Abstract

The kDB system, consisting of the propagation vector k and the plane perpendicular to k and containing the field vectors D and B, is developed to treat electromagnetic waves in anisotropic and bianisotropic media. In using the kDB system, we first transform from the original coordinate systems to the kDB, determine the characteristics of the normal modes, and then transform the results back to the original coordinates.

© 1974 Optical Society of America

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References

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  1. A. Sommerfeld, Optics (Academic, New York, 1964), Ch. 4.
  2. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Ch. 14.
  3. J. A. Kong, J. Opt. Soc. Am. 61, 49 (1971).
    [Crossref]
  4. W. P. Mason, Crystal Physics and Interaction Processes (Academic, New York, 1966), Ch. 7.
  5. J. A. Kong, Proc. IEEE 60, 1036 (1972).
    [Crossref]
  6. H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, Mass., 1965), p. 109.
  7. I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 37, 881 (1959) [Sov. Phys.—JETP 10, 628 (1960)].
  8. B. D. H. Tellegen, Philips Res. Rept. 3, 81 (1948).
  9. C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw–Hill, New York, 1965), Ch. 7.

1972 (1)

J. A. Kong, Proc. IEEE 60, 1036 (1972).
[Crossref]

1971 (1)

1959 (1)

I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 37, 881 (1959) [Sov. Phys.—JETP 10, 628 (1960)].

1948 (1)

B. D. H. Tellegen, Philips Res. Rept. 3, 81 (1948).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Ch. 14.

Dzyaloshinskii, I. E.

I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 37, 881 (1959) [Sov. Phys.—JETP 10, 628 (1960)].

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, Mass., 1965), p. 109.

Kong, J. A.

Mason, W. P.

W. P. Mason, Crystal Physics and Interaction Processes (Academic, New York, 1966), Ch. 7.

Papas, C. H.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw–Hill, New York, 1965), Ch. 7.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1964), Ch. 4.

Tellegen, B. D. H.

B. D. H. Tellegen, Philips Res. Rept. 3, 81 (1948).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Ch. 14.

J. Opt. Soc. Am. (1)

Philips Res. Rept. (1)

B. D. H. Tellegen, Philips Res. Rept. 3, 81 (1948).

Proc. IEEE (1)

J. A. Kong, Proc. IEEE 60, 1036 (1972).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 37, 881 (1959) [Sov. Phys.—JETP 10, 628 (1960)].

Other (5)

H. Goldstein, Classical Mechanics (Addison–Wesley, Reading, Mass., 1965), p. 109.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw–Hill, New York, 1965), Ch. 7.

W. P. Mason, Crystal Physics and Interaction Processes (Academic, New York, 1966), Ch. 7.

A. Sommerfeld, Optics (Academic, New York, 1964), Ch. 4.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Ch. 14.

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Figures (2)

Fig. 1
Fig. 1

kDB system.

Fig. 2
Fig. 2

(a) k surfaces for a moving isotropic medium with a = b = 1 and n = 2 in its rest frame; (b) k surfaces for a moving uniaxial medium with a = b = 2 and n = 2 in its rest frame.

Tables (1)

Tables Icon

Table I Field components for the two types of characteristic waves.

Equations (54)

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E = κ ¯ · D + χ ¯ · B ,
H = ν ¯ · B + γ ¯ · D .
k × E = ω B ,
k × H = - ω D ,
k × B = 0 ,
k × D = 0.
A k = T ¯ · A ,
T ¯ = [ t 1 x t 1 y t 1 z t 2 x t 2 y t 2 z t 3 x t 3 y t 3 z ] = [ sin ϕ - cos ϕ 0 cos θ cos ϕ cos θ sin ϕ - sin θ sin θ cos ϕ sin θ sin ϕ cos θ ]
κ ¯ k = T ¯ · κ ¯ · T ¯ t ,
ν ¯ k = T ¯ · ν ¯ · T ¯ t ,
χ ¯ k = T ¯ · χ ¯ · T ¯ t ,
γ ¯ k = T ¯ · γ ¯ · T ¯ t .
[ u + χ 21 χ 22 - χ 11 u - χ 12 ] [ B 1 B 2 ] = [ - κ 21 - κ 22 κ 11 κ 12 ] [ D 1 D 2 ] ,
[ u - γ 21 - γ 22 γ 11 u + γ 12 ] [ D 1 D 2 ] = [ ν 21 ν 21 - ν 11 - ν 12 ] [ B 1 B 2 ] ,
u = ω / k
E = [ κ κ κ z ] · D + [ χ χ χ z ] · B ,
H = [ ν ν ν z ] · B + [ γ γ γ z ] · D ,
κ ¯ k = [ κ 0 0 0 κ cos 2 θ + κ z sin 2 θ ( κ - κ z ) sin θ cos θ 0 ( κ - κ z ) sin θ cos θ κ sin 2 θ + κ z cos 2 θ ] .
[ κ θ ( u 2 ν + ν θ χ γ - κ ν ν θ ) u κ θ ( ν θ χ - ν γ θ ) u κ ( ν θ γ - ν χ θ ) κ ( u 2 ν θ + ν χ θ γ θ - κ θ ν ν θ ) ] [ D 1 D 2 ] = 0 ,
κ θ = κ cos 2 θ + κ z sin 2 θ ,
ν θ = ν cos 2 θ + ν z sin 2 θ ,
χ θ = χ cos 2 θ + χ z sin 2 θ ,
γ θ = γ cos 2 θ + γ z sin 2 θ .
E = κ · D
H = ν · B ,
κ = [ κ κ κ z ]
ν = [ ν ν ν z ] .
E = γ ( E + v × B ) + ( 1 - γ ) E · v v 2 v ,
D = γ ( D + 1 c 2 v × H ) + ( 1 - γ ) D · v v 2 v ,
H = γ ( H - v × D ) + ( 1 - γ ) H · v v 2 v ,
B = γ ( B - 1 c 2 v × E ) + ( 1 - γ ) B · v v 2 v ,
κ ¯ = [ κ κ κ z ] ,
ν ¯ = [ ν ν ν z ] ,
χ ¯ = γ ¯ + = [ 0 χ 0 - χ 0 0 0 0 0 ] ,
κ = c 2 ( 1 - β 2 ) / ν ( n 2 - β 2 ) ,
κ z = κ z ,
ν = ν n 2 ( 1 - β 2 ) / ( n 2 - β 2 ) ,
ν z = ν z ,
χ = c β ( n 2 - 1 ) / ( n 2 - β 2 ) ,
n 2 = c 2 / κ ν .
κ ¯ k = [ κ 0 0 0 κ cos 2 θ + κ z sin 2 θ ( κ - κ z ) sin θ cos θ 0 ( κ - κ z ) sin θ cos θ κ sin 2 θ + κ z cos 2 θ ] ,
ν ¯ k = [ ν 0 0 0 ν cos 2 θ + ν z sin 2 θ ( ν - ν z ) sin θ cos θ 0 ( ν - ν z ) sin θ cos θ ν sin 2 θ + ν z cos 2 θ ] ,
χ ¯ k = γ ¯ k + = [ 0 χ cos θ χ sin θ - χ cos θ 0 0 - χ sin θ 0 0 ] .
( 1 - ( u - χ cos θ ) 2 / κ ( ν cos 2 θ + ν z sin 2 θ )             0 0             1 - ( u - χ cos θ ) 2 / ν ( κ cos 2 θ + κ z sin 2 θ ) ) ( D 1 D 2 ) = 0.
u = χ cos θ ± [ κ ( ν cos 2 θ + ν z sin 2 θ ) ] .
k x 2 + k y 2 + b 1 - n 2 β 2 1 - β 2 ( k z + β n 2 - 1 1 - n 2 β 2 ω c ) 2 - b n 2 ( 1 - β 2 ) 1 - n 2 β 2 ω 2 c 2 = 0 ,
u = χ cos θ ± [ ν ( κ cos 2 θ + κ z sin 2 θ ) ]
k x 2 + k y 2 + a 1 - n 2 β 2 1 - β 2 ( k z + β n 2 - 1 1 - n 2 β 2 ω c ) 2 - a n 2 ( 1 - β 2 ) 1 - n 2 β 2 ω 2 c 2 = 0 ,
k x 2 + k y 2 + b 1 - n 2 β 2 1 - β 2 ( k z - n + β n β + 1 ω c ) × ( k z - n - β n β - 1 ω c ) = 0 ,
k x 2 + k y 2 + a 1 - n 2 β 2 1 - β 2 ( k z - n + β n β + 1 ω c ) × ( k z - n - β n β - 1 ω c ) = 0.
k = x ˆ k x = ± x ˆ b ω c ( n 2 - β 2 1 - β 2 ) 1 2 ,
k = x ˆ k x = ± x ˆ a ω c ( n 2 - β 2 1 - β 2 ) 1 2 .
k = z ˆ n + β n β + 1 ω c ,
k = z ˆ n - β n β - 1 ω c .