Review of models of mode instability in fiber amplifiers

Transcription

1 Review of models of mode instability in fiber amplifiers Arlee V. Smith and Jesse J. Smith AS-Photonics, LLC, 8500 Menaul Blvd. NE, Suite B335, Albuquerque, NM USA Abstract: We compare several published models of mode instability for fiber amplifiers Optical Society of America OCIS codes: ( ) Fiber optics amplifiers and oscillators; ( ) Nonlinear optics, fibers; ( ) Thermal effects; ( ) Stimulated scattering, modulation, etc 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. Exp. 19, (2011), 2. M. Karow, H. Tünnermann, J. Neumann, D. Kracht, and P. Wessels, Beam quality degradation of a single-frequency Yb-doped photonic crystal fiber amplifier with low mode instability threshold, Opt. Lett. 37, (2012), 3. 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, (2012), 4. B. Ward, C. Robin, and I. Dajani, Origin of thermal modal instabilities in large mode area fiber amplifiers, Opt. Express 20, (2012), 5. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, Highspeed modal decomposition of mode instabilities in high-power fiber lasers, Opt. Lett. 36, (2011), 6. A. V. Smith and J. J. Smith, Mode instability in high power fiber amplifiers, Opt. Express 19, (2011), 7. A. V. Smith and J. J. Smith, Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers, Opt. Express 20, (2012), 8. A. V. Smith and J. J. Smith, Steady-periodic method for modeling mode instability in fiber amplifiers, Opt. Express 21, (2013) 9. A. V. Smith and J. J. Smith, Maximizing the mode instability threshold of a fiber amplifier, arxiv: [physics.optics] (2013), A. V. Smith and J. J. Smith, Frequency dependence of mode coupling gain in Yb doped fiber amplifiers due to stimulated thermal Rayleigh scattering, arxiv: [physics.optics] (2013), A. V. Smith and J. J. Smith, Modeled fiber amplifier performance near the mode instability threshold, arxiv: [physics.optics] (2013), A. V. Smith and J. J. Smith, Increasing mode instability thresholds of fiber amplifiers by gain saturation, Opt. Express 21, (2013),

2 13. A. V. Smith and J. J. Smith, Spontaneous Rayleigh seeding of stimulated Rayleigh scattering in high power fiber amplifiers, arxiv: [physics.optics] (2013), K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, Thermally induced mode coupling in rare-earth doped fiber amplifiers, Opt. Lett. 37, (2012), K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, Theoretical analysis of mode instability in high-power fiber amplifiers, Opt. Express 21, (2013), L. Dong, Stimulated thermal rayleigh scattering in optical fibers, Opt. Express 21, (2013), 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, (2012), I-Ning Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers, Photonics West Conf. Fiber Lasers X, paper (2013) 19. M. D. Feit and J. A. Fleck, Computation of mode properties in optical fiber waveguides by a propagating beam method, Applied Optics 19, (1980), M. D. Feit and J. A. Fleck, Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method, Applied Optics 19, (1980), K. D. Cole, Steady-periodic Green s functions and thermal-measurement applications in rectangular coordinates, J. of Heat Trans. 128, (2006); DOI: / K. D. Cole and P. E. Crittenden, Steady-periodic Heating of a cylinder, J. of Heat Trans. 131, (2009); DOI: / S. Naderi, I. Dajani, T. Madden, B. Ward, C. Robin, and J. Grosek, Numerical studies of modal instabilities in high-power fiber amplifiers, Photonics West Conf. Fiber Lasers X, paper (2013). 24. C. Jauregui, J. Limpert, and A. Tünnermann, Origin of thermal modal instabilities in large mode area fiber amplifiers, Spotlight on Optics, June 11, 2012, S. Naderi, I. Dajani, T. Madden, and C. Robin, Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations, Opt. Exp. 21, (2013), B.G. Ward, Modeling of transient modal instability in fiber amplifiers, Opt. Exp. 21, (2013), M.M. Johansen, K.R. Hansen, M. Laurila, T.T. Alkeskjold, and J. Laegsgaard, Estimating modal instability threshold for photonic crystal rod fiber amplifiers, Opt. Exp. 21, , 1. Introduction Mode instability refers to the threshold-like degradation of the output beam profile from large mode area fiber amplifiers. The instability has been observed by several researchers [1 5] in Yb-doped amplifiers pumped at 976 nm and operating in the range nm. Reported instability thresholds fall in the range of 100 W to 2500 W. The instability has been attributed to stimulated thermal Rayleigh scattering (STRS) by some [6 12, 14 16, 18] and to perhaps alternative thermal mode coupling processes by others [4,17,23,25,26]. In all models the mode coupling process responsible for the instability is thermal. Regions of the core are heated by the quantum defect fraction of the absorbed pump light. The resulting temperature profile creates a refractive index grating via the thermo optic effect, and this index grating scatters light from mode LP 01 into higher order

3 modes, primarily into LP 11. In all the models that are compared here, the critical model components are calculations of the quantum defect heating associated with signal amplification, calculations of the resulting temperature and refractive index profiles, and calculations of the rate of power transfer from LP 01 to LP 11. It is generally recognized that such mode coupling requires a phase shift between the irradiance grating produced by modal interference and the index grating it creates. However, the cause of that phase shift is assumed to be a frequency offset between modes in the STRS models [6 12,14 16,18], and perhaps to other effects in the remaining models [4,17,23,25,26]. In the following sections we describe the philosophies and the approximations of each model. When possible we compare predictions of the different models with one another and with measured performance of laboratory amplifiers. 2. Model approximations and methods All physical models necessarily involve numerous approximations of the actual physics. The key to a useful model is selecting the appropriate approximations for the application of interest. Two different philosophies have been adopted based on two different assumptions about the cause of the instability. One model type starts with a transient model and runs to convergence. The other model type starts with the steady-periodic assumption and integrates over one period of the beat between two optical fields that are offset in frequency. In this type model a small perturbation is added to the steady state solution of amplification to determine whether it grows exponentially and, if so, whether threshold is reached. The physics of heating and mode coupling is nearly the same in the two model types, so we expect they would converge if periodic fields were used as inputs to the transient model. Other choices that differ among models include the method of beam propagation and the method of solving the thermal diffusion equation. A beam propagation model (BPM) includes all transverse modes automatically. A coupled mode model requires that the user specify which modes to include as well as defining the modes. A large number of included modes would require integrating a large number of mode coupling equations. The steady-periodic assumption allows the use of steady-periodic Green s functions to solve the temperature equation. Alternatively, Dong [16] deals with the heat equation by describing the temperature in terms of a set of temperature modes derived in cylindrical coordinates. The alternating direction explicit/implicit (ADI) method can also be used to solve the temperature equation. Below we list some of the approximations that have been used in mode instability models. The choice of approximations can strongly influence the computational speed, with some choices being advantageous in exploiting parallel computing methods Number of modes included A coupled mode treatment requires explicit inclusion of all modes of interest. A coupling equation is added to the model for each coupled mode pair. If only two modes are of interest this can be an advantage, but if many modes are important, it can be a disadvantage. Usually only two modes are of primary interest in computing thresholds, but for above threshold operation many high order modes might be important. A beam propagation model (BPM) automatically includes all modes because a single signal field is propagated without regard to modal content.

4 2.2. Thermal boundary condition There are several choices to make for the time dependent thermal model. The first is whether to use a two dimensional (x,y,t) or three dimensional (x,y,z,t) solver. A two dimensional solver is applied to each z position, independent of adjacent z positions. The two dimensional solver is probably faster than the three dimensional solver, but it does not allow longitudinal flow of heat. The second choice is what to use as the thermal boundary condition. If the average temperature of the core, rather than the much smaller temperature variation near the core, is expected to be important, the model boundary should coincide with the outer diameter of the fiber. If a polymer coating is used, it should be included. A further complication may arise if the fiber under simulation is a photonic crystal fiber (that is, it has a complicated structure of thermal conductivity and density). If slow transient effects are of interest the model boundary should match the physical boundary. However, if only transverse temperature variations about the average and with a length scale comparable to the core diameter are expected to be important, a boundary removed from the fiber center by a few times the core diameter can be used. Based on thermal diffusion rates, frequencies of a few hundred hertz to a few kilohertz imply length scales of a few tens of microns. If the steady-periodic assumption is imposed, the closer boundary can be used. Fast temperature oscillations within the core region are not affected by a nearby boundary because the heating is expected to have a frequency corresponding to thermal diffusion across the core. A third choice for the model boundary is its shape. It should be round to best match the physical boundary if the full fiber is to be included. However, a square boundary at approximately the same distance may be acceptable and may allow much faster solution. Similarly, if only the core region is to be included, a square boundary may allow fast solution with acceptable error. A fourth choice is whether the boundary is held at a fixed temperature or whether some other thermal boundary condition is imposed. If the fiber is cooled asymmetrically, it may be more realistic to fix the temperature only on the cooled portion and allow the other portions to float, meaning the temperature slope is fixed and small so there is low heat flow there Spatial grid Models can be based on a rectangular or a cylindrical coordinate system. Both coupled mode and BPM modeling is possible using cylindrical coordinates. Cylindrical coordinates fit better to a cylindrical thermal boundary. However, if only r dependence is included, the optical field is limited to LP 0n modes. The LP 11 mode that is usually observed in the laboratory is not accommodated. Of course, if the φ dependence is included, all modes can be treated. Rectangular coordinates also permit all modes, and they allow BPM using fast Fourier transform (FFT) methods [19, 20]. If necessary a cylindrical thermal boundary can be step wise approximated on a rectangular grid in some temperature solvers Heat deposition Some models assume the heat deposition profile matches the signal irradiance profile [14 16]. They assume that when the signal irradiance grows by an amount δ I(x,y) on

5 propagation by z, the heat deposition is equal to δ I(x,y) multiplied by the quantum defect factor. This means the profile of deposited heat matches the signal irradiance profile within a multiplication factor. Other models [4, 6 12, 23, 25, 26] calculate the population inversion [n u (x,y) n l (x,y)] at each z location and use that to compute the absorbed pump power over one z step, assuming the pump power is uniformly distributed over the pump cladding. The profile of the deposited heat computed from the pump absorption is proportional to [σ a pn l (x,y) σ e pn u (x,y)] which can strongly differ from the signal irradiance profile if the population inversion is depleted by transverse spatial hole burning, as was shown in [10,12]. It is an important point that efficient fiber amplifiers, operating above a few watts, generally have strong transverse hole burning at some or all positions along the fiber. If photodarkening or linear absorption is included in the model, an absorption profile α(x,y,z) can be added, with the assumption that all absorbed light is converted to heat. This can increase the mode coupling gain. Photodarkening is usually claimed to be strongest in regions of high population inversion and to change slowly in time, so a photodarkening contribution to α(x, y, z) might be computed from the time-averaged local population inversions Method of propagating beam If a rectangular grid is used, and if the amplifier is an index guiding fiber rather than a photonic fiber, FFT methods can be used to propagate the signal field [19, 20]. Alternatively, if cylindrical symmetry is assumed, fast Hankel transforms can be used to propagate the field. If a photonic crystal fiber (PCF) is used, it can be approximated as a step index fiber and FFT or Hankel methods can be used, or the actual index profile can be accommodated in a finite difference beam propagation model. Propagation using finite element methods is also possible. The finite element and finite difference methods have the advantage that a conformal grid can be used to handle the air holes without a high density grid. Full vectorial FFT methods that can handle the large index steps of a PCF are also available, but the square gridding means a dense grid is necessary to properly account for the small air holes. Recently Ward [26] reported a beam propagation method that is a hybrid of finite element method, finite differences, and Fourier angular expansion. However, a BPM model is probably slower than a coupled mode model because its z step must be a small fraction of the modal beat length, usually a few microns, whereas several times larger z steps are usually possible in coupled mode models. Coupled mode models do not propagate fields directly. They require integration in z of a set of equations describing the gain of each fixed profile mode due to four processes, laser gain, thermally induced self phase modulation, thermally induced self phase modulation, and interference induced mode coupling. One equation is necessary for each mode involved, and it includes coupling terms between each of the modes included including itself. Coupled mode models force the light to remain in only those modes that are specifically included. One consequence is that if only LP 01 and LP 11 are included, LP 01 isn t allowed to thermally lense into a profile that includes LP 02 or other radially symmetric modes. Additionally, for operation above threshold where a large number of modes may be important, will not be realistically modeled. The advantage of a coupled mode treatment is that the z-step can be larger than in a BPM model. This is a limited advantage because the thermal equation must be solved just as often for either model,

6 and the thermal solver usually occupies substantially more time than the beam propagation model, even in the BPM models. Another advantage of coupled modes is that the mode profiles can be those of a PCF or other complex index structure, whereas the modes for a split-step FFT BPM model are usually those of an index guiding fiber with low index contrast Temperature solver The choice of solution method for the time-dependent temperature equation may be the most important choice in a mode instability model because most of the computation time is usually spent on this task. The choice depends largely on whether the model uses the steady-periodic assumption or not. If so, steady-periodic Green s function solutions are available for a rectangular grid and rectangular thermal boundary conditions [21], and for cylindrical grid and cylindrical boundary conditions [22]. Additionally, Dong [16] developed a steady-periodic temperature solver based on temperature modes. Other methods are available for transient models, including relaxation methods which involve a relatively simple finite-difference explicit integration in time. Relaxation methods are general and can be used in arbitrary boundary conditions, but are very slow because they are limited in the size of time step. Paradoxically, larger time steps are allowed only in cases where the transverse spatial step is large (i.e. fine structure is unimportant). Implicit methods are faster. They are unconditionally stable, so larger time steps are possible, but accuracy suffers if the time step is too large. The alternating-direction implicit method (ADI) combines explicit and implicit integration to improve accuracy and maintain unconditional stability. Finite-difference and finiteelement methods can also be used. They are computationally intensive, but general enough to handle arbitrary boundary conditions and conformal meshes Feedback effects A physical system with gain, time delay, and negative feedback can form an oscillator if the gain is sufficient. The oscillation frequency is determined by the time delay. Fiber amplifiers probably do not form such oscillators, but the presence of time delayed feedback combined with amplifier gain could have an influence on the amplifier stability. In the case of a fiber amplifier the feedback would be caused by changes in the signal or pump power near the output end of the fiber leading to altered conditions earlier in the fiber. This can happen only in counter-pumped amplifiers where the pump light can carry feedback information toward the signal input end. For example, if the pump absorption is smaller when LP 11 is populated near the output end, the pump power early in the fiber is increased when mode switching into LP 11 begins. This would tend to move the grating upstream, decreasing the power transfer into LP 11, a negative feedback response. It would be strongest in fibers with confined doping where LP 01 overlaps the dopant profile better than LP 11. However, there is also a response with the opposite sign; higher pump power early in the fiber tends to increase the degree of power switching if the grating motion is not altered, as would be the case near the signal input end. The net effect is unclear. It may be that feedback plays a role in the large excursions in modal power content sometimes observed in counter-pumped amplifiers operated above threshold [1, 3]. However, mode instability has been documented in both co- and counter-pumped amplifiers at similar powers, and confined doping is known to raise the threshold, so feedback must play a

7 secondary role in modal instability Slinky effects This effect is named for the toy coil spring that flip-flops its way down stairways. Imagine the coils representing the relative phases of the two interacting modes. In a transient model of mode coupling, when the power balance of the two modes changes, the profile of the temperature in the core changes in a way that either pulls the propagation constants of the two modes closer together, in which case the slinky expands, or pushes them farther apart, in which case the slinky contracts. The thermal grating lags the moving light grating and this produces the phase shift that allows power transfer between the modes. The direction of power transfer depends on the direction of movement of the local grating or slinky coils. If motion is slow enough that the phase lag between irradiance and temperature gratings is less than 90, downstream motion leads to transfer power into LP 11 ; upstream motion leads to transfer power into LP 01. Slinky motions can be complicated with upstream motion in one zone and downstream motion in another Thermal effects All current models of mode instability include quantum defect heating and use it to compute the temperature profile and from that the refractive index changes due to the thermo optic effect. They all ignore other temperature dependent effects such as the temperature dependence of the Yb absorption and emission cross sections, or thermallyinduced strain leading to changes in the refractive index due to the photo elastic effect. Temperature variation of thermal conductivity and heat capacity are ignored. The neglected effects are expected to be insignificant compared with the thermo optic effect Masking effect When LP 11 (0) is seeded with much more than the quantum noise level, it beats with LP 01 to produce a stationary irradiance and temperature grating that masks the much weaker moving gratings produced by beating between LP 01 and weak LP 11 (ν =0) light. The stationary gratings can be removed by subtracting the DC part of the gratings. When this is done the weak moving gratings are sometimes clearly revealed, or unmasked. 3. AS-Photonics Model The earliest AS-Photonics paper [6] on mode instability describes the physics of STRS. The numerical model we developed was presented in full detail in another paper [8]. Other papers present model results from studies of various physical effects and operating conditions [7, 9 13]. Assumptions: Refractive index guiding - usually a step index, steady-periodic heating, heating computed from the absorbed pump using the (x, y) dependent population inversion, no longitudinal heat flow, square thermal boundary, square grid. The steady-periodic assumption implies the frequency spectrum consists of regularly spaced frequencies with a spacing equal to the inverse of the period. Methods: Green s function steady-periodic temperature solver, FFT beam propagation.

8 Features: Highly parallelizable in both the BPM and the thermal solver; highly numerical to minimize assumptions about fiber or pump or seed; run times minutes per meter. Notes: In Fig. 1 (from [10]) we show the computed mode coupling gain (blue curve), total heating (red curve), and laser gain (green curve) for a co-pumped amplifier, computed by this model. The amplifier in this case has d core = d dope = 80 µm, L = 1.2 m, A clad = µm 2, P pump = 600 W, P 01 = 10 W, P 11 = W. The shape of mode coupling gain is affected by transverse spatial hole burning. Early in the fiber the pump is strong and the signal weak so transverse hole burning is weak and mode coupling gain is relatively high. Late in the fiber the pump is weak and the signal is strong so transverse hole burning is strong and mode coupling gain is relatively weak Gain [db/m] Heat [norm.] Z [m] Fig. 1. Heat (red), mode coupling gain (blue), and laser gain (green) for a copumped amplifier operating near the mode instability threshold. We further explore the effect of transverse hole burning on the frequency of maximum gain in [10]. The frequency is slightly altered by changes in the shape of the transverse heating profile. The mode coupling strength is strongly affected by population saturation [10, 12]. In [7] we demonstrate that the threshold of instability can be strongly affected by amplitude modulation of the pump or signal inputs. The gain of higher order modes LP 11 and LP 02 are shown in Fig. 1 of [7]. We show the magnitude of gain is strongly dependent on the frequency offset and has a dispersion-like curve, and the peak gain is much larger for LP 01 coupling into LP 11 than for LP 01 coupling into LP 02. This accounts for the universal observation of transfer into LP 11 rather than into LP 02 in experiments. In [9] we show the relationship the threshold has with amplitude modulation of the input signal and pump has logarithmic dependence upon modulation depth. We also show how thresholds are affected by photodarkening and by mode specific

9 loss (or mode discrimination). For purposes of illustration we treat photodarkening using a simple model of uniform linear absorption in doped region of the fiber. In [11], we attempt to match the experimental results presented by Otto et al. in [3]. We show that the presence of harmonics in the mode content at the output is indicative of amplitude modulation of the input pump or signal. Power switching between modes LP 01 and LP 11 on the millisecond time scale is shown to occur in the presence of amplitude modulated input signal. The model using the parameters described in [11] does not produce full power switching between modes, but we view this as inconsequential since the parameters of the fiber are not fully described in [3] (e.g. mode discrimination, V parameter, and photodarkening), and the spectra of pump and input signal noise are likewise unspecified. In [13] we develop the theory of spontaneous thermal Rayleigh scattering and show it is stronger than quantum noise so it should be used to seed the modes in most cases in place of quantum noise. 4. DTU Model This model was first described in a short paper [14], and later expanded in a longer paper [15]. Later a study of thermal lensing and its impact on thresholds was added [27]. Assumptions: Arbitrary but fixed mode profiles, frequency-space treatment, broadband or narrow band signal, heating assumed to follow the signal irradiance profile, no population saturation / transverse spatial hole burning, mode coupling gain is related directly to laser gain rather than pump power, no longitudinal heat flow, cylindrical thermal boundary, thermal lensing assumes a uniform deposition of the quantum defect heat across the full core to find a temperature profile to include when computing thermally lensed modes that are constant in z Methods: Broad band Green s function temperature solver, coupled mode propagation method. Features: Uses coupled mode equations for two modes, LP 01 and LP 11 ; the mode coupling gain is related to the laser gain; the mode instability threshold is found to be determined by the laser gain, without direct dependence on other amplifier parameters; in [14] the threshold for a 1030 nm signal is around 400 W, independent of pump direction, core and cladding size, etc. (there is a weak dependence on the V parameter); no run time is reported. Notes: The frequency dependence of mode coupling gain is studied for different fiber core sizes. The mode coupling gain vanishes for zero frequency offset. Table 1 of [14] lists predicted mode instability thresholds for quantum noise initiation of LP 11. Threshold signal powers are 440 W for a 20 µm diameter core; 458 W for a 40 µm diameter core; 479 W for an 80 µm diameter core. Each case used V = 3. There is no variation with fiber length or degree of gain saturation. Increasing V from 3.0 to 3.5 for the 40 µm diameter core reduced the threshold from 458 W to 401 W. The lesson is that the threshold is always near 400 W unless V is reduced to near the single mode limit of 2.4 or unless the doped diameter is made smaller than the core. For example, with a core diameter of 40 µm and a

10 doping diameter of 20 µm the threshold more than doubles, from 458W to 1035 W. In a subsequent publication [15], a more comprehensive description of the model is presented, and they apply it in two new ways - using an input signal with a Gaussian relative intensity noise, and a narrow-band of amplitude modulated signal (similar to [7]). They make the claim that an amplitude modulated signal is very likely the case in any experiment, and it causes the threshold of modal instability to be much lower than for quantum noise seeding. It also leads to periodic oscillations in output modal power. This is a claim remarkably similar to [7]. They vary the convection coefficient of the boundary cooling from h q = 10 W/ (m 2 K) to h q = 10 4 W/(m 2 K) and find the mode coupling gain coefficient is unchanged, in apparent contradiction to [4]. However, they consider only symmetric cooling. They vary input signal power: 1 W, 10 W and 50 W. Thresholds increase from 448 W to 480 W and 537 W. This is not too surprising since there will be less heat deposited per unit length for a fiber that has a gain of 20 db vs. one that has a gain of 27 db, and because in this model the mode coupling gain is proportional to the signal laser gain. They vary the amplitude modulation of the LP 11 seed light and show an approximately logarithmic reduction in threshold with modulation amplitude. Consequently only modest improvements on threshold can be expected by reducing amplitude modulation. They show that for higher V parameters, the frequency of maximum coupling increases slightly, but, more importantly, the magnitude of the peak of the coupling gain increases substantially. For example, for V = 3, peak χ 0.07 W 1 ; for V = 5, peak χ 0.09 W 1. They also show that LP 01 -LP 11 coupling is roughly twice as large as LP 01 -LP 02 coupling, which they say explains why LP 02 isn t the mode that mixes with the fundamental in laboratory studies. This agrees with the claims of [7]. They can show modal power switching well above threshold. Below and near threshold, the spectrum of LP 11 at the output end is very narrow and has one peak for quantum seeding. The spectrum broadens a bit as gain exceeds threshold. For amplitude modulation above threshold, this spectrum also shows harmonics, and well above threshold broadens significantly. Thus, the presence of harmonics in the spectrum of LP 11 is indicative of some starting amplitude modulation, as also claimed in [7]. In ref. [27] the effect of thermal lensing on modal profiles and resulting effect on mode coupling is considered. The thermally lensed mode calculation assumed a uniform distribution of heat across the core. The calculation could have been made self consistent by iterating the mode and temperature calculations to reach a converged solution under the assumption that the heat profile is that for the fundamental mode. The fibers considered had a high degree of index structure and the guidance properties were sensitive to temperature. The modes modified by thermal lensing were used to compute the mode coupling gain near the output

11 end of a counter pumped amplifier. In interpreting the conclusions it should be remembered that thermal lensing confines the light to a smaller region of the core, accentuating the influence of population saturation which is not included in the DTU models. The DTU results, apart from the thermal lensing results, are all in good qualitative agreement with the AS-Photonics results under conditions where direct comparisons are possible. 5. Clemson Model This model is described in a paper by Liang Dong [16]. Assumptions: Arbitrary but fixed mode profiles, steady-periodic heating, heating profile from irradiance profile, includes longitudinal heat flow, cylindrical thermal boundary, cylindrical grid. Methods: Cylindrical coordinates, semi-analytic temperature solver using temperature modes, coupled mode method. Features: The temperature solver uses a cylindrical thermal boundary and includes longitudinal heat flow, thresholds and gains are compared with the DTU model, no run time is reported but it should be very fast. Notes: Dong considers only single-frequency inputs (though he does discuss spectral considerations in the latter part of his text). Longitudinal heat flow is included in this model, but it is shown to contribute negligibly to mode coupling. The threshold reported here is suspect because the starting power in the parasitic mode is W which is far below the level of quantum noise, and his starting power isn t sufficiently justified in the text. The thresholds are near 400 W but they should be closer to 300 W if the starting power were adjusted to quantum noise levels of W. This lower threshold would be consistent with the DTU results if the 50% higher quantum defect associated with 1060 nm signal of this model compared with the 1030 nm DTU model were used to adjust the threshold. Dong shows that if the frequency offset is zero, the mode coupling coefficient disappears. Dong attempts to describe behavior above threshold by re-seeding the signal light so once the signal has transferred into LP 11 it can couple back to LP 01. One result of this scheme is shown in his Fig. 6, which is clearly far above threshold. There, he s turned the amplifier gain up to 34 db, so his 19.2 W signal is amplified to somewhere around 50 kw, and notes that the spatial frequency at which the power transfer occurs is increasing as you progress along the fiber. Then he claims this behavior and some external perturbation or an unspecified feedback effect might account for the chaotic behavior seen in experiment in [3]. This claim is unconvincing. 6. Air Force Models There are three versions of the Air Force model. Each is applied to transient cases only, where the boundary conditions are suddenly changed at time zero. The first model was

12 described in a 2012 paper [4]. A modified formulation of the model was later developed by Shadi Naderi et al. at AFRL, and is described in a Photonics West paper [23] and a full article [25]. These two implementations are very similar, but the Naderi model is reformulated in cylindrical coordinates so true circular boundary conditions can be applied. A second modified formulation using cylindrical coordinates was published by Ward [26] in which a numerical beam propagation method is used in place of the original coupled mode method. Assumptions: Arbitrary but fixed mode profiles except in Ward revision where arbitrary profiles are allowed; transient heating; heating from population inversion; longitudinal heat flow considered first version; square grid in first version, cylindrical grid in Naderi revision [25] and Ward revision [26]; in the original paper [4] the pump feedback effect is turned off by artificially fixing the pump longitudinal profile to the initial steady state profile in the initial version; the fiber is co-pumped in the Naderi version and counter-pumped in the original and Ward revision; the slinky effect is strong because of the strong transient at time zero in all versions. Methods: The original version develops the coupled mode method, with coupled mode equations that include terms for temperature-induced mode coupling and also for changes in the propagation constant for the individual modes that are computed from the temperature profile within the core. The temperature solver is unspecifically described as a Crank-Nicolson scheme in the first version; in the Ward revision the temperature solver is described as a hybrid of finite element and finite differences; the temperature is solved using an ADI method in the Naderi revision. The Ward revision develops a new beam propagation method to replace the coupled mode treatment. Features: The original version uses PCF modes and a counter-pumped configuration. That model begins with steady state solution for one set of input pump and signal powers. At t=0 the input conditions are suddenly changed, with 2.5 W of the total input signal power of 50 W suddenly transferred from the LP 01 mode to the LP 11 mode. The Naderi revision starts with a cold, (step index?) fiber and turns on the full pump power at t=0, with W of seed in LP 01 plus 5% of that as unshifted seed light in LP 11. The Ward revision treats a step index fiber, and at time zero the pump power is ramped from zero to the set point power over 10 ms. There are no frequency offsets of the input signal in the original version. In Naderi s revision the input signal is sometimes amplitude modulated, producing frequency side bands in the 500 Hz to 2 khz range. Run time is several days in the Naderi revision because the full transverse profile is used in the thermal solver, and because the model must run long enough for the initial transient to decay away. Run time of the Ward revision is unspecified, but it is designed to run on a high power, multi processor computer. Interpretation (original version): The original model produces mode coupling without introducing a frequency offset for light in LP 11. However, it is difficult to discern a distinct threshold from the presented data. The authors do not offer an explanation of the physical mechanism responsible for initiating or sustaining this mode coupling. Clearly, at early times, before thermal equilibrium is

13 reached, there are moving gratings as expected from the slinky effect. However, after many milli seconds equilibrium should be approached, the gratings should cease to move, and power transfer to LP 11 should cease. This behavior appears to be present in the Naderi revision when operated below threshold (see movie1). Above threshold mode coupling does not cease after the settling time. Does this mean there is a physical process that causes mode coupling, or is a frequency shift somehow introduced by the numerics of the model? How can sustained oscillations arise in response to only an initial perturbation? Perhaps the authors believe the conceptual model of Jena [17] can explain this? We described above in section 3 how numerical truncation errors can form a broad band noise background similar to quantum noise, and the noise within the gain band could be amplified to threshold in the Air Force models. There are multiple tests of the numerical model that can detect this effect. The first is a comparison of the threshold computed by this model with the thresholds computed by the AS-Photonics or the DTU or the Clemson models. Is the gain computed using those models sufficient to amplify truncation noise to threshold when the threshold power levels of the Air Force model are used? Another test is Fourier transforming the field amplitudes to search for frequency offsets of the amplified LP 11 light at various points along the fiber. This paper claims that cooling conditions at the thermal boundary change the threshold, but in fact there seems to be no such effect on the mode coupling equations if the overall temperature of the fiber is changed - only the difference in propagation constants should impact the transient slinky effect. It is true the mode profiles will change with heating when only one side is cooled, but this effect is not included in the model since the modes are not allowed to change to accommodate an asymmetric temperature profile. Interpretation (Naderi revision): Unlike the original version, the Nadari version seems to produce well defined thresholds, at least in some cases. Naderi et al. claim once again that the whole fiber, out to the cooled surface, should be included in the thermal model in order to correctly account for thermal boundary conditions. While this is technically true, it makes the run times exceedingly long, and the paper never demonstrates that including the full fiber makes a significant difference in the threshold in any specific cases. The thermal diffusion time for the 250 µm radius of the cooled boundary is approximately 70 ms but the model appears to produce stable outputs after only 20 ms suggesting the distant thermal boundary is not involved in an important way. The impact of the outer boundary location was not tested by varying its diameter. The Naderi et al. paper contains a number of interesting peculiarities. For one, it is claimed that no frequency offsets need be involved in creating a mode instability. However, when the input seed is amplitude modulated, the threshold is reduced by a factor of two or so. It is obvious that such seed modulation produces the required frequency shifted side band that is amplified to threshold. The threshold without this modulation is almost certainly due to amplification of frequency offset numerical truncation noise, as explained above. Another peculiarity is the claim that quantum noise level seeding does not trig-

14 ger mode instability. However, they test this by seeding only the zero frequency (unshifted) higher order mode with varying powers and, not surprisingly, see no change in pump threshold power. It is not explained how one can turn off quantum noise at shifted frequencies in the laboratory, so this treatment violates fundamental physical laws. If the shifted frequencies were seeded at the quantum level, we expect the threshold would fall by a factor of approximately two, just as it does for the case of amplitude modulated seed light. This paper also contains the claim that steady-periodic models cannot incorporate multiple frequencies. This is wrong. It also claims that the steady-periodic models must find the frequency of maximum gain by trial and error, when it is actually quite easy to predict the frequency [10] within 50 Hz. In any case, the modulation frequency was varied in this paper to locate, by trial and error, the frequency offset of maximum gain. The field amplitudes (rather than irradiances) are not Fourier transformed to look at the spectra of the modes. Unfortunately this paper, like too many in the field, does not present the full list of amplifier parameters necessary to perform model comparisons or to replicate the model. Missing information includes: pump wavelength, signal wavelength, emission and absorption cross sections for pump and signal, numerical aperture of the fiber, upper state lifetime, thermal conductivity, heat capacity, mass density, time, and lengths of the confined doping fibers. The model is used to compute thresholds for different distributions of Yb 3+ dopant. One variant is a simple confinement of the ions to the center of the core. Another is a three-fold symmetric doping profile. Unfortunately the orientation of this profile relative to the mode symmetry is not given. Both of the special doping profiles are found to raise the instability threshold. Interpretation (Ward revision): The model update by Ward [26] replaces the coupled mode treatment with a beam propagation method based on a one dimensional finite element treatment combined with an angular expansion method. A new thermal solver is also developed and applied. Cylindrical coordinates are used. The model is still applied to a transient situation with integrations over the range ms. The fiber has a step index profile and step doping profile. The fiber is again counter pumped only. Longitudinal heat flow is allowed, and the temperature is computed over the full fiber diameter of 400 µm, but the integration time of 100 ms not sufficient to reach full thermal equilibration with the outer fiber boundary. There are no Fourier transforms of the field amplitudes and thus no unambiguous information about spectral shifts. Frequency shifts of the input signal are included. Ward sees the threshold power is a factor of three lower than in the original coupled mode version and attributes this to the inclusion of all possible transverse modes in place of just two. This seems an unlikely explanation to us but we cannot give a better one. However, there are no really clear thresholds identified in either case. Ward recognizes that quantum noise initiation might be important, but doesn t add noise to the model to test the idea. This paper strongly restates the original claim that instability occurs only near the output end of the amplifier. This is probably just the masking effect since the seed

15 includes strong unshifted LP 11 power. He also says the sudden onset of instability late in the fiber might explain the lower thresholds for larger core diameters, although he does not offer any support for this claim or for this threshold trend. The Ward revision still presents plots of the Fourier-transformed irradiance, rather than Fourier-transformed fields, discarding valuable spectral information. Additionally, the Fourier transformed data, as presented, may includes the strong initial transient response to the perturbation, perhaps hiding what may be more important near-steady-state responses. Ward offers a good plan for future studies at the end of section 6 of this paper, and we look forward to these results. 7. Jena Model This model is described in a 2012 paper [17]. Assumptions: Transient heating; heating from population inversion; cylindrical boundary; cylindrical grid; cylindrically symmetric fields LP 01 and LP 02. Methods: Cylindrically symmetric BPM; temperature solver unspecified. Features: This is actually more a conceptual model than an actual numerical model. They do develop a numerical model of a train of 1 ns pulses, but they run the model for only 1 ms [17] which is shorter than the time for heat diffusion across the core. However, they endorse the Air Force numerical model [24] as the most... so evidently they believe the Air Force model is a good implementation of their ideas. The conceptual model is based on zero frequency offset between input signal modes at the fiber input. Phase shifts are said to develop between the irradiance grating and the temperature grating due to the rapid temperature rise with z in a counterpumped configuration. Longitudinal heat flow is said to enhance this but is not necessary. This grating shift, or phase shift, is sufficient to cause power transfer between modes as the irradiance and temperature gratings chase one another downstream. The resulting Doppler shift is said to cause frequency offsets. Then the feedback effect accentuates the power transfer and perhaps leads to oscillatory behavior. Notes: It is unclear how the proposed mechanism of mode coupling translates to copumped fibers where the temperature gradients are far different from counterpumped fibers, yet reportedly have similar instability thresholds. No convincing evidence is given for the claimed phase shifts, nor is any citation of such an effect given. The Air Force model has the feedback effect artificially turned off (the absorbed pump power distribution in z is forced to remain unchanged regardless of the relative strengths of LP 01 and LP 11 ), so that model does not fully incorporate these ideas. Jena seems to hedge on the claim of spontaneous breaking of the steady state by saying perturbations of some sort may be necessary to initiate the mode coupling process (no further information is provided about the nature these proposed perturbations or their source), but they cannot decide whether their conceptual model would settle into a steady state or be chaotic. A periodic

16 perturbation is not considered. Cooling conditions should make a difference in this model because poorer cooling leads to larger longitudinal temperature gradients, and it is the large temperature gradient that is claimed to start the process. 8. Michigan Model This model is described in a 2013 paper [18]. Assumptions: Time dependence need not be steady-periodic. The steady-periodic case is labeled the static case in ref. [18]. The general case is labeled timedependent. Frequency offset between LP 01 and LP 11 is included, unlike in the Air Force model. The heat profile given by Eq. (5) includes a population inversion but it does not have transverse profile. The transverse heat profile appears to be that of the irradiance. Longitudinal heat flow is not included. Single frequency LP 01 and LP 11. Fiber is counter-pumped. Methods: Coupled mode method, thermal equation solved by separation of variables in cylindrical coordinates, integration of coupled mode equations is semi-analytical, pump is turned on at time zero it appears. Features: When operating slightly above threshold the time-dependent model produces strong temporal oscillation in the higher order mode even though it is seeded at a single frequency. In other words, another frequency appears in LP 11 and it is detuned to the blue side of the LP 01 frequency. As pump power increases to ten times threshold more equally spaced frequency sidebands appear and the temporal oscillation become complicated. A red cascade occurs and shifts the weight of the LP 11 spectrum to several times the maximum gain point. When the detuning of the seed is not near the point of highest gain the spectra of LP 11 become more continuous and lose most of the discrete sideband structure. Notes: The red cascade and broadening noted in this paper are qualitatively similar to the behavior noted by Hansen [15] for multifrequency seed, and by Smith [11] for amplitude modulated seed or pump, and to a lessor extent for unmodulated inputs. The threshold is lower because the seed power is 10 5 W rather than the W used by Hansen and by Smith. One question is how the harmonic (and non harmonic) offset frequencies arise in this model? No physical explanation is given. At low powers (150 W and 500 W) the oscillations are periodic after the initial transients, so the steady-periodic models should give the same results. Do they? Another question regards the very high maximum pump level of ten times threshold. At that point numerical truncation errors can be amplified to threshold. Is that important here? 9. Comments Our first comment on alternative models of mode instability is that the AS-Photonics, DTU, and Clemson models are not wrong. They all start with frequency shifted light in LP 11 and find it grows exponentially to threshold. There can be little doubt that this gain process is real. Furthermore the three models listed produce thresholds that are in close agreement when LP 11 is seeded with frequency shifted light at the quantum noise level and when the same assumptions about the heat profile are used. We think