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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24545–24558
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Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers

Arlee V. Smith and Jesse J. Smith  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24545-24558 (2012)
http://dx.doi.org/10.1364/OE.20.024545


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Abstract

Using numerical simulations of thermally induced mode coupling we show how the instability threshold can be substantially reduced if the pump or injected signal is modulated in the kHz range. We also show how the mode coupling gain varies with the frequency offset of the parasitic mode. We model thresholds when the source of detuned light is quantum background, amplitude modulation of the pump power, and amplitude modulation of the signal seed. We suggest several key experimental and modeling tests of our model.

© 2012 OSA

1. Introduction

This raises the important question of the origin of the LP′11 seed light, and whether its strength is sensitive to operating conditions. Here we present a detailed discussion of this question, showing how our model can account for observed threshold powers as well as a sensitivity to operating conditions. Because our concern is coherent beam combining, we define threshold as 5% of the output signal in higher order modes, this being the usual limit set for beam combining systems. In this paper we do not model behavior above this threshold, although our model is capable of that in many cases. We will also briefly compare our model with similar models published recently by other authors [4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

, 6

6. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20, 12912–12925 (2012). [CrossRef] [PubMed]

]. Finally, we will suggest straightforward experiments that can critically test our proposed mode coupling mechanism and our model.

2. Mode coupling mechanism

We briefly review the gain mechanism built into our model. The LP01 and LP′11 modes interfere to produce an irradiance grating that moves along the fiber with a velocity proportional to Δν, the frequency detuning between them. This irradiance grating creates a moving temperature grating which in turn produces a moving refractive index grating via the thermo optic effect. The temperature and index gratings have a phase lag relative to the irradiance grating because of the time lag introduced by thermal diffusion. The moving, phase-shifted index grating couples light from the strong LP01 mode into the weak LP′11 mode.

This mode coupling mechanism is a form of stimulated (near) forward thermal Rayleigh scattering. It is categorized as such because it involves a diffusive temperature wave rather than a propagating wave such as the acoustic wave in stimulated Brillouin scattering (SBS), and it does not involve a material resonance like stimulated Raman scattering (SRS) [8

8. R.W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).

]. However, like SBS or SRS it can produce the large exponential gain usually associated with a sharp threshold. Like other stimulated Rayleigh processes, the one we propose has a dispersion-like gain profile centered at zero offset frequency [8

8. R.W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).

, 9

9. O. Okusaga, J. Cahill, A. Docherty, W. Zhou, and C. R. Menyuk, “Guided entropy mode Rayleigh scattering in optical fibers,” Opt. Lett. 37, 683–685 (2012). [CrossRef] [PubMed]

]. The time scale, and thus the inverse of the frequency offset at maximum gain, is approximated by the thermal diffusion time across the core radius, (r2ρC/K) where r is the radius, ρ the density, C the specific heat, and K the thermal conductivity. For typical fiber amplifiers this time falls in the range 0.3–3 ms. An approximate semi-analytic version of this model was recently published by Hansen et al [10

10. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012). [CrossRef] [PubMed]

]. They illustrated the variation of gain with frequency offset for several fiber designs, and also estimated the instability thresholds.

One perhaps subtle point is that the signal spectrum can be broad compared with the 1 kHz scale frequency shifts incorporated into our model. In fact most signal seeds will be much broader than 1 kHz, for example when the signal consists of a train of short pulses or when it is cw light that is phase modulated to suppress SBS. The important point is that any signal spectrum, broad or narrow, will be shifted in its entirety by approximately 1 kHz when it scatters from the moving index grating. Further, interference between a broad signal spectrum and its frequency shifted replica will produce the strong mode beats responsible for creating the moving temperature gratings. Therefore, spectrally broad signals behave exactly like single frequency signals in those respects that matter to our model. If the two replicas acquire a time offset comparable to their inverse line width due to differing group velocities for the two modes, the interference will weaken. Mode dispersion delays are on the order of 1 ps/m so the line width limit is in the range of several hundred GHz, much broader than the line width limit imposed by coherent beam combining. Hence our model simulates equally well both narrow and broad signal spectra, provided they are frequency shifted and have limited line widths.

3. Model details

The methods we use for this paper are conceptually identical to those described earlier [7

7. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011). [CrossRef] [PubMed]

]. We use a fast Fourier transform (FFT) beam propagation method (BPM) [11

11. M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” Appl. Opt. 19, 2240–2246 (1980). [CrossRef] [PubMed]

, 12

12. A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995). [CrossRef]

] in order to make the model as general as possible. This allows us to include a wide variety of physical effects. For example, it automatically includes all transverse modes, and it accounts for all thermal lensing. The split-step propagation procedure first updates the time varying optical field, which includes all frequency components and all modes, by adding the time dependent propagation phases due to the guiding index step and due to the thermo optic phase computed from the transverse, time varying temperature profile. The laser gain or loss is also added to the fields in this part of the split step. In the second part of the split step each time slice of the wave is propagated by Fourier transforming the field, adding the appropriate propagation phases, and performing the reverse Fourier transformation. At each z location we solve the time-dependent temperature equation over one beat cycle of duration 1/Δν. The heat Q(x, y,t) at each location is computed by finding the absorbed pump power from the steady state solutions to the population/gain equations. The resulting time-dependent temperature profile is converted to a time-dependent refractive index change Δn(x, y,t) by multiplying by the thermo optic coefficient. This index change is used in the beam propagation integration. When we analyze the signal field into modes for each time slice we use the low power modes as the basis set. The time varying amplitude of each mode is further analyzed into frequencies (0, ±Δν, ±2Δν, ...) by Fourier transformation. This model can accommodate mode-specific losses due to leakage from the core or due to scattering. It can also include heating due to photo darkening or other types of absorption, but they are not included here.

In computing the laser gain and the deposited heat we use the steady state solution for the upper state population fraction
nu=σpaIp/hνp+σsaIs/hνs(σpa+σpe)Ip/hνp+(σsa+σse)Is/hνs+1/τ
(1)
to find the heat deposition due to pump absorption
Q=[σpa(σpa+σpe)nu]NYbIp[1νsνp].
(2)
The σs are absorption and emission cross sections for the pump and signal, νp and νs are the pump and signal optical frequencies, Is(x, y,t) and Ip(x, y,t) are irradiances, and τ is the upper state lifetime. The two dimensional heat equation
ρCdTdt=Q+K(2Tx2+2Ty2)
(3)
is solved numerically to find T(x, y,t).

We make certain approximations for the sake of computational speed. As mentioned, we use a steady-state population/heating model in which the population instantaneously follows the local signal and pump irradiances. We justify this by noting that at the power levels of interest the effective lifetime of the upper state Yb3+ ions is in the range of 10 μs, which is much shorter than the 1 ms beat times of interest. We also compute the temperature in two dimensions rather than three. This means we cannot account for longitudinal heat flow, but we believe this is a small effect. Additionally, we place the thermal boundary approximately two core diameters from the core center rather than at the actual fiber edge. We justify this placement by noting that moving the thermal boundary out by a factor of three does not change our model results significantly but slows the model by a large factor. We also set the temperature rise at the boundary to zero. In reality, thermal transport from this boundary to the physical boundary results in a temperature rise at the model boundary, and this adds a nearly uniform temperature rise in the vicinity of the core. Such a uniform rise would affect neither thermal lensing nor mode coupling directly, but it could still have a small influence on amplifier performance because the Yb3+ absorption and emission cross sections as well as the upper state lifetime change slightly with temperature over the range of interest [13

13. S. W. Moore, T. Barnett, T. A. Reichardt, and R. L. Farrow, “Optical properties of Yb3+-doped fibers and fiber lasers at high temperature,” Opt. Commun. 284, 5774–5780 (2011). [CrossRef]

, 14

14. T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007). [CrossRef]

]. The absorption and emission cross sections at 1064 nm are relatively insensitive to temperature. However, the pump cross sections at 976 nm fall by 10–20% over 0–400°C, and the lifetime falls by 10% over 0–400°C, and much faster above 500°C. Temperature dependent cross sections and lifetimes could be included in our model, but presently they are not.

3.1. Using the ADI method

In the results presented here based on computations using the ADI method we use a single cycle of the time-dependent heating. We string together copies of the heat cycle to make a periodic time sequence containing many cycles. This is used as the heat term in the ADI computation. We then run the ADI integration until there is negligible change in the temperature response from one cycle to the next, and we keep the results for the last cycle. This guarantees a steady-periodic response is being modeled. This allows periodic modulation of the signal and pump inputs with frequency components (0, Δν, 2Δν, ...). If desired, this method can also be used to study transient effects that follow sudden changes in the injected signal or pump light. In such cases we would not enforce the periodicity or stability criteria.

3.2. Using the Green’s function method

Unlike the ADI method, the Green’s function method is restricted to steady-periodic heating. As described in Appendix B, Green’s functions are calculated for sinusoidal heating at frequencies ωm = 2π(0, Δν, 2Δν, ...). In applying this method, the heat at each grid point is analyzed into frequency, amplitude, and phase, and the appropriate Green’s function is used to find the 2D temperature response. This is repeated at all desired frequencies for all heated (all Yb3+-doped) grid points. Because mode coupling is directly driven predominantly by the component of temperature oscillations at frequency 1Δν, usually the threshold computations can be performed using only the zero frequency Green’s function to account for thermal lensing, plus the 1Δν Green’s function to handle mode coupling. For computations of above threshold behavior it is sometimes necessary to include additional, higher frequency Green’s Functions.

4. Gain spectrum

We have mentioned that the gain of a stimulated Rayleigh process such as we propose is expected to have a dispersion-like gain profile. Figure 1 illustrates this with gains we compute using the Green’s function method for a co-pumped amplifier with 30 μm core diameter. Further details of this amplifier are given in [7

7. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011). [CrossRef] [PubMed]

] except here we use 1 kW of pump and compute the gains over 15 mode beat lengths at the input end of the amplifier. These curves show the total gain of LP′11 including laser gain and mode coupling gain. The laser gain, which accounts for the gain at zero frequency shift, is relatively small compared with the peak mode coupling gain. The shape of the gain curve changes with z position due to evolving gain saturation effects and due to thermal lensing. However, the exact details of the fiber are not important here as our point is to illustrate a typical dispersion-like gain profile, demonstrating gain on the Stokes side and loss on the anti-Stokes side. Note that in this example the mode coupling gain of LP′11 is approximately twice that of LP′02. In fact, we find the LP′11 gain is generally greater than that of any other mode, explaining why, to our knowledge, LP′11 is the intruding mode that is always observed in the laboratory.

Fig. 1 Total modal gains (laser gain plus mode coupling gain) computed for LP′11 and LP′02 for the amplifier design studied in [7]. The detuning Δν is the frequency shift of the mode of interest relative to the strong driving mode LP01.

5. Model parameters

The parameters we used in the modeling presented below are chosen to be nearly equal to those used by Ward et al [4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

], and are listed in Table 1. The spatial grid is 64 × 64, the time cycle is gridded into 64 points, and the propagation step is 6 μm. Unlike Ward et al [4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

] we make the doping diameter equal to the core diameter, and we shorten the fiber to 0.8 m because there is little gain in the first half of their 1.6 m version.

Table 1. Parameters of test amplifier.

table-icon
View This Table

6. Sources of LP′11 seed light

6.1. Quantum noise

One possibility is that LP′11 is populated solely by quantum noise. If this noise is added at the input to the final amplifier stage, this would give the lowest possible starting level and thus the highest possible instability threshold. Quantum noise provides approximately one photon per beat cycle, corresponding to a power of (Δν) which, for a 1 kHz offset and linewidth, is approximately 2 × 10−16 W. We use 10−16 W in simulations aimed at finding this highest possible instability threshold.

In practice the laser and preamplifiers supplying the signal seed will almost certainly have a higher noise level than this, in which case the threshold will be somewhat lower. Some of the noisy signal input will find its way into LP11 either through imperfect mode matching or through scattering at fiber imperfections. The amount of light injected into LP′11, and thus the threshold, will be sensitive to the level of seed noise and to the launch conditions.

6.2. Spontaneous thermal Rayleigh scattering

A second possible source of LP′11 seed light is thermally induced fluctuations in the refractive index near the input end of the amplifier. Such fluctuations could scatter light from LP01 or LP11 into LP′11. This is a type of spontaneous thermal Rayleigh scattering. However, based on measured attenuations of less than 1 dB/km for temperature independent Rayleigh scattering, combined with the fact that a temperature dependent contribution is unobservable in the background of the temperature independent part [17

17. P. C. Wait and T. P. Newson, “Landau-Placzek ratio applied to distributed fiber sensing,” Opt. Commun. 122, 141–146 (1996). [CrossRef]

], we estimate that this process provides at most only slightly more seed light than does quantum noise. A signature of this contribution would be a reduction in the threshold with an increase in the temperature of the signal input end of the amplifier.

6.3. Pump modulation

A third possible source of LP′11 seed light, and one we think may dominate, is pump modulation. If the pump light is slightly amplitude or spectrum modulated in the 1 kHz band, and if the signal is launched with a fraction of its power in LP11 at the fiber input, gain modulation of the signal due to the pump modulation produces frequency side bands on the signal, including the portion in LP11. Two symmetric frequency sidebands are created. The blue shifted one creates an optical interference and temperature wave that moves upstream; the red shifted one creates an optical interference and temperature wave that moves downstream. Each sideband interacts strongly only with the temperature wave that co-travels with its interference wave. The phase shift between the temperature wave and the optical interference wave is opposite for the two sidebands, leading to amplification for the red shifted sideband (LP′11) and deamplification for the blue shifted sideband as illustrated in Fig. 1. The threshold would thus be reduced in proportion to the degree of pump modulation and in proportion to the power inadvertently launched into or scattered into LP11.

6.4. Signal modulation

Similarly, if the injected signal is slightly amplitude or phase modulated with a side band in the −1 kHz range at input to the fiber, the side band on the fraction that is inadvertently injected into LP11 will populate LP′11. The strong carrier light in LP01 will act as the pump for the weak LP′11 light, as usual, again leading to a reduced threshold.

7. Threshold calculations

7.1. Quantum noise

Figure 2 shows model results when quantum noise is used as the seed to the amplifier specified in Table 1. The LP′11 mode is seeded with 1 × 10−16 W and mode LP11 is unseeded. The counter pump input power is 353 W of which 7 W is unabsorbed. The input LP01 power is 50 W; the total signal out including light in all modes is 356 W with 323 W in LP01 and 8.4 W in LP′11. This case yields the highest possible threshold at 356 W of signal. At this power level there is significant thermal lensing. The effective area of LP01 changes from 2770 μm2 at the signal input end to approximately 1700 μm2 at the pumped end. Our model includes this effect.

Fig. 2 Results for quantum noise seeding. Power in the pump, and two signal modes for Δν = −525 Hz. The brackets around Power indicate powers are time averaged over a beat cycle.

7.2. Pump modulation

Fig. 3 Results for 300 ppm pump modulation at 525 Hz. The pump power in is 288 W; the signal output power is 302 W in all modes with 12 W in LP′11.

7.3. Signal modulation

The case of signal amplitude modulation without pump input modulation is difficult to model precisely since modulation of the signal induces modulation of the pump. Because we integrate from the pump output end to the pump input end we must precompensate by specifying the pump output modulation that will lead to an unmodulated input. We have not succeeded in perfect precompensation but we have come close. The results are shown in Fig. 4. The input signal is sinusoidally amplitude modulated at 30 parts per million peak-to-peak. In this simulation we apply this amplitude modulation to the input light of both the 49.75 W in LP01 and the 0.25 W in LP11. The AM sideband of LP11 forms the 10−12 W input LP′11 which is amplified to threshold by the carrier LP01 light. We find the threshold signal output power is 275 W total, with 5.1 W in LP′11, corresponding to a pump input power of 257 W and output power of 1.4 W. This is a large reduction from the 353 W pump threshold for quantum noise alone.

Fig. 4 Results for 30 ppm signal modulation. Power in the pump, and three signal modes for Δν = −525 Hz. The pump power is 257 W in and 1.4 W out; the total output signal power is 275 W with 5.1 W in LP′11.

8. Comparison with measured thresholds

We have computed the highest threshold for one particular amplifier, as well as demonstrating that substantial threshold reductions can be caused by the introduction of light into LP′11 by modulating the pump or the signal seed, in combination with a slight population of LP11. It is interesting to compare these results with some reported measured thresholds.

Several experimental studies have reported instability threshold powers. They seem to fall in the range 100–300 W for amplifiers ≈ 1 m long with core diameters of 40 μm or greater [1

1. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011). [CrossRef] [PubMed]

, 18

18. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011). [CrossRef] [PubMed]

, 19

19. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011). [CrossRef] [PubMed]

], and in the range 500–1500 W for ≈ 10 m long amplifiers with core diameters less than 40 μm [1

1. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011). [CrossRef] [PubMed]

, 20

20. N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tüunnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012). [CrossRef] [PubMed]

]. The thresholds computed above for one example of a short, large diameter fiber ranged from 275 to 350 W of signal, in reasonable agreement with the measured values. Furthermore, the model qualitatively matches the observation of sharp power thresholds [1

1. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011). [CrossRef] [PubMed]

, 4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

]. An increase of 10% in pump power is usually sufficient to fully switch the output signal from LP01 to LP′11.

Our model results account for LP′11 having the lowest threshold, in agreement with the universal observation of only LP01 and LP11 above threshold. Finally, we have explained how instability thresholds can be sensitive to the operating conditions, specifically pump or signal modulations combined with slight initial population of LP11.

9. Suggested experiments

Here we propose some critical experimental tests of the gain mechanism and of our model.

9.1. Look for frequency shifts

The most direct way to look for the frequency shifts that are essential to our model is to study in detail the behavior of the output beam near the modulation instability threshold. One could look at the output beam simultaneously at two symmetric locations along the line of symmetry. They could be positioned near the peak irradiance points of the LP11 mode. The pump power should be turned up gradually until mode distortion is barely noticeable. Out-of-phase amplitude modulation would indicate beating between frequency offset modes with even and odd symmetry in the plane of the detectors - presumably modes LP01 and LP′11. This measurement should determine whether or not LP′11 is present. Similar experiments have been performed using a single detector, and they showed sinusoidal modulation [4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

]. Better documentation, including a thorough catalog of fiber parameters and detector placement and aperture, plus detection with dual detectors would be an important improvement.

9.2. Measure/modify the pump modulation

As we illustrated above, a small amount of pump modulation could significantly reduce the instability threshold. One could measure the degree of pump power modulation in the 1 kHz band. To see whether this is an important contributor, the pump power could be deliberately modulated. One could also alter the position of the seed beam to vary the fraction injected into LP11 as a further test of this idea. If power modulation is found to affect the threshold, the modulation frequency could be varied to map the instability threshold and the mode coupling gain.

It may be that pump spectral modulation rather than power modulation is important. In that case one could look for fluorescence modulation to determine whether pump modulation near 1 kHz is significant.

9.3. Measure/modify the signal modulation

We also described how a small sideband on the injected signal, combined with inadvertent seeding of LP11 will lead to the population and amplification of LP′11. One could look for amplitude or phase modulation of the signal before injection, or one could deliberately add such modulation, to test this possibility. Tuning the modulation frequency would map the gain line shape.

9.4. Numerical experiments

The light transit time through the amplifier is much shorter than the mode beat time so the order of integration in our model (t then z) can be reversed. Rather than integrating over the full t range followed by stepping in z, one could integrate over the full z range followed by stepping in t. Use of this reversed order has the advantage that the heat diffusion equation can be integrated in three dimensions rather than two dimensions as we have done. This would make it possible to test whether longitudinal heat flow has any impact on mode instability. Two recent papers [4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

, 6

6. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20, 12912–12925 (2012). [CrossRef] [PubMed]

] described such a model. The physics included in those models is nearly identical to that in our model. However, they were applied only to studies of transient mode coupling following abrupt turn on of the pump [6

6. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20, 12912–12925 (2012). [CrossRef] [PubMed]

] or abrupt changes to the seed light [4

4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

]. Applying them to a case with a steady-periodic perturbation would allow a test of the role of longitudinal heat flow in the instability mechanism we are touting.

10. Conclusions

We have presented a detailed model of mode instability that is capable of predicting modulation frequencies and threshold powers for step index fibers. Replacing the FFT-based BPM with a more general BPM would make the model suitable for accurate modeling of photonic crystal fibers as well. The model predictions in every case that we are aware of match qualitatively the laboratory results. Unfortunately, detailed quantitative comparisons are hampered by incomplete documentation of the experiments. We hope that in the future a more thorough catalog of fiber parameters will be included, and that this standard will be applied to numerical models as well.

In this study we used narrow linewidth perturbations to compute the gain spectrum. The chosen frequency offsets were those giving the highest gain. In the laboratory, the width of the gain profile will depend on the nature of the input signal and pump light. If the source of LP′11 is quantum noise, the Stokes spectrum should be broad. If it is pump or signal modulation the Stokes spectrum will reflect the relevant modulation spectrum. By showing how small amounts of modulation of the pump or the signal seed can substantially reduce the instability threshold we hope to motivate measurements of such modulations and studies of their influence on thresholds. If such modulations are present, the path to maximizing thresholds is clearly their minimization.

Although our model has been successful in qualitative comparison with experiments, stronger confirmation is desired. We suggested key experiments to test the model. If it is validated, we can proceed with parametric studies to learn how to best suppress the mode instability while still combating other nonlinearities such as SBS and SRS. This will require further speed up of the model. We anticipate that the Green’s function approach in particular can made much faster using a computer with a large number of processors in place of our single processor model.

11. Appendix A: ADI method

ADI is short for alternating direction implicit. This method is outlined in Numerical Recipes [15

15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1992).

], and we present our implementation here. In this method the temperature is updated over Δt in two steps of Δt/2 each. The temperature on the boundary is maintained at zero, so we operate on the (Nx − 2) × (Ny − 2) interior points of the T grid. In the first half-step we use explicit integration in the y–direction and implicit integration in the x–direction. This yields the first half-step difference equation
Tx,yt+Δt/2Tx,yt=(Tx+Δx,yt+Δt/22Tx,yt+Δt/2+TxΔx,yt+Δt/2)λx2+(Tx,y+Δyt2Tx,yt+Tx,yΔyt)λy2
(4)
+Δt2ρCQt+Δt/4
(5)
which can be rewritten in matrix form as
Tt+Δt/2Ax+ByTt+Δt2ρCQt+Δt/4
(6)
Tt+Δt/2=ByTtAx1+Δt2ρCQt+Δt/4Ax1
(7)
where Ax is the (Nx − 2)×(Nx − 2) tridiagonal matrix with diagonal elements (1 + λx) and off-diagonal elements −λx/2, and By is the (Ny − 2) × (Ny − 2) tridiagonal matrix with diagonal elements (1 − λy) and off-diagonal elements λy/2, where
λx=KΔtρCΔx2
(8)
λy=KΔtρCΔy2.
(9)

The second half-step in time reverses the implicit/explicit order to yield a similar difference equation,
Tx,yt+ΔtTx,yt+Δt/2=(Tx+Δx,yt+Δt/22Tx,yt+Δt/2+TxΔx,yt+Δt/2)λx2+(Tx,y+Δyt+Δt2Tx,yt+Δt+Tx,yΔyt+Δt)λy2
(10)
+Δt2ρCQt+3Δt/4
(11)
which we write
AyTt+Δt=Tt+Δt/2Bx+Δt2ρCQt+3Δt/4
(12)
Tt+Δt=Ay1Tt+Δt/2Bx+Δt2ρCAy1Qt+3Δt/4
(13)
where Ay and Bx are the same as above except redimensioned as appropriate, and with λx and λy exchanged.

Combining these two half-step equations gives the expression used to update the temperature over the full time step
Tt+Δt=Ay1ByTtAx1Bx+Δt2ρCAy1(Qt+Δt/4Ax1Bx+Qt+3Δt/4).
(14)
We find Qtt/4 and Qt+3Δt/4 from the heats at t and t + Δt by linear interpolation.

12. Appendix B: GF method

The idea of the Green’s function method is that the two-dimensional (x, y) temperature profile due to one unit of heat at a chosen grid point (x′, y′) can be computed once for (x′, y′) point and used repeatedly with the proper weighting and phase throughout the integration. The temperature profile due to an actual heating profile is constructed by summing the properly weighted and phased Green’s function values from all heated grid points. This procedure can be implemented in two dimensions for steady state heating, but also for steady-periodic heating [16

16. K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” Mech. Eng. Faculty Pubs., Univ. Nebraska, Lincoln, Paper 49, http://digitalcommons.unl.edu/mechengfacpub/49.

] where the heat at each grid point oscillates with a frequency ω, phase ϕ, and amplitude Q.

The steady-periodic Green’s function for a clamped temperature (T = 0) on the (0 = y = W) and (0 = x = L) boundaries is [16

16. K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” Mech. Eng. Faculty Pubs., Univ. Nebraska, Lincoln, Paper 49, http://digitalcommons.unl.edu/mechengfacpub/49.

]
G(x,y,ω|x,y)=n=0Fn(y,y)Pn(x,x,ω)
(15)
where
Fn(y,y)=1WKsin(nπWy)sin(nπWy)
(16)
Pn(x,x,ω)={exp[σn(2L|xx|)]exp[σn(2Lxx)]+exp[σn|xx|]exp[σn(x+x)}/σn(1exp[2σnL]),
(17)
with
σn2=(nπW)2+iωρCK.
(18)

In our model we use a fixed time interval of (1/Δν) so steady-periodic heating can have frequencies of ωm = 2πmΔν where m = 0, 1, 2.... The functions G(x, y, ωm|x′, y′) are computed for each (x′, y′) location that will be heated, which in our case means all that are Yb doped, and for each ωm of interest. Calculation of G(x, y, ωm|x′, y′) for each ωm of interest is performed once before starting the model run.

In applying the GF we Fourier analyze the heat at each grid location to find complex Fourier coefficients q(x′, y′, ωm) and use them as inputs to the appropriate GF. We compute T (x, y,t) using
T(x,y,t)=Real[m=0maxx,yq(x,y,ωm)G(x,y,ωm|x,y)exp(iωmt)]
(19)
where q(x′, y′, ωm) are the complex Fourier coefficients of the heat at point (x′, y′), and max is the highest frequency component ωmax to be included.

Acknowledgments

We thank Shadi Naderi of Air Force Research Laboratory for pointing out an important improvement in the ADI code. This effort was supported under funding from the Air Force Research Laboratory Directed Energy Directorate.

References and links

1.

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011). [CrossRef] [PubMed]

2.

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35, 94–96 (2010). [CrossRef] [PubMed]

3.

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “Impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011). [CrossRef] [PubMed]

4.

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]

5.

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20, 440–451 (2012). [CrossRef] [PubMed]

6.

C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20, 12912–12925 (2012). [CrossRef] [PubMed]

7.

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011). [CrossRef] [PubMed]

8.

R.W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).

9.

O. Okusaga, J. Cahill, A. Docherty, W. Zhou, and C. R. Menyuk, “Guided entropy mode Rayleigh scattering in optical fibers,” Opt. Lett. 37, 683–685 (2012). [CrossRef] [PubMed]

10.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012). [CrossRef] [PubMed]

11.

M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” Appl. Opt. 19, 2240–2246 (1980). [CrossRef] [PubMed]

12.

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995). [CrossRef]

13.

S. W. Moore, T. Barnett, T. A. Reichardt, and R. L. Farrow, “Optical properties of Yb3+-doped fibers and fiber lasers at high temperature,” Opt. Commun. 284, 5774–5780 (2011). [CrossRef]

14.

T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007). [CrossRef]

15.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1992).

16.

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” Mech. Eng. Faculty Pubs., Univ. Nebraska, Lincoln, Paper 49, http://digitalcommons.unl.edu/mechengfacpub/49.

17.

P. C. Wait and T. P. Newson, “Landau-Placzek ratio applied to distributed fiber sensing,” Opt. Commun. 122, 141–146 (1996). [CrossRef]

18.

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011). [CrossRef] [PubMed]

19.

F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011). [CrossRef] [PubMed]

20.

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tüunnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012). [CrossRef] [PubMed]

21.

H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20, 15710–15722 (2012). [CrossRef] [PubMed]

22.

M. Laurila, T. T. Alkeskjold, M. M. Jørgensen, S. R. Petersen, J. Broeng, and J. Laegsgaard, “Highly efficient high power single-mode fiber amplifier utilizing the distributed mode filtering bandgap rod fiber,” in Fiber Lasers IX: Technology, Systems, and Applications, E.C. Honea and S.T. Hendow eds., Proc. SPIE 8237, 8237–8265 (2012).

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(140.6810) Lasers and laser optics : Thermal effects
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(290.5870) Scattering : Scattering, Rayleigh

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 22, 2012
Revised Manuscript: October 2, 2012
Manuscript Accepted: October 9, 2012
Published: October 12, 2012

Citation
Arlee V. Smith and Jesse J. Smith, "Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers," Opt. Express 20, 24545-24558 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24545


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References

  1. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011). [CrossRef] [PubMed]
  2. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35, 94–96 (2010). [CrossRef] [PubMed]
  3. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “Impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011). [CrossRef] [PubMed]
  4. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012). [CrossRef] [PubMed]
  5. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20, 440–451 (2012). [CrossRef] [PubMed]
  6. C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20, 12912–12925 (2012). [CrossRef] [PubMed]
  7. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011). [CrossRef] [PubMed]
  8. R.W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).
  9. O. Okusaga, J. Cahill, A. Docherty, W. Zhou, and C. R. Menyuk, “Guided entropy mode Rayleigh scattering in optical fibers,” Opt. Lett. 37, 683–685 (2012). [CrossRef] [PubMed]
  10. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012). [CrossRef] [PubMed]
  11. M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” Appl. Opt. 19, 2240–2246 (1980). [CrossRef] [PubMed]
  12. A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995). [CrossRef]
  13. S. W. Moore, T. Barnett, T. A. Reichardt, and R. L. Farrow, “Optical properties of Yb3+-doped fibers and fiber lasers at high temperature,” Opt. Commun. 284, 5774–5780 (2011). [CrossRef]
  14. T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007). [CrossRef]
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1992).
  16. K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” Mech. Eng. Faculty Pubs., Univ. Nebraska, Lincoln, Paper 49, http://digitalcommons.unl.edu/mechengfacpub/49 .
  17. P. C. Wait and T. P. Newson, “Landau-Placzek ratio applied to distributed fiber sensing,” Opt. Commun. 122, 141–146 (1996). [CrossRef]
  18. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011). [CrossRef] [PubMed]
  19. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011). [CrossRef] [PubMed]
  20. N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tüunnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012). [CrossRef] [PubMed]
  21. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20, 15710–15722 (2012). [CrossRef] [PubMed]
  22. M. Laurila, T. T. Alkeskjold, M. M. Jørgensen, S. R. Petersen, J. Broeng, and J. Laegsgaard, “Highly efficient high power single-mode fiber amplifier utilizing the distributed mode filtering bandgap rod fiber,” in Fiber Lasers IX: Technology, Systems, and Applications, E.C. Honea and S.T. Hendow eds., Proc. SPIE 8237, 8237–8265 (2012).

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