Abstract

For low-symmetry crystals that belong to 2 and m point groups, two kinds of nonlinear optical coefficients matrix were derived after the positive directions of the optical coordinate axes were defined with the aid of the piezoelectric coordinate axes. These two kinds of nonlinear optical coefficient matrix are the cause of the nonsymmetry of the effective nonlinear optical coefficients about the x, y, and z optical coordinate planes. For the corresponding wave vector k(θ,φ) in the Nth quadrant, the effective nonlinear optical coefficient expressions are deffI=aie2[dijk(ω3,ω2,ω1)]N+2aje1ake1 (Type I) and deffII=aie2[dijk(ω3,ω2,ω1)]N+1aje1ake2 (Type II), where N is the number of the quadrant. Only one kind of nonlinear optical coefficient matrix is the maximum effective nonlinear optical coefficient. It is necessary to find out which quadrant, in which the wave vector k(θ,φ) is located, corresponds to the maximum effective nonlinear optical coefficient.

© 2005 Optical Society of America

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References

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  1. H. Rabin and C. L. Tang, Nonlinear Optics, Vol. 1 of Quantum Electronics (Academic, 1975), pp. 209-281.
  2. M. V. Hobden, "Phase matched second-harmonic generation in biaxial crystals," J. Appl. Phys. 38, 4365-4372 (1967).
    [Crossref]
  3. H. Ito, H. Naito, and H. Inaba, "New phase-matchable nonlinear optical crystals," IEEE J. Quantum Electron. QE-10, 247-252 (1974).
    [Crossref]
  4. H. Ito, "Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals," J. Appl. Phys. 49, 3992-3998 (1975).
    [Crossref]
  5. G. Aka, A. Kahn-harari, F. Mougel, and D. Vivien, "Linear and nonlinear-optical properties of a new gadolinium calcium oxoborate crystal, Ca4GDO(BO3)3," J. Opt. Soc. Am. B 14, 2238-2247 (1997).
    [Crossref]
  6. M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
    [Crossref]
  7. Z. Wang, G. Xu, J. Liu, D. Hu, X. Xu, J. Wang, and Z. Shaso, "Nonlinear second-harmonic generation of BiB3O6," J. Opt. Soc. Am. B 21, 1348-1353 (2004).
    [Crossref]
  8. J. Minghua, Crystal Physics (Shandong Science and Technology Press, 1982; in Chinese), pp. 28-36.
  9. "Standards on piezoelectric crystals," Proc. IRE 37, 1378-1390 (1949).
    [Crossref]
  10. J. D. Biertein and C. B. Arweller, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917-919 (1986).
    [Crossref]

2004 (1)

1997 (2)

G. Aka, A. Kahn-harari, F. Mougel, and D. Vivien, "Linear and nonlinear-optical properties of a new gadolinium calcium oxoborate crystal, Ca4GDO(BO3)3," J. Opt. Soc. Am. B 14, 2238-2247 (1997).
[Crossref]

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

1986 (1)

J. D. Biertein and C. B. Arweller, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917-919 (1986).
[Crossref]

1975 (1)

H. Ito, "Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals," J. Appl. Phys. 49, 3992-3998 (1975).
[Crossref]

1974 (1)

H. Ito, H. Naito, and H. Inaba, "New phase-matchable nonlinear optical crystals," IEEE J. Quantum Electron. QE-10, 247-252 (1974).
[Crossref]

1967 (1)

M. V. Hobden, "Phase matched second-harmonic generation in biaxial crystals," J. Appl. Phys. 38, 4365-4372 (1967).
[Crossref]

1949 (1)

"Standards on piezoelectric crystals," Proc. IRE 37, 1378-1390 (1949).
[Crossref]

Aka, G.

Arweller, C. B.

J. D. Biertein and C. B. Arweller, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917-919 (1986).
[Crossref]

Biertein, J. D.

J. D. Biertein and C. B. Arweller, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917-919 (1986).
[Crossref]

Furuya, H.

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

Hobden, M. V.

M. V. Hobden, "Phase matched second-harmonic generation in biaxial crystals," J. Appl. Phys. 38, 4365-4372 (1967).
[Crossref]

Hu, D.

Inaba, H.

H. Ito, H. Naito, and H. Inaba, "New phase-matchable nonlinear optical crystals," IEEE J. Quantum Electron. QE-10, 247-252 (1974).
[Crossref]

Ito, H.

H. Ito, "Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals," J. Appl. Phys. 49, 3992-3998 (1975).
[Crossref]

H. Ito, H. Naito, and H. Inaba, "New phase-matchable nonlinear optical crystals," IEEE J. Quantum Electron. QE-10, 247-252 (1974).
[Crossref]

Iwai, M.

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

Kahn-harari, A.

Kobayashi, T.

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

Liu, J.

Minghua, J.

J. Minghua, Crystal Physics (Shandong Science and Technology Press, 1982; in Chinese), pp. 28-36.

Mori, Y.

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

Mougel, F.

Naito, H.

H. Ito, H. Naito, and H. Inaba, "New phase-matchable nonlinear optical crystals," IEEE J. Quantum Electron. QE-10, 247-252 (1974).
[Crossref]

Rabin, H.

H. Rabin and C. L. Tang, Nonlinear Optics, Vol. 1 of Quantum Electronics (Academic, 1975), pp. 209-281.

Sasaki, T.

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

Shaso, Z.

Tang, C. L.

H. Rabin and C. L. Tang, Nonlinear Optics, Vol. 1 of Quantum Electronics (Academic, 1975), pp. 209-281.

Vivien, D.

Wang, J.

Wang, Z.

Xu, G.

Xu, X.

Appl. Phys. Lett. (1)

J. D. Biertein and C. B. Arweller, "Electro-optic and dielectric properties of KTiOPO4," Appl. Phys. Lett. 49, 917-919 (1986).
[Crossref]

IEEE J. Quantum Electron. (1)

H. Ito, H. Naito, and H. Inaba, "New phase-matchable nonlinear optical crystals," IEEE J. Quantum Electron. QE-10, 247-252 (1974).
[Crossref]

J. Appl. Phys. (2)

H. Ito, "Generalized study on angular dependence of induced second-order nonlinear optical polarizations and phase matching in biaxial crystals," J. Appl. Phys. 49, 3992-3998 (1975).
[Crossref]

M. V. Hobden, "Phase matched second-harmonic generation in biaxial crystals," J. Appl. Phys. 38, 4365-4372 (1967).
[Crossref]

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

Jpn. J. Appl. Phys. Part 1 (1)

M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki, "Crystal growth and optical characterization of rare-earth (Re) calcium oxyborate ReCa4O(BO3)3 (Re=Y or Gd) as new nonlinear optical material," Jpn. J. Appl. Phys. Part 1 36, L276-L279 (1997).
[Crossref]

Proc. IRE (1)

"Standards on piezoelectric crystals," Proc. IRE 37, 1378-1390 (1949).
[Crossref]

Other (2)

J. Minghua, Crystal Physics (Shandong Science and Technology Press, 1982; in Chinese), pp. 28-36.

H. Rabin and C. L. Tang, Nonlinear Optics, Vol. 1 of Quantum Electronics (Academic, 1975), pp. 209-281.

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Figures (2)

Fig. 1
Fig. 1

Right-hand rotation rule for rectangular coordinate system.

Fig. 2
Fig. 2

Phase-matched quadrants that correspond to wave vector k ( θ , φ ) .

Tables (3)

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Table 1 Signs of Coordinate Axes in the Eight Quadrants

Tables Icon

Table 2 Matrices of NLO Coefficients in Eight Quadrants for Point Group 2 ( y b )

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Table 3 Matrices of NLO coefficients in Eight Quadrants for Point Group m ( y b )

Equations (72)

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I 2 ω ( ω 2 l 2 d eff 2 n 1 ω n 2 ω n 2 2 ω ) ( sin 2 [ Δ k ( θ , φ ) l 2 ] [ Δ k ( θ , φ ) l 2 ] 2 ) ( p ω 2 A 2 ) ,
sin 2 [ Δ k ( θ , φ ) l 2 ] [ Δ k ( θ , φ ) l 2 ] 2 = 1 .
n 2 2 ω = n 1 ω ( Type I phase match ) ,
n 2 2 ω = 1 2 ( n 1 ω + n 2 ω ) ( Type II phase match ) .
sin 2 θ cos φ n 2 n x 2 + sin 2 θ cos φ n 2 n y 2 + cos 2 n 2 n z 2 = 0 ,
d eff I = a i e 2 d i j k a j e 1 a k e 1 ( Type I ) ,
d eff II = a i e 2 d i j k a j e 1 a k e 2 ( Type II ) ,
a i e 2 = ( cos θ cos φ sin δ sin φ cos δ cos θ sin φ sin δ + cos φ cos δ sin θ sin δ ) ,
a k e 1 = ( cos θ cos φ cos δ sin φ sin δ cos θ sin φ cos δ + cos φ sin δ sin θ cos δ ) ,
cot 2 δ = cot 2 Ω cos 2 θ cos 2 θ cos 2 φ + sin 2 φ cos θ sin 2 φ ,
x + , y + , z + .
1 + , 2 + , 3 + .
d 311 , d 322 , d 333 , d 113 , d 223 .
x , y , z + .
d 311 d 3 ( 1 ) ( 1 ) d 311 , d 322 d 3 ( 2 ) ( 2 ) d 322 ,
d 333 d 333 , d 113 d ( 1 ) ( 1 ) 3 d 113 ,
d 223 d ( 2 ) ( 2 ) 3 d 223 .
x , y + , z + ,
d 311 d 3 ( 1 ) ( 1 ) d 311 , d 322 d 322 ,
d 333 d 333 , d 113 d ( 1 ) ( 1 ) 3 d 113 , d 223 d 223 .
x + , y + , z ,
d 311 d 11 ( 3 ) d 311 , d 322 d ( 3 ) 22 d 322 ,
d 333 d ( 3 ) ( 3 ) ( 3 ) d 333 , d 113 d 11 ( 3 ) d 113 ,
d 223 d 22 ( 3 ) d 223 .
[ 0 0 0 d 14 0 d 16 d 21 d 22 d 23 0 d 25 0 0 0 0 d 34 0 d 36 ] ( y b ) ,
[ 0 0 0 d 14 d 15 0 0 0 0 d 24 d 25 0 d 31 d 32 d 33 0 0 d 36 ] ( z b ) ,
[ d 11 d 12 d 13 d 14 0 0 0 0 0 0 d 25 d 26 0 0 0 0 d 35 d 36 ] ( x b ) ,
11 1 , 22 2 , 33 3 , 11 1 ,
23 4 , 31 5 , 12 6 .
d 211 , d 222 , d 233 , d 231 ,
d 123 , d 112 , d 323 , d 312 .
x + , y + , z ,
d 211 d 211 , d 222 d 222 ,
d 233 d 2 ( ) ( 3 ) d 233 , d 231 d 2 ( 3 ) 1 d 231 ,
d 123 d 12 ( 3 ) d 123 , d 112 d 112 ,
d 323 d ( 3 ) 2 ( 3 ) d 323 , d 312 d ( 3 ) 12 d 312 .
x + , y , z + ,
d 211 d ( 2 ) 11 d 211 , d 222 d ( 2 ) ( 2 ) ( 2 ) d 222 ,
d 233 d ( 2 ) 33 d 233 ,
d 231 d ( 2 ) 31 d 231 , d 123 d 1 ( 2 ) 3 d 123 ,
d 112 d 11 ( 2 ) d 112 , d 323 d 3 ( 2 ) 3 d 323 ,
d 312 d 31 ( 2 ) d 312 .
[ d 11 d 12 d 13 0 d 15 0 0 0 0 d 24 0 d 26 d 31 d 32 d 33 0 d 35 0 ] ( y b ) ,
[ d 11 d 12 d 13 0 0 d 16 d 21 d 22 d 23 0 0 d 26 0 0 0 d 34 d 35 0 ] ( z b ) ,
[ 0 0 0 0 d 15 d 16 d 21 d 22 d 23 d 24 0 0 d 31 d 32 d 33 d 34 0 0 ] ( x b ) ,
x + , y + , z + ,
d 111 , d 122 , d 133 , d 131 , d 223 ,
d 212 , d 311 , d 322 , d 333 , d 331 .
x , y + , z ,
d 111 d ( 1 ) ( 1 ) ( 1 ) d 111 , d 122 d ( 1 ) 22 d 122 ,
d 133 d ( 1 ) 33 d 133 , d 131 d ( 1 ) ( 3 ) ( 1 ) d 131 ,
d 223 d 22 ( 3 ) d 223 , d 212 d 2 ( 1 ) 2 d 212 ,
d 311 d ( 3 ) ( 1 ) ( 1 ) d 311 , d 322 d ( 3 ) 22 d 322 ,
d 333 d ( 3 ) ( 3 ) ( 3 ) d 333 , d 331 d ( 3 ) ( 3 ) 1 d 311 .
x + , y + , z ,
d 111 d 111 , d 122 d 122 , d 133 d 1 ( 3 ) ( 3 ) d 133 ,
d 131 d 1 ( 3 ) 1 d 131 , d 223 d 22 ( 3 ) d 223 ,
d 212 d 212 d 212 ,
d 311 d ( 3 ) 11 d 311 , d 322 d ( 3 ) 22 d 322 ,
d 333 d ( 3 ) ( 3 ) ( 3 ) d 333 ,
d 331 d ( 3 ) ( 3 ) 1 d 331 .
x + , y , z + .
d 111 d 111 , d 122 d 1 ( 2 ) ( 2 ) d 122 ,
d 133 d 133 , d 131 d 131 ,
d 223 d ( 2 ) ( 2 ) 3 d 223 ,
d 212 d ( 2 ) 1 ( 2 ) d 212 , d 311 d 311 ,
d 322 d 3 ( 2 ) ( 2 ) d 322 ,
d 333 d 333 , d 331 d 331 .
d eff I = a i e 2 [ d i j k ( ω 3 , ω 2 , ω 1 ) ] III a j e 1 a k e 1 ,
d eff II = a i e 2 [ d i j k ( ω 3 , ω 2 , ω 1 ) ] II a j e 1 a k e 2 ,
d eff I = a i e 2 [ d i j k ( ω 3 , ω 2 , ω 1 ) ] N + 2 a j e 1 a k e 1 ,
d eff II = a i e 2 [ d i j k ( ω 3 , ω 2 , ω 1 ) ] N + 1 a j e 1 a k e 2

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