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Abstract
In this work we investigate transverse mode instability (TMI) in the presence of pump intensity noise and a controlled perturbation of the input coupling for a rod-type fiber amplifier using spatially and temporally resolved imaging (ST). We show that inherent pump intensity noise from the power supply can define significant peaks in the resulting TMI spectrum. ST measurements show that the TMI in the transition region consists of different orientations of LP11. This finding indicates that the simple picture of TMI being seeded by the combination of a static initial fraction of LP11 and pump or signal intensity noise is not valid for our measurements. Furthermore we present seeding of TMI by perturbing the input coupling dynamically. ST measurements of the resulting TMI as a function of perturbation frequency provides quantitative information regarding the frequency response of the non-linear coupling coefficient. Finally, ST measurements of the resulting TMI as a function of signal power shows that the TMI experiences an exponential gain long before visible beam fluctuations appear.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For many years, single mode fiber amplifiers have dominated the field of high power and high beam quality coherent light sources. The large surface-to-volume ratio of fibers provides excellent heat dissipation and high efficiencies arise from the long interaction length between the light and active material [1,2]. Several intensity-dependent non-linear effects such as Raman scattering and self-phase modulation limiting in particular the peak power of ultrafast systems have been mitigated by increasing core size and reducing fiber length. This strategy has proven very effective but maintaining single mode behaviour for larger core sizes requires better control of the refractive indices which limits the simple step-index-fiber (SIF) design and has increased the dissipated heat per unit length. The demand for single-mode (SM) fibers with larger cores has led to the development of multiple different non-flexible rod-type fibers [3–5]. In these fibers, the first observations of transverse mode instability (TMI) were found [6,7] and it is now the main limiting factor in average power scaling of both flexible large-mode-area fiber amplifiers and rod fiber amplifiers.
TMI is a thermally induced non-linear effect where heat generated from the amplification process affects the refractive index of the fiber and can lead to dynamical transfer of energy between modes. For a specific amplifier system this dynamic effect is usually only seen above a certain signal power called the TMI threshold. The physics of TMI has been widely accepted but many models of TMI are still used which have some differences with respect to describing the inner workings of the instabilities [8–11]. Besides numerical models to improve the understanding and mitigation of this limitation, comprehensive experimental work has been performed to describe TMI [12–15]. Some early work utilized high-speed cameras to show transfer of power between different transverse modes [16,17] and since then the simpler photodiode (PD) method proposed by Otto et. al. [17] has been standard when measuring TMI. This method consists of a PD measuring the signal power after a pinhole cropping part of the light and mainly measures the TMI threshold. Comparisons of experimental studies are difficult to perform for an effect like TMI since the measured threshold will depend on many parameters. Mode properties of the fiber, coupling, and pump intensity noise are difficult to measure accurately and vary from laboratory to laboratory. The combination of large uncertainties of the input parameters and the limited data from the PD method creates many fitting parameters for numerical models.
We have recently developed a more quantitative and qualitative method called spatially and temporally resolved imaging [18], ST in short. This method accurately captures fluctuations using the transverse spatial dimension. In this work we will use the ST method to show TMI dynamics in a new perspective and obtain information regarding the seeding mechanisms which is not described by current models. In this work we define TMI as the temporal instabilities of the output signal beam shape caused by interaction between the fundamental and higher-order modes due to thermo-optic effects. The TMI gain is the amplification of these instabilities for increasing signal power and will depend strongly on the frequency.
This paper has three main sections. Section 2 is dedicated to describing the ST method. Here the data acquisition and analysis are explained including application of the technique on simple numerical and experimental data. In Section 3, TMI is measured in the presence of inherent pump intensity noise from the pump power supply. In Section 4, TMI is seeded using a controllable dynamical coupling into the main amplifier. Interesting fluctuations are observed and explained using simple mode simulations.
2. Spatially and temporally resolved imaging
In this section, application of spatially and temporally resolved imaging [18] is outlined. The ST method is applied on both numerical and experimental data.
An ST measurement is performed with a high-speed camera to capture a movie of the beam dynamics. The movie is Fourier transformed in time to achieve spatial resolution of the power spectral densities,
where $I(x, y, t)$ is the intensity captured by the movie and $T$ is the length of the movie in time. The resulting data structure, $\tilde {I} (x,y,\nu )$, will be a complex function of space and frequency holding information regarding the fluctuations of captured beam dynamics. The absolute value and angle represents the strength and phase of the fluctuations, respectively. The ST measurement achieves a single accurate power spectral density (PSD) for any beam dynamic by integrating the PSD in space:Application on numerical beam dynamics
In this section, two simple beam dynamics are investigated numerically with the ST method to create improved interpretation of the results of the method. The two beam dynamics, which are simulated, are a beating in time between two modes (LP$_{01}$ and LP$_{11}$) and a simple transverse harmonic movement of a fundamental mode (FM).
The power in each mode is constant while the phase difference changes linearly for beating in time between two modes due to different optical frequencies. The spatial intensity of such a dynamic is given by:
The ST data of the mode beating is seen in Fig. 1. This dynamic has a single oscillating term at the beating frequency which is obvious from Eq. (3). The DC peak is the static term and here resembles the dominating mode which is the LP$_{01}$. For positive intensities the DC phase is always flat which is true when no background subtraction is applied. The peak at the beating frequency resembles an LP$_{11}$ mode in both intensity and phase but is in fact the product between the two electrical fields. Using Eq. (3), the input modes and their respective power relation could be found from a ST measurement, replicating the S$^2$ measurement but in time.
Fig. 1. ST method applied on simulated mode beating with 99 % power in the FM. a) The PSD as a function of frequency expressed in decibels relative to the carrier per Hertz (dBc/Hz). b) Spatial distributions of the intensity and phase for 0 and 300 Hz.
The interpretation is quite clear in the case of mode beating but as the next simulation shows, different dynamics may result in similar fluctuations. Such a dynamic could be given by a transverse harmonic movement of a FM. The resulting intensity may be described by:
The ST data of the movement of the FM is seen in Fig. 2. The integrated PSD plot shows oscillation power at the movement frequency but also at higher harmonics. The DC peak which is not shown resembles the LP$_{01}$ since the movement amplitude is small. For larger amplitudes, the beam will be stretched in the movement direction. The peak at 300 Hz is the main fluctuation and has a two-lobed structure in intensity and a $ {\mathrm{\pi}} $-shift in phase. The spatial distributions of this peak can be understood as the first derivative in the movement direction of the input beam with the intensity and phase corresponding to the magnitude and sign of the derivative, respectively. The second harmonic at 600 Hz is in the same way defined by the double derivative and so on for higher harmonics. In these two cases, it is easy to distinguish the underlying effects due to the presence of higher order harmonics of the beam movement but for small amplitudes the picture changes. The power in the second harmonic depends on the main fluctuation and will drown in noise if the power in the main fluctuation is smaller than half of the noise level in the dBc/Hz scale. The main fluctuation peak for the moving FM has many similarities to the peak seen in Fig. 1 originating from mode beating and with no higher harmonic, they can be difficult to distinguish.
Fig. 2. ST method applied on simulated mode movement with an amplitude of 1/10 of the core radius. a) The PSD as a function of frequency. b) Spatial distributions of the intensity and phase for 300, and 600 Hz.
Application on experimental data
Application of the ST method on numerical data showed how the two different fluctuations translate into Fourier space. Here the ST method is applied to experimental data of TMI. In Fig. 3 an ST measurement of TMI in the transition region for an aeroGAIN-ROD-PM85 [20] is seen. For this and all other ST measurement in this work a Basler high-speed camera of type acA640-750um [21] is used. This camera can achieve 4000 frames per second for a reduced area which is sufficient for the investigated rods. The PSD in Fig. 3(a) is representative of TMI for these types of rods and the spatial distribution of the fluctuations are shown for four different frequencies. The lowest frequency, 46 Hz, which is shown in red originates from air flow around components heating up and causing small deflections of the signal beam. This conclusion was supported by investigation of the system at high power operation below observable TMI with and without a clean air flow above our setup. With the air flow, low frequency instabilities are observed up to around 100 Hz for large signal powers. By turning off the air flow, these instabilities are strongly reduced and can therefore be attributed to the air flow. For all measurements shown in this work, the system is operated with air flow to reduce particles on the optics and instabilities observed below 100 Hz are not assumed to be TMI.
Fig. 3. ST measurement of TMI in the transition region for a rod fiber [20]. a) shows the PSD with colored dots indicating the frequencies for which the spatial distributions of the intensity and phase are shown in b). c) shows a still from a movie of the beam as function of time (see Visualization 1).
The spatial distributions of the three TMI peaks above 100 Hz show the same dynamics with different variations. A recent work regarding dual-pass rod-type fiber amplifiers [22] has shown that vectorial modes are needed to fully describe the mode properties of these large core rods, but in this work we use the LP-modes for simplicity. All the TMI fluctuations in Fig. 3 are interaction between the FM and a superposition of the two LP$_{11}$-modes with some offset in phase. The two-lobed structures are caused by no phase offset while the doughnut shapes requires both a phase offset and power in both LP$_{11}$-modes. For no phase difference the resulting centroid of the intensity will always move on a line in the transverse dimension while a non-zero phase difference between the two LP$_{11}$-modes can lead to elliptical movement of the centroid. The latter is observed for the peak at 299 Hz while both the peaks at 413 Hz and 601 Hz show the simple two-lobed distribution. As mentioned previously, this PSD of the TMI is representative for these fibers. The instability strength is defined by the seeding and the frequency-dependent gain. The decrease in magnitude of the instabilities for higher frequencies are mainly related to the frequency-dependent gain which decreases for higher frequencies [9,23]. The gain and seeding will lead to significant peaks seen in the TMI and the origin of some of these is discussed in the next section. The movie seen in Fig. 3 is part of the beam measurement used to perform the ST analysis. The scanning area of the camera captures the important core light and the inner row of high-index inclusions in the rod [20] but not the full cladding. This configuration is used for all measurements in this work and does not cause a problem since TMI is investigated in the transition region which ensures small fractions of higher-order-modes and consequently small amounts of cladding light.
3. Transverse mode instability and pump intensity noise
The seeding mechanism of TMI has been discussed extensively. Recent studies by Stihler et al. [14,24] have shown large impact of pump intensity noise on the TMI threshold of fiber amplifiers. In these studies, the TMI threshold was investigated when adding a controlled homogeneous white noise in some frequency range on the pump intensity. In this work, we will show that even inherent noise from the power supply of the pump impacts the resulting TMI. The PSD of the output signal is measured as a function of signal power for two different pump power supplies using the ST measurement.
In Fig. 4, a sketch of the backward-pumped amplifier is shown. The system consists of an 80 cm long aeroGAIN-ROD-PM85 [20] with a MFD of 65 µm. The seed has a full spectral width at half maximum of 1 nm centered at 1030 nm and for all measurements in this section the seed power is 15 W. The wavelength of the pump is power dependent and the center wavelength is 976 nm at full power.
Fig. 4. Sketch of the setup investigating the impact of pump intensity noise originating from the pump power supply.
In Fig. 5 the PSD from an ST measurement of the output signal is shown in the same plot as the PSD of the pump diode for the two different power supplies. The measurement of the intensity noise of the pump diode shown by red curves are taken over 5 minutes using a photo diode before implementation in the setup shown in Fig. 4. The intensity noise of the pump diode from Fig. 5 depends on pump power. It is shown for the pump power resulting in the signal power of the output signal analyzed with the ST measurement which is shown by blue curves. For all ST measurements shown in this work an acquisition time of 0.5 seconds has been used. The larger noise level of the ST measurement can in part be explained by the large difference in acquisition time but will also depend on inherent noise parameters of the high speed camera compared to the photo diode. The sample time of the ST measurement was hardware limited and will be improved for future work.
Fig. 5. Noise power spectral densities of the pump and from an ST measurement of the output signal as a function of frequency where a) shows the plot achieved with power supply 1 at 375 W of signal power and b) shows the plot for power supply 2 at 365 W of signal power.
For power supply 1 seen in Fig. 5(a) the PSD from the ST measurement has two significant TMI peaks at 300 Hz and 600 Hz where peaks in the pump intensity noise are also found. The pump intensity noise also shows more peaks with similar strength which are not observed in the measured TMI. The TMI gain is highest for frequencies between 200 Hz and 800 Hz and consequently, the expected TMI peaks at pump intensity noise peaks above 1000 Hz will not exceed the noise level of the ST measurement for this signal power. Another factor is the time duration of the measurements. In the brief time of the ST measurement the output signal might not experience the pump intensity noise averaged over 5 minutes. While the propagation of pump noise into the TMI is observed for power supply 1, the same measurements for power supply 2 seen in Fig. 5(b) shows a much stronger correlation. The pump intensity noise for power supply 2 has significant peaks at each 100 Hz which are clearly seen in the PSD of the ST-measurement. The magnitude of the peaks in the ST-measurements somewhat reveals the shape of the TMI gain as a function of frequency and shows highest gain around 400 Hz. Quantitative information regarding the TMI gain is difficult to retrieve since the pump intensity noise might not be representative for the 0.5 seconds sampling time for the ST-measurement as mentioned previously. For all measurements, the TMI is in the transition region [17] and the DC spatial distribution of the ST-measurement resembles a Gaussian.
The spectrum of TMI could also have been measured using a photodiode behind a pinhole but by using the ST method we gain knowledge regarding the specific fluctuations observed. This becomes very clear when visualizing the spatial distribution of the fluctuation at 100 Hz for different signal powers for the system with power supply 2 which is seen in Fig. 6. For the smallest signal power, 199 W, the only significant peak in the PSD is at 100 Hz. The flat-phased Gaussian spatial distribution shows that this fluctuation is intensity noise which directly originates from the pump and not TMI. As the signal power is increased, the spatial distributions at 100 Hz show the change of the fluctuation from the initial intensity noise to the regular TMI interaction found in Fig. 3. The intensity noise is still visible for the highest signal power showing in the different magnitudes of the intensity lobes.
Fig. 6. ST measurements for different signal powers for power supply 2. a) shows the signal PSD for the signal powers from plot b), where the spatial distributions of the intensity and phase are shown for 100 Hz. Black dots represent the 100 Hz peaks in the PSDs.
Another interesting use of the of ST measurement can be seen in Fig. 7 where the spatial distribution of four different significant fluctuations are shown. This ST measurement is the same as the one found in Fig. 5(b) which shows that the significant peaks are caused by strong pump intensity noise at these frequencies. Figure 7 displays that the peaks at 100 Hz, 200 Hz, 401 Hz, and 900 Hz all look like expected TMI due to interactions between the LP$_{01}$ and LP$_{11}$-modes. It is clear that the spatial distributions for the different frequencies do not have same the orientation. The three peaks at 100 Hz, 200 Hz, and 900 Hz all have a similar two-lobed distribution which happens when the two orientations of LP$_{11}$ are in phase. Rotation of this two-lobed distribution occurs when the relative power between the two orientations of LP$_{11}$-modes changes. The peak at 401 Hz does not have the same pure two-lobed structure but rather a doughnut with a dominating direction. This doughnut structure is caused by two orthogonal LP$_{11}$ with a non-zero phase difference.
Fig. 7. ST measurement with power supply 2 at 365 W of signal power. a) shows the PSD with colored dots indicating the frequencies for which the spatial distributions of the intensity and phase are shown in b). c) shows a still from a movie of the beam as function of time (see Visualization 2).
In many models [8–10,23,25], the seeding of TMI is due to some initial excitation of the LP$_{11}$-modes usually caused by alignment tolerances in the input coupling. Signal amplitude noise coupled into the LP$_{11}$ in this way can then be exponentially amplified, ultimately causing beam fluctuations. We will now discuss what is required for this seeding mechanism to produce the observed fluctuations. Assuming seeding by a static misalignment and intensity noise, the different orientations can only be explained by some coupling between the two orientations of LP$_{11}$ through the rod. Without this coupling, the modal interference pattern will follow the initial excitation leading to gain in only the misalignment orientation. The fluctuation at 401 Hz in Fig. 7 can as mentioned only be observed for a phase difference between the two LP$_{11}$ orientations while the peak at 100 Hz, 200 Hz, and 900 Hz can only be observed for no phase difference. The initial excitation will achieve in-phase LP$_{11}$-modes. To achieve the resulting fluctuations the coupling of power between the two orientations happens both with different strengths and phase seemingly randomly for different frequencies. It is of course difficult to completely dismiss the static misalignment and intensity noise as the seeding mechanism but a simpler seeding mechanism can also explain the observed orientations of LP$_{11}$. In the next section we show that a dynamic misalignment of the input signal in the transverse direction can seed TMI. For this seeding mechanism the orientations of the observed fluctuations will depend on the pointing stability of the beam which will inherently have some randomness in orientation. We obviously still record a strong correlation between the resulting significant frequencies in the ST measurement and the intensity noise of the pump diode. This correlation can be explained by the intensity noise stimulating the TMI gain for these frequencies which was also concluded by Stihler et al. [14].
4. Seeding of transverse mode instability by perturbation of input coupling
In this section, the signal light is investigated when inducing a controlled perturbation of the input coupling. A sketch of the setup is seen in Fig. 8. The perturbation of the input coupling is generated with a kinematic mirror mount [26] with a piezo actuator. We define the input perturbation as the harmonic perturbation in the transverse direction of the seed signal at the input-facet of the rod in time. In this work, only vertical perturbations of the mirror are induced. The piezo actuator is controlled by a frequency generator and since only small amplitudes are used no amplification of the signal from the frequency generator is used. Control of the input coupling has previously been attempted for mitigating TMI by Otto et al. [27] where an acousto-optic deflector was used in a feedback loop. In this work, we oppositely try to seed TMI by periodically changing input coupling with a frequency which in the following section is called the perturbation frequency (PF). The response of TMI for different system parameters can be measured by seeding with a known magnitude. The ST method combined with simple simulations of the dynamics achieves accurate quantitative information regarding the frequency response of the amplifier and TMI gain as a function of signal power.
Fig. 8. Sketch of the amplifier system with input perturbation. Small perturbations are induced by a kinematic mirror mount [26] on the input side.
Power supply 1 from the previous section is used to drive the pump diode and a PSD resembling the one from Fig. 5(a) is expected. A pump noise of this magnitude will not influence the measurements since the controlled perturbations will be orders of magnitude larger and dominate which will be evident from Fig. 9. The Yb-doped rod in the main amplifier is specifically made to exhibit few-mode behaviour at 1030 nm guiding both the LP$_{01}$ and LP$_{11}$-modes. The few-mode behaviour is used to effectively propagate the input perturbation which allows small seeding perturbations to be visible in the ST measurements for low signal powers and consequently tracking of the instability at these powers. The rod has a MFD of around 65 µm and is 80 cm long. For all measurements in this section the input signal power is 10 W and the peak-to-peak voltage of the sinusoidal electrical signal from the function generator is 2 V corresponding to a transverse beam movement of around 0.2 µm at the fiber input-facet. PFs from 25 Hz to 950 Hz will be used to seed the system.
Fig. 9. ST measurement for 162 W of signal power with a input perturbation at 225 Hz. a) shows the PSD with colored dots indicating the frequencies for which the spatial distributions of the intensity and phase are shown in b). c) shows a still from a movie of the beam as function of time (see Visualization 3).
The transverse position of the input seed light is translated a very small amount when the piezo actuator is driven by the function generator. The small translation of the coupling adds a dynamic to the static input content of the LP$_{01}$ and LP$_{11}$-modes. Below it is shown that this dynamic can be a seeding mechanism of TMI. ST measurement of the signal without amplification shows a PSD peak at the PF just above −60 dBc/Hz for this perturbation strength. For larger pump powers, this peak experiences strong gain. An ST measurement of the amplified signal @ 162 W and a PF of 225 Hz is seen in Fig. 9. For this power the TMI is still in the transition region and the DC component looks Gaussian. This ensures spatial distributions that are understandable. The PF of 225 Hz is chosen to avoid overlapping with peaks in the pump intensity noise and to be within the region of high TMI gain of the rod.
The PSD clearly shows a lot of gain at the PF compared to the −60 dBc/Hz of seeding. The spatial distribution of the fluctuation at this frequency is similar to that observed for other ST measurements of TMI in the transition region seen in Fig. 7 and Fig. 3. The same structure is also observed for a "natural" TMI fluctuation at 501 Hz i.e. TMI we do not excite with our perturbation. The orientation of this fluctuation is orthogonal to the strong seeding which again shows that the orientation of the TMI peaks not seeded on purpose seem random. At the second harmonic of the PF, 449 Hz, a new interesting spatial distribution is observed. Both the intensity and phase have three lobes on a line. The center-lobe is the strongest and has a flat phase which ${\mathrm{\pi}}$-shifted compared to the outer lobes. This structure looks similar to the second harmonic of the moving Gaussian from Fig. 2 except for smoother lobes. The perturbation of the coupling results in exactly a moving Gaussian at the input of the rod. To fully understand the nature of the second harmonic of the perturbation a simple simulation of the beam dynamic is performed.
Simulation of perturbation dynamic
In the following section, a simple simulation of the resulting output signal induced by the perturbation is presented and used to gain quantitative information regarding mode content. More specifically, the goal of the simulations is to understand the experiments and to extract the fractions of HOM for a given ST measurement of TMI with input perturbation. The scope of this section is not to simulate the TMI but to find the mode dynamic achieving the observed ST measurements.
For simplicity the simulation has been performed with the modes of a SIF of normalized frequency V=2.5 which only guides the LP$_{01}$ and LP$_{11}$-modes. Only one LP$_{11}$-mode is assumed since we only perturb in one axis. The intensity at a given position inside the fiber can be defined through the two modes,
where $\psi _{01}$ and $\psi _{11}$ are the two normalized modes of the electrical field, respectively, and $a_e$ and $b_e$ are their complex weight. We assume perfect initial coupling and the shape of the input light is consequently proportional to $\psi _{01}$. The time dependence is described by a sinusoidal translation and the input electric field becomesThe amplitude of the perturbation i.e. $A$ in Eq. (6) was measured experimentally by placing a high-speed camera at the position of the rod. The resulting ST measurement was compared to simulations of a moving LP$_{01}$-mode with different amplitudes. The response of the piezo actuator depended somewhat on the PF but for most frequencies an amplitude of around 0.25 % of the MFD was found which is equal to 0.16 µm for this rod.
To visually show the initial excitation of the modes in the rod, a movie of the electrical fields of the fiber modes, the electrical field of the input coupling, and the resulting overlaps are seen in Fig. 10. Here the amplitude of the input coupling is increased to 30 % of the MFD of the LP$_{01}$-mode to visualize the dynamics. When the input coupling is translated the modal weights change. For the two extrema of the translation, the fraction of power in the LP$_{11}$-mode is at its maximum but $\pi$-shifted in phase with respect to each other. Both of these points are experienced once every period of the fluctuation. Between the two extrema of the translation the input light only couples to the LP$_{01}$-mode and at this exact point, the phase of the LP$_{11}$ jumps $\pi$.
Fig. 10. The upper three plots show the electrical field of the LP$_{01}$-mode, LP$_{11}$-mode, and input coupling from left to right. For this purely explanatory simulation the amplitude was exaggerated to 30 % of the MFD compared to the 0.25 % measured. The lower plot shows the electrical field weights of the two modes as a function of time, $a_{e0}(t)$ and $b_{e0}(t)$ (see Visualization 4).
Larger fractions of LP$_{11}$ are needed to realize the goal of mapping simulations onto the experimental data and revealing the second harmonic. Simulating the TMI amplification process is outside of the scope of this work. A simple amplification scheme was decided which is described by the following formulas:
where $g_{11}$ is the power gain of the LP$_{11}$-mode due to TMI. This is a very primitive amplification of the modal weight but it will keep the phase of the initial excitation. The second formula shows that the excitation of LP$_{01}$ does not depend on the initial excitation, but is chosen to keep the total power constant and in that way account for the amplification of the LP$_{11}$-mode. In this model the initial power transfer between the modes, $a_{e0}$ and $b_{e0}$, are purely caused by the input perturbation. It is easy to show that this amplification scheme does not immediately follow the established theory of TMI. For an initial excitation of the LP$_{11}$-mode following a periodic translation the mode content can be written as an up-shifted and down-shifted part:In Fig. 11 the ST measurement for a simulation is shown where the initial excitation of the LP$_{11}$ is amplified 40 dB. This results in a maximum power fraction of the LP$_{11}$-mode of around 5 %. The PSD immediately shows that the simulated dynamic leads to higher harmonics. The spatial distributions for the DC and main fluctuation have the usual structures seen for all investigated dynamics. The second harmonic has almost exactly the distribution seen from the TMI with input perturbation shown in Fig. 9.
Fig. 11. ST measurement of simulated intensity inside the rod for a dynamically translated coupling. a) The PSD as a function of frequency. b) Spatial resolution of the intensity and phase for 225 and 450 Hz.
The simulation shows that this second harmonic may be understood purely from the modal weights of the two modes. For the small sinusoidal translation the modal weights are described by
Inserting into Eq. (5) yields the intensity which defines the ST measurement:The intensity written here consists of three terms: The first static term of the intensity of the LP$_{01}$-mode, the second term consisting of a static part and a part oscillating with a frequency of double the PF, and the cross term. Equation (11) reveals that the spatial distribution of the second harmonic is the difference in the square of the fields and that it has its origin from the transfer of power between the modes. The cross term gives rise to the main peak of the fluctuation.
As we have shown, this simple simulation captures a lot of the physics, resembles the experimental data quite well, and gives confidence in the resulting mode dynamics from the simulation. On the other hand, there are some obvious problems with the simulation which should be commented. The difference in phase velocities between the modes was set to zero to achieve resemblance to the experiments and simplify the model, but will in most cases be non-zero. A non-zero phase velocity difference would lead to a constant phase difference between the modes caused by the difference in accumulated phase through the rod. It is actually possible to completely suppress the cross term and therefore the main fluctuation which has not been observed. The model also has another problem which is of a more practical nature. In the simulation we assume perfect alignment of the input coupling leading to excitation on both sides of the core and both phases of the LP$_{11}$. As mentioned previously the maximum translation of the input is in the order of 0.16 µm. This margin of error is not possible with our coupling scheme and in most cases all translations will excite the same phase of the LP$_{11}$. This is also not observed experimentally.
Frequency and signal power response of seeded TMI
In this section the strength of the main fluctuation of the ST measurement for different operations of the system shown in Fig. 8 are investigated. The strength of the fluctuation for controlled seeding is defined through the root-mean-square (RMS) value at the PF,
Fig. 12. a) shows the RMS-value at the PF of the ST measurement on the output signal (blue curve) for a signal power of 134 W and the input seeding at the rod facet (red curve) as a function of PF. b) shows the RMS-value of the main fluctuation peak of the ST measurement as a function of signal power for PF of 225 Hz. c) shows the relationship between the resulting RMS-value of the main fluctuation peak and the maximum fraction of LP$_{11}$ calculated from simulations for two different V-parameters of the SIF.
The results from the mapping of experimental RMS-values to fractions of LP$_{11}$ are shown in Fig. 13 for both values of the V-parameter. Here the gain of the LP$_{11}$-mode through the rod as a function of frequency and the fraction of LP$_{11}$ as a function of signal power are plotted. The gain of the LP$_{11}$ clearly shows a strong dependence on frequency with no difference between the two V-parameters. This mapping allows us to compare almost directly to numerical models of the frequency dependent gain of TMI by Hansen et al. [9,28] and Dong [23]. Both models show comparable results with respect to frequency and there is in general great similarity between our measured gain shape and the predicted gain shape. For the lowest frequency recorded in Fig. 13(a), the relative gain of the LP$_{11}$ is negative which is caused by the differential gain between the LP$_{01}$ and LP$_{11}$-modes. This differential gain will suppress the LP$_{11}$-modes compared to the LP$_{01}$-mode and reduce the measured TMI gain by some factor for all frequencies investigated.
Fig. 13. a) shows the relative gain of the LP$_{11}$-mode as a function of perturbation frequency with respect to the LP$_{01}$-mode. The relative gain shown for simulations with different V-parameters. b) shows the fraction of LP$_{11}$ as a function of signal power.
The frequency sweep describes the frequency response of the TMI gain and can for example be used to optimize amplifier systems with respect to noise. The gain of the maximum fraction of LP$_{11}$-mode, seen in Fig. 13(b), does not have the same practical application but is also interesting. For small signal powers the mapping predicts mode fractions below $10^{^-5}$ which requires coupling accuracy below initial translation, $\approx {0.2}$ µm, and is therefore very unlikely. On the other hand the RMS-values of Fig. 12(b) are calculated directly from a real signal. A better way to think about the fraction of power described in Fig. 13(b) could be the amplitude of the dynamic part of power in the LP$_{11}$. The plot shows that the dynamic fraction of power in the LP$_{11}$ does not change significantly for two different V-parameters. For small signal powers the dynamic fraction decreases which again is caused by the differential gain of the two modes. For signal powers above 40 W, the dynamic fraction of power experiences an exponential growth which is seen by the linear tendency on a log-scale. Without intentional seeding of the system, instabilities are first observed after around 120 W of signal power. This shows that TMI is an effect which is present at all times and that TMI experiences exponential gain long before detectable fluctuations.
The largest source of error of the results from Fig. 13 is probably the imposed fit of the measurements to the crude model. The qualitative analysis of the ST measurement in Fig. 9 seem very similar to the crude model, but quantitative estimation of this error is difficult. What the mapping does show is that the results from Fig. 13 do not depend strongly on the confinement of the LP$_{11}$-mode indicating small errors from mode differences between the investigated rod and the SIF simulations.
We have shown with the ST method that the seeded instability seen in Fig. 9 behaves in many ways as TMI without induced seeding from Fig. 3, Fig. 7, and Fig. 9. Contrary to TMI without seeding we also observe a second harmonic of the main fluctuation but this does not truly distinguish the seeded instability. For TMI without controlled seeding the fluctuations are broad in frequency and any potential second harmonics will be dominated by a main TMI fluctuation. Consequently it is not expected to observe the second harmonic of TMI without an intentional strong seeding of TMI at a specific frequency. Following the indication that the TMI is seeded by beam pointing stability can explain experiments by Qiuhui Chu et al. [29], where they showed that larger misalignment actually enhanced the TMI threshold. For a larger misalignment the same beam pointing stability would lead to a smaller dynamic part of the power in the modes of the fiber.
5. Conclusion
In this work we have investigated the behaviour of TMI in the presence of pump intensity noise and perturbation of the input coupling with the spatially and temporally resolved imaging. Application of the ST method on numerical beam dynamics showed its potential to describe small fluctuations and the importance of careful interpretation. Experimental application showed that the significant frequencies found in TMI spectra can originate from pump intensity noise caused by the pump power supply. Different power supplies have different noise levels and can influence the level of TMI of amplifier systems. It was also shown how an ST measurement can be used to determine the fluctuation using the spatial component and distinguish between intensity noise and TMI. For a given signal power and input coupling, we observe different orientations of TMI in the transition region. This result indicates that the combination of a static initial fraction of LP$_{11}$ and pump intensity noise does not act as the seeding mechanism. We have managed to directly seed TMI by harmonic misalignment of the input coupling in the transverse direction using a kinematic mirror mount. The strong seeding combined with the ST-measurement revealed an interesting fluctuation at double the perturbation frequency not observed previously. Through simple numerical simulations it was shown that the harmonic misalignment leads to dynamic power transfer between the LP$_{01}$ and LP$_{11}$-modes. The simulated dynamic achieves a very similar ST-measurement as the induced TMI and is used to estimate the fraction of LP$_{11}$ for experimental TMI. The induced TMI was investigated for different perturbation frequencies achieving measurements of the TMI gain as a function frequency with great compliance with existing models of the non-linear coupling coefficient. Lastly, induced TMI at 225 Hz was investigated for increasing signal power which showed exponential gain long before visible beam fluctuations.
Funding
Danmarks Tekniske Universitet and NKT Photonics A/S.
Disclosures
Simon L. Christensen: NKT Photonics A/S (F), Mette M. Johansen: NKT Photonics A/S (E), Mattia Michieletto: NKT Photonics A/S (E), Marco Triches: NKT Photonics A/S (E), Martin D. Maack: NKT Photonics A/S (E), and Jesper Lægsgaard: NKT Photonics A/S (C).
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