Abstract

A radiation-balanced Yb:YAG disk laser is demonstrated in an intracavity pumping geometry. Detailed analysis of the data reveals the feasibility of using the multi-kilowatt level “athermal” disk lasers with minimal modal instabilities, which arise from thermal lensing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
    [Crossref]
  2. Y.-C. Jeong, A. J. Boyland, J. K. Sahu, S.-H. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Korea 13(4), 416–422 (2009).
    [Crossref]
  3. S.-S. Schad, V. Kuhn, T. Gottwald, V. Negoita, A. Killi, and K. Wallmeroth, “Near fundamental mode high-power thin-disk laser,” SPIE Proc. 8959, 89590U (2014).
  4. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
    [Crossref] [PubMed]
  5. K. Schuhmann, K. Kirch, F. Nez, R. Pohl, and A. Antognini, “Thin-disk laser scaling limit due to thermal lens induced misalignment instability,” Appl. Opt. 55(32), 9022–9032 (2016).
    [Crossref] [PubMed]
  6. A. Diebold, F. Saltarelli, I. J. Graumann, C. J. Saraceno, C. R. Phillips, and U. Keller, “Gas-lens effect in kW-class thin-disk lasers,” Opt. Express 26(10), 12648–12659 (2018).
    [Crossref] [PubMed]
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    [Crossref]
  8. S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35(1), 115–122 (1999).
    [Crossref]
  9. S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
    [Crossref]
  10. S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
    [Crossref]
  11. S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. 41(12), 1510–1517 (2005).
    [Crossref]
  12. G. Nemova and R. Kashyap, “Thin-disk athermal laser system,” Opt. Commun. 319, 100–105 (2014).
    [Crossref]
  13. W. Zhao, G. Zhu, Y. Chen, B. Gu, M. Wang, and J. Dong, “Numerical analysis of a multi-pass pumping Yb:YAG thick-disk laser with minimal heat generation,” Appl. Opt. 57(18), 5141–5149 (2018).
    [Crossref] [PubMed]
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  15. K. Schuhmann, T. W. Hänsch, K. Kirch, A. Knecht, F. Kottmann, F. Nez, R. Pohl, D. Taqqu, and A. Antognini, “Thin-disk laser pump schemes for large number of passes and moderate pump source quality,” Appl. Opt. 54(32), 9400–9408 (2015).
    [Crossref] [PubMed]
  16. M. Ghasemkhani, A. R. Albrecht, S. D. Melgaard, D. V. Seletskiy, J. G. Cederberg, and M. Sheik-Bahae, “Intra-cavity cryogenic optical refrigeration using high power vertical external-cavity surface-emitting lasers (VECSELs),” Opt. Express 22(13), 16232–16240 (2014).
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  19. S. R. Bowman, “Optimizing average power in low quantum defect lasers,” Appl. Opt. 54(31), F78–F84 (2015).
    [Crossref] [PubMed]
  20. D. S. Sumida and T. Y. Fan, “Effect of radiation trapping on fluorescence lifetime and emission cross section measurements in solid-state laser media,” Opt. Lett. 19(17), 1343–1345 (1994).
    [Crossref] [PubMed]
  21. D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986).
    [Crossref]
  22. D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
    [Crossref]
  23. S. R. Bowman and C. E. Mungan, “New materials for optical cooling,” Appl. Phys. B 71(6), 807–811 (2000).
    [Crossref]
  24. C. Vorholt and U. Wittrock, “Intra-cavity pumped Yb:YAG thin-disk laser with 1.74% quantum defect,” Opt. Lett. 40(20), 4819–4822 (2015).
    [Crossref] [PubMed]
  25. Z. Yang, A. R. Albrecht, J. Meng, and M. Sheik-Bahae, “Investigation of radiation balanced disk lasers,” Proc. SPIE 10550, 1055010 (2018).

2018 (3)

2016 (1)

2015 (3)

2014 (3)

2011 (1)

2010 (1)

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

2009 (1)

2006 (1)

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

2005 (2)

S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. 41(12), 1510–1517 (2005).
[Crossref]

D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
[Crossref]

2000 (2)

S. R. Bowman and C. E. Mungan, “New materials for optical cooling,” Appl. Phys. B 71(6), 807–811 (2000).
[Crossref]

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

1999 (1)

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35(1), 115–122 (1999).
[Crossref]

1995 (1)

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

1994 (2)

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

D. S. Sumida and T. Y. Fan, “Effect of radiation trapping on fluorescence lifetime and emission cross section measurements in solid-state laser media,” Opt. Lett. 19(17), 1343–1345 (1994).
[Crossref] [PubMed]

1992 (1)

1986 (1)

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986).
[Crossref]

Albrecht, A. R.

Antognini, A.

Asmerom, Y.

Balembois, F.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

Biswal, S.

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. 41(12), 1510–1517 (2005).
[Crossref]

Bowman, S. R.

S. R. Bowman, “Optimizing average power in low quantum defect lasers,” Appl. Opt. 54(31), F78–F84 (2015).
[Crossref] [PubMed]

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. 41(12), 1510–1517 (2005).
[Crossref]

S. R. Bowman and C. E. Mungan, “New materials for optical cooling,” Appl. Phys. B 71(6), 807–811 (2000).
[Crossref]

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35(1), 115–122 (1999).
[Crossref]

Boyland, A. J.

Brauch, U.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

Brown, D. C.

D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
[Crossref]

Buchwald, M. I.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

Cederberg, J. G.

Chen, Y.

Chénais, S.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

Chung, S.-H.

Condon, N. J.

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

Cone, R. L.

D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
[Crossref]

Contag, K.

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

Diebold, A.

Dong, J.

Druon, F.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

Edwards, B. C.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

Eggleston, J. M.

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986).
[Crossref]

Eidam, T.

Epstein, R. I.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

Equall, R. W.

D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
[Crossref]

Erhard, S.

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

Fan, T. Y.

Forget, S.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

Georges, P.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

Ghasemkhani, M.

Giesen, A.

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

Gosnell, T. R.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

Graumann, I. J.

Gu, B.

Hänsch, T. W.

Hügel, H.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

Jansen, F.

Jauregui, C.

Jeong, Y.-C.

Karszewski, M.

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

Kashyap, R.

G. Nemova and R. Kashyap, “Thin-disk athermal laser system,” Opt. Commun. 319, 100–105 (2014).
[Crossref]

Keller, U.

Kirch, K.

Knecht, A.

Kottmann, F.

Limpert, J.

Lowenthal, D. D.

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986).
[Crossref]

Melgaard, S.

Melgaard, S. D.

Meng, J.

Z. Yang, A. R. Albrecht, J. Meng, and M. Sheik-Bahae, “Investigation of radiation balanced disk lasers,” Proc. SPIE 10550, 1055010 (2018).

Mungan, C. E.

S. R. Bowman and C. E. Mungan, “New materials for optical cooling,” Appl. Phys. B 71(6), 807–811 (2000).
[Crossref]

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

Nemova, G.

G. Nemova and R. Kashyap, “Thin-disk athermal laser system,” Opt. Commun. 319, 100–105 (2014).
[Crossref]

Nez, F.

Nilsson, J.

O’Connor, S. P.

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. 41(12), 1510–1517 (2005).
[Crossref]

Opower, H.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

Otto, H. J.

Payne, D. N.

Phillips, C. R.

Pohl, R.

Polyak, V.

Rosenberg, A.

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

Sahu, J. K.

Salin, F.

Saltarelli, F.

Saraceno, C. J.

Schmidt, O.

Schreiber, T.

Schuhmann, K.

Seletskiy, D.

Seletskiy, D. V.

Sheik-Bahae, M.

Squier, J.

Stewen, C.

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

Stutzki, F.

Sumida, D. S.

Taqqu, D.

Tünnermann, A.

Vorholt, C.

Voss, A.

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

Wang, M.

Wirth, C.

Wittig, K.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

Wittrock, U.

Yang, Z.

Z. Yang, A. R. Albrecht, J. Meng, and M. Sheik-Bahae, “Investigation of radiation balanced disk lasers,” Proc. SPIE 10550, 1055010 (2018).

Yongchen Sun,

D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
[Crossref]

Zhao, W.

Zhu, G.

Appl. Opt. (4)

Appl. Phys. B (2)

S. R. Bowman and C. E. Mungan, “New materials for optical cooling,” Appl. Phys. B 71(6), 807–811 (2000).
[Crossref]

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58(5), 365 (1994).
[Crossref]

IEEE J. Quantum Electron. (4)

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35(1), 115–122 (1999).
[Crossref]

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46(7), 1076–1085 (2010).
[Crossref]

S. R. Bowman, S. P. O’Connor, and S. Biswal, “Ytterbium laser with reduced thermal loading,” IEEE J. Quantum Electron. 41(12), 1510–1517 (2005).
[Crossref]

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986).
[Crossref]

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

D. C. Brown, R. L. Cone, Yongchen Sun, and R. W. Equall, “Yb:YAG absorption at ambient and cryogenic temperatures,” IEEE J. Sel. Top. Quantum Electron. 11(3), 604–612 (2005).
[Crossref]

in ASSL. OSA TOPS (1)

S. Erhard, M. Karszewski, C. Stewen, A. Giesen, K. Contag, and A. Voss, “Pumping schemes for multi-kW thin disk lasers,” in ASSL. OSA TOPS 34, 78–84 (2000).

J. Opt. Soc. Korea (1)

Nature (1)

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescence cooling of a solid,” Nature 377(6549), 500–503 (1995).
[Crossref]

Opt. Commun. (1)

G. Nemova and R. Kashyap, “Thin-disk athermal laser system,” Opt. Commun. 319, 100–105 (2014).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Proc. SPIE (1)

Z. Yang, A. R. Albrecht, J. Meng, and M. Sheik-Bahae, “Investigation of radiation balanced disk lasers,” Proc. SPIE 10550, 1055010 (2018).

Prog. Quantum Electron. (1)

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state-lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
[Crossref]

Other (1)

S.-S. Schad, V. Kuhn, T. Gottwald, V. Negoita, A. Killi, and K. Wallmeroth, “Near fundamental mode high-power thin-disk laser,” SPIE Proc. 8959, 89590U (2014).

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Figures (6)

Fig. 1
Fig. 1 Energy diagram of Yb:YAG. λf, λP, and λL represent the mean fluorescence wavelength, the wavelength of pump and laser photon, respectively.
Fig. 2
Fig. 2 (a) Measurement (circles) and fitting (solid curve) of the laser cooling efficiency ηc of the Yb:YAG disk at room temperature. The blue-shaded area represents the cooling regime. (b) Absorption α 0 ( λ ) (dotted curve) and  α( λ )=γ(λ,  I P =20  kW/c m 2 , I L =0) (solid curve) spectra of 5% Yb:YAG. The small signal gain spectrum is calculated with 20 kW/cm2 incident pump intensity at 1030 nm. The yellow-shaded area represents the regime with optical gain.
Fig. 3
Fig. 3 Schematic of the intracavity-pumped radiation balanced disk laser setup. On the right, the mounting of the Yb:YAG disk is shown where it is glued onto two bare fibers, which are in turn supported by a glass slide to reduce the heat load.
Fig. 4
Fig. 4 Side-scattered fluorescence spectrum from the Yb:YAG disk laser. The zero-crossing wavelength at 1021 nm along with the scattered VECSEL pump at 1030 nm and the laser line at 1050 nm are also shown. Inset is the measured beam profile at the RBL condition.
Fig. 5
Fig. 5 (a) and (b) are the thermal images of the mounted gain disk at room temperature and radiation balancing condition after 30 minutes, with darker shades representing lower temperatures. The red lines in (a) represent the outline of the disk. (c) The line-integrated time-evolution of the temperature change along the vertical white dash line in (a) after the VECSEL cavity is unblocked at t = 1 minute. The black dashed lines represent the pump beam position. The estimated Gaussian profiles of the pump (green) and laser (red) beams are depicted on the right.
Fig. 6
Fig. 6 The five data points represent the measured intracavity Yb:YAG disk laser powers at λL = 1050 nm versus the intracavity VECSEL pump power at λP = 1030 nm. Open-squares are in the self-cooling regime (H<0), solid-square is in the heating regime (H>0) and the solid-circle is the RBL point (H∼0). The theoretical curves given by Eqs. (6) and (7) show an excellent agreement with the data; indicating the RBL point at the crossing of the two curves. For comparison, the RBL condition for plane-waves (or top-hat beams) as given by Eq. (5), is also shown.

Equations (7)

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γ(λ)= α 0 (λ) i P ( β P β L 1)1 1+ i P + i L ,
γ( λ L )= T 2 + l i 2d ,
i L =(1+1/θ)[ i P i P th 1 ],
Q=( β P β L ) N T h( ν P ν L ) τ 2 i L i P i L min i P i L i P min 1+ i P + i L ,
i P min i P + i L min i L =1,
Hπd Q(r)d r 2 =0 .
r( λ L ) i L (r)d r 2 / i L (r)d r 2 =( T 2 + l i ) /2d.

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