Abstract

The behavior of the critical angles between a high-index isotropic medium and a biaxial crystal with arbitrary orientation of the optical tensor has been theoretically analyzed and numerically modeled. The results indicate that, as the biaxial crystal is rotated around an axis perpendicular to the interface, two critical angles appear, corresponding to the excitation of two eigen modes, which periodically vary with a period of π. An optical procedure for fully characterizing the optical tensor of a biaxial crystal is suggested on the basis of the twist-angle dependence of these critical angles. This procedure simply requires the measurement of the p- to s-conversion reflectivity against the sample rotation angle, with just one polished surface of a biaxial crystal.

© 2002 Optical Society of America

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References

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  1. D. Riviere, Y. Levy, C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
    [CrossRef]
  2. K. Ediner, G. Mayer, R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
    [CrossRef]
  3. F. Bloisi, L. Vicari, F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45–55 (1990).
  4. M. Warenghem, M. Ismaili, D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).
  5. F. Yang, J. R. Sambles, “Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
    [CrossRef]
  6. F. Yang, J. R. Sambles, “Optical characterization of a uniaxial material by the polarization-conversion reflectivity technique,” J. Opt. Soc. Am. B 11, 605–617 (1994).
    [CrossRef]
  7. M. C. Simon, I. Diaz, “Total reflection in biaxial crystals: calculation of the limiting angle for incidence from an isotropic medium,” Optik 102, 1–8 (1996).
  8. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

1996 (1)

M. C. Simon, I. Diaz, “Total reflection in biaxial crystals: calculation of the limiting angle for incidence from an isotropic medium,” Optik 102, 1–8 (1996).

1994 (1)

1993 (1)

F. Yang, J. R. Sambles, “Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[CrossRef]

1992 (1)

M. Warenghem, M. Ismaili, D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

1990 (1)

F. Bloisi, L. Vicari, F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45–55 (1990).

1986 (1)

K. Ediner, G. Mayer, R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[CrossRef]

1978 (1)

D. Riviere, Y. Levy, C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[CrossRef]

Bloisi, F.

F. Bloisi, L. Vicari, F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45–55 (1990).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Diaz, I.

M. C. Simon, I. Diaz, “Total reflection in biaxial crystals: calculation of the limiting angle for incidence from an isotropic medium,” Optik 102, 1–8 (1996).

Ediner, K.

K. Ediner, G. Mayer, R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[CrossRef]

Hector, D.

M. Warenghem, M. Ismaili, D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Imbert, C.

D. Riviere, Y. Levy, C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[CrossRef]

Ismaili, M.

M. Warenghem, M. Ismaili, D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Levy, Y.

D. Riviere, Y. Levy, C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[CrossRef]

Mayer, G.

K. Ediner, G. Mayer, R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[CrossRef]

Riviere, D.

D. Riviere, Y. Levy, C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[CrossRef]

Sambles, J. R.

F. Yang, J. R. Sambles, “Optical characterization of a uniaxial material by the polarization-conversion reflectivity technique,” J. Opt. Soc. Am. B 11, 605–617 (1994).
[CrossRef]

F. Yang, J. R. Sambles, “Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[CrossRef]

Schuster, R.

K. Ediner, G. Mayer, R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[CrossRef]

Simon, M. C.

M. C. Simon, I. Diaz, “Total reflection in biaxial crystals: calculation of the limiting angle for incidence from an isotropic medium,” Optik 102, 1–8 (1996).

Simoni, F.

F. Bloisi, L. Vicari, F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45–55 (1990).

Vicari, L.

F. Bloisi, L. Vicari, F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45–55 (1990).

Warenghem, M.

M. Warenghem, M. Ismaili, D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Yang, F.

F. Yang, J. R. Sambles, “Optical characterization of a uniaxial material by the polarization-conversion reflectivity technique,” J. Opt. Soc. Am. B 11, 605–617 (1994).
[CrossRef]

F. Yang, J. R. Sambles, “Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[CrossRef]

J. Mod. Opt. (1)

F. Yang, J. R. Sambles, “Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. III (Paris) (1)

M. Warenghem, M. Ismaili, D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Mol. Cryst. Liq. Cryst. (1)

F. Bloisi, L. Vicari, F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45–55 (1990).

Opt. Commun. (1)

D. Riviere, Y. Levy, C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[CrossRef]

Optik (1)

M. C. Simon, I. Diaz, “Total reflection in biaxial crystals: calculation of the limiting angle for incidence from an isotropic medium,” Optik 102, 1–8 (1996).

Phys. Lett. A (1)

K. Ediner, G. Mayer, R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

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Figures (9)

Fig. 1
Fig. 1

Geometrical frame systems for the isotropic prism—biaxial material geometry.

Fig. 2
Fig. 2

Relation between the refracted waves and the refractive-index ellipsoid of the biaxial medium.

Fig. 3
Fig. 3

Critical angles β C1 and β C2 and related angles of refraction α C1 and α C2, against the sample twist angle Φ.

Fig. 4
Fig. 4

(a) to (h), Incident angles β against the refractive angles α for two eigen modes when the OY axis is in the plane ZOY′ with different ε Y values (see text).

Fig. 5
Fig. 5

Critical angles β C1 and β C2 and related angles of refraction α C1 and α C2 against the sample twist angle Φ.

Fig. 6
Fig. 6

Angle-dependent reflectivity signals (a) R SS and (b) R SP , illustrating the third pseudocritical angle for a special orientation of the biaxial optical tensor.

Fig. 7
Fig. 7

Angle-dependent p- to s-conversion reflectivity, R PS , for three different sample twist angles.

Fig. 8
Fig. 8

Critical angles β C1 and β C2 against the sample twist Φ.

Fig. 9
Fig. 9

Angle-dependent p- to s-conversion reflectivity, R PS , for two different groups of the optical tensor parameters at the same sample twist angle Φ = 125.26°.

Equations (51)

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X=J1X+J2Y+J3Z,
Y=K1X+K2Y+K3Z,
Z=L1X+L2Y+L3Z,
J1=cos γ cos Φ-cos θ sin γ sin Φ,
J2=-cos γ sin Φ-cos θ sin γ cos Φ,
J3=sin θ sin γ,
K1=sin γ cos Φ+cos θ cos γ sin Φ,
K2=-sin γ sin Φ+cos θ cos γ cos Φ,
K3=-sin θ cos γ,
L1=sin θ sin Φ,
L2=sin θ cos Φ,
L3=cos θ.
cos ψ=cos θ cos α-sin θ sin Φ sin α.
XSX+YSY+ZSZ=0,
X2/εX+Y2/εY+Z2/εZ=0.
m±=b±b2-4ac1/2/2a,
a=εXSX2+εYSY2+εZSZ2,
b=εXεY+εZSX2+εYεX+εZSY2+εZεX+εYSZ2,
c=εXεYεZ.
SX=cos α sin θ+sin α cos θ sin Φsin γ-sin α cos Φ cos γ,
SY=-cos α sin θ+sin α cos θ sin Φcos γ-sin α cos Φ sin γ,
SZ=cos α cos θ-sin α sin θ sin Φ.
a=a1 sin2 α+a2 sin α cos α+a3,
a1=εXcos Φ cos γ-cos θ sin Φ sin γ2-sin2 θ sin2 γ+εYcos Φ sin γ+cos θ sin Φ cos γ2-sin2 θ cos2 γ+εZsin2 θ sin2 Φ-cos2 θ,
a2=εXsin 2θ sin Φ sin2 γ-sin θ cos Φ sin 2γ+εYsin 2θ sin Φ cos2 γ+sin θ cos Φ sin 2γ-εZ sin 2θ sin Φ,
a3=εX sin2 θ sin2 γ+εY sin2 θ cos2 γ+εZ cos2 θ,
b=b1 sin2 α+b2 sin α cos α+b3,
b1=εXεYcos2 θ-sin2 θ sin2 Φ+εXεZsin2 θ cos2 γ-cos Φ sin γ+cos θ sin Φ cos γ2+εYεZsin2 θ sin2 γ-cos Φ cos γ-cos θ sin Φ sin γ2
b2=εXεY sin 2θ sin Φ-εXεZsin θ cos Φ sin 2γ+sin 2θ sin Φ cos2 γ-εYεZsin 2θ sin Φ sin2 γ-sin θ cos Φ sin 2γ,
b3=εXεY sin2 θ+εXεZ1-sin2 θ cos2 γ+εYεZ1-sin2 θ sin2 γ.
ε1 sin2 β=m1,2 sin2 α1,2,
ε1 sin2 βC1,2=maxm1,2 sin2 α1,2,
b2a1+a3b2-b1+b3a2+a22c=0.
cot Φ=cos θ tan γ.
m=b-b2-4ac1/2/2a=εX,
2B/2α|α=π/2=2m/2α-2m|α=π/2 =2εXab1-εYεZ+a3b-εYεZ/ab2-4ac1/2.
cot Φ=-cos θ cot γ.
m=b-b2-4ac1/2/2a=εY.
εYεZεX/εX+εZ-εXsin2 θ sin2 Φ.
2B/2α|α=π/2=2m/2α-2m|α=π/2 =2εYab1-εXεZ+a3b-εXεZ/ab2-4ac1/2.
m=b+b2-4ac1/2/2a=εY.
εY>εZεX/εX+εZ-εXsin2 θ sin2 Φ=εY.
B1=2B/2α|α=π/2=2m/2α-2m|α=π/2 =2εYaεXεZ-b1+a3εXεZ-b/ab2-4ac1/2,
εY=-G+G2+4εXεZ sin2 θ cos2 θ cos2 Φ1/22 sin2 θ cos2 Φ,
G=εZ cos2 θ-εZ-εX1-sin2 θ sin2 Φ2-εX sin2 θ cos2 Φ,
m sin2 α=b+b2-4ac1/2/2asin2 α=εY.
b1+a3εY-εXεZtan2 α+b2 tan α+a3εY=0απ/2.
Δ=b22-4a3εYb1+a3εY-εXεZ=0.
T1εY3+T2εY2+T3εY+T4=0,
T1=-sin4 θ cos4 Φ,T2=sin2 θ cos2 ΦεXsin4 θ sin2 Φ cos2 Φ-cos2 θ+εZsin2 θ cos2 Φ-cos2 θ-sin4 θ sin2 Φ cos2 Φ,T3=εXεZ cos2 θsin2 θ cos2 Φ-cos2 θ+εX2 sin4 θ sin2 Φ cos2 Φ cos2 θ+εZ2 sin2 θ cos2 Φ cos2 θ1-sin2 θ sin2 ΦT4=εXεZ cos4 θεX sin2 θ sin2 Φ+εZ1-sin2 θ sin2 Φ.
y3+py+q=0,

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