHz in all these cases.
HOM is likely mostly absent from the output in all these cases due to the combination of high HOM loss in the fiber and the use of cladding light stripper (CLS). The observation of mostly FM at the output even when TMI threshold is reached by noting a significant increase of amplitude noise and a saturation of FM output power provides some evidence for this in Ref. [22]. In the total absence of HOM at the output, the FM-HOM beating at 4-6kHz cannot be observed in these cases. The observation of amplitude noise at much lower frequencies in Ref. [21–24,27] may be explained by FM-FM beating at Ω1−Ω2 and related high order effects. It is worth noting that the exact values of Ω1 and Ω2 are highly sensitive to precise oscillator conditions and can be altered by pumping powers and pumping arrangements due to the constrains of FM and HOM longitudinal modes, which may explain the differences observed in co-pumping and counter-pumping cases in Ref. [21–24,27].
Gain saturation in an oscillator can be similarly evaluated at each location both across the core and along the fiber as in Ref. [10] by simply replacing the signal intensity by the sum of local forward and backward signal intensities. STRS coupling coefficients χ can also be similarly evaluated at each location. The reader can refer to Ref. [10] for details, which are not repeated here for brevity and clarity. The nonlinear coupled mode equations, however, need to be revised to account for the bidirectional nature and the additional HOM frequencies following the similar process described in Ref. [4]. Considering the phases of the FM and HOM are identical at the beginning of each forward and backward pass due to oscillator requirements, we have for forward-propagation three coupled-mode equations for FM at f0 and HOM at f0−Ω1 at f0 + Ω2,
Similarly, for counter-propagation
New terms related to the coupling for HOM at f0 + Ω2 are added in addition to all the terms already described in Ref. [4]. These new coupling terms are due to a six-wave-mixing process involving two co-propagating optical waves, two counter-propagting opical waves and the related thermal waves. Superscript + and − indicates power propagating in forward and counter propagating direction, respectively. Subscript 01 and 11 indicates LP01 and LP11 modes respectively. Note local FM gain g01, LP11 mode gain g11, and STRS coupling coefficient χ are evaluated based on local inversion and mode distribution and are the same for both propagation directions. α01 and α11 are FM loss and LP11 mode loss respectively.
The HOM gain is dominated by the gain at f0−Ω1 during the forward pass. HOM gain at f0 + Ω2 depends also on how close Ω1 and Ω2 are and is much higher if both Ω1 and Ω2 are close to their respective STRS gain peaks, noting this is highly dependent on precise oscillator conditions and can be altered by pumping powers and pumping arrangements due to the constrains of FM and HOM longitudinal modes. The round-trip gain is, therefore, expected to be larger at f0−Ω1 while the round-trip gain at f0 + Ω2 is expected to be at similar orders of magnitude. In the following analysis, we are mainly concerned with the HOM frequency which reaches the threshold first and will focus our study entirely on HOM at f0−Ω1, which has the largest round-trip gain,
When the threshold condition of G11 = 1 is met, a HOM laser oscilation establishes in the oscillator at f0−Ω1, clamping its round-trip gain to G11 = 1. Any further increase of FM output power is consequently much constrained! This is very different from the amplifier case where FM output power can still increase albeit at a slower rate. This FM output clamping has indeed been observed in mutiple cases [21–24,27].
In the following analysis, we use the maximum STRS gain to evaluate the round trip gain and define the TMI threshold as when the HOM threshold condition is met at f0−Ω1. The frequency separation between corresponding longitudinal mode of FM and HOM changes by the difference of the longitudinal mode spacings of FM and HOM when the longitudinal mode order changes by 1, see bottom of Fig. 1. In our following example, the difference of longitudinal mode spacings of FM and HOM is ∼2kHz. Our analysis therefore provides the lowest possible TMI threhold for the given oscillator.
3. Simulation
The quasi-3D fiber amplifier model described in Ref. [10,12,30] was used first to model the FM power in both direction in the oscillator considering the reflection at the two fiber Bragg gratings (FBG). 100 cylindrical layers are used for the core. HOM is ignored at this point as we are only interested in the region below the HOM threshold where the HOM power is negligible. 3D gain saturation is considered in the way described in Ref. [10] considering powers in both directions. STRS coupling coefficient χ is then evaluated for each location based on the simulated oscillator data across its spectrum. Peak round-trip HOM gain is then evaluated using the maximum in the HOM round-trip gain spectrum in the fiber according to Eq. (3) and the reflectivity of the FBGs. If HOM threshold is not reached, pump power is increased. Once HOM threshold condition is met, the correspond FM output is noted as threshold power.
Diffraction-limited ∼3.05 kW was achieved in a bidirectional pumped oscillator at ∼976 nm in 2018 [24]. Co and counter pumping TMI threshold were measured to be 1.45 kW and 1.93 kW respectively. When co-pumping and counter-pumping powers were 1.55 kW and 2.6 kW respectively, TMI threshold was observed to be ∼2.88 kW. Fraction of counter pumping power is calculated as the fraction of counter pumping power in total pump power and was 0.627 in this case. Finally, when co and counter pumping powers were 1.85 kW and 2.5 kW respectively, i.e. fraction of counter pumping power of 0.575, the maximum power of 3.05 kW was achieved. TMI was not observed in this case. The many carefully measured TMI thresholds in this work provide many opportunities to validate our model.
The gain fiber was an ytterbium-doped double-clad fiber with 21µm core diameter of 0.065 NA and 400µm cladding diameter of 0.46 NA in Ref. [24]. Pump absorption at 976 nm was 1.45 dB/m. We have assumed doped area occupies 80% of the core diameter as this has provided the best fit for a similar fiber in Ref. [10]. 18 m fiber was used, coiled in ∼12 cm coil diameters.
High reflectivity (HR) FBG had a reflectivity of 99.9% at 1080 nm with a 3 dB bandwidth of ∼3 nm and output coupler (OC) FBG has a reflectivity of 8.7% at 1080 nm with a 3 dB bandwidth of ∼1 nm. Based on the reflectively and bandwidth of the HR FBG, it is estimated to have a coupling coefficient κ of 10.05mm-1 for the FM and a length of 0.413 mm. Assuming the FBG occupies the entire core uniformly, the overlap of LP01 and LP11 modes with FBG can be calculated as 0.9485 and 0.8469 respectively. The same FBG for the LP11 mode can then be evaluated as having a coupling coefficient κ of (0.8469/0.9485) × 10.05mm-1 = 8.973mm-1. Based on the simulated difference of LP01 and LP11 mode indices of 5.24 × 10−4, the LP11 mode is calculated to have a peak reflection at (5.24 × 10−4/1.45) × 1080nm = 0.39 nm below 1080 nm. The HR FBG’s reflectivity for LP01 and LP11 modes are shown in Fig. 2(a). Similarly, the OC FBG’s reflectivity for LP01 and LP11 modes are shown in Fig. 2(b). The OC FBG has a length of 0.37 mm and κ of 0.8216mm-1 and 0.7337mm-1 for LP01 and LP11 modes respectively. To summarize, $R_1^{FM} = 99.9\%$, $R_1^{HOM} = 99.7\%$, $R_2^{FM} = 8.7\%$, and $R_2^{HOM} = 4.6\%$ at the lasing wavelength of 1080 nm.
Fig. 2. Calculated reflectivity for the LP01 and LP11 modes for (a) HR FBG and (b) OC FBG in Ref. [24].
The FM longitudinal mode spacing is 5.75 MHz and the difference between LP01 and LP11 longitudinal mode spacings is 2.08kHz in this case. 3 dB spectral bandwidth of ∼3.6 nm, i.e. ∼926 GHz, was measured when the output power was 3.05 kW [24], corresponding to the number of longitudinal modes of 926 GHz/5.75MHz≈1.6 × 105!
The TMI threshold versus fraction of counter pumping power is then simulated for the case in Ref. [24] for three cases of LP11 mode loss of 1.19 dB/m, 1.23 dB/m and 1.35 dB/m. The results are given in Fig. 3(a) along with measured TMI thresholds. TMI was not observed at the maximum power of 3.05 kW. TMI threshold for an amplifier with the same fiber at LP11 mode loss of 1.23 dB/m was also given in Fig. 3(a) with seed power of 50W and seed power ratio of LP11 and LP01 modes of 3 × 10−12 same as used in Ref. [12]. The TMI threshold peaks just below a fraction of counter pumping power of 0.6, in contrast to that of just above 0.8 in the amplifier. The pump and laser powers in both directions at fraction of counter pumping power of 0.55 is given in Fig. 3 (b). The dominance of forward laser power is clearly visible, which contributes mostly to the HOM gain at f0−Ω1 during the forward pass.
Fig. 3. (a) Simulated oscillator TMI threshold versus fraction of counter pumping power for LP11 mode loss of 1.19 dB/m, 1.23 dB/m and 1.35 dB/m. Measured data are from Ref. [24]. TMI for an amplifier with the same fiber at LP11 mode loss of 1.23 dB/m was also simulated with seed power of 50W and seed power ratio of LP11 and LP01 modes of 3 × 10−12 same as used in Ref. [12]. (b) Simulated pump and laser powers in both directions in an oscillator at fraction of counter pumping power of 0.55.
The LP11 mode loss of ∼1.23 dB/m is plausible given the fiber and coil diameter. The fit with measured TMI threshold is reasonable, especially considering the dependence on the fraction of counter pumping power. It is also worth noting that our simulated threshold is based on the peak STRS gain. The real STRS gain depends on how close Ω1 is to the STRS gain peak, which is highly sensitive to precise oscillator conditions such as pumping powers and pumping arrangements due to the constrains of FM and HOM longitudinal modes.
The frequency separation between corresponding longitudinal mode of FM and HOM changes by the difference of the longitudinal mode spacings of FM and HOM, i.e. ∼2kHz in this case, when the longitudinal mode order changes by 1, see bottom of Fig. 1. Ω1 can therefore be ∼2kHz/2=∼1kHz away from the STRS gain peak. Our simulated TMI threshold, therefore, provides the lowest case scenario.
The heat load and STRS coupling coefficient χ are also simulated for various fractions of counter pumping power at TMI threshold for LP11 mode loss of 1.23 dB/m in Fig. 4. The heat load peaks at both fiber ends for bi-directional pumping due to the large total signal power in the fiber from two directions. It is easy to see that the heat load is almost symmetric at fraction of counter pumping power of ∼0.6 and so is the STRS coupling coefficient, providing a minimum overall STRS coupling over the length of the oscillator. This symmetry of the χ in an oscillator is in strong contrast with the asymmetry it exhibits in an amplifier in Ref. [10], where a TMI threshold peaks at fraction of counter pumping power of ∼0.8.
Fig. 4. Simulated heat load (a) and STRS coupling coefficient χ (b) for the oscillator for various fractions of counter pumping power at TMI threshold for LP11 mode loss of 1.23 dB/m.
The simulated fundamental mode and normalized LP11 mode power inside the oscillator in both forward and backward directions for fractions of counter pumping power of 1, 0.55 and 0 at TMI threshold for LP11 mode loss of 1.23 dB/m are plotted in Fig. 5. The LP11 mode power is normalized to 1 at the HR reflector end. The fundamental mode power is plotted in linear scale while the normalized LP11 mode power in logarithmic scale to reflect its large change of over four orders of magnitude during a round-trip within the LP11 mode oscillator. It is shown that the FM power (Fig. 5(a)) follows the expected smooth evolution, with the forward and backward propagating modes experiencing the same inversion-induced gain. The LP11 mode power evolution (Fig. 5(b)), on the other hand, shows much more pronounced variations in each propagation direction, experiencing much stronger local gain and attenuation, depending on the relative values of inversion-induced and STRS gains counter-balancing the bend-induced LP11 mode losses. Although, the inversion-induced gain and the bend-induced LP11 mode losses are the same in both directions, the STRS gain is substantially lower in the backward direction and, as a result, the backward-propagating LP11 mode suffers net losses in all cases. The OC FBG sees the highest power and can suffer from thermal distortion, which is not considered in our simulation.
Fig. 5. Simulated (a) FM power and (b) normalized LP11 mode power in the oscillator in both directions for fractions of counter pumping power of 1, 0.55 and 0 at TMI threshold for LP11 mode loss of 1.23 dB/m.
The simulated TMI threshold versus LP11 mode loss and FM OC peak reflectivity at LP11 mode loss of 1.23 dB/m are given for fraction of counter pumping power of 0.55 in Fig. 6. Both show near linear dependence. The TMI threshold increase with the LP11 mode loss is easy to understand. Its increase with OC peak reflectivity is due to an increase of intracavity power and, consequently, stronger gain saturation. It is worth noting that $R_2^{HOM}$ also increases with an R2, see Fig. 6(b), which has the tendency to lower TMI threshold. This is, however, a weaker effect. The nominal pump wavelength is 976nm and the nominal laser wavelength is 1080nm unless otherwise stated.
Fig. 6. Simulated TMI threshold versus (a) LP11 mode loss and (b) FM OC peak reflectivity R2 at LP11 mode loss of 1.23 dB/m for fraction of counter pumping power of 0.55. Associated change in $R_2^{HOM}$ is also given.
For fraction of counter pumping power of 0.55, TMI threshold was simulated versus fiber length at LP11 mode loss of 1.23 dB/m and core diameter at LP11 mode loss of 0 dB/m (Fig. 7). The TMI threshold increase with length is due to an increase of total LP11 mode loss and its increase at smaller core diameter is due to stronger gain saturation due to an increase of signal intensity in the fiber.
Fig. 7. Simulated TMI threshold versus (a) fiber length at LP11 mode loss of 1.23 dB/m and (b) core diameter at LP11 mode loss of 0 dB/m both for fraction of counter pumping power of 0.55.
Fig. 8. Simulated TMI threshold versus (a) laser wavelength and (b) pump wavelength at LP11 mode loss of 1.23 dB/m and fraction of counter pumping power of 0.55. For (b) the small-signal pump absorption is kept constant by adjusting ytterbium concentration, also shown in the figure.
In Fig. 8, TMI threshold was simulated versus laser wavelength and pump wavelength at LP11 mode loss of 1.23 dB/m and fraction of counter pumping power of 0.55. For the pump wavelength dependence, the small-signal pump absorption was kept constant by adjusting ytterbium concentration, which is also shown in the Fig. 8 (b). The increase of TMI threshold towards shorter laser wavelength and away from 976 nm is similar to that in amplifiers [10], both due to the increased gain saturation. The magnitude of the increase is, however, much more pronounced than that in amplifiers. This is due to that the oscillators are more gain-saturated to start with.
Interestingly, with this same fiber at LP11 mode loss of 1.23 dB/m and fraction of counter pumping power of 0.55, TMI threshold of ∼14 kW can be achieved just by moving the laser wavelength to ∼1030nm! TMI threshold of >25 kW is possible with a pump wavelength of ∼950 nm and TMI threshold of >12 kW is possible with a pump wavelength of 915 nm. This, however, comes at the cost of much longer fiber if ytterbium concentration is kept constant, unlike the change of laser wavelength.
4. Conclusions
We performed the first theoretical study of TMI threshold in fiber oscillators based on a STRS model and found that LP11 mode lasing action can take place in an oscillator, first noted in Ref. [16]. The HOM lasing is dominated at a frequency slightly lower than that of the corresponding fundamental mode (FM). The HOM can also be amplified at a frequency slightly higher than that of the corresponding FM mode. Other higher order effects can also happen, generating HOM at frequencies of harmonics and mixing frequencies, but they are much weaker.
We identified paths for FM at additional mixing frequencies to appear at the output of the oscillator through HOM mode to FM coupling. This can generate amplitude noise at the FM-FM beat frequencies of up to few kHz, which could explain the observed amplitude noise above TMI threshold in oscillators at frequencies from tens of Hertz to few kilohertz [21–24,27], much below what is expected from the FM-HOM beat frequency.
The TMI threshold in an oscillator can be characterized by the lasing action at the dominant HOM frequency. Once the threshold of the HOM oscillator is reached, the growth of FM output power is much limited with any further increase of pump power mostly fueling the HOM growth, a behavior markedly different from that in an amplifier, but consistent with many observations [21–24,27].
Using the numerical model based on STRS and quasi-3D fiber amplifier, we numerically studied this new TMI phenomena in oscillators and find that our model is quantitatively consistent with the measured TMI thresholds in oscillators using HOM losses consistent with the fibers and coiling arrangements used in the experiments. This is especially true considering our simulated threshold is based on the maximum STRS gain which provides the lowest case scenario. Due to the constrains of FM and HOM longitudinal modes of the oscillators, the real STRS may happen at ∼1kHz away from the STRS gain peak in our case and is also expected to be highly sensitive to precise oscillator conditions such as pumping powers and pumping arrangements.
Our model also predicts the observed TMI behaviors with regard to bidirectional pumping and pumping wavelength, and also shows that the benefits of bidirectional and off-resonant pumping are much more significant for oscillators than amplifiers.
Funding
Army Research Office (W911NF1910409).
Disclosures
The authors declare that there are 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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