Abstract

We develop a theoretical framework to analyze the mechanism of refractive index changes (RIC) in double-clad Yb3+ doped optical fibers under resonant core or clad pumping, and with signal amplification. The model describes and compares thermal and electronic contributions to the phase shifts induced on the amplified signal at 1064 nm and the probe signal at 1550 nm, i.e. located inside and outside of the fiber amplification band, respectively. The ratio between the thermal and electronic phase shifts is evaluated as a function of the pump pulse duration, the gain saturation, the amplified beam power and for a variety of fiber parameters.

© 2013 Optical Society of America

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2013 (1)

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
[CrossRef]

2012 (2)

2011 (6)

2010 (2)

R. Soulard, R. Moncorgé, A. Zinoviev, K. Petermann, O. Antipov, and A. Brignon, “Nonlinear spectroscopic properties of Yb3+-doped sesquioxides Lu2O3 and Sc2O3,” Opt. Express18, 11173–11180 (2010).
[CrossRef] [PubMed]

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B27, 63–92 (2010).
[CrossRef]

2009 (2)

A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

A. Fotiadi, N. G. Zakharov, O. Antipov, and P. Mégret, “All-fiber coherent combining of Er-doped amplifiers through refractive index control in Yb-doped fibers,” Opt. Lett.34, 3574–3576 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

2005 (3)

H. Bruesselbach, S. Wang, M. Minden, D. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B22, 347–353 (2005).
[CrossRef]

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett.30, 13–15 (2005).
[CrossRef]

T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron.11, 567–577 (2005).
[CrossRef]

2003 (1)

O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
[CrossRef]

2001 (1)

D. Brown and H. 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]

1998 (2)

1997 (1)

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: A review,” Opt. Fiber Technol.3, 44–64 (1997).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

Antipov, O.

R. Soulard, R. Moncorgé, A. Zinoviev, K. Petermann, O. Antipov, and A. Brignon, “Nonlinear spectroscopic properties of Yb3+-doped sesquioxides Lu2O3 and Sc2O3,” Opt. Express18, 11173–11180 (2010).
[CrossRef] [PubMed]

A. Fotiadi, N. G. Zakharov, O. Antipov, and P. Mégret, “All-fiber coherent combining of Er-doped amplifiers through refractive index control in Yb-doped fibers,” Opt. Lett.34, 3574–3576 (2009).
[CrossRef] [PubMed]

A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

A. Fotiadi, O. Antipov, and P. Mégret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express16, 12658–12663 (2008).
[CrossRef] [PubMed]

O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
[CrossRef]

A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193–230.
[CrossRef]

A. Fotiadi, O. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” in Frontiers in Guided Wave Optics and Optoelectronics, B. Pal, ed. (INTECH, 2010), chap. 11, pp. 209–234.

A. Fotiadi, O. Antipov, I. Bufetov, E. Dianov, and P. Mégret, “Comparative study of pump-induced refractive index changes in aluminum and phosphate silicate Yb-doped fibers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, (Optical Society of America, Washington, D.C., 2009), paper JWA9.
[CrossRef]

Arkwright, J. W.

Atkins, G. R.

Barty, C.

Bass, M.

M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, Handbook for Optics, 2nd ed. (MGH, 1995).

Beach, R. J.

Bednyakova, A.

Bredikhin, D.

O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
[CrossRef]

Brignon, A.

Brown, D.

D. Brown and H. 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]

Bruesselbach, H.

Bufetov, I.

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

A. Fotiadi, O. Antipov, I. Bufetov, E. Dianov, and P. Mégret, “Comparative study of pump-induced refractive index changes in aluminum and phosphate silicate Yb-doped fibers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, (Optical Society of America, Washington, D.C., 2009), paper JWA9.
[CrossRef]

Carslow, H.

H. Carslow and J. Jaeger, Conduction of Heat in Solids (Science, Moscow, 1964).

Clarkson, W. A.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B27, 63–92 (2010).
[CrossRef]

Davis, M.

M. Davis, M. Digonnet, and R. Pantell, “Thermal effects in doped fibers,” J. Lightwave Tech.16, 1013–1023 (1998).
[CrossRef]

Dawson, J. W.

Desurvire, E.

E. Desurvire, Erbium-doped Fiber Amplifiers: Principles and Applications (Willey, New York, 1994).

Dianov, E.

A. Fotiadi, O. Antipov, I. Bufetov, E. Dianov, and P. Mégret, “Comparative study of pump-induced refractive index changes in aluminum and phosphate silicate Yb-doped fibers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, (Optical Society of America, Washington, D.C., 2009), paper JWA9.
[CrossRef]

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

Digonnet, M.

M. Davis, M. Digonnet, and R. Pantell, “Thermal effects in doped fibers,” J. Lightwave Tech.16, 1013–1023 (1998).
[CrossRef]

Digonnet, M. F.

Digonnet, M. J. F.

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: A review,” Opt. Fiber Technol.3, 44–64 (1997).
[CrossRef]

Eidam, T.

Elango, P.

Eremeykin, O.

O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
[CrossRef]

Fan, T.

T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron.11, 567–577 (2005).
[CrossRef]

Fedoruk, M.

Fotiadi, A.

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
[CrossRef]

S. Turitsyn, A. Bednyakova, M. Fedoruk, A. Latkin, A. Fotiadi, A. Kurkov, and E. Sholokhov, “Modeling of cw Yb-doped fiber lasers with highly nonlinear cavity dynamics,” Opt. Express19, 8394–8405 (2011).
[CrossRef] [PubMed]

A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

A. Fotiadi, N. G. Zakharov, O. Antipov, and P. Mégret, “All-fiber coherent combining of Er-doped amplifiers through refractive index control in Yb-doped fibers,” Opt. Lett.34, 3574–3576 (2009).
[CrossRef] [PubMed]

A. Fotiadi, O. Antipov, and P. Mégret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express16, 12658–12663 (2008).
[CrossRef] [PubMed]

S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064 nm,” Opt. Express15, 8832–8837 (2007).
[CrossRef] [PubMed]

A. Fotiadi, O. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” in Frontiers in Guided Wave Optics and Optoelectronics, B. Pal, ed. (INTECH, 2010), chap. 11, pp. 209–234.

A. Fotiadi, O. Antipov, I. Bufetov, E. Dianov, and P. Mégret, “Comparative study of pump-induced refractive index changes in aluminum and phosphate silicate Yb-doped fibers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, (Optical Society of America, Washington, D.C., 2009), paper JWA9.
[CrossRef]

A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193–230.
[CrossRef]

Gainov, V.

V. Gainov and O. Ryabushkin, “Effect of optical pumping on the refractive index and temperature in the core of active fibre,” Quantum Electronics41, 809–814 (2011).
[CrossRef]

Heebner, J. E.

Hoffman, H.

D. Brown and H. 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]

Jaeger, J.

H. Carslow and J. Jaeger, Conduction of Heat in Solids (Science, Moscow, 1964).

Jansen, F.

Jauregui, C.

Jones, D.

Jones, D. C.

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett.30, 13–15 (2005).
[CrossRef]

Kinet, D.

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
[CrossRef]

Kracht, D.

Kravtsov, K.

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

Kurkov, A.

Kuznetsov, M.

A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
[CrossRef]

A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193–230.
[CrossRef]

Latkin, A.

Limpert, J.

López-Mercado, C.

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
[CrossRef]

Luikov, A.

A. Luikov, Analytical Heat Diffusion Theory (Academic Press, 1968).

Mangir, M.

Mangir, M. S.

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett.30, 13–15 (2005).
[CrossRef]

Mégret, P.

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
[CrossRef]

A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

A. Fotiadi, N. G. Zakharov, O. Antipov, and P. Mégret, “All-fiber coherent combining of Er-doped amplifiers through refractive index control in Yb-doped fibers,” Opt. Lett.34, 3574–3576 (2009).
[CrossRef] [PubMed]

A. Fotiadi, O. Antipov, and P. Mégret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express16, 12658–12663 (2008).
[CrossRef] [PubMed]

S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064 nm,” Opt. Express15, 8832–8837 (2007).
[CrossRef] [PubMed]

A. Fotiadi, O. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” in Frontiers in Guided Wave Optics and Optoelectronics, B. Pal, ed. (INTECH, 2010), chap. 11, pp. 209–234.

A. Fotiadi, O. Antipov, I. Bufetov, E. Dianov, and P. Mégret, “Comparative study of pump-induced refractive index changes in aluminum and phosphate silicate Yb-doped fibers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, (Optical Society of America, Washington, D.C., 2009), paper JWA9.
[CrossRef]

A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193–230.
[CrossRef]

Melkumov, M.

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

Messerly, M. J.

Minden, M.

Moncorgé, R.

Neumann, J.

Nilsson, J.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B27, 63–92 (2010).
[CrossRef]

Otto, H.-J.

Panajotov, K.

A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

Pantell, R.

M. Davis, M. Digonnet, and R. Pantell, “Thermal effects in doped fibers,” J. Lightwave Tech.16, 1013–1023 (1998).
[CrossRef]

Pantell, R. H.

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: A review,” Opt. Fiber Technol.3, 44–64 (1997).
[CrossRef]

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D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B27, 63–92 (2010).
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V. Gainov and O. Ryabushkin, “Effect of optical pumping on the refractive index and temperature in the core of active fibre,” Quantum Electronics41, 809–814 (2011).
[CrossRef]

Sadowski, R. W.

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: A review,” Opt. Fiber Technol.3, 44–64 (1997).
[CrossRef]

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S. Stepanov and M.P. Sánchez, “Phase population gratings recorded in ytterbium doped fiber at 1064 nm,” in 22nd Congress of the International Commission for Optics: Light for the Development of the World, R. Rodríguez-Vera and R. Díaz-Uribe, eds., Proc. SPIE 8011, 801153 (2011).
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O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
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M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: A review,” Opt. Fiber Technol.3, 44–64 (1997).
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Shubin, A.

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

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Smith, J.

Soulard, R.

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V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
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Sridharan, A. K.

Stappaerts, E. A.

Stepanov, S.

S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064 nm,” Opt. Express15, 8832–8837 (2007).
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S. Stepanov and M.P. Sánchez, “Phase population gratings recorded in ytterbium doped fiber at 1064 nm,” in 22nd Congress of the International Commission for Optics: Light for the Development of the World, R. Rodríguez-Vera and R. Díaz-Uribe, eds., Proc. SPIE 8011, 801153 (2011).
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O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
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Zinoviev, A.

Zolotovskiy, I.

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
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[CrossRef]

O. Antipov, O. Eremeykin, A. Savikin, V. Vorob’ev, D. Bredikhin, and M. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron.39, 910–918 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron.11, 567–577 (2005).
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A. Fotiadi, O. Antipov, M. Kuznetsov, K. Panajotov, and P. Mégret, “Rate equation for the nonlinear phase shift in Yb-doped optical fibers under resonant diode-laser pumping,” J. Hologr. and Speckle5, 1–4 (2009).

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

H. Bruesselbach, S. Wang, M. Minden, D. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B22, 347–353 (2005).
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D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B27, 63–92 (2010).
[CrossRef]

Laser Phys. Lett. (1)

V. Spirin, C. López-Mercado, D. Kinet, P. Mégret, I. Zolotovskiy, and A. Fotiadi, “Single longitudinal-mode brillouin fiber laser passively stabilized at pump resonance frequency with dynamic population inversion grating,” Laser Phys. Lett.10, 015102 (2013).
[CrossRef]

Opt. Express (10)

S. Stepanov, A. Fotiadi, and P. Mégret, “Effective recording of dynamic phase gratings in Yb-doped fibers with saturable absorption at 1064 nm,” Opt. Express15, 8832–8837 (2007).
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A. Fotiadi, O. Antipov, and P. Mégret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express16, 12658–12663 (2008).
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R. Soulard, R. Moncorgé, A. Zinoviev, K. Petermann, O. Antipov, and A. Brignon, “Nonlinear spectroscopic properties of Yb3+-doped sesquioxides Lu2O3 and Sc2O3,” Opt. Express18, 11173–11180 (2010).
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C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers„” Opt. Express19, 3258–3271 (2011).
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S. Turitsyn, A. Bednyakova, M. Fedoruk, A. Latkin, A. Fotiadi, A. Kurkov, and E. Sholokhov, “Modeling of cw Yb-doped fiber lasers with highly nonlinear cavity dynamics,” Opt. Express19, 8394–8405 (2011).
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A. Smith and J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19, 10180–10192 (2011).
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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. Express19, 13218–13224 (2011).
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C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express20, 440–451 (2012).
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H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “Gain dynamics and refractive index changes in fiber amplifiers: a frequency domain approach,” Opt. Express20, 13539–13550 (2012).
[CrossRef] [PubMed]

Opt. Fiber Technol. (1)

M. J. F. Digonnet, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: A review,” Opt. Fiber Technol.3, 44–64 (1997).
[CrossRef]

Opt. Lett. (3)

Quantum Electronics (1)

V. Gainov and O. Ryabushkin, “Effect of optical pumping on the refractive index and temperature in the core of active fibre,” Quantum Electronics41, 809–814 (2011).
[CrossRef]

Other (13)

M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, Handbook for Optics, 2nd ed. (MGH, 1995).

V. Privalko, Handbook for Physical Chemistry of Polymers (Naukova Dumka, Kiev, 1984).

M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, Cross Sections of Absorption and Stimulated Emission of Yb3+ Ions in Silica Fibers Doped with P2O5 and Al2O3 (FORC, Moscow, 2004).

A. Fotiadi, O. Antipov, I. Bufetov, E. Dianov, and P. Mégret, “Comparative study of pump-induced refractive index changes in aluminum and phosphate silicate Yb-doped fibers,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, (Optical Society of America, Washington, D.C., 2009), paper JWA9.
[CrossRef]

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

H.-G. Unger, Planar Optical Waveguides and Fibres (Oxford University Press, 1977).

E. Tugolukov, Solution of Problems of Thermal Conductivity by Finite Integral Transforms: a Tutorial (TSTU, Tambov, 2005).

H. Carslow and J. Jaeger, Conduction of Heat in Solids (Science, Moscow, 1964).

S. Stepanov and M.P. Sánchez, “Phase population gratings recorded in ytterbium doped fiber at 1064 nm,” in 22nd Congress of the International Commission for Optics: Light for the Development of the World, R. Rodríguez-Vera and R. Díaz-Uribe, eds., Proc. SPIE 8011, 801153 (2011).
[CrossRef]

A. Luikov, Analytical Heat Diffusion Theory (Academic Press, 1968).

E. Desurvire, Erbium-doped Fiber Amplifiers: Principles and Applications (Willey, New York, 1994).

A. Fotiadi, O. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” in Frontiers in Guided Wave Optics and Optoelectronics, B. Pal, ed. (INTECH, 2010), chap. 11, pp. 209–234.

A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in Coherent Laser Beam Combining, A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193–230.
[CrossRef]

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Figures (8)

Figure 1
Figure 1

Scheme of the cross-section of the fiber light guide. r1 is the glass radius, r0 is the core radius activated by Yb3+-ions, red part denotes the layer of plastic r1 < r < r2.

Figure 2
Figure 2

The difference of Yb3+-ion polarizabilities in excited and ground states for phosphate silicate fibers (non resonant part in cyan, resonant part in green and total part in blue) and for aluminum-silicate fibers (non resonant part in gray, resonant part in red and total part in orange).

Figure 3
Figure 3

Distribution of refractive index changes (RIC) at λ = 1550nm and z0 = 20cm in the transverse coordinate shown on logarithmic scales. The heat source is assumed to be uniformly distributed in the fiber core (pumping in the clad of 145 mW). Orange curve is the steady-state electronic contribution to RIC, green curve is the thermal contribution to RIC at t = 0.5s, and red curve is the steady-state thermal contribution to RIC.

Figure 4
Figure 4

Time dependence of the refractive index change Δn(t, z0) at 1550 nm for z0 = 20cm in the case of cladding pumping of 145 mW(a) and 100 W (b). The total refractive index change is in black, the electronic refractive index change in orange and the thermal refractive index change in green. Curves are shown for input signal at 1060 nm with powers of 0 mW (solid curves), 10 mW (dashed curves), and 100 mW (dashed-dotted curves).

Figure 5
Figure 5

Time dependence of the phase shift of the probe beam at 1550 nm without signal at 1060 nm and for a pump power of 145 mW. Orange curve is the electronic contribution to the phase shift, green curve is the thermal contribution to the phase shift due to thermo-optic coefficient, blue curve is the thermal contribution to the phase shift associated with the elongation of the fiber, red curve is the thermal contribution to the phase shift associated with the lateral expansion of the fiber, and dashed-black curve is the total phase shift. Axes are plotted on logarithmic scales

Figure 6
Figure 6

Time dependence of the phase shift at 1060 nm for cladding pumping of 100 W. The total phase shift is in black, the electronic phase shift in orange and the thermal phase shift in green. Curves are shown for input signal powers of 0 mW (solid curves), 10 mW (dashed curves), and 100 mW (dashed-dotted curves).

Figure 7
Figure 7

The dependence of the alignment time of total contribution to the the phase shift on the input signal power at 1060 nm for cladding pumping at 145 mW for the curves in orange and blue, and for cladding pumping at 100 W for the curves in red, green and black. The fiber length is equal to 2 m (orange, red and black curves), 20 cm (blue curve), and 10 m (green curve). The heat transfer coefficient η is equal to 0.000118 cal/(cm2 K s) at curves blue, orange, green, and red, and 0.2388 cal/(cm2 K s) at curve black

Figure 8
Figure 8

Time dependence of the phase shift (a) in the amplified rectangular pulse train at 1060 nm, and (b) its zoom corresponding to 20th pulse. 10 W cladding pumping is used and the signal input power is 100 mW. The pulse train period is 10 ms, the pulse duration is 20 μs. Curve in black is the total phase shift, curve is green is the thermal component and curve in orange is the electronic component.

Tables (1)

Tables Icon

Table 1 Parameters values used in calculations

Equations (44)

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Δ φ = 0 l Δ β ( z ) d z = k o 0 l 0 Δ n ( z , r , t ) | G ( r ) | 2 r d r d z 0 | G ( r ) | 2 r d r
G ( r ) = A J 0 ( u r / r 0 ) J 0 ( u ) if r r 0 G ( r ) = A K 0 ( w r / r 0 ) K 0 ( w ) if r > r 0
u 2 = r 0 2 ( n 0 2 k 0 2 β 2 ) ; w 2 = r 0 2 ( β 2 n 1 2 k 0 2 ) , V 2 = u 2 + w 2 = k 0 2 r 0 2 [ n 0 2 n 1 2 ]
u J 1 ( u ) J 0 ( u ) = w K 1 ( w ) K 0 ( w )
δ n T ( z , r , t ) = n T δ T
T t a i 2 2 T = Q ( z , r , t )
Q ( z , r , t ) = α A n r I A + α P n r I P ρ 1 c p 1 + h ν B 1 N 2 ( z , r , t ) ρ 1 c p 1 τ 1 + ν B L N 2 ( z , r , t ) I L σ 21 ( ν L ) ρ 1 c p 1 ν L + ν B 3 { [ σ 21 ( ν A ) + σ 12 ( ν A ) ] N 2 ( z , r , t ) σ 12 ( ν A ) N 0 ( z , r , t ) } I A ρ 1 c p 1 ν A
K 2 T ( r , t ) r | r 2 + η [ T ( r , t ) T air ] | r 2 = 0
δ T ( z , r , t ) = n = 1 1 Z n K 1 a 1 2 J 0 ( μ n r a 1 ) 0 r 0 J 0 ( μ n r a 1 ) 0 t exp [ μ n 2 ( t t ) ] Q ( z , r , t ) d t r d r
Z n = K 1 r 1 2 2 a 1 2 [ J 0 2 ( Ψ n , 1 , 1 ) + J 1 2 ( Ψ n , 1 , 1 ) ] + K 2 a 2 2 { 0.5 r 2 2 C 2 2 ( μ n ) [ J 0 2 ( Ψ n , 2 , 2 ) + J 1 2 ( Ψ n , 2 , 2 ) ] + r 2 2 C 2 ( μ n ) D 2 ( μ n ) [ J 0 ( Ψ n , 2 , 2 ) Y 0 ( Ψ n , 2 , 2 ) + J 1 ( Ψ n , 2 , 2 ) Y 1 ( Ψ n , 2 , 2 ) ] + 0.5 r 2 2 D 2 2 ( μ n ) [ Y 0 2 ( Ψ n , 2 , 2 ) + Y 1 2 ( Ψ n , 2 , 2 ) ] 0.5 r 1 2 C 2 2 ( μ n ) [ J 0 2 ( Ψ n , 1 , 2 ) + J 1 2 ( Ψ n , 1 , 2 ) ] 0.5 r 1 2 C 2 ( μ n ) D 2 ( μ n ) [ J 0 ( Ψ n , 1 , 2 ) Y 0 ( Ψ n , 1 , 2 ) + J 1 ( Ψ n , 1 , 2 ) Y 1 ( Ψ n , 1 , 2 ) ] 0.5 r 1 2 D 2 2 ( μ n ) [ Y 0 2 ( Ψ n , 1 , 2 ) + Y 1 2 ( Ψ n , 1 , 2 ) ] }
C 2 ( μ n ) [ J 0 ( Ψ n , 2 , 2 ) μ n K 2 a 2 η J 1 ( Ψ n , 2 , 2 ) ] + D 2 ( μ n ) [ Y 0 ( Ψ n , 2 , 2 ) μ n K 2 a 2 η Y 1 ( Ψ n , 2 , 2 ) ] = 0
C 2 ( μ n ) = J 0 ( Ψ n , 1 , 1 ) D 2 ( μ n ) Y 0 ( Ψ n , 1 , 2 ) J 0 ( Ψ n , 1 , 2 )
D 2 ( μ n ) = ( K 1 a 2 / K 2 a 1 ) J 0 ( Ψ n , 1 , 2 ) J 1 ( Ψ n , 1 , 1 ) J 1 ( Ψ n , 1 , 2 ) J 0 ( Ψ n , 1 , 1 ) J 0 ( Ψ n , 1 , 2 ) Y 1 ( Ψ n , 1 , 2 ) J 1 ( Ψ n , 1 , 2 ) Y 0 ( Ψ n , 1 , 2 )
δ φ 1 = k 0 n T 0 l 0 δ T ( z , r , t ) | G ( r ) | 2 r d r d z 0 | G ( r ) | 2 r d r
δ φ 2 = 0 l β r 0 r 0 T a v T a v ( z , t ) d z = β r 0 δ T ( 1 + b ) r 0 0 l 2 r 0 2 0 r 0 δ T ( r , z , t ) r d r d z
T a v ( z , t ) = 2 r 0 2 0 r 0 δ T ( r , z , t ) r d r
β = k 0 2 n 0 2 u 2 r 0 2
β r 0 = { J 1 ( u ) K 0 ( w ) w K 1 ( w ) } 2 r 0 2 π ( n 0 n 1 ) / λ [ J 0 2 ( u ) + J 1 2 ( u ) ] r 0 2 2 u 2 + { [ K 0 2 ( w r 1 / r 0 ) K 1 2 ( w r 1 / r 0 ) ] r 1 2 + [ K 1 2 ( w ) K 0 2 ( w ) ] r 0 2 } J 1 2 ( u ) 2 w 2 K 1 2 ( w )
δ φ 3 = δ T β 0 l 2 r 1 2 0 r 1 δ T ( r , z , t ) r d r d z
δ φ 2 < δ φ 3 δ φ 1 δ T 0 r 1 δ T ( r , z , t ) r d r 0 r 1 | G ( r ) | 2 r d r n T r 1 2 0 r 1 δ T ( r , z , t ) | G ( r ) | 2 r d r
δ n N = n N δ N
Δ p ( λ ) = n 0 4 π 3 F L 2 0 λ 2 λ 2 λ 2 [ σ 12 ( λ ) + σ 21 ( λ ) σ esa ( λ ) ] d λ
N ¯ 2 ( z ) = 2 r 0 2 0 r 0 N 2 ( z , r ) r d r
( N ¯ 2 t + N ¯ 2 τ 1 ) S core = σ 12 ( ν p ) ( Γ p 0 N 0 Γ p 2 N ¯ 2 ) P p h ν p σ 21 ( ν p ) Γ p 2 N ¯ 2 P p h ν p σ 21 ( ν L ) Γ L 2 N ¯ 2 P L h ν L + σ 12 ( ν L ) ( Γ L 0 N 0 Γ L 2 N ¯ 2 ) P L h ν L σ 21 ( ν A ) Γ A 2 N ¯ 2 P A h ν A + σ 12 ( ν A ) ( Γ A 0 N 0 Γ A 2 N ¯ 2 ) P A h ν A
P p z = [ ( Γ p 0 N 0 Γ p 2 N ¯ 2 ) σ 12 ( ν p ) Γ p 2 N ¯ 2 σ 21 ( ν p ) ] P p α p n r P p
P L z = Γ L 2 N ¯ 2 σ 21 ( ν L ) P L + N ¯ 2 ζ ( Γ L 0 N 0 Γ L 2 N ¯ 2 ) σ 12 ( ν L ) P L α L n r P L
P A z = Γ A 2 N ¯ 2 σ 21 ( ν A ) P A ( Γ A 0 N 0 Γ A 2 N ¯ 2 ) σ 12 ( ν A ) P A α A n r P A
ζ = h ν L τ 1 S core 4 π ( 2 r 0 l ) 2
Γ p , L , A ; j ( z ) = r 0 2 2 0 r 0 N j ( z , r ) I p , L , A ( r ) r d r 0 I p , L , A ( r ) r d r 0 r 0 N j ( z , r ) r d r
0 l 0 δ n T ( z , r , t m ) | G ( r ) | 2 r d r d z = 0 l 0 δ n N ( z , r , t m ) | G ( r ) | 2 r d r d z
t m = r 2 K 2 ( 1 p ) 2 a 2 2 η ln { 1 4 π F L 2 Δ p τ 1 r 2 η ξ ( 1 + p ) n 0 n T r 0 2 h ν B 1 ( 1 p ) }
p = r 1 2 r 2 2 [ ρ 1 c p 1 ρ 2 c p 2 1 ]
ξ = r 0 2 2 0 l 0 r 0 N 2 ( z , r ) | G ( r ) | 2 r d r d z 0 | G ( r ) | 2 r d r 0 l 0 r 0 N 2 ( z , r ) r d r d z
ε = 4 π F L 2 Δ p τ 1 ξ { n 0 n T r 0 2 h ν B 1 [ 1 η r 2 + ln ( r 2 / r 1 ) K 2 + 1 / 2 + ln ( r 1 / r 0 ) K 1 ] } 1 < 1
η cr = { r 2 [ 4 π F L 2 Δ p τ 1 ξ n 0 r 0 2 h ν B 1 n / T ] ln ( r 2 / r 1 ) K 2 1 / 2 + ln ( r 1 / r 0 ) K 1 } 1
δ φ N = 4 π 2 F L 2 Δ p ξ λ n 0 r 0 2 / 2 0 l 0 N 2 ( z , r ) r d r d z = [ P p ( 0 ) P p ( l ) ] 4 π F L 2 Δ p ξ τ 1 λ n 0 r 0 2 h ν p S core
δ T ( t ) = 2 r 0 r 1 τ 1 Q n = 1 J 1 ( μ n r 0 r 1 ) μ n 3 J 1 2 ( μ n ) S n ( t , τ p , τ 1 )
S n ( t , τ p , τ 1 ) = { [ 1 exp ( μ n 2 t τ 1 ) ] if t τ p [ 1 exp ( μ n 2 τ p τ 1 ) ] exp ( μ n 2 t τ p τ 1 ) if t τ p
δ T ( t ) = 2 r 0 r 1 τ 1 Q n = 1 J 1 ( μ n r 0 r 1 ) μ n 3 J 0 2 ( μ n ) S n ( t , τ p , τ 1 ) + r 0 2 r 1 2 Q U ( t , τ p )
U ( t , τ p ) = { t if t τ p τ p if t τ p
δ T st = { [ K 2 η r 2 + ln r 2 r 1 ] K 1 K 2 + ln r 1 r 0 } 1 a 1 2 0 r 0 r Q ( r ) d r + 1 a 1 2 0 r 0 1 r 0 r r Q ( r ) d r d r
δ T st = Q r 0 2 2 a 1 2 { [ K 2 η r 2 + ln r 2 r 1 ] K 1 K 2 + ln r 1 r 0 + 1 2 }
δ T ( t ) = n = 1 1 Z n K 1 a 1 2 μ n 2 0 r 0 J 0 ( μ n r a 1 ) Q ( r ) r d r S n ( t , τ p , 1 )
δ T ( t ) = n = 1 1 Z n K 1 r 0 a 1 μ n 3 Q J 1 ( μ n r 0 a 1 ) S n ( t , τ p , 1 )

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