Abstract

A theoretical model of the self-sum-frequency mixing (self-SFM) laser generated by a single crystal is proposed in which the spatial distribution of the pump and circulating fundamental lasers with arbitrary beam waists is taken into account. The model is then applied to two kinds of crystal of current interest, Nd:YAl3(BO3)4 and Nd:Ca4GdO(BO3)3. Numerical analyses of the self-SFM laser’s properties predict and confirm some experimental results. Several ways to improve the self-SFM laser output are discussed and concluded. The model not only is applicable to self-SFM lasers but is also effective for general analyses of fundamental or nonlinear laser generation with Gaussian beams.

© 2001 Optical Society of America

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References

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  1. T. Kellner, F. Heine, and G. Huber, “Efficient laser performance of Nd:YAG at 946 nm and intracavity frequency doubling with LiIO3, beta-BaB2O4, and LiB3O5,” Appl. Phys. B 65, 789–792 (1997).
    [CrossRef]
  2. I. D. Lindsay and M. Ebrahimzadeh, “Efficient continuous-wave and Q-switched operation of a 946-nm Nd:YAG laser pumped by an injection-locked broad-area diode laser,” Appl. Opt. 37, 3961–3970 (1998).
    [CrossRef]
  3. D. Fluck and P. Gunter, “Efficient generation of cw blue light by sum-frequency mixing of laser diodes in KNbO3,” Opt. Commun. 136, 257–260 (1997).
    [CrossRef]
  4. W. P. Risk and W. Lenth, “Diode laser pumped blue-light source based on intracavity sum frequency generation,” Appl. Phys. Lett. 54, 789–791 (1989).
    [CrossRef]
  5. D. Jaque, J. Capmany, F. Molero, and J. Garcia Sole, “Blue-light laser source by sum-frequency mixing in Nd:YAl3(BO3)4,” Appl. Phys. Lett. 73, 3659–3661 (1998).
    [CrossRef]
  6. D. Jaque, J. Capmany, and J. Garcia Sole, “Red, green, and blue laser light from a single Nd:YAl3(BO3)4 crystal based on laser oscillation at 1.3 μm,” Appl. Phys. Lett. 75, 325–327 (1999).
    [CrossRef]
  7. A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
    [CrossRef]
  8. A. Brenier and G. Boulon, “Self-frequency summing NYAB laser for tunable UV generation,” J. Lumin. 86, 125–128 (2000).
    [CrossRef]
  9. F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
    [CrossRef]
  10. D. Jaque, J. Capmany, and J. Garcia Sole, “Red, blue and green laser-light generation from the NYAB nonlinear crystal,” in Solid State Lasers VIII, Proc. SPIE 3613, 140–150 (1999).
    [CrossRef]
  11. W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5, 1412–1423 (1988).
    [CrossRef]
  12. A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25, 760–766 (1989).
    [CrossRef]
  13. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  14. W. Koechner, “Laser oscillator,” in Solid-State Laser Engineering, 3rd ed., A. L. Schawlow, K. Shimoda, A. E. Siegman, and T. Tamir, eds. (Springer-Verlag, Berlin, 1992), Chap. 3.
  15. D. Findlay and R. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966).
    [CrossRef]
  16. D. Shen, A. Liu, J. Song, and K. I. Ueda, “Efficient operation of an intracavity-doubled Nd:YVO4/KTP laser end pumped by a high-brightness laser diode,” Appl. Opt. 37, 7785–7788 (1998).
    [CrossRef]
  17. D. Jaque, J. Capmany, J. Garcia Sole, Z. D. Luo, and A. D. Jiang, “Continuous-wave laser properties of the self-frequency-doubling YAl3(BO3)4:Nd crystal,” J. Opt. Soc. Am. B 15, 1656–1662 (1998).
    [CrossRef]
  18. Z.-d. Luo, “The optimum neodymium concentration of self-frequency-doubling laser crystal NYAB,” Progr. Nat. Sci. 4, 504–508 (1994).
  19. F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
    [CrossRef]

2000 (1)

A. Brenier and G. Boulon, “Self-frequency summing NYAB laser for tunable UV generation,” J. Lumin. 86, 125–128 (2000).
[CrossRef]

1999 (4)

F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
[CrossRef]

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, blue and green laser-light generation from the NYAB nonlinear crystal,” in Solid State Lasers VIII, Proc. SPIE 3613, 140–150 (1999).
[CrossRef]

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, green, and blue laser light from a single Nd:YAl3(BO3)4 crystal based on laser oscillation at 1.3 μm,” Appl. Phys. Lett. 75, 325–327 (1999).
[CrossRef]

A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
[CrossRef]

1998 (4)

1997 (3)

T. Kellner, F. Heine, and G. Huber, “Efficient laser performance of Nd:YAG at 946 nm and intracavity frequency doubling with LiIO3, beta-BaB2O4, and LiB3O5,” Appl. Phys. B 65, 789–792 (1997).
[CrossRef]

D. Fluck and P. Gunter, “Efficient generation of cw blue light by sum-frequency mixing of laser diodes in KNbO3,” Opt. Commun. 136, 257–260 (1997).
[CrossRef]

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

1994 (1)

Z.-d. Luo, “The optimum neodymium concentration of self-frequency-doubling laser crystal NYAB,” Progr. Nat. Sci. 4, 504–508 (1994).

1989 (2)

A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25, 760–766 (1989).
[CrossRef]

W. P. Risk and W. Lenth, “Diode laser pumped blue-light source based on intracavity sum frequency generation,” Appl. Phys. Lett. 54, 789–791 (1989).
[CrossRef]

1988 (1)

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1966 (1)

D. Findlay and R. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966).
[CrossRef]

Aka, G.

F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
[CrossRef]

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Alfrey, A. J.

A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25, 760–766 (1989).
[CrossRef]

Benitez, J. M.

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Boulon, G.

A. Brenier and G. Boulon, “Self-frequency summing NYAB laser for tunable UV generation,” J. Lumin. 86, 125–128 (2000).
[CrossRef]

A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Brenier, A.

A. Brenier and G. Boulon, “Self-frequency summing NYAB laser for tunable UV generation,” J. Lumin. 86, 125–128 (2000).
[CrossRef]

A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
[CrossRef]

Capmany, J.

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, green, and blue laser light from a single Nd:YAl3(BO3)4 crystal based on laser oscillation at 1.3 μm,” Appl. Phys. Lett. 75, 325–327 (1999).
[CrossRef]

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, blue and green laser-light generation from the NYAB nonlinear crystal,” in Solid State Lasers VIII, Proc. SPIE 3613, 140–150 (1999).
[CrossRef]

D. Jaque, J. Capmany, F. Molero, and J. Garcia Sole, “Blue-light laser source by sum-frequency mixing in Nd:YAl3(BO3)4,” Appl. Phys. Lett. 73, 3659–3661 (1998).
[CrossRef]

D. Jaque, J. Capmany, J. Garcia Sole, Z. D. Luo, and A. D. Jiang, “Continuous-wave laser properties of the self-frequency-doubling YAl3(BO3)4:Nd crystal,” J. Opt. Soc. Am. B 15, 1656–1662 (1998).
[CrossRef]

Clay, R.

D. Findlay and R. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966).
[CrossRef]

Ebrahimzadeh, M.

Findlay, D.

D. Findlay and R. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966).
[CrossRef]

Fluck, D.

D. Fluck and P. Gunter, “Efficient generation of cw blue light by sum-frequency mixing of laser diodes in KNbO3,” Opt. Commun. 136, 257–260 (1997).
[CrossRef]

Garcia Sole, J.

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, blue and green laser-light generation from the NYAB nonlinear crystal,” in Solid State Lasers VIII, Proc. SPIE 3613, 140–150 (1999).
[CrossRef]

Gunter, P.

D. Fluck and P. Gunter, “Efficient generation of cw blue light by sum-frequency mixing of laser diodes in KNbO3,” Opt. Commun. 136, 257–260 (1997).
[CrossRef]

Heine, F.

T. Kellner, F. Heine, and G. Huber, “Efficient laser performance of Nd:YAG at 946 nm and intracavity frequency doubling with LiIO3, beta-BaB2O4, and LiB3O5,” Appl. Phys. B 65, 789–792 (1997).
[CrossRef]

Huber, G.

T. Kellner, F. Heine, and G. Huber, “Efficient laser performance of Nd:YAG at 946 nm and intracavity frequency doubling with LiIO3, beta-BaB2O4, and LiB3O5,” Appl. Phys. B 65, 789–792 (1997).
[CrossRef]

Hubert, H.

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Jaque, D.

A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
[CrossRef]

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, blue and green laser-light generation from the NYAB nonlinear crystal,” in Solid State Lasers VIII, Proc. SPIE 3613, 140–150 (1999).
[CrossRef]

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, green, and blue laser light from a single Nd:YAl3(BO3)4 crystal based on laser oscillation at 1.3 μm,” Appl. Phys. Lett. 75, 325–327 (1999).
[CrossRef]

D. Jaque, J. Capmany, F. Molero, and J. Garcia Sole, “Blue-light laser source by sum-frequency mixing in Nd:YAl3(BO3)4,” Appl. Phys. Lett. 73, 3659–3661 (1998).
[CrossRef]

D. Jaque, J. Capmany, J. Garcia Sole, Z. D. Luo, and A. D. Jiang, “Continuous-wave laser properties of the self-frequency-doubling YAl3(BO3)4:Nd crystal,” J. Opt. Soc. Am. B 15, 1656–1662 (1998).
[CrossRef]

Jiang, A. D.

Kahn-Harari, A.

F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
[CrossRef]

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Kellner, T.

T. Kellner, F. Heine, and G. Huber, “Efficient laser performance of Nd:YAG at 946 nm and intracavity frequency doubling with LiIO3, beta-BaB2O4, and LiB3O5,” Appl. Phys. B 65, 789–792 (1997).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Lenth, W.

W. P. Risk and W. Lenth, “Diode laser pumped blue-light source based on intracavity sum frequency generation,” Appl. Phys. Lett. 54, 789–791 (1989).
[CrossRef]

Lindsay, I. D.

Liu, A.

Luo, Z. D.

Luo, Z.-d.

Z.-d. Luo, “The optimum neodymium concentration of self-frequency-doubling laser crystal NYAB,” Progr. Nat. Sci. 4, 504–508 (1994).

Molero, F.

D. Jaque, J. Capmany, F. Molero, and J. Garcia Sole, “Blue-light laser source by sum-frequency mixing in Nd:YAl3(BO3)4,” Appl. Phys. Lett. 73, 3659–3661 (1998).
[CrossRef]

Mougel, F.

F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
[CrossRef]

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Risk, W. P.

W. P. Risk and W. Lenth, “Diode laser pumped blue-light source based on intracavity sum frequency generation,” Appl. Phys. Lett. 54, 789–791 (1989).
[CrossRef]

W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses,” J. Opt. Soc. Am. B 5, 1412–1423 (1988).
[CrossRef]

Shen, D.

Sole, J. Garcia

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, green, and blue laser light from a single Nd:YAl3(BO3)4 crystal based on laser oscillation at 1.3 μm,” Appl. Phys. Lett. 75, 325–327 (1999).
[CrossRef]

A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
[CrossRef]

D. Jaque, J. Capmany, J. Garcia Sole, Z. D. Luo, and A. D. Jiang, “Continuous-wave laser properties of the self-frequency-doubling YAl3(BO3)4:Nd crystal,” J. Opt. Soc. Am. B 15, 1656–1662 (1998).
[CrossRef]

D. Jaque, J. Capmany, F. Molero, and J. Garcia Sole, “Blue-light laser source by sum-frequency mixing in Nd:YAl3(BO3)4,” Appl. Phys. Lett. 73, 3659–3661 (1998).
[CrossRef]

Song, J.

Ueda, K. I.

Vivien, D.

F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
[CrossRef]

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

T. Kellner, F. Heine, and G. Huber, “Efficient laser performance of Nd:YAG at 946 nm and intracavity frequency doubling with LiIO3, beta-BaB2O4, and LiB3O5,” Appl. Phys. B 65, 789–792 (1997).
[CrossRef]

Appl. Phys. Lett. (3)

W. P. Risk and W. Lenth, “Diode laser pumped blue-light source based on intracavity sum frequency generation,” Appl. Phys. Lett. 54, 789–791 (1989).
[CrossRef]

D. Jaque, J. Capmany, F. Molero, and J. Garcia Sole, “Blue-light laser source by sum-frequency mixing in Nd:YAl3(BO3)4,” Appl. Phys. Lett. 73, 3659–3661 (1998).
[CrossRef]

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, green, and blue laser light from a single Nd:YAl3(BO3)4 crystal based on laser oscillation at 1.3 μm,” Appl. Phys. Lett. 75, 325–327 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. J. Alfrey, “Modeling of longitudinally pumped cw Ti:sapphire laser oscillators,” IEEE J. Quantum Electron. 25, 760–766 (1989).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

J. Lumin. (1)

A. Brenier and G. Boulon, “Self-frequency summing NYAB laser for tunable UV generation,” J. Lumin. 86, 125–128 (2000).
[CrossRef]

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

Opt. Commun. (1)

D. Fluck and P. Gunter, “Efficient generation of cw blue light by sum-frequency mixing of laser diodes in KNbO3,” Opt. Commun. 136, 257–260 (1997).
[CrossRef]

Opt. Mater. (3)

F. Mougel, G. Aka, A. Kahn-Harari, and D. Vivien, “CW blue laser generation by self-sum frequency mixing in Nd:GdCOB single crystal,” Opt. Mater. 13, 293–297 (1999).
[CrossRef]

A. Brenier, G. Boulon, D. Jaque, and J. Garcia Sole, “Self-frequency-summing NYAB laser for tunable blue generation,” Opt. Mater. 13, 311–317 (1999).
[CrossRef]

F. Mougel, G. Aka, A. Kahn-Harari, H. Hubert, J. M. Benitez, and D. Vivien, “Infrared laser performance and self-frequency doubling of Nd:GdCOB,” Opt. Mater. 8, 161–173 (1997).
[CrossRef]

Phys. Lett. (1)

D. Findlay and R. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966).
[CrossRef]

Proc. SPIE (1)

D. Jaque, J. Capmany, and J. Garcia Sole, “Red, blue and green laser-light generation from the NYAB nonlinear crystal,” in Solid State Lasers VIII, Proc. SPIE 3613, 140–150 (1999).
[CrossRef]

Progr. Nat. Sci. (1)

Z.-d. Luo, “The optimum neodymium concentration of self-frequency-doubling laser crystal NYAB,” Progr. Nat. Sci. 4, 504–508 (1994).

Other (1)

W. Koechner, “Laser oscillator,” in Solid-State Laser Engineering, 3rd ed., A. L. Schawlow, K. Shimoda, A. E. Siegman, and T. Tamir, eds. (Springer-Verlag, Berlin, 1992), Chap. 3.

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Figures (14)

Fig. 1
Fig. 1

Schematic of the propagating beams along the z axis in a plano–convex cavity.

Fig. 2
Fig. 2

Collinear beam geometry in a plano–concave cavity.

Fig. 3
Fig. 3

Fundamental laser output as a function of the pump power incident upon a NYAB crystal for (a) different atomic concentrations at a fixed crystal length L=1.7 mm and fixed beam waists wp0=5 µm and wc0=30 µm and (b) a crystal length at a fixed atomic concentration N=5.5 at. % and fixed beam waists wp0=15 µm and wc0=30 µm.

Fig. 4
Fig. 4

Self-SFM laser threshold of NYAB as a function of (a) Nd concentration for different crystal lengths, (b) crystal length for different Nd concentrations at fixed beam waists wp0=5 µm and wc0=30 µm, (c) cavity mode size for different pump beam waists.

Fig. 5
Fig. 5

Self-SFM laser output as a function of Nd concentration and crystal length for an input pump power of 500 mW in a NYAB crystal.

Fig. 6
Fig. 6

Typical curves of self-SFM laser output as functions of (a) Nd concentration for different crystal lengths, (b) crystal length for different Nd concentrations, taken down from Fig. 5.

Fig. 7
Fig. 7

(a) Optimum Nd concentration as a function of crystal length, (b) optimum crystal length as a function of Nd concentration. Filled squares, numerical results; solid curves, fitted exponential decay curves.

Fig. 8
Fig. 8

(a) Calculated blue (458-nm) and infrared (1062-nm) laser output power for several absorbed pump powers. The experimental results are given in (b).

Fig. 9
Fig. 9

Dependence of the self-SFM laser output on the walk-off angle. Dotted curves, output at the walk-off angle of 2.4° for NYAB.

Fig. 10
Fig. 10

Self-SFM laser output as a function of cavity mode size (a) for different pump beam waists in simulating a Ti:sapphire pump, and (b) at a fixed pump waist (60 µm) simulating a LD pump. N=5.5 at. % and L=1.7 mm for the NYAB crystal; the absorbed power is 200 mW.

Fig. 11
Fig. 11

Blue laser power at 458 nm as a function of phase mismatch Δk for several cavity mode sizes at a fixed pump waist (wp0=5 µm) in a NYAB crystal.

Fig. 12
Fig. 12

(a) Qualitative estimation of blue laser output as a function of the pump wavelength for NYAB. (b) Experimental result from Ref. 5.

Fig. 13
Fig. 13

Fundamental laser output as a function of pump power incident upon the Nd:GdCOB crystal for samples A (N=5 at. % and L=8 mm) and B (N=7 at. % and L=5 mm).

Fig. 14
Fig. 14

Calculated blue and infrared (1061-nm) laser output power as a function of absorbed pump power (a) for sample A (N=5 at. % and L=8 mm) and (b) for sample B (N=7 at. % and L=5 mm). (c), (d) Experimental results for (a) and (b), respectively, from Ref. 9.

Tables (2)

Tables Icon

Table 1 Optical and Spectroscopic Data of NYAB

Tables Icon

Table 2 Optical and Spectroscopic Data of Nd:GdCOB

Equations (40)

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

wp2(z)=wp021+λ12z2π2wp04n12,
wc2(z)=wc021+λ22z2π2wc04n22,
I(r, z)=P(z)f(r, z),
f(r, z)=2πw2(z) exp[-2r2/w2(z)],
dIcdz=g0(r, z)1+2Ic/Is-αsIc,
g0(r, z)=σeΔN0(r, z)=σeτfWp(r, z)Ntot.
Wp(r, z)Ntot=cn2σ12 Ntot q(r, z)=αpλ1hcIp(r, z),
Pp(z)=Pp0 exp(-αpz),
g0(r, z)=2σeτfαpλ1Pp0πhcwp2(z) exp[-αpz-2r2/wp2(z)].
dIcdz=2σeτfαpλ1Pp0πhcwp2(z)×exp[-αpz-2r2/wp2(z)]1+4Pcπwc2(z)Is exp[-2r2/wc2(z)]-αcIc,
dPcdz=2σeτfαp exp(-αpz)λ1Pp0πhc[wp2(z)+wc2(z)]Q(z)-αcIc,
Q(z)=011+4Pcπwc2(z)Isywp2(z)/wp2(z)+wc2(z)-1dy,
y=exp-21wp2(z)+1wc2(z)r2.
2 01 2σeτfαp exp(-αpz)λ1Pp0πhc[wp2(z)+wc2(z)]Q(z)dz=2αcL.
αc=δ-12L ln R+12L2KPp0.
R=r+R1R21+rR1R22,
Pp0=2δL-ln R0L 4σeτfαp exp(-αpz)λ1πhc[wp2(z)+wc2(z)]Q(z)dz-2K.
w¯i2=0Lwi2(z)dz/L(i=p, c).
Q(z)=011+4Pcπw¯c2Isyw¯p2/w¯p2+w¯c2-1dy1+4Pcqπw¯c2Is-1,
Pc=πw¯c2Is4q 2G2δL-ln R+2KPp0Pp0-1,
G=2σeτfλ1[1-exp(-αpL)]πhc(w¯p2+w¯c2).
Pth=2δL-ln R0L 4σeτfαp exp(-αpz)λ1πhc[wp2(z)+wc2(z)]dz-2K.
Pth=πhc(w¯p2+w¯c2)(2δL-ln R)4σeτfλ1[1-exp(-αpL)]-2πhc(w¯p2+w¯c2)K.
P3=2ω02χeff2 P1P2πε0c3n02n3Lk0 exp(-αL)×(1-ζ2)(1-γ2)1+γζh(σ, β, κ, ξ, μ),
k0=k1+k22,
b0=k1b1+k2b2k1+k2 orw02=2(k12w102+k22w202)(k1+k2)2.
ω1=ω0(1-γ),ω2=ω0(1+γ),
n1=n0(1-ζ),n2=n0(1+ζ),
P3=2ω02χeff2P1P2πε0c3n02n3Lk0 exp(-αI)×(1-ζ2)(1-γ2)1+γζh(σ, β, κ, ξ, μ),
h(σ, β, κ, ξ, μ)=exp(μαI)4ξβ1β2 -ξ(1-μ)ξ(1+μ) dτdτ
×(1+iτ)(1-iτ)exp[-κ(τ+τ)-iσ(τ-τ)+β2(τ-τ)2]1+i τβ11-i τβ11+i τβ21-i τβ2,
β1=b1/b0,β2=b2/b0,
α=½(α1+α2-α3),
α=½(α1+α2-α3).
Δk=k1+k2-k3,ξ=L/b0,σ=b0Δk/2,
μ=(L-2 f)/L,κ=αb0/2,β=ρk0w0/2,
PSFM=KPp0Pc,
K=2ω02χeff2πε0c3n02n3Lk0 exp(-αpL/2)×(1-ζ2)(1-γ2)1+γζh(σ, β, κ, ξ, μ).
Pf=-ln R-ln(R1R2)(1-R2)Pc,
Ps=KPcPp0T,

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