Abstract

A new approach for finding configurations for measuring quadratic electro-optic effect in noncentrosymmetric crystals without any background of a linear electro-optic response is presented. This approach utilizes a generalized form of Fresnel’s wave vectors equation. The method allows the modulation of the fast and slow waves in the crystal to be considered separately. An example is given for the lithium niobate crystal.

© 2008 Optical Society of America

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References

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  1. G. Alexakis, N. Theofanous, A. Arapoyianni, M. Aillerie, C. Carabatos-Nedelec, and M. Fontana, "Measurement of quadratic electrooptic coefficients in LiNbO3 using a variation of the FDEOM method," Opt. Quantum Electron. 26, 1043-1059 (1994).
    [CrossRef]
  2. M. J. Gunning, R. E. Raab, and W. Kucharczyk, "Magnitude and nature of the quadratic electro-optic effect in potassium dihydrogen phosphate and ammonium dihydrogen phosphate crystals," J. Opt. Soc. Am. B 18, 1092-1098 (2001).
    [CrossRef]
  3. R. Ledzion, M. Izdebski, K. Bondarczuk, and W. Kucharczyk, "Experimental investigation of the anisotropy of quadratic electrooptic effect in ADP," Opto-electronics (London) 12, 449-451 (2004).
  4. M. Melnichuk and L. T. Wood, "Method for measuring off-diagonal Kerr coefficients by using polarized light transmission," J. Opt. Soc. Am. A 22, 377-384 (2005).
    [CrossRef]
  5. M. Melnichuk and L. T. Wood, "Method for measuring off-diagonal Kerr coefficients by using polarized light transmission: errata," J. Opt. Soc. Am. A 24, 2843 (2007).
    [CrossRef]
  6. M. Melnichuk and L. T. Wood, "Determining selected quadratic coefficients in noncentrosymmetric crystals," J. Opt. Soc. Am. A 23, 1236-1242 (2006).
    [CrossRef]
  7. M. J. Gunning and R. E. Raab, "Algebraic determination of the principal refractive indices and axes in the electro-optic effect," Appl. Opt. 37, 8438-8447 (1998).
    [CrossRef]
  8. M. J. Gunning and R. E. Raab, "Systematic eigenvalue approach to crystal optics: an analytic alternative to the geometric ellipsoid model," J. Opt. Soc. Am. A 15, 2199-2207 (1998).
    [CrossRef]
  9. M. Izdebski, W. Kucharczyk, and R. E. Raab, "On relationships between electro-optic coefficients for impermeability and nonlinear electric susceptibilities," J. Opt. A, Pure Appl. Opt. 6, 421-424 (2004).
    [CrossRef]
  10. T. A. Maldonaldo and T. K. Gaylord, "Electrooptic effect calculations: simplified procedure for arbitrary cases," Appl. Opt. 27, 5051-5565 (1988).
    [CrossRef]
  11. M. Izdebski, "Application of the general Jacobi diagonalization method to the optical properties of a medium perturbed by an external field," Appl. Opt. 45, 8262-8272 (2006).
    [CrossRef] [PubMed]
  12. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).

2007

2006

2005

2004

R. Ledzion, M. Izdebski, K. Bondarczuk, and W. Kucharczyk, "Experimental investigation of the anisotropy of quadratic electrooptic effect in ADP," Opto-electronics (London) 12, 449-451 (2004).

M. Izdebski, W. Kucharczyk, and R. E. Raab, "On relationships between electro-optic coefficients for impermeability and nonlinear electric susceptibilities," J. Opt. A, Pure Appl. Opt. 6, 421-424 (2004).
[CrossRef]

2001

1998

1994

G. Alexakis, N. Theofanous, A. Arapoyianni, M. Aillerie, C. Carabatos-Nedelec, and M. Fontana, "Measurement of quadratic electrooptic coefficients in LiNbO3 using a variation of the FDEOM method," Opt. Quantum Electron. 26, 1043-1059 (1994).
[CrossRef]

1988

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

M. Izdebski, W. Kucharczyk, and R. E. Raab, "On relationships between electro-optic coefficients for impermeability and nonlinear electric susceptibilities," J. Opt. A, Pure Appl. Opt. 6, 421-424 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Quantum Electron.

G. Alexakis, N. Theofanous, A. Arapoyianni, M. Aillerie, C. Carabatos-Nedelec, and M. Fontana, "Measurement of quadratic electrooptic coefficients in LiNbO3 using a variation of the FDEOM method," Opt. Quantum Electron. 26, 1043-1059 (1994).
[CrossRef]

Opto-electronics (London)

R. Ledzion, M. Izdebski, K. Bondarczuk, and W. Kucharczyk, "Experimental investigation of the anisotropy of quadratic electrooptic effect in ADP," Opto-electronics (London) 12, 449-451 (2004).

Other

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).

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Equations (43)

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B 11 x 2 + B 22 y 2 + B 33 z 2 + 2 B 23 y z + 2 B 13 x z + 2 B 12 x y = 1 ,
B i j = δ i j n i 2 + r i j k E k + g i j k l E k E l + ,
ε i j = δ i j n i 2 + χ i j k E k + χ i j k l E k E l + .
n 2 ( σ 1 2 1 ) + n 1 2 + T 11 n 2 σ 1 σ 2 + T 12 n 2 σ 1 σ 3 + T 13 n 2 σ 2 σ 1 + T 21 n 2 ( σ 2 2 1 ) + n 2 2 + T 22 n 2 σ 2 σ 3 + T 23 n 2 σ 3 σ 1 + T 31 n 2 σ 3 σ 2 + T 32 n 2 ( σ 3 2 1 ) + n 3 2 + T 33 = 0 ,
T i j = χ i j k E k + χ i j k l E k E l + .
σ 1 2 1 n 2 1 ε 11 + σ 2 2 1 n 2 1 ε 22 + σ 3 2 1 n 2 1 ε 33 = 0 .
1 n f 2 = 1 2 b + 1 2 b 2 4 a ,
1 n s 2 = 1 2 b 1 2 b 2 4 a ,
a = ( B 22 B 33 B 23 2 ) σ 1 2 + ( B 11 B 33 B 13 2 ) σ 2 2 + ( B 11 B 22 B 12 2 ) σ 3 2 + 2 ( B 13 B 12 B 11 B 23 ) σ 2 σ 3 + 2 ( B 23 B 12 B 22 B 13 ) σ 1 σ 3 + 2 ( B 23 B 13 B 33 B 12 ) σ 1 σ 2 ,
b = B 11 ( σ 2 2 + σ 3 2 ) + B 22 ( σ 1 2 + σ 3 2 ) + B 33 ( σ 1 2 + σ 2 2 ) 2 B 23 σ 2 σ 3 2 B 13 σ 1 σ 3 2 B 12 σ 1 σ 2 .
n 4 b n 2 + a = 0 .
[ ( B 22 B 33 B 23 2 ) n 4 ( B 22 + B 33 ) n 2 + 1 ] σ 1 2 + [ ( B 11 B 33 B 13 2 ) n 2 ( B 11 + B 33 ) n 2 + 1 ] σ 2 2 + [ ( B 11 B 22 B 12 2 ) n 4 ( B 11 + B 22 ) n 2 + 1 ] σ 3 2 + 2 [ ( B 13 B 12 B 11 B 23 ) n 4 + B 23 n 2 ] σ 2 σ 3 + 2 [ ( B 23 B 12 B 22 B 13 ) n 4 + B 13 n 2 ] σ 1 σ 3 + 2 [ ( B 23 B 13 B 33 B 12 ) n 2 + B 12 n 2 ] σ 1 σ 2 = 0 .
1 n σ , o 2 = 1 2 b 1 2 sgn ( n o 2 n e 2 ) b 2 4 a ,
1 n σ , e 2 = 1 2 b + 1 2 sgn ( n o 2 n e 2 ) b 2 4 a ,
B 11 E = 0 = B 22 E = 0 = 1 n o 2 ,
B 33 E = 0 = 1 n e 2 ,
B i j E = 0 = 0 for i j ,
1 n σ , o 2 E = 0 = 1 n o 2 ,
1 n σ , e 2 E = 0 = σ 3 2 n o 2 + σ 1 2 + σ 2 3 n e 2 .
1 n σ , o 2 = ( n σ , o E = 0 + n σ , o E E = 0 E + ) 2 = 1 n σ , o 2 E = 0 2 n σ , o 3 n σ , o E E = 0 E + ,
1 n σ , e 2 = ( n σ , e E = 0 + n σ , e E E = 0 E + ) 2 = 1 n σ , e 2 E = 0 2 n σ , e 3 n σ , e E E = 0 E + .
( 1 n σ , o 2 ) E E = 0 = 0 ,
( 1 n σ , e 2 ) E E = 0 = 0 .
h 11 ( 1 ) σ 2 2 + h 22 ( 1 ) σ 1 2 2 h 12 ( 1 ) σ 1 σ 2 σ 1 2 + σ 2 2 = 0 .
h i j ( 1 ) = r i j k E k E .
( h 11 ( 1 ) σ 1 2 + h 22 ( 1 ) σ 2 2 + 2 h 12 ( 1 ) σ 1 σ 2 ) σ 3 2 σ 1 2 + σ 2 2 + ( σ 1 2 + σ 2 2 ) h 33 ( 1 ) 2 σ 2 σ 3 h 23 ( 1 ) 2 σ 1 σ 3 h 13 ( 1 ) = 0 .
b 2 4 a = ( B 11 B 22 ) 2 + 4 B 12 2 = E ( h 11 ( 1 ) h 22 ( 1 ) ) 2 + 4 h 12 ( 1 ) 2 + .
( 1 n f 2 ) E E = 0 = 1 2 ( h 11 ( 1 ) + h 22 ( 1 ) ) + 1 2 ( h 11 ( 1 ) h 22 ( 1 ) ) 2 + 4 h 12 ( 1 ) 2 = 0 ,
( 1 n s 2 ) E E = 0 = 1 2 ( h 11 ( 1 ) + h 22 ( 1 ) ) 1 2 ( h 11 ( 1 ) h 22 ( 1 ) ) 2 + 4 h 12 ( 1 ) 2 = 0 .
h 12 ( 1 ) 2 = h 11 ( 1 ) h 22 ( 1 ) .
( h 11 ( 1 ) + h 22 ( 1 ) ) { < 0 , fast wave = 0 , both waves > 0 , slow wave .
( r 222 E 2 + r 113 E 3 ) σ 2 2 + ( r 222 E 2 + r 113 E 3 ) σ 1 2 + 2 r 222 E 1 σ 1 σ 2 = 0 .
[ ( r 222 E 2 + r 113 E 3 ) σ 1 2 + ( r 222 E 2 + r 113 E 3 ) σ 2 2 2 r 222 E 1 σ 1 σ 2 ] σ 3 2 + [ r 333 E 3 ( σ 1 2 + σ 2 2 ) 2 r 232 E 2 σ 2 σ 3 2 r 232 E 1 σ 1 σ 3 ] ( σ 1 2 + σ 2 2 ) = 0 .
E = ( E , 0 , 0 ) , σ 1 σ 2 = 0 , any σ 3 ,
E = ( 0 , E , 0 ) , σ 1 = σ 2 or σ 1 = σ 2 , any σ 3 ,
E = ( E , E , 0 ) 2 , σ 1 = ( 2 1 ) σ 2 or σ 1 = ( 2 + 1 ) σ 2 , any σ 3 ,
E = ( E , 0 , 0 ) , σ 1 σ 3 = 0 , any σ 2 ,
E = ( 0 , E , 0 ) , σ 3 = 0 , any σ 1 and σ 2 ,
E = ( E , E , 0 ) 2 , σ 3 = 0 , any σ 1 and σ 2 .
n f = 2 1 n o 2 + 1 n e 2 + ( g 1111 + g 3311 ) E 2 + [ 1 n e 2 1 n o 2 + ( g 3311 g 1111 ) E 2 ] 2 + 4 r 232 2 E 2 ,
n s = 2 1 n o 2 + 1 n e 2 + ( g 1111 + g 3311 ) E 2 [ 1 n e 2 1 n o 2 + ( g 3311 g 1111 ) E 2 ] 2 + 4 r 232 2 E 2 .
n f = n e 1 2 n e 3 ( g 3311 + n o 2 n e 2 n o 2 n e 2 r 232 2 ) E 2 ,
n s = n o 1 2 n o 3 ( g 1111 n o 2 n e 2 n o 2 n e 2 r 232 2 ) E 2 .

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