Abstract

The most widespread approach to anisotropic media is dyadic analysis. However, to get a geometrical picture of a dielectric tensor, one has to resort to a coordinate system for a matrix form in order to obtain, for example, the index-ellipsoid, thereby obnubilating the deeper coordinate-free meaning of anisotropy itself. To overcome these shortcomings we present a novel approach to anisotropy: using geometric algebra we introduce a direct geometrical interpretation without the intervention of any coordinate system. By applying this new approach to biaxial crystals we show the effectiveness and insight that geometric algebra can bring to the optics of anisotropic media.

© 2007 Optical Society of America

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References

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  1. J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol.  2, p. 443.
  2. M. Born and E. Wolf, Principles of Optics, 7th expanded ed., (Cambridge University Press, Cambridge, 1999) pp. 790-852.
  3. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).
  4. I. Richter, P. C. Sun, F. Xu, and Y. Fainman, "Design considerations of form birefringence microstructures," Appl. Opt. 34, 2421-2429 (1995). http://www.opticsinfobase.org/abstract.cfm?URI=ao--34-14-2421
    [CrossRef] [PubMed]
  5. U. Levy, C. H. Tsai, L. Pang, and Y. Fainman, "Engineering space-time variant inhomogeneous media for polarization control," Opt. Lett. 29, 1718-1720 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-15-1718
    [CrossRef] [PubMed]
  6. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
    [CrossRef] [PubMed]
  7. I. V. Lindell, Differential Forms in Electromagnetics, (IEEE Press, Piscataway, 2004) pp. 123-161.
  8. I. V. Lindell, Methods for Electromagnetic Field Analysis, (IEEE Press, Piscataway, 2nd ed., 1995) pp. 17-52.
  9. D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Publishers, Dordrecht, 2nd ed., 1999).
  10. P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2nd ed., 2001).
    [CrossRef]
  11. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).
  12. L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).
  13. D. Hestenes, "Oersted Medal Lecture 2002: Reforming the mathematical language of physics," Am. J. Phys. 71, 104-121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
    [CrossRef]
  14. P. Puska, "Covariant isotropic constitutive relations in Clifford’s geometric algebra," Progress in Electromagnetics Research - PIER 32, 413-428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf.
    [CrossRef]
  15. C. R. Paiva and M. A. Ribeiro, "Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach," J. Electromagn. Waves Appl. 20, 941-953 (2006). http://dx.doi.org/10.1163/156939306776149806.
    [CrossRef]
  16. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63-136.
  17. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach, (McGraw-Hill, Singapore, 1985) pp. 215-216.
  18. M. A. Ribeiro, S. A. Matos, and C. R. Paiva, "A geometric algebra approach to anisotropic media," in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032-4035.
  19. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, (Oxford University Press, Oxford, 1985) pp. 24-25.

2006

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
[CrossRef] [PubMed]

C. R. Paiva and M. A. Ribeiro, "Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach," J. Electromagn. Waves Appl. 20, 941-953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[CrossRef]

2004

2003

D. Hestenes, "Oersted Medal Lecture 2002: Reforming the mathematical language of physics," Am. J. Phys. 71, 104-121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[CrossRef]

1995

1954

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol.  2, p. 443.

Fainman, Y.

Hestenes, D.

D. Hestenes, "Oersted Medal Lecture 2002: Reforming the mathematical language of physics," Am. J. Phys. 71, 104-121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[CrossRef]

Levy, U.

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol.  2, p. 443.

Paiva, C. R.

C. R. Paiva and M. A. Ribeiro, "Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach," J. Electromagn. Waves Appl. 20, 941-953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[CrossRef]

Pang, L.

Pendry, J. B.

Ribeiro, M. A.

C. R. Paiva and M. A. Ribeiro, "Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach," J. Electromagn. Waves Appl. 20, 941-953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[CrossRef]

Richter, I.

Schurig, D.

Smith, D. R.

Sun, P. C.

Tsai, C. H.

Xu, F.

Am. J. Phys.

D. Hestenes, "Oersted Medal Lecture 2002: Reforming the mathematical language of physics," Am. J. Phys. 71, 104-121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[CrossRef]

Appl. Opt.

J. Electromagn. Waves Appl.

C. R. Paiva and M. A. Ribeiro, "Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach," J. Electromagn. Waves Appl. 20, 941-953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[CrossRef]

New York

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol.  2, p. 443.

Opt. Express

Opt. Lett.

Other

M. Born and E. Wolf, Principles of Optics, 7th expanded ed., (Cambridge University Press, Cambridge, 1999) pp. 790-852.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).

I. V. Lindell, Differential Forms in Electromagnetics, (IEEE Press, Piscataway, 2004) pp. 123-161.

I. V. Lindell, Methods for Electromagnetic Field Analysis, (IEEE Press, Piscataway, 2nd ed., 1995) pp. 17-52.

D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Publishers, Dordrecht, 2nd ed., 1999).

P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2nd ed., 2001).
[CrossRef]

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).

L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63-136.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach, (McGraw-Hill, Singapore, 1985) pp. 215-216.

M. A. Ribeiro, S. A. Matos, and C. R. Paiva, "A geometric algebra approach to anisotropic media," in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032-4035.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, (Oxford University Press, Oxford, 1985) pp. 24-25.

P. Puska, "Covariant isotropic constitutive relations in Clifford’s geometric algebra," Progress in Electromagnetics Research - PIER 32, 413-428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf.
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (495 KB)     
» Media 2: MOV (77 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(Color online) The electrical anisotropy of a medium is characterized by the bivector F = ED = β F̂ = ϱst, where β = ϱsin(θ), ϱ = ED (with E = |E|, D = |D|) and F̂ = sr is a unit bivector as sr = 0 (r, s and t are unit vectors). Angle θ is such that cos(θ) = st = ε 0 ε s E/D, where ε S = s ∙ ε(s) is the permittivity along s and ε(s) is the dielectric function. One has D = D + D with D = D s and D = D r , where D = Dcos(θ) and D = Dsin(θ). For an isotropic medium β = 0, i.e., θ = 0 for all possible directions S.

Fig. 2.
Fig. 2.

(Color online) In a biaxial crystal the two unit vectors (d 1,d 2) characterize the dielectric function ε, whereas the two unit vectors (c 1,c 2) characterize the impermeability function η. One should stress that (c 1,c 2) are the optic axes of the crystal – not (d 1,d 2) . Orthonormal basis {e 1,e 2,e 3} contains the three principal dielectric axes of the crystal. One has τ 1 τ 3 = ε 3 ε 1 ( γ 1 γ 3 ) . .

Fig. 3.
Fig. 3.

(496 kB) The 3D refractive index surfaces, α 0 n ± (), for the two eigenwaves of a lossless nonmagnetic crystal. [Media 1]

Fig. 4.
Fig. 4.

(77 kB) The two refractive index surfaces, α 0 n ± (), on the plane c 1c 2 : surface α 0 n - () is in red and corresponds to the ordinary wave only in the uniaxial case; surface α 0 n + () is in green and corresponds to the extraordinary wave only in the uniaxial case. [Media 2]

Fig. 5.
Fig. 5.

(Color online) Normalized energy velocities v (±) e /v 2 and phase velocities v (±) p /v 2 for the two eigenwaves of a biaxial crystal. One has v 2 = α 0 = c/ ε 2 .

Equations (34)

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u = ED = E D + E D = α + F ,
α = E D = ( u + u ˜ ) 2 , F = E D = ( u + u ˜ ) 2 ,
u 2 = u u ˜ = EDDE = E 2 D 2 = ( α + F ) ( α F ) = α 2 F 2 = α 2 + β 2 = ϱ 2 0 .
u = α + β F ̂ = ϱ exp ( θ F ̂ ) = ϱ cos ( θ ) + F ̂ ϱ sin ( θ ) , ϱ = E D = α 2 + β 2 ,
Cℓ 3 = 3 2 3 3 3 .
{ 1 schalars , e 1 , e 2 , e 3 vectors , e 12 , e 31 , e 23 bivectors , e 123 trivectors }
Cen ( Cℓ 3 ) = 3 3 .
d 1 = γ 1 e 1 + γ 3 e 3 , d 2 = γ 1 e 1 + γ 3 e 3 , γ 1 = ε 2 ε 1 ε 3 ε 1 , γ 3 = ε 3 ε 2 ε 3 ε 1 ,
d 2 = r ϕ d 1 r ˜ ϕ , r ϕ = exp ( ϕ e 31 2 ) = cos ( ϕ 2 ) + e 31 sin ( ϕ 2 ) ,
ε ( E ) = ε 2 E + [ ( ε 3 ε 1 ) 2 ] [ ( E d 1 ) d 2 + ( E d 2 ) d 1 ] .
ε ( E ) = ε E + ( ε ε ) ( E c ) c .
D = ε 0 ε ( E ) + ξ ( H ) , B = μ 0 μ ( H ) + ζ ( E )
c 1 = τ 1 e 1 + τ 3 e 3 , c 1 = τ 1 e 1 + τ 3 e 3 , τ 1 = ε 3 ε 2 γ 1 , τ 3 = ε 1 ε 2 γ 3 ,
c 2 = r δ c 1 r ˜ δ , r δ = exp ( δ e 31 2 ) = cos ( δ 2 ) + e 31 sin ( δ 2 ) .
c 1 c 2 = α γ γβ γβ γ d 1 d 2 , β = ε 3 ε 1 ε 3 + ε 1 , γ = 1 1 β 2 .
η ( D ) = η 2 D + [ ( η 3 η 1 ) 2 ] [ ( D c 1 ) c 2 + ( D c 2 ) c 1 ] .
n E = c Be 123 , n H = c De 123 .
n 2 E = ε ( E ) , E = E E = E ( E k ̂ ) k ̂ .
k ̂ w = 0 , w = n 2 η ( E ) E ,
( n 2 α 0 1 ) ( k ̂ E ) + n 2 β 0 [ ( c 1 E ) ( k ̂ c 2 ) + ( c 2 E ) ( k ̂ c 1 ) ] = 0
u = ( k ̂ c 1 ) e 123 , v = ( k ̂ c 2 ) e 123
c 1 E = n 2 β 0 u 2 1 n 2 ( α 0 + β 0 u v ) ( c 2 E ) , c 2 E = n 2 β 0 v 2 1 n 2 ( α 0 + β 0 u v ) ( c 1 E ) .
{ 1 n 4 β 0 2 u 2 v 2 [ 1 n 2 ( α 0 + β 0 u v ) ] 2 } ( c 1 E ) = 0 .
1 n ± 2 = α 0 + β 0 ( u v ± u 2 v 2 ) .
ζ = ε 2 ε 1 , κ = ε 3 ε 2 ,
b ± = k ̂ c 2 c 2 ± k ̂ c 1 c 2
D ± = ε 0 n ± 2 E 0 [ k ̂ ( k ̂ b ± ) ] , E ± = α 0 ε 0 D ± + ( 1 α 0 n ± 2 ) E 0 b ± ,
B ± = μ 0 H ± = n ± c E 0 ( k ̂ b ± ) e 123 .
k ̂ b ± = k ̂ c 2 ( k ̂ c 1 ) ± k ̂ c 1 ( k ̂ c 2 )
D ± = ε 0 α 0 E ± , E ± = E 0 [ k ̂ ( k ̂ b ± ) ] = E 0 [ b ± ( k ̂ b ± ) k ̂ ] .
D ± = ε 0 n ± 2 E ± , E ± = E 0 b ± .
cos 2 ( θ ± ) = b ± 2 ( k ̂ b ± ) 2 b ± 2 α 0 n ± 2 ( 2 α 0 n ± 2 ) ( k ̂ b ± ) 2 .
v e (±) = [ b ± 2 α 0 n ± 2 ( k ̂ b ± ) 2 ] v p (±) ( k ̂ b ± ) ( 1 α 0 n ± 2 ) v p (±) b ± b ± 2 ( k ̂ b ± ) 2
k ̂ v e ( ± ) = v p ( ± ) .

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