Abstract

A relatively simple derivation of the paraxial equation for extraordinary electromagnetic waves in uniaxial media is presented. The propagation of the waves is described with both Abraham’s and Minskowski’s definitions of the electromagnetic momentum. A Gaussian beam is worked out to illustrate the sidewalk effect of extraordinary rays.

© 2010 Optical Society of America

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References

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  1. J. A. Fleck, Jr., and M. D. Felt, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920-926 (1983).
    [CrossRef]
  2. D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystal. I: Narrowband pump field,” J. Mod. Opt. 42, 1037-1067 (1995).
    [CrossRef]
  3. E. Cojocaru, “Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,” Appl. Opt. 36, 302-306 (1997).
    [CrossRef] [PubMed]
  4. M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial theory for birefringent lenses,” J. Opt. Soc. Am. A 22, 881-890 (2005).
    [CrossRef]
  5. A. Nisbet, “Electromagnetic potential in a hetereogeneous non-conducting medium,” Proc. R. Soc. London, Ser. A 240, 375-381 (1957).
    [CrossRef]
  6. S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A, Pure Appl. Opt. 11, 085204 (2009).
    [CrossRef]
  7. M. G. Burt and R. Peierl, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London, Ser. A 333, 149-156 (1973).
    [CrossRef]
  8. R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005).
    [CrossRef]

2009

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A, Pure Appl. Opt. 11, 085204 (2009).
[CrossRef]

2005

R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005).
[CrossRef]

M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial theory for birefringent lenses,” J. Opt. Soc. Am. A 22, 881-890 (2005).
[CrossRef]

1997

1995

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystal. I: Narrowband pump field,” J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

1983

1973

M. G. Burt and R. Peierl, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London, Ser. A 333, 149-156 (1973).
[CrossRef]

1957

A. Nisbet, “Electromagnetic potential in a hetereogeneous non-conducting medium,” Proc. R. Soc. London, Ser. A 240, 375-381 (1957).
[CrossRef]

Auerbach, J. M.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystal. I: Narrowband pump field,” J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

Avendaño-Alejo, M.

Barnett, S. M.

R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Baxter, C.

R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Burt, M. G.

M. G. Burt and R. Peierl, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London, Ser. A 333, 149-156 (1973).
[CrossRef]

Cojocaru, E.

Eimerl, D.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystal. I: Narrowband pump field,” J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

Felt, M. D.

Fleck, J. A.

Hacyan, S.

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A, Pure Appl. Opt. 11, 085204 (2009).
[CrossRef]

Jáuregui, R.

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A, Pure Appl. Opt. 11, 085204 (2009).
[CrossRef]

Loudon, R.

R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Milonni, P. W.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystal. I: Narrowband pump field,” J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

Nisbet, A.

A. Nisbet, “Electromagnetic potential in a hetereogeneous non-conducting medium,” Proc. R. Soc. London, Ser. A 240, 375-381 (1957).
[CrossRef]

Peierl, R.

M. G. Burt and R. Peierl, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London, Ser. A 333, 149-156 (1973).
[CrossRef]

Rosete-Aguilar, M.

Appl. Opt.

J. Mod. Opt.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystal. I: Narrowband pump field,” J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A, Pure Appl. Opt. 11, 085204 (2009).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev. A

R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A

A. Nisbet, “Electromagnetic potential in a hetereogeneous non-conducting medium,” Proc. R. Soc. London, Ser. A 240, 375-381 (1957).
[CrossRef]

M. G. Burt and R. Peierl, “The momentum of a light wave in a refracting medium,” Proc. R. Soc. London, Ser. A 333, 149-156 (1973).
[CrossRef]

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Equations (22)

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ϵ ̂ = ϵ 1 + Δ ϵ a a ,
B = i ω a × ψ ,
E = 1 ϵ ( a ψ ) ω 2 c 2 ψ a ,
D = ( a ψ ) + 2 ψ a ,
ϵ 2 ψ + Δ ϵ ( a ) 2 ψ + ϵ ϵ ω 2 c 2 ψ = 0 .
ψ ( r ) = A ( r ) e i k r ,
k ( θ ) = n ( θ ) ω c ,
1 n 2 ( θ ) = sin 2 θ ϵ + cos 2 θ ϵ ,
ϵ 2 A + Δ ϵ ( a ) 2 A + 2 i k ϵ ̂ A = 0 .
k 1 ϵ ϵ ̂ k = k + Δ ϵ ϵ ( k a ) a .
a = ( 0 , sin θ , cos θ ) ,
[ ϵ 2 x 2 + ( ϵ + Δ ϵ sin 2 θ ) 2 y 2 + 2 i ϵ | k | z ] A = 0 .
k k = ϵ ω 2 c 2 , k a = ϵ ϵ k ( θ ) cos θ ,
n ( θ ) = n ( θ ) ( sin 2 θ + ϵ 2 ϵ 2 cos 2 θ ) 1 2 .
tan θ = ϵ ϵ tan θ .
[ ϵ 2 x 2 + ( ϵ n ( θ ) ) 2 2 y 2 + 2 i ϵ ω c n ( θ ) z ] A = 0 ,
E × H = ω c ϵ k 2 ( θ ) sin 2 θ ψ 2 k
D × B = ω c k 2 ( θ ) sin 2 θ ψ 2 k .
( sin 2 θ ϵ 2 + cos 2 θ ϵ 2 ) 2 ,
A ( r ) = A 0 1 u ( z ) v ( z ) exp { 1 2 ( x 2 u 2 ( z ) + y 2 v 2 ( z ) ) } ,
u 2 ( z ) = u 0 2 + i c n ω z ,
v 2 ( z ) = v 0 2 + i c ϵ 2 n 3 ϵ ω z ,

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