Abstract

It is essential to understand the pump-induced lensing and aberration effects in solid-state lasers, such as Alexandrite, since these set limits on laser power scaling whilst maintaining high spatial TEM00 beam quality. In this work, we present direct wavefront measurements of pump-induced lensing and spherical aberration using a Shack-Hartmann wavefront sensor, for the first time, in a diode-pumped Alexandrite laser, and under both non-lasing and lasing conditions. The lens dioptric power is found to be weakly sub-linear with respect to the absorbed pump power, and under lasing, the lensing power is observed to decrease to 60 % of its non-lasing value. The results are inconsistent with a thermal lens model but a fuller theoretical formulation is made of a combined thermal and population lens model giving good quantitative agreement to the observed pump power dependence of the induced-lensing under non-lasing conditions and the reduced lensing under lasing conditions. The deduced value for the difference in excited to ground state polarizability is consistent with prior measurement estimates for other chromium-doped gain media. The finding of this paper provide new insight into pump-induced lensing in Alexandrite and also provides a basis for a fast saturable population lens mechanism to account for self-Q-switching observed recently in Alexandrite laser systems.

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References

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    [Crossref]

2019 (1)

2018 (7)

2017 (2)

M. J. Damzen, G. M. Thomas, and A. Minassian, “Diode-side-pumped alexandrite slab lasers,” Opt. Express 25(10), 11622–11636 (2017).
[Crossref]

L. Cini and J. I. Mackenzie, “Analytical thermal model for end-pumped solid-state lasers,” Appl. Phys. B: Lasers Opt. 123(12), 273 (2017).
[Crossref]

2016 (4)

2015 (1)

W. R. Kerridge-Johns and M. J. Damzen, “Analysis of pump excited state absorption and its impact on laser efficiency,” Laser Phys. Lett. 12(12), 125002 (2015).
[Crossref]

2014 (2)

2013 (1)

2012 (1)

2010 (1)

2006 (4)

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped yb:yag laser crystals,” Opt. Lett. 31(6), 763–765 (2006).
[Crossref]

N. Passilly, E. Haouas, V. Ménard, R. Moncorgé, and K. Aït-Ameur, “Population lensing effect in cr:lisaf probed by z-scan technique,” Opt. Commun. 260(2), 703–707 (2006).
[Crossref]

S. A. Amarande and M. J. Damzen, “Measurement of the thermal lens of grazing-incidence diode-pumped nd:yvo4 laser amplifier,” Opt. Commun. 265(1), 306–313 (2006).
[Crossref]

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

2004 (2)

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
[Crossref]

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
[Crossref]

2001 (1)

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

2000 (1)

1998 (2)

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” J. Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in nd:yag,” IEEE J. Quantum Electron. 34(3), 560–572 (1998).
[Crossref]

1997 (1)

V. Pilla, P. R. Impinnisi, and T. Catunda, “Measurement of saturation intensities in ion doped solids by transient nonlinear refraction,” Appl. Phys. Lett. 70(7), 817–819 (1997).
[Crossref]

1995 (1)

B. Neuenschwander, R. Weber, and H. P. Weber, “Determination of the thermal lens in solid-state lasers with stable cavities,” IEEE J. Quantum Electron. 31(6), 1082–1087 (1995).
[Crossref]

1990 (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[Crossref]

1983 (1)

M. Shand and H. Jenssen, “Temperature dependence of the excited-state absorption of alexandrite,” IEEE J. Quantum Electron. 19(3), 480–484 (1983).
[Crossref]

1980 (1)

J. Walling, O. Peterson, H. Jenssen, R. Morris, and E. O’Dell, “Tunable Alexandrite lasers,” IEEE J. Quantum Electron. 16(12), 1302–1315 (1980).
[Crossref]

1970 (1)

Ait-Ameur, K.

Aït-Ameur, K.

N. Passilly, E. Haouas, V. Ménard, R. Moncorgé, and K. Aït-Ameur, “Population lensing effect in cr:lisaf probed by z-scan technique,” Opt. Commun. 260(2), 703–707 (2006).
[Crossref]

Akbari, R.

Amarande, S. A.

S. A. Amarande and M. J. Damzen, “Measurement of the thermal lens of grazing-incidence diode-pumped nd:yvo4 laser amplifier,” Opt. Commun. 265(1), 306–313 (2006).
[Crossref]

Anashkina, E.

Antipov, O.

Antipov, O. L.

Aschoff, H. E.

J. W. Kuper, T. Chin, and H. E. Aschoff, “Extended tuning range of alexandrite at elevated temperatures,” in Advanced Solid State Lasers, (Optical Society of America, 1990, p. CL3.

Balembois, F.

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

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
[Crossref]

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
[Crossref]

Beecher, S. J.

Beyatli, E.

Blows, J. L.

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” J. Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

Bredikhin, D. V.

Brown, D. C.

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in nd:yag,” IEEE J. Quantum Electron. 34(3), 560–572 (1998).
[Crossref]

Burnham, D. C.

Cante, S.

Catunda, T.

Chenais, S.

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

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
[Crossref]

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
[Crossref]

Chin, T.

J. W. Kuper, T. Chin, and H. E. Aschoff, “Extended tuning range of alexandrite at elevated temperatures,” in Advanced Solid State Lasers, (Optical Society of America, 1990, p. CL3.

Cihan, C.

Cini, L.

L. Cini and J. I. Mackenzie, “Analytical thermal model for end-pumped solid-state lasers,” Appl. Phys. B: Lasers Opt. 123(12), 273 (2017).
[Crossref]

Clarkson, W. A.

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

Cruz, R. A.

Damzen, M. J.

G. Tawy and M. J. Damzen, “Tunable, dual wavelength and self-q-switched alexandrite laser using crystal birefringence control,” Opt. Express 27(13), 17507–17520 (2019).
[Crossref]

U. Parali, X. Sheng, A. Minassian, G. Tawy, J. Sathian, G. M. Thomas, and M. J. Damzen, “Diode-pumped alexandrite laser with passive sesam q-switching and wavelength tunability,” Opt. Commun. 410, 970–976 (2018).
[Crossref]

W. R. Kerridge-Johns and M. J. Damzen, “Temperature effects on tunable cw alexandrite lasers under diode end-pumping,” Opt. Express 26(6), 7771–7785 (2018).
[Crossref]

X. Sheng, G. Tawy, J. Sathian, A. Minassian, and M. J. Damzen, “Unidirectional single-frequency operation of a continuous-wave alexandrite ring laser with wavelength tunability,” Opt. Express 26(24), 31129–31136 (2018).
[Crossref]

M. J. Damzen, G. M. Thomas, and A. Minassian, “Diode-side-pumped alexandrite slab lasers,” Opt. Express 25(10), 11622–11636 (2017).
[Crossref]

G. M. Thomas, A. Minassian, X. Sheng, and M. J. Damzen, “Diode-pumped alexandrite lasers in q-switched and cavity-dumped q-switched operation,” Opt. Express 24(24), 27212–27224 (2016).
[Crossref]

W. R. Kerridge-Johns and M. J. Damzen, “Analytical model of tunable alexandrite lasing under diode end-pumping with experimental comparison,” J. Opt. Soc. Am. B 33(12), 2525–2534 (2016).
[Crossref]

W. R. Kerridge-Johns and M. J. Damzen, “Analysis of pump excited state absorption and its impact on laser efficiency,” Laser Phys. Lett. 12(12), 125002 (2015).
[Crossref]

A. Teppitaksak, A. Minassian, G. M. Thomas, and M. J. Damzen, “High efficiency >26w diode end-pumped alexandrite laser,” Opt. Express 22(13), 16386–16392 (2014).
[Crossref]

S. A. Amarande and M. J. Damzen, “Measurement of the thermal lens of grazing-incidence diode-pumped nd:yvo4 laser amplifier,” Opt. Commun. 265(1), 306–313 (2006).
[Crossref]

Dawes, J. M.

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” J. Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

Demirbas, U.

Doualan, J.-L.

Druon, F.

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

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
[Crossref]

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
[Crossref]

Eremeykin, O. N.

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[Crossref]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[Crossref]

Forget, S.

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

Fromager, M.

Georges, P.

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

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
[Crossref]

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
[Crossref]

Ghanbari, S.

Godin, T.

Graf, T.

Haouas, E.

N. Passilly, E. Haouas, V. Ménard, R. Moncorgé, and K. Aït-Ameur, “Population lensing effect in cr:lisaf probed by z-scan technique,” Opt. Commun. 260(2), 703–707 (2006).
[Crossref]

Hoffmann, H.-D.

Höffner, J.

Impinnisi, P. R.

V. Pilla, P. R. Impinnisi, and T. Catunda, “Measurement of saturation intensities in ion doped solids by transient nonlinear refraction,” Appl. Phys. Lett. 70(7), 817–819 (1997).
[Crossref]

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[Crossref]

Ivakin, E. V.

Jenssen, H.

M. Shand and H. Jenssen, “Temperature dependence of the excited-state absorption of alexandrite,” IEEE J. Quantum Electron. 19(3), 480–484 (1983).
[Crossref]

J. Walling, O. Peterson, H. Jenssen, R. Morris, and E. O’Dell, “Tunable Alexandrite lasers,” IEEE J. Quantum Electron. 16(12), 1302–1315 (1980).
[Crossref]

Jungbluth, B.

Kerridge-Johns, W. R.

Kocabas, C.

Koechner, W.

W. Koechner, Solid-state laser engineering (Springer, 2006, 6 ed.).

Kuper, J. W.

J. W. Kuper, T. Chin, and H. E. Aschoff, “Extended tuning range of alexandrite at elevated temperatures,” in Advanced Solid State Lasers, (Optical Society of America, 1990, p. CL3.

Kurt, A.

Loiko, P.

Lübken, F.-J.

Lucas-Leclin, G.

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
[Crossref]

S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
[Crossref]

Mackenzie, J. I.

S. Cante, S. J. Beecher, and J. I. Mackenzie, “Characterising energy transfer upconversion in nd-doped vanadates at elevated temperatures,” Opt. Express 26(6), 6478–6489 (2018).
[Crossref]

L. Cini and J. I. Mackenzie, “Analytical thermal model for end-pumped solid-state lasers,” Appl. Phys. B: Lasers Opt. 123(12), 273 (2017).
[Crossref]

Major, A.

Matrosov, V.

Ménard, V.

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N. Passilly, E. Haouas, V. Ménard, R. Moncorgé, and K. Aït-Ameur, “Population lensing effect in cr:lisaf probed by z-scan technique,” Opt. Commun. 260(2), 703–707 (2006).
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[Crossref]

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U. Parali, X. Sheng, A. Minassian, G. Tawy, J. Sathian, G. M. Thomas, and M. J. Damzen, “Diode-pumped alexandrite laser with passive sesam q-switching and wavelength tunability,” Opt. Commun. 410, 970–976 (2018).
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B. Neuenschwander, R. Weber, and H. P. Weber, “Determination of the thermal lens in solid-state lasers with stable cavities,” IEEE J. Quantum Electron. 31(6), 1082–1087 (1995).
[Crossref]

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B. Neuenschwander, R. Weber, and H. P. Weber, “Determination of the thermal lens in solid-state lasers with stable cavities,” IEEE J. Quantum Electron. 31(6), 1082–1087 (1995).
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Appl. Opt. (1)

Appl. Phys. B: Lasers Opt. (1)

L. Cini and J. I. Mackenzie, “Analytical thermal model for end-pumped solid-state lasers,” Appl. Phys. B: Lasers Opt. 123(12), 273 (2017).
[Crossref]

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M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[Crossref]

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D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in nd:yag,” IEEE J. Quantum Electron. 34(3), 560–572 (1998).
[Crossref]

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S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part i: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004).
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S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-part ii: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40(9), 1235–1243 (2004).
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J. Walling, O. Peterson, H. Jenssen, R. Morris, and E. O’Dell, “Tunable Alexandrite lasers,” IEEE J. Quantum Electron. 16(12), 1302–1315 (1980).
[Crossref]

M. Shand and H. Jenssen, “Temperature dependence of the excited-state absorption of alexandrite,” IEEE J. Quantum Electron. 19(3), 480–484 (1983).
[Crossref]

J. Appl. Phys. (1)

J. L. Blows, J. M. Dawes, and T. Omatsu, “Thermal lensing measurements in line-focus end-pumped neodymium yttrium aluminium garnet using holographic lateral shearing interferometry,” J. Appl. Phys. 83(6), 2901–2906 (1998).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. D: Appl. Phys. (1)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D: Appl. Phys. 34(16), 2381–2395 (2001).
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W. R. Kerridge-Johns and M. J. Damzen, “Analysis of pump excited state absorption and its impact on laser efficiency,” Laser Phys. Lett. 12(12), 125002 (2015).
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N. Passilly, E. Haouas, V. Ménard, R. Moncorgé, and K. Aït-Ameur, “Population lensing effect in cr:lisaf probed by z-scan technique,” Opt. Commun. 260(2), 703–707 (2006).
[Crossref]

U. Parali, X. Sheng, A. Minassian, G. Tawy, J. Sathian, G. M. Thomas, and M. J. Damzen, “Diode-pumped alexandrite laser with passive sesam q-switching and wavelength tunability,” Opt. Commun. 410, 970–976 (2018).
[Crossref]

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Opt. Express (8)

Opt. Lett. (3)

Opt. Mater. Express (3)

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S. Chenais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: The case of ytterbium-doped materials,” Prog. Quantum Electron. 30(4), 89–153 (2006).
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W. Koechner, Solid-state laser engineering (Springer, 2006, 6 ed.).

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Figures (10)

Fig. 1.
Fig. 1. Energy levels of Alexandrite showing electronic transitions and their respective cross sections with broad bands representing the vibrational levels. Non-radiative transitions are shown as dashed lines.
Fig. 2.
Fig. 2. (a) Schematic of the experimental pump-induced lensing measurement system of the end-pumped Alexandrite rod in a laser cavity (formed by mirrors BM and OC). $f_1$ and $f_2$ form an afocal magnifying telescope to relay-image a probe beam wavefront at the Alexandrite crystal onto a Shack-Hartmann wavefront sensor (SH-WFS). DM1 and DM2 are dichroic mirrors to separate pump, probe and laser wavelengths. (b) Measured radial intensity profile of the pump beam at focus (red line) together with a Gaussian ($n=2$) fit (dashed blue line). Inset shows 2-D pump beam profile.
Fig. 3.
Fig. 3. Measured thermal lens dioptric power under non-lasing conditions (blue), lasing conditions (red) and laser power (green) as a function of the absorbed pump power.
Fig. 4.
Fig. 4. (a) Laser power as a function of absorbed pump power. Inset shows laser wavelength spectrum and beam profile at maximum power. (b) 2-D wavefront at maximum pump power under non-lasing and lasing conditions.
Fig. 5.
Fig. 5. Effective heating factor (fractional heating factor calculated from FEA model) under non-lasing (blue), lasing conditions (red) and laser power (green) as a function of the absorbed pump power.
Fig. 6.
Fig. 6. Fractional heating factor as a function of absorbed pump power.
Fig. 7.
Fig. 7. Measured (blue-solid) and analytical (blue-dashed) lens dioptric power as a function of the absorbed pump power under non-lasing conditions. Thermal (orange-dashed) and population (green-dashed) lens components of the total lens are also shown.
Fig. 8.
Fig. 8. Measured (red-solid) and analytical (red-dashed) lens dioptric power as a function of the absorbed pump power under lasing conditions. Thermal (orange-dashed) and population (green-dashed) lens components of the total lens are also shown.
Fig. 9.
Fig. 9. Wavefront measured by the SH-WFS (shown in red) as a function of the radial coordinate, $r$, at the pump region at maximum pump power (8.3 W). A theoretical fitting applied to the data (shown in blue) is used to determine the lens dioptric power, $D$ and quartic aberration coefficient, $w_{40}$, under non-lasing (a) and lasing conditions (b).
Fig. 10.
Fig. 10. (a) Quartic aberration coefficient as a function of absorbed pump power under non lasing and lasing conditions. (b) Measured (solid) and fitted thermal lens dioptric power (dashed) as a function of absorbed pump power.

Tables (1)

Tables Icon

Table 1. Alexandrite and pump parameters used for the FEA model. Temperature dependent parameters have been evaluated at T = 16 °C. Thermo-optic ceofficient (zero-strain) and refractive index is that for light polarised to the crystal b-axis at probe wavelength 532 nm.

Equations (28)

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1 r r ( r T r ) + 2 T z 2 = Q ( r , z ) K c
D T = 1 f T = P a b s χ η h 2 π w p 2 K c
D T ( r ) = 1 f T ( r ) = P a b s χ η h π w p 2 K c [ 1 exp ( 2 r 2 / w p 2 ) ] 2 r 2 / w p 2 .
η h = 1 η s η p .
η p , e s a = σ 0 n 0 σ 0 n 0 + σ 1 n 1 = 1 f 1 + ( γ 1 ) f
Δ n e ( r , z ) = 2 π n f L 2 N Δ α p n 1 ( r , z ) N = C n 1 ( r , z ) N
W ( r ) = 0 l Δ n e ( r , z ) d z = C 0 l n 1 ( r , z ) N d z = C F ( r )
D P = 2 r 2 Δ W = 2 r 2 C ( F ( 0 ) F ( r ) ) .
η p , e s a = 1 T e σ 1 N F 1 T σ 1 N F e σ 1 N F 1
I I s = 1 1 T e σ 1 N F e σ 1 N F 1 γ
η p , e s a 1 1 2 σ 1 N F ,
F 1 σ 0 N [ I I s γ 2 ( I I s ) 2 ] .
η h = 1 η s η p , e s a 1 η s ( 1 γ 2 [ P 0 P s γ 2 ( P 0 P s ) 2 ] )
D T ( 0 ) = χ π w p 2 K c [ η h 0 P 0 + η s γ 2 P 0 2 P s ]
D P ( 0 ) = 4 C α 0 w p 2 [ P 0 P s γ ( P 0 P s ) 2 ] .
D NL ( 0 ) = [ χ η h 0 π w p 2 K c + 4 C α 0 w p 2 P s ] P 0 + [ χ π w p 2 K c γ 2 η s P s γ 4 C α 0 w p 2 P s 2 ] P 0 2 .
D NL ( 0 ) = χ π w p 2 K c [ η h 0 + η s 2 ] P 0 = P 0 χ η h π w p 2 K c
w p 2 = 0 l α 0 e α 0 z w 2 ( z ) d z 0 l α 0 e α 0 z d z
η h = 1 η s ( 1 γ 2 [ P t h P s γ 2 P t h 2 P s 2 ] ) ( 1 γ l )
D L ( 0 ) = P 0 χ π w p 2 K c [ η h 0 + η s γ 2 P t h P s ] ( 1 γ l ) + 4 C α 0 w p 2 [ P t h P s γ P t h 2 P s 2 ]
Δ W = r 2 2 D w 40 r 4
η h = 1 η s η p , e s a 1 η s ( 1 1 2 σ 1 N F )
η h = 1 η s η p , e s a 1 η s ( 1 γ 2 [ I I s γ 2 ( I I s ) 2 ] ) .
η h = 1 η s η p , e s a 1 η s ( 1 γ 2 [ P 0 P s γ 2 ( P 0 P s ) 2 ] ) .
F ( r ) 1 σ 0 N [ I ( r ) I s γ 2 ( I ( r ) I s ) 2 ] .
F ( r ) = γ σ 1 N [ P 0 P s e 2 r 2 / w p 2 γ 2 ( P 0 P s e 2 r 2 / w p 2 ) 2 ] .
F ( r ) = 1 σ 0 N [ ( P 0 P s γ 2 ( P 0 P s ) 2 ) 2 r 2 w p 2 ( P 0 P s γ ( P 0 P s ) 2 ) ] .
D P ( 0 ) = 4 C α 0 w p 2 [ P 0 P s γ ( P 0 2 P s ) 2 ] .

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