Abstract

Thermal refraction focusing in very-large-core step-index antiguided fibers is investigated theoretically. An analytical model based on a zero-field approximation is presented for treating the combined effects of index-antiguiding and thermal focusing for both the linearly polarized fundamental and first-high-order radial modes. At low pumping power, the modes are antiguided by the amplifying core boundary whereas at high pumping power they narrow due to thermal focusing. The same analysis is also applicable for very-large-core graded-index fibers in which the fields are truncated at the boundary.

© 2013 Optical Society of America

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References

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    [Crossref]
  2. A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24, 1677–1682 (2007).
    [Crossref]
  3. A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
    [Crossref]
  4. W. Hageman, Y. Chen, X. Wang, L. Gao, G. U. Kim, M. Richardson, and M. Bass, “Scalable side-pumped, gain-guided index-antiguided fiber laser,” J. Opt. Soc. Am. B 27, 2451–2459 (2010).
    [Crossref]
  5. K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
    [Crossref]
  6. Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
    [Crossref]
  7. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
    [Crossref]
  8. J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
    [Crossref]
  9. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11, 2982–2990 (2003).
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  12. M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
    [Crossref]
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  16. 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, 510–512 (2013).
    [Crossref]
  17. L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, 1960), p. 67, Eq. (4.4.8).
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2013 (2)

2012 (1)

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

2010 (3)

W. Hageman, Y. Chen, X. Wang, L. Gao, G. U. Kim, M. Richardson, and M. Bass, “Scalable side-pumped, gain-guided index-antiguided fiber laser,” J. Opt. Soc. Am. B 27, 2451–2459 (2010).
[Crossref]

K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
[Crossref]

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

2007 (2)

2006 (1)

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

2003 (2)

2001 (1)

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

1976 (1)

1973 (1)

1965 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Sections 13.1.1 and 13.1.10, p. 504.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Section 6.1, p. 255.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Table 22.10, p. 799.

Ballato, J.

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

Bass, M.

Brown, D. C.

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

Casperson, L. W.

Chen, Y.

Cherin, A. H.

A. H. Cherin, Introduction to Optical Fibers (McGraw-Hill, 1983), p. 128.

Dai, S.

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

Dittli, A.

Foy, P.

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

Gao, L.

Guyenot, V.

Hageman, W.

Hawkins, W.

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

Her, T.-H.

Hoffman, H. J.

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

Jansen, F.

Jauregui, C.

Kim, G. U.

Kogelnik, H.

Li, W.

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

Liem, A.

Limpert, J.

Lu, M.

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

McComb, T.

Nolte, S.

Otto, H.-J.

Peng, B.

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
[Crossref]

Peschel, T.

Richardson, M.

Richardson, M. C.

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

Schreiber, T.

Siegman, A. E.

A. E. Siegman, “Gain-guided, index-antiguided fiber lasers,” J. Opt. Soc. Am. B 24, 1677–1682 (2007).
[Crossref]

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20, 1617–1628 (2003).
[Crossref]

Slater, L. J.

L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, 1960), p. 67, Eq. (4.4.8).

L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, 1960), p. 95, Eq. (5.5.2).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Table 22.10, p. 799.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Section 6.1, p. 255.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Sections 13.1.1 and 13.1.10, p. 504.

Stutzki, F.

Sudesh, V.

Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. 32, 2505–2507 (2007).
[Crossref]

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

Tünnermann, A.

Wang, X.

Wei, W.

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
[Crossref]

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

Wu, J. W.

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

Yan, K. L.

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

Yan, K.-L.

K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
[Crossref]

Zellmer, H.

Zhou, E. Y.

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

Zhou, E.-Y.

K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
[Crossref]

Zou, K.

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

M. Lu, W. Li, K. Zou, S. Dai, W. Wei, and B. Peng, “Yb3+-doped 200  μm diameter core, gain guided index-antiguided fiber,” Appl. Phys. B 98, 301–304 (2010).
[Crossref]

Appl. Phys. Lett. (1)

A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. 89, 251101 (2006).
[Crossref]

IEEE J. Quantum Electron. (1)

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

J. Mod. Opt. (1)

K.-L. Yan, E.-Y. Zhou, W. Wei, and B. Peng, “Efficiency of pump absorption in gain-guided and index-antiguided (GG-IAG) fiber,” J. Mod. Opt. 57, 480–484 (2010).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Laser Technol. (1)

J. W. Wu, E. Y. Zhou, K. L. Yan, W. Wei, and B. Peng, “Comparison of end-pumped and multi-point pumped Yb3+-doped gain guided and index antiguided fiber laser,” Opt. Laser Technol. 44, 2371–2376 (2012).
[Crossref]

Opt. Lett. (3)

Other (6)

L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, 1960), p. 67, Eq. (4.4.8).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Section 6.1, p. 255.

L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, 1960), p. 95, Eq. (5.5.2).

A. H. Cherin, Introduction to Optical Fibers (McGraw-Hill, 1983), p. 128.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Sections 13.1.1 and 13.1.10, p. 504.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, in National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, 1970), Table 22.10, p. 799.

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Figures (4)

Fig. 1.
Fig. 1.

Radial profile of the index of refraction in a step-IAG fiber with a positive temperature coefficient of refractive index dn/dT. The horizontal axis is in units of the fiber radius ro.

Fig. 2.
Fig. 2.

Radial intensity distribution of the fundamental mode as a function of the normalized radial distance r/r0 from the axis to the antiguiding core–cladding boundary, for various values of the unbounded Gaussian spot size also normalized to r0. The circles in the figure represent the limiting form J02(2.405r/r0), as shown below in Eq. (23).

Fig. 3.
Fig. 3.

Plot of the boundary-fitting parameter a as a function of the normalized Gaussian spot size w/r0 where r0 is the fiber core radius. The curve labeled 0 represents the lowest order linearly polarized modes having no nodes for radii less than r0, and the curve labeled 1 represents modes having one node for radii less than r0.

Fig. 4.
Fig. 4.

Radial intensity distribution of the azimuthally symmetric next-to-lowest-order radial mode as a function of the normalized radial distance r/r0 from the axis to the antiguiding core–cladding boundary, for various values of the unbounded Gaussian spot size also normalized to fiber radius r0. The circles in the figure represent the limiting form J02(5.520r/r0) as shown below in Eq. (31).

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

×E=iωμH,
×H=iωεE,
××Eω2μεE=(μ/μ)××E.
2E+ω2μεE=[(ε/ε)·E](μ/μ)××E,
2E+k2E=0,
2E+k2E=0.
k2(r)=k0(k0k2xr2).
2Er2+1rEr+1r2Eϕ2+2Ez2+k02Ek0k2r2E=0.
2Er2+1rEr+2Ez2+k02Ek0k2r2E=0.
E(r,z)=R(r)Z(z).
d2Rdr2+1rdRdr+(ark0k2r2)R=0,
d2Zdz2+(k02ar)Z=0,
R(r)=A(r)exp(r2/w2).
d2Adr2+(1r4rw2)dAdr+(4r2w42w2)A+(ark0k2r2)A=0.
w4=4(k0k2)1,
d2Adr2+(1r4rw2)dAdr2w2A+arA=0.
w=21/2(β0β2)1/4.
d2Adr2+(1r2r)dAdr(1+2a)A=0,
a=ar2(β0β2)1/2
ρd2Adρ2+(1ρ)dAdρ1+2a4A=0.
A(ρ)=F11(a2+14;1;ρ).
R(r)=exp(r2w2)F11(a2+14;1;2r2w2),
R(r)=J0(2.405r/ro),
limcF11[bc;b;x/c]=Γ(b)x1/2b/2Jb1(2x),
limcF11[1c;1;x/c]=J0(2x),
xc=2r2w2=2(r/r0)2(w/r0)2,
1c=a2+14.
x=(1.5a)(r/r0)2(w/r0)2.
R(r)=lima,w/r0J0[2(1.5a)1/2r/r0w/r0].
R(r)J0(3.261r/r0).
R(r)=J0(5.520r/ro),
Lν(μ)(z)=Γ(1+μ+ν)Γ(1+μ)Γ(1+ν)F11[ν;1+μ;z],
Lν(0)(z)=Γ(1+ν)Γ(1)Γ(1+ν)F11[ν;1;z].
Lν(0)(z)=F11[ν;1;z].
ν=(a2+14).
R(r)=exp(r2/w2)L0(0)(2r2/w2).
R(r)=exp(r2/w2).
R(r)=exp(r2/w2)L1(0)(2r2/w2).
R(r)=exp(r2/w2)(12r2/w2).

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