Abstract

Firm evidence of the biaxial nematic phase in liquid crystals, not induced by a magnetic or electric field, has been established only recently. The discovery of these biaxially anisotropic liquid crystals has opened up new areas of both fundamental and applied research. The advances in biaxial liquid-crystal-related topics call for a good overview on the propagation of waves through biaxially anisotropic media. Although the literature sporadically discusses biaxial interfaces, the propagation of waves through inhomogeneous biaxially anisotropic bulk materials has never been fully addressed. For this reason, we present a novel ray-tracing method for inhomogeneous biaxially anisotropic media. In the geometrical-optics approach, we clearly show how to assess the optical properties of inhomogeneous biaxially anisotropic media in three dimensions.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  35. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the hart of crystal optics,” Prog. Opt. 50, 13-50 (2007).
    [CrossRef]
  36. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).
  37. W. PressS. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge U. Press, 1992).
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2008

2007

2006

J. G. Lunney and D. Weaire, “The ins and outs of conical refraction,” Europhys. News 37, 26-29 (2006).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. London, Ser. A 462, 1629-1642 (2006).
[CrossRef]

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic-anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067-1073 (2006).
[CrossRef]

2005

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

2004

B. R. Acharya, A. Primak, and S. Kumar, “Biaxial nematic phase in bent-core thermotropic mesogens,” Phys. Rev. Lett. 92, 145506 (2004).
[CrossRef] [PubMed]

L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, “Thermotropic biaxial nematic liquid crystals,” Phys. Rev. Lett. 92, 145505 (2004).
[CrossRef] [PubMed]

G. R. Luckhurst, “Liquid crystals: a missing phase found at last?” Nature 430, 413-414 (2004).
[CrossRef] [PubMed]

M. V. Berry, “Conical diffraction asymptotics: fine structure of Pogendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6, 289-300 (2004).
[CrossRef]

2003

G. Panasyuk, J. R. Kelly, E. C. Gartland, and D. W. Allender, “Geometrical-optics approach in liquid crystal films with three-dimensional director variations,” Phys. Rev. E 67, 041702 (2003).
[CrossRef]

2002

2001

1998

1996

1993

1992

1991

1987

1986

1982

1975

1962

1911

A. Sommerfeld and J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optiek,” Ann. Phys. 35, 277-298 (1911).
[CrossRef]

1841

W. R. Hamilton, “On a mode of deducing the equation of Fresnel's wave,” Philos. Mag. 19, 381-383 (1841).

Acharya, B. R.

B. R. Acharya, A. Primak, and S. Kumar, “Biaxial nematic phase in bent-core thermotropic mesogens,” Phys. Rev. Lett. 92, 145506 (2004).
[CrossRef] [PubMed]

Allender, D. W.

G. Panasyuk, J. R. Kelly, E. C. Gartland, and D. W. Allender, “Geometrical-optics approach in liquid crystal films with three-dimensional director variations,” Phys. Rev. E 67, 041702 (2003).
[CrossRef]

Avendaño-Alejo, M.

Bellver-Cebreros, C.

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic-anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067-1073 (2006).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the hart of crystal optics,” Prog. Opt. 50, 13-50 (2007).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. London, Ser. A 462, 1629-1642 (2006).
[CrossRef]

M. V. Berry, “Conical diffraction asymptotics: fine structure of Pogendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6, 289-300 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Braat, J. J. M.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Courier Dover, 1993).

Chipman, R. A.

de Boer, D. K. G.

Dingemans, T. J.

L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, “Thermotropic biaxial nematic liquid crystals,” Phys. Rev. Lett. 92, 145505 (2004).
[CrossRef] [PubMed]

Echarri, R. M.

Flannery, B. P.

W. PressS. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge U. Press, 1992).

Fletcher, L.

L. Fletcher, The Optical Indicatrix and the Transmission of Light in Crystals (Oxford U. Press, 1892).

Gao, H.

Gartland, E. C.

G. Panasyuk, J. R. Kelly, E. C. Gartland, and D. W. Allender, “Geometrical-optics approach in liquid crystal films with three-dimensional director variations,” Phys. Rev. E 67, 041702 (2003).
[CrossRef]

Gaylord, T. K.

Ghatak, A. K.

Glytsis, E. N.

Griffiths, L. W.

L. W. Griffiths, Introduction to the Theory of Equations (Wiley, 1946).

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).

Hai, M.

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

Hamilton, W. R.

W. R. Hamilton, “On a mode of deducing the equation of Fresnel's wave,” Philos. Mag. 19, 381-383 (1841).

Hikmet, R. A. M.

Hillman, L. W.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the hart of crystal optics,” Prog. Opt. 50, 13-50 (2007).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. London, Ser. A 462, 1629-1642 (2006).
[CrossRef]

Jenkins, C.

Jianping, X.

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Kelly, J. R.

G. Panasyuk, J. R. Kelly, E. C. Gartland, and D. W. Allender, “Geometrical-optics approach in liquid crystal films with three-dimensional director variations,” Phys. Rev. E 67, 041702 (2003).
[CrossRef]

Kline, M.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Kraan, T. C.

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Kumar, D. V.

Kumar, S.

B. R. Acharya, A. Primak, and S. Kumar, “Biaxial nematic phase in bent-core thermotropic mesogens,” Phys. Rev. Lett. 92, 145506 (2004).
[CrossRef] [PubMed]

S. Kumar, “Biaxial liquid crystal electro-optic devices,” U.S. patent WO/2007/025111 (2007).

Luckhurst, G. R.

G. R. Luckhurst, “Liquid crystals: a missing phase found at last?” Nature 430, 413-414 (2004).
[CrossRef] [PubMed]

Lunney, J. G.

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. London, Ser. A 462, 1629-1642 (2006).
[CrossRef]

J. G. Lunney and D. Weaire, “The ins and outs of conical refraction,” Europhys. News 37, 26-29 (2006).
[CrossRef]

Madsen, L. A.

L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, “Thermotropic biaxial nematic liquid crystals,” Phys. Rev. Lett. 92, 145505 (2004).
[CrossRef] [PubMed]

Maldonado, T. A.

McClain, S. C.

Nakata, M.

L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, “Thermotropic biaxial nematic liquid crystals,” Phys. Rev. Lett. 92, 145505 (2004).
[CrossRef] [PubMed]

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Panasyuk, G.

G. Panasyuk, J. R. Kelly, E. C. Gartland, and D. W. Allender, “Geometrical-optics approach in liquid crystal films with three-dimensional director variations,” Phys. Rev. E 67, 041702 (2003).
[CrossRef]

Pei, W.

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

Peijun, Y.

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

Press, W.

W. PressS. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge U. Press, 1992).

Primak, A.

B. R. Acharya, A. Primak, and S. Kumar, “Biaxial nematic phase in bent-core thermotropic mesogens,” Phys. Rev. Lett. 92, 145506 (2004).
[CrossRef] [PubMed]

Rodriguez-Danta, M.

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic-anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067-1073 (2006).
[CrossRef]

Runge, J.

A. Sommerfeld and J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optiek,” Ann. Phys. 35, 277-298 (1911).
[CrossRef]

Sambles, J. R.

Samulski, E. T.

L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, “Thermotropic biaxial nematic liquid crystals,” Phys. Rev. Lett. 92, 145505 (2004).
[CrossRef] [PubMed]

Schaefer, Clemens

Clemens Schaefer, Einführung in die Theoretische Physik (de Gruyter, 1932).

Schubert, M.

Sharma, A.

Simon, M. C.

Sithambaranathan, G. S.

Sluijter, M.

Sommerfeld, A.

A. Sommerfeld and J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optiek,” Ann. Phys. 35, 277-298 (1911).
[CrossRef]

Stamnes, J. J.

Stavroudis, O. N.

Swindell, W.

Teukolsky, S. A.

W. PressS. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge U. Press, 1992).

van Bommel, T.

Vetterling, W. T.

W. PressS. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge U. Press, 1992).

Walker, D. B.

Weaire, D.

J. G. Lunney and D. Weaire, “The ins and outs of conical refraction,” Europhys. News 37, 26-29 (2006).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Yang, F.

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).

Yonghua, L.

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

Zhang, W.

Ann. Phys.

A. Sommerfeld and J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optiek,” Ann. Phys. 35, 277-298 (1911).
[CrossRef]

Appl. Opt.

Europhys. News

J. G. Lunney and D. Weaire, “The ins and outs of conical refraction,” Europhys. News 37, 26-29 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

M. V. Berry, “Conical diffraction asymptotics: fine structure of Pogendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6, 289-300 (2004).
[CrossRef]

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic-anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067-1073 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

C. Jenkins, “Ray equation for a spatially variable uniaxial crystal and its use in the optical design of liquid-crystal lenses,” J. Opt. Soc. Am. A 24, 2089-2096 (2007).
[CrossRef]

T. C. Kraan, T. van Bommel, and R. A. M. Hikmet, “Modeling liquid-crystal gradient-index lenses,” J. Opt. Soc. Am. A 24, 3467-3477 (2007).
[CrossRef]

M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260-1273 (2008).
[CrossRef]

J. J. Stamnes and G. S. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A 18, 3119-3129 (2001).
[CrossRef]

M. Avendaño-Alejo and O. N. Stavroudis, “Huygens's principle and rays in uniaxial anisotropic media. II. Crystal axis orientation arbitrary,” J. Opt. Soc. Am. A 19, 1674-1679 (2002).
[CrossRef]

D. B. Walker, E. N. Glytsis, and T. K. Gaylord, “Surface mode at isotropic-uniaxial and isotropic-biaxial interfaces,” J. Opt. Soc. Am. A 15, 248-260 (1998).
[CrossRef]

M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201-2204 (1987).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383-2392 (1993).
[CrossRef]

M. Schubert, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” J. Opt. Soc. Am. A 13, 1930-1940 (1996).
[CrossRef]

Nature

G. R. Luckhurst, “Liquid crystals: a missing phase found at last?” Nature 430, 413-414 (2004).
[CrossRef] [PubMed]

Opt. Commun.

L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429-435 (2005).
[CrossRef]

Philos. Mag.

W. R. Hamilton, “On a mode of deducing the equation of Fresnel's wave,” Philos. Mag. 19, 381-383 (1841).

Phys. Rev. E

G. Panasyuk, J. R. Kelly, E. C. Gartland, and D. W. Allender, “Geometrical-optics approach in liquid crystal films with three-dimensional director variations,” Phys. Rev. E 67, 041702 (2003).
[CrossRef]

Phys. Rev. Lett.

B. R. Acharya, A. Primak, and S. Kumar, “Biaxial nematic phase in bent-core thermotropic mesogens,” Phys. Rev. Lett. 92, 145506 (2004).
[CrossRef] [PubMed]

L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, “Thermotropic biaxial nematic liquid crystals,” Phys. Rev. Lett. 92, 145505 (2004).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. London, Ser. A 462, 1629-1642 (2006).
[CrossRef]

Prog. Opt.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the hart of crystal optics,” Prog. Opt. 50, 13-50 (2007).
[CrossRef]

Other

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

W. PressS. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing (Cambridge U. Press, 1992).

L. W. Griffiths, Introduction to the Theory of Equations (Wiley, 1946).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Courier Dover, 1993).

S. Kumar, “Biaxial liquid crystal electro-optic devices,” U.S. patent WO/2007/025111 (2007).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

L. Fletcher, The Optical Indicatrix and the Transmission of Light in Crystals (Oxford U. Press, 1892).

Clemens Schaefer, Einführung in die Theoretische Physik (de Gruyter, 1932).

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).

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Figures (11)

Fig. 1
Fig. 1

Schematic presentation of the biaxial optical indicatrix. (a) shows one octant of the biaxial optical indicatrix in the principal coordinate system. (b)–(d) show the intersections of the optical indicatrix with the principal planes ( v w , u v , and u w planes, respectively). The two concentric shells touch each other in their common points of intersection in the x z plane. The lines that go through these points and the origin are called the optical axes. The optical axes are indicated by the dashed lines. The angle between the optical axes and the w axis is indicated by the angle ϑ. In this case, we assumed n w > n v > n u .

Fig. 2
Fig. 2

Octant of the biaxial optical indicatrix as in Fig. 1a, but now with the electric polarization vectors indicated by the arrows. Both the electric and magnetic polarization vectors are tangent with respect to the biaxial optical indicatrix. As a consequence, for each arbitrary direction of propagation p, the Poynting vector is in the direction of p H , perpendicular to the optical indicatrix.

Fig. 3
Fig. 3

Locally, a biaxially anisotropic medium is characterized by two position-dependent optical axes, indicated by d ̂ 1 and d ̂ 2 . The corresponding principal coordinate system is defined by the unit vectors u ̂ , v ̂ , and w ̂ , which can be expressed in terms of the local optical axes d ̂ 1 and d ̂ 2 .

Fig. 4
Fig. 4

When an unpolarized beam of light is refracted along the optical axis of a biaxially anisotropic medium, the light beam is transformed to a hollow cone of light; see (a). This phenomenon is known as internal conical refraction. The refracted light rays of the beam inside the biaxial medium have a common wave normal. In the case when the refracted light rays of the beam have a common Poynting vector, we observe external conical refraction; see (b).

Fig. 5
Fig. 5

Unpolarized beam of light incident to a biaxial medium with one of the optical axes aligned with the vertical z axis; see (a). Internal conical refraction occurs and the incident beam is transformed to a cone of light with semiangle ν. At z = 100 the light distribution is calculated; see (b).

Fig. 6
Fig. 6

Light intensity distribution at z = 100 for different values of the solid angle d Ω . For (a)–(c) the solid angles are 1 × 10 3 sr , 4 × 10 3 sr , and 9 × 10 3 sr , respectively. The unpolarized beam of light enters the biaxial medium at the origin. Apparently, the disk edge increases with increasing solid angle. In addition, the intensity decreases with increasing solid angle.

Fig. 7
Fig. 7

Light intensity distribution at z = 100 for different orientations of the optical axis. In (a)–(c) the angle between the optical axis and the vertical z axis in the x z plane is 2.7°, 1.0°, and 0.0°, respectively. (a) shows double refraction, whereas (c) shows internal conical refraction. (b) is an intermediate state and shows how the two light beams are transformed to a hollow cone of light.

Fig. 8
Fig. 8

Point charge q at a distance a above the origin. The plane z = 0 is defined a grounded conducting plate. As a result, there is an electric field in the half-space z 0 .

Fig. 9
Fig. 9

Principal unit vector w ̂ (i.e., the normalized electric field due to the point charge q) in the x z plane for a = 50 , x [ 50 , 50 ] , and z [ 0 , 100 ] . The corresponding principal unit vectors u ̂ and v ̂ are also indicated.

Fig. 10
Fig. 10

Ray paths of two refracted rays incident at the position ( x 0 , y 0 , z 0 ) = ( 5 , 9 , 0 ) . (a) and (b) show the image projections in the x z and y z planes, respectively. Likewise, (c) shows the top view of the two ray paths and (d) shows the ray paths in three-dimensional space. Apparently, both ray paths are curved and they are drawn away from the region above the point charge.

Fig. 11
Fig. 11

Intensity distribution I at z = 100 for x [ 50 , 50 ] and y [ 50 , 50 ] . (a)–(c) show the light distribution for a uniaxial director profile ( n o = 1.5 and n e = 1.7 ). (d)–(f) show the light distribution for a biaxial director profile ( n u = 1.3 , n v = 1.5 , and n w = 1.7 ). The square (white) indicates the boundary in which the initial positions of the incident rays lie. The square is moved along the line x = y .

Equations (57)

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E ( r , t ) = E ̃ ( r ) e i ( k 0 ψ ( r ) ω t ) ,
H ( r , t ) = H ̃ ( r ) e i ( k 0 ψ ( r ) ω t ) ,
E ̃ ( r ) = A ( r ) e i δ ( r ) E ̂ ( r ) ,
ψ × H ̃ + c ε 0 ε ͇ E ̃ = 1 i k 0 × H ̃ ,
ψ × E ̃ c μ 0 H ̃ = 1 i k 0 × E ̃ ,
ψ ε ͇ E ̃ = 1 i k 0 ε ͇ E ̃ ,
ψ H ̃ = 1 i k 0 H ̃ .
ε ͇ E ̃ k 0 1 .
p × ( p × E ̃ ) + ε ͇ E ̃ = 0 .
ε ͇ ( r ) = ( ε u ( r ) 0 0 0 ε v ( r ) 0 0 0 ε w ( r ) ) ,
A ( p ) E ̃ = 0 ,
Null ( A ) = { E ̃ C 3 A E ̃ = 0 } .
H = ( ε u p u 2 + ε v p v 2 + ε w p w 2 ) ( p u 2 + p v 2 + p w 2 ) ε u p u 2 ( ε v + ε w ) ε v p v 2 ( ε u + ε w ) ε w p w 2 ( ε u + ε v ) + ε u ε v ε w = 0 .
p = p p ̂ ,
H = ( ε u p ˆ u 2 + ε v p ˆ v 2 + ε w p ˆ w 2 ) p 4 ( ε u p ˆ u 2 ( ε v + ε w ) + ε v p ˆ v 2 ( ε u + ε w ) + ε w p ˆ w 2 ( ε u + ε v ) ) p 2 + ε u ε v ε w = 0 .
tan ( ϑ ) = ε w ( ε v ε u ) ε u ( ε w ε v ) .
p ̂ = ± ε w ( ε v ε u ) ε v ( ε w ε u ) u ̂ + 0 v ̂ ± ε u ( ε w ε v ) ε v ( ε w ε u ) w ̂ ,
p 2 E ̃ u ε u E ̃ u = ( E ̃ p ) p u ,
p 2 E ̃ v ε v E ̃ v = ( E ̃ p ) p v ,
p 2 E ̃ w ε w E ̃ w = ( E ̃ p ) p w .
E ̂ i = C ( E ̃ p ) p i p 2 ε i , i = u , v , w ,
E ̂ = ( ± 1 , 0 , 0 ) .
p H ( r , p ) = H ( r , p ) p u u ̂ ( r ) + H ( r , p ) p v v ̂ ( r ) + H ( r , p ) p w w ̂ ( r ) .
p H E ̂ = C ( E ̃ p ) f ( p u , p v , p w ) ( p 2 ε u ) ( p 2 ε v ) ( p 2 ε w ) H ,
f ( p u , p v , p w ) = 2 p 4 p u 2 ( ε v + ε w ) p v 2 ( ε u + ε w ) p w 2 ( ε u + ε v ) .
S = 1 2 Re ( E × H ) ,
S p H .
d r d τ = S ( r ( τ ) ) ,
d ̂ 1 = ε w ( ε v ε u ) ε v ( ε w ε u ) u ̂ + 0 v ̂ + ε u ( ε w ε v ) ε v ( ε w ε u ) w ̂ ,
d ̂ 2 = ε w ( ε v ε u ) ε v ( ε w ε u ) u ̂ + 0 v ̂ + ε u ( ε w ε v ) ε v ( ε w ε u ) w ̂ .
u ̂ = d ̂ 1 d ̂ 2 d ̂ 1 d ̂ 2 ,
v ̂ = d ̂ 2 × d ̂ 1 d ̂ 2 × d ̂ 1 ,
w ̂ = d ̂ 1 + d ̂ 2 d ̂ 1 + d ̂ 2 .
D = ε 0 ε u ( E u ̂ ) u ̂ + ε 0 ε v ( E v ̂ ) v ̂ + ε 0 ε w ( E w ̂ ) w ̂ .
( ε x x + p x 2 p 2 ε x y + p x p y ε x z + p x p z ε y x + p y p x ε y y + p y 2 p 2 ε y z + p y p z ε z x + p z p x ε z y + p z p y ε z z + p z 2 p 2 ) E ̃ = 0 .
H ( x , y , z , p x , p y , p z ) = [ ε u ( p u ̂ ) 2 + ε v ( p v ̂ ) 2 + ε w ( p w ̂ ) 2 ] p 2 + ε u ε v [ ( u ̂ p × v ̂ ) 2 p 2 ] + ε u ε w [ ( u ̂ p × w ̂ ) 2 p 2 ] + ε v ε w [ ( v ̂ p × w ̂ ) 2 p 2 ] + ε u ε v ε w = 0 ,
d i d τ = α H p i , i = x , y , z ,
d p i d τ = α H i , i = x , y , z ,
d ( x , y , z ) d τ = α p H ,
d ( p x , p y , p z ) d τ = α r H ,
H i = 2 p 2 [ ε u ( p u ̂ ) ( p u ̂ i ) + ε v ( p v ̂ ) ( p v ̂ i ) + ε w ( p w ̂ ) ( p w ̂ i ) ] + 2 ε u ε v [ u ̂ i ( p × v ̂ ) + u ̂ ( p × v ̂ i ) ] u ̂ ( p × v ̂ ) + 2 ε u ε w [ u ̂ i ( p × w ̂ ) + u ̂ ( p × w ̂ i ) ] u ̂ ( p × w ̂ ) + 2 ε v ε w [ v ̂ i ( p × w ̂ ) + v ̂ ( p × w ̂ i ) ] v ̂ ( p × w ̂ ) + h ( ε u i , ε v i , ε w i ) , i = x , y , z ,
h = p 2 [ ε u i ( p u ̂ ) 2 + ε v i ( p v ̂ ) 2 + ε w i ( p w ̂ ) 2 ] + [ u ̂ ( p × v ̂ ) ] 2 i ( ε u ε v ) + [ u ̂ ( p × w ̂ ) ] 2 i ( ε u ε w ) + [ v ̂ ( p × w ̂ ) ] 2 i ( ε v ε w ) + i ( ε u ε v ε w ) , i = x , y , z .
H p i = 2 p 2 [ ε u ( p u ̂ ) u ̂ i + ε v ( p v ̂ ) v ̂ i + ε w ( p w ̂ ) w ̂ i ] + 2 p i [ ε u ( p u ̂ ) 2 + ε v ( p v ̂ ) 2 + ε w ( p w ̂ ) 2 ε u ε v ε u ε w ε v ε w ] 2 ε u ε v ( u ̂ × v ̂ ) i u ̂ ( p × v ̂ ) 2 ε u ε w ( u ̂ × w ̂ ) i u ̂ ( p × w ̂ ) 2 ε v ε w ( v ̂ × w ̂ ) i v ̂ ( p × w ̂ ) , i = x , y , z .
d r ( τ ) d τ = p H ( d ̂ ) ,
d p ( τ ) d τ = r H ( d ̂ ) ,
r ( τ 0 ) = ( x 0 , y 0 , z 0 ) ,
p ( τ 0 ) = p 0 .
i ( τ N + Δ τ ) = i ( τ N ) + Δ τ H p i ( τ N ) ,
p i ( τ N + Δ τ ) = p i ( τ N ) Δ τ H i ( τ N ) , i = x , y , z ,
p i = n S ̂ i ,
p i × n ̂ = p × n ̂ ,
p tn = p i ( p i n ̂ ) n ̂ .
p = p tn + ξ n ̂ , ξ 0 ,
p = ± ε w ( ε v ε u ) ε w ε u u ̂ + 0 v ̂ ± ε u ( ε w ε v ) ε w ε u w ̂ .
ν = ( n w n v ) ( n v n u ) n v ,
Φ ( x , y , z ) = q 4 π ε 0 1 x 2 + y 2 + ( z + a ) 2 q 4 π ε 0 1 x 2 + y 2 + ( z a ) 2 .
w ̂ ( x , y , z ) = E ( x , y , z ) E ( x , y , z ) , z 0 .

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