Abstract

A theoretical model as well as the experimental verification of the combined guiding mechanism for the transverse mode formation in the end-pumped laser resonator are investigated. The nonlinear Schrödinger-type wave equation in the gain medium is derived, in which the combined guiding mechanism: the thermal induced refractive index guiding effect as well as the gain guiding effect, is taken into account. The gain saturation and spatial hole burning are considered. The split step Fourier method is used to solve the nonlinear wave equation. A high power end-pumped Nd:YVO4 laser resonator is built up. After establishing the pump absorption model of our laser resonator, the temperature distribution in the gain medium is obtained by the numerical solving of the heat diffusion equation. The combined guiding effect is first observed in the end-pumped Nd:YVO4 laser resonator, and the experimental transverse mode profiles well agree with the theoretical prediction from the derived nonlinear Schrödinger-type wave equation. The geometric design criterion of the TEM00 mode laser is compared with our wave theory. The experimental- and theoretical- results show that our wave theory with the combined guiding mechanism dominates the transverse mode formation in high power end-pumped laser resonator.

© 2011 OSA

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    [CrossRef]
  13. A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
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  14. S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
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  15. C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
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  20. T. R. Taha and X. Xiangming, “Parallel split-step Fourier methods for the coupled nonlinear Schrodinger type equations,” J. Supercomput. 32, 5–23 (2005).
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    [CrossRef]
  24. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
    [CrossRef]
  25. H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).
  26. J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
    [CrossRef]
  27. P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002).
    [CrossRef]
  28. B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995).
    [CrossRef]
  29. P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
    [CrossRef]
  30. Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
    [CrossRef]
  31. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001).
    [CrossRef]

2006

Y. L. Bogomolov and A. D. Yunakovsky, “Split-step Fourier method for nonlinear Schrodinger equation,” in International Conference Days on Diffraction 2006 , Proceedings of the International Conference ’Days on Diffraction’ 2006, DD (Inst. of Elec. and Elec. Eng. Computer Society, 2006), 34–42.
[CrossRef]

2005

G. M. Muslu and H. A. Erbay, “Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation,” Math. Comput. Simulat. 67, 581–595 (2005).
[CrossRef]

T. R. Taha and X. Xiangming, “Parallel split-step Fourier methods for the coupled nonlinear Schrodinger type equations,” J. Supercomput. 32, 5–23 (2005).
[CrossRef]

2004

2003

J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, “Gain and thermal guiding effects in diode-pumped lasers,” SPIE 5120, 164(2003)
[CrossRef]

2002

O. Denchev, S. Kurtev, and P. Petrov, “Experimental investigation of saturable gain-guided modes,” Appl. Opt. 41, 1677–1684 (2002).
[CrossRef] [PubMed]

P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002).
[CrossRef]

2001

1999

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

1998

J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
[CrossRef]

1996

D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996).
[CrossRef]

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
[CrossRef]

1995

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995).
[CrossRef]

C. A. Schrama, D. Bouwmeester, G. Nienhuis, and J. P. Woerdman, “Mode dynamics in optical cavities,” Phys. Rev. A 51, 641–645 (1995)
[CrossRef] [PubMed]

1994

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
[CrossRef] [PubMed]

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

1993

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

1992

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992).
[CrossRef]

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992).
[CrossRef] [PubMed]

1991

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

1983

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

1982

B. N. Perry, P. Rabinowitz, and M. Newstein, “Exact solution of the scalar wave equation with focused gaussian gain,” Phys. Rev. Lett. 49, 1921–1924 (1982).
[CrossRef]

1973

H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).

1968

A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968).
[CrossRef]

1966

Albers, P.

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992).
[CrossRef]

Albrecht, G. F.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995).
[CrossRef]

Asundi, A.

P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002).
[CrossRef]

Balmer, P.

H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).

Beach, R. J.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995).
[CrossRef]

Blatte, M.

H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).

Blows, J. L.

J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
[CrossRef]

Bogomolov, Y. L.

Y. L. Bogomolov and A. D. Yunakovsky, “Split-step Fourier method for nonlinear Schrodinger equation,” in International Conference Days on Diffraction 2006 , Proceedings of the International Conference ’Days on Diffraction’ 2006, DD (Inst. of Elec. and Elec. Eng. Computer Society, 2006), 34–42.
[CrossRef]

Boon, J. R.

D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996).
[CrossRef]

Bouwmeester, D.

C. A. Schrama, D. Bouwmeester, G. Nienhuis, and J. P. Woerdman, “Mode dynamics in optical cavities,” Phys. Rev. A 51, 641–645 (1995)
[CrossRef] [PubMed]

Brussard, M.

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

Cerullo, G.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

Chen, Y. F.

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
[CrossRef]

Clarkson, W. A.

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001).
[CrossRef]

Comaskey, B.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995).
[CrossRef]

Conroy, R. S.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996).
[CrossRef]

Danielmeyer, H. G.

H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).

Dawes, J.

J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
[CrossRef]

Denchev, O.

Druten, N. J.

N. J. Druten, S. S. R. Oemrawsingh, Y. Lien, C. Serrat, M. P. van Exter, and J. P. Woerdman, “Observation of transverse modes in a microchip laser with combined gain and index guiding,” J. Opt. Soc. Am. B 18, 1793–1804 (2001).
[CrossRef]

C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Erbay, H. A.

G. M. Muslu and H. A. Erbay, “Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation,” Math. Comput. Simulat. 67, 581–595 (2005).
[CrossRef]

Exter, M. P.

C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Fan, T. Y.

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
[CrossRef] [PubMed]

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

Firth, W. J.

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

Fox, A.

A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968).
[CrossRef]

Frauchiger, J.

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992).
[CrossRef]

Friel, G. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Harkness, G. K.

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

Huang, T. M.

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
[CrossRef]

Jabczynski, J. K.

J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, “Gain and thermal guiding effects in diode-pumped lasers,” SPIE 5120, 164(2003)
[CrossRef]

Kao, C. F.

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
[CrossRef]

Kemp, A. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Koechner, W.

W. Koechner, Solid-tate Laser Engineering , 6th ed. (Springer-Verlag Publications, 2006).

Kogelnik, H.

Kurtev, S.

Kwiatkowski, J.

J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, “Gain and thermal guiding effects in diode-pumped lasers,” SPIE 5120, 164(2003)
[CrossRef]

Laporta, P.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

Lei, X.

P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002).
[CrossRef]

Li, T.

Liao, T. S.

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
[CrossRef]

Lien, Y.

Lin, K. H.

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996).
[CrossRef]

Longhi, S.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

Maes, C. F.

Magni, V.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

Matthews, D. G.

D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996).
[CrossRef]

Moran, B. D.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995).
[CrossRef]

Muslu, G. M.

G. M. Muslu and H. A. Erbay, “Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation,” Math. Comput. Simulat. 67, 581–595 (2005).
[CrossRef]

Newstein, M.

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

B. N. Perry, P. Rabinowitz, and M. Newstein, “Exact solution of the scalar wave equation with focused gaussian gain,” Phys. Rev. Lett. 49, 1921–1924 (1982).
[CrossRef]

Nienhuis, G.

C. A. Schrama, D. Bouwmeester, G. Nienhuis, and J. P. Woerdman, “Mode dynamics in optical cavities,” Phys. Rev. A 51, 641–645 (1995)
[CrossRef] [PubMed]

Oemrawsingh, S. S. R.

Omatsu, T.

J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
[CrossRef]

Pask, H.

J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
[CrossRef]

Perry, B. N.

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

B. N. Perry, P. Rabinowitz, and M. Newstein, “Exact solution of the scalar wave equation with focused gaussian gain,” Phys. Rev. Lett. 49, 1921–1924 (1982).
[CrossRef]

Petrov, P.

Rabinowitz, P.

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

B. N. Perry, P. Rabinowitz, and M. Newstein, “Exact solution of the scalar wave equation with focused gaussian gain,” Phys. Rev. Lett. 49, 1921–1924 (1982).
[CrossRef]

Salin, F.

Schrama, C. A.

C. A. Schrama, D. Bouwmeester, G. Nienhuis, and J. P. Woerdman, “Mode dynamics in optical cavities,” Phys. Rev. A 51, 641–645 (1995)
[CrossRef] [PubMed]

Serrat, C.

N. J. Druten, S. S. R. Oemrawsingh, Y. Lien, C. Serrat, M. P. van Exter, and J. P. Woerdman, “Observation of transverse modes in a microchip laser with combined gain and index guiding,” J. Opt. Soc. Am. B 18, 1793–1804 (2001).
[CrossRef]

C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers , (Univ. Sci. Books, 1986), pp. 323.

Sinclair, B. D.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996).
[CrossRef]

Squier, J.

Taccheo, S.

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N. J. Druten, S. S. R. Oemrawsingh, Y. Lien, C. Serrat, M. P. van Exter, and J. P. Woerdman, “Observation of transverse modes in a microchip laser with combined gain and index guiding,” J. Opt. Soc. Am. B 18, 1793–1804 (2001).
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[CrossRef]

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[CrossRef] [PubMed]

Wright, E. M.

Xiangming, X.

T. R. Taha and X. Xiangming, “Parallel split-step Fourier methods for the coupled nonlinear Schrodinger type equations,” J. Supercomput. 32, 5–23 (2005).
[CrossRef]

Xiaoyuan, P.

P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002).
[CrossRef]

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J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, “Gain and thermal guiding effects in diode-pumped lasers,” SPIE 5120, 164(2003)
[CrossRef]

Appl. Opt.

Appl. Phys. A

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Appl. Phys. Lett.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

IEEE J. Quantum Electron

A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968).
[CrossRef]

IEEE J. Quantum Electron.

C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

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[CrossRef]

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992).
[CrossRef]

P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998).
[CrossRef]

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[CrossRef]

Other

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Figures (12)

Fig. 1
Fig. 1

The sketch of a general laser oscillator. HR: high reflectivity mirror, OC: output coupler, E ±: the forward- and backward- propagating waves respectively, and L 1,2: the cavity length parameters.

Fig. 2
Fig. 2

The periodic cell-train model of the laser resonator. C k : the k–th cell, Jk : the junction between the cell C k and C k +1 while J 0 is the initial reference, E k 1 ± : the forward- and backward- propagating waves for the first pass through of the gain medium in the unfolded model, and E k 2 ± denotes that of the second pass.

Fig. 3
Fig. 3

The sketch for the combined guiding in the laser gain medium. ΔnT : the profile of the refractive index induced by the temperature gradient(the real refractive index), ΔnG : the profile of the gain induced refractive index(the imaginary refractive index).

Fig. 4
Fig. 4

The experimental setup of the dual-end pumped composite Nd:YVO4 laser.

Fig. 5
Fig. 5

The pump absorption model of the gain medium. z 0: the location of the pump waist, ωp (z): the pump beam radius at the location z, and ωp 0: the pump waist radius at the pump waist location of z 0, i.e., ωp 0=ωp (z0). θp : the half far-field divergence angle of the pump beam. I p ± : the forward- and backward- pump beam.

Fig. 6
Fig. 6

3D profile of the temperature distribution of the gain medium pumped at P 0 = 45 W with ωp 0 = 0.4mm, z 0 = 2mm, and α eff =2.0 cm−1.

Fig. 7
Fig. 7

The temperature distribution on the optical axis in the gain medium varies with the effective absorption coefficient α eff increased from 1.0 cm–1 to 6.0 cm−1 with the increment of Δα eff = 0.5 cm−1.

Fig. 8
Fig. 8

3D small signal gain profile of the gain medium pumped at P 0 = 45 W with ωp 0 = 0.4mm, z 0 = 2mm, and α eff =2.0 cm−1.

Fig. 9
Fig. 9

small signal gain (solid line) and the saturated gain with I 0 = I sat exp[ 2 ( x 2 + y 2 ) / ω p 0 2 ] (dash-dot line) and I 0 =4 I sat exp[2 ( x 2 + y 2 ) / ω p 0 2 ] (dash line) for one pass through the gain medium, P 0 = 45 W, z 0 = 2mm, ωp 0 = 0.4mm, and α eff =2.0 cm−1.

Fig. 10
Fig. 10

The transverse mode profile varied with the number of iteration cycle of the Fox-Li iteration.

Fig. 11
Fig. 11

The experimental- and theoretical- transverse mode profiles for α eff increased from 1.5 cm−1 to 7.0 cm−1. Exp. denotes the experimental results(blue solid line), CG denotes the theoretical results with the combined guiding effect considered(red solid line), GG and IG denote theoretical results with only gain guiding effect/thermal induced refraction index guiding effect considered respectively (dash dot line/dash line). Mode i corresponds to the transverse mode with α eff = 1.5 + (i − 1)×0.5 cm−1. P 0 = 45 W, ωp 0 = 0.4mm, z 0 = 2mm.

Fig. 12
Fig. 12

the equivalent pump radius (solid line), the radius of TEM00 mode calculated by the geometrical theory of the diffraction optics (GTDO) (dot line), and the transverse mode size calculated by the nonlinear Schrödinger-type wave equation (NStWE) (dash line). The dash-dot lines show the geometrical TEM00 mode region. P 0 = 45 W, ωp 0 = 0.4mm, z 0 = 2mm.

Equations (45)

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E k 1 = E k 1 2 + E k 2 = E k 1 + ,
( 2 ɛ μ 2 t 2 ) E ( x , y , z , ω , t ) = μ 2 t 2 P ( x , y , z , ω , t ) ,
E ( x , y , z , ω , t ) = 1 2 e ^ [ E ( x , y , z ) e i ( k z ω t ) + c . c . ] = 1 2 e ^ [ E ( x , y , z , ω ) e i ω t + c . c . ] ,
P ( x , y , z , ω , t ) = 1 2 e ^ [ P ( x , y , z , ω ) e i ω t + c . c . ] ,
( 2 + k 0 2 ) E ( x , y , z , ω ) = μ ω 2 2 t 2 P ( x , y , z , ω ) ,
E ( x , y , z , ω ) E ( x , y , z , ω ) e ik 0 z n b ( z ) d z .
P ( x , y , z , ω ) ɛ 0 χ ( x , y , z , ω ) E ( x , y , z , ω ) e ik 0 z n b ( z ) d z ,
n 2 ( x , y , z , ω ) = 1 + χ ( x , y , z , ω ) .
n ( x , y , z , ω ) = n b ( z ) + Δ n ( x , y , z , ω ) ,
χ ( x , y , z , ω ) n b 2 ( z ) + 2 n b ( z ) Δ n ( x , y , z , ω ) 1.
[ 2 + 2 ik 0 n b z + k 0 2 k 0 2 n b 2 ] E ( x , y , z , ω ) k 0 2 [ n b 2 + 2 n b Δ n ( x , y , z , ω ) 1 ] E ( x , y , z , ω ) .
| k 0 2 n b 2 E | | 2 k 0 n b E z | | 2 E z 2 | ,
[ 2 + 2 ik 0 n b z + 2 k 0 2 n b Δ n ( x , y , z , ω ) ] E ( x , y , z , ω ) = 0.
Δ n ( x , y , z , ω ) = Δ n T ( x , y , z ) + Δ n G ( x , y , z , ω ) ,
Δ n T ( x , y , z ) = d n d T ( T ( x , y , z ) T r ) ,
[ K x 2 x 2 + K y 2 y 2 + K z 2 z 2 ] T ( x , y , z ) = η h P abs ( x , y , z ) ,
Δ n G ( x , y , z , ω ) = G ( x , y , z , ω ) 2 k i Δ ω / Ω 1 + ( Δ ω / Ω ) 2 ,
Δ n G ( x , y , z ) = i G ( x , y , z , ω c ) 2 k .
G ( x , y , z ) = G 0 ( x , y , z , ω c ) 1 + I ( x , y , z , ω c ) I sat ,
G 0 ( x , y , z ) = P abs ( x , y , z ) λ p σ 21 τ f h c ,
I sat = ( W r + γ ) h c λ l σ 21 τ f ,
G ( x , y , z ) = P abs ( x , y , z ) h c λ p σ 21 τ f + λ l c ɛ 0 ɛ r 2 λ p | E + ( x , y , z ) + E ( x , y , z ) | 2 .
z E ± ( x , y , z ) = [ i 2 k 0 n b 2 + ik 0 Δ n T ( x , y , z ) + G ( x , y , z ) 2 ] E ± ( x , y , z ) ,
i ψ z = ( D ^ + N ^ ) ψ ,
D ^ = 2 2 k 0 n b = 1 2 k 0 n b ( 2 x 2 + 2 y 2 ) ,
N ^ = k 0 Δ n T ( x , y , z ) + i G ( x , y , z ) 2 .
ψ ( x , y , z + Δ z ) = e i ( D ^ + N ^ ) Δ z ψ ( x , y , z ) .
ψ ( x , y , z + Δ z ) e i ( D ^ Δ z ) e i ( N ^ Δ z ) ψ ( x , y , z ) .
ψ ( x , y , z + Δ z ) e i ( D ^ Δ z ) ψ ( x , y , z ) .
exp ( i D ^ Δ z ) ψ ( x , y , z ) = exp ( i D ^ Δ z ) exp [ i ( ω x x + ω x y ) ] ψ ˜ ( ω x , ω y , z ) d ω x d ω y ,
exp ( i D ^ Δ z ) exp [ i ( ω x x + ω x y ) ] = { n 1 n ! [ i Δ z 2 k 0 n b ( 2 x 2 ) ] n exp ( i ω x x ) } { n 1 n ! [ i Δ z 2 k 0 n b ( 2 y 2 ) ] n exp ( i ω y y ) } = [ exp ( i ω x x ) n 1 n ! ( i Δ z 2 k 0 n b ω x 2 ) n ] [ exp ( i ω y y ) n 1 n ! ( i Δ z 2 k 0 n b ω y 2 ) n ] = exp [ i Δ z 2 k 0 n b ( ω x 2 + ω y 2 ) ] exp [ i ( ω x x + ω x y ) ] ,
ψ ( x , y , z + Δ z ) = exp [ i Δ z 2 k 0 n b ( ω x 2 + ω y 2 ) ] exp [ i ( ω x x + ω x y ) ] ψ ˜ ( ω x , ω y , z ) d ω x d ω y = 1 { [ ψ ( x , y , z ) ] exp [ i Δ z 2 k 0 n b ( ω x 2 + ω y 2 ) ] } .
ψ ( x , y , z + Δ z ) e i ( N ^ Δ z ) ψ ( x , y , z ) .
ψ ( x , y , z + Δ z ) exp [ i z z + Δ z N ^ ( x , y , z ) d z ] ψ ( x , y , z ) .
z z + Δ z N ^ ( x , y , z ) d z Δ z 2 [ N ^ ( x , y , z ) + N ^ ( x , y , z + Δ z ) ] .
exp [ i ( D ^ + N ^ ) Δ z ] exp [ i D ^ Δ z / 2 ] exp [ i N ^ Δ z ] exp [ i D ^ Δ z / 2 ] .
I P + ( x , y , z ) = C 0 π ω p 2 ( z ) exp [ 2 ( x 2 + y 2 ) N ω p N ( z ) ] exp ( α eff z ) ,
C 0 = P 0 z = 0 1 π ω p 2 ( z ) exp [ 2 ( x 2 + y 2 ) N ω p N ( z ) ] exp ( α eff z ) d x d y ,
ω p ( z ) = ω p 0 1 + [ θ p ( z z 0 ) ω p 0 ] 2 ,
I p ( x , y , z ) = I P + ( x , y , z ) + I P ( x , y , z ) ,
P abs ( x , y , z ) = I p ( x , y , z ) z α eff I P ( x , y , z ) .
T ( x , y , z ) x | x = ± w 2 = h c K x [ T c T ( ± w 2 , y , z ) ] T ( x , y , z ) y | y = ± h 2 = h c K y [ T c T ( x , ± h 2 , z ) ] T ( x , y , z ) z | z = ± l 2 = h a K z [ T r T ( x , y , ± l 2 ) ] ,
ω equ = P abs ( x , y , z ) ω eff ( z ) d x d y d z P abs ( x , y , z ) d x d y d z ,
x 2 + y 2 ω eff 2 ( z ) P abs ( x , y , z ) d x d y P abs ( x , y , z ) d x d y = 1 1 e 2 0.865.
1 < ω equ ω TEM 00 < 1.73.

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