Abstract

Aimed for generating high-average power ultraviolet (UV) lasers via third-harmonic generation (THG) consisting of frequency doubling and tripling stages, we numerically and experimentally demonstrate a novel frequency tripling scheme capable of supporting temperature-insensitive phase-matching (PM). Two cascaded tripling crystals, with opposite signs of the temperature derivation of phase-mismatch, are proposed and theoretically studied for improving the temperature-acceptance of PM. The proof-of-principle tripling experiment using two crystals of LBO and BBO shows that the temperature acceptance can be ~1.5 times larger than that of using a single tripling crystal. In addition, the phase shift caused by air dispersion, along with its influence on the temperature-insensitive PM, are also discussed. To illustrate the potential applications of proposed two-crystal tripling design in the high-average-power regime, full numerical simulations for the tripling process, are implemented based on the realistic crystals. The demonstrated two-crystal tripling scheme may provide a promising route to high-average-power THG in the UV region.

© 2014 Optical Society of America

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2013 (1)

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Two-crystal design and numerical simulations for high-average-power second-harmonic generation,” Chin. Phys. Lett. 30(1), 014208 (2013).
[CrossRef]

2012 (1)

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Versatile temperature-insensitive second-harmonic generation by compensating thermally induced phase-mismatch in a two-crystal design,” Laser Phys. Lett. 9(6), 434–439 (2012).
[CrossRef]

2010 (1)

C. Stolzenburg, W. Schüle, I. Zawischa, A. Killi, D. Sutter, “700 W intracavity-frequency doubled Yb: YAG thin-disk laser at 100 kHz repetition rate,” Proc. SPIE 7578, 75780A (2010).
[CrossRef]

2009 (1)

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (1)

2003 (2)

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

H. Kitano, K. Sato, N. Ushiyama, M. Yoshimura, Y. Mori, T. Sasaki, “Efficient 355-nm generation in CsB3O5 crystal,” Opt. Lett. 28(4), 263–265 (2003).
[CrossRef] [PubMed]

2000 (1)

T. Sasaki, Y. Mori, M. Yoshimura, Y. K. Yap, T. Kamimura, “Recent development of nonlinear optical borate crystals: key materials for generation of visible and UV light,” Mater. Sci. Eng. Rep. 30(1–2), 1–54 (2000).
[CrossRef]

1998 (1)

1997 (2)

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron. 33(8), 1424–1429 (1997).
[CrossRef]

D. J. Armstrong, W. J. Alford, T. D. Raymond, A. V. Smith, M. S. Bowers, “Parametric amplification and oscillation with walkoff-compensating crystals,” J. Opt. Soc. Am. B 14(2), 460–474 (1997).
[CrossRef]

1996 (1)

1994 (1)

D. N. Nikogosyan, “Lithium triborate (LBO) - A review of its properties and applications,” Appl. Phys. A Mater. Sci. Process. 58(3), 181–190 (1994).
[CrossRef]

1990 (1)

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

1987 (1)

D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. 23(5), 575–592 (1987).
[CrossRef]

1971 (1)

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18(3), 70–73 (1971).
[CrossRef]

1966 (1)

B. Edlén, “The refractive index of air,” Metrologia 2(2), 71–80 (1966).
[CrossRef]

Alford, W. J.

Allee, E. S.

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

Ando, M.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

Armstrong, D. J.

Armstrong, J.

Barty, C.

Bayramian, A.

Beer, G.

Bowers, M. S.

Caird, J.

Campbell, R.

Chai, B.

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron. 33(8), 1424–1429 (1997).
[CrossRef]

Cheung, E. C.

Cross, R.

Deki, K.

Diart, R.

Du, K.

Dudley, D. R.

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

Ebbers, C.

Edlén, B.

B. Edlén, “The refractive index of air,” Metrologia 2(2), 71–80 (1966).
[CrossRef]

Eimerl, D.

D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. 23(5), 575–592 (1987).
[CrossRef]

Epp, P.

Erlandson, A.

Falk, J.

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18(3), 70–73 (1971).
[CrossRef]

Fei, Y.

Fields, R. A.

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

Freitas, B.

Goodno, G. D.

Haas, R.

Hitz, C. B.

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18(3), 70–73 (1971).
[CrossRef]

Hodgson, N.

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

Howland, D.

Hu, P.

Huang, T. M.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron. 33(8), 1424–1429 (1997).
[CrossRef]

Inagaki, M.

Injeyan, H.

Innocenzi, M. E.

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

Kamimura, T.

T. Sasaki, Y. Mori, M. Yoshimura, Y. K. Yap, T. Kamimura, “Recent development of nonlinear optical borate crystals: key materials for generation of visible and UV light,” Mater. Sci. Eng. Rep. 30(1–2), 1–54 (2000).
[CrossRef]

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron. 33(8), 1424–1429 (1997).
[CrossRef]

Kato, K.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

Kent, R.

Killi, A.

C. Stolzenburg, W. Schüle, I. Zawischa, A. Killi, D. Sutter, “700 W intracavity-frequency doubled Yb: YAG thin-disk laser at 100 kHz repetition rate,” Proc. SPIE 7578, 75780A (2010).
[CrossRef]

Kitano, H.

Kitatochi, N.

Komine, H.

Li, D.

Long, W.

Loosen, P.

Ma, Z.

McClellan, M.

McNaught, S. J.

Mehl, O.

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

Menapace, J.

Molander, W.

Mori, Y.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

H. Kitano, K. Sato, N. Ushiyama, M. Yoshimura, Y. Mori, T. Sasaki, “Efficient 355-nm generation in CsB3O5 crystal,” Opt. Lett. 28(4), 263–265 (2003).
[CrossRef] [PubMed]

T. Sasaki, Y. Mori, M. Yoshimura, Y. K. Yap, T. Kamimura, “Recent development of nonlinear optical borate crystals: key materials for generation of visible and UV light,” Mater. Sci. Eng. Rep. 30(1–2), 1–54 (2000).
[CrossRef]

Y. K. Yap, K. Deki, N. Kitatochi, Y. Mori, T. Sasaki, “Alleviation of thermally induced phase mismatch in CsLiB6O10 crystal by means of temperature-profile compensation,” Opt. Lett. 23(13), 1016–1018 (1998).
[CrossRef] [PubMed]

Y. K. Yap, M. Inagaki, S. Nakajima, Y. Mori, T. Sasaki, “High-power fourth- and fifth-harmonic generation of a Nd:YAG laser by means of a CsLiB6O10,” Opt. Lett. 21(17), 1348–1350 (1996).
[CrossRef] [PubMed]

Nakajima, S.

Nikogosyan, D. N.

D. N. Nikogosyan, “Lithium triborate (LBO) - A review of its properties and applications,” Appl. Phys. A Mater. Sci. Process. 58(3), 181–190 (1994).
[CrossRef]

Pang, H. Y.

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

Qian, L. J.

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Two-crystal design and numerical simulations for high-average-power second-harmonic generation,” Chin. Phys. Lett. 30(1), 014208 (2013).
[CrossRef]

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Versatile temperature-insensitive second-harmonic generation by compensating thermally induced phase-mismatch in a two-crystal design,” Laser Phys. Lett. 9(6), 434–439 (2012).
[CrossRef]

Raymond, T. D.

Redmond, S.

Sasaki, T.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

H. Kitano, K. Sato, N. Ushiyama, M. Yoshimura, Y. Mori, T. Sasaki, “Efficient 355-nm generation in CsB3O5 crystal,” Opt. Lett. 28(4), 263–265 (2003).
[CrossRef] [PubMed]

T. Sasaki, Y. Mori, M. Yoshimura, Y. K. Yap, T. Kamimura, “Recent development of nonlinear optical borate crystals: key materials for generation of visible and UV light,” Mater. Sci. Eng. Rep. 30(1–2), 1–54 (2000).
[CrossRef]

Y. K. Yap, K. Deki, N. Kitatochi, Y. Mori, T. Sasaki, “Alleviation of thermally induced phase mismatch in CsLiB6O10 crystal by means of temperature-profile compensation,” Opt. Lett. 23(13), 1016–1018 (1998).
[CrossRef] [PubMed]

Y. K. Yap, M. Inagaki, S. Nakajima, Y. Mori, T. Sasaki, “High-power fourth- and fifth-harmonic generation of a Nd:YAG laser by means of a CsLiB6O10,” Opt. Lett. 21(17), 1348–1350 (1996).
[CrossRef] [PubMed]

Sato, K.

Schaffers, K.

Schell, A.

Schüle, W.

C. Stolzenburg, W. Schüle, I. Zawischa, A. Killi, D. Sutter, “700 W intracavity-frequency doubled Yb: YAG thin-disk laser at 100 kHz repetition rate,” Proc. SPIE 7578, 75780A (2010).
[CrossRef]

Shi, P.

Siders, C.

Simon, J.

Simpson, R.

Smith, A. V.

Sollee, J.

Stolzenburg, C.

C. Stolzenburg, W. Schüle, I. Zawischa, A. Killi, D. Sutter, “700 W intracavity-frequency doubled Yb: YAG thin-disk laser at 100 kHz repetition rate,” Proc. SPIE 7578, 75780A (2010).
[CrossRef]

Sutter, D.

C. Stolzenburg, W. Schüle, I. Zawischa, A. Killi, D. Sutter, “700 W intracavity-frequency doubled Yb: YAG thin-disk laser at 100 kHz repetition rate,” Proc. SPIE 7578, 75780A (2010).
[CrossRef]

Sutton, S.

Suzuki, K.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

Takaoka, E.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

Tassano, J.

Telford, S.

Umemura, N.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

Ushiyama, N.

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron. 33(8), 1424–1429 (1997).
[CrossRef]

Wang, G. Y.

D. R. Dudley, O. Mehl, G. Y. Wang, E. S. Allee, H. Y. Pang, N. Hodgson, “Q-switched diode pumped Nd: YAG rod laser with output power of 420 W at 532 nm and 160 W at 355 nm,” Proc. SPIE 7193, 71930Z (2009).
[CrossRef]

Wang, S. C.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron. 33(8), 1424–1429 (1997).
[CrossRef]

Weber, M.

Weiss, S. B.

Yap, Y. K.

Yarborough, J. M.

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18(3), 70–73 (1971).
[CrossRef]

Yoshimura, M.

N. Umemura, M. Ando, K. Suzuki, E. Takaoka, K. Kato, M. Yoshimura, Y. Mori, T. Sasaki, “Temperature-insensitive second-harmonic generation at 0.5321 μm in YCa4O(BO3)3,” Jpn. J. Appl. Phys. 42(8), 5040–5042 (2003).
[CrossRef]

H. Kitano, K. Sato, N. Ushiyama, M. Yoshimura, Y. Mori, T. Sasaki, “Efficient 355-nm generation in CsB3O5 crystal,” Opt. Lett. 28(4), 263–265 (2003).
[CrossRef] [PubMed]

T. Sasaki, Y. Mori, M. Yoshimura, Y. K. Yap, T. Kamimura, “Recent development of nonlinear optical borate crystals: key materials for generation of visible and UV light,” Mater. Sci. Eng. Rep. 30(1–2), 1–54 (2000).
[CrossRef]

Yuan, P.

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Two-crystal design and numerical simulations for high-average-power second-harmonic generation,” Chin. Phys. Lett. 30(1), 014208 (2013).
[CrossRef]

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Versatile temperature-insensitive second-harmonic generation by compensating thermally induced phase-mismatch in a two-crystal design,” Laser Phys. Lett. 9(6), 434–439 (2012).
[CrossRef]

Yura, H. T.

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

Zawischa, I.

C. Stolzenburg, W. Schüle, I. Zawischa, A. Killi, D. Sutter, “700 W intracavity-frequency doubled Yb: YAG thin-disk laser at 100 kHz repetition rate,” Proc. SPIE 7578, 75780A (2010).
[CrossRef]

Zhong, H. Z.

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Two-crystal design and numerical simulations for high-average-power second-harmonic generation,” Chin. Phys. Lett. 30(1), 014208 (2013).
[CrossRef]

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Versatile temperature-insensitive second-harmonic generation by compensating thermally induced phase-mismatch in a two-crystal design,” Laser Phys. Lett. 9(6), 434–439 (2012).
[CrossRef]

Zhu, H. Y.

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Two-crystal design and numerical simulations for high-average-power second-harmonic generation,” Chin. Phys. Lett. 30(1), 014208 (2013).
[CrossRef]

H. Z. Zhong, P. Yuan, H. Y. Zhu, L. J. Qian, “Versatile temperature-insensitive second-harmonic generation by compensating thermally induced phase-mismatch in a two-crystal design,” Laser Phys. Lett. 9(6), 434–439 (2012).
[CrossRef]

Appl. Phys. A Mater. Sci. Process. (1)

D. N. Nikogosyan, “Lithium triborate (LBO) - A review of its properties and applications,” Appl. Phys. A Mater. Sci. Process. 58(3), 181–190 (1994).
[CrossRef]

Appl. Phys. Lett. (2)

J. M. Yarborough, J. Falk, C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,” Appl. Phys. Lett. 18(3), 70–73 (1971).
[CrossRef]

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

Chin. Phys. Lett. (1)

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup for THG, including both the SHG and the tripling stages. (a) Conventional single-crystal tripling design. (b) The proposed two-crystal tripling design (L = ~60 cm). BS, beam splitter; DM, dichroic mirror; F1, F2, short-pass filters; M1, M2, M3 silver mirrors.

Fig. 2
Fig. 2

Measured temperature-dependent conversion efficiency for Type-I frequency-tripling at the wavelength of 1064 nm. Square symbol, 2-mm-long BBO and 2.4-mm-long LBO; circle symbol, 8-mm-long LBO crystal; triangle symbol, 3-mm-long BBO crystal. For each curve, all the efficiencies are normalized to the maximal value at the initially set PM temperature of ~110 °C for both BBO and LBO crystals.

Fig. 3
Fig. 3

Dependence of the small-signal tripling efficiency on crystal lengths. Solid line, ∆T = 0 °C; Dashed line, ∆T = 2 °C; Dotted line, ∆T = 4 °C; Dashed-dotted line, ∆T = 6 °C. All the efficiencies are normalized to the maximal value of the solid curve. The vertical dashed line guides the maximum for each case, while the solid one guides the situation of L1 = 2.4 mm and L2 = 2 mm.

Fig. 4
Fig. 4

Temperature-dependent conversion efficiency for Type-I frequency-tripling at the wavelength of 1064 nm, for a specific two-crystal design (i.e., 2-mm-long BBO and 2.4-mm-long LBO). Square symbol, under the optimum situation; Triangle symbol, with a phase shift of ~0.5 π caused by the dispersion of air.

Fig. 5
Fig. 5

Calculated temperature-dependent tripling efficiency in the small-signal regime, for a specific two-crystal design (i.e., 2-mm-long BBO and 2.4-mm-long LBO) with several distinct phase shifts. Solid line, 0 π; Dashed line, 0.25 π; Dotted line, 0.5 π; Dashed-dotted line, 0.75 π. All the efficiencies are normalized to the maximal value of the solid curve.

Fig. 6
Fig. 6

Calculated THG conversion efficiency versus the 1064nm input power. Dashed line, the ideal situation without thermal effects; Solid line, 13.5-mm-long LBO and 14.5-mm-long CLBO; Dotted line, 24-mm-long LBO; Dashed-dotted line, 33-mm-long CLBO.

Tables (2)

Tables Icon

Table 1 The first temperature derivations of phase-mismatch (δk) for the frequency-tripling stage of THG at 1064 nm

Tables Icon

Table 2 Nonlinear Optical Crystal Parameters for Type-I tripling process at 1064 nm

Equations (4)

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A FH (z,t) z = i ω FH d eff n FH c A TH (z,t) A SH (z,t)exp[iΔk(T)z]
A SH (z,t) z = i ω SH d eff n SH c A TH (z,t) A FH (z,t)exp[iΔk(T)z]
A TH (z,t) z = i ω TH d eff n TH c A FH (z,t) A SH (z,t)exp[iΔk(T)z],
T(r,z)= T 0 + P h 4πκ m=1 (1) m mm! ( 2 ω p 2 ) m ( r 2m r b 2m ),r r b ,

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