Abstract

We obtain a formula for the change of the reflectivity from a symmetric metallic grating due to the excitation of surface polaritons. We find that reflectivity measurements at near-normal incidence can show a k gap in the dispersion curves of the surface polaritons even when direct coupling between two polaritons at the zone boundary is present. We discuss the limit in which this can occur and examine the reflectivity surface as the direct coupling is varied. The formula permits a simple discussion of the k gaps at 2nπ/a for any n.

© 1988 Optical Society of America

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References

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  1. F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
    [CrossRef]
  2. N. E. Glass, M. G. Weber, D. L. Mills, Phys. Rev. B 29, 6548 (1984).
    [CrossRef]
  3. V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, Phys. Rev. B (to be published).
  4. H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
    [CrossRef]
  5. D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
    [CrossRef]
  6. G. C. Brown, V. Celli, M. Haller, Surf. Sci. 136, 381 (1984).
    [CrossRef]
  7. G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
    [CrossRef]
  8. P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  9. M. G. Weber, D. L. Mills, Phys. Rev. B 34, 2893 (1986).
    [CrossRef]

1987

H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
[CrossRef]

D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

1986

M. G. Weber, D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

1985

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

1984

G. C. Brown, V. Celli, M. Haller, Surf. Sci. 136, 381 (1984).
[CrossRef]

N. E. Glass, M. G. Weber, D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

1977

F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

1972

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Brown, G. C.

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

G. C. Brown, V. Celli, M. Haller, Surf. Sci. 136, 381 (1984).
[CrossRef]

Celli, V.

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

G. C. Brown, V. Celli, M. Haller, Surf. Sci. 136, 381 (1984).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, Phys. Rev. B (to be published).

Christy, R. W.

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Ellis, T. H.

H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
[CrossRef]

Glass, N. E.

N. E. Glass, M. G. Weber, D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

Haller, M.

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

G. C. Brown, V. Celli, M. Haller, Surf. Sci. 136, 381 (1984).
[CrossRef]

Heitmann, D.

D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Jónsson, H.

H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
[CrossRef]

Kroo, N.

D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

Maradudin, A. A.

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, Phys. Rev. B (to be published).

Marvin, A.

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

Mills, D. L.

M. G. Weber, D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

N. E. Glass, M. G. Weber, D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, Phys. Rev. B (to be published).

Schulz, Ch.

D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

Scoles, G.

H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
[CrossRef]

Szentirmay, Zs.

D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

Tran, P.

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, Phys. Rev. B (to be published).

Weare, J. H.

H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
[CrossRef]

Weber, M. G.

M. G. Weber, D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

N. E. Glass, M. G. Weber, D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

Phys. Rev. B

F. Toigo, A. Marvin, V. Celli, N. R. Hill, Phys. Rev. B 15, 5618 (1977); B. Laks, D. L. Mills, A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

N. E. Glass, M. G. Weber, D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

D. Heitmann, N. Kroo, Ch. Schulz, Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, Phys. Rev. B 31, 4993 (1985).
[CrossRef]

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

M. G. Weber, D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

Surf. Sci.

H. Jónsson, J. H. Weare, T. H. Ellis, G. Scoles, Surf. Sci. 180, 353 (1987).
[CrossRef]

G. C. Brown, V. Celli, M. Haller, Surf. Sci. 136, 381 (1984).
[CrossRef]

Other

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, Phys. Rev. B (to be published).

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Figures (4)

Fig. 1
Fig. 1

Plots, in units of γ, of the reflectivity minima for several values of z/γ at w/γ = 4.

Fig. 2
Fig. 2

Plots of the reflectivity surface R(x, y) from Eq. (6) for parameters corresponding to Fig. 1: (a) z/γ = 0.0, (b) z/γ = 3.0, (c) z/γ = 6.87, (d) z/γ = 10.0.

Fig. 3
Fig. 3

Symmetric grating with period a; h is fixed and δ is varied to change the direct coupling between the ±1 SPO branches; δ = 0 corresponds to a symmetric sawtooth, and the first-order coupling vanishes.

Fig. 4
Fig. 4

Reflectivity calculation using the reduced Rayleigh equation from Ref. 6 with nine channels. Curves 1, 2, 3, and 4 correspond to an unperturbed SPO at 1, 0.999, 0.998, and 0.997 (2π/a), respectively. (a) δ = 0.1, (b) δ = 0.2.

Equations (7)

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T ( k , k o ) = V ( k , k o ) + k V ( k , k ) G ( k ) T ( k , k o ) .
V ( k , k o ) = ζ ˆ ( k k o ) 1 ( k k o β β o / ) .
V ( k , k o ) = V ( k , k o ) + k V ( k , k ) G ( k ) V ( k , k o ) ,
T ( k , k o ) = V ( k , k o ) + k V ( k , k ) G ( k ) T ( k , k o ) ,
A o R o R o = G a 1 G b 1 V a b 2 G o A V a o 2 ( G a 1 + G b 1 + 2 V a b ) G a 1 G b 1 V a b 2 G o V a o 2 ( G a 1 + G b 1 + 2 V a b ) ,
| A 0 | 2 | R o | 2 = ( y 2 x 2 z 2 γ 2 + 2 w γ ) 2 + 4 [ y γ w ( y + z ) ] 2 ( y 2 x 2 z 2 γ 2 2 w γ ) 2 + 4 [ y γ + w ( y + z ) ] 2 .
x 2 = [ ( y + z ) 2 + γ 2 ] + 2 { [ ( y + z ) 2 + γ 2 ] ( w 2 + y 2 ) } 1 / 2 .

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