Experimental and theoretical investigation of a volume-Bragg-grating-locked Yb:KYW laser at selected wavelengths

Björn Jacobsson

doi:10.1364/OE.16.006443

Experimental and theoretical investigation of a volume-Bragg-grating-locked Yb:KYW laser at selected wavelengths

Björn Jacobsson

Optics Express
Vol. 16,
Issue 9,
pp. 6443-6454
(2008)
•https://doi.org/10.1364/OE.16.006443

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Björn Jacobsson, "Experimental and theoretical investigation of a volume-Bragg-grating-locked Yb:KYW laser at selected wavelengths," Opt. Express 16, 6443-6454 (2008)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Volume gratings (050.7330)
- Lasers, ytterbium (140.3615)

About this Article
History
- Original Manuscript: February 21, 2008
- Revised Manuscript: April 18, 2008
- Manuscript Accepted: April 18, 2008
- Published: April 22, 2008

Accessible

Open Access

Abstract

Laser action is demonstrated in Yb:KYW at wavelengths of 990 nm, 997 nm and 1066 nm, when pumped at 980 nm by a Ti:sapphire laser, with a lowest laser quantum defect of 1.0%. The laser output powers at the various wavelengths were 70mW, 160mW and 140mW, respectively. Locking of the laser wavelength and the spectrally close spacing of pump and laser were achieved by the use of a volume Bragg grating as an input coupler. A theoretical model is also presented that accurately describes the laser at various wavelengths, by solving the laser rate equations at spatial points throughout the laser crystal.

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References

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|
Year
|
Author
|
Publication

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
[Crossref] [PubMed]
B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Single-longitudinal-mode Nd-laser with a Bragg grating Fabry-Perot cavity,” Opt. Express 14, 9284–9292 (2006).
[Crossref] [PubMed]
T. Chung, A. Rapaport, V. Smirnov, L. B. Glebov, M. C. Richardson, and M. Bass, “Solid-state laser spectral narrowing using a volumetric photothermal refractive Bragg grating cavity mirror,” Opt. Lett. 31, 229–231 (2006).
[Crossref] [PubMed]
I. Häggström, B. Jacobsson, and F. Laurell, “Monolithic Bragg-locked Nd:GdVO4 laser,” Opt. Express 15(18), 11,589–11,594 (2007).
[Crossref]
B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]
B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]
O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]
J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]
J. Petit, P. Goldner, B. Viana, J. Didierjean, F. Balembois, F. P. Druon, and P. Georges, “Quest of athermal solid-state laser: case of Yb:CaGdAlO4,” in Advanced Solid-State Photonics, (Optical Society of America, 2006), WD1.
F. Balembois, F. Falcoz, F. Kerboull, F. Druon, P. Georges, and A. Brun, “Theoretical and experimental investigations of small-signal gain for a diode-pumped Q-switched Cr:LiSAF laser,” IEEE J. Quantum Electron. 33, 269–278 (1997).
[Crossref]
F. Auge, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, “Theoretical and experimental investigations of a diode-pumped quasi-three-level laser: the Yb3+-doped Ca4GdO(BO3)3 (Yb:GdCOB) laser,” IEEE J. Quantum Electron. 36, 598–606 (2000).
[Crossref]
S. Yiou, F. Balembois, and P. Georges, “Numerical modeling of a continuous-wave Yb-doped bulk crystal laser emitting on a three-level laser transition near 980 nm,” J. Opt. Soc. Am. B 22, 572–581 (2005).
[Crossref]
S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]
G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer methods for mathematical computations (Prentice-Hall, 1976).
K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[Crossref]
A. A. Demidovich, A. N. Kuzmin, G. I. Ryabtsev, M. B. Danailov, W. Strek, and A. N. Titov, “Influence of Yb concentration on Yb: KYW laser properties,” J. Alloys Comp. 300–301, 238–241 (2000).
[Crossref]
N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]
B. Aull and H. Jenssen, “Vibronic interactions in Nd:YAG resulting in nonreciprocity of absorption and stimulated emission cross sections,” IEEE J. Quantum Electron. 18, 925–930 (1982).
[Crossref]
G. Métrat, M. Boudeulle, N. Muhlstein, A. Brenier, and G. Boulon, “Nucleation, morphology and spectroscopic properties of Yb3+-doped KY(WO4)2 crystals grown by the top nucleated floating crystal method,” J. Cryst. Growth 197, 883–888 (1999).
[Crossref]

2008 (1)

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]

2007 (3)

J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]

I. Häggström, B. Jacobsson, and F. Laurell, “Monolithic Bragg-locked Nd:GdVO4 laser,” Opt. Express 15(18), 11,589–11,594 (2007).
[Crossref]

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]

2006 (3)

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
[Crossref] [PubMed]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Single-longitudinal-mode Nd-laser with a Bragg grating Fabry-Perot cavity,” Opt. Express 14, 9284–9292 (2006).
[Crossref] [PubMed]

T. Chung, A. Rapaport, V. Smirnov, L. B. Glebov, M. C. Richardson, and M. Bass, “Solid-state laser spectral narrowing using a volumetric photothermal refractive Bragg grating cavity mirror,” Opt. Lett. 31, 229–231 (2006).
[Crossref] [PubMed]

2005 (2)

K. Petermann, D. Fagundes-Peters, J. Johannsen, M. Mond, V. Peters, J. J. Romero, S. Kutovoi, J. Speiser, and A. Giesen, “Highly Yb-doped oxides for thin-disc lasers,” J. Cryst. Growth 275, 135–140 (2005).
[Crossref]

S. Yiou, F. Balembois, and P. Georges, “Numerical modeling of a continuous-wave Yb-doped bulk crystal laser emitting on a three-level laser transition near 980 nm,” J. Opt. Soc. Am. B 22, 572–581 (2005).
[Crossref]

2000 (2)

F. Auge, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, “Theoretical and experimental investigations of a diode-pumped quasi-three-level laser: the Yb3+-doped Ca4GdO(BO3)3 (Yb:GdCOB) laser,” IEEE J. Quantum Electron. 36, 598–606 (2000).
[Crossref]

A. A. Demidovich, A. N. Kuzmin, G. I. Ryabtsev, M. B. Danailov, W. Strek, and A. N. Titov, “Influence of Yb concentration on Yb: KYW laser properties,” J. Alloys Comp. 300–301, 238–241 (2000).
[Crossref]

1999 (2)

G. Métrat, M. Boudeulle, N. Muhlstein, A. Brenier, and G. Boulon, “Nucleation, morphology and spectroscopic properties of Yb3+-doped KY(WO4)2 crystals grown by the top nucleated floating crystal method,” J. Cryst. Growth 197, 883–888 (1999).
[Crossref]

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

1997 (2)

F. Balembois, F. Falcoz, F. Kerboull, F. Druon, P. Georges, and A. Brun, “Theoretical and experimental investigations of small-signal gain for a diode-pumped Q-switched Cr:LiSAF laser,” IEEE J. Quantum Electron. 33, 269–278 (1997).
[Crossref]

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

1992 (1)

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

1982 (1)

B. Aull and H. Jenssen, “Vibronic interactions in Nd:YAG resulting in nonreciprocity of absorption and stimulated emission cross sections,” IEEE J. Quantum Electron. 18, 925–930 (1982).
[Crossref]

Aka, G.

Auge, F.

Aull, B.

Balembois, F.

J. Petit, P. Goldner, B. Viana, J. Didierjean, F. Balembois, F. P. Druon, and P. Georges, “Quest of athermal solid-state laser: case of Yb:CaGdAlO4,” in Advanced Solid-State Photonics, (Optical Society of America, 2006), WD1.

Bass, M.

Boudeulle, M.

Boulon, G.

Brenier, A.

Brun, A.

Chase, L. L.

Chung, T.

Danailov, M. B.

Demidovich, A. A.

Didierjean, J.

Druon, F.

Druon, F. P.

Efimov, O.

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

Fagundes-Peters, D.

Falcoz, F.

Forsythe, G. E.

G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer methods for mathematical computations (Prentice-Hall, 1976).

Georges, P.

Giesen, A.

Glebov, L.

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

Glebov, L. B.

Glebova, L.

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

Goldner, P.

Häggström, I.

I. Häggström, B. Jacobsson, and F. Laurell, “Monolithic Bragg-locked Nd:GdVO4 laser,” Opt. Express 15(18), 11,589–11,594 (2007).
[Crossref]

Hellström, J. E.

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]

J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]

Huber, G.

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

Jacobsson, B.

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]

J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]

I. Häggström, B. Jacobsson, and F. Laurell, “Monolithic Bragg-locked Nd:GdVO4 laser,” Opt. Express 15(18), 11,589–11,594 (2007).
[Crossref]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
[Crossref] [PubMed]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Single-longitudinal-mode Nd-laser with a Bragg grating Fabry-Perot cavity,” Opt. Express 14, 9284–9292 (2006).
[Crossref] [PubMed]

Jenssen, H.

Johannsen, J.

Kerboull, F.

Krupke, W. F.

Kuleshov, N. V.

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

Kutovoi, S.

Kuzmin, A. N.

Kway, W. L.

Lagatsky, A. A.

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

Laurell, F.

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]

J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]

I. Häggström, B. Jacobsson, and F. Laurell, “Monolithic Bragg-locked Nd:GdVO4 laser,” Opt. Express 15(18), 11,589–11,594 (2007).
[Crossref]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
[Crossref] [PubMed]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Single-longitudinal-mode Nd-laser with a Bragg grating Fabry-Perot cavity,” Opt. Express 14, 9284–9292 (2006).
[Crossref] [PubMed]

Malcolm, M. A.

G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer methods for mathematical computations (Prentice-Hall, 1976).

Métrat, G.

Mikhailov, V. P.

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

Moler, C. B.

G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer methods for mathematical computations (Prentice-Hall, 1976).

Mond, M.

Mougel, F.

Muhlstein, N.

Pasiskevicius, V.

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]

J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Single-longitudinal-mode Nd-laser with a Bragg grating Fabry-Perot cavity,” Opt. Express 14, 9284–9292 (2006).
[Crossref] [PubMed]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
[Crossref] [PubMed]

Payne, S. A.

Petermann, K.

Peters, V.

Petit, J.

Podlipensky, A. V.

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

Rapaport, A.

Richardson, K.

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

Richardson, M. C.

Romero, J. J.

Ryabtsev, G. I.

Smirnov, V.

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

Smith, L. K.

Speiser, J.

Strek, W.

Titov, A. N.

Viana, B.

Vivien, D.

Yiou, S.

Appl. Opt. (1)

O. Efimov, L. Glebov, L. Glebova, K. Richardson, and V. Smirnov, “High-efficiency Bragg gratings in photother-morefractive glass,” Appl. Opt. 38, 619–627 (1999).
[Crossref]

Appl. Phys. B (1)

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Tunable Yb:KYW laser using volume Bragg grating in s-polarization,” Appl. Phys. B 91, 85–88 (2008).
[Crossref]

IEEE J. Quantum Electron. (4)

J. Alloys Comp. (1)

J. Cryst. Growth (2)

J. Opt. Soc. Am. B (1)

Opt. Express (4)

I. Häggström, B. Jacobsson, and F. Laurell, “Monolithic Bragg-locked Nd:GdVO4 laser,” Opt. Express 15(18), 11,589–11,594 (2007).
[Crossref]

J. E. Hellström, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Quasi-two-level Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 13,930–13,935 (2007).
[Crossref]

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Single-longitudinal-mode Nd-laser with a Bragg grating Fabry-Perot cavity,” Opt. Express 14, 9284–9292 (2006).
[Crossref] [PubMed]

B. Jacobsson, J. E. Hellström, V. Pasiskevicius, and F. Laurell, “Widely tunable Yb:KYW laser with a volume Bragg grating,” Opt. Express 15, 1003–1010 (2007).
[Crossref] [PubMed]

Opt. Lett. (3)

B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Tunable single-longitudinal-mode ErYb:glass laser locked by a bulk glass Bragg grating,” Opt. Lett. 31, 1663–1665 (2006).
[Crossref] [PubMed]

N. V. Kuleshov, A. A. Lagatsky, A. V. Podlipensky, V. P. Mikhailov, and G. Huber, “Pulsed laser operation of Yb-doped KY(WO4)2 and KGd(WO4)2,” Opt. Lett. 22, 1317–1319 (1997).
[Crossref]

Other (2)

G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer methods for mathematical computations (Prentice-Hall, 1976).

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Fig. 1. Setup in the laser crystal, showing pump (dashed blue) and laser (solid red) parameters.

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Fig. 2. Yb:KYW gain cross sections at various inversions N_u /N (see legend), corresponding to emission cross section σ_e for N_u /N=1 and (negative) absorption cross section -σ_a for N_u /N=0 (all for the n_m direction). The experimental laser spectra and (negative) pump spectrum are also given (thick black lines).

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Fig. 3. Laser setup, showing the laser beam (solid red) and the pump beam (dashed blue). Mirrors M₂ and M₃ are highly reflective with a radius of curvature of 50 mm and mirror M₄ is flat with variable reflectivity R.

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Fig. 4. Pump-induced saturation of pump absorption for various situations, showing experimental measurements (points) and theoretical simulations (lines).

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Fig. 5. Laser threshold for various output coupler reflectivities, experimental data (points) and theoretical simulations (lines).

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Fig. 6. Laser power, comparison of experiments (points) and theory (lines) for various wavelengths. The legend gives the experimental slope efficiency.

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Equations (15)

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\frac{d N_{l}}{d t} = - \frac{d N_{u}}{d t} = (\frac{1}{τ} + σ_{el} \frac{λ_{l}}{hc} I + σ_{ep} \frac{λ_{p}}{hc} I_{p}) N_{u} - (σ_{al} \frac{λ_{l}}{hc} I + σ_{ap} \frac{λ_{p}}{hc} I_{p}) N_{l} .

N_{l} = N - N_{u} = N \frac{σ_{ep} \frac{λ_{p}}{hc} I_{p} + σ_{el} \frac{λ_{l}}{hc} I + \frac{1}{τ}}{(σ_{ep} + σ_{ap}) \frac{λ_{p}}{hc} I_{p} + (σ_{el} + σ_{al}) \frac{λ_{l}}{hc} I + \frac{1}{τ}} .

g (λ) = - α (λ) = σ_{e} (λ) N_{u} - σ_{a} (λ) N_{l},

α_{p} = N \frac{τ (σ_{el} σ_{ap} - σ_{al} σ_{ep}) \frac{λ_{l}}{hc} I + σ_{ap}}{τ (σ_{ep} + σ_{ap}) \frac{λ_{p}}{hc} I_{p} + τ (σ_{el} + σ_{al}) \frac{λ_{l}}{hc} I + 1}

g = N \frac{τ (σ_{el} σ_{ap} - σ_{al} σ_{ep}) \frac{λ_{p}}{hc} I_{p} - σ_{al}}{τ (σ_{ep} + σ_{ap}) \frac{λ_{p}}{hc} I_{p} + τ (σ_{el} + σ_{al}) \frac{λ_{l}}{hc} I + 1} .

I_{p} (0) = P_{in} \frac{2}{π w_{p}^{2}} \exp (\frac{- 2 r^{2}}{w_{p}^{2}}) .

I (z) = \frac{1 + R}{1 - R} P_{out} \frac{2}{π w_{l}^{2} (z)} \exp (\frac{- 2 r^{2}}{w_{l}^{2}} (z)) .

\frac{d I_{p}^{+}}{d z} = - α (I_{p}^{+} + I_{p}^{-}) I_{p}^{+}

\frac{d I_{p}^{-}}{d z} = + α (I_{p}^{+} + I_{p}^{-}) I_{p}^{-}

I_{p}^{+} (0) = I_{p} (0)

I_{p}^{-} (d) = R_{p} I_{p}^{+} (d) .

I_{p}^{+} (z) I_{p}^{-} (z) = constant = R_{p} I_{p}^{+ 2} (d) .

\frac{d I_{p}^{+}}{d z} = - α (I_{p}^{+} + \frac{R_{p} I_{p}^{+ 2} (d)}{I_{p}^{+}}) I_{p}^{+},

G = \frac{1}{R (1 - L)} .

G = {(1 + \int_{0}^{d} d z \int_{0}^{\infty} r d r g (r, z) \frac{4}{w_{l}^{2} (z)} \exp (\frac{- 2 r^{2}}{w_{l}^{2}} (z)))}^{2} .