Abstract

Amplified spontaneous emission (ASE) is important to power scaling of the large-scale, high-gain thin disk laser. In this paper, spectral properties of ASE in Yb:YAG thin disk lasers are deeply studied in both theory and experiment. The experimental results show that the ASE strength is much stronger when emitted from the edge surface than the pumping area. And the spectrum of ASE emitted from the coarsened edge surface is angle independent. Meanwhile, the reabsorption effect in the Yb:YAG crystal on spectral linewidth is analyzed and corrected. Finally, ASE spectral linewidths have been measured. We demonstrate that the spectral linewidths can evaluate ASE strength effectively.

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

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References

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2018 (2)

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

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]

2017 (1)

2016 (2)

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

2015 (3)

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

Y. Huang, X. Zhu, G. Zhu, J. Shang, H. Wang, L. Qi, C. Zhu, and F. Guo, “A multi-pass pumping scheme for thin disk lasers with good anti-disturbance ability,” Opt. Express 23(4), 4605–4613 (2015).
[Crossref] [PubMed]

2013 (3)

2012 (3)

H. Su and X. Wang, “Amplified spontaneous emission in thin-disk lasers,” Proc. SPIE 8235, 82351Q (2012).
[Crossref]

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

J. Koerner, C. Vorholt, H. Liebetrau, M. Kahle, D. Kloepfel, R. Seifert, J. Hein, and M. C. Kaluza, “Measurement of temperature-dependent absorption and emission spectra of Yb:YAG, Yb:LuAG, and Yb:CaF2 between 20 °C and 200 °C and predictions on their influence on laser performance,” J. Opt. Soc. Am. B 29(9), 2493–2502 (2012).
[Crossref]

2011 (2)

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

D. C. Brown and V. A. Vitali, “Yb:YAG Kinetics Model Including Saturation and Power Conservation,” IEEE J. Quantum Electron. 47(1), 3–12 (2011).
[Crossref]

2010 (1)

E. M. Calzado, P. G. Boj, and M. A. Díaz-García, “Amplified Spontaneous Emission Properties of Semiconducting Organic Materials,” Int. J. Mol. Sci. 11(6), 2546–2565 (2010).
[Crossref] [PubMed]

2009 (4)

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

J. Speiser, “Thin disk laser — energy scaling,” Laser Phys. 19(2), 274–280 (2009).
[Crossref]

J. Speiser, “Scaling of thin-disk lasers—influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009).
[Crossref]

2007 (1)

A. Giesen and J. Speiser, “Fifteen Years of Work on Thin-Disk Lasers: Results and Scaling Laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

2004 (1)

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

2003 (1)

J. Dong and P. Deng, “The effect of Cr concentration on emission cross-section and fluorescence lifetime in Cr,Yb:YAG crystal,” J. Lumin. 104, 151–158 (2003).
[Crossref]

1995 (1)

1991 (1)

Ackermann, M.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

Adachi, C.

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

Aggarwal, R. L.

Amaro, F. D.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Antognini, A.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Bauer, D.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

Bencheikh, F.

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

Biraben, F.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Bisson, J.-F.

D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

Boj, P. G.

E. M. Calzado, P. G. Boj, and M. A. Díaz-García, “Amplified Spontaneous Emission Properties of Semiconducting Organic Materials,” Int. J. Mol. Sci. 11(6), 2546–2565 (2010).
[Crossref] [PubMed]

Brauch, U.

Brown, D. C.

D. C. Brown and V. A. Vitali, “Yb:YAG Kinetics Model Including Saturation and Power Conservation,” IEEE J. Quantum Electron. 47(1), 3–12 (2011).
[Crossref]

Calzado, E. M.

E. M. Calzado, P. G. Boj, and M. A. Díaz-García, “Amplified Spontaneous Emission Properties of Semiconducting Organic Materials,” Int. J. Mol. Sci. 11(6), 2546–2565 (2010).
[Crossref] [PubMed]

Carson, T.

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

Chen, Y.

Choi, H. K.

Chosrowjan, H.

Copeland, D. A.

D. A. Copeland, “Amplified spontaneous emission (ASE) models and approximations for thin-disk laser modeling,” Proc. SPIE 8599, 85991P (2013).
[Crossref]

Dax, A.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

de Lima Filho, E. S.

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Deng, P.

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

J. Dong and P. Deng, “The effect of Cr concentration on emission cross-section and fluorescence lifetime in Cr,Yb:YAG crystal,” J. Lumin. 104, 151–158 (2003).
[Crossref]

Díaz-García, M. A.

E. M. Calzado, P. G. Boj, and M. A. Díaz-García, “Amplified Spontaneous Emission Properties of Semiconducting Organic Materials,” Int. J. Mol. Sci. 11(6), 2546–2565 (2010).
[Crossref] [PubMed]

Dong, J.

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]

J. Dong and P. Deng, “The effect of Cr concentration on emission cross-section and fluorescence lifetime in Cr,Yb:YAG crystal,” J. Lumin. 104, 151–158 (2003).
[Crossref]

Fan, T. Y.

Fujita, M.

Fukunaga, T.

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

Furuse, H.

Gavrielides, A.

Giesen, A.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

A. Giesen and J. Speiser, “Fifteen Years of Work on Thin-Disk Lasers: Results and Scaling Laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

U. Brauch, A. Giesen, M. Karszewski, C. Stewen, and A. Voss, “Multiwatt diode-pumped Yb:YAG thin disk laser continuously tunable between 1018 and 1053 nm,” Opt. Lett. 20(7), 713–715 (1995).
[Crossref] [PubMed]

Gottwald, T.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

Graf, T.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Gu, B.

Guo, F.

Hänsch, T. W.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Hein, J.

Huang, Y.

Indelicato, P.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Izawa, Y.

Julien, L.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Kahle, M.

Kaluza, M. C.

Kao, C.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Karszewski, M.

Kashyap, R.

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Kawanaka, J.

Killi, A.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

Kloepfel, D.

Knowles, P. E.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Koerner, J.

Kottmann, F.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Kouznetsov, D.

D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

Krishnaiah, K. V.

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Kuhn, V.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

Lacovara, P.

Latham, W. P.

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Le Bigot, E.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Ledemi, Y.

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Liebetrau, H.

Lin, Z.

Liu, Y.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Lucas, T.

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

Ludhova, L.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Mamada, M.

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

Messaddeq, Y.

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Miyanaga, N.

Moschüring, N.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Mulhauser, F.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Nebel, T.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Nemova, G.

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Newell, T. C.

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Nez, F.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Peterson, P.

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Pohl, R.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Qi, L.

Qiao, Y.

Rabinowitz, P.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Ryba, T.

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

Sandanayaka, A. S. D.

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

Schad, S. S.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

Scharun, M.

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

Schuhmann, K.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Schwob, C.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Seifert, R.

Shang, J.

Y. Huang, X. Zhu, G. Zhu, J. Shang, H. Wang, L. Qi, C. Zhu, and F. Guo, “A multi-pass pumping scheme for thin disk lasers with good anti-disturbance ability,” Opt. Express 23(4), 4605–4613 (2015).
[Crossref] [PubMed]

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

Song, P.

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

Speiser, J.

J. Speiser, “Scaling of thin-disk lasers—influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009).
[Crossref]

J. Speiser, “Thin disk laser — energy scaling,” Laser Phys. 19(2), 274–280 (2009).
[Crossref]

A. Giesen and J. Speiser, “Fifteen Years of Work on Thin-Disk Lasers: Results and Scaling Laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

Stewen, C.

Stolzenburg, C.

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

Su, H.

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

H. Su and X. Wang, “Amplified spontaneous emission in thin-disk lasers,” Proc. SPIE 8235, 82351Q (2012).
[Crossref]

Tang, C.

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

Taqqu, D.

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

Ueda, K.

D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

Vitali, V. A.

D. C. Brown and V. A. Vitali, “Yb:YAG Kinetics Model Including Saturation and Power Conservation,” IEEE J. Quantum Electron. 47(1), 3–12 (2011).
[Crossref]

Vorholt, C.

Voss, A.

Vretenar, N.

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Wang, C. A.

Wang, H.

Wang, M.

Wang, X.

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

H. Su and X. Wang, “Amplified spontaneous emission in thin-disk lasers,” Proc. SPIE 8235, 82351Q (2012).
[Crossref]

Xu, J.

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

Xu, X.

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

Yu, Y.

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

Zhao, W.

Zhao, Z.

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

Zhu, C.

Zhu, G.

Zhu, X.

Adv. Funct. Mater. (1)

M. Mamada, T. Fukunaga, F. Bencheikh, A. S. D. Sandanayaka, and C. Adachi, “Low amplified spontaneous emission threshold from organic dyes based on Bis-stilbene,” Adv. Funct. Mater. 28(32), 1802130 (2018).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

D. C. Brown and V. A. Vitali, “Yb:YAG Kinetics Model Including Saturation and Power Conservation,” IEEE J. Quantum Electron. 47(1), 3–12 (2011).
[Crossref]

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

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

A. Giesen and J. Speiser, “Fifteen Years of Work on Thin-Disk Lasers: Results and Scaling Laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

Int. J. Mol. Sci. (1)

E. M. Calzado, P. G. Boj, and M. A. Díaz-García, “Amplified Spontaneous Emission Properties of Semiconducting Organic Materials,” Int. J. Mol. Sci. 11(6), 2546–2565 (2010).
[Crossref] [PubMed]

J. Alloys Compd. (1)

X. Xu, Z. Zhao, P. Song, J. Xu, and P. Deng, “Growth of high-quality single crystal of 50 at.% Yb:YAG and its spectral properties,” J. Alloys Compd. 364, 311–314 (2004).
[Crossref]

J. Lumin. (1)

J. Dong and P. Deng, “The effect of Cr concentration on emission cross-section and fluorescence lifetime in Cr,Yb:YAG crystal,” J. Lumin. 104, 151–158 (2003).
[Crossref]

J. Opt. (1)

H. Su, X. Wang, J. Shang, Y. Yu, and C. Tang, “A coupling model for amplified spontaneous emission in laser resonators,” J. Opt. 17(10), 105801 (2015).
[Crossref]

J. Opt. Soc. Am. B (3)

Laser Phys. (1)

J. Speiser, “Thin disk laser — energy scaling,” Laser Phys. 19(2), 274–280 (2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. Mater. (1)

D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

Proc. SPIE (5)

V. Kuhn, T. Gottwald, C. Stolzenburg, S. S. Schad, A. Killi, and T. Ryba, “Latest advances in high brightness disk lasers,” Proc. SPIE 9342, 93420Y (2015).
[Crossref]

S. S. Schad, T. Gottwald, V. Kuhn, M. Ackermann, D. Bauer, M. Scharun, and A. Killi, “Recent development of disk lasers at TRUMPF,” Proc. SPIE 9726, 972615 (2016).
[Crossref]

D. A. Copeland, “Amplified spontaneous emission (ASE) models and approximations for thin-disk laser modeling,” Proc. SPIE 8599, 85991P (2013).
[Crossref]

H. Su and X. Wang, “Amplified spontaneous emission in thin-disk lasers,” Proc. SPIE 8235, 82351Q (2012).
[Crossref]

N. Vretenar, T. Carson, T. C. Newell, T. Lucas, W. P. Latham, and P. Peterson, “Yb:YAG Thin-Disk Laser Performance at Room and Cryogenic Temperatures,” Proc. SPIE 8235, 82350S (2012).
[Crossref]

Sci. Rep. (1)

K. V. Krishnaiah, E. S. de Lima Filho, Y. Ledemi, G. Nemova, Y. Messaddeq, and R. Kashyap, “Development of ytterbium-doped oxyfluoride glasses for laser cooling applications,” Sci. Rep. 6(1), 21905 (2016).
[Crossref] [PubMed]

Other (1)

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice Hall, 1995), Chap. 8.

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Figures (15)

Fig. 1
Fig. 1 Yb:YAG energy level diagram at 300K.
Fig. 2
Fig. 2 Iteration diagram of solving ASE flux by dichotomy.
Fig. 3
Fig. 3 (a) The normalized ASE spectrum in different effective gain length at Ip = 3kW/cm2. (b) The spectral linewidth and the effective lifetime as a function of effective gain length at Ip = 3kW/cm2.
Fig. 4
Fig. 4 (a) The normalized ASE spectrum in different pump power density at S = 4cm. (b) The spectral linewidth and the effective lifetime as a function of pump power density at S = 4cm.
Fig. 5
Fig. 5 The relation between the ASE spectral linewidth and the effective lifetime at S = 4cm or Ip = 3kW/cm2.
Fig. 6
Fig. 6 A schematic diagram of ASE photons propagation path in a thin disk crystal.
Fig. 7
Fig. 7 The fluorescence distribution of the Yb:YAG thin disk pumped by a 940nm laser diode.
Fig. 8
Fig. 8 The effective gain length of the photon emitted from different positions in different angles.
Fig. 9
Fig. 9 The experimental scheme for measuring the ASE spectrum and gain coefficients.
Fig. 10
Fig. 10 The spectrum of the photons emitted at different pump powers from the pump area (a) and edge surface (b).
Fig. 11
Fig. 11 The spectra measured from the edge surface at pump powers of 57W (a) and 400W (b) at different measurement angles. (c) The spectral linewidths of the measured from the edge surface at different pump powers and different measurement angles.
Fig. 12
Fig. 12 (a) The gain coefficients at different pump power. (b) The spectra of the photons emitted from the pump area at low pump power. (c) The spectra of the photons emitted from the edge surface at the low pump power. (d) The linewidth ratios of the spectral linewidth measured in edge surface and the pump area below the threshold of the ASE effect.
Fig. 13
Fig. 13 Corrected spectral linewidth at different pump powers.
Fig. 14
Fig. 14 The iteration diagram of calculating the effective lifetime by the corrected spectral linewidth.
Fig. 15
Fig. 15 The effective lifetime calculated by the gain coefficient and spectral linewidth, respectively.

Tables (3)

Tables Icon

Table 1 The fluorescence branch ratio of Yb:YAG

Tables Icon

Table 2 The effective emission cross section of Yb:YAG

Tables Icon

Table 3 Parameters in simulation

Equations (22)

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f 1i = exp( hc E 1i kT ) p=1 3 exp( hc E 1p kT ) , f 0j = exp( hc E 0j kT ) p=1 4 exp( hc E 0p kT ) ,
L( ν, ν ij )= Δν 2π ( ν ν ij ) 2 + ( Δν/2 ) 2 ,
{ g s ( ν )= g s ( ν ) [ g s ( ν )δν ] g s ( ν )= i=1 3 j=1 4 f 1i β 0j 1i j=1 4 β 0j 1i L( ν, ν ij ) ,
Φ SE ( ν )= N 2 dV 4π r 2 τ 0 g s ( ν )δν,
{ d Φ ASE ( ν )= N 2 4π r 2 τ 0 g s ( ν )δνexp[ G( ν )r ]dV G( ν )= i=1 3 j=1 4 [ G ij L( ν, ν ij )δν ] ,
G ij =( N 2 f 1i N 1 f 0j ) σ ij spe ,
Φ ASE ( ν )= 0 S N 2 τ 0 g s ( ν )δνexp[ G( ν )r ]dr = N 2 τ 0 exp[ G( ν )S ]1 G( ν ) g s ( ν )δν.
Φ={ Φ SE ( ν ), S=0 exp[ G( ν )S ]1 G( ν )S Φ SE ( ν ),S>0 .
Δ n L t =( f 1i + f 1j )( I pabs Lh ν p Δ n L σ ij spe I L h ν L N 2 τ 0 [ G( ν ) Φ ASE ( ν ) ] ),
τ eff 1 = τ 0 1 + [ G( ν ) Φ ASE ( ν ) ] N 2 .
I pabs Lh ν p N 2 τ eff =0.
Φ upper = N 2 2 τ 0 0 α c 0 L/( 2cosθ ) exp( γr )drsinθdθ + N 2 2 τ 0 π α c π 0 3L/( 2cosθ ) exp( γr )drsinθdθ ,
Φ edge = N 2 2 τ 0 α c Θ 1 0 3L/( 2cosθ ) N p ( θ ) exp( γr )drsinθdθ + N 2 2 τ 0 Θ 1 π/2 0 R 2 +9 L 2 /4 N p ( θ ) exp( γr )drsinθdθ + N 2 2 τ 0 π/2 πΘ 0 R 2 + L 2 /4 N p ( θ ) exp( γr )drsinθdθ + N 2 2 τ 0 πΘ π α c 0 L/( 2cosθ ) N p ( θ ) exp( γr )drsinθdθ ,
Φ= N 2 2 τ 0 exp( γr )drsinθdθ = N 2 τ 0 exp( γS )1 γ .
S edge = 1 γ ln[ γ 2 α c π/2 exp( γr )drsinθdθ π/2 α c +1 ],
S total = 1 γ ln[ γ 2 0π/2 exp( γr )drsinθdθ π/2 +1 ].
d[ I ASE ( ν,z )dν ] dz =α( ν ) I ASE ( ν,z )dν+ Δ Ω z 4π A 21 n 2 h ν 0 g s ( ν, ν 0 )dν,
Δ Ω z S pump L 2 + R 2 ,
I ASE ( ν,z )= I ASE ( ν,0 )exp( α 0 ( ν ) Z abs ),
α 0 ( ν )= α 0 ( ν 0 ) ( Δ ν H 2 ) 2 ( ν ν 0 ) 2 + ( Δ ν H 2 ) 2 ,
Δ ν Re =Δ ν H α 0 ( ν 0 ) Z abs ln exp( α 0 ( ν 0 ) Z abs )+1 2 1 .
η Re = Δ ν H Δ ν Re = Δ λ Re Δ λ H =constant,

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