Noise generation and experimental setup
To investigate the impact of noise on the TMI threshold, we have artificially imprinted noise onto the pump and seed radiation, respectively. In order to do this, we have generated white noise with a computer, since it represents a homogeneous distribution of the noise energy among all frequencies. Afterwards, this white noise has been transferred, via an arbitrary waveform generator (AWG, Rigol DG4202), to the seed or the pump of the fiber amplifier.
In case of the pump intensity-noise, the AWG was connected to the driver (Delta Elektronika SM 120–50) of the pump laser-diode (Dilas D4F4Q22–976.3-2000C-IS45.8), which pumped the main-amplifier fiber at 976 nm in a counter-propagating configuration, as can be seen in Fig. 1 (blue dashed box). The main-amplifier fiber was an Yb-doped Large-Pitch Fiber (LPF) [15] with a core diameter of ~ 65 μm, a similar mode-field diameter and a length of 1.07 m. To measure the pump intensity-noise, a small reflection of the pump beam was directed to a photodiode (Thorlabs PDA20CS-EC, blue solid box), which was terminated with 50 Ω and connected to a 12 bit oscilloscope (Teledyne Lecroy HDO6104). The noise was acquired during 10 s with a sample rate of 106 samples/s and a 240 kHz low-pass filter was used to avoid aliasing. Before the experiments were carried out, the noise trace was optimized with the help of the photodiode signal, in order to get a spectrum as flat as possible. This procedure allowed us to pre-compensate for any spectrally-dependent noise attenuation, which could be introduced by the electronics of the driver and/or the laser diode.
In order to transfer the noise to the seed signal, the AWG modulated the driver (Delta Elektronika SM 120–50) of the laser diode that pumped the pre-amplifier fiber, as depicted in Fig. 1 (black dashed box). This, in turn, resulted in a modulation of the seed radiation of the main amplifier fiber. The active fiber was seeded by a source delivering fs-pulses (stretched to 1 ns) with a repetition rate of 19 MHz and an average signal power of 5 W centered at 1030 nm (with a 3 dB-bandwidth of 7 nm). Finally, the leakage through one of the input-coupling mirrors was used to measure the seed intensity-noise with the same photodiode (black solid box) that was used for the pump-noise measurements.
Frequency region of interest
Former investigations on TMI have revealed that the beam fluctuations (which are related to the modal energy transfer) close to the TMI threshold occur at a dominant frequency typically below 10 kHz [5]. This frequency depends on the fiber design and, particularly, on the mode-field diameter and is related to the thermal diffusion time of the fiber core. Thus, it is expected that some noise frequencies will have a stronger impact on the TMI threshold than others. To verify this, noise traces with different bandwidths have been generated and imprinted onto the pump and seed radiation, respectively, and the corresponding TMI threshold was measured.
Pump intensity-noise
To determine the frequency region of interest, white noise was first generated from 1 Hz up to a cut-off frequency of 6 kHz. This frequency band contains all relevant frequencies of the TMI-induced beam fluctuations for the active fiber used in the experiments, which typically lie below 2 kHz. To generate noise traces with different bandwidths, the cut-off frequency was progressively decreased, which means that all frequency components above this cut-off value were deleted from the trace. By doing this, noise traces with cut-off frequencies from 6 kHz down to 75 Hz were created. All these noise traces were then imprinted onto the pump radiation consecutively and the corresponding TMI thresholds were measured according to the guidelines and definitions in [5].
Figure 2 shows selected examples of the pump-noise traces measured with the photodiode (blue solid box in Fig. 1). Their power spectral density is depicted in Fig. 2a and the integrated noise (RIN - relative intensity noise) is shown in Fig. 2b. The RIN was calculated by integrating the power spectral density from 500 kHz (given by the sample rate of the oscilloscope) down to 1 Hz (determined by the acquisition time) for each measurement.
It is important to stress that the power per frequency segment was kept constant when reducing the bandwidth of the noise trace. This resulted in the same power-spectral-density distribution up to the respective cut-off frequency in each trace (see Fig. 2a). That also implies that the integrated noise (i.e. the RIN) becomes smaller the lower the cut-off frequency is (see Fig. 2b) (since the overall noise energy is reduced). For ideal white noise, the RIN in Fig. 2b should follow a linear behavior since each frequency noise component should contribute equally to the integrated noise. As can be seen, we have not been able to fully achieve this behavior, since, even though we optimized the noise traces, some frequency components are slightly more pronounced than others. Nevertheless, especially for lower cut-off frequencies (which will become more important in the following), the dependence of the RIN on the noise frequency approaches a linear function.
A reduction of the TMI threshold has been observed when imprinting the noise traces onto the pump radiation. The amplitude of the pump intensity-noise, which can be inferred from Fig. 2, was chosen in such a way that the broadest noise trace (with a cut-off frequency 6 kHz) caused a reduction of the TMI threshold sufficiently strong to extract a clear statement from the subsequent measurements.
The decrease of the TMI threshold changed depending on which frequencies components were contained in the pump intensity-noise, as can be seen in Fig. 3. When imprinting the pump-noise trace with frequency components from 1 Hz up to 6 kHz (cut-off frequency = 6 kHz, RIN = 0.455%), the TMI threshold was reduced from the initial 260 W (no artificial noise, red circle) to a value of 211 W. A similar decrease of the TMI threshold was observed for cut-off frequencies of 4 kHz and 2 kHz. This finding implies that pump-noise frequencies above 2 kHz have no significant influence on the TMI threshold of the used fiber (indicated by the red shading in Fig. 3).
However, when decreasing the cut-off frequency further and, thus, cutting more of the high pump-noise frequencies, the reduction of the TMI threshold became less pronounced. This means that the frequencies that have been cut, were responsible for the former reduction of the TMI threshold. This trend continues for cut-off frequencies of a few hundreds of Hz until it saturates below 100 Hz. In general, it can be said that pump-noise frequencies below 2 kHz have an impact on the TMI threshold of the used active fiber and, thus, this frequency band represents the region of interest (green shading in Fig. 3) for the subsequent investigations. Hence, the noise frequencies that influence the TMI threshold seem to be similar to the frequencies of the beam fluctuations during TMI. This statement will be confirmed in the next paragraph. Please note that the frequency region of interest will differ from fiber to fiber since it is related to the design parameters, such as e.g. the mode-field diameter. However, the frequency region of interest will typically not exceed 20 kHz even in high-power fiber amplifiers with smaller mode-field diameters (e.g. 20 μm), since all TMI-related beam fluctuations in fiber amplifiers reported so far occurred with frequencies below 20 kHz [5, 16,17,18].
For the active fiber used in these experiments, the pump-noise frequencies between 125 Hz and 500 Hz have the strongest impact on the TMI threshold (indicated by the blue shading in Fig. 3). This corresponds to the main beam-fluctuation frequency of 241 Hz in the active fiber, which was measured slightly above the TMI threshold according to [5]. Thus, noise frequencies around the main beam-fluctuation frequency of a fiber most likely induce the strongest phase shift between the MIP and the RIG, which results in a strong modal energy transfer. This is because the phase-shift introduction is linked to a heat-load change and, thus, directly to the thermal diffusion time of the fiber [9].
Seed intensity-noise
The same investigations have been done to determine the frequency region of interest for the seed intensity-noise. Therefore, exactly the same noise traces as before were imprinted onto the pump radiation of the pre-amplifier, which have then been converted into intensity noise of the seed radiation for the main amplifier. This noise was measured with the photodiode (black solid box in Fig. 1) placed behind one of the input-coupling mirrors. Selected pump-noise traces are depicted in Fig. 4, where the power spectral density is illustrated in Fig. 4a and the RIN (integrated from 500 kHz down to 1 Hz) is shown in Fig. 4b. Note that the traces in Fig. 4a have inconsistent noise floors above their corresponding cut-off frequency, which was caused by an imperfect noise imprinting. However, this did not influence the subsequent measurements since the noise floor of each trace was still ~ 30 dB below its imprinted signal.
As in the previous experiment, the applied noise resulted in a decrease of the TMI threshold, which was initially measured to be 295 W (no artificial noise). Note that the initial TMI thresholds of the pump- and the seed-noise measurement differ slightly, which is most likely due to maintenance-related adjustments in between the measurements.
The measured TMI thresholds as a function of the cut-off frequency of the corresponding seed-noise trace are illustrated in Fig. 5 and show a behavior similar to that of the pump-noise investigations. Consequently, it can be concluded that also seed-noise frequencies above 2 kHz have no significant impact on the TMI threshold of the used active fiber (red shading in Fig. 5), whereas frequencies below 2 kHz result in a decrease of the TMI threshold (green shaded area in Fig. 5). Similar to the pump-intensity-noise case, seed-noise frequencies between 125 Hz and 500 Hz have the strongest influence on the TMI threshold (blue shaded area in Fig. 5), which corresponds to the main beam-fluctuation frequency of the fiber.
To double-check the results of the frequency dependence of pump and seed intensity-noise, a trace with noise content contained only within a narrow frequency band was generated. The frequency band of the noise trace has then been tuned across the interesting frequency region and the corresponding TMI thresholds have been measured. The results obtained in this experiment have confirmed the conclusions extracted from Figs. 3 and 5.