Abstract

We present a simple theoretical model of transverse mode instability in high-power rare-earth doped fiber amplifiers. The model shows that efficient power transfer between the fundamental and higher-order modes of the fiber can be induced by a nonlinear interaction mediated through the thermo-optic effect, leading to transverse mode instability. The temporal and spectral characteristics of the instability dynamics are investigated, and it is shown that the instability can be seeded by both quantum noise and signal intensity noise, while pure phase noise of the signal does not induce instability. It is also shown that the presence of a small harmonic amplitude modulation of the signal can lead to generation of higher harmonics in the output intensity when operating near the instability threshold.

© 2013 OSA

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References

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  1. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011).
    [Crossref] [PubMed]
  2. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
    [Crossref] [PubMed]
  3. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011).
    [Crossref] [PubMed]
  4. 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]
  5. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011).
    [Crossref] [PubMed]
  6. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers,” Opt. Express 19, 23965–23980 (2011).
    [Crossref] [PubMed]
  7. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
    [Crossref] [PubMed]
  8. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
    [Crossref] [PubMed]
  9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
    [Crossref] [PubMed]
  10. K. D. Cole and P. E. Crittenden, “Steady-Periodic Heating of a Cylinder,” ASME J. Heat Transfer 131, 091301 (2009).
    [Crossref]
  11. F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20, 3997–4008 (2012).
    [Crossref] [PubMed]
  12. R. G. Smith, “Optical Power Handling Capacity of Low Loss Optical Fibers as Determined by Stimulated Raman and Brillouin Scattering,” Appl. Opt. 11, 2489–2494 (1972).
    [Crossref] [PubMed]
  13. P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, “VODE: A Variable Coefficient ODE Solver,” SIAM J. Sci. Stat. Comput. 10, 1038–1051 (1989).
    [Crossref]
  14. 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 power,” Opt. Lett. 37, 4242–4244 (2012).
    [Crossref] [PubMed]
  15. J. Chen, J. W. Sickler, E. P. Ippen, and F. X. Kärtner, “High repetition rate, low jitter, low intensity noise, fundamentally mode-locked 167 fs soliton Er-fiber laser,” Opt. Lett. 32, 1566–1568 (2007).
    [Crossref] [PubMed]
  16. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20, 5742–5753 (2012).
    [Crossref] [PubMed]

2012 (6)

2011 (6)

2009 (1)

K. D. Cole and P. E. Crittenden, “Steady-Periodic Heating of a Cylinder,” ASME J. Heat Transfer 131, 091301 (2009).
[Crossref]

2007 (1)

1989 (1)

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, “VODE: A Variable Coefficient ODE Solver,” SIAM J. Sci. Stat. Comput. 10, 1038–1051 (1989).
[Crossref]

1972 (1)

Alkeskjold, T. T.

Broeng, J.

Brown, P. N.

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, “VODE: A Variable Coefficient ODE Solver,” SIAM J. Sci. Stat. Comput. 10, 1038–1051 (1989).
[Crossref]

Byrne, G. D.

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, “VODE: A Variable Coefficient ODE Solver,” SIAM J. Sci. Stat. Comput. 10, 1038–1051 (1989).
[Crossref]

Chen, J.

Cole, K. D.

K. D. Cole and P. E. Crittenden, “Steady-Periodic Heating of a Cylinder,” ASME J. Heat Transfer 131, 091301 (2009).
[Crossref]

Crittenden, P. E.

K. D. Cole and P. E. Crittenden, “Steady-Periodic Heating of a Cylinder,” ASME J. Heat Transfer 131, 091301 (2009).
[Crossref]

Dajani, I.

Eidam, T.

Gaida, C.

Hansen, K. R.

Hindmarsh, A. C.

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, “VODE: A Variable Coefficient ODE Solver,” SIAM J. Sci. Stat. Comput. 10, 1038–1051 (1989).
[Crossref]

Ippen, E. P.

Jansen, F.

Jauregui, C.

Jørgensen, M. M.

Karow, M.

Kärtner, F. X.

Kracht, D.

Lægsgaard, J.

Laurila, M.

Liem, A.

Limpert, J.

Neumann, J.

Otto, H.-J.

Robin, C.

Schmidt, O.

Schreiber, T.

Sickler, J. W.

Smith, A. V.

Smith, J. J.

Smith, R. G.

Steinmetz, A.

Stutzki, F.

Tünnermann, A.

Tünnermann, H.

Ward, B.

Weßels, P.

Wirth, C.

Appl. Opt. (1)

ASME J. Heat Transfer (1)

K. D. Cole and P. E. Crittenden, “Steady-Periodic Heating of a Cylinder,” ASME J. Heat Transfer 131, 091301 (2009).
[Crossref]

Opt. Express (8)

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

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

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers,” Opt. Express 19, 23965–23980 (2011).
[Crossref] [PubMed]

F. Jansen, F. Stutzki, H.-J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20, 3997–4008 (2012).
[Crossref] [PubMed]

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20, 5742–5753 (2012).
[Crossref] [PubMed]

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

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

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]

Opt. Lett. (5)

SIAM J. Sci. Stat. Comput. (1)

P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, “VODE: A Variable Coefficient ODE Solver,” SIAM J. Sci. Stat. Comput. 10, 1038–1051 (1989).
[Crossref]

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Figures (20)

Fig. 1
Fig. 1

Nonlinear coupling coefficient χ for LP01 − LP11 coupling as a function of Ω for Fiber A.

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Fig. 2
Fig. 2

(a) Output HOM content ξ as a function of output power in the FM P1 and (b) output PSD of the HOM S2 of Fiber A. The input power in the FM P1(0) = 1 W.

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Fig. 3
Fig. 3

Output HOM content ξ as a function of FM output power P1 for intensity noise seeding of Fiber A with a RIN of 10−13 Hz−1, 10−12 Hz−1 and 10−11 Hz−1. Quantum noise seeding is shown for comparison. The input power in the FM P0,1 = 1 W and the initial HOM content ξ(0) = 0.01.

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Fig. 4
Fig. 4

Nonlinear coupling coefficient χ for LP01 − LP11 coupling and LP01 − LP02 coupling for a SIF with V = 5 and all other parameters the same as Fiber A. The peak value is seen to be higher for LP01 − LP11 coupling, leading to a lower threshold power for this process.

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Fig. 5
Fig. 5

Output PSD Sn of the light in LP01 and LP11 of 1 m of Fiber A with g = ln(500)/Γ1 m−1. The input signal is a CW signal with a linewidth due to phase noise of 1 Hz (a,b), 1 kHz (c,d) and 10 kHz (e,f), and the input power in the FM is 1 W. Quantum noise acts as a seed for TMI, which is seen as the presence of the redshifted light in the HOM.

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Fig. 6
Fig. 6

(a) Average mode power Pn of the FM (blue curve) and HOM (green curve), and (b) HOM content ξ as a function of z for the fiber amplifier described in Fig. 5 with an input signal linewidth of 10 kHz. The results for the 1 Hz and 1 kHz cases are indistinguishable from the 10 kHz case.

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Fig. 7
Fig. 7

Instantaneous mode power at the fiber output |pn(L,t)|2 as a function of time for the fiber amplifier described in Fig. 5. The signal power is seen to fluctuate between the FM (blue curve) and HOM (green curve) in a chaotic fashion on a ms timescale.

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Fig. 8
Fig. 8

(a) Average mode power Pn of the FM (blue curve) and HOM (green curve), and (b) HOM content ξ as a function of z for Fiber A with g = ln(1000)/Γ1 m−1. The input signal is a CW signal with a linewidth due to phase noise of 1 kHz, and the input power in the FM / HOM is 1 W / 0 W, with quantum noise added to both modes. The HOM content is seen to converge to 0.5 as the signal power increases beyond the TMI threshold.

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Fig. 9
Fig. 9

Output PSD Sn of the light in (a) LP01 and (b) LP11 of the SIF amplifier descibed in Fig. 8. Quantum noise acts as a seed for TMI, and the multiple power flow reversals between the modes result in an additional redshift and spectral broadening of the output signal.

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Fig. 10
Fig. 10

Instantaneous mode power at the fiber output |pn(L,t)|2 as a function of time for the fiber amplifier described in Fig. 8. The signal power is seen to fluctuate between the FM (blue curve) and HOM (green curve) in a chaotic fashion on a ms timescale. Note that a full transfer of power between the modes occurs on a sub-ms timescale.

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Fig. 11
Fig. 11

Output HOM content ξ as a function of FM output power P1 for Fiber A. The TMI is seeded by a sinusoidal modulation of the input mode amplitude with a modulation frequency Ωm/2π = 1 kHz and modulation depth a of 10−4, 10−5 and 10−6. The input HOM content ξ(0) = 0.01 and the input FM power P0,1 = 1 W.

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Fig. 12
Fig. 12

(a) Average mode power Pn and (b) HOM content ξ as a function of z for 1 m of Fiber A with g = ln(350)/Γ1 m−1. The input signal is an amplitude modulated signal with a linewidth due to phase noise of 1 Hz, a modulation depth a = 10−4 and a modulation frequency Ωm/(2π) = 1 kHz. The input power in the FM / HOM is 0.99 W / 0.01 W.

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Fig. 13
Fig. 13

Output PSD Sn of the light in (a) LP01 and (b) LP11 of the SIF amplifier described in Fig. 12 with an amplitude modulated input signal. The first Stokes sideband of the HOM acts as a seed for TMI and experiences nonlinear gain, while the anti-Stokes side band of the FM is depleted by coupling to the HOM carrier. The seed for the second Stokes sideband of the HOM is generated by an intra-modal FWM process between the initial frequency components.

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Fig. 14
Fig. 14

Spectrum of the output intensity I at a point located at r = 20 μm, ϕ = 0 of the SIF amplifier described in Fig. 12. The intensity fluctuations are harmonic with a strong component at 1 kHz and a much weaker component at 2 kHz. The peaks are due to interference between the FM carrier and the first and second Stokes sidebands of the HOM.

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Fig. 15
Fig. 15

(a) Average mode power Pn and (b) HOM content ξ as a function of z for 1 m of Fiber A with g = ln(400)/Γ1 m−1. The input signal is an amplitude modulated signal with a linewidth due to phase noise of 1 Hz, a modulation depth a = 10−4 and a modulation frequency Ωm/(2π) = 1 kHz. The input power in the FM / HOM is 0.99 W / 0.01 W.

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Fig. 16
Fig. 16

Output PSD Sn of the light in (a) LP01 and (b) LP11 of the SIF amplifier described in Fig. 15 with an amplitude modulated input signal. The additional Stokes sidebands are generated by intra-modal FWM, and experience nonlinear gain in a cascade process.

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Fig. 17
Fig. 17

Spectrum of the output intensity I at a point located at r = 20 μm, ϕ = 0 of the SIF amplifier described in Fig. 15. The second, third and fourth harmonics of the modulation frequency of 1 kHz are clearly visible, and are due to interference between different spectral components of the FM and HOM.

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Fig. 18
Fig. 18

(a) Average mode power Pn and (b) HOM content ξ as a function of z for 1 m of Fiber A with g = ln(700)/Γ1 m−1. The input signal is an amplitude modulated signal with a linewidth due to phase noise of 1 Hz, a modulation depth a = 10−4 and a modulation frequency Ωm/(2π) = 1 kHz. The input power in the FM / HOM is 0.99 W / 0.01 W.

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Fig. 19
Fig. 19

Output PSD Sn of the light in (a) LP01 and (b) LP11 of the SIF amplifier described in Fig. 18. At this power level, quantum noise seeded TMI results in broad output spectra, without the discrete sidebands seen at lower power.

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Fig. 20
Fig. 20

Spectrum of the output intensity I at a point located at r = 20 μm, ϕ = 0 of the SIF amplifier described in Fig. 19 operating well above the TMI threshold. Discrete spectral components are no longer seen at this power level, since quantum noise seeding is dominating. The broad spectrum reflects the chaotic nature of the mode fluctuations.

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Tables (1)

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Table 1 Parameters of Fiber A.

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Equations (58)

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E ( r , t ) = 1 2 u ( E ( r , t ) e i ω 0 t + c . c . ) ,
∇ 2 E ( r , ω − ω 0 ) + ω 2 c 2 ε ( r ) E ( r , ω − ω 0 ) = − μ 0 ω 2 P N L ( r , ω − ω 0 ) ,
ε ( r ) = ε f ( r ⊥ ) − i g ( r ) ε f ( r ⊥ ) k 0 ,
P N L ( r , t ) = ε 0 η Δ T ( r , t ) E ( r , t ) ,
Δ ε ( r , t ) = η Δ t ( r , t ) .
∇ 2 E ( r , Ω ) + k 2 ε ( r ) E ( r , Ω ) = − η k 2 2 π ∫ − ∞ ∞ Δ T ( r , Ω ′ ) E ( r , Ω − Ω ′ ) d Ω ′ ,
ρ C ∂ Δ T ∂ t − κ ∇ ⊥ 2 Δ T ( r , t ) = Q ( r , t ) ,
Q ( r , t ) = ( λ s λ p − 1 ) g ( r ) I ( r , t ) ,
∇ ⊥ 2 Δ T ( r , ω ) − q ( ω ) Δ T ( r , ω ) = − Q ( r , ω ) κ ,
Δ T ( r , ω ) = 1 κ ∬ G ( r ⊥ , r ⊥ ′ , ω ) Q ( r ′ , ω ) d 2 r ⊥ ′ ,
∇ ⊥ 2 G ( r ⊥ , r ⊥ ′ , ω ) − q ( ω ) G ( r ⊥ , r ⊥ ′ , ω ) = − δ ( r ⊥ − r ⊥ ′ ) .
I ( r , t ) = 1 2 ε f ( r ⊥ ) ε 0 c E ( r , t ) E ( r , t ) * ,
Δ T ( r , ω ) = n c ε 0 c 4 π κ ( λ s λ p − 1 ) g ( z ) ∬ S d G ( r ⊥ , r ⊥ ′ , ω ) ∫ − ∞ ∞ E ( r ⊥ ′ , ω + ω ′ ) E ( r ′ , ω ′ ) * d ω ′ d r ′ ⊥ .
E ( r , Ω ) = ∑ m A m ( z , Ω ) ψ m ( r ⊥ ) e − i β 0 , m z ,
∇ ⊥ 2 ψ m ( r ⊥ ) + k 2 ε f ( r ⊥ ) ψ m ( r ⊥ ) = β m ( ω ) 2 ψ m ( r ⊥ ) ,
2 i β 0 , n ∂ A n ∂ z = β 1 , n Ω A n ( z , Ω ) + i k 0 ∑ m α n m A m ( z , Ω ) e i Δ β n m z + B ∑ k l m e i ( Δ β n m − Δ β k l ) z × ∫ − ∞ ∞ A m ( z , Ω − Ω ′ ) G n m k l ( Ω ′ ) ∫ − ∞ ∞ A k ( z , Ω ′ + Ω ″ ) A l ( z , Ω ″ ) * d Ω ″ d Ω ′ .
α n m = n c g ( z ) ∬ S d ψ n ( r ⊥ ) * ψ m ( r ⊥ ) d 2 r ⊥
B ( z ) = η k 0 2 n c ε 0 c 8 π 2 κ ( λ s λ p − 1 ) g ( z ) .
G n m k l ( Ω ) = ∬ ψ n ( r ⊥ ) * ψ m ( r ⊥ ) ∬ S d G ( r ⊥ , r ⊥ ′ , Ω ) ψ k ( r ⊥ ′ ) ψ l ( r ⊥ ′ ) * d r ⊥ ′ d r ⊥ ,
κ ∂ Δ T ∂ r + h q Δ T = 0 ,
G ( r ⊥ , r ⊥ ′ , Ω ) = 1 2 π ∑ m = − ∞ ∞ g m ( r , r ′ , Ω ) e i m ( ϕ − ϕ ′ ) .
g n ( r , r ′ , Ω ) = { I n ( q r ) [ C n I n ( q r ′ ) + K n ( q r ′ ) ] for   0 ≤ r ≤ r ′ I n ( q r ′ ) [ C n I n ( q r ) + K n ( q r ) ] for   r ′ ≤ r ≤ R ,
C n = K n + 1 ( q R ) + K n − 1 ( q R ) − a K n ( q R ) I n + 1 ( q R ) + I n − 1 ( q R ) + a I n ( q R ) ,
∂ p 1 ∂ z = ( n c Γ 1 g 2 n eff , 1 − i Ω v g , 1 ) p 1 ( z , Ω ) − i K 1 g × ( ∫ − ∞ ∞ p 1 ( z , Ω − Ω ′ ) G 1111 ( Ω ′ ) C 11 ( z , Ω ′ ) d Ω ′ + ∫ − ∞ ∞ p 1 ( z , Ω − Ω ′ ) G 1122 ( Ω ′ ) C 22 ( z , Ω ′ ) d Ω ′ + ∫ − ∞ ∞ p 2 ( z , Ω − Ω ′ ) G 1212 ( Ω ′ ) C 12 ( z , Ω ′ ) d Ω ′ ) ,
∂ p 2 ∂ z = ( n c Γ 2 g 2 n eff , 2 − i Ω v g , 2 ) p 2 ( z , Ω ) − i K 2 g × ( ∫ − ∞ ∞ p 2 ( z , Ω − Ω ′ ) G 2222 ( Ω ′ ) C 22 ( z , Ω ′ ) d Ω ′ + ∫ − ∞ ∞ p 2 ( z , Ω − Ω ′ ) G 2211 ( Ω ′ ) C 11 ( z , Ω ′ ) d Ω ′ + ∫ − ∞ ∞ p 1 ( z , Ω − Ω ′ ) G 2121 ( Ω ′ ) C 21 ( z , Ω ′ ) d Ω ′ ) ,
Γ n = ∬ S d ψ n ( r ⊥ ) * ψ n ( r ⊥ ) d 2 r ⊥ .
K n = η ( λ s − λ p ) 4 π κ n eff , n λ s λ p ,
C i j ( z , Ω ) = ∫ − ∞ ∞ p i ( z , Ω + Ω ′ ) p j ( z , Ω ′ ) * d Ω ′ .
p n ( z , Ω ) = 2 π P 0 , n exp ( n c Γ n 2 n eff , n ∫ 0 z g ( z ′ ) d z ′ ) e i Φ n ( z ) δ ( Ω ) ,
Φ 1 ( z ) = Φ 1 ( 0 ) − [ γ 11 ( P 1 ( z ) − P 0 , 1 ) + γ 12 ( P 2 ( z ) − P 0 , 2 ) ] ,
Φ 2 ( z ) = Φ 2 ( 0 ) − [ γ 22 ( P 2 ( z ) − P 0 , 2 ) + γ 21 ( P 1 ( z ) − P 0 , 1 ) ] ,
P n ( z ) = P 0 , n exp ( n c Γ n n eff , n ∫ 0 z g ( z ′ ) d z ′ )
γ 11 = 4 π 2 K 1 n eff , 1 n c Γ 1 G 1111 ( 0 ) ,                                                 γ 22 = 4 π 2 K 2 n eff , 2 n c Γ 2 G 2222 ( 0 ) ,
γ 12 = 4 π 2 K 1 n eff , 2 n c Γ 2 [ G 1122 ( 0 ) + G 1212 ( 0 ) ] ,                   γ 21 = 4 π 2 K 2 n eff , 1 n c Γ 1 [ G 2211 ( 0 ) + G 2121 ( 0 ) ] .
p n ( z , t ) = P 0 , n exp ( n c Γ n 2 n eff , n ∫ 0 z g ( z ′ ) d z ′ ) e i [ Φ n ( z ) + θ ( t ) ] ,
∂ p 1 ∂ z = Γ 1 2 g ( z ) p 1 ( z , Ω ) − i K g ( z ) ∫ − ∞ ∞ p 1 ( z , Ω − Ω ′ ) G 1111 ( Ω ′ ) C 11 ( z , Ω ′ ) d Ω ′ ,
∂ p 2 ∂ z = Γ 2 2 g ( z ) p 2 ( z , Ω ) − i K g ( z ) ( ∫ − ∞ ∞ p 2 ( z , Ω − Ω ′ ) G 2211 ( Ω ′ ) C 11 ( z , Ω ′ ) d Ω ′ + ∫ − ∞ ∞ p 1 ( z , Ω − Ω ′ ) G 2121 ( Ω ′ ) C 21 ( z , Ω ′ ) d Ω ′ ) ,
∂ | p 2 | 2 ∂ z = [ Γ 2 + χ ( Ω ) P 1 ( z ) ] g ( z ) | p 2 ( z , Ω ) | 2 ,
| p 2 ( L , Ω ) | 2 = | p 2 ( 0 , Ω ) | 2 exp ( Γ 2 g a v L ) exp [ χ ( Ω ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ] ,
g a v = 1 L ∫ 0 L g ( z ) d z .
S 2 ( L , Ω ) = S 2 ( 0 , Ω ) exp ( Γ 2 g a v L ) exp [ χ ( Ω ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ] .
P 2 ( L ) = exp ( Γ 2 g a v L ) ∫ − ∞ ∞ h ¯ ( ω 0 + Ω ) exp [ χ ( Ω ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ] d Ω .
G 2121 ( Ω ) = π ∫ 0 R R 1 ( r ) R 2 ( r ) r ∫ 0 R c R 1 ( r ′ ) R 2 ( r ′ ) g 1 ( r , r ′ , Ω ) r ′ d r ′ d r .
ξ ( L ) ≈ h ¯ ω 0 P 0 , 1 exp ( − Δ Γ g a v L ) ∫ − ∞ ∞ exp [ χ ( Ω ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ] d Ω ,
ξ ( L ) ≈ h ¯ ω 0 2 π Γ 1 | χ ″ ( Ω p ) | P 1 ( L ) ( Γ 2 / Γ 1 − 3 / 2 ) P 0 , 1 Γ 2 / Γ 1 exp [ χ ( Ω p ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ] ,
p n ( 0 , t ) = P 0 , n ( 1 + ε N ( t ) ) e i Φ n ( 0 ) ≈ P 0 , n ( 1 + 1 2 ε N ( t ) ) e i Φ n ( 0 ) ,
S 2 ( 0 , Ω ) = 1 2 π ∫ − ∞ ∞ 〈 p 2 ( 0 , t ) p 2 ( 0 , t + t ′ ) * 〉 e − i Ω t ′ d t ′ ,
S 2 ( 0 , Ω ) = P 0 , 2 δ ( Ω ) + 1 4 R N ( Ω ) P 0 , 2 ,
R N ( Ω ) = 1 2 π ∫ − ∞ ∞ 〈 ε N ( t ) ε N ( t + t ′ ) 〉 e − i Ω t ′ d t ′ .
S 2 ( L , Ω ) = P 0 , 2 exp ( Γ 2 g a v L ) δ ( Ω ) + 1 4 P 0 , 2 exp ( Γ 2 g a v L ) R N ( Ω ) exp ( Δ P 1 Γ 1 χ ( Ω ) ) ,
ξ ( L ) = ξ ( 0 ) exp ( − Δ Γ g a v L ) ( 1 + 1 4 ∫ − ∞ ∞ R N ( Ω ) exp ( Δ P 1 Γ 1 χ ( Ω ) ) d Ω ) .
ξ ( L ) ≈ ξ ( 0 ) ( P 0 , 1 P 1 ( L ) ) 1 − Γ 2 Γ 1 [ 1 + 1 4 R N ( Ω p ) 2 π Γ 1 P 1 ( L ) | χ ″ ( Ω p ) | exp ( Δ P 1 Γ 1 χ ( Ω p ) ) ] ,
p n ( 0 , t ) = P 0 , n e i ( Φ n ( 0 ) + θ ( t ) ) ,
p n ( 0 , t ) = P 0 , n [ 1 + a sin ( Ω m t ) ] ,
S 2 ( 0 , Ω ) = P 0 , 2 δ ( Ω ) + P 0 , 2 a 2 4 [ δ ( Ω − Ω m ) + δ ( Ω + Ω m ) ] .
P 2 ( L ) ≈ P 0 , 2 exp ( Γ 2 g a v L ) [ 1 + a 4 4 exp ( χ ( − Ω m ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ) ] ,
ξ ( L ) ≈ ξ ( 0 ) ( P 0 , 1 P 1 ( L ) ) 1 − Γ 2 Γ 1 [ 1 + a 2 4 exp ( χ ( − Ω m ) Γ 1 ( P 1 ( L ) − P 0 , 1 ) ) ] .
p n ( 0 , t ) = P 0 , n [ 1 + a sin ( Ω m t ) ] e i [ Φ n ( 0 ) + θ ( t ) ] .

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