Abstract

We numerically investigate the power and energy scaling potential of cryogenic Yb:YLF regenerative amplifiers in rod geometry. Our approach is based on solving the coupled set of equations describing thermal behavior of the material and its effect on spectroscopic properties, gain, and overall amplification. The approach is first benchmarked with earlier experimental data. By carefully analyzing the sensitivity of the system to operation parameters, we see that the relatively low gain nature of the Yb:YLF and the onset of thermal effects are the main factors that limited the performance in earlier experimental work. We show that usage of dual-rod geometry promises much improved performance. Specifically, we demonstrate that sub-250 fs pulses with an average power of up to 270 W and a peak power above 500 GW can be extracted directly from a single-stage Yb:YLF regenerative amplifier employing dual Yb:YLF rods. We further show that by adjusting the spot size in the regenerative amplifier, one can operate the amplifier in either high-energy mode (${\gt}{100}\;{\rm mJ}$ at 1 kHz) or high-average-power mode (${\gt}{25}\;{\rm mJ}$ at 10 kHz, with ${\gt}{250}\;{\rm W}$). We also discuss pros and cons of Yb:YLF with respect to Yb:YAG, and underline the need for measurement of population and photo-elastic-effect-induced lensing in Yb:YLF to obtain a better understanding of Yb:YLF systems. The findings presented in this work can be used for the design and development of next-generation high-average and peak-power Yb:YLF amplifier systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (2)

2019 (3)

2018 (5)

D. F. Zhang, A. Fallahi, M. Hemmer, X. J. Wu, M. Fakhari, Y. Hua, H. Cankaya, A. L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kartner, “Segmented terahertz electron accelerator and manipulator (STEAM),” Nat. Photonics 12, 336–342 (2018).
[Crossref]

I. Tamer, S. Keppler, M. Hornung, J. Korner, J. Hein, and M. C. Kaluza, “Spatio-temporal characterization of pump-induced wavefront aberrations in Yb3+-doped materials,” Laser Photon. Rev. 12, 1700211 (2018).
[Crossref]

Y. Hua, W. Liu, M. Hemmer, L. E. Zapata, G. J. Zhou, D. N. Schimpf, T. Eidam, J. Limpert, A. Tunnermann, F. X. Kartner, and G. Q. Chang, “87-W 1018-nm Yb-fiber ultrafast seeding source for cryogenic Yb: yttrium lithium fluoride amplifier,” Opt. Lett. 43, 1686–1689 (2018).
[Crossref]

J. Manni, D. Harris, and T. Y. Fan, “High-gain (43 dB), high-power (40 W), highly efficient multipass amplifier at 995 nm in Yb:LiYF4,” Opt. Commun. 417, 54–56 (2018).
[Crossref]

C. J. Saraceno, “Mode-locked thin-disk lasers and their potential application for high-power terahertz generation,” J. Opt. 20, 4 (2018).
[Crossref]

2017 (3)

T. Nubbemeyer, M. Kaumanns, M. Ueffing, M. Gorjan, A. Alismail, H. Fattahi, J. Brons, O. Pronin, H. G. Barros, Z. Major, T. Metzger, D. Sutter, and F. Krausz, “1 kW, 200 mJ picosecond thin-disk laser system,” Opt. Lett. 42, 1381–1384 (2017).
[Crossref]

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

H. Hu, C. Xu, Y. Zhao, K. J. Ziegler, and J. N. Chung, “Boiling and quenching heat transfer advancement by nanoscale surface modification,” Sci. Rep. 7, 1 (2017).
[Crossref]

2016 (5)

2015 (2)

Z. L. Zhang, Q. Liu, M. M. Nie, E. C. Ji, and M. L. Gong, “Experimental and theoretical study of the weak and asymmetrical thermal lens effect of Nd:YLF crystal for sigma and pi polarizations,” Appl. Phys. B 120, 689–696 (2015).
[Crossref]

L. E. Zapata, H. Lin, A. L. Calendron, H. Cankaya, M. Hemmer, F. Reichert, W. R. Huang, E. Granados, K. H. Hong, and F. X. Kartner, “Cryogenic Yb:YAG composite-thin-disk for high energy and average power amplifiers,” Opt. Lett. 40, 2610–2613 (2015).
[Crossref]

2014 (1)

2013 (6)

U. Demirbas, “Modelling and optimization of tapered-diode pumped Cr:LiCAF regenerative amplifiers,” Opt. Commun. 311, 90–99 (2013).
[Crossref]

X. Delen, Y. Zaouter, I. Martial, N. Aubry, J. Didierjean, C. Honninger, E. Mottay, F. Balembois, and P. Georges, “Yb:YAG single crystal fiber power amplifier for femtosecond sources,” Opt. Lett. 38, 109–111 (2013).
[Crossref]

D. E. Miller, J. R. Ochoa, and T. Y. Fan, “Cryogenically cooled, 149 W, Q-switched, Yb:LiYF4 laser,” Opt. Lett. 38, 4260–4261 (2013).
[Crossref]

X. Fu, K. H. Hong, L. J. Chen, and F. X. Kartner, “Performance scaling of high-power picosecond cryogenically cooled rod-type Yb:YAG multipass amplification,” J. Opt. Soc. Am. B 30, 2798–2809 (2013).
[Crossref]

C. Kruse, T. Anderson, C. Wilson, C. Zuhlke, D. Alexander, G. Gogos, and S. Ndao, “Extraordinary shifts of the Leidenfrost temperature from multiscale micro/nanostructured surfaces,” Langmuir 29, 9798–9806 (2013).
[Crossref]

O. Slezak, A. Lucianetti, M. Divoky, M. Sawicka, and T. Mocek, “Optimization of wavefront distortions and thermal-stress induced birefringence in a cryogenically-cooled multislab laser amplifier,” IEEE J. Quantum Electron. 49, 960–966 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (2)

V. V. Zelenogorskii and E. A. Khazanov, “Influence of the photoelastic effect on the thermal lens in a YLF crystal,” Quantum Electron. 40, 40–44 (2010).
[Crossref]

C. J. M. Lasance, “How to estimate heat spreading effects in practice,” J. Electron. Packag. 132, 031004 (2010).
[Crossref]

2007 (2)

M. Vannini, G. Toci, D. Alderighi, D. Parisi, F. Cornacchia, and M. Tonelli, “High efficiency room temperature laser emission in heavily doped Yb : YLF,” Opt. Express 15, 7994–8002 (2007).
[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, 598–609 (2007).
[Crossref]

2006 (3)

A. Sugiyama, M. Katsurayama, Y. Anzai, and T. Tsuboi, “Spectroscopic properties of Yb doped YLF grown by a vertical Bridgman method,” J. Alloys Comp. 408, 780–783 (2006).
[Crossref]

A. Sennaroglu, U. Demirbas, S. Ozharar, and F. Yaman, “Accurate determination of saturation parameters for Cr4+-doped solid-state saturable absorbers,” J. Opt. Soc. Am. B 23, 241–249 (2006).
[Crossref]

A. Isemann, P. Wessels, and C. Fallnich, “Directly diode-pumped Colquiriite regenerative amplifiers,” Opt. Commun. 260, 211–222 (2006).
[Crossref]

2005 (2)

R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAIO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80–300 K temperature range,” J. Appl. Phys. 98, 103514 (2005).
[Crossref]

J. Kawanaka, S. Tokita, H. Nishioka, M. Fujita, K. Yamakawa, K. Ueda, and Y. Izawa, “Dramatically improved laser characteristics of diode-pumped Yb-doped materials at low temperature,” Laser Phys. 15, 1306–1312 (2005).

2004 (1)

A. Bensalah, Y. Guyot, M. Ito, A. Brenier, H. Sato, T. Fukuda, and G. Boulon, “Growth of Yb3+-doped YLiF4 laser crystal by the Czochralski method. Attempt of Yb3+ energy level assignment and estimation of the laser potentiality,” Opt. Mater. 26, 375–383 (2004).
[Crossref]

2003 (1)

2002 (1)

2000 (1)

1998 (1)

S. Kim and H. Ledbetter, “Low-temperature elastic coefficients of polycrystalline indium,” Mater. Sci. Eng. A 252, 139–143 (1998).
[Crossref]

1997 (2)

V. Pilla, P. R. Impinnisi, and T. Catunda, “Measurement of saturation intensities in ion doped solids by transient nonlinear refraction,” Appl. Phys. Lett. 70, 817–819 (1997).
[Crossref]

H. W. Bruesselbach, D. S. Sumida, R. A. Reeder, and R. W. Byren, “Low-heat high-power scaling using InGaAs-diode-pumped Yb:YAG lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 105–116 (1997).
[Crossref]

1996 (1)

N. Uehara, K. Ueda, and Y. Kubota, “Spectroscopic measurements of a high-concentration Yb3+:LiYF4 crystal,” Jpn. J. Appl. Phys. 35, L499–L501 (1996).
[Crossref]

1994 (1)

A. Giesen, H. Hugel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
[Crossref]

1993 (1)

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

1989 (1)

1981 (1)

M. Kida, Y. Kikuchi, O. Takahashi, and I. Michiyoshi, “Pool-boiling heat-transfer in liquid-nitrogen,” J. Nucl. Sci. Technol. 18, 501–513 (1981).
[Crossref]

1963 (1)

L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[Crossref]

1962 (1)

R. W. Powell, M. J. Woodman, and R. P. Tye, “Thermal conductivity and electrical resistivity of indium,” Philos. Mag. 7, 1183–1186 (1962).
[Crossref]

Aggarwal, R. L.

R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAIO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80–300 K temperature range,” J. Appl. Phys. 98, 103514 (2005).
[Crossref]

Agounoun, R.

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

Alderighi, D.

Alexander, D.

C. Kruse, T. Anderson, C. Wilson, C. Zuhlke, D. Alexander, G. Gogos, and S. Ndao, “Extraordinary shifts of the Leidenfrost temperature from multiscale micro/nanostructured surfaces,” Langmuir 29, 9798–9806 (2013).
[Crossref]

Alismail, A.

Anderson, T.

C. Kruse, T. Anderson, C. Wilson, C. Zuhlke, D. Alexander, G. Gogos, and S. Ndao, “Extraordinary shifts of the Leidenfrost temperature from multiscale micro/nanostructured surfaces,” Langmuir 29, 9798–9806 (2013).
[Crossref]

Anzai, Y.

A. Sugiyama, M. Katsurayama, Y. Anzai, and T. Tsuboi, “Spectroscopic properties of Yb doped YLF grown by a vertical Bridgman method,” J. Alloys Comp. 408, 780–783 (2006).
[Crossref]

Aubry, N.

Avizonis, P. V.

Balembois, F.

Barros, H. G.

Baudouy, B.

B. Baudouy, “Heat transfer and cooling techniques at low temperature,” arXiv: 1501.07153 (2015).

Bauer, D.

Beach, R. J.

Bensalah, A.

A. Bensalah, Y. Guyot, M. Ito, A. Brenier, H. Sato, T. Fukuda, and G. Boulon, “Growth of Yb3+-doped YLiF4 laser crystal by the Czochralski method. Attempt of Yb3+ energy level assignment and estimation of the laser potentiality,” Opt. Mater. 26, 375–383 (2004).
[Crossref]

Bouazaoui, K.

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J. Manni, D. Harris, and T. Y. Fan, “High-gain (43 dB), high-power (40 W), highly efficient multipass amplifier at 995 nm in Yb:LiYF4,” Opt. Commun. 417, 54–56 (2018).
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Z. L. Zhang, Q. Liu, M. M. Nie, E. C. Ji, and M. L. Gong, “Experimental and theoretical study of the weak and asymmetrical thermal lens effect of Nd:YLF crystal for sigma and pi polarizations,” Appl. Phys. B 120, 689–696 (2015).
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Ochoa, J. R.

D. E. Miller, J. R. Ochoa, and T. Y. Fan, “Cryogenically cooled, 149 W, Q-switched, Yb:LiYF4 laser,” Opt. Lett. 38, 4260–4261 (2013).
[Crossref]

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Pergament, M.

Pervak, V.

Pilla, V.

V. Pilla, P. R. Impinnisi, and T. Catunda, “Measurement of saturation intensities in ion doped solids by transient nonlinear refraction,” Appl. Phys. Lett. 70, 817–819 (1997).
[Crossref]

Powell, R. W.

R. W. Powell, M. J. Woodman, and R. P. Tye, “Thermal conductivity and electrical resistivity of indium,” Philos. Mag. 7, 1183–1186 (1962).
[Crossref]

Pronin, O.

Rahmoune, M.

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

Rand, D.

Reeder, R. A.

H. W. Bruesselbach, D. S. Sumida, R. A. Reeder, and R. W. Byren, “Low-heat high-power scaling using InGaAs-diode-pumped Yb:YAG lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 105–116 (1997).
[Crossref]

Reichert, F.

Ripin, D. J.

D. E. Miller, L. E. Zapata, D. J. Ripin, and T. Y. Fan, “Sub-picosecond pulses at 100 W average power from a Yb:YLF chirped-pulse amplification system,” Opt. Lett. 37, 2700–2702 (2012).
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D. Rand, D. Miller, D. J. Ripin, and T. Y. Fan, “Cryogenic Yb3+-doped materials for pulsed solid-state laser applications [Invited],” Opt. Mater. Express 1, 434–450 (2011).
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R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAIO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80–300 K temperature range,” J. Appl. Phys. 98, 103514 (2005).
[Crossref]

Saadani, R.

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

Sanjeewa, L.

D. C. Brown, S. Tornegard, J. Kolis, C. McMillen, C. Moore, L. Sanjeewa, and C. Hancock, “The application of cryogenic laser physics to the development of high average power ultra-short pulse lasers,” Appl. Sci. 6, 23 (2016).
[Crossref]

Saraceno, C. J.

C. J. Saraceno, “Mode-locked thin-disk lasers and their potential application for high-power terahertz generation,” J. Opt. 20, 4 (2018).
[Crossref]

Sato, H.

A. Bensalah, Y. Guyot, M. Ito, A. Brenier, H. Sato, T. Fukuda, and G. Boulon, “Growth of Yb3+-doped YLiF4 laser crystal by the Czochralski method. Attempt of Yb3+ energy level assignment and estimation of the laser potentiality,” Opt. Mater. 26, 375–383 (2004).
[Crossref]

Sawicka, M.

O. Slezak, A. Lucianetti, M. Divoky, M. Sawicka, and T. Mocek, “Optimization of wavefront distortions and thermal-stress induced birefringence in a cryogenically-cooled multislab laser amplifier,” IEEE J. Quantum Electron. 49, 960–966 (2013).
[Crossref]

Sbai, K.

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

Schimpf, D. N.

Seletskiy, D. V.

Sennaroglu, A.

Sheik-Bahae, M.

Skidmore, J. A.

Slezak, O.

O. Slezak, A. Lucianetti, M. Divoky, M. Sawicka, and T. Mocek, “Optimization of wavefront distortions and thermal-stress induced birefringence in a cryogenically-cooled multislab laser amplifier,” IEEE J. Quantum Electron. 49, 960–966 (2013).
[Crossref]

Speiser, J.

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598–609 (2007).
[Crossref]

Sugiyama, A.

A. Sugiyama, M. Katsurayama, Y. Anzai, and T. Tsuboi, “Spectroscopic properties of Yb doped YLF grown by a vertical Bridgman method,” J. Alloys Comp. 408, 780–783 (2006).
[Crossref]

Sumida, D. S.

H. W. Bruesselbach, D. S. Sumida, R. A. Reeder, and R. W. Byren, “Low-heat high-power scaling using InGaAs-diode-pumped Yb:YAG lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 105–116 (1997).
[Crossref]

Sutter, D.

Sutton, S. B.

Takahashi, O.

M. Kida, Y. Kikuchi, O. Takahashi, and I. Michiyoshi, “Pool-boiling heat-transfer in liquid-nitrogen,” J. Nucl. Sci. Technol. 18, 501–513 (1981).
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Tamer, I.

I. Tamer, S. Keppler, M. Hornung, J. Korner, J. Hein, and M. C. Kaluza, “Spatio-temporal characterization of pump-induced wavefront aberrations in Yb3+-doped materials,” Laser Photon. Rev. 12, 1700211 (2018).
[Crossref]

Thesinga, J.

Toci, G.

Tokita, S.

J. Kawanaka, S. Tokita, H. Nishioka, M. Fujita, K. Yamakawa, K. Ueda, and Y. Izawa, “Dramatically improved laser characteristics of diode-pumped Yb-doped materials at low temperature,” Laser Phys. 15, 1306–1312 (2005).

Tonelli, M.

Tornegard, S.

D. C. Brown, S. Tornegard, J. Kolis, C. McMillen, C. Moore, L. Sanjeewa, and C. Hancock, “The application of cryogenic laser physics to the development of high average power ultra-short pulse lasers,” Appl. Sci. 6, 23 (2016).
[Crossref]

Tsuboi, T.

A. Sugiyama, M. Katsurayama, Y. Anzai, and T. Tsuboi, “Spectroscopic properties of Yb doped YLF grown by a vertical Bridgman method,” J. Alloys Comp. 408, 780–783 (2006).
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Tunnermann, A.

Tye, R. P.

R. W. Powell, M. J. Woodman, and R. P. Tye, “Thermal conductivity and electrical resistivity of indium,” Philos. Mag. 7, 1183–1186 (1962).
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J. Kawanaka, S. Tokita, H. Nishioka, M. Fujita, K. Yamakawa, K. Ueda, and Y. Izawa, “Dramatically improved laser characteristics of diode-pumped Yb-doped materials at low temperature,” Laser Phys. 15, 1306–1312 (2005).

J. Kawanaka, K. Yamakawa, H. Nishioka, and K. Ueda, “30-mJ, diode-pumped, chirped-pulse Yb : YLF regenerative amplifier,” Opt. Lett. 28, 2121–2123 (2003).
[Crossref]

J. Kawanaka, K. Yamakawa, H. Nishioka, and K. Ueda, “Improved high-field laser characteristics of a diode-pumped Yb:LiYF4 crystal at low temperature,” Opt. Express 10, 455–460 (2002).
[Crossref]

N. Uehara, K. Ueda, and Y. Kubota, “Spectroscopic measurements of a high-concentration Yb3+:LiYF4 crystal,” Jpn. J. Appl. Phys. 35, L499–L501 (1996).
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Ueffing, M.

Uehara, N.

N. Uehara, K. Ueda, and Y. Kubota, “Spectroscopic measurements of a high-concentration Yb3+:LiYF4 crystal,” Jpn. J. Appl. Phys. 35, L499–L501 (1996).
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Vanherzeele, H.

Vannini, M.

Voss, A.

A. Giesen, H. Hugel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
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A. Isemann, P. Wessels, and C. Fallnich, “Directly diode-pumped Colquiriite regenerative amplifiers,” Opt. Commun. 260, 211–222 (2006).
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C. Kruse, T. Anderson, C. Wilson, C. Zuhlke, D. Alexander, G. Gogos, and S. Ndao, “Extraordinary shifts of the Leidenfrost temperature from multiscale micro/nanostructured surfaces,” Langmuir 29, 9798–9806 (2013).
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Wittig, K.

A. Giesen, H. Hugel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
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R. W. Powell, M. J. Woodman, and R. P. Tye, “Thermal conductivity and electrical resistivity of indium,” Philos. Mag. 7, 1183–1186 (1962).
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D. F. Zhang, A. Fallahi, M. Hemmer, X. J. Wu, M. Fakhari, Y. Hua, H. Cankaya, A. L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kartner, “Segmented terahertz electron accelerator and manipulator (STEAM),” Nat. Photonics 12, 336–342 (2018).
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Xu, C.

H. Hu, C. Xu, Y. Zhao, K. J. Ziegler, and J. N. Chung, “Boiling and quenching heat transfer advancement by nanoscale surface modification,” Sci. Rep. 7, 1 (2017).
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Yamakawa, K.

Yaman, F.

Ye, H.

Zaouter, Y.

Zapata, L. E.

H. Cankaya, U. Demirbas, Y. Hua, M. Hemmer, L. E. Zapata, M. Pergament, and F. X. Kärtner, “190-mJ cryogenically-cooled Yb:YLF amplifier system at 1019.7 nm,” OSA Continuum 2, 3547–3553 (2019).
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D. F. Zhang, A. Fallahi, M. Hemmer, H. Ye, M. Fakhari, Y. Hua, H. Cankaya, A. L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kartner, “Femtosecond phase control in high-field terahertz-driven ultrafast electron sources,” Optica 6, 872–877 (2019).
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D. F. Zhang, A. Fallahi, M. Hemmer, X. J. Wu, M. Fakhari, Y. Hua, H. Cankaya, A. L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kartner, “Segmented terahertz electron accelerator and manipulator (STEAM),” Nat. Photonics 12, 336–342 (2018).
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Y. Hua, W. Liu, M. Hemmer, L. E. Zapata, G. J. Zhou, D. N. Schimpf, T. Eidam, J. Limpert, A. Tunnermann, F. X. Kartner, and G. Q. Chang, “87-W 1018-nm Yb-fiber ultrafast seeding source for cryogenic Yb: yttrium lithium fluoride amplifier,” Opt. Lett. 43, 1686–1689 (2018).
[Crossref]

L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kartner, “250 W average power, 100 kHz repetition rate cryogenic Yb:YAG amplifier for OPCPA pumping,” Opt. Lett. 41, 492–495 (2016).
[Crossref]

L. E. Zapata, H. Lin, A. L. Calendron, H. Cankaya, M. Hemmer, F. Reichert, W. R. Huang, E. Granados, K. H. Hong, and F. X. Kartner, “Cryogenic Yb:YAG composite-thin-disk for high energy and average power amplifiers,” Opt. Lett. 40, 2610–2613 (2015).
[Crossref]

D. E. Miller, L. E. Zapata, D. J. Ripin, and T. Y. Fan, “Sub-picosecond pulses at 100 W average power from a Yb:YLF chirped-pulse amplification system,” Opt. Lett. 37, 2700–2702 (2012).
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V. V. Zelenogorskii and E. A. Khazanov, “Influence of the photoelastic effect on the thermal lens in a YLF crystal,” Quantum Electron. 40, 40–44 (2010).
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Zhang, D. F.

D. F. Zhang, A. Fallahi, M. Hemmer, H. Ye, M. Fakhari, Y. Hua, H. Cankaya, A. L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kartner, “Femtosecond phase control in high-field terahertz-driven ultrafast electron sources,” Optica 6, 872–877 (2019).
[Crossref]

D. F. Zhang, A. Fallahi, M. Hemmer, X. J. Wu, M. Fakhari, Y. Hua, H. Cankaya, A. L. Calendron, L. E. Zapata, N. H. Matlis, and F. X. Kartner, “Segmented terahertz electron accelerator and manipulator (STEAM),” Nat. Photonics 12, 336–342 (2018).
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Zhang, Z. L.

Z. L. Zhang, Q. Liu, M. M. Nie, E. C. Ji, and M. L. Gong, “Experimental and theoretical study of the weak and asymmetrical thermal lens effect of Nd:YLF crystal for sigma and pi polarizations,” Appl. Phys. B 120, 689–696 (2015).
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Zhao, Y.

H. Hu, C. Xu, Y. Zhao, K. J. Ziegler, and J. N. Chung, “Boiling and quenching heat transfer advancement by nanoscale surface modification,” Sci. Rep. 7, 1 (2017).
[Crossref]

Zhou, C.

Zhou, G. J.

Ziegler, K. J.

H. Hu, C. Xu, Y. Zhao, K. J. Ziegler, and J. N. Chung, “Boiling and quenching heat transfer advancement by nanoscale surface modification,” Sci. Rep. 7, 1 (2017).
[Crossref]

Zoubir, A.

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

Zuhlke, C.

C. Kruse, T. Anderson, C. Wilson, C. Zuhlke, D. Alexander, G. Gogos, and S. Ndao, “Extraordinary shifts of the Leidenfrost temperature from multiscale micro/nanostructured surfaces,” Langmuir 29, 9798–9806 (2013).
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Appl. Opt. (2)

Appl. Phys. B (2)

Z. L. Zhang, Q. Liu, M. M. Nie, E. C. Ji, and M. L. Gong, “Experimental and theoretical study of the weak and asymmetrical thermal lens effect of Nd:YLF crystal for sigma and pi polarizations,” Appl. Phys. B 120, 689–696 (2015).
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A. Giesen, H. Hugel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994).
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V. Pilla, P. R. Impinnisi, and T. Catunda, “Measurement of saturation intensities in ion doped solids by transient nonlinear refraction,” Appl. Phys. Lett. 70, 817–819 (1997).
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Appl. Sci. (1)

D. C. Brown, S. Tornegard, J. Kolis, C. McMillen, C. Moore, L. Sanjeewa, and C. Hancock, “The application of cryogenic laser physics to the development of high average power ultra-short pulse lasers,” Appl. Sci. 6, 23 (2016).
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IEEE J. Sel. Top. Quantum Electron. (2)

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598–609 (2007).
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H. W. Bruesselbach, D. S. Sumida, R. A. Reeder, and R. W. Byren, “Low-heat high-power scaling using InGaAs-diode-pumped Yb:YAG lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 105–116 (1997).
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Int. J. Mech. Mechatronics Eng. (1)

K. Bouazaoui, R. Agounoun, K. Sbai, A. Zoubir, I. Kadiri, M. Rahmoune, and R. Saadani, “Experimental and numerical study of pool boiling heat transfer of liquid nitrogen LN2: application to the brass ribbon cooling in horizontal position,” Int. J. Mech. Mechatronics Eng. 17, 74–82 (2017).

J. Alloys Comp. (1)

A. Sugiyama, M. Katsurayama, Y. Anzai, and T. Tsuboi, “Spectroscopic properties of Yb doped YLF grown by a vertical Bridgman method,” J. Alloys Comp. 408, 780–783 (2006).
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R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAIO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80–300 K temperature range,” J. Appl. Phys. 98, 103514 (2005).
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M. Kida, Y. Kikuchi, O. Takahashi, and I. Michiyoshi, “Pool-boiling heat-transfer in liquid-nitrogen,” J. Nucl. Sci. Technol. 18, 501–513 (1981).
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C. J. Saraceno, “Mode-locked thin-disk lasers and their potential application for high-power terahertz generation,” J. Opt. 20, 4 (2018).
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Jpn. J. Appl. Phys. (1)

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Langmuir (1)

C. Kruse, T. Anderson, C. Wilson, C. Zuhlke, D. Alexander, G. Gogos, and S. Ndao, “Extraordinary shifts of the Leidenfrost temperature from multiscale micro/nanostructured surfaces,” Langmuir 29, 9798–9806 (2013).
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I. Tamer, S. Keppler, M. Hornung, J. Korner, J. Hein, and M. C. Kaluza, “Spatio-temporal characterization of pump-induced wavefront aberrations in Yb3+-doped materials,” Laser Photon. Rev. 12, 1700211 (2018).
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Laser Phys. (1)

J. Kawanaka, S. Tokita, H. Nishioka, M. Fujita, K. Yamakawa, K. Ueda, and Y. Izawa, “Dramatically improved laser characteristics of diode-pumped Yb-doped materials at low temperature,” Laser Phys. 15, 1306–1312 (2005).

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Opt. Commun. (3)

U. Demirbas, “Modelling and optimization of tapered-diode pumped Cr:LiCAF regenerative amplifiers,” Opt. Commun. 311, 90–99 (2013).
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A. Isemann, P. Wessels, and C. Fallnich, “Directly diode-pumped Colquiriite regenerative amplifiers,” Opt. Commun. 260, 211–222 (2006).
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J. Manni, D. Harris, and T. Y. Fan, “High-gain (43 dB), high-power (40 W), highly efficient multipass amplifier at 995 nm in Yb:LiYF4,” Opt. Commun. 417, 54–56 (2018).
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Opt. Express (7)

M. Vannini, G. Toci, D. Alderighi, D. Parisi, F. Cornacchia, and M. Tonelli, “High efficiency room temperature laser emission in heavily doped Yb : YLF,” Opt. Express 15, 7994–8002 (2007).
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J. Kawanaka, K. Yamakawa, H. Nishioka, and K. Ueda, “Improved high-field laser characteristics of a diode-pumped Yb:LiYF4 crystal at low temperature,” Opt. Express 10, 455–460 (2002).
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D. V. Seletskiy, S. D. Melgaard, R. I. Epstein, A. Di Lieto, M. Tonelli, and M. Sheik-Bahae, “Local laser cooling of Yb:YLF to 110 K,” Opt. Express 19, 18229–18236 (2011).
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A. L. Calendron, H. Cankaya, and F. X. Kartner, “High-energy kHz Yb:KYW dual-crystal regenerative amplifier,” Opt. Express 22, 24752–24762 (2014).
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U. Demirbas, H. Cankaya, J. Thesinga, F. X. Kartner, and M. Pergament, “Efficient, diode-pumped, high-power >300 W) cryogenic Yb:YLF laser with broad-tunability (995–1020.5 nm): investigation of E//a-axis for lasing,” Opt. Express 27, 36562–36579 (2019).
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U. Demirbas, H. Cankaya, Y. Hua, J. Thesinga, M. Pergament, and F. X. Kaertner, “20-mJ, sub-ps pulses at up to 70 W average power from a cryogenic Yb:YLF regenerative amplifier,” Opt. Express 28, 2466–2479 (2020).
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H. Cankaya, A. L. Calendron, C. Zhou, S. H. Chia, O. D. Mucke, G. Cirmi, and F. X. Kartner, “40-µJ passively CEP-stable seed source for ytterbium-based high-energy optical waveform synthesizers,” Opt. Express 24, 25169–25180 (2016).
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Opt. Lett. (11)

U. Demirbas, J. Thesinga, H. Cankaya, M. Kellert, F. X. Kartner, and M. Pergament, “High-power passively mode-locked cryogenic Yb:YLF laser,” Opt. Lett. 45, 2050–2053 (2020).
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Y. Hua, W. Liu, M. Hemmer, L. E. Zapata, G. J. Zhou, D. N. Schimpf, T. Eidam, J. Limpert, A. Tunnermann, F. X. Kartner, and G. Q. Chang, “87-W 1018-nm Yb-fiber ultrafast seeding source for cryogenic Yb: yttrium lithium fluoride amplifier,” Opt. Lett. 43, 1686–1689 (2018).
[Crossref]

L. E. Zapata, H. Lin, A. L. Calendron, H. Cankaya, M. Hemmer, F. Reichert, W. R. Huang, E. Granados, K. H. Hong, and F. X. Kartner, “Cryogenic Yb:YAG composite-thin-disk for high energy and average power amplifiers,” Opt. Lett. 40, 2610–2613 (2015).
[Crossref]

L. E. Zapata, F. Reichert, M. Hemmer, and F. X. Kartner, “250 W average power, 100 kHz repetition rate cryogenic Yb:YAG amplifier for OPCPA pumping,” Opt. Lett. 41, 492–495 (2016).
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J. Brons, V. Pervak, D. Bauer, D. Sutter, O. Pronin, and F. Krausz, “Powerful 100-fs-scale Kerr-lens mode-locked thin-disk oscillator,” Opt. Lett. 41, 3567–3570 (2016).
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D. E. Miller, L. E. Zapata, D. J. Ripin, and T. Y. Fan, “Sub-picosecond pulses at 100 W average power from a Yb:YLF chirped-pulse amplification system,” Opt. Lett. 37, 2700–2702 (2012).
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D. E. Miller, J. R. Ochoa, and T. Y. Fan, “Cryogenically cooled, 149 W, Q-switched, Yb:LiYF4 laser,” Opt. Lett. 38, 4260–4261 (2013).
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J. Kawanaka, K. Yamakawa, H. Nishioka, and K. Ueda, “30-mJ, diode-pumped, chirped-pulse Yb : YLF regenerative amplifier,” Opt. Lett. 28, 2121–2123 (2003).
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E. C. Honea, R. J. Beach, S. C. Mitchell, J. A. Skidmore, M. A. Emanuel, S. B. Sutton, S. A. Payne, P. V. Avizonis, R. S. Monroe, and D. G. Harris, “High-power dual-rod Yb : YAG laser,” Opt. Lett. 25, 805–807 (2000).
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Opt. Mater. (1)

A. Bensalah, Y. Guyot, M. Ito, A. Brenier, H. Sato, T. Fukuda, and G. Boulon, “Growth of Yb3+-doped YLiF4 laser crystal by the Czochralski method. Attempt of Yb3+ energy level assignment and estimation of the laser potentiality,” Opt. Mater. 26, 375–383 (2004).
[Crossref]

Opt. Mater. Express (1)

Optica (1)

OSA Continuum (1)

Philos. Mag. (1)

R. W. Powell, M. J. Woodman, and R. P. Tye, “Thermal conductivity and electrical resistivity of indium,” Philos. Mag. 7, 1183–1186 (1962).
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Quantum Electron. (1)

V. V. Zelenogorskii and E. A. Khazanov, “Influence of the photoelastic effect on the thermal lens in a YLF crystal,” Quantum Electron. 40, 40–44 (2010).
[Crossref]

Sci. Rep. (1)

H. Hu, C. Xu, Y. Zhao, K. J. Ziegler, and J. N. Chung, “Boiling and quenching heat transfer advancement by nanoscale surface modification,” Sci. Rep. 7, 1 (2017).
[Crossref]

Other (2)

B. Baudouy, “Heat transfer and cooling techniques at low temperature,” arXiv: 1501.07153 (2015).

R. J. Corruccini and J. J. Gniewek, Thermal Expansion of Technical Solids at Low Temperatures: A Compilation from the Literature (United States Government Printing Office, 1961).

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Figures (17)

Fig. 1.
Fig. 1. Comparison of effective gain profile of cryogenic Yb:YLF with Yb:YAG in frequency domain. The centers of the gain spectra are shifted to base band for better comparison. The graph contains the broadband 1016 nm emission band of Yb:YLF (${E}{/\!/}{a}$, 80 K), the narrowband 995 nm emission band of Yb:YLF (${E}{/\!/}{c}$, 80 K), and 1030 nm band of Yb:YAG at room and cryogenic temperatures. An inversion level of 25% is assumed for room-temperature Yb:YAG, and all curves are shown in normalized units.
Fig. 2.
Fig. 2. Schematics of the cryogenic Yb:YLF regenerative amplifiers that have been modeled in the simulations. (a) Ring cavity with a single Yb:YLF crystal and (b) standing wave cavity employing two Yb:YLF crystals. DM, dichroic mirror; f1-f5, lens; W, antireflection coated window; PC, Pockels cell; QWP, quarter-wave plate; HWP, half-wave plate; TFP, thin-film polarizer; PBS, polarizing beam splitter cube; FR, Faraday rotator; HR, high-reflector mirror; BD, beam dump.
Fig. 3.
Fig. 3. (a) Yb:YLF crystal bonded to the heat sink via indium bonding. The heat sink is designed in pyramid structure to improve heat extraction efficiency via heat spreading. The top part of the heat sink is in direct contact with liquid nitrogen. (b) Simplified geometry used in thermal simulations, which contains only the Yb:YLF crystal and the indium bonding layer.
Fig. 4.
Fig. 4. Pump beam profile incident on the Yb:YLF crystal [length (L): ${20} + {3} + {3}\;{\rm mm}$, height (H): 10 mm, width (W): 15 mm]. The pump beam waist is taken as 1 mm. The dimensions are in mm, and the pump density is in units of ${\rm kW}/{{\rm cm}^3}$. The calculation was performed for a total absorbed pump power of 400 W.
Fig. 5.
Fig. 5. Estimated mode-matching factor (MMF) between the pump and resonator beams as a function of pump/seed spot size employed in the regen. The estimation was performed for a 2 cm long Yb:YLF gain element assuming homogeneous distribution of inversion along the crystal length. The beam qualities (${{\rm M}^2}$ factors) of the pump and laser beams are taken as 250 and one, respectively. As an example, inset shows the mode-mismatch between the pump and seed beams as they propagate through the Yb:YLF crystal for a sample beam waist of 1 mm (corresponds to a minimum beam diameter of 2 mm at the center of the crystal).
Fig. 6.
Fig. 6. Calculated temperature profile of the Yb:YLF crystal at an absorbed pump power of 400 W. The pump beam waist has a size of 1 mm and is positioned at the center of the crystal. The dimensions are in mm, and the temperatures are in K. (a) Full 3D profile of temperature; (b), (c) 2D temperature profiles at the planes of ($x,y, z = {\rm LL}/{2}$), and ($x,y = {\rm H}/{2},z$), respectively.
Fig. 7.
Fig. 7. Calculated variation of optical path difference (OPD) along the Yb:YLF crystal surface due to thermo-optic effect (variation of index of refraction with temperature). The dimensions are in mm, and the OPD is specified in units of waves. The calculation was performed for a 1016 nm Yb:YLF regenerative amplifier crystal, using a pump beam waist of 1 mm and an absorbed pump power of 400 W.
Fig. 8.
Fig. 8. Calculated temperature-induced deformation on the (a) front and (b) back surfaces of the Yb:YLF crystal. The dimensions are in mm, and the path differences are specified in units of waves. The calculation was performed for a 1016 nm Yb:YLF regenerative amplifier crystal, using a pump beam waist of 1 mm and an absorbed pump power of 400 W.
Fig. 9.
Fig. 9. (a) Calculated OPD due to the variation of refractive index with temperature (thermo-optic effect) in the $y$ axis at $x = {\rm W}/{2}$. The spherical component has a focal length of around ${-}{2}\;{\rm m}$. (b) Calculated optical path difference due to deformation/bulging of the front and back surfaces of the Yb:YLF rod in the $y$ axis. The total OPD due to bulging is also shown. The spherical component of the deformation lens has a focal length of around 12 m. (c) Estimated variation of total (${\rm bulging} + dn/dT$) OPD along the $y$ axis. The spherical component of the total lens has an estimated focal length of around ${-}{2.5}\;{\rm m}$. The remaining aspherical component and its 40 times magnified version are also shown. The calculation was performed for a 2 cm long 1% Yb-doped Yb:YLF regenerative amplifier crystal containing 3 mm un-doped wedges on both surfaces. We have assumed a pump beam waist of 1 mm and an absorbed pump power of 400 W.
Fig. 10.
Fig. 10. Calculated variation of single-crystal 1016 nm Yb:YLF cryogenic regenerative amplifier output energy as a function of number of round trips at different repetition rates between 1 Hz and 5 kHz. Due to LIDT, the obtainable pulse energy is limited to around 35 mJ (shown with the horizontal gray line). Simulation parameters: ${{w}_p} = {1}\;{\rm mm}$, ${\rm MMF} \cong {70}\%$, ${\rm loss} \cong {10}\%$, ${\tau _s} = {2}\;{\rm ns}$, ${{P}_{\text{abs}}} = {350}\;{\rm W }+ $ ½ ${{P}_{\text{out}}}$.
Fig. 11.
Fig. 11. Calculated variation of single-crystal 1016 nm Yb:YLF cryogenic regenerative amplifier performance at 5 kHz repetition rate at different output energy levels. The maximum extractable output energy is limited to around 17 mJ in this configuration (corresponds to 85 W average power). Simulation parameters: ${{w}_p} = {1}\;{\rm mm}$, ${\rm MMF} \cong {70}\%$, ${\rm loss} \cong {10}\%$, ${\tau _s} = {2}\;{\rm ns}$, ${{P}_{\text{abs}}} = {350}\;{\rm W +}$ ½ ${{P}_{\text{out}}}$.
Fig. 12.
Fig. 12. Estimated performance of single-crystal 1016 nm Yb:YLF cryogenic regenerative amplifier as a function of repetition rate. Due to LIDT, the obtainable pulses are limited to around 35 mJ at low repetition rates. Empty spots with energies up to 250 mJ are not feasible, but are shown on purpose to illustrate the extraction limits of the system for a cw seed. At higher repetition rates, obtainable energies are limited by the achievable average power level (${\sim}85\; {\rm W}$) from the system. Simulation parameters: ${{w}_p} = {1}\;{\rm mm}$, $ {\rm MMF}\cong 70\%$, $ {\rm loss}\cong 10\%$, ${\tau _s} = {2}\;{\rm ns}$, ${{P}_{\text{abs}}} = {350}\;{\rm W +}$ ½ ${{P}_{\text{out}}}$.
Fig. 13.
Fig. 13. Estimated effect of (a) cavity losses (L) and (b) absorbed pump power level ($P_{\rm abs}$) on the single-crystal cryogenic 1016 nm Yb:YLF regenerative amplifier. Simulation parameters: ${{w}_p} = {1}\;{\rm mm}$, $ {\rm MMF }\cong 70\%$.
Fig. 14.
Fig. 14. Compared performance of dual-crystal and single-crystal 1016 nm cryogenic Yb:YLF regenerative amplifier. Simulation parameters: ${{w}_p} = {1}\;{\rm mm}$, ${\rm MMF}\cong 70\%$, $ {\rm loss}\cong 10\%$, ${\tau _s} = {2} \; {\rm ns}$.
Fig. 15.
Fig. 15. Estimated effect of pump spot size (${{w}_p}$) on the dual-crystal cryogenic 1016 nm Yb:YLF regenerative amplifier. Simulation parameters: ${\rm MMF} \cong 70 \%$, $ {\rm loss} \cong 10\% $, ${\tau _s}= {2} \; {\rm ns}$, ${{P}_{\text{abs}}} = {700}\;{\rm W +}$ ½ ${{P}_{\text{out}}}$.
Fig. 16.
Fig. 16. Estimated effect of absorbed pump power ($P_{\rm abs}$) on the dual-crystal cryogenic 1016 nm Yb:YLF regenerative amplifier. Simulation parameters: ${{w}_p} = {1}\;{\rm mm}$, $ {\rm MMF}\cong 70 \%$, $ {\rm loss}\cong 10\%$, ${\tau _s} = {2}\; {\rm ns}$.
Fig. 17.
Fig. 17. Comparison of the estimated performance of the Yb:YLF regenerative amplifier with literature in terms of peak power that could be achieved as a function of repetition rate.

Equations (29)

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P a b s , max = 350 W + 1 2 P out ,
[ k ( x , y , z , T ) T ( x , y , z ) ] + Q ( x , y , z ) = 0 ,
k Y L F , a ( T ) 21800 T 1.48 ,
k Y L F , c ( T ) 33000 T 1.49 ,
k 1 % Y b : Y L F , a ( T ) 14300 T 1.41 ,
k 1 % Y b : Y L F , c ( T ) 21200 T 1.42 ,
k i n d ( T ) 120.6 0.01465 T + 0.0001 T 2 .
Q ( x , y , z ) = 2 ( F T L ) P in α eff π w p 0 2 ( z ) E x p [ 2 ( x 2 + y 2 ) w p 0 2 ( z ) ] × E x p [ α eff z ] ,
ω p 0 ( z ) = ω 0 1 + ( z L / 2 z r ) 2 ,
z r = 1 M 2 π ω 0 2 λ p .
F T L = 1.5 ( Q D ) = 1.5 ( 1 λ p λ l ) ,
T ( x , y , z ) y | y = h ind = H htc k ind ( T LN T ) .
2 U + 1 1 2 υ ( U ) = 2 ( 1 + ν ) 1 2 υ α T ,
α Y L F , a ( T ) 8 + 0.118 T 0.000146 T 2 ,
α Y L F , c ( T ) 4.75 + 0.094 T 0.00015 T 2 ,
α ind ( T ) 16.24 + 0.085 T 0.000108 T 2 .
O P D ( x , y ) = ( n o 1 ) Δ L ( x , y ) + n T 0 L [ T ( x , y , z ) T LN ] d z ,
d n d T Y L F , a ( T ) 0.104 + 0.00105 T 0.000053 T 2 ,
d n d T Y L F , c ( T ) 0.039 0.00155 T 0.0000217 T 2 .
J out = J sat L n [ 1 + e g 0 ( e J in J sat 1 ) ] ,
J sat = h c λ l ( σ em + σ ab ) = E λ l ( σ em + σ ab ) ,
J sto = E a b s A eff λ p λ ι ,
g 0 = J sto J a b s E a b s E λ l λ p λ ι M M F ( σ em + σ ab ) A eff ,
σ em ( T ) 0.75 × 10 20 [ 1 3.3 × 10 3 ( T 77 ) + 7.2 × 10 6 ( T 77 ) 2 ] ,
σ ab ( T ) 0.04 × 10 20 [ 0.03 + 0.000025 ( T 77 ) 2 ] .
β g 0 L g ( N Yb ) σ em ,
E a b s , s = P abs T [ 1 e T τ ] τ T = P abs τ [ 1 e T τ ] ,
E abs = E a b s , s + m = 1 N ( E a b s , s E used ) E x p ( m τ f rep ) ,
L s a , 1016 ( T ) 0.034 0.001 ( T 77 ) + 0.000025 ( T 77 ) 2 .

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