Abstract

Average powers from fiber lasers have reached the point that a quantitative understanding of thermal lensing and its impact on transverse mode instability is becoming critical. Although thermal lensing is well known qualitatively, there is a general lack of a simple method for quantitative analysis. In this work, we first conduct a study of thermal lensing in optical fibers based on a perturbation technique. The perturbation technique becomes increasingly inaccurate as thermal lensing gets stronger. It, however, provides a basis for determining a normalization factor to use in a more accurate numerical study. A simple thermal lensing threshold condition is developed. The impact of thermal lensing on transverse mode instability is also studied.

© 2016 Optical Society of America

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References

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  1. T. Katsouleas, “Accelerator physics: Electrons hang ten on laser wake,” Nature 431(7008), 515–516 (2004).
    [Crossref] [PubMed]
  2. J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993).
    [Crossref] [PubMed]
  3. J. Hecht, “Half a century of laser weapons,” Opt. Photonics News 20(2), 14–21 (2009).
    [Crossref]
  4. 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(14), 13218–13224 (2011).
    [Crossref] [PubMed]
  5. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
    [Crossref] [PubMed]
  6. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013).
    [Crossref] [PubMed]
  7. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
    [Crossref] [PubMed]
  8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19(24), 23965–23980 (2011).
    [Crossref] [PubMed]
  9. 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(4), 3997–4008 (2012).
    [Crossref] [PubMed]
  10. C. Jauregui, H. J. Otto, F. Stutzki, J. Limpert, and A. Tünnermann, “Simplified modelling the mode instability threshold of high power fiber amplifiers in the presence of photodarkening,” Opt. Express 23(16), 20203–20218 (2015).
    [Crossref] [PubMed]
  11. F. Jansen, F. Stutzki, H. J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “High-power thermally guiding index-antiguiding-core fibers,” Opt. Lett. 38(4), 510–512 (2013).
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  12. L. Kong, J. Cao, S. Guo, Z. Jiang, and Q. Lu, “Thermal-induced transverse-mode evolution in thermally guiding index-antiguided-core fiber,” Appl. Opt. 55(5), 1183–1189 (2016).
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  13. L. Dong, “Approximate treatment of nonlinear waveguide equation in the regime of nonlinear self-focus,” J. Lightwave Technol. 26(20), 3476–3485 (2008).
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  14. L. Dong, “Formulation of a complex mode solver for arbitrary circular acoustic waveguides,” J. Lightwave Technol. 28, 3162–3175 (2010).
  15. L. Dong, “Limits of stimulated Brillouin scattering suppression in optical fibers with transverse acoustic waveguide designs,” J. Lightwave Technol. 28, 3156–3161 (2010).
  16. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978).
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  17. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
    [Crossref]
  18. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

2016 (1)

2015 (1)

2013 (2)

2012 (1)

2011 (3)

2010 (2)

2009 (1)

J. Hecht, “Half a century of laser weapons,” Opt. Photonics News 20(2), 14–21 (2009).
[Crossref]

2008 (2)

2004 (1)

T. Katsouleas, “Accelerator physics: Electrons hang ten on laser wake,” Nature 431(7008), 515–516 (2004).
[Crossref] [PubMed]

2001 (1)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

1993 (1)

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993).
[Crossref] [PubMed]

1978 (1)

Alkeskjold, T. T.

Barty, C. P. J.

Beach, R. J.

Broeng, J.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Cao, J.

Dawson, J. W.

Dong, L.

Eidam, T.

Gordon, C. L.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993).
[Crossref] [PubMed]

Guo, S.

Hansen, K. R.

Hecht, J.

J. Hecht, “Half a century of laser weapons,” Opt. Photonics News 20(2), 14–21 (2009).
[Crossref]

Heebner, J. E.

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Jansen, F.

Jauregui, C.

Jiang, Z.

Katsouleas, T.

T. Katsouleas, “Accelerator physics: Electrons hang ten on laser wake,” Nature 431(7008), 515–516 (2004).
[Crossref] [PubMed]

Kmetec, J. D.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993).
[Crossref] [PubMed]

Kong, L.

Lægsgaard, J.

Liem, A.

Limpert, J.

Lu, Q.

Macklin, J. J.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993).
[Crossref] [PubMed]

Marom, E.

Messerly, M. J.

Otto, H. J.

Pax, P. H.

Schmidt, O.

Schreiber, T.

Shverdin, M. Y.

Siders, C. W.

Smith, A. V.

Smith, J. J.

Sridharan, A. K.

Stappaerts, E. A.

Stutzki, F.

Tünnermann, A.

Wirth, C.

Yariv, A.

Yeh, P.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. (1)

Nature (1)

T. Katsouleas, “Accelerator physics: Electrons hang ten on laser wake,” Nature 431(7008), 515–516 (2004).
[Crossref] [PubMed]

Opt. Express (7)

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(14), 13218–13224 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013).
[Crossref] [PubMed]

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[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(24), 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(4), 3997–4008 (2012).
[Crossref] [PubMed]

C. Jauregui, H. J. Otto, F. Stutzki, J. Limpert, and A. Tünnermann, “Simplified modelling the mode instability threshold of high power fiber amplifiers in the presence of photodarkening,” Opt. Express 23(16), 20203–20218 (2015).
[Crossref] [PubMed]

Opt. Lett. (1)

Opt. Photonics News (1)

J. Hecht, “Half a century of laser weapons,” Opt. Photonics News 20(2), 14–21 (2009).
[Crossref]

Phys. Rev. Lett. (1)

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993).
[Crossref] [PubMed]

Other (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Figures (8)
Fig. 1
Fig. 1 Mode collapse under the influence of thermal lensing. Equation (3) is evaluated for normalized frequency V = 3-8. A numerical simulation is also performed for comparison for fibers with core diameters of 10μm, 20μm and 30μm and NA of 0.06. The results are plotted as MFD for near-field MFD and eMFD for effective MFD, i.e. 2(Aeff/π) ½.

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Fig. 2
Fig. 2 Comparison of the temperature profile from the numerical method used in this work and the parabolic solution for a uniform heat load. ΔT is set to be 0K at the cladding boundary. Core diameter is 20μm; cladding diameter is 400μm and NA is 0.06. The total heat load is 64w/m in both cases. The laser wavelength is at 1.03μm. The normalized heat load profile, i.e. normalized mode intensity profile, is also shown.

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Fig. 3
Fig. 3 Typical run for the case in Fig. 2. Core diameter is 20μm and NA is 0.06. The total heat load is 64w/m. The laser wavelength is at 1.03μm. Relative changes are shown on the vertical axis on the right. The heat load is gradually increased over 30 steps.

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Fig. 4
Fig. 4 Refractive index profiles under various heat loads. Core diameter is 20μm and NA is 0.06. The laser wavelength is at 1.03μm.

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Fig. 5
Fig. 5 Simulated MFD for the fibers in [9]. Quantum defect heating is used to convert heat load to extracted power. Average heat load is used.

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Fig. 6
Fig. 6 Simulated TMI nonlinear coupling coefficient for fibers with core diameters of 10μm, 15μm, 20μm, 25μm and 30μm respectively. The fiber NA is 0.06 and cladding diameter is 400μm. This study is conducted at a wavelength of 1.06μm. Contour lines indicate constant thermal load.

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Fig. 7
Fig. 7 The LP01 and LP11 modes at 1.06μm in a fiber with a NA of 0.06 and a core diameter of 20μm without heat load and with a heat load of 1000w/m, i.e. ξQ0w02 = 2.19. The cladding diameter is 400μm.

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Fig. 8
Fig. 8 Illustration of the scheme for solving the heat conduction equation with an arbitrary heat load profile with cylindrical symmetry.

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Equations (36)

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n(r)=n 0 (r) +k T ΔT(r)
ΔT(r)= Q 0 4πκ ρ 2 ( ρ 2 − r 2 )
1 2 ξ Q 0 w 0 2 γ 6 +( U 2 +3ξ Q 0 ρ 2 + 9 2 ξ Q 0 w 0 2 ) γ 4 +3 ξ Q 0 ρ 2 γ 2 − U 2 =0
ξ= k T k 0 2 n co 2 4πκ
∇ t ψ(R)+{ w 2 k 0 2 n 0w 2 (R)[ 1+2 k T ΔT(R) ]− w 2 β 2 − m 2 R 2 }ψ(R)=0
∇ t ψ 0 (R)+{ w 0 2 k 0 2 n 0w0 2 (R)− w 0 2 β 0 2 − m 2 R 2 } ψ 0 (R)=0
∬ A∞ [ w 0 2 ( k 0 2 n 0w0 2 − β 0 2 ) ]− w 2 ( k 0 2 n 0w 2 − β 2 )ψ ψ 0 dA=2 k T w 2 k 0 2 ∬ A∞ n 0w 2 ΔTdA+ ∬ A∞ ψ 0 ∇ t ψ−ψ ∇ t ψ 0 dA
w 2 β 2 − w 0 2 β 0 2 =2 k T w 2 k 0 2 n T 2 − k 0 2 ( w 0 2 n a 2 − w 2 n b 2 )
w 2 = w 0 2 ( k 0 2 n a 2 − β 0 2 )−2 k T w 2 k 0 2 n T 2 k 0 2 n b 2 − β 2
n a 2 = ∬ A∞ n 0w0 2 (R)ψ(R) ψ 0 (R)dA ∬ A∞ ψ (R)ψ 0 (R)dA   n b 2 = ∬ A∞ n 0w 2 (R)ψ(R) ψ 0 (R)dA ∬ A∞ ψ(R) ψ 0 (R)dA   n T 2 = ∬ A∞ n 0w 2 (R)ΔT(R)ψ(R) ψ 0 (R)dA ∬ A∞ ψ (R)ψ 0 (R)dA
∇ t ψ(r)+{ k 0 2 n 0 2 (r)[ 1+2 k T ΔT(r) ]− β 2 − m 2 r 2 }ψ(R)=0
∇ t ψ 0 (r)+{ k 0 2 n 0 2 (r)− β 0 2 − m 2 R 2 } ψ 0 (r)=0
β 2 − β 0 2 =2 k T k 0 2 n Tr 2
n Tr 2 = ∬ A∞ n 0 2 (r)ΔT(r)ψ(r) ψ 0 (r)dA ∬ A∞ ψ(r) ψ 0 (r)dA
ψ(r)= e − 1 2 r 2 w 2
n a 2 ≈ n co 2 n b 2 ≈ n co 2 n T 2 ≈ n co 2 Q 0 8πκ ρ 2 ( ρ 2 + w 2 2 )
n Tr 2 ≈ n co 2 Q 0 4πκ ρ 2 ( ρ 2 + 2 1 w 2 + 1 w 0 2 )
β 2 − β 0 2 = k T k 0 2 n co 2 Q 0 2πκ ρ 2 ( ρ 2 + 2 1 w 2 + 1 w 0 2 )
1 2 ξ Q 0 w 0 2 γ 6 +( U 2 +3ξ Q 0 ρ 2 + 9 2 ξ Q 0 w 0 2 ) γ 4 +3 ξ Q 0 ρ 2 γ 2 − U 2 =0
ξ= k T k 0 2 n co 2 4πκ
∂ 2 T(r) ∂r 2 + 1 r ∂T(r) ∂r + Q(r) κ =0
T(r)= C 1 + C 2 ln(r)
T(r)=C− Q 4κ r 2
T 1 (r)= T c − Q 1 4κ r 1 2 r 1a 2 r 2
T 1a (r)= T c − Q 1 4κ r 1 2 − Q 1 2κ r 1 2 ln( r r 1a )
T 2 (r)= T 1a ( r 1 )+ Q 2 4κ r 2 2 − r 1 2 r 2a 2 − r 1 2 ( r 1 2 − r 2 )
T 2a (r)= T 1a ( r 1 )+ Q 2 4κ ( r 1 2 − r 2 2 )− Q 2 2κ r 2 2 − r 1 2 r 2a 2 − r 1 2 r 2a 2 ln( r r 2a )
r 2a 2 = r 1 2 + Q 2 Q 1 ( r 2 2 − r 1 2 )
T 3 (r)= T 2a ( r 2 )+ Q 3 4κ r 3 2 − r 2 2 r 3a 2 − r 2 2 ( r 2 2 − r 2 )
T 3a (r)= T 2a ( r 2 )+ Q 3 4κ ( r 2 2 − r 3 2 )− Q 3 2κ r 3 2 − r 2 2 r 3a 2 − r 2 2 r 3a 2 ln( r r 3a )
r 3a 2 = r 2 2 + Q 3 Q 2 ( r 3 2 − r 2 2 ) r 2a 2 − r 1 2 r 2 2 − r 1 2 r 2 2 r 2a 2
T N (r)= T (N−1)a ( r N−1 )+ Q N 4κ r N 2 − r N−1 2 r Na 2 − r N−1 2 ( r N−1 2 − r 2 )
T Na (r)= T (N−1)a ( r N−1 )+ Q N 4κ ( r N−1 2 − r N 2 )− Q N 2κ r N 2 − r N−1 2 r Na 2 − r N−1 2 r Na 2 ln( r r Na )
r Na 2 = r N−1 2 + Q N Q N−1 ( r N 2 − r N−1 2 ) r (N−1)a 2 − r N−2 2 r N−1 2 − r N−2 2 r N−1 2 r (N−1)a 2
T(r)= T (N−1)a ( r N−1 )+ Q N 4κ ( r N−1 2 − r N 2 )− Q N 2κ r N 2 − r N−1 2 r Na 2 − r N−1 2 r Na 2 ln( r r Na )
T 0 = T (N−1)a ( r N−1 )+ Q N 4κ ( r N−1 2 − r N 2 )− Q N 2κ r N 2 − r N−1 2 r Na 2 − r N−1 2 r Na 2 ln( b r Na )

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