Abstract

The optical properties of pure LuAl3(BO3)4 (abbreviated as LuAB) crystals were investigated for the first time. Large UV-transparent LuAl3(BO3)4 crystals were grown by a high-temperature top-seeding method with Li2WO4O7-B2O3 as the flux. The refractive indices of LuAl3(BO3)4 at several wavelengths covering ultraviolet-visible and near-infrared regions were measured by the auto-collimation method. The parameters of Sellmeier’s dispersion equation were determined from the experimental data. The phase-matching curve of second harmonic generation was measured. The nonlinear optical coefficient d11 of LuAB crystal was determined to be 1.10 pm/V by a phase-matching method. The UV cut-off wavelength of the LuAl3(BO3)4 crystal was shorter than 190 nm.

© 2013 OSA

1. Introduction

The family of alumino-borates RAl3(BO3)4 (R = Y, La–Lu) with the huntite structure shows very promising optical properties [1,2]. For example, YAl3(BO3)4 and GdAl3(BO3)4 doped with Nd3+ and Yb3+ ions have proven to be good laser materials [35]. In recent years, it has been reported that the undoped YAl3(BO3)4 crystals are a type of excellent UV frequency doubling material [6]. YAl3(BO3)4 crystals are also used as non-linear optical (NLO) crystals for fourth harmonic generation when a new flux system was used to control the absorption in the UV region [7]. In the past decades, LuAl3(BO3)4 crystals (abbreviated as LuAB), a type of promising nonlinear material as well, has been mainly investigated as a host material for laser application, with Ce3+, Er3+ and Yb3+ as the doping ions, or as scintillation crystals [810]. No thorough investigation has been conducted on LuAB crystals as nonlinear optical UV frequency doubling crystals. This may be attributed to the commonly used flux compound polymolybdate. Since polymolybdate contains molybdenum oxide, which often results in Mo contamination of the crystals [11,12]. Short-wavelength applications of the Mo-containing crystals are limited by an interfering peak from Mo at around 300 nm in the UV absorbtion spectrum [1]. In our present work, high-quality large-bulk LuAB crystals were successfully grown using the top-seeded solution growth (TSSG) method with the Li2WO4-B2O3 flux [13]. The UV transmittance spectrum showed a cutoff edge of about 178 nm, which made it an attractive candidate for a wide range of frequency conversion applications in the UV spectral regions. In this letter, the UV transmittance spectrum, the refractive dispersion and phase-matching conditions of LuAB were studied in detail. Phase-matching angles were also investigated.

The LuAB crystal is a negative uniaxial crystal and have the structure in the space group R32 with cell parameters in the parentheses (a = b = 0.916 nm, c = 0.71 nm). The LuAB crystal used in our experiment was grown from the Li2WO4-B2O3 flux system. The cutoff wavelength of the as-grown LuAB crystal was measured using a McPherson Vulvas 2000 vacuum UV spectrometer. The transmittance spectra of the as-grown LuAB crystal from 185 to 800 nm and 170-220 nm are shown in Fig. 1. The UV cut-off wavelength is at about 178 nm. To the best of our knowledge, the transmittance spectrum of the LuAB crystal has not been reported elsewhere. As a comparison, the cut-off edge of YAB crystals grown from the K2Mo3O10–B2O3 and PbO2–B2O3 ñux systems was reported to be around 300 nm [1], much higher than that in the Li2WO4–B2O3 ñux system [6]. This result indicates that the UV cut-off wavelength of LuAB crystal grown from a Li2WO4-B2O3 ñux is shorter than the K2Mo3O10–B2O3 ñux systems. However, the transmittance drops sharply below 300 nm, presumably due to the contamination by unknown trace impurities such as Fe3+ possibly from raw Al2O3. This phenomenon was also reported in the growth of the YAB [14], and KABO [15] crystals.

 

Fig. 1 The UV transmittance curve of LuAB crystals.

Download Full Size | PPT Slide | PDF

2. Crystal growth

Figure 2 shows the phase diagram of the quasi-ternary system LuAB-Li2WO4-B2O3 with an excess weight of 20% Al2O3. The crystallization region of LuAB crystals is clearly distinguishable. In our previous study, the additional Al2O3 proved to be useful for the growth of LuAB in our previous experiments [13]. A series of large high-quality bulk LuAB crystals with their weights over 40 g were successfully grown along [001] and [100] crystallographic directions by TSSG method. The as-grown LuAB crystals with seed orientations along [001] and [100] directions are pictured in Fig. 3. The crystals appear to be colorless, transparent and with few defects.

 

Fig. 2 Crystallization region for LuAB crystals with an excess of 20 wt% Al2O3; ■—LuAB + LuLiW2O8; ○—dose not melt at 1100 þC; ●—LuAB; □—LuLiW2O8; ♦ —LuBO3.

Download Full Size | PPT Slide | PDF

 

Fig. 3 LuAB crystal grown along [100] direction (a) ; and along [001] direction (b).

Download Full Size | PPT Slide | PDF

3. Measurements of the refractive indices

The auto-collimation method was used to measure the refractive indices of the crystals at 25 °C. LuAB in space group R32 [16, 17] had two corresponding principal refractive indices, i.e., no and ne. Therefore only one sample was employed. A Littrow prism as shown in Fig. 4 was cut from a LuAB crystal. The two right-angle sides of the Littrow prism were parallel to X and Y axes. The X-Y plane was perpendicular to the optical axis along which the refractive indices were all uniform. The inclined plane and the right angle side of the Littrow prism, which were adjacent to the acute angle, were well polished. The polished right angle side was coated with an aurum reflective film. The light sources used in this measurement were intense stable 1.064 μm and 1.338 μm Nd:YAG laser, 0.6328 μm He-Ne laser, as well as 0.532 μm, 0.473 μm, 0.355 μm and 0.266 μm LD pumped laser. When the incident angle of monochromatic parallel light equals the minimum deviation angle, the reflective beam from the aurum face of prism goes back along the direction of incident light, at that time

n=sinχ/sinξ
where n is the certain principal refractive index, ξ is the top angle of the Littrow prism and χ is the auto-collimation angle.

 

Fig. 4 Sketch of the Littow prism

Download Full Size | PPT Slide | PDF

The refractive indices were fit to the sellmeier Eqs.:

ni2=Ai+Biλ2CiDiλ2,(i=o,e)

Based on the measured values, the sellmeier constants Ai, Bi, Ci and Di may be determined through nonlinear curve fit, as shown in Table 1. The measured results of refractive index and the values calculated by Eq. (2) are all given in Table 2. The calculated and the measured values agrees well, the deviation between which is no more than 3 × 10−4. This small deviation falls in the error zone of our measurement. It is therefore concluded that the values of refractive indices generated by the sellmeier Eqs. in the region from 266 nm to 1338 nm are reliable. The dispersion curves generated using the Sellmeier are shown in Fig. 5. The birefringence values (no - ne) are located at 0.067-0.088 from the near IR to the UV regions.

Tables Icon

Table 1. Constants of sellmeier Eqs. for the LuAB crystal

Tables Icon

Table 2. The measured and calculated values of the principal refractive indices

 

Fig. 5 Refractive indices of LuAB.

Download Full Size | PPT Slide | PDF

4.Phase-matching properties for SHG

The phase-matching angles of the LuAB crystal for type I and type II phase-matching may be calculated using the following equations [18], respectively.

θmI=sin1[(ne2ωnoω)2(no2ω)2(noω)2(no2ω)2(ne2ω)2]1/2
[cos2(θmII)(no2ω)2+sin2(θmII)(ne2ω)2]=12{noω+[cos2(θmII)(noω)2+sin2(θmII)(neω)2]1/2}

Figure 6 gives the experimental phase-matching angles of type I (PM-I) and the calculated PM curves for different wavelengths for type I (PM-I) and type II (PM-II). The phase-matching angles of LuAB for second harmonic generation (SHG) were measured with an optical parametric amplifier which produced an output pulse wavelength tunable from 2.0 μm to 410 nm. Three high optical quality LuAB crystal samples cut at θ = 30þ, θ = 40þ and θ = 70.5þ were employed. The experimental data once again agreed with the calculated data.

 

Fig. 6 Phase-matching curves and measured phase-matching angles (circles) of LuAB.

Download Full Size | PPT Slide | PDF

It may be deduced from this Fig. that LuAB is phase matchable in the region from 0.49~1.4 µm for a PM-I and 0.65~1.4 µm for a PM-II, respectively. For such wavelengths, the angular acceptance is in the range from 1.38 to 0.239 mrad.cm shown in Fig. 7(a). Spatial walk-off angle is an important parameter which effectively reduces the gain length for SHG, and therefore affects the attainment of maximum SHG output power and efficiency. In Fig. 7(b), the variation curve of walk-off angle for PM-I indicates that the walk-off angle for PM-I varies between ~5 and ~45 mrad for fundamental wavelengths between 0.49 and 0.68 µm.

 

Fig. 7 Variation of walk-off angle and angular acceptance for type I as a function of fundamental wavelength.

Download Full Size | PPT Slide | PDF

5. The NLO Coefficients

The second-order NLO coefficient was determined by the SHG technique [17, 19] of measuring the harmonic power outputP2ω. The second-order susceptibility of crystal may be expressed by [20]

P2ω(l)=32π2c3ω2n12n2deff2Pω(0)ω02l2
where P2ω(l) and Pω(0) are the intensities. ω is angular frequency of the fundamental wave. n1 and n2 are the refractive indices of the fundamental and the SH beams, respectively. ω0 is the radius of the fundamental wave. l is the length of the crystal in the direction of the laser beam. A crystal (A) with known nonlinear coefficient and phase matching direction was used as standard sample. The intensities of the SH beams of crystal A and crystal X were determined in the same conditions. The effective nonlinear coefficient of crystal X may be expressed as follows [20]:

deff(X)=[PX2ωPA2ωn12(X)n2(X)n12(A)n2(A)]1/2lAlXdeff(A)

In our study, a 1.064 µm output of a Q-switched Nd:YAG laser (repetition rate = 150 Hz; pulse width = 15 ns) was used as the fundamental beam, and an LBO crystal was used as the standard sample. LBO and LuAB crystals were all cut and polished in the phase matching direction with a length of 2 mm.

The measured effective nonlinear optical coefficient (deff) value of LuAB was 1.03 pm/V. Figure 8 shows the effective nonlinear coefficients of LuAB as a function of the fundamental wavelength. The relevant nonlinear coefficients are given by the following Eqs.:

 

Fig. 8 Effective nonlinear coefficient as a function of fundamental wavelength for type I and type II SHG in LuAB.

Download Full Size | PPT Slide | PDF

For type I

deff=d11cosθcos3φ

For type II

deff=d11cos2θsin3φ
where d11 equals 1.10 pm/V, as obtained according to Eq. (6). As aforementioned, deff is equal to 1.03 pm/V. The deff denotes the effective nonlinear susceptibility, which may be simplified significantly for a particular symmetric crystal. It may be calculated from the independent components and incident angle.

In comparison to currently available NLO crystals designed for the UV and visible region, LuAB has excellent balancing combination properties including linear, nonlinear optical and physical properties. It has moderate birefringence that results in a smaller walk-off angle and a larger acceptance angle than those of BBO’s. Although the LuAB crystal has relatively small d11 coefficient than that of LBO, it has a wider range of phase-matching angle in the UV region. As a derivative member of YAB, LuAB was one of the most robust against thermal strike, moisture or acidic environment. It may be concluded that the LuAB crystal is a very promising UV and visible NLO material, despite the difficulty to grow large crystal with good optical quality. The twin structures have been observed in some grown crystals and detailed investigation are undergoing.

6.Conclusion

Fluxes of Li2WO4-B2O3 were introduced to grow LuAB crystals. UV transparent LuAB crystals were obtained. The cut-off edge of the as-grown LuAB crystal was 178 nm. The nonlinear optical properties of LuAB single crystal was investigated. The crystal has moderate birefringence located at 0.067-0.088. The phase-matching conditions were studied. The coefficient of frequency doubling d11 of LuAB crystal was measured to be 1.10 pm/V by a phase-matching method.

Acknowledgments

This research was supported by National Science Foundation of China (Grant No. 60608018, and 50872132) and National High Technology Research and Development Program of China (Grant No. 2006AA030107).

References and links

1. N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31, 179–278 (1995). [CrossRef]  

2. N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31(3-4), 279–312 (1995). [CrossRef]  

3. P. Becker, “Borate materials in nonlinear optics,” Adv. Mater. 10(13), 979–992 (1998). [CrossRef]  

4. D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999). [CrossRef]  

5. A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

6. H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011). [CrossRef]  

7. Q. Liu, X. P. Yan, M. L. Gong, H. Liu, G. Zhang, and N. Ye, “High-power 266 nm ultraviolet generation in yttrium aluminum borate,” Opt. Lett. 36(14), 2653–2655 (2011). [CrossRef]   [PubMed]  

8. O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001). [CrossRef]  

9. Y. J. Chen, Y. F. Lin, J. H. Huang, X. H. Gong, Z. D. Luo, and Y. D. Huang, “Spectroscopic and laser properties of Er3+:Yb3+:LuAl3(BO3)4 crystal at 1.5-1.6 microm,” Opt. Express 18(13), 13700–13707 (2010). [CrossRef]   [PubMed]  

10. J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009). [CrossRef]  

11. A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

12. N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005). [CrossRef]  

13. S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011). [CrossRef]  

14. Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010). [CrossRef]  

15. L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009). [CrossRef]  

16. A. D. Mills, “Crystallographic data for new rare earth borate compounds, RX3(BO3)4,” Inorg. Chem. 1(4), 960–961 (1962). [CrossRef]  

17. E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981). [CrossRef]  

18. R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, Inc, 1996), Chap. 2.

19. J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154(3), 851–860 (1967). [CrossRef]  

20. R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, Inc, 1996), Chap. 4.

References

  • View by:
  • |
  • |
  • |

  1. N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31, 179–278 (1995).
    [Crossref]
  2. N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31(3-4), 279–312 (1995).
    [Crossref]
  3. P. Becker, “Borate materials in nonlinear optics,” Adv. Mater. 10(13), 979–992 (1998).
    [Crossref]
  4. D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999).
    [Crossref]
  5. A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).
  6. H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
    [Crossref]
  7. Q. Liu, X. P. Yan, M. L. Gong, H. Liu, G. Zhang, and N. Ye, “High-power 266 nm ultraviolet generation in yttrium aluminum borate,” Opt. Lett. 36(14), 2653–2655 (2011).
    [Crossref] [PubMed]
  8. O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
    [Crossref]
  9. Y. J. Chen, Y. F. Lin, J. H. Huang, X. H. Gong, Z. D. Luo, and Y. D. Huang, “Spectroscopic and laser properties of Er3+:Yb3+:LuAl3(BO3)4 crystal at 1.5-1.6 microm,” Opt. Express 18(13), 13700–13707 (2010).
    [Crossref] [PubMed]
  10. J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
    [Crossref]
  11. A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).
  12. N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
    [Crossref]
  13. S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011).
    [Crossref]
  14. Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010).
    [Crossref]
  15. L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
    [Crossref]
  16. A. D. Mills, “Crystallographic data for new rare earth borate compounds, RX3(BO3)4,” Inorg. Chem. 1(4), 960–961 (1962).
    [Crossref]
  17. E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
    [Crossref]
  18. R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, Inc, 1996), Chap. 2.
  19. J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154(3), 851–860 (1967).
    [Crossref]
  20. R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, Inc, 1996), Chap. 4.

2011 (3)

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Q. Liu, X. P. Yan, M. L. Gong, H. Liu, G. Zhang, and N. Ye, “High-power 266 nm ultraviolet generation in yttrium aluminum borate,” Opt. Lett. 36(14), 2653–2655 (2011).
[Crossref] [PubMed]

S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011).
[Crossref]

2010 (2)

2009 (2)

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

2007 (1)

A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

2005 (1)

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

2001 (1)

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

1999 (1)

D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999).
[Crossref]

1998 (1)

P. Becker, “Borate materials in nonlinear optics,” Adv. Mater. 10(13), 979–992 (1998).
[Crossref]

1995 (2)

N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31, 179–278 (1995).
[Crossref]

N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31(3-4), 279–312 (1995).
[Crossref]

1981 (1)

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

1979 (1)

A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

1967 (1)

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154(3), 851–860 (1967).
[Crossref]

1962 (1)

A. D. Mills, “Crystallographic data for new rare earth borate compounds, RX3(BO3)4,” Inorg. Chem. 1(4), 960–961 (1962).
[Crossref]

Aka, G.

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Aloui-Lebbou, O.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Azizov, A. V.

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

Becker, P.

P. Becker, “Borate materials in nonlinear optics,” Adv. Mater. 10(13), 979–992 (1998).
[Crossref]

Belokoneva, E. L.

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

Belov, N. V.

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

Bjorkholm, J. E.

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154(3), 851–860 (1967).
[Crossref]

Brenier, A.

A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

Capmany, J.

D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999).
[Crossref]

Chen, C.

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Chen, X.

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Chen, Y. J.

Cohen-Adad, M. T.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Dujardin, C.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Fan, J. D.

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

Fang, S. H.

S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011).
[Crossref]

Florian, P.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Gong, M. L.

Gong, X. H.

Goutaudier, C.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Han, S. J.

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

Hruschka, M. A.

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Hu, Z. G.

Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010).
[Crossref]

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Huang, J. H.

Huang, L. X.

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Huang, Y. D.

Jaque, D.

D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999).
[Crossref]

Keszler, D. A.

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Kong, W.

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Kubota, S.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Leoniuk, N. I.

A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

Leonyuk, N. I.

N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31, 179–278 (1995).
[Crossref]

N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31(3-4), 279–312 (1995).
[Crossref]

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

Li, J.

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

Li, J. F.

A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

Li, R. K.

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Lin, Y. F.

Liu, C.

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Liu, H.

S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011).
[Crossref]

Q. Liu, X. P. Yan, M. L. Gong, H. Liu, G. Zhang, and N. Ye, “High-power 266 nm ultraviolet generation in yttrium aluminum borate,” Opt. Lett. 36(14), 2653–2655 (2011).
[Crossref] [PubMed]

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Liu, L.

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Liu, Q.

Luo, Z. D.

Massiot, D.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Mills, A. D.

A. D. Mills, “Crystallographic data for new rare earth borate compounds, RX3(BO3)4,” Inorg. Chem. 1(4), 960–961 (1962).
[Crossref]

Pedrini, C.

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Siegman, A. E.

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154(3), 851–860 (1967).
[Crossref]

Simonov, M. A.

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

Sole, J. G.

D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999).
[Crossref]

Stone-Sundberg, J. L.

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Timchenko, T. I.

A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

Tu, C. Y.

A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

Wang, J. Y.

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

Wang, X.

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Xu, G. G.

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

Xu, X.

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Yan, X. P.

Yao, J. Y.

Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010).
[Crossref]

Ye, N.

S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011).
[Crossref]

Q. Liu, X. P. Yan, M. L. Gong, H. Liu, G. Zhang, and N. Ye, “High-power 266 nm ultraviolet generation in yttrium aluminum borate,” Opt. Lett. 36(14), 2653–2655 (2011).
[Crossref] [PubMed]

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Yu, X. S.

Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010).
[Crossref]

Yue, Y. C.

Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010).
[Crossref]

Zhang, G.

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Q. Liu, X. P. Yan, M. L. Gong, H. Liu, G. Zhang, and N. Ye, “High-power 266 nm ultraviolet generation in yttrium aluminum borate,” Opt. Lett. 36(14), 2653–2655 (2011).
[Crossref] [PubMed]

Zhu, Z. J.

A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

Adv. Mater. (1)

P. Becker, “Borate materials in nonlinear optics,” Adv. Mater. 10(13), 979–992 (1998).
[Crossref]

Appl. Phys. Lett. (2)

D. Jaque, J. Capmany, and J. G. Sole, “Continuous wave laser radiation at 669 nm from a self-frequency-doubled laser of YAl3(BO3) 4: Nd3+,” Appl. Phys. Lett. 75, 325–327 (1999).
[Crossref]

A. Brenier, C. Y. Tu, Z. J. Zhu, and J. F. Li, “Diode pumped passive Q switching of Yb3+-doped GdAl3(BO3)4 nonlinear laser crystal,” Appl. Phys. Lett. 90(071103), 1–3 (2007).

Chem. Mater. (1)

N. Ye, J. L. Stone-Sundberg, M. A. Hruschka, G. Aka, W. Kong, and D. A. Keszler, “Nonlinear optical crystal YxLayScz(BO3)4 (x plus y plus z = 4),” Chem. Mater. 17(10), 2687–2692 (2005).
[Crossref]

Cryst. Growth Des. (1)

S. H. Fang, H. Liu, and N. Ye, “Growth and thermophysical properties of nonlinear optical crystal LuAl3(BO3)4,” Cryst. Growth Des. 11(11), 5048–5052 (2011).
[Crossref]

Dokl. Akad. Nauk SSSR (1)

A. V. Azizov, N. I. Leoniuk, T. I. Timchenko, and N. V. Belov, “Crystallization of yttrium-aluminium borate from solution in melt on the base of potassium trimolybdate,” Dokl. Akad. Nauk SSSR 246, 91–93 (1979).

Inorg. Chem. (1)

A. D. Mills, “Crystallographic data for new rare earth borate compounds, RX3(BO3)4,” Inorg. Chem. 1(4), 960–961 (1962).
[Crossref]

J. Cryst. Growth (2)

Z. G. Hu, X. S. Yu, Y. C. Yue, and J. Y. Yao, “YAl3 (BO3) 4: crystal growth and characterization,” J. Cryst. Growth 312(20), 3029–3033 (2010).
[Crossref]

J. Li, G. G. Xu, S. J. Han, J. D. Fan, and J. Y. Wang, “Growth and optical properties of self-frequency-doubling laser crystal Yb:LuAl3(BO3)4,” J. Cryst. Growth 311(17), 4251–4254 (2009).
[Crossref]

J. Struct. Chem. (1)

E. L. Belokoneva, A. V. Azizov, N. I. Leonyuk, M. A. Simonov, and N. V. Belov, “Crystal structure of YAl3(BO3)4,” J. Struct. Chem. 22(3), 476–478 (1981).
[Crossref]

Mater. Res. Innov. (1)

H. Liu, X. Chen, L. X. Huang, X. Xu, G. Zhang, and N. Ye, “Growth and optical propertiesof UV transparent YAB crystals,” Mater. Res. Innov. 15(2), 140–144 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. (1)

O. Aloui-Lebbou, C. Goutaudier, S. Kubota, C. Dujardin, M. T. Cohen-Adad, C. Pedrini, P. Florian, and D. Massiot, “Structural and scintillation properties of new Ce3+-doped alumino-borate,” Opt. Mater. 16(1-2), 77–86 (2001).
[Crossref]

Phys. Rev. (1)

J. E. Bjorkholm and A. E. Siegman, “Accurate cw measurements of optical second-harmonic generation in ammonium dihydrogen phosphate and calcite,” Phys. Rev. 154(3), 851–860 (1967).
[Crossref]

Prog. Cryst. Growth Ch. (2)

N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31, 179–278 (1995).
[Crossref]

N. I. Leonyuk, “Recent developments in the growth of RM3(BO3)4 crystals for science and modern applications,” Prog. Cryst. Growth Ch. 31(3-4), 279–312 (1995).
[Crossref]

Solid State Sci. (1)

L. Liu, C. Liu, X. Wang, Z. G. Hu, R. K. Li, and C. Chen, “Impact of Fe3+ on UV absorption of K2Al2B2O7 crystals,” Solid State Sci. 11(4), 841–844 (2009).
[Crossref]

Other (2)

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, Inc, 1996), Chap. 4.

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, Inc, 1996), Chap. 2.

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 (8)

Fig. 1
Fig. 1

The UV transmittance curve of LuAB crystals.

Fig. 2
Fig. 2

Crystallization region for LuAB crystals with an excess of 20 wt% Al2O3; ■—LuAB + LuLiW2O8; ○—dose not melt at 1100 þC; ●—LuAB; □—LuLiW2O8; ♦ —LuBO3.

Fig. 3
Fig. 3

LuAB crystal grown along [100] direction (a) ; and along [001] direction (b).

Fig. 4
Fig. 4

Sketch of the Littow prism

Fig. 5
Fig. 5

Refractive indices of LuAB.

Fig. 6
Fig. 6

Phase-matching curves and measured phase-matching angles (circles) of LuAB.

Fig. 7
Fig. 7

Variation of walk-off angle and angular acceptance for type I as a function of fundamental wavelength.

Fig. 8
Fig. 8

Effective nonlinear coefficient as a function of fundamental wavelength for type I and type II SHG in LuAB.

Tables (2)

Tables Icon

Table 1 Constants of sellmeier Eqs. for the LuAB crystal

Tables Icon

Table 2 The measured and calculated values of the principal refractive indices

Equations (8)

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

n=sinχ/sinξ
n i 2 = A i + B i λ 2 C i D i λ 2 ,(i=o,e)
θ m I = sin 1 [ ( n e 2ω n o ω ) 2 ( n o 2ω ) 2 ( n o ω ) 2 ( n o 2ω ) 2 ( n e 2ω ) 2 ] 1/2
[ cos 2 ( θ m II ) ( n o 2ω ) 2 + sin 2 ( θ m II ) ( n e 2ω ) 2 ]= 1 2 { n o ω + [ cos 2 ( θ m II ) ( n o ω ) 2 + sin 2 ( θ m II ) ( n e ω ) 2 ] 1 /2 }
P 2ω (l)= 32 π 2 c 3 ω 2 n 1 2 n 2 d eff 2 P ω (0) ω 0 2 l 2
d eff (X)= [ P X 2ω P A 2ω n 1 2 (X) n 2 (X) n 1 2 (A) n 2 (A) ] 1/2 l A l X d eff (A)
d eff = d 11 cosθcos3φ
d eff = d 11 cos 2 θsin3φ

Metrics