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.

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References

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  1. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19, 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. Express20, 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. Express20, 15710–15722 (2012).
    [CrossRef] [PubMed]
  4. 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]
  5. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20, 11407–11422 (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. Express20, 12912–12925 (2012).
    [CrossRef] [PubMed]
  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]
  9. G. P. Agrawal, Nonlinear fiber optics, second ed. (Academic Press, 1995).
  10. D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. (Academic Press, 1991).
  11. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am.66, 216–220 (1976).
    [CrossRef]
  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]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1992).
  14. P. W. Milonni, The quantum vacuum: an introduction to quantum electronics (Academic Press, 1994).
  15. M. Frigo and S. G. Johnson, “FFTW Home Page,” http://www.fftw.org/ .

2012

2011

2006

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

1980

1976

Agrawal, G. P.

G. P. Agrawal, Nonlinear fiber optics, second ed. (Academic Press, 1995).

Alkeskjold, T. T.

Broeng, J.

Cole, K. D.

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

Dajani, I.

Eidam, T.

Feit, M. D.

Flannery, B. P.

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

Fleck, J. A.

Hansen, K. R.

Jansen, F.

Jauregui, C.

Laegsgaard, J.

Limpert, J.

Marcuse, D.

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

D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. (Academic Press, 1991).

Milonni, P. W.

P. W. Milonni, The quantum vacuum: an introduction to quantum electronics (Academic Press, 1994).

Otto, H.-J.

Press, W. H.

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

Robin, C.

Smith, A. V.

Smith, J. J.

Stutzki, F.

Teukolsky, S. A.

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

Tünnermann, A.

Vetterling, W. T.

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

Ward, B.

Appl. Opt.

J. Heat Trans.

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

J. Opt. Soc. Am.

Opt. Express

Opt. Lett.

Other

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

P. W. Milonni, The quantum vacuum: an introduction to quantum electronics (Academic Press, 1994).

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

G. P. Agrawal, Nonlinear fiber optics, second ed. (Academic Press, 1995).

D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. (Academic Press, 1991).

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Figures (2)

Fig. 1
Fig. 1

Flow diagram for split-step FFT propagation using alternating direction implicit integration (ADI) or steady-periodic Greens function method for temperature calculation.

Fig. 2
Fig. 2

Power in LP11 versus time (T) and distance (Z) at the input end of the amplifier similar to the one specified in [2]. This illustrates the growth of the low power mode over one beat cycle and over one beat length. The pump is unmodulated in this example.

Tables (1)

Tables Icon

Table 1 Model input parameters

Equations (54)

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E s ( x , y , z , t ) z = i 2 k c 2 E s ( x , y , z , t ) + i [ k 2 ( x , y , z , t ) k c 2 ] 2 k c E s ( x , y , z , t )
n ( x , y , z , t ) = n core ( x , y , z , t ) + d n d T T ( x , y , z , t ) ,
E ( x , y , z , t ) z = i 2 k c 2 E ( x , y , z , t ) + i [ k 2 ( x , y , z , t ) k c 2 ] 2 k c E ( x , y , z , t ) + g ( x , y , z , t ) E ( x , y , z , t ) .
E s ( x , y , z , t ) z = i 2 k c 2 E s ( x , y , z , t ) .
E s ( x , y , z , t ) z = i [ k 2 ( x , y , z , t ) k c 2 ] 2 k c E s ( x , y , z , t ) + g ( x , y , z , t ) E s ( x , y , z , t ) .
Δ x = L x / ( N x 1 )
Δ y = L y / ( N y 1 )
Δ t = ϒ / N t .
ϕ μ ν = 4 L x L y sin ( π L x μ x ) sin ( π L y ν y )
ψ ( x , y , z ) = e i β z μ = 1 ν = 1 C μ ν ϕ μ ν ( x , y )
2 ψ x 2 + 2 ψ y 2 + 2 ψ z 2 + n 2 ( x , y ) k 2 ψ = 0
μ = 1 M x ν = 1 M y C μ ν [ n 2 ( x , y ) k 2 β 2 π 2 ( μ 2 L x 2 + ν 2 L y 2 ) ] ϕ μ ν ( x , y ) = 0.
μ = 1 M x ν = 1 M y A μ ν , μ ν C μ ν = ( n eff 2 n clad 2 ) C μ ν .
A μ ν , μ ν = 0 L x o L y d x d y { [ n 2 ( x , y ) n clad 2 ] ϕ μ ν ( x , y ) ϕ μ ν ( x , y ) } π 2 k 2 ( μ 2 L x 2 + ν 2 L y 2 ) δ μ , μ δ ν , ν .
A C = n ¯ 2 C
n eff ( m , n ) = n ¯ m , n 2 + n clad 2 .
n ( x , y ) = n clad + ( n core n clad ) × exp ( ln 2 { [ 2 ( x x 0 ) / d core ] 2 + [ 2 ( y y 0 ) / d core ] 2 } S ) ,
n bend ( x , y ) = n ( x , y ) ( x x 0 ) / R bend .
c ε 0 2 d x d y n ( x , y ) [ N m , n ψ m , n ( x , y ) ] 2 = 1 W
u m , n ( x , y ) = N m , n ψ m , n ( x , y ) .
ρ C T t = Q + K ( 2 T x 2 + 2 T y 2 )
G ( x , y , ω | x , y ) = n = 0 F n ( y , y ) P n ( x , x , ω )
F n ( y , y ) = 1 W K sin ( n π W y ) sin ( n π W y )
P n ( x , x , ω ) = { exp [ σ n ( 2 H | x x | ) ] exp [ σ n ( 2 H x x ) ] + exp [ σ n | x x | ] exp [ σ n ( x + x ) ] } / σ n ( 1 exp [ 2 σ n H ] )
σ n 2 = ( n π W ) 2 + i ω ρ C K ,
T x , y t + Δ t / 2 T x , y t = ( T x + Δ x , y t + Δ t / 2 2 T x , y t + Δ t / 2 + T x Δ x , y t + Δ t / 2 ) λ x 2 + ( T x , y + Δ y t 2 T x , y t + T x , y Δ y t ) λ y 2 + Δ t 2 ρ C Q t + Δ t / 4
T t + Δ t / 2 A x = B y T t + Δ t 2 ρ C Q t + Δ t / 4
T t + Δ t / 2 = B y T t A x 1 + Δ t 2 ρ C Q t + Δ t / 4 A x 1 ,
A x = [ ( 1 + λ x ) λ x / 2 0 0 0 0 λ x / 2 ( 1 + λ x ) λ x / 2 0 0 0 0 0 0 λ x / 2 ( 1 + λ x ) λ x / 2 0 0 0 0 λ x / 2 ( 1 + λ x ) ] ,
B y = [ ( 1 λ y ) λ y / 2 0 0 0 0 λ y / 2 ( 1 λ y ) λ y / 2 0 0 0 0 0 0 λ y / 2 ( 1 λ y ) λ y / 2 0 0 0 0 λ y / 2 ( 1 λ y ) ] .
λ x = K Δ t ρ C Δ x 2
λ y = K Δ t ρ C Δ y 2 .
T x , y t + Δ t T x , y t + Δ t / 2 = ( T x + Δ x , y t + Δ t / 2 2 T x , y t + Δ t / 2 + T x Δ x , y t + Δ t / 2 ) λ x 2 + ( T x , y + Δ y t + Δ t 2 T x , y t + Δ t + T x , y Δ y t + Δ t ) λ y 2 + Δ t 2 ρ C Q t + 3 Δ t / 4 .
A y T t + Δ t = T t + Δ t / 2 B x + Δ t 2 ρ C Q t + 3 Δ t / 4 T t + Δ t = A y 1 T t + Δ t / 2 B x + Δ t 2 ρ C A y 1 Q t + 3 Δ t / 4 ,
T t + Δ t = A y 1 B y T t A x 1 B x + Δ t 2 ρ C A y 1 ( Q t + Δ t / 4 A x 1 B x + Q t + 3 Δ t / 4 ) ,
E s ( x , y , z , t ) = 1 2 π E s ( k x , k y , z , t ) e i k x x e i k y y d k x d k y .
E s ( k x , k y , z , t ) z = i k x 2 2 k c E s ( k x , k y , z , t ) i k y 2 2 k c E s ( k x , k y , z , t ) .
ϕ ( k x , k y ) = Δ z 2 ( k x 2 + k y 2 ) 2 k c .
E s ( x , y , z , t ) | z = 0 = modes P m , n s u m , n ( x , y ) M m , n s ( t )
n u ( x , y , t ) = P p σ p a / h ν p A p + I s σ s a / h ν s P p ( σ p a + σ p e ) / h ν p A p + I s ( σ s a + σ s e ) / h ν s + 1 / τ
E s ( x , y , t ) z = 1 2 [ σ s a + ( σ s a + σ s e ) n u ( x , y , t ) ] N Yb ( x , y ) E s ( x , y , t ) 1 2 α ( x , y ) E s ( x , y , t ) ,
P p = P f p + P b p .
d P p d z = P p A p [ ( σ p a + σ p e ) n u ( x , y ) σ p a ] N Yb ( x , y ) d x d y .
d P f p d z = d P p d z P f p P p
d P b p d z = d P p d z P b p P p .
Q ( x , y , t ) = N Yb ( x , y ) [ ν p ν s ν p ] [ σ p a ( σ p a + σ p e ) n u ( x , y , t ) ] P p ( t ) A p + α ( x , y ) I s ( x , y , t ) ,
q ( x , y , ω m ) = Δ x Δ y i = 0 N t 1 Q ( x , y , t i ) exp ( i ω m t i ) .
T ( x , y , t ) = Real [ m = 0 max x , y q ( x , y , ω m ) G ( x , y , ω m | x , y ) exp ( i ω m t ) ] .
ϕ ( x , y , t ) = Δ z k 2 ( x , y , t ) k c 2 2 k c .
ϕ ( x , y , t ) = Δ z ω c c ( [ n ( x , y , t ) n clad ] + [ n ( x , y , t ) n clad ] 2 2 n clad ) .
F m , n ( z , t ) = d x d y E s ( x , y , z , t ) u m , n ( x , y ) d x d y [ u m , n ( x , y ) ] 2 ,
P m , n ( z , t ) = | F m , n ( z , t ) | 2 .
A eff ( z , t ) = [ | E s ( x , y , z , t ) | 2 d x d y ] 2 | E s ( x , y , z , t ) | 4 d x d y .
P = h ν Δ ν = ( 6.6 × 10 34 ) ( 2.8 × 10 14 ) Δ ν = 1.8 × 10 19 Δ ν

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