Abstract

We present a giant-pulse generation laser realized by the emission cross-section control of a gain medium in a passively Q-switched Nd:YVO4 microchip laser with a Cr4+:YAG saturable absorber. Up to 1.17 MW peak power and 1.03 mJ pulse energy were obtained with a 100 Hz repetition rate. By combining the Nd:YVO4 crystal with a Sapphire plate, lower temperature difference between a pump region in the gain crystal and a crystal holder was obtained which helped to keep the cavity in stability zone at elevated temperatures and allowed the achievement of the high peak power for this laser system.

© 2016 Optical Society of America

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Corrections

9 February 2016: A correction was made to Fig. 7.


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References

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  1. T. Taira, “Domain-controlled laser ceramics toward giant micro-photonics [Invited],” Opt. Mater. Express 1(5), 1040–1050 (2011).
    [Crossref]
  2. M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
    [Crossref]
  3. J. Gao, T. Sakamoto, R. Bhandari, T. Taira, S. Ishiuchi, and Y. Furukawa, “Investigation of post-ionizations of sputtered atoms in TOF-MS using a palm-top-size megawatt microlaser,” presented at the 2nd Laser Ignition Conference, Pacifico Yokohama, Japan, 22–24 April, 2014. Paper LIC4–2.
  4. R. Bhandari and T. Taira, “Palm-top size megawatt peak power ultraviolet microlaser,” Opt. Eng. 52(7), 076102 (2013).
    [Crossref]
  5. T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO(4) microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
    [Crossref] [PubMed]
  6. Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82(6), 844–846 (2003).
    [Crossref]
  7. Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO(4), YVO(4), and Y(3)Al(5)O(12) measured by quasi-one-dimensional flash method,” Opt. Express 14(22), 10528–10536 (2006).
    [Crossref] [PubMed]
  8. T. Taira and T. Kobayashi, “Q-switching and frequency doubling of solid-state lasers by a single intracavity KTP crystal,” IEEE J. Quantum Electron. 30(3), 800–804 (1994).
    [Crossref]
  9. T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
    [Crossref]
  10. U. Brauch and J. Muckenschnabel, “Temperature dependence of flashlamp-pumped Nd:YAG and Nd:Cr:GSGG lasers,” Opt. Commun. 73(1), 62–66 (1989).
    [Crossref]
  11. M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
    [Crossref]
  12. O. Kimmelma, I. Tittonen, and S. C. Buchter, “Thermal tuning of laser pulse parameters in passively Q-switched Nd:YAG lasers,” Appl. Opt. 47(23), 4262–4266 (2008).
    [Crossref] [PubMed]
  13. G. Turri, H. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
    [Crossref]
  14. X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
    [Crossref]
  15. Y. Sato and T. Taira, “Temperature dependencies of stimulated emission cross section for Nd-doped solid-state laser materials,” Opt. Mater. Express 2(8), 1076–1087 (2012).
    [Crossref]
  16. S. Joly and T. Taira, “Novel method for pulse control in Nd:YVO4/Cr4+:YAG passively Q-switched microchip laser,” in CLEO/Europe and EQEC 2011 Conference Digest, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CA8_1.
  17. M. Nie, Q. Liu, E. Ji, and M. Gong, “End-pumped temperature-dependent passively Q-switched lasers,” Appl. Opt. 54(28), 8383–8387 (2015).
    [Crossref] [PubMed]
  18. N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAGCr(4+):YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
    [Crossref] [PubMed]
  19. Y.-F. Chen and Y.P. Lan, “Comparison between c-cut and a-cut Nd:YVO4 lasers passively Q-switched with a Cr4+:YAG saturable absorber,” Appl. Phys. B 74(4–5), 415–418 (2002).
  20. N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
    [Crossref]
  21. J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
    [Crossref]
  22. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd 3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
    [Crossref] [PubMed]
  23. W. Koechner, Solid State Laser Engineering, 6th ed. (Springer, 2006).
  24. Y. Sato and T. Taira, “Highly accurate interferometric evaluation of thermal expansion and dn/dT of optical materials,” Opt. Mater. Express 4(5), 876–888 (2014).
    [Crossref]
  25. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

2015 (1)

2014 (1)

2013 (1)

R. Bhandari and T. Taira, “Palm-top size megawatt peak power ultraviolet microlaser,” Opt. Eng. 52(7), 076102 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (2)

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAGCr(4+):YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[Crossref] [PubMed]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

2009 (1)

2008 (2)

2007 (1)

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[Crossref]

2006 (1)

2003 (2)

M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
[Crossref]

Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82(6), 844–846 (2003).
[Crossref]

2002 (1)

Y.-F. Chen and Y.P. Lan, “Comparison between c-cut and a-cut Nd:YVO4 lasers passively Q-switched with a Cr4+:YAG saturable absorber,” Appl. Phys. B 74(4–5), 415–418 (2002).

2001 (1)

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
[Crossref]

1995 (1)

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
[Crossref]

1994 (1)

T. Taira and T. Kobayashi, “Q-switching and frequency doubling of solid-state lasers by a single intracavity KTP crystal,” IEEE J. Quantum Electron. 30(3), 800–804 (1994).
[Crossref]

1991 (1)

1989 (1)

U. Brauch and J. Muckenschnabel, “Temperature dependence of flashlamp-pumped Nd:YAG and Nd:Cr:GSGG lasers,” Opt. Commun. 73(1), 62–66 (1989).
[Crossref]

Ando, A.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Balembois, F.

Bass, M.

G. Turri, H. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
[Crossref]

M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
[Crossref]

Bhandari, R.

R. Bhandari and T. Taira, “Palm-top size megawatt peak power ultraviolet microlaser,” Opt. Eng. 52(7), 076102 (2013).
[Crossref]

Brauch, U.

U. Brauch and J. Muckenschnabel, “Temperature dependence of flashlamp-pumped Nd:YAG and Nd:Cr:GSGG lasers,” Opt. Commun. 73(1), 62–66 (1989).
[Crossref]

Brickeen, B. K.

M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
[Crossref]

Buchter, S. C.

Chen, Y.-F.

Y.-F. Chen and Y.P. Lan, “Comparison between c-cut and a-cut Nd:YVO4 lasers passively Q-switched with a Cr4+:YAG saturable absorber,” Appl. Phys. B 74(4–5), 415–418 (2002).

Cornacchia, F.

Degnan, J. J.

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
[Crossref]

Délen, X.

Georges, P.

Gong, M.

Inohara, T.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Jenssen, H.

Ji, E.

Kan, H.

Kanehara, K.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Kido, N.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Kimmelma, O.

Kobayashi, T.

T. Taira and T. Kobayashi, “Q-switching and frequency doubling of solid-state lasers by a single intracavity KTP crystal,” IEEE J. Quantum Electron. 30(3), 800–804 (1994).
[Crossref]

T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO(4) microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
[Crossref] [PubMed]

Kurimura, S.

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
[Crossref]

Lan, Y.P.

Y.-F. Chen and Y.P. Lan, “Comparison between c-cut and a-cut Nd:YVO4 lasers passively Q-switched with a Cr4+:YAG saturable absorber,” Appl. Phys. B 74(4–5), 415–418 (2002).

Liu, Q.

Lupei, V.

Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82(6), 844–846 (2003).
[Crossref]

Muckenschnabel, J.

U. Brauch and J. Muckenschnabel, “Temperature dependence of flashlamp-pumped Nd:YAG and Nd:Cr:GSGG lasers,” Opt. Commun. 73(1), 62–66 (1989).
[Crossref]

Mukai, A.

Nie, M.

Nozawa, Y.

Pavel, N.

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAGCr(4+):YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[Crossref] [PubMed]

Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82(6), 844–846 (2003).
[Crossref]

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
[Crossref]

Saikawa, J.

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
[Crossref]

Sakai, H.

Sato, Y.

Taira, T.

Y. Sato and T. Taira, “Highly accurate interferometric evaluation of thermal expansion and dn/dT of optical materials,” Opt. Mater. Express 4(5), 876–888 (2014).
[Crossref]

R. Bhandari and T. Taira, “Palm-top size megawatt peak power ultraviolet microlaser,” Opt. Eng. 52(7), 076102 (2013).
[Crossref]

Y. Sato and T. Taira, “Temperature dependencies of stimulated emission cross section for Nd-doped solid-state laser materials,” Opt. Mater. Express 2(8), 1076–1087 (2012).
[Crossref]

T. Taira, “Domain-controlled laser ceramics toward giant micro-photonics [Invited],” Opt. Mater. Express 1(5), 1040–1050 (2011).
[Crossref]

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAGCr(4+):YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[Crossref] [PubMed]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd 3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
[Crossref] [PubMed]

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[Crossref]

Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO(4), YVO(4), and Y(3)Al(5)O(12) measured by quasi-one-dimensional flash method,” Opt. Express 14(22), 10528–10536 (2006).
[Crossref] [PubMed]

Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82(6), 844–846 (2003).
[Crossref]

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
[Crossref]

T. Taira and T. Kobayashi, “Q-switching and frequency doubling of solid-state lasers by a single intracavity KTP crystal,” IEEE J. Quantum Electron. 30(3), 800–804 (1994).
[Crossref]

T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO(4) microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
[Crossref] [PubMed]

Tittonen, I.

Tonelli, M.

Tsunekane, M.

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAGCr(4+):YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[Crossref] [PubMed]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Turri, G.

Vigil, S.

M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
[Crossref]

Weichman, L. S.

M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

Y.-F. Chen and Y.P. Lan, “Comparison between c-cut and a-cut Nd:YVO4 lasers passively Q-switched with a Cr4+:YAG saturable absorber,” Appl. Phys. B 74(4–5), 415–418 (2002).

Appl. Phys. Lett. (1)

Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82(6), 844–846 (2003).
[Crossref]

IEEE J. Quantum Electron. (4)

T. Taira and T. Kobayashi, “Q-switching and frequency doubling of solid-state lasers by a single intracavity KTP crystal,” IEEE J. Quantum Electron. 30(3), 800–804 (1994).
[Crossref]

M. Bass, L. S. Weichman, S. Vigil, and B. K. Brickeen, “The temperature dependence of Nd3+ doped solid-state lasers,” IEEE J. Quantum Electron. 39(6), 741–748 (2003).
[Crossref]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
[Crossref]

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

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[Crossref]

J. Opt. Soc. Am. B (2)

Jpn. J. Appl. Phys. (1)

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(1), 1253–1259 (2001).
[Crossref]

Opt. Commun. (1)

U. Brauch and J. Muckenschnabel, “Temperature dependence of flashlamp-pumped Nd:YAG and Nd:Cr:GSGG lasers,” Opt. Commun. 73(1), 62–66 (1989).
[Crossref]

Opt. Eng. (1)

R. Bhandari and T. Taira, “Palm-top size megawatt peak power ultraviolet microlaser,” Opt. Eng. 52(7), 076102 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Opt. Mater. Express (3)

Other (4)

W. Koechner, Solid State Laser Engineering, 6th ed. (Springer, 2006).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

J. Gao, T. Sakamoto, R. Bhandari, T. Taira, S. Ishiuchi, and Y. Furukawa, “Investigation of post-ionizations of sputtered atoms in TOF-MS using a palm-top-size megawatt microlaser,” presented at the 2nd Laser Ignition Conference, Pacifico Yokohama, Japan, 22–24 April, 2014. Paper LIC4–2.

S. Joly and T. Taira, “Novel method for pulse control in Nd:YVO4/Cr4+:YAG passively Q-switched microchip laser,” in CLEO/Europe and EQEC 2011 Conference Digest, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CA8_1.

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Figures (7)

Fig. 1
Fig. 1 Passively Q-switched output parameter dependence on temperature. Output energy, pulse duration, and peak power are normalized to its maximum value.
Fig. 2
Fig. 2 Experimental setup.
Fig. 3
Fig. 3 Q-switch output for 100 Hz repetition rate operation for both the single Nd:YVO4 (black square) and composite Sapphire/Nd:YVO4 (red circle) crystals. Pulse energy (a), peak power, (b) and pulse duration (c) are shown.
Fig. 4
Fig. 4 Q-switch output for the 1 kHz repetition rate operation. (a) Output energy for the single Nd:YVO4 crystal. (b) Output energy for the composite Sapphire/Nd:YVO4 crystal. The caption on both graphs shows the beam profile of the last measured point. The multiple beam peak pattern of the profile is visible.
Fig. 5
Fig. 5 Experimental data compared with the Q-switched model discussed in this work at 100 Hz repetition rate for the composite Sapphire/Nd:YVO4 crystal. The black circle represents the measured data, the red dashed line shows model results for the case when the mode size on the gain material is constant and the output energy depends only on the emission cross-section, which changes with temperature. The green dotted and dashed line shows results for an additional mode size change on the crystal due to higher order mode generation.
Fig. 6
Fig. 6 Experimental results for the M2 values at 100 Hz repetition rate for the composite Sapphire/Nd:YVO4 crystal. Vertical and horizontal values are measured.
Fig. 7
Fig. 7 Experimental results for the temperature measurement of the output surface of the composite Sapphire/Nd:YVO4 crystal. (a) The measurement setup: the center peak temperature Tp was measured with an infrared thermal camera and close up lens with 100-μm spatial resolution. (b) Experimental data: the difference ΔT between the center peak Tp and holder Th temperatures was measured. For 100 Hz operation, this difference is almost flat over a wide operational range. For 1 kHz, the temperature difference keeps increasing for the same pump conditions.

Tables (2)

Tables Icon

Table 1 Q-switched parameters used for calculation

Tables Icon

Table 2 Thermal lens and beam size parameters

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

E p = h ν A g 2 σ g γ g ln ( 1 R ) ln ( n g i n g f ) ,
P p = h ν A g l g γ g t r ln ( 1 R ) n g i [ ( 1 n g t n g i ) + 1 α p ( 1 δ ) ( 1 n g t α n g i α ) + [ 1 p ( 1 δ ) ] ln ( n g t n g i ) ] .
τ p = E p P p .
σ e m ( T ) = σ e m ( T 0 ) ( e 0 e 1 T + e 2 T 2 ) .
f t h = π K ( T ) P i n η a b s η h [ 1 2 d n d T + α w p ( n 1 ) l g ] 1 .
w 01 = λ L * π f t h 2 ( f t h d 2 ) d 2 ,
w l = M 2 w 01 .
d ϕ d t = ϕ t r [ 2 σ g n g l g 2 σ S A n S A l S A 2 σ E S A ( n S A i n S A ) l S A ( L ln R ) ] ,
d n g d t = W p n g τ g γ g σ g c ϕ n g ,
d n S A d t = n S A i n S A τ S A γ S A σ S A c ϕ n S A A g A S A ,
δ = σ E S A σ S A .
α = γ S A γ g σ S A σ g A g A S A
p = ln T 0 2 ln R + L ln T 0 2 .
n g i = ln R + L ln T 0 2 2 σ g l g .
1 n g t n g i p ( 1 δ ) ( 1 n g t α n g i α ) = 0.
( 1 n g f n g i ) 1 α p ( 1 δ ) ( 1 n g f α n g i α ) + [ 1 p ( 1 δ ) ] ln ( n g f n g i ) = 0.
E p = h ν A g 2 σ g γ g ln ( 1 R ) ln ( n g i n g f )
P p = h ν A g l g γ g t r ln ( 1 R ) n g i [ ( 1 n g t n g i ) + 1 α p ( 1 δ ) ( 1 n g t α n g i α ) + [ 1 p ( 1 δ ) ] ln ( n g t n g i ) ] .
g i * = g i d j f t h ( 1 1 ρ i ) ,
g i = 1 d 1 + d 2 ρ i ( 1 1 ρ i ) ,
L * = d 1 + d 2 d 1 d 2 f t h ,
w i , j 2 = λ L * π g j , i * g i , j * ( 1 g 1 g 2 ) .
w L 2 = w 1 2 [ ( 1 d 1 ρ 1 ) 2 + ( d 1 L * ) 2 g 1 * ( 1 g 1 * g 2 * ) g 2 * ] .

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