Abstract

We report on the dispersion of surface waves in a bounded gyrotropic medium. Numerical results are obtained for quartz and sodium chlorate. An attenuated-total-reflection (ATR) experiment for surface waves is formulated for a vacuum–medium interface. We also compute the ATR spectra for quartz and sodium chlorate.

© 1982 Optical Society of America

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References

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  1. D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magneto-electric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
    [Crossref]
  2. D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
    [Crossref]
  3. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1966), paragraph 83.
  4. S. Chandrasekhar, “Optical rotarory dispersion of crystals,” Proc. R. Soc. London 259, 531–553 (1981); V. M. Agranovich and G. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Interscience, New York, 1966), paragraph 6.
  5. M. F. Bishop and A. A. Maradudin, “Linear wave vector effects on the optical properties of semi-infinite crystals,” Solid State Commun. 23, 507–510 (1977).
    [Crossref]
  6. H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523–534 (1973).
    [Crossref]

1981 (2)

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magneto-electric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[Crossref]

S. Chandrasekhar, “Optical rotarory dispersion of crystals,” Proc. R. Soc. London 259, 531–553 (1981); V. M. Agranovich and G. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Interscience, New York, 1966), paragraph 6.

1977 (1)

M. F. Bishop and A. A. Maradudin, “Linear wave vector effects on the optical properties of semi-infinite crystals,” Solid State Commun. 23, 507–510 (1977).
[Crossref]

1974 (1)

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

1973 (1)

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523–534 (1973).
[Crossref]

Birman, J. L.

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magneto-electric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[Crossref]

Bishop, M. F.

M. F. Bishop and A. A. Maradudin, “Linear wave vector effects on the optical properties of semi-infinite crystals,” Solid State Commun. 23, 507–510 (1977).
[Crossref]

Burstein, E.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, “Optical rotarory dispersion of crystals,” Proc. R. Soc. London 259, 531–553 (1981); V. M. Agranovich and G. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Interscience, New York, 1966), paragraph 6.

Falge, H. J.

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523–534 (1973).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1966), paragraph 83.

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1966), paragraph 83.

Maradudin, A. A.

M. F. Bishop and A. A. Maradudin, “Linear wave vector effects on the optical properties of semi-infinite crystals,” Solid State Commun. 23, 507–510 (1977).
[Crossref]

Mills, D. L.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Otto, A.

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523–534 (1973).
[Crossref]

Pattanayak, D. N.

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magneto-electric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[Crossref]

Phys. Rev. B (1)

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magneto-electric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[Crossref]

Phys. Status Solidi B (1)

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523–534 (1973).
[Crossref]

Proc. R. Soc. London (1)

S. Chandrasekhar, “Optical rotarory dispersion of crystals,” Proc. R. Soc. London 259, 531–553 (1981); V. M. Agranovich and G. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Interscience, New York, 1966), paragraph 6.

Rep. Prog. Phys. (1)

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Solid State Commun. (1)

M. F. Bishop and A. A. Maradudin, “Linear wave vector effects on the optical properties of semi-infinite crystals,” Solid State Commun. 23, 507–510 (1977).
[Crossref]

Other (1)

L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1966), paragraph 83.

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Figures (6)

Fig. 1
Fig. 1

Real part of refractive index n(ω) versus reduced frequency (ωω0)/ω0 near resonance. Parameters used are appropriate to quartz: b = 2.35, 4πα0 = 1.57, k2 = 0.399 × π × 10−8 cm, λ0 = 926 Å. [See text for definitions, Eqs. (15)(17)].

Fig. 2
Fig. 2

Reduced frequency (ωω0)/ω0 versus surface-wave dispersion u in stopgap region. Parameters for quartz as in Fig. 1.

Fig. 3
Fig. 3

Reduced frequency (ωω0)/ω0 versus surface-wave dispersion u in stopgap region. Parameters used are appropriate to sodium chlorate: b = 2.8866, 4πα0 = 1.18, k2 = 0.06 × π × 10−8 cm, λ0 = 900 Å. (See text for definition of parameters).

Fig. 4
Fig. 4

Geometry of attenuated-total-reflection experiment.

Fig. 5
Fig. 5

Reduced frequency (ωω0)/ω0 versus attenuated-total-reflection coefficient R(ω) for various angles of incidence θ. Parameters for quartz as in Fig. 1.

Fig. 6
Fig. 6

Reduced frequency (ωω0)/ω0 versus attenuated-total-reflection coefficient R(ω) for various angles of incidence θ. Parameters for sodium chlorate (NaClO3) as in Fig. 3.

Equations (38)

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n + ( w + + w 0 ) ( w - + n - 2 w 0 ) + n - ( w - + w 0 ) ( w + + n + 2 w 0 ) = 0.
w ± ( k 0 2 n ± 2 - u 2 ) 1 / 2 ,             Im w ± > 0 ,
w 0 ( k 0 2 - u 2 ) 1 / 2 ,             Im w 0 > 0 ,
n ± = ± μ ( ω ) 2 + [ ( ω ) + μ 2 ( ω ) ] 1 / 2 ,
i j ( ω , k ) = 0 ( ω ) δ i j + i 1 ( ω ) e i j l k l .
( ω ) = 1 + 4 π χ e e ( ω ) , μ ( ω ) = χ e h ( ω ) ,
χ e e = 0 ( ω ) - 1 4 π ,             χ e h = k 0 1 ( ω ) 4 π .
u 2 > k 0 2 ,
u 2 > Re ( k 0 2 n ± 2 ) .
u 4 - ¼ [ ( n + + n - ) 2 ( 1 + n + n - ) + ( 1 + n + n - ) ] k 0 2 u 2 + ¼ n + n - k 0 4 = 0.
μ ( ω ) = ( n + - n - ) .
ϕ ( λ ) = k 2 λ 2 ( λ 2 - λ 0 2 ) 2 ,
μ ( ω ) = 2 c ω ϕ ( ω ) .
( ω ) = b + 4 π α 0 ω 0 2 ( ω 0 2 - ω 2 - i ω Γ ) ,
E total p ( r , ω ) = Ê inc ( r , ω ) + Ê refl ( r , ω ) ,
E inc ( r , ω ) = ( x ^ E x i + z ^ E z i ) e i w p z e i u x d u ,
E refl ( r , ω ) = ( x ^ E R x + y ^ E R y + z ^ E R z ) e - i w p z e i u x d u ,
w p = ( p k 0 2 - u 2 ) 1 / 2 ,
H total p ( r , ω ) = { - x ^ ( E z i e i w p z - E R z e - i w p z ) + y ^ [ ( E x i e i w p z - E R x e - i w p z ) - u ω p ( E z i e i w p z + E R z e - i w p z ) ] + z ^ u ω p E R y e - i w p z } d u .
E total air ( r , ω ) = { Ê ( V ) exp [ i ( u x - w 0 z ) ] + E ¯ ( V ) exp [ - i ( u x + w 0 z ) ] } d u ,
w 0 = ( k 0 2 - u 2 ) 1 / 2 ,
H ( V ) ( r , ω ) = { [ x ^ ( Ê y ( V ) e - i w 0 z - E ¯ y ( V ) e i w 0 z ) - y ^ ( Ê x ( V ) e - i w 0 z - E ¯ ^ x ( V ) e i w 0 z ) ] w 0 k 0 + z ^ ( Ê x ( V ) e - i w 0 z - E ¯ ^ x ( V ) e i w 0 z ) × ( u 2 + w 0 2 ) k 0 } e i u x d u .
η 1 · Ê ( V ) = 0 ,
η 2 · Ê ( V ) = 0 ,
η 1 ( u , 0 , - w 0 )
η 2 ( u , 0 , w 0 ) .
Ê ( r , ω ) = { Ê + exp [ i ( u x + w + z ) ] + Ê - exp [ i ( u x - w - z ) ] } d u ,
Ê + = [ x ^ + i ( k 0 n + w + + i u v ) ( k 0 2 n + 2 - u 2 ) y ^ + u w + + i k 0 n + v u 2 - k 0 2 n + 2 z ^ ] Ê + x ( u , v ) ,
Ê - ( u , v ) = [ x ^ - i ( k 0 n - w - - i u v ) ( k 0 2 n - 2 - u 2 ) y ^ + ( u w - - i k 0 n - v ) k 0 2 n - 2 - u 2 z ^ ] Ê - x ( u , v ) ,
H = - i k 0 × E .
z ^ × E ( - Δ ) p = z ^ × E ( - Δ ) V , ( z ^ × H ) ( - Δ ) p = ( z ^ × H ) ( - Δ ) V , z ^ × E 0 V - = z ^ × E 0 gyro + , z ^ × H 0 V - = z ^ × H 0 gyro + ,
E R x , E R y , Ê x V , Ê y V , E ¯ ^ x V , E ¯ ^ y V , Ê + x , Ê - x .
[ e i w p Δ 0 - e i w 0 Δ 0 - e - i w 0 Δ 0 0 0 0 - e + i w p Δ 0 e i w 0 Δ 0 e - i w 0 Δ 0 0 - u w p e i w p Δ 0 0 - w 0 w p e i w 0 Δ 0 w 0 w p e - i w 0 Δ 0 0 - ( u 2 + w p 2 ) w p 2 e i w p Δ 0 - w 0 w p e i w 0 Δ 0 w 0 w p e - i w 0 Δ 0 0 0 0 0 1 0 0 0 - 1 - 1 0 0 0 1 0 0 - a - c 0 0 0 w 0 k 0 0 - w 0 k 0 - α + - α - 0 0 w 0 k 0 0 - w 0 k 0 0 α + a α - c ]     [ E R x E R y Ê x V Ê y V E ¯ ^ x V E ¯ ^ y V Ê + x Ê - x ] = [ - E x i e - i w p Δ 0 - E z i e - i w p Δ - u w p e - i w p Δ E z i - E x i e - i w p Δ 0 0 0 0 ]
a = - i k 0 n + w + k 0 2 n + 2 - u 2 ,
b = i k 0 n - w - k 0 2 n - 2 - u 2 ,
u = p ω c sin θ ,
α + = i n + ,             α - = i n - .
R ( ω ) = ( w p 2 + u 2 w p 2 ) E R x 2 + E R y 2 .