Abstract

This work presents a compact LiNbO3 (lithium niobate, LN) electro-optic (EO) Q-switch with a lower driving voltage than the conventional LN Q-switches. By using non-direct cuts of a certain crystallographic orientation, a LN crystal is used both as a quarter-wave plate (QWP) and a pockels cell in a laser cavity. Through theoretical calculations and experiments, we have determined the optimized crystal orientations with low quarter-wave voltages (QWV). A set of compact LN EO Q-switches were prepared and used successfully in the pulse-on mode in a Nd:YAG laser. The Q-switched laser outputs are comparable to those obtained by using a conventional Z-cut LN Q-switch with a QWP. The QWV of the Q-switch with the optimized crystal orientation is 600V lower than that of the Z-cut LN Q-switch.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. H. Ma, D. Z. Lu, H. H. Yu, H. J. Zhang, X. K. Han, Q. M. Lu, C. Q. Ma, and J. Y. Wang, “High repetition rates optically active langasite electro-optically Q-switched laser at 1.34 μm,” Opt. Express 25(20), 24007–24014 (2017).
    [Crossref]
  2. D. Li, H. Xue, Y. Wang, M. Qi, W. Kim, C. Li, J. Riikonen, Z. Ren, J. Bai, H. Lipsanen, and Z. Sun, “Active synchronization and modulation of fiber lasers with a graphene electro-optic modulator,” Opt. Lett. 43(15), 3497–3500 (2018).
    [Crossref]
  3. D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
    [Crossref]
  4. J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
    [Crossref]
  5. L. A. Rios, C. E. Minor, N. A. Barboza, and R. S. Cudney, “Q-switching and quasi-phase-matching using a domain structured LiNbO3­ crystal,” Opt. Express 26(13), 17591–17596 (2018).
    [Crossref]
  6. B. Bai, Y. Bai, D. Li, Y. Sun, J. Li, and J. Bai, “Double Q-switched 946 nm laser with MgO:LN electro-optic crystal and MoSe2 saturable absorber,” Chin. Opt. Lett. 16(3), 031402 (2018).
    [Crossref]
  7. Z. Cui, X. M. Duan, B. Q. Yao, H. Y. Yang, J. Li, J. H. Yuan, T. Y. Dai, C. Y. Li, and Y. B. Pan, “Doubly Q-switched Ho:LuAG laser with acoustic-optic modulator and Cr2+:ZnS saturable absorber,” Appl. Opt. 54(34), 10272–10276 (2015).
    [Crossref]
  8. Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
    [Crossref]
  9. J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).
  10. P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
    [Crossref]
  11. S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
    [Crossref]
  12. Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
    [Crossref]
  13. H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
    [Crossref]
  14. Y. Wang, J. Yao, D. Xu, P. Zhao, and P. Wang, “Precise calculation of the KTP crystal used as both an intracavity electro-optic Q-switch and a second harmonic generator,” Chin. Opt. Lett. 4(7), 419–421 (2006).
  15. D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
    [Crossref]
  16. Z. J. Liu, Q. P. Wang, X. Y. Zhang, Z. J. Liu, A. J. Wei, J. Chang, F. F. Su, and G. F. Jin, “Pulse-off electro-optic Q-switch made of La3Ga5SiO14,” Opt. Express 13(18), 7086–7090 (2005).
    [Crossref]
  17. M. Roth, M. Tseitlin, and N. Angert, “Oxide crystals for electro-optic Q-switching of lasers,” Glass Phys. Chem. 31(1), 86–95 (2005).
    [Crossref]
  18. P. C. Amundsen and G. Wang, “Low-loss LiNbO3 Q switches: compensation of acoustically-induced refractive index variations,” IEEE J. Quantum Electron. 23(12), 2252–2257 (1987).
    [Crossref]
  19. W. D. Fountain, “Comments on: transient elastooptic effects and Q-switching performance in lithium niobate and KDP pockels cells,” Appl. Opt. 10(4), 972–973 (1971).
    [Crossref]
  20. J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
    [Crossref]
  21. J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
    [Crossref]
  22. A. M. Prokhorov and Y. S. Kuzminov, Physics and Chemistry of Crystalline Lithium Niobate (Adam Hilger, 1990).
  23. W. P. Mason, “Optical properties and the electro-optic and photoelastic effects in crystals expressed in tensor form,” Bell Syst. Tech. J. 29(2), 161–188 (1950).
    [Crossref]
  24. W. Koechner, Solid-state Laser Engineering (Springer, 2000).
  25. K. Zhao and X. Zhong, Optics (Peking University, 2001), Chap. 7.
  26. J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26(4), 449–458 (2004).
    [Crossref]
  27. IEEE Standard on Piezoelectricity, ANSI/IEEE Std 176-1987.
  28. J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

2020 (2)

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

2018 (4)

2017 (3)

S. H. Ma, D. Z. Lu, H. H. Yu, H. J. Zhang, X. K. Han, Q. M. Lu, C. Q. Ma, and J. Y. Wang, “High repetition rates optically active langasite electro-optically Q-switched laser at 1.34 μm,” Opt. Express 25(20), 24007–24014 (2017).
[Crossref]

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

2015 (2)

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Z. Cui, X. M. Duan, B. Q. Yao, H. Y. Yang, J. Li, J. H. Yuan, T. Y. Dai, C. Y. Li, and Y. B. Pan, “Doubly Q-switched Ho:LuAG laser with acoustic-optic modulator and Cr2+:ZnS saturable absorber,” Appl. Opt. 54(34), 10272–10276 (2015).
[Crossref]

2014 (1)

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

2013 (1)

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

2011 (1)

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

2010 (1)

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

2006 (2)

2005 (2)

2004 (2)

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26(4), 449–458 (2004).
[Crossref]

1987 (1)

P. C. Amundsen and G. Wang, “Low-loss LiNbO3 Q switches: compensation of acoustically-induced refractive index variations,” IEEE J. Quantum Electron. 23(12), 2252–2257 (1987).
[Crossref]

1971 (1)

1950 (1)

W. P. Mason, “Optical properties and the electro-optic and photoelastic effects in crystals expressed in tensor form,” Bell Syst. Tech. J. 29(2), 161–188 (1950).
[Crossref]

Abarkan, M.

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26(4), 449–458 (2004).
[Crossref]

Amundsen, P. C.

P. C. Amundsen and G. Wang, “Low-loss LiNbO3 Q switches: compensation of acoustically-induced refractive index variations,” IEEE J. Quantum Electron. 23(12), 2252–2257 (1987).
[Crossref]

Angert, N.

M. Roth, M. Tseitlin, and N. Angert, “Oxide crystals for electro-optic Q-switching of lasers,” Glass Phys. Chem. 31(1), 86–95 (2005).
[Crossref]

Bai, B.

Bai, J.

Bai, J. T.

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Bai, Y.

B. Bai, Y. Bai, D. Li, Y. Sun, J. Li, and J. Bai, “Double Q-switched 946 nm laser with MgO:LN electro-optic crystal and MoSe2 saturable absorber,” Chin. Opt. Lett. 16(3), 031402 (2018).
[Crossref]

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Barboza, N. A.

Chang, J.

Chen, X. Y.

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

Cudney, R. S.

Cui, Z.

Dai, T. Y.

Deng, J.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Dong, C. Y.

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

Dou, F. F.

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

Duan, X. M.

Fan, J. J.

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Fontana, M. D.

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26(4), 449–458 (2004).
[Crossref]

Fountain, W. D.

Ge, Y.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Han, X. K.

Hao, H. S.

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

Helena, J.

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

Huang, J. H.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Jan, S.

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

Jin, G. F.

Jin, G. Y.

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

Ju, Y. L.

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

Kim, W.

Koechner, W.

W. Koechner, Solid-state Laser Engineering (Springer, 2000).

Kong, Y. F.

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

Kuzminov, Y. S.

A. M. Prokhorov and Y. S. Kuzminov, Physics and Chemistry of Crystalline Lithium Niobate (Adam Hilger, 1990).

Lang, H.

D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
[Crossref]

Li, C.

Li, C. Y.

Li, D.

Li, G.

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

Li, J.

Li, J. H.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Li, Q. L.

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Lin, W. X.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Lipsanen, H.

Liu, H. G.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Liu, L.

D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
[Crossref]

Liu, S. G.

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

Liu, Z. J.

Lu, B. L.

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Lu, D. Z.

Lu, Q. M.

Ma, C. Q.

Ma, S. H.

Mason, W. P.

W. P. Mason, “Optical properties and the electro-optic and photoelastic effects in crystals expressed in tensor form,” Bell Syst. Tech. J. 29(2), 161–188 (1950).
[Crossref]

Meng, P. B.

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

Michal, N.

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

Minor, C. E.

Miroslav, C.

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

Pan, W.

D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
[Crossref]

Pan, Y. B.

Petr, K.

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

Prokhorov, A. M.

A. M. Prokhorov and Y. S. Kuzminov, Physics and Chemistry of Crystalline Lithium Niobate (Adam Hilger, 1990).

Qi, M.

Ren, Z.

Ren, Z. Y.

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Riikonen, J.

Rios, L. A.

Roth, M.

M. Roth, M. Tseitlin, and N. Angert, “Oxide crystals for electro-optic Q-switching of lasers,” Glass Phys. Chem. 31(1), 86–95 (2005).
[Crossref]

Ruan, K. M.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Salvestrini, J. P.

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26(4), 449–458 (2004).
[Crossref]

Shang, J. F.

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Shi, F.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Shi, Y.

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Su, F. F.

Sun, J.

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Sun, Y.

Sun, Z.

Tseitlin, M.

M. Roth, M. Tseitlin, and N. Angert, “Oxide crystals for electro-optic Q-switching of lasers,” Glass Phys. Chem. 31(1), 86–95 (2005).
[Crossref]

Wang, C.

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

Wang, G.

P. C. Amundsen and G. Wang, “Low-loss LiNbO3 Q switches: compensation of acoustically-induced refractive index variations,” IEEE J. Quantum Electron. 23(12), 2252–2257 (1987).
[Crossref]

Wang, J.

D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
[Crossref]

Wang, J. Y.

S. H. Ma, D. Z. Lu, H. H. Yu, H. J. Zhang, X. K. Han, Q. M. Lu, C. Q. Ma, and J. Y. Wang, “High repetition rates optically active langasite electro-optically Q-switched laser at 1.34 μm,” Opt. Express 25(20), 24007–24014 (2017).
[Crossref]

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Wang, P.

Wang, Q. P.

Wang, Y.

Wang, Y. Z.

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

Wei, A. J.

Weng, W.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Wu, C. T.

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

Wu, H. C.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Wu, J.

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

Xu, D.

Xu, J. J.

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Xue, H.

Yang, H. Y.

Yang, J. F.

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Yao, B. Q.

Z. Cui, X. M. Duan, B. Q. Yao, H. Y. Yang, J. Li, J. H. Yuan, T. Y. Dai, C. Y. Li, and Y. B. Pan, “Doubly Q-switched Ho:LuAG laser with acoustic-optic modulator and Cr2+:ZnS saturable absorber,” Appl. Opt. 54(34), 10272–10276 (2015).
[Crossref]

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

Yao, J.

Yu, H. H.

Yu, Y. J.

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

Yuan, J. H.

Zhang, C.

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

Zhang, H. J.

Zhang, L.

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Zhang, X. Y.

Zhang, Y. J.

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

Zhao, D.

D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
[Crossref]

Zhao, K.

K. Zhao and X. Zhong, Optics (Peking University, 2001), Chap. 7.

Zhao, P.

Zhao, X.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Zheng, D. H.

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

Zheng, H.

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Zhong, X.

K. Zhao and X. Zhong, Optics (Peking University, 2001), Chap. 7.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

W. P. Mason, “Optical properties and the electro-optic and photoelastic effects in crystals expressed in tensor form,” Bell Syst. Tech. J. 29(2), 161–188 (1950).
[Crossref]

Chin. Opt. Lett. (2)

Glass Phys. Chem. (1)

M. Roth, M. Tseitlin, and N. Angert, “Oxide crystals for electro-optic Q-switching of lasers,” Glass Phys. Chem. 31(1), 86–95 (2005).
[Crossref]

Guangzi Xuebao (1)

J. F. Shang, J. Sun, Q. L. Li, J. Wu, L. Zhang, F. F. Dou, C. Y. Dong, and J. J. Xu, “High-repetition-rate LiNbO3 electro-optic Q-switched Nd:YVO4 laser,” Guangzi Xuebao 47(5), 514001 (2018).
[Crossref]

IEEE J. Quantum Electron. (1)

P. C. Amundsen and G. Wang, “Low-loss LiNbO3 Q switches: compensation of acoustically-induced refractive index variations,” IEEE J. Quantum Electron. 23(12), 2252–2257 (1987).
[Crossref]

J. Synth. Cryst. (2)

J. F. Shang, J. Sun, Y. J. Zhang, J. F. Yang, Q. L. Li, L. Zhang, and J. J. Xu, “A method to measure electro-optic coefficients of crystals by combining conoscopic interference and near optical axis electro optic modulation,” J. Synth. Cryst. 44(11), 2925–2930 (2015).

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Study on light leakage phenomenon of lithium niobate electro-optic Q-switching crystals,” J. Synth. Cryst. 49(6), 71–76 (2020).

Laser Phys. (3)

P. B. Meng, B. Q. Yao, G. Li, Y. L. Ju, and Y. Z. Wang, “Comparison of RTP electro-optic Q-switch and acousto-optic Q-switch in Tm, Ho:GdVO4 laser,” Laser Phys. 21(2), 348–351 (2011).
[Crossref]

Y. Bai, C. Zhang, J. J. Fan, Y. Shi, B. L. Lu, Z. Y. Ren, and J. T. Bai, “High repetition rate intracavity frequency doubled LD side-pumped ceramic Nd:YAG green laser based on BBO electro-optical Q-switch,” Laser Phys. 20(7), 1585–1589 (2010).
[Crossref]

H. Zheng, J. H. Huang, X. Zhao, J. Y. Wang, F. Shi, J. H. Li, J. Deng, H. G. Liu, W. Weng, Y. Ge, K. M. Ruan, H. C. Wu, and W. X. Lin, “An LD-pumped Q-switched Nd:YAG laser using La3Ga5SiO14 for the electro-optic modulator and optical activity compensation,” Laser Phys. 24(4), 045802 (2014).
[Crossref]

Laser Phys. Lett. (1)

S. Jan, J. Helena, K. Petr, N. Michal, and C. Miroslav, “LiNbO3 pockels cell for Q-switch of Er:YAG laser,” Laser Phys. Lett. 1(2), 59–64 (2004).
[Crossref]

Opt. Commun. (1)

Y. J. Yu, X. Y. Chen, C. Wang, C. T. Wu, and G. Y. Jin, “High repetition rate 880 nm diode-directly-pumped electro-optic Q-switched Nd:GdVO4 laser with a double-crystal RTP electro-optic modulator,” Opt. Commun. 304, 39–42 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Opt. Mater. (1)

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26(4), 449–458 (2004).
[Crossref]

Optik (2)

J. F. Shang, J. F. Yang, H. S. Hao, Q. L. Li, L. Zhang, and J. Sun, “Research on optimum slowly-rising driver for electro-optic Q-switched laser,” Optik 202, 163553 (2020).
[Crossref]

D. Zhao, L. Liu, J. Wang, H. Lang, and W. Pan, “Laser ranging based on electro-optic switch,” Optik 117(9), 443–448 (2006).
[Crossref]

Sci. Rep. (1)

J. F. Shang, J. Sun, Q. L. Li, J. F. Yang, L. Zhang, and J. J. Xu, “Single-block pulse-on electro-optic Q-switch made of LiNbO3,” Sci. Rep. 7(1), 4651 (2017).
[Crossref]

Sci. Sin. Tech. (1)

D. H. Zheng, J. Wu, J. F. Shang, L. Zhang, S. G. Liu, J. Sun, and Y. F. Kong, “Progress on electro-optic crystals for Q-switches,” Sci. Sin. Tech. 47(1), 80–88 (2017).
[Crossref]

Other (4)

A. M. Prokhorov and Y. S. Kuzminov, Physics and Chemistry of Crystalline Lithium Niobate (Adam Hilger, 1990).

IEEE Standard on Piezoelectricity, ANSI/IEEE Std 176-1987.

W. Koechner, Solid-state Laser Engineering (Springer, 2000).

K. Zhao and X. Zhong, Optics (Peking University, 2001), Chap. 7.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. Eigen polarization directions of the wave in crystal for an electrical field applied. X, Y, and Z axes along each crystallographic axis respectively, k represents wave normal direction, P­1 and P2 are eigen polarization directions.
Fig. 2.
Fig. 2. Experimental setup of the Nd:YAG laser for measuring dynamic QWVs in LN crystals.
Fig. 3.
Fig. 3. Illustration of the $({\textrm{XZtw}} )- \frac{\theta }{{\sqrt 2 }}/\frac{\theta }{{\sqrt 2 }}$ cut LN crystals.
Fig. 4.
Fig. 4. Experimental setup for measuring dynamic extinction ratio of compact LN Q-switches.
Fig. 5.
Fig. 5. Pulse shape of the compact LN EO Q-switched laser.

Tables (4)

Tables Icon

Table 1. Theoretical and experimental QWVs in different directions in LN crystals.

Tables Icon

Table 2. Dynamic QWVs in various directions with different angles θ.

Tables Icon

Table 3. Dynamic extinction ratios of the LN EO Q-switches.

Tables Icon

Table 4. Output results of the compact LN EO Q-switched Nd:YAG lasers.

Equations (6)

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

( 1 n o 2 γ 22 E γ 51 E γ 22 E 1 n o 2 0 γ 51 E 0 1 n e 2 )
Δ n = n o 3 2 K 2
K 2 = ( β 33 β 22 ) 2 sin 4 θ + 2 ( β 33 β 22 ) sin 2 θ [ β 12 sin 2 φ ( 1 + cos 2 θ ) β 13 sin 2 θ cos φ ] + ( 2 β 12 ) 2 ( sin 4 θ sin 2 φ cos 2 φ + cos 2 θ ) 4 β 12 β 13 sin 2 θ sin φ [ cos 2 θ cos 2 φ + sin 2 φ ] + ( 2 β 13 ) 2 sin 2 θ ( cos 2 θ cos 2 φ + sin 2 φ )
tan 2 ψ = 2 β 12 cos θ cos 2 φ + 2 β 13 sin θ sin φ β 11 [ cos 2 θ cos 2 φ sin 2 φ ] + β 12 ( 1 + cos 2 θ ) sin 2 φ β 13 sin 2 θ cos φ + β 22 ( cos 2 θ sin 2 φ cos 2 φ ) + β 33 sin 2 θ
Γ 0 = l Δ n o 3 sin 2 θ 2 = λ 4 + k λ 2 k = 0 , ± 1 , ± 2
Γ 1  =  n o 3 K 2 l 2 = k λ 2 k = 0 , ± 1 , ± 2

Metrics