Abstract

We present a coupled-mode model of transverse mode instability in high-power fiber amplifiers, which takes the effect of gain saturation into account. The model provides simple semi-analytical formulas for the mode instability threshold, which are valid also for highly saturated amplifiers. The model is compared to recently published detailed numerical simulations of mode instability, and we find reasonably good agreement with our simplified coupled-mode model.

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  1. T. Eidam, C. Wirth, C. jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, 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. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011).
    [CrossRef] [PubMed]
  3. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20, 15710–15722 (2012).
    [CrossRef] [PubMed]
  4. M. M. Johansen, M. Laurila, M. D. Maack, D. Noordegraaf, C. Jakobsen, T. T. Alkeskjold, J. Lægsgaard, “Frequency resolved transverse mode instability in rod fiber amplifiers,” Opt. Express 21, 21847–21856 (2013).
    [CrossRef] [PubMed]
  5. C. Jauregui, T. Eidam, J. Limpert, A. Tünnermann, “Impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011).
    [CrossRef] [PubMed]
  6. A. V. Smith, J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
    [CrossRef] [PubMed]
  7. A. V. Smith, 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]
  8. B. Ward, C. Robin, I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
    [CrossRef] [PubMed]
  9. B. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
    [CrossRef] [PubMed]
  10. S. Naderi, I. Dajani, T. Madden, C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21, 16111–16129 (2013).
    [CrossRef] [PubMed]
  11. K. R. Hansen, T. T. Alkeskjold, J. Broeng, J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
    [CrossRef] [PubMed]
  12. K. R. Hansen, T. T. Alkeskjold, J. Broeng, J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21, 1944–1971 (2013).
    [CrossRef] [PubMed]
  13. A. V. Smith, J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
    [CrossRef] [PubMed]

2013

2012

2011

Alkeskjold, T. T.

Broeng, J.

Dajani, I.

Eidam, T.

Gaida, C.

Hansen, K. R.

Jakobsen, C.

Jansen, F.

Jauregui, C.

Johansen, M. M.

Lægsgaard, J.

Laurila, M.

Limpert, J.

Maack, M. D.

Madden, T.

Naderi, S.

Noordegraaf, D.

Otto, H.-J.

Robin, C.

Schmidt, O.

Schreiber, T.

Smith, A. V.

Smith, J. J.

Stutzki, F.

Tünnermann, A.

Ward, B.

Wirth, C.

Opt. Express

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

M. M. Johansen, M. Laurila, M. D. Maack, D. Noordegraaf, C. Jakobsen, T. T. Alkeskjold, J. Lægsgaard, “Frequency resolved transverse mode instability in rod fiber amplifiers,” Opt. Express 21, 21847–21856 (2013).
[CrossRef] [PubMed]

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

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

A. V. Smith, 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]

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

B. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
[CrossRef] [PubMed]

S. Naderi, I. Dajani, T. Madden, C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21, 16111–16129 (2013).
[CrossRef] [PubMed]

T. Eidam, C. Wirth, C. jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, 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]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21, 1944–1971 (2013).
[CrossRef] [PubMed]

A. V. Smith, J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[CrossRef] [PubMed]

Opt. Lett.

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Figures (6)

Fig. 1
Fig. 1

Nonlinear gain coefficient χ as a function of P/Psat for a fixed frequency difference Δν = 1 kHz. The solid curves are calculated using the exact expression for A, while the dashed curves are calculated using the approximate expression for A. The red curves are the results for Fiber A with a doped radius of 20 μm, while the blue curves are for Fiber B with a doped radius of 15 μm. As expected, the nonlinear gain coefficient decreases with increasing signal power due to the effect of gain saturation. The approximate calculation of χ remains reasonable as the signal power is increased, especially for Fiber B with the reduced doped area.

Fig. 2
Fig. 2

HOM content ξ as a function of signal output power Ps for Fiber A for counter-pumped (blue curve) and co-pumped (red curve) operation. The dashed curve shows the result when the gain oscillations are neglected as in [11, 12].

Fig. 3
Fig. 3

HOM content ξ as a function of signal output power Ps for Fiber B for counter-pumped (blue curve) and co-pumped (red curve) operation. The dashed curve shows the result when the gain oscillations are neglected as in [11, 12].

Fig. 4
Fig. 4

Nonlinear gain coefficient χ as a function of z for Fiber C. The blue curve is for counter-pumped operation with a pump power of 493 W, and the red curve is for co-pumped operation with a pump power of 525 W. The dashed curve shows the value of χ0.

Fig. 5
Fig. 5

Nonlinear gain coefficient χ as a function of z for Fiber D. The blue curve is for counter-pumped operation with a pump power of 1350 W, and the red curve is for co-pumped operation with a pump power of 1200 W. The dashed curve shows the value of χ0.

Fig. 6
Fig. 6

Nonlinear gain coefficient χ at a fixed frequency offset Δν = 1 kHz as a function of P1/Psat for Fiber C and D. The solid curve is calculated using the exact expression, and the dashed curved is calculated using the approximate expression.

Tables (2)

Tables Icon

Table 1 Fiber parameters for Fiber A and B.

Tables Icon

Table 2 Fiber parameters for Fiber C and D used for comparison with [13].

Equations (34)

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E ( r , t ) = a 1 ( z ) ψ 1 ( r ) e i ( β 1 z ω 1 t ) + a 2 ( z ) ψ 2 ( r ) e i ( β 2 z ω 2 t ) .
2 E z 2 + 2 E ε ( r , t ) c 2 2 E t 2 = 0 ,
ε ( r , t ) = ε f ( r ) + i g ( r ) ε f k + Δ ε ( r , t ) .
ρ C Δ T t κ 2 Δ T ( r , t ) = Q ( r , t ) ,
κ Δ T r + h q Δ T ( r , t ) = 0 ,
Q ( r , t ) = ( λ s λ p 1 ) g ( I ) I ( r , t ) ,
g ( I ) = g 0 1 + I / I sat ,
I ( r , t ) = I 0 ( r ) + I ˜ ( r , t ) ,
I 0 ( r ) = | p 1 ( z ) | 2 | ψ 1 ( r ) | 2 + | p 2 ( z ) | 2 | ψ 2 ( r ) | 2 ,
I ˜ ( r , t ) = p 1 ( z ) p 2 ( z ) * ψ 1 ( r ) ψ 2 ( r ) * e i ( Δ β z Δ ω t ) + c . c .
p i ( z ) = 1 2 n c ε 0 c a i ( z ) ,
Q ( r , t ) ( λ s λ p 1 ) g 0 1 + I 0 / I sat [ I 0 + I sat ( I ˜ I 0 + I sat ( I ˜ I 0 + I sat ) 2 ) ] .
Q ( r , t ) ( λ s λ p 1 ) g 0 1 + I 0 / I sat ( I 0 + I ˜ 1 + I 0 / I sat ) .
Δ T ( r , ω ) = 1 κ G ( r , r , ω ) Q ( r , ω ) d 2 r .
P 1 z = g 0 ( z ) χ ( z , Δ ω ) P 2 P 1 + Γ 1 g ˜ ( z ) P 1 ,
P 2 z = g 0 ( z ) χ ( z , Δ ω ) P 2 P 1 + Γ 2 g ˜ ( z ) P 2 .
g ˜ = g 0 1 + Γ 1 P 1 / P sat ,
Γ i = A d | ψ i ( r ) | 2 d 2 r .
χ ( z , Δ ω ) = η k 2 κ β 1 ( 1 λ s λ p ) Im [ A ( z , Δ ω ) ] ,
A ( z , Δ ω ) = ψ 1 ( r ) * ψ 2 ( r ) A d G ( r , r , Δ ω ) ψ 1 ( r ) ψ 2 ( r ) * ( 1 + P 1 ( z ) | ψ 1 ( r ) | 2 / I sat ) 2 d 2 r d 2 r ,
| ψ 1 ( r ) | 2 1 A d A d | ψ 1 ( r ) | 2 d 2 r
A ( z , Δ ω ) ( g ˜ ( z ) g 0 ( z ) ) 2 ψ 1 ( r ) * ψ 2 ( r ) A d G ( r , r , Δ ω ) ψ 1 ( r ) ψ 2 ( r ) * d 2 r d 2 r .
P 1 z = g ˜ ( z ) χ ˜ ( z , Δ ω ) P 2 P 1 + Γ 1 g ˜ ( z ) P 1 ,
P 2 z = g ˜ ( z ) χ ˜ ( z , Δ ω ) P 2 P 1 + Γ 2 g ˜ ( z ) P 2 ,
χ ˜ ( z , Δ ω ) = g ˜ ( z ) g 0 ( z ) χ 0 ( Δ ω )
χ 0 = η k 2 κ β 1 ( 1 λ s λ p ) Im [ A 0 ( z , Δ ω ) ] ,
A 0 ( Δ ω ) = ψ 1 ( r ) * ψ 2 ( r ) A d G ( r , r , Δ ω ) ψ 1 ( r ) ψ 2 ( r ) * d 2 r d 2 r .
P p z = ± A d A cl g ˜ p ( P p , P 1 ) P p ,
P 1 z = Γ 1 g ˜ ( P p , P 1 ) P 1 ,
P 2 z = g ˜ ( P p , P 1 ) χ ˜ ( z , Δ ω ) P 1 P 2 + Γ 2 g ˜ ( P p , P 1 ) P 2 ,
P 2 ( L ) = S 2 ( 0 , Δ ω ) ( P 1 ( L ) P 1 ( 0 ) ) Γ 2 / Γ 1 exp ( χ 0 ( Δ ω ) 0 L g ˜ 2 g 0 P 1 d z ) ,
S 2 ( 0 , Δ ω ) = P 2 ( 0 ) δ ( Δ ω ) + 1 4 R N ( Δ ω ) P 2 ( 0 ) ,
ξ ( L ) ξ ( 0 ) ( P 1 ( 0 ) P 1 ( L ) ) 1 Γ 2 / Γ 1 [ 1 + R N ( Ω ) 4 2 π | χ 0 ( Ω ) | M exp ( χ 0 ( Ω ) M ) ] ,
M = 0 L g ˜ 2 g 0 P 1 d z .

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