## Steady-periodic method for modeling mode instability in fiber amplifiers |

Optics Express, Vol. 21, Issue 3, pp. 2606-2623 (2013)

http://dx.doi.org/10.1364/OE.21.002606

Acrobat PDF (1599 KB)

### Abstract

We present a detailed description of the methods used in our model of mode instability in high-power, rare earth-doped, large-mode-area fiber amplifiers. Our model assumes steady-periodic behavior, so it is appropriate to operation after turn on transients have dissipated. It can be adapted to transient cases as well. We describe our algorithm, which includes propagation of the signal field by fast-Fourier transforms, steady-state solutions of the laser gain equations, and two methods of solving the time-dependent heat equation: alternating-direction-implicit integration, and the Green’s function method for steady-periodic heating.

© 2013 OSA

## 1. Introduction

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

2. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express **20**, 24545–24558 (2012). [CrossRef] [PubMed]

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**, 15710–15722 (2012). [CrossRef] [PubMed]

_{11}. Assuming most of the signal seed light is injected into the fundamental mode LP

_{01}with a small amount populating a higher order mode, usually mode LP

_{11}, these two modes interfere along the length of the fiber. Because they have different propagation constants their interference creates a signal irradiance pattern that oscillates along the length of the fiber. Pump light is preferentially absorbed in regions of higher signal irradiance, and because a fraction of the absorbed pump is converted to heat, this creates a heating pattern that resembles the irradiance pattern. The heat pattern is converted to a similar temperature pattern, and the temperature pattern creates a refractive index pattern via the thermo optic effect. If the interference pattern is stationary, there is little or no phase shift between the thermally-induced index pattern and the irradiance pattern, so there is almost zero net transfer of power between modes. However, if the light in the higher order mode is slightly detuned in frequency from that in LP

_{01}, the irradiance pattern moves along the fiber - down stream for a red detuning and up stream for a blue detuning. The temperature pattern also moves, but it lags the irradiance pattern, and this lag produces the phase shift necessary for power transfer between modes. Red detuning of LP

_{11}leads to power transfer from LP

_{01}to LP

_{11}. The detuning that maximizes the mode coupling is set largely by the thermal diffusion time across the fiber core. This is approximately 1 ms, implying an optimum frequency detuning of approximately 1 kHz.

_{01}is transferred to LP

_{11}by deflection from the moving grating it experiences a frequency shift equal to the frequency offset, due to a Doppler effect [1

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

_{11}. This mechanism can cause a substantial transfer of the power from LP

_{01}to LP

_{11}if the pump power exceeds a sharp mode instability threshold. This gain process can be categorized as near-forward stimulated thermal Rayleigh scattering.

## 2. The scalar, paraxial beam propagation equation

7. M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. **19**, 1154–1164 (1980). [CrossRef] [PubMed]

8. 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]

*E*is a complex envelope function that represents all spatial and temporal modulation of a monochromatic, plane carrier wave. The direction of propagation is

_{s}*z*. This equation is appropriate for small index contrasts typical of step index, large mode area fiber amplifiers. where

*k*=

_{c}*ω*

_{c}n_{clad}/

*c*is the wave number of the carrier wave in the cladding, and

*k*(

*x*,

*y*,

*z*,

*t*) =

*ω*(

_{c}n*x*,

*y*,

*z*,

*t*)/

*c*. Here

*n*(

*x*,

*y*,

*z*,

*t*) is the guiding index including the thermo-optic contribution, where d

*n*/d

*T*is the thermo optic coefficient given in Table 1. The first term on the right hand side of Eq. (1) describes diffraction in a homogeneous medium with a refractive index equal to the cladding index

*n*. Here

_{c}*x*and

*y*. The second term adds a correction to account for the core guidance, including the usual core refractive index step plus the thermally induced index change. In general

*k*(

*x*,

*y*,

*z*,

*t*) is imaginary to account for gain or loss, but we will use

*k*(

*x*,

*y*,

*z*,

*t*) to represent only the real part, and write the much smaller imaginary part as a separate term with the coefficient

*g*(

*x*,

*y*,

*z*,

*t*),

## 3. Integrating the propagation equation

*k*(

*x*,

*y*,

*z*,

*t*) for use in the

*z*integration.

*t*= 0 based on an initial temperature profile

*T*(

*x*,

*y*,

*z*, 0). The temperature profile is then updated based on the heat deposited during the time increment Δ

*t*, plus the previous temperature profile and the thermal boundary conditions. The new temperature profile is used to integrate the field at time Δ

*t*along the full length of the fiber. This iteration of full

*z*integration of

*E*alternating with stepping

_{s}*T*by Δ

*t*is repeated for successive time steps to find the temperature and field histories, both over the domain (

*x*,

*y*,

*z*,

*t*).

*z*= 0 can be used to compute

*T*(

*x*,

*y*,

*t*) at the input and that time varying temperature can be used to propagate the time varying field by Δ

*z*. This is repeated for successive

*z*steps. One limitation is that the temperature is solved in two dimensions rather than three so longitudinal heat flow is not accounted for.

### 3.1. Steady-periodic condition

*T*(

*x*,

*y*,

*t*) for a single period at one

*z*position, then we step the signal and pump fields by Δ

*z*.

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**, 15710–15722 (2012). [CrossRef] [PubMed]

### 3.2. Transverse heat flow approximation

*μ*m. The ratio of longitudinal to transverse length is of order 100:1.

### 3.3. Narrow line width approximation

### 3.4. Split-step integration in t and z

*z*on a set of time samples of the field that are evenly spaced over one period. At the fiber input we specify

*E*(

_{s}*x*,

*y*, 0,

*t*) and the pump power. Using this information we compute the temperature profiles for each sampling time over the full period. We describe two methods for the integration of the heat equation below. The temperature profile for each time sample is then used to propagate the corresponding time sample of the signal field by Δ

*z*. This is repeated until the end of fiber is reached. This method is used because it makes efficient use of limited computer resources, and because it can be made to run fast.

*z*integrations. Split-step methods have been employed in numerical simulations of CW optical diffraction, including guiding in optical fibers, at least since the 1970s [7

7. M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. **19**, 1154–1164 (1980). [CrossRef] [PubMed]

8. 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]

*N*time steps. We use the split-step method by applying it individually to each of the field time slices. Related FFT methods for dispersive propagation are also widely used to model propagation of short light pulses propagating in a single fiber mode [9]. Although we are also propagating pulses consisting of one beat time, it is best to think of the time slices as widely separated samples that are unaffected by dispersion. We will say more about this in Sec. 17.

_{t}*z*in three steps. In the first step, linear diffractive propagation is applied to advance the field by Δ

*z*/2 in a homogeneous medium with a refractive index equal to the cladding index. In this step, the beam propagation equation is used, keeping only the diffractive term on the right-hand-side In the second step, the field is advanced by Δ

*z*without diffraction, keeping only the phases induced by the guiding and thermally induced index, along with the laser gain and linear loss contained in

*g*(

*x*,

*y*,

*z*,

*t*). The propagation equation for this step becomes The gain term will be described in more detail in the next section. In the third step, linear diffractive propagation is again applied to advance the field by Δ

*z*/2. This method produces errors of order 𝒪(Δ

*z*

^{3}).

## 4. Algorithm

## 5. Read problem parameters and fiber parameters

*K*= 1.38 W/m-K; specific heat

*C*= 703 J/kg-K; density

*ρ*= 2201 kg/m

^{3}; thermo-optic coefficient d

*n*/d

*T*= 1.2 × 10

^{−5}K

^{−1}. Usually the pump wavelength is

*λ*= 976 nm, and the upper state ion lifetime is

_{p}*τ*= 901

*μ*s.

*z*-location on an (

*x*,

*y*,

*t*) grid. The grid is equally spaced in (

*x*,

*y*,

*t*), with number of points typically (

*N*= 64,

_{x}*N*= 64,

_{y}*N*= 64). The spatial grid spans a domain of size

_{t}*L*×

_{x}*L*with the core at its center, where

_{y}*L*and

_{x}*L*are typically ≈3 ×

_{y}*d*

_{core}. We choose as the thermal boundary condition a fixed temperature on the domain boundary. We assume there is a fixed frequency offset between modes equal to Δ

*ω*. The period ϒ is defined as one cycle of the beat frequency, ϒ = 2

*π*/Δ

*ω*. The grid step sizes are then

*N*

_{Yb}(

*x*,

*y*), the refractive index profile

*n*

_{core}(

*x*,

*y*) and the linear loss

*α*(

*x*,

*y*) vary with (

*x*,

*y*) position. We usually use super-Gaussian profiles for these, with super-Gaussian coefficient of order 40. The FWHM values for the super-Gaussian diameters are

*d*

_{core}and

*d*

_{dopant}. The pump is confined to the pump cladding of diameter

*d*

_{clad}. The absorption and emission cross sections for pump and signal are

*M*(

^{p}*t*). Similarly, each signal mode LP

*with time averaged power*

_{m,n}*γ*are mode specific losses (non heating).

_{m,n}*L*, and the fiber is bent to a radius of

*R*

_{bend}. The integration step size Δ

*z*is typically set to a few microns. We store information about the field every Δ

_{sample}integration steps. Typically the stored information includes the local time sequences for the mode content, total signal power, effective area of the signal, and pump power.

## 6. Calculate modes

### 6.1. Refractive index profile

*n*

_{clad}and

*n*

_{core}are the refractive indices of the cladding and core, (

*x*

_{0},

*y*

_{0}) are the coordinates of the core center,

*d*

_{core}is the core diameter, and

*S*is the super-Gaussian coefficient (we typically use

*S*= 40).

*n*

_{bend}to

*n*(

*x*,

*y*) in the plane of the bend so the index is higher on the outside of the bend. If bending is in the

*x*̂ plane, this added ramp can be written Bend losses are calculated using the method of Marcuse [11

11. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. **66**, 216–220 (1976). [CrossRef]

### 6.2. Normalization

*N*for each mode constructed from Eq. (10), where

_{m,n}*N*is the value which satisfies The normalized mode

_{m,n}*u*(

_{m,n}*x*,

*y*) is defined by

## 7. Set up temperature solver

*Q*. In an isotropic homogeneous medium, the heat equation is where

*T*=

*T*(

*x*,

*y*,

*t*) is the temperature,

*Q*=

*Q*(

*x*,

*y*,

*t*) is the heating source,

*K*is the scalar thermal conductivity,

*ρ*is the density and

*C*is the specific heat capacity. No

*∂*

^{2}/

*∂z*

^{2}term is included in Eq. (21) because we don’t allow longitudinal heat flow. We consider only the thermal boundary condition of a fixed temperature on the (

*x*,

*y*) grid boundary. Alternative boundary conditions can be dealt with by modifying our procedure, but we do not discuss them here.

*T*. The first method, the Green’s function method for steady-periodic heating, involves computing the temperature over the (

*x*,

*y*) grid as a sum of independent contributions, one from each heated grid point. The second method, the alternating direction implicit (ADI) integration, involves stepping in time via successive matrix multiplications involving all pixels contributing together.

### 7.1. Calculate Green’s functions for steady-periodic heating

*x*,

*y*) grid due to the periodically heated point (

*x*′,

*y*′). These functions describe the temperature rise due to a steady-periodic heat source at a single modulation frequency. Formulations of the Green’s functions for different boundary conditions are detailed in [12

12. K. D. Cole, “Steady-periodic Green’s functions and
thermal-measurement applications in rectangular coordinates,” J.
Heat Trans. **128**, 706–716 (2006); DOI:
[CrossRef] .

*ω*is the frequency of the heat, (0 ≤

*x*≤

*H*), and (0 ≤

*y*≤

*W*).

*G*(

*x*,

*y*,

*ω*′,

_{m}|x*y*′) for

*ω*= (0, 1, 2, 3,...) × Δ

_{m}*ω*where Δ

*ω*is the specified frequency offset. The use of this set of Green’s functions to compute

*T*(

*x*,

*y*,

*t*) is described in Sec. 13.1.

### 7.2. Calculate ADI matrices

*t*is split into two halves. We first integrate explicitly in

*y*and implicitly in

*x*for one half time step Δ

*t*/2, then integrate explicitly in

*x*and implicitly in

*y*for another half time step Δ

*t*/2.

*x*/explicit-

*y*integration

*A*is the matrix for implicit integration in the x-direction, and

_{x}*B*is the matrix for explicit integration in the y-direction. These matrices are tridiagonal, and

_{y}*A*has size (

_{x}*N*− 2) × (

_{x}*N*− 2) while

_{x}*B*has size (

_{y}*N*− 2) × (

_{y}*N*− 2). The matrices are Matrix

_{y}*A*has diagonal elements (1 +

_{x}*λ*) and off-diagonal elements −

_{x}*λ*/2, while matrix

_{x}*B*has diagonal elements (1 −

_{y}*λ*) and off-diagonal elements

_{y}*λ*/2, where

_{y}*A*and

_{y}*B*follow a definition similar to Eqs. 28 and 29, with

_{x}*λ*swapped with

_{x}*λ*and redimensioned as appropriate.

_{y}*t*becomes where, because we only have

*Q*defined on integral numbers of time steps, we use linear interpolation to find

*Q*

^{t+Δt/4}and

*Q*

^{t+3Δt/4}from the heat at

*t*and

*t*+ Δ

*t*. This method is converged to 𝒪(Δ

*t*

^{2}).

## 8. Construct propagation phase array

*k*,

_{x}*k*,

_{y}*z*,

*t*) space. Equation (4) can be transformed to this space by inserting the following form of

*E*(

_{s}*x*,

*y*,

*z*,

*t*) The transformed equation is Advancing the field

*E*(

*k*,

_{x}*k*,

_{y}*z*,

*t*) by Δ

*z*/2 consists of shifting the phase of each (

*k*,

_{x}*k*) plane wave component by The inverse two-dimensional Fourier transform is used to convert

_{y}*E*(

_{s}*k*,

_{x}*k*,

_{y}*z*,

*t*) back to

*E*(

_{s}*x*,

*y*,

*z*,

*t*). In practice, fast Fourier transforms (FFTs) are used to efficiently convert fields between (

*x*,

*y*,

*z*,

*t*) and (

*k*,

_{x}*k*,

_{y}*z*,

*t*) spaces.

## 9. Construct input signal field and pump powers

### 9.1. Signal

*u*(

_{m,n}*x*,

*y*) defined in Eq. (20) to construct the input signal field where

*m*,

*n*)

^{th}mode and

_{01}populated by unshifted light, and LP

_{11}populated by light detuned by Δ

*ω*, we set

*a*is the small peak-to-peak power modulation. We can also use more complicated modulations, as long as they are periodic. For more complicated modulation we must include an adequate number of harmonics in the Green’s function temperature solver.

### 9.2. Pump

## 10. Propagate signal field

*E*(

_{s}*x*,

*y*,

*z*,

*t*) to

*E*(

_{s}*k*,

_{x}*k*,

_{y}*z*,

*t*) using 2D-FFTs. The phase of each plane wave component is advanced by the propagation phase

*ϕ*(

*k*,

_{x}*k*) given in Eq. (37), and the inverse 2D-FFTs of

_{y}*E*(

_{s}*k*,

_{x}*k*,

_{y}*z*,

*t*) gives the updated field

*E*(

_{s}*x*,

*y*,

*z*,

*t*). If the propagation step is not the first or last half-step in the fiber, the two consecutive half-step propagations are combined by advancing the phase by 2

*ϕ*(

*k*,

_{x}*k*).

_{y}## 11. Update the field for laser gain

*x*,

*y*,

*t*) from the signal irradiance profile, pump power, and doping profile, and use it to compute the signal gain and pump loss. The (steady-state) upper state population is where

*I*=

_{s}*I*(

_{s}*x*,

*y*,

*t*) is the signal irradiance,

*ν*and

_{s}*ν*are the signal and pump frequencies,

_{p}*τ*is the ion upper-state lifetime,

*P*is the pump power, and

_{p}*A*is the area of the pump cladding. We assume that the cross sections and lifetime are independent of temperature. The effective ion lifetime at typical amplifier powers is a few micro seconds so at modulation frequencies of order 1 kHz the steady state solution is appropriate.

_{p}*N*

_{Yb}(

*x*,

*y*) is the Yb

^{3+}doping profile,

*α*(

*x*,

*y*) is the linear absorption coefficient. This method is general enough to treat an arbitrarily-shaped doping profile, but we typically consider super-Gaussian doping profiles similar to the one defined in Eq. (17). The linear absorption coefficient

*α*can be non-uniform in (

*x*,

*y*) to accommodate a photodarkening model. We assume that all the power absorbed due to

*α*(

*x*,

*y*) is turned into heat. Referring to Eq. (5), the laser gain and loss term

*g*(

*x*,

*y*,

*t*)

*E*(

_{s}*x*,

*y*,

*t*) is given by the right hand side of Eq. (40).

## 12. Update pump powers

*P*is assumed to be uniformly distributed across the pump cladding. The change in the pump power is computed directly from the ion inversion, rather than from the signal increment. This allows us to include linear signal loss and fluorescence loss correctly. The total change in pump power is given by The loss is apportioned between the forward and backward pumps according to

_{p}## 13. Compute T(x,y,t) and thermo-optic index

*T*(

*x*,

*y*,

*t*) we use either the Green’s function method or the ADI method. We calculate the heat deposition rate from the absorbed pump and the quantum defect according to where the upper state population

*n*(

_{u}*x*,

*y*,

*t*) is given by Eq. (39).

### 13.1. Green’s function method

*x*′,

*y*′) pixel is resolved into its Fourier components

*ω*= (0, 1, 2, 3...)×Δ

_{m}*ω*by performing a temporal Fourier transform on

*Q*(

*x*′,

*y*′,

*t*) Here,

*q*(

*x*′,

*y*′,

*ω*) is a complex coefficient that includes the phase as well as the amplitude of the heat deposition. These

_{m}*q*(

*x*′,

*y*′,

*ω*) values are used to weight the Green’s functions in computing the steady-periodic temperature over the entire transverse grid. We always include frequencies (0, 1)×Δ

_{m}*ω*, and we add higher frequency terms as needed for convergence. The time grid must be fine enough to resolve the highest frequency. Higher frequency terms are usually necessary only above the mode instability threshold.

### 13.2. ADI Method

*t*. For steady-periodic heating, we enforce the steady-state criterion by integrating for several periods, reusing the heat

*Q*(

*x*,

*y*,

*t*) from one period to the next. We terminate this process when the difference in temperatures between periods has reached an acceptably small residual.

## 14. Add thermo-optic and guiding index phases to the field

*z*using The phase can be rewritten using Eq. (2) as We write it in this form merely to show that the phase is (

*ω*Δ

_{c}*z*Δ

*n*/

*c*) plus a small correction from the quadratic term.

## 15. Resolve time-dependent modal power contents

*E*(

_{s}*x*,

*y*,

*z*,

*t*) into fiber modes. We compute the inner product of the field and the normalized mode from where

*F*(

_{m,n}*z*,

*t*) is a complex quantity. The mode amplitude

*F*(

_{m,n}*z*,

*t*) is converted to power in watts using

### 15.1. Spectral analysis

*m*,

*n*) we Fourier transform

*F*(

_{m,n}*z*,

*t*) from time to frequency. The allowed frequencies for the periodic function

*F*(

_{m,n}*z*,

*t*) are the

*ω*.

_{m}## 16. An example computation

_{11}over one LP

_{01}-LP

_{11}beat length at the input end of an amplifier versus

*t*and

*z*. The motion of the surface ridges reflect the change in phase of LP

_{11}relative to LP

_{01}due to the difference in propagation constants for the two modes. In this example the LP

_{11}seed light is Stokes shifted by 600 Hz relative to LP

_{01}, but the details of the fiber are unimportant for this illustration since all large mode fiber amplifiers produce qualitatively similar surfaces. A small fraction of the gain is due to laser gain but most is due to thermally induced mode coupling.

## 17. A narrow band width model for broad band light?

## 18. Sources of frequency offset HOM seed light

## 19. Desktop computer implementation

### 19.1. Memory requirements

*L*/Δ

*z*] array of the signal field would require on the order of 1 – 10 terabytes in a meter long amplifier with step sizes of a few microns. Therefore, we do not store

*E*fields at each step. We only store computed properties such as modal content

*F*(

_{m,n}*z*,

*t*), total signal power

*I*(

*z*,

*t*), and effective area

*A*

_{eff}(

*z*,

*t*) at positions separated by (Δ

_{sample}× Δ

*z*). This reduces the amount of memory required to at most a few gigabytes.

### 19.2. Parallel computing

### 19.3. Execution speed considerations

15. M. Frigo and S. G. Johnson, “FFTW Home Page,” http://www.fftw.org/.

*z*-grid than the FFT propagation problem. This can dramatically reduce the model run time as well. Care in choosing this grid is important because a grid that is too coarse can reduce the computed gain and increase the instability threshold power.

*z*step size and the number of harmonics included in the Green’s function. Larger cores use larger

*z*steps, and near-threshold runs require only a single harmonic, permitting times near 0.25 hour/m.

## 20. Approximations of model

- Single
*λ*pump with single absorption and emission cross sections - Pump power is uniformly distributed across the pump cladding
- All signal light is identically polarized
- Steady-periodic heating is required for application of Green’s function
- Fixed period ϒ, which allows only signal frequency offsets 1/ϒ × (0, ±1, ±2,...)
- Thermal boundary condition is fixed temperature on square boundary
- Thermal boundary size approximately three times core diameter
- Heat equation solved in two dimensions (no longitudinal heat flow)
- Thermal properties assumed uniform and isotropic
- Temperature dependence of cross-sections, d
*n*/d*T*, and*τ*not included - No refractive index dependence on
*n*_{u} - Signal bandwidth < 100 GHz
- Low contrast refractive index profile,
*e.g.*no air-holes as in PCF - Upper state population
*n*follows_{u}*I*instantaneously (steady-state expression)_{s}

## 21. Attributes of model

- Highly numeric - general and simple to add additional physical effects
- Model a variety of refractive index, linear absorption, and doping profiles
- Steady-periodic eliminates long integration times before steady state
- Steady-periodic model produces well defined thresholds
- The ADI method can be used to study transient behavior if desired
- All transverse modes are automatically included
- Thermal lensing automatically included
- Comparatively short run times and minimal memory requirements
- Variety of thermal boundary conditions possible using Green’s functions
- Green’s function method offers large speedup using multiple processors

## Acknowledgments

## References and links

1. | A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express |

2. | A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express |

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 |

4. | K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. |

5. | B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express |

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 |

7. | M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. |

8. | M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” Appl. Opt. |

9. | G. P. Agrawal, |

10. | D. Marcuse, |

11. | D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. |

12. | K. D. Cole, “Steady-periodic Green’s functions and
thermal-measurement applications in rectangular coordinates,” J.
Heat Trans. |

13. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

14. | P. W. Milonni, |

15. | M. Frigo and S. G. Johnson, “FFTW Home Page,” http://www.fftw.org/. |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(140.6810) Lasers and laser optics : Thermal effects

(190.2640) Nonlinear optics : Stimulated scattering, modulation, etc.

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 22, 2012

Revised Manuscript: January 16, 2013

Manuscript Accepted: January 17, 2013

Published: January 28, 2013

**Citation**

Arlee V. Smith and Jesse J. Smith, "Steady-periodic method for modeling mode instability in fiber amplifiers," Opt. Express **21**, 2606-2623 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2606

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### References

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