Abstract

Uniaxial crystals may become biaxial in an external field. The general and exact law of double refraction on the boundary of an isotropic medium and a biaxial crystal lead to an unwieldy set of involved formulas. Rotations of the principal axes of the optical permittivity tensor in the crystal subjected to the field lead to further difficulties in ray and wave tracing. Therefore, we propose a calculus in which the directions of refracted rays and waves in an unperturbed uniaxial crystal are taken as the first approximation, and then small perturbations of rays and waves due to the applied field are considered. Our approach is based on Huygens’s principle and a generalized form of Fresnel’s ray equation. As an example, the method is applied to the electro-optic modulation of ray and wave directions in a BaTiO3 crystal of 4mm symmetry.

© 2008 Optical Society of America

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References

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2008

2007

2006

2005

2004

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

2002

2000

C.-L. Lin and J.-J. Wu, “Abnormal total external refraction for waves propagating from an isotropic medium to an anisotropic medium,” Chin. J. Phys. (Taipei) 38, 24-35 (2000).

1998

1997

1991

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461-466(1991).
[CrossRef]

1990

1986

1983

1975

1965

1962

Avendaño-Alejo, M.

Beyerle, G.

Boyain y Goitia, A. R.

Chipman, R. A.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461-466(1991).
[CrossRef]

Cojocaru, E.

Echarri, R. M.

Izdebski, M.

Johnson, A. R.

Kaminow, I. P.

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

Kucharczyk, W.

Liang, Q.-T.

Lin, C.-L.

C.-L. Lin and J.-J. Wu, “Abnormal total external refraction for waves propagating from an isotropic medium to an anisotropic medium,” Chin. J. Phys. (Taipei) 38, 24-35 (2000).

McDermid, I. S.

Melnichuk, M.

Pockels, F. C. A.

F. C. A. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner, 1904).

Raab, R. E.

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

M. Izdebski, W. Kucharczyk, and R. E. Raab, “Analysis of accuracy of measurement of quadratic-electro optic coefficients in uniaxial crystals: a case study of KDP,” J. Opt. Soc. Am. A 19, 1417-1421 (2002).
[CrossRef]

Shaskolskaya, M. P.

Yu. I.Sirotin and M. P. Shaskolskaya, Fundamentals of Crystal Physics, English transl. (Mir, 1982).

Simon, M. C.

Sirotin, Yu. I.

Yu. I.Sirotin and M. P. Shaskolskaya, Fundamentals of Crystal Physics, English transl. (Mir, 1982).

Stavroudis, O. N.

Swindell, W.

Trolinger, J. D.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461-466(1991).
[CrossRef]

Weingart, J. M.

Wilson, D. K.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461-466(1991).
[CrossRef]

Wood, L. T.

Wu, J.-J.

C.-L. Lin and J.-J. Wu, “Abnormal total external refraction for waves propagating from an isotropic medium to an anisotropic medium,” Chin. J. Phys. (Taipei) 38, 24-35 (2000).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Appl. Opt.

Chin. J. Phys. (Taipei)

C.-L. Lin and J.-J. Wu, “Abnormal total external refraction for waves propagating from an isotropic medium to an anisotropic medium,” Chin. J. Phys. (Taipei) 38, 24-35 (2000).

J. Opt. A

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Eng.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461-466(1991).
[CrossRef]

Other

Yu. I.Sirotin and M. P. Shaskolskaya, Fundamentals of Crystal Physics, English transl. (Mir, 1982).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

F. C. A. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner, 1904).

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

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Figures (1)

Fig. 1
Fig. 1

Huygens’s principle applied for double refraction in a uniaxial crystal. The directions of the ordinary and extraordinary refracted rays P o and P e are drawn to the points of tangency of Fresnel’s double surface of ray propagation with the appropriate plane; the directions of refracted waves S o and S e are normal to the planes.

Equations (98)

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[ x y z ] = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] [ x y z ] .
sin α = σ 1 2 + σ 2 2 .
r = [ cos ξ sin ξ 0 sin ξ cos ξ 0 0 0 1 ] ,
sin ξ = { σ 2 / sin α for     sin α 0 0 for     sin α = 0 ,
cos ξ = { σ 1 / sin α for     sin α 0 1 for     sin α = 0 .
b = ra .
x = v iso t / sin α , y R , z = 0.
A x + B z A v iso t / sin α = 0 , y R .
x = v 1 t , y = v 2 t , z = v 3 t .
g ( v 1 , v 2 , v 3 ) = D v 1 sin α + v 3 D v iso = 0 ,
ε i j x i x j = 1.
n s 2 = 1 2 b + 1 2 b 2 4 a ,
n f 2 = 1 2 b 1 2 b 2 4 a ,
a = ( ε 22 ε 33 ε 23 2 ) p 1 2 + ( ε 11 ε 33 ε 13 2 ) p 2 2 + ( ε 11 ε 22 ε 12 2 ) p 3 2 + 2 ( ε 13 ε 12 ε 11 ε 23 ) p 2 p 3 + 2 ( ε 23 ε 12 ε 22 ε 13 ) p 1 p 3 + 2 ( ε 23 ε 13 ε 33 ε 12 ) p 1 p 2 ,
b = ε 11 ( p 2 2 + p 3 2 ) + ε 22 ( p 1 2 + p 3 2 ) + ε 33 ( p 1 2 + p 2 2 ) 2 ε 23 p 2 p 3 2 ε 13 p 1 p 3 2 ε 12 p 1 p 2 .
f s ( v 1 , v 2 , v 3 ) = c 2 1 2 B 1 2 B 2 4 A v 2 = 0 ,
f f ( v 1 , v 2 , v 3 ) = c 2 1 2 B 1 2 B 2 4 A v 2 = 0 ,
A = a v 2 = ( ε 22 ε 33 ε 23 2 ) v 1 2 + ( ε 11 ε 33 ε 13 2 ) v 2 2 + ( ε 11 ε 22 ε 12 2 ) v 3 2 + 2 ( ε 13 ε 12 ε 11 ε 23 ) v 2 v 3 + 2 ( ε 23 ε 12 ε 22 ε 13 ) v 1 v 3 + 2 ( ε 23 ε 13 ε 33 ε 12 ) v 1 v 2 ,
B = b v 2 = ε 11 ( v 2 2 + v 3 2 ) + ε 22 ( v 1 2 + v 3 2 ) + ε 33 ( v 1 2 + v 2 2 ) 2 ε 23 v 2 v 3 2 ε 13 v 1 v 3 2 ε 12 v 1 v 2 .
f v 1 g v 2 = f v 2 g v 1 ,
f v 1 g v 3 = f v 3 g v 1 .
v 3 0.
v = bv .
u 1 D u 3 sin α = 0.
u 1 = D 2 v iso sin α 1 + D 2 sin α ,
u 2 = 0 ,
u 3 = D v iso 1 + D 2 sin 2 α .
f o ( v 1 , v 2 , v 3 ) = c 2 1 2 B sign ( n o 2 n e 2 ) 1 2 B 2 4 A v 2 = 0 ,
f e ( v 1 , v 2 , v 3 ) = c 2 1 2 B sign ( n o 2 n e 2 ) 1 2 B 2 4 A v 2 = 0.
ε 11 ( 0 ) = ε 22 ( 0 ) = n o 2 ,
ε 33 ( 0 ) = n e 2 ,
ε i j ( 0 ) = 0 for     i j .
f o ( v 1 , v 2 , v 3 ) = c 2 n o 2 v ( 0 ) 2 = 0 ,
f e ( v 1 , v 2 , v 3 ) = c 2 n o 2 v 3 ( 0 ) 2 n e 2 ( v 1 ( 0 ) 2 + v 2 ( 0 ) 2 ) = 0.
ε i j ( E ) = ε i j ( 0 ) + χ i j k E k = ε i j ( 0 ) + h i j ( 1 ) E ,
χ i j k = n i 2 n j 2 r i j k .
h i j ( 1 ) = χ i j k E k / E .
v i ( E ) = v i ( 0 ) + v i ( 1 ) E ,
D ( E ) = D ( 0 ) + D ( 1 ) E .
g E | E = 0 = ( b 11 v 1 ( 0 ) + b 12 v 2 ( 0 ) + b 13 v 3 ( 0 ) ) D ( 1 ) sin α + ( b 31 + b 11 D ( 0 ) sin α ) v 1 ( 1 ) + ( b 32 + b 12 D ( 0 ) sin α ) v 2 ( 1 ) + ( b 33 + b 13 D ( 0 ) sin α ) v 3 ( 1 ) D ( 1 ) v iso = 0 .
f E | E = 0 = c 2 B E + A E v 2 + A v 2 E 2 c 2 B | E = 0 = 0 .
E ( f v 1 g v 2 ) | E = 0 = E ( f v 2 g v 1 ) | E = 0 ,
E ( f v 1 g v 3 ) | E = 0 = E ( f v 3 g v 1 ) | E = 0 ,
g v i = b 3 i + b 1 i D sin α ,
f v i = c 2 B v i + A v i v 2 + A v 2 v i 2 c 2 B .
v 1 ( 0 ) = c 2 n o 2 v iso sin α ,
v 2 ( 0 ) = 0 ,
v 3 ( 0 ) = c n o 2 v iso n o 2 v iso 2 c 2 sin 2 α ,
D ( 0 ) = c n o 2 v iso 2 c 2 sin 2 α .
sin β r ( 0 ) = v 1 ( 0 ) v 1 ( 0 ) 2 + v 3 ( 0 ) 2 = c n o v iso sin α = n iso n o sin α .
v 1 ( 1 ) = v 1 ( 0 ) v iso v ( 0 ) 2 b 11 sin α n o 2 G o F o + v 2 ( 0 ) v 3 ( 0 ) 2 H o v 1 ( 0 ) n o 2 h 22 ( 1 ) + v 2 ( 0 ) n o 2 h 12 ( 1 ) ,
v 2 ( 1 ) = v 2 ( 0 ) v iso v ( 0 ) 2 b 12 sin α n o 2 G o F o v 1 ( 0 ) v 3 ( 0 ) 2 H o v 2 ( 0 ) n o 2 h 11 ( 1 ) + v 1 ( 0 ) n o 2 h 12 ( 1 ) ,
v 3 ( 1 ) = ( v 3 ( 0 ) v iso v ( 0 ) 2 b 13 sin α G o v 3 ( 0 ) ) F o n o 2 ,
D ( 1 ) = D ( 0 ) v iso v ( 0 ) 2 c 2 G o F o ,
F o = v 2 ( 0 ) 2 h 11 ( 1 ) + v 1 ( 0 ) 2 h 22 ( 1 ) 2 v 1 ( 0 ) v 2 ( 0 ) h 12 ( 1 ) v 1 ( 0 ) 2 + v 2 ( 0 ) 2 ,
G o = 2 [ v iso ( v 1 ( 0 ) b 11 + v 2 ( 0 ) b 12 + v 3 ( 0 ) b 13 ) sin α ] ,
H o = v 1 ( 0 ) v 2 ( 0 ) ( h 11 ( 1 ) h 22 ( 1 ) ) + ( v 2 ( 0 ) 2 v 1 ( 0 ) 2 ) h 12 ( 1 ) n o 2 ( v 1 ( 0 ) 2 + v 2 ( 0 ) 2 ) 2 .
v 1 ( 0 ) = n o 2 ( b 31 + b 11 D ( 0 ) sin α ) D ( 0 ) v iso / M ,
v 2 ( 0 ) = n o 2 ( b 32 + b 12 D ( 0 ) sin α ) D ( 0 ) v iso / M ,
v 3 ( 0 ) = n e 2 ( b 33 + b 13 D ( 0 ) sin α ) D ( 0 ) v iso / M ,
M = n o 2 ( b 31 + b 11 D ( 0 ) sin α ) 2 + n o 2 ( b 32 + b 12 D ( 0 ) sin α ) 2 + n e 2 ( b 33 + b 13 D ( 0 ) sin α ) 2 ,
D ( 0 ) = b b 2 4 a c 2 a ,
a = ( n o 2 b 11 2 + n o 2 b 12 2 + n e 2 b 13 2 ) c 2 sin 2 α n o 2 n e 2 v iso 2 ,
b = 2 ( n o 2 b 11 b 31 + n o 2 b 12 b 32 + n e 2 b 13 b 33 ) c 2 sin α ,
c = ( n o 2 b 31 2 + n o 2 b 32 2 + n e 2 b 33 2 ) c 2 .
v 1 ( 1 ) = n e 2 v 1 ( 0 ) v iso / c 2 b 11 sin α n e 2 G e F e v 2 ( 0 ) v 3 ( 0 ) 2 H e v 1 ( 0 ) h 33 ( 1 ) n e 2 + v 3 ( 0 ) h 13 ( 1 ) n e 2 ,
v 2 ( 1 ) = n e 2 v 2 ( 0 ) v iso / c 2 b 12 sin α n e 2 G e F e + v 1 ( 0 ) v 3 ( 0 ) 2 H e v 2 ( 0 ) h 33 ( 1 ) n e 2 + v 3 ( 0 ) h 23 ( 1 ) n e 2 ,
v 3 ( 1 ) = n o 2 v 3 ( 0 ) v iso / c 2 b 13 sin α n o 2 G e F e ( v 1 ( 0 ) 2 h 11 ( 1 ) + v 2 ( 0 ) 2 h 22 ( 1 ) + 2 v 1 ( 0 ) v 2 ( 0 ) h 12 ( 1 ) ) v 3 ( 0 ) n o 2 ( v 1 ( 0 ) 2 + v 2 ( 0 ) 2 ) + v 1 ( 0 ) h 13 ( 1 ) + v 2 ( 0 ) h 23 ( 1 ) n o 2 ,
D ( 1 ) = M n o 2 n e 2 D ( 0 ) v iso G e F e ,
F e = ( v 1 ( 0 ) 2 h 11 ( 1 ) + v 2 ( 0 ) 2 h 22 ( 1 ) + 2 v 1 ( 0 ) v 2 ( 0 ) h 12 ( 1 ) ) v 3 ( 0 ) 2 v 1 ( 0 ) 2 + v 2 ( 0 ) 2 + ( v 1 ( 0 ) 2 + v 2 ( 0 ) 2 ) h 33 ( 1 ) 2 v 2 ( 0 ) v 3 ( 0 ) h 23 ( 1 ) 2 v 1 ( 0 ) v 3 ( 0 ) h 13 ( 1 ) ,
G e = 2 [ v iso ( b 11 v 1 ( 0 ) + b 12 v 2 ( 0 ) + b 13 v 3 ( 0 ) ) sin α ] ,
H e = v 1 ( 0 ) v 2 ( 0 ) ( h 11 ( 1 ) h 22 ( 1 ) ) + ( v 2 ( 0 ) 2 v 1 ( 0 ) 2 ) h 12 ( 1 ) n e 2 ( v 1 ( 0 ) 2 + v 2 ( 0 ) 2 ) 2 .
u 1 = D ( 0 ) 2 v iso sin α 1 + D ( 0 ) 2 sin 2 α + 2 D ( 0 ) D ( 1 ) v iso sin α ( 1 + D ( 0 ) 2 sin 2 α ) 2 E ,
u 2 = 0 ,
u 3 = D ( 0 ) v iso 1 + D ( 0 ) 2 sin 2 α + D ( 1 ) ( 1 D ( 0 ) 2 sin 2 α ) v iso ( 1 + D ( 0 ) 2 sin 2 α ) 2 E .
[ σ 1 σ 2 σ 3 ] = [ 1 0 0 0 cos β sin β 0 sin β cos β ] [ cos γ 0 sin γ 0 1 0 sin γ 0 cos γ ] [ 0 0 1 ] = [ sin γ sin β cos γ cos β cos γ ] .
sin α = sin 2 γ + sin 2 β cos 2 γ .
a = [ 0 0 1 0 1 0 1 0 0 ] .
b = ra = [ 0 sin β cos γ / sin α sin γ / sin α 0 sin γ / sin α sin β cos γ / sin α 1 0 0 ] .
h 13 ( 1 ) = χ 232 ,
D ( 0 ) = n o c n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) c 2 .
v 1 ( 0 ) = c n e 1 c 2 n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) ,
v 2 ( 0 ) = c 2 n e 2 v iso sin β cos γ ,
v 3 ( 0 ) = c 2 n o 2 v iso sin γ .
v 1 ( 1 ) = 0 ,
v 2 ( 1 ) = 0 ,
v 3 ( 1 ) = c n e r 232 1 c 2 n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) .
v 1 = v 1 ( 0 ) + v 1 ( 1 ) E = c n e 1 c 2 n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) ,
v 2 = v 2 ( 0 ) + v 2 ( 1 ) E = c 2 n e 2 v iso sin β cos γ ,
v 3 = v 3 ( 0 ) + v 3 ( 1 ) E c 2 n o 2 v iso sin γ c n e r 232 E .
D ( 1 ) = n o 2 n e 2 c 2 r 232 sin γ n o 2 n e 2 v iso c 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) .
u 1 = n o 2 c 2 v iso sin α n o 2 n e 2 v iso 2 + c 2 ( n o 2 n e 2 ) sin 2 γ + 2 n o 3 n e 2 c 3 v iso sin ( γ ) sin ( α ) n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) c 2 [ n o 2 n e 2 v iso 2 + c 2 ( n o 2 n e 2 ) sin 2 γ ] 2 r 232 E ,
u 2 = 0 ,
u 3 = n o c v iso n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) c 2 n o 2 n e 2 v iso 2 + c 2 ( n o 2 n e 2 ) sin 2 γ + n o 2 n e 2 v iso 2 2 n o 2 c 2 sin 2 β cos 2 γ c 2 ( n e 2 + n e 2 ) c 2 sin 2 γ [ n o 2 n e 2 v iso 2 + c 2 ( n o 2 n e 2 ) sin 2 γ ] 2 c 2 v iso n o 2 n e 2 sin ( γ ) r 232 E .
u 1 n o 2 c 2 v iso sin α n o 2 n e 2 v iso 2 + c 2 ( n o 2 n e 2 ) sin 2 γ + 2 c 3 sin γ sin α n e v iso 2 r 232 E ,
u 2 = 0 ,
u 3 n o c v iso n o 2 n e 2 v iso 2 ( n o 2 sin 2 β cos 2 γ + n e 2 sin 2 γ ) c 2 n o 2 n e 2 v iso 2 + c 2 ( n o 2 n e 2 ) sin 2 γ + c 2 sin γ v iso r 232 E .
h 11 ( 1 ) = h 22 ( 1 ) , h 12 ( 1 ) = 0 .

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