Abstract

An end-pumped Nd:YLF laser was demonstrated, which delivered 60.3 W continuous-wave and more than 52 W Q-switched average power for all repetition rates from 5 to 30 kHz. To achieve this, an analytical solution to estimate and optimize the unsaturated gain in an end-pumped laser gain medium was derived. The approach presented here should open up the route for scaling end-pumped lasers to even higher power and energy levels.

© 2010 OSA

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2009 (2)

2008 (3)

D. Li, Z. Ma, R. Haas, A. Schell, P. Zhu, P. Shi, and K. Du, “Diode-end-pumped double Nd:YLF slab laser with high energy, short pulse width, and diffraction-limited quality,” Opt. Lett. 33(15), 1708–1710 (2008).
[CrossRef] [PubMed]

N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

2007 (1)

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

2006 (4)

2005 (3)

2001 (1)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D Appl. Phys. 34(16), 2381–2395 (2001).
[CrossRef]

1999 (3)

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

P. J. Hardman, W. A. Clarkson, G. J. Friel, M. Pollnau, and D. C. Hanna, “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals,” IEEE J. Quantum Electron. 35(4), 647–655 (1999).
[CrossRef]

J. D. Zuegel and W. Seka, “Upconversion and reduced 4F3/2 upper-state lifetime in intensely pumped Nd:YLF,” Appl. Opt. 38(12), 2714–2723 (1999).
[CrossRef]

1998 (3)

W. A. Clarkson, P. J. Hardman, and D. C. Hanna, “High-power diode-bar end-pumped Nd:YLF laser at 1.053 microm,” Opt. Lett. 23(17), 1363–1365 (1998).
[CrossRef]

M. Pollnau, P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “Upconversion, lifetime quenching, and ground-state bleaching in Nd3+:LiYF4,” Opt. Commun. 147(1-3), 203–211 (1998).
[CrossRef]

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(12), 1727–1729 (1998).
[CrossRef]

1992 (2)

R. Lavi, S. Jackel, S. Tsadka, O. Levi, and R. Lallouz, “Comparison between Nd:YAG and Nd:YLF laser oscillators, end pumped by high brightness diode laser arrays,” Proc. SPIE 1971, 326–336 (1992).
[CrossRef]

J. R. Ryan and R. Beach, “Optical absorption and stimulated emission of neodymium in yttrium lithium fluoride,” J. Opt. Soc. Am. B 9(10), 1883–1887 (1992).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[CrossRef]

Asundi, A.

Bagnoud, V.

Baldochi, S. L.

N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

Balmer, J. E.

Beach, R.

Bernhardi, E. H.

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

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(12), 1727–1729 (1998).
[CrossRef]

Bollig, C.

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

Botha, L. R.

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

Byer, R. L.

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[CrossRef]

Camargo, F. A.

N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

Chen, H.

Clarkson, W. A.

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D Appl. Phys. 34(16), 2381–2395 (2001).
[CrossRef]

P. J. Hardman, W. A. Clarkson, G. J. Friel, M. Pollnau, and D. C. Hanna, “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals,” IEEE J. Quantum Electron. 35(4), 647–655 (1999).
[CrossRef]

M. Pollnau, P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “Upconversion, lifetime quenching, and ground-state bleaching in Nd3+:LiYF4,” Opt. Commun. 147(1-3), 203–211 (1998).
[CrossRef]

W. A. Clarkson, P. J. Hardman, and D. C. Hanna, “High-power diode-bar end-pumped Nd:YLF laser at 1.053 microm,” Opt. Lett. 23(17), 1363–1365 (1998).
[CrossRef]

Coluccelli, N.

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(12), 1727–1729 (1998).
[CrossRef]

Dergachev, A.

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

Diart, R.

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

Du, K.

D. Li, Z. Ma, R. Haas, A. Schell, P. Zhu, P. Shi, and K. Du, “Diode-end-pumped double Nd:YLF slab laser with high energy, short pulse width, and diffraction-limited quality,” Opt. Lett. 33(15), 1708–1710 (2008).
[CrossRef] [PubMed]

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

Dumas, P.

Esser, M. J. D.

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

Fan, T. Y.

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid-state lasers,” IEEE J. Quantum Electron. 24(6), 895–912 (1988).
[CrossRef]

Flint, J. H.

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

Forbes, A.

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

Fornasiero, L.

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

Friel, G. J.

P. J. Hardman, W. A. Clarkson, G. J. Friel, M. Pollnau, and D. C. Hanna, “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals,” IEEE J. Quantum Electron. 35(4), 647–655 (1999).
[CrossRef]

Fu, X.

Galzerano, G.

Gong, M.

Guardalben, M. J.

Haas, C. R.

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

Haas, R.

Hanna, D. C.

P. J. Hardman, W. A. Clarkson, G. J. Friel, M. Pollnau, and D. C. Hanna, “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals,” IEEE J. Quantum Electron. 35(4), 647–655 (1999).
[CrossRef]

M. Pollnau, P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “Upconversion, lifetime quenching, and ground-state bleaching in Nd3+:LiYF4,” Opt. Commun. 147(1-3), 203–211 (1998).
[CrossRef]

W. A. Clarkson, P. J. Hardman, and D. C. Hanna, “High-power diode-bar end-pumped Nd:YLF laser at 1.053 microm,” Opt. Lett. 23(17), 1363–1365 (1998).
[CrossRef]

Hardman, P. J.

P. J. Hardman, W. A. Clarkson, G. J. Friel, M. Pollnau, and D. C. Hanna, “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals,” IEEE J. Quantum Electron. 35(4), 647–655 (1999).
[CrossRef]

M. Pollnau, P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “Upconversion, lifetime quenching, and ground-state bleaching in Nd3+:LiYF4,” Opt. Commun. 147(1-3), 203–211 (1998).
[CrossRef]

W. A. Clarkson, P. J. Hardman, and D. C. Hanna, “High-power diode-bar end-pumped Nd:YLF laser at 1.053 microm,” Opt. Lett. 23(17), 1363–1365 (1998).
[CrossRef]

Huber, G.

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

Isyanova, Y.

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

Jackel, S.

R. Lavi, S. Jackel, S. Tsadka, O. Levi, and R. Lallouz, “Comparison between Nd:YAG and Nd:YLF laser oscillators, end pumped by high brightness diode laser arrays,” Proc. SPIE 1971, 326–336 (1992).
[CrossRef]

Jacobs, C.

E. H. Bernhardi, C. Bollig, M. J. D. Esser, A. Forbes, L. R. Botha, and C. Jacobs, “A single-element plane-wave solid-state laser rate equation model,” S. Afr. J. Sci. 104, 389–393 (2008).

Kellner, T.

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

Knappe, R.

Kück, S.

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

Lallouz, R.

R. Lavi, S. Jackel, S. Tsadka, O. Levi, and R. Lallouz, “Comparison between Nd:YAG and Nd:YLF laser oscillators, end pumped by high brightness diode laser arrays,” Proc. SPIE 1971, 326–336 (1992).
[CrossRef]

Laporta, P.

Lavi, R.

R. Lavi, S. Jackel, S. Tsadka, O. Levi, and R. Lallouz, “Comparison between Nd:YAG and Nd:YLF laser oscillators, end pumped by high brightness diode laser arrays,” Proc. SPIE 1971, 326–336 (1992).
[CrossRef]

Levi, O.

R. Lavi, S. Jackel, S. Tsadka, O. Levi, and R. Lallouz, “Comparison between Nd:YAG and Nd:YLF laser oscillators, end pumped by high brightness diode laser arrays,” Proc. SPIE 1971, 326–336 (1992).
[CrossRef]

Li, D.

D. Li, Z. Ma, R. Haas, A. Schell, P. Zhu, P. Shi, and K. Du, “Diode-end-pumped double Nd:YLF slab laser with high energy, short pulse width, and diffraction-limited quality,” Opt. Lett. 33(15), 1708–1710 (2008).
[CrossRef] [PubMed]

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

Liu, Q.

Ma, Z.

McDonagh, L.

Meyn, J. P.

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

Möbert, P. E. A.

L. Fornasiero, T. Kellner, S. Kück, J. P. Meyn, P. E. A. Möbert, and G. Huber, “Excited state absorption and stimulated emission of Nd3+ in crystals III: LaSc3(BO3)4, CaWO4, andYLiF4,” Appl. Phys. B 68(1), 67–72 (1999).
[CrossRef]

Mooney, T.

Moulton, P. F.

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

Nebel, A.

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(12), 1727–1729 (1998).
[CrossRef]

Parisi, D.

Paschotta, R.

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(12), 1727–1729 (1998).
[CrossRef]

Pati, B.

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

Peng, X.

Pollnau, M.

P. J. Hardman, W. A. Clarkson, G. J. Friel, M. Pollnau, and D. C. Hanna, “Energy-transfer upconversion and thermal lensing in high-power end-pumped Nd:YLF laser crystals,” IEEE J. Quantum Electron. 35(4), 647–655 (1999).
[CrossRef]

M. Pollnau, P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “Upconversion, lifetime quenching, and ground-state bleaching in Nd3+:LiYF4,” Opt. Commun. 147(1-3), 203–211 (1998).
[CrossRef]

Puth, J.

Ranieri, I. M.

N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

Ryan, J. R.

Schell, A.

Seka, W.

Shell, A.

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

Shi, P.

D. Li, Z. Ma, R. Haas, A. Schell, P. Zhu, P. Shi, and K. Du, “Diode-end-pumped double Nd:YLF slab laser with high energy, short pulse width, and diffraction-limited quality,” Opt. Lett. 33(15), 1708–1710 (2008).
[CrossRef] [PubMed]

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

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A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

Sousa, E. C.

N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

Tateda, M.

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(12), 1727–1729 (1998).
[CrossRef]

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[CrossRef]

Vanherzeele, H.

Wall, K. F.

A. Dergachev, J. H. Flint, Y. Isyanova, B. Pati, E. V. Slobodtchikov, K. F. Wall, and P. F. Moulton, “Review of Multipass Slab Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 13(3), 647–660 (2007).
[CrossRef]

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N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

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H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

Zhu, P.

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Appl. Opt. (4)

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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(12), 1727–1729 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

N. U. Wetter, E. C. Sousa, F. A. Camargo, I. M. Ranieri, and S. L. Baldochi, “Efficient and compact diode-side-pumped Nd:YLF laser operating at 1053 nm with high beam quality,” J. Opt. A, Pure Appl. Opt. 10(10), 104013 (2008).
[CrossRef]

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[CrossRef]

H. Zhang, D. Li, P. Shi, R. Diart, A. Shell, C. R. Haas, and K. Du, “Efficient, high power, Q-switched Nd:YLF slab laser end-pumped by diode stack,” Opt. Commun. 250(1-3), 157–162 (2005).
[CrossRef]

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Proc. SPIE (1)

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Figures (8)

Fig. 1
Fig. 1

Theoretical Nd:YLF unsaturated round-trip gain G vs pump and laser beam radius w for 135 W of absorbed pump power, assuming that the pump and laser beams are of equal sizes and that no diffraction takes place in the laser material. Upconversion and ASE are neglected (see text).

Fig. 2
Fig. 2

Solid line: Theoretical minimum single-pass Q-switch loss required to prevent lasing on the σ-polarisation under full pump power (135W absorbed). Dotted line: Theoretical threshold pump power as percentage of full pump power versus pump and laser beam radius w for an output coupler transmission of 20%.

Fig. 3
Fig. 3

Simplified diagram of the pump scheme which generates a bell-shaped transverse intensity distribution in the waist of the pump beam. Also shown are the transverse intensity distributions at the different positions along the pump beam. See text for details.

Fig. 4
Fig. 4

Pump setup to pump two Nd:YLF crystals.

Fig. 5
Fig. 5

Resonator setup for the CW and the Q-switched experiments.

Fig. 6
Fig. 6

Output power of the CW laser versus pump power from the laser diode. The red dots are measured experimental values, the solid black line is a linear regression for output powers above 10 W and the dashed line shows the theoretical values for ηp = 0.75 according to [26].

Fig. 7
Fig. 7

Beam profile of the CW laser at full power (60.3 W output), detected with a 1/2” LaserCam IIID beam diagnostics camera from Coherent (6.4 x 4.8 mm2 CCD sensor with 736 x 484 pixels resolution).

Fig. 8
Fig. 8

Average power (red, left y-axis) and pulse energy (blue, right y-axis) vs. pulse repetition rate. The dots are measured experimental values while the dashed line represents calculated theoretical values and the solid line numerical simulation results from a plane-wave rate equation model.

Equations (9)

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G r = exp ( 2 σ Δ N L ) ,
G r = exp ( 2 σ L s 0 ( r , z ) Δ N ( r , z ) d V ) .
G r = exp ( 2 σ e L η p P a τ h ν p s 0 ( r , z ) r ( r , z ) d V ) ,
s 0 ( r , z ) r ( r , z ) d V = 2 π ( w 0 2 + w p 2 ) L ,
G r = exp ( 4 P a π ( w 0 2 + w p 2 ) K ) ,
K = h ν p σ e η p τ ,
G r = exp ( 2 P a π w 2 K ) for round-trip gain ,
G s = exp ( P a π w 2 K ) for single-pass gain .
E = P c w τ ( 1 e 1 f τ ) ξ , with ξ = ( n n f n n t h ) ,

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