Abstract

Uniaxial optical anisotropy in the geometrical-optics approach is a classical problem, and most of the theory has been known for at least fifty years. Although the subject appears frequently in the literature, wave propagation through inhomogeneous anisotropic media is rarely addressed. The rapid advances in liquid-crystal lenses call for a good overview of the theory on wave propagation via anisotropic media. Therefore, we present a novel polarized ray-tracing method, which can be applied to anisotropic optical systems that contain inhomogeneous liquid crystals. We describe the propagation of rays in the bulk material of inhomogeneous anisotropic media in three dimensions. In addition, we discuss ray refraction, ray reflection, and energy transfer at, in general, curved anisotropic interfaces with arbitrary orientation and/or arbitrary anisotropic properties. The method presented is a clear outline of how to assess the optical properties of uniaxially anisotropic media.

© 2008 Optical Society of America

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References

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    [CrossRef]
  3. M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (2006).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [PubMed]
  38. SHINTECH.Inc, http://www.shintech.jp.
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2007 (1)

2006 (3)

M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (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]

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

2005 (1)

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 (1)

G. Panasyuk, J. R. Kelly, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

2003 (2)

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]

E. Hällstig, J. Stigwall, M. Lindgren, and L. Sjöqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13-23 (2003).
[CrossRef]

2002 (2)

2001 (1)

1999 (1)

1998 (1)

1997 (1)

1995 (1)

1993 (4)

1992 (1)

1991 (2)

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-465 (1991).
[CrossRef]

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503-518 (1991).
[CrossRef]

1990 (1)

1989 (1)

1986 (1)

1983 (1)

1975 (1)

1962 (1)

1911 (1)

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

Allender, D. W.

G. Panasyuk, J. R. Kelly, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

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]

Beyerle, G.

Born, M.

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

Bos, P.

G. Panasyuk, J. R. Kelly, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

Chipman, R. A.

Chumakov, S. M.

Cojocaru, E.

de Boer, D. K. G.

M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (2006).
[CrossRef]

de Smet, D. J.

D. J. de Smet, “Brewster's angle and optical anisotropy,” Am. J. Phys. 62, 246-248 (1993).
[CrossRef]

de Zwart, S. T.

M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (2006).
[CrossRef]

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” in SID Symposium Digest of Technical Papers (Society for Information Display, 2005), Vol. 36, pp. 98-101.
[CrossRef]

Dekker, T.

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” in SID Symposium Digest of Technical Papers (Society for Information Display, 2005), Vol. 36, pp. 98-101.
[CrossRef]

Echarri, R. M.

Flannery, B. P.

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

Fletcher, L.

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

Gartland, E. C.

G. Panasyuk, J. R. Kelly, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

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]

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]

Hällstig, E.

E. Hällstig, J. Stigwall, M. Lindgren, and L. Sjöqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13-23 (2003).
[CrossRef]

Hikmet, R. A. M.

Hillman, L. W.

IJzerman, W. L.

M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (2006).
[CrossRef]

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” in SID Symposium Digest of Technical Papers (Society for Information Display, 2005), Vol. 36, pp. 98-101.
[CrossRef]

Jackson, J. D.

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

Jen, Y.-J.

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, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

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]

Lee, C.-C.

Lekner, J.

J. Lekner, “Reflection by uniaxial crystals: polarizing angle and Brewster angle,” J. Opt. Soc. Am. A 16, 2763-2766 (1999).
[CrossRef]

J. Lekner, “Brewster angles in reflection by uniaxial crystals,” Am. J. Phys. 10, 2059-2064 (1993).

Liang, Q. T.

Lindgren, M.

E. Hällstig, J. Stigwall, M. Lindgren, and L. Sjöqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13-23 (2003).
[CrossRef]

Lunney, J. G.

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

McClain, S. C.

McDermid, I. S.

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, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

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]

Perez, L. I.

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503-518 (1991).
[CrossRef]

Press, W.

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

Rivera, A. L.

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 Optik,” Ann. Phys. 35, 277-298 (1911).
[CrossRef]

Simon, M. C.

Sjöqvist, L.

E. Hällstig, J. Stigwall, M. Lindgren, and L. Sjöqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13-23 (2003).
[CrossRef]

Sluijter, M.

M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (2006).
[CrossRef]

Sommerfeld, A.

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

Stavroudis, O. N.

Stigwall, J.

E. Hällstig, J. Stigwall, M. Lindgren, and L. Sjöqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13-23 (2003).
[CrossRef]

Swindell, W.

Teukolsky, S. A.

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

Trolinger, J. D.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-465 (1991).
[CrossRef]

van Bommel, T.

Vetterling, W. T.

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

Weaire, D.

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

Wilson, D. K.

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-465 (1991).
[CrossRef]

Wolf, E.

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

Wolf, K. B.

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.-Q.

Am. J. Phys. (2)

D. J. de Smet, “Brewster's angle and optical anisotropy,” Am. J. Phys. 62, 246-248 (1993).
[CrossRef]

J. Lekner, “Brewster angles in reflection by uniaxial crystals,” Am. J. Phys. 10, 2059-2064 (1993).

Ann. Phys. (1)

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

Appl. Opt. (7)

Europhys. News (1)

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

J. Mod. Opt. (1)

M. C. Simon and L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503-518 (1991).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

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. (1)

J. Opt. Soc. Am. A (7)

Liq. Cryst. (1)

G. Panasyuk, J. R. Kelly, P. Bos, E. C. Gartland, and D. W. Allender, “The geometrical-optics approach for multidimensional liquid crystal cells,” Liq. Cryst. 31, 1503-1515 (2004).
[CrossRef]

Opt. Commun. (1)

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]

Opt. Eng. (Bellingham) (1)

J. D. Trolinger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-465 (1991).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

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]

Proc. SPIE (2)

E. Hällstig, J. Stigwall, M. Lindgren, and L. Sjöqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13-23 (2003).
[CrossRef]

M. Sluijter, W. L. IJzerman, D. K. G. de Boer, and S. T. de Zwart, “Residual lens effects in 2D mode of auto-stereoscopic lenticular-based switchable 2D/3D displays,” Proc. SPIE 6196, 61960I (2006).
[CrossRef]

Other (10)

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

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” in SID Symposium Digest of Technical Papers (Society for Information Display, 2005), Vol. 36, pp. 98-101.
[CrossRef]

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

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

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

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

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

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

SHINTECH.Inc, http://www.shintech.jp.

AUTRONIC MELCHERS GmbH, http://www.autronic-melchers.com.

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Figures (9)

Fig. 1
Fig. 1

Octant of the uniaxial optical indicatrix in the principal coordinate system. The two surfaces, sphere and ellipsoid, touch each other in their common points of intersection with the z axis. Here we assumed positive birefringence, i.e., n e > n o .

Fig. 2
Fig. 2

Octant of the optical indicatrix in the principal coordinate system. The electric polarization vectors of the ordinary waves are indicated by the arrows on the sphere surface. The electric polarization vectors of the extraordinary waves are indicated by the arrows on the ellipsoid surface. The polarization vectors of both the ordinary and extraordinary waves are tangent to the optical indicatrix.

Fig. 3
Fig. 3

Refraction and reflection at an anisotropic–anisotropic interface. The transmitted ordinary wave and extraordinary wave are indicated by T o and T e , respectively. The reflected ordinary wave and extraordinary wave are indicated by R o and R e , respectively.

Fig. 4
Fig. 4

Transmittance and reflectance factors as a function of the angle of incidence θ i for an air–calcite interface. The optical axis is at 45 ° with the plane of incidence; T o and T e are the ordinary and extraordinary transmittance factors, respectively; R s and R p are the reflectance factors for s- and p-polarized light, respectively; T t is the sum of these factors and should result in 1 for any value of θ i . The Brewster angle θ B is the angle where R p vanishes and reads 59.76 ° .

Fig. 5
Fig. 5

Brewster angle θ B for an air–calcite interface as a function of γ 2 . The optical axis is in the plane of incidence. The parameter γ is the cosine of the angle Γ between the optical axis o ̂ and the surface normal n ̂ .

Fig. 6
Fig. 6

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

Fig. 7
Fig. 7

Director profile (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 profile has azimuthal symmetry.

Fig. 8
Fig. 8

Ray paths of several extraordinary waves at normal incidence to the plane z = 0 , where the x z plane is the plane of incidence. Note the “curtainlike” behavior, allowing no light in the region above the point charge.

Fig. 9
Fig. 9

Intensity distribution I at z = 100 for x [ 50 , 50 ] and y [ 50 , 50 ] . The square (white) indicates the boundary in which the initial positions of the incident rays lie. This boundary is moved along the line x = y .

Equations (85)

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2 E ε μ c 2 2 E t 2 + ( ln μ ) × × E + ( E ln ε ) = 0 ,
2 E ε μ c 2 2 E t 2 = 0 .
E ( r , t ) = E ̃ ( r ) e i ( k 0 ψ ( r ) ω t ) ,
E ̃ ( r ) = A ( r ) e i δ ( r ) E ̂ ,
1 k 0 E ̃ E ̃ 1 , 1 k 0 k k 1 .
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 ͇ = ( ε x 0 0 0 ε y 0 0 0 ε z ) ,
A ( p ) E ̃ = 0 ,
H = ( p x 2 + p y 2 n e 2 + p z 2 n o 2 1 ) ( p 2 n o 2 1 ) = 0 .
( p ̂ x 2 p ̂ x p ̂ y p ̂ x p ̂ z p ̂ y p ̂ x p ̂ y 2 p ̂ y p ̂ z p ̂ z p ̂ x p ̂ z p ̂ y n e 2 n o 2 p ̂ x 2 p ̂ y 2 ) E o = 0 ,
E o = ( p ̂ y p ̂ x 0 ) .
E ̂ o = p o × o ̂ p o × o ̂ .
E ̂ e = ( p e × o ̂ ) × p H e ( p e × o ̂ ) × p H e ,
E ̃ i = ( E ̃ p ) p i p 2 ε i , i = x , y , z .
p H e E ̃ e = 2 ( E ̃ e p e ) p e 2 ( p e 2 n o 2 ) ( p e 2 n e 2 ) H e ,
S = 1 2 Re ( E × H * ) ,
S p H .
d r d τ = C S ( r ( τ ) ) ,
D = ε 0 ε ( E d ̂ ) d ̂ + ε 0 ε ( E ( E d ̂ ) d ̂ ) .
D = ε 0 ε E + ε 0 ε ( E d ̂ ) d ̂ .
ε i j = ε δ i j + ε d ̂ i d ̂ j , i , j = x , y , z ,
( ε + ε d ̂ x 2 + p x 2 p 2 ε d ̂ x d ̂ y + p x p y ε d ̂ x d ̂ z + p x p z ε d ̂ y d ̂ x + p y p x ε + ε d ̂ y 2 + p y 2 p 2 ε d ̂ y d ̂ z + p y p z ε d ̂ z d ̂ x + p z p x ε d ̂ z d ̂ y + p z p y ε + ε d ̂ z 2 + p z 2 p 2 ) E = 0 .
H ( x , y , z , p x , p y , p z )
= ( ε p 2 + ε ( p d ̂ ) 2 ε ( ε + ε ) ) ( p 2 ε ) = 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 ,
p H = H o p H e + H e p H o ,
r H = H o r H e + H e r H o .
d ( x , y , z ) d τ = α p H o ,
d ( p o x , p o y , p o z ) d τ = α r H o ,
H o i = ε i , H o p o i = 2 p o i , i = x , y , z .
H e i = 2 ( ε ε ) ( p e d ̂ ) ( p e x d ̂ x i + p e y d ̂ y i + p e z d ̂ z i ) + ε ε i + ( ε + p e 2 ) ε i ,
H e p e i = 2 ε p e i + 2 ( ε ε ) ( p e d ̂ ) d ̂ i , i = x , y , z .
H o i = 0 , H o p o i = 2 p o i , i = x , y , z .
H e i = 2 ( ε ε ) ( p e d ̂ ) ( p e x d ̂ x i + p e y d ̂ y i + p e z d ̂ z i ) ,
H e p e i = 2 ε p e i + 2 ( ε ε ) ( p e d ̂ ) d ̂ i , i = x , y , z .
d ( x , y , z ) d τ = α ( p x , p y , p z ) , d ( p x , p y , p z ) d τ = 0 ,
S ̂ i = p H e p H e .
p i e = ( n e 2 S ̂ i x , n e 2 S ̂ i y , n o 2 S ̂ i z ) n e 2 + ( n o 2 n e 2 ) S ̂ i z 2 .
p i × n ̂ = p × n ̂ ,
p t n = p i ( p i n ̂ ) n ̂ .
p o = p t n ± n o 2 p t n 2 n ̂ ,
p e = p t n + ξ n ̂ .
ξ = B ± B 2 4 A C 2 A ,
A = n ̂ z 2 n o 2 + n ̂ x 2 + n ̂ y 2 n e 2 ,
B = 2 p t n z n ̂ z n o 2 + 2 p t n x n ̂ x + 2 p t n y n ̂ y n e 2 ,
C = p t n z 2 n o 2 + p t n x 2 + p t n y 2 n e 2 1 .
t s = p i × n ̂ , t p = n ̂ × t s .
t s ( a t o E ̂ t o + a t e E ̂ t e ) = t s ( E ̃ i + a r o E ̂ r o + a r e E ̂ r e ) ,
t p ( a t o E ̂ t o + a t e E ̂ t e ) = t p ( E ̃ i + a r o E ̂ r o + a r e E ̂ r e ) ,
t s ( a t o H t o + a t e H t e ) = t s ( H ̃ i + a r o H r o + a r e H r e ) ,
t p ( a t o H t o + a t e H t e ) = t p ( H ̃ i + a r o H r o + a r e H r e ) ,
( t s E ̂ t o t s E ̂ t e t s E ̂ r o t s E ̂ r e t p E ̂ t o t p E ̂ t e t p E ̂ r o t p E ̂ r e t s H t o t s H t e t s H r o t s H r e t p H t o t p H t e t p H r o t p H r e ) ( a t o a t e a r o a r e ) = ( t s E ̃ i t p E ̃ i t s H ̃ i t p H ̃ i ) .
E ̂ r s = p r × n ̂ p r × n ̂ ,
E ̂ r p = E ̂ r s × p r E ̂ r s × p r .
E r = ( a r s E ̂ r s + a r p E ̂ r p ) e i ( k 0 ψ r ω t ) ,
H r = 1 c μ 0 ( a r s H r s + a r p H r p ) e i ( k 0 ψ r ω t ) ,
E t e = a t e E ̂ t e e i ( k 0 ψ t e ω t ) , H t e = 1 c μ 0 a t e H t e e i ( k 0 ψ t e ω t ) .
n ̂ S t o + n ̂ S t e n ̂ S r o n ̂ S r e = n ̂ S i .
n ̂ S t o n ̂ S i + n ̂ S t e n ̂ S i n ̂ S r o n ̂ S i n ̂ S r e n ̂ S i = 1 .
T o + T e + R o + R e = 1 ,
T o = n ̂ S t o n ̂ S i , T e = n ̂ S t e n ̂ S i ,
R o = n ̂ S r o n ̂ S i , R e = n ̂ S r e n ̂ S i .
E ̂ t o = p t o × o ̂ 2 p t o × o ̂ 2 ,
E ̂ t e = ( p t e × o ̂ 2 ) × p H e p = p t e ( p t e × o ̂ 2 ) × p H e p = p t e ,
E ̂ r o = p r o × o ̂ 1 p r o × o ̂ 1 ,
E ̂ r e = ( p r e × o ̂ 1 ) × p H e p = p r e ( p r e × o ̂ 1 ) × p H e p = p r e .
E i = E ̂ i e i ( k 0 ψ i ω t ) , H i = 1 c μ 0 H i e i ( k 0 ψ i ω t ) ,
E t o = a t o E ̂ t o e i ( k 0 ψ t o ω t ) , H t o = 1 c μ 0 a t o H t o e i ( k 0 ψ t o ω t ) ,
E t e = a t e E ̂ t e e i ( k 0 ψ t e ω t ) , H t e = 1 c μ 0 a t e H t e e i ( k 0 ψ t e ω t ) .
d r ( τ ) d τ = p H e ( d ̂ ) ,
d p e ( τ ) d τ = r H e ( d ̂ ) ,
H e i = 2 ( ε ε ) ( p e d ̂ ) ( p e x d ̂ x i + p e y d ̂ y i + p e z d ̂ z i ) ,
H e p e i = 2 ε p e i + 2 ( ε ε ) ( p e d ̂ ) d ̂ i , i = x , y , z .
r ( τ 0 ) = ( x 0 , y 0 , z 0 ) ,
p e ( τ 0 ) = p t e .
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 .
d ̂ ( x , y , z ) = E ( x , y , z ) E ( x , y , z ) , z 0 .

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