Abstract

A new Green-function formalism is developed for calculating fields generated by sources in the presence of a multilayer geometry. The approach is to formulate the problem immediately in terms of s- and p-polarized waves generated, so that the calculation of the effect of interfaces proceeds by the introduction of Fresnel reflection and transmission coefficients and parallels the simple physical picture of light progressing through the structure.

© 1987 Optical Society of America

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References

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  1. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  2. Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984).
  3. J. E. Sipe, Surf. Sci 105, 489 (1981). I take this opportunity to correct some typographical errors in that paper. Equation (8) should read asF↔(k)=2πω˜2w0-1[s^s^+p^0+p^0+kz-w0-iη-s^s^+p^0-p^0-kz+w0+iη]-4πz^z^;Eq. (11) should read asF↔(κ;Z)=2πiω˜2w0-1(s^s^+p^0+p^0+)θ(Z)exp(iw0Z)+2πiω˜2w0-1(s^s^+p^0-p^0-)θ(-Z)exp(-i-w0Z)-4πz^z^δ(Z); and Eq. (16) should read as g(κ;Z)=2πiw0-1[θ(Z)exp(iw0Z)+θ(-Z)exp(-iw0Z)].
    [Crossref]
  4. G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
    [Crossref]
  5. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
    [Crossref]
  6. J. E. Sipe and G. I. Stegeman, in Surface Polaritons, D. L. Mills and V. M. Agranovich, eds. (North-Holland, Amsterdam, 1982), pp. 661–701.
  7. M. Cardona, Am. J. Phys. 39, 1277 (1971).
    [Crossref]
  8. H. Morawitz and M. R. Philpott, Phys. Rev. B 12, 4869 (1974).
  9. R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2184 (1974).
    [Crossref]
  10. J. E. Sipe, Phys. Rev. B 22, 1589 (1980).
    [Crossref]
  11. J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
    [Crossref]
  12. J. A. Litwin, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 31, 5543 (1985).
    [Crossref]
  13. G. W. Ford and W. H. Weber, Surf. Sci. 109, 451 (1981).
    [Crossref]
  14. W. H. Weber and G. W. Ford, Phys. Rev. Lett. 44, 1774 (1980).
    [Crossref]
  15. W. Lukosz, Phys. Rev. B 22, 3030 (1980).
    [Crossref]
  16. W. Lukosz, J. Opt. Soc. Am. 71, 744 (1981).
    [Crossref]
  17. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).
  18. Z. Knittl, Optics of Thin Films: An Optical Multilayer Theory (Wiley, New York, 1976).
  19. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
    [Crossref]
  20. H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983).
    [Crossref]
  21. P. Feibelman, Prog. Surf. Sci. 12, No. 4 (1982).
    [Crossref]
  22. T. F. Heinz, Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1982) (unpublished).
  23. J. E. Sipe, H. M. van Driel, and J. F. Young, Can. J. Phys. 63, 104 (1985).
    [Crossref]

1985 (2)

J. A. Litwin, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 31, 5543 (1985).
[Crossref]

J. E. Sipe, H. M. van Driel, and J. F. Young, Can. J. Phys. 63, 104 (1985).
[Crossref]

1984 (1)

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[Crossref]

1983 (2)

J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
[Crossref]

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983).
[Crossref]

1982 (1)

P. Feibelman, Prog. Surf. Sci. 12, No. 4 (1982).
[Crossref]

1981 (3)

J. E. Sipe, Surf. Sci 105, 489 (1981). I take this opportunity to correct some typographical errors in that paper. Equation (8) should read asF↔(k)=2πω˜2w0-1[s^s^+p^0+p^0+kz-w0-iη-s^s^+p^0-p^0-kz+w0+iη]-4πz^z^;Eq. (11) should read asF↔(κ;Z)=2πiω˜2w0-1(s^s^+p^0+p^0+)θ(Z)exp(iw0Z)+2πiω˜2w0-1(s^s^+p^0-p^0-)θ(-Z)exp(-i-w0Z)-4πz^z^δ(Z); and Eq. (16) should read as g(κ;Z)=2πiw0-1[θ(Z)exp(iw0Z)+θ(-Z)exp(-iw0Z)].
[Crossref]

W. Lukosz, J. Opt. Soc. Am. 71, 744 (1981).
[Crossref]

G. W. Ford and W. H. Weber, Surf. Sci. 109, 451 (1981).
[Crossref]

1980 (4)

W. H. Weber and G. W. Ford, Phys. Rev. Lett. 44, 1774 (1980).
[Crossref]

W. Lukosz, Phys. Rev. B 22, 3030 (1980).
[Crossref]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
[Crossref]

J. E. Sipe, Phys. Rev. B 22, 1589 (1980).
[Crossref]

1974 (2)

H. Morawitz and M. R. Philpott, Phys. Rev. B 12, 4869 (1974).

R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2184 (1974).
[Crossref]

1971 (1)

M. Cardona, Am. J. Phys. 39, 1277 (1971).
[Crossref]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
[Crossref]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
[Crossref]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).

Cardona, M.

M. Cardona, Am. J. Phys. 39, 1277 (1971).
[Crossref]

Chance, R.

R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2184 (1974).
[Crossref]

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
[Crossref]

Feibelman, P.

P. Feibelman, Prog. Surf. Sci. 12, No. 4 (1982).
[Crossref]

Ford, G. W.

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[Crossref]

G. W. Ford and W. H. Weber, Surf. Sci. 109, 451 (1981).
[Crossref]

W. H. Weber and G. W. Ford, Phys. Rev. Lett. 44, 1774 (1980).
[Crossref]

Fukui, M.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
[Crossref]

Heinz, T. F.

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983).
[Crossref]

T. F. Heinz, Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1982) (unpublished).

Jha, S. S.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
[Crossref]

Knittl, Z.

Z. Knittl, Optics of Thin Films: An Optical Multilayer Theory (Wiley, New York, 1976).

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
[Crossref]

Litwin, J. A.

J. A. Litwin, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 31, 5543 (1985).
[Crossref]

Lukosz, W.

W. Lukosz, J. Opt. Soc. Am. 71, 744 (1981).
[Crossref]

W. Lukosz, Phys. Rev. B 22, 3030 (1980).
[Crossref]

Morawitz, H.

H. Morawitz and M. R. Philpott, Phys. Rev. B 12, 4869 (1974).

Philpott, M. R.

H. Morawitz and M. R. Philpott, Phys. Rev. B 12, 4869 (1974).

Preston, J. S.

J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
[Crossref]

Prock, A.

R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2184 (1974).
[Crossref]

Shen, Y. R.

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983).
[Crossref]

Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984).

Silbey, R.

R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2184 (1974).
[Crossref]

Sipe, J. E.

J. A. Litwin, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 31, 5543 (1985).
[Crossref]

J. E. Sipe, H. M. van Driel, and J. F. Young, Can. J. Phys. 63, 104 (1985).
[Crossref]

J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
[Crossref]

J. E. Sipe, Surf. Sci 105, 489 (1981). I take this opportunity to correct some typographical errors in that paper. Equation (8) should read asF↔(k)=2πω˜2w0-1[s^s^+p^0+p^0+kz-w0-iη-s^s^+p^0-p^0-kz+w0+iη]-4πz^z^;Eq. (11) should read asF↔(κ;Z)=2πiω˜2w0-1(s^s^+p^0+p^0+)θ(Z)exp(iw0Z)+2πiω˜2w0-1(s^s^+p^0-p^0-)θ(-Z)exp(-i-w0Z)-4πz^z^δ(Z); and Eq. (16) should read as g(κ;Z)=2πiw0-1[θ(Z)exp(iw0Z)+θ(-Z)exp(-iw0Z)].
[Crossref]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
[Crossref]

J. E. Sipe, Phys. Rev. B 22, 1589 (1980).
[Crossref]

J. E. Sipe and G. I. Stegeman, in Surface Polaritons, D. L. Mills and V. M. Agranovich, eds. (North-Holland, Amsterdam, 1982), pp. 661–701.

So, V. C. Y.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
[Crossref]

Stegeman, G. I.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
[Crossref]

J. E. Sipe and G. I. Stegeman, in Surface Polaritons, D. L. Mills and V. M. Agranovich, eds. (North-Holland, Amsterdam, 1982), pp. 661–701.

Tom, H. W. K.

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983).
[Crossref]

van Driel, H. M.

J. A. Litwin, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 31, 5543 (1985).
[Crossref]

J. E. Sipe, H. M. van Driel, and J. F. Young, Can. J. Phys. 63, 104 (1985).
[Crossref]

J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
[Crossref]

Weber, W. H.

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[Crossref]

G. W. Ford and W. H. Weber, Surf. Sci. 109, 451 (1981).
[Crossref]

W. H. Weber and G. W. Ford, Phys. Rev. Lett. 44, 1774 (1980).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).

Young, J. F.

J. E. Sipe, H. M. van Driel, and J. F. Young, Can. J. Phys. 63, 104 (1985).
[Crossref]

J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
[Crossref]

Am. J. Phys. (1)

M. Cardona, Am. J. Phys. 39, 1277 (1971).
[Crossref]

Can. J. Phys. (1)

J. E. Sipe, H. M. van Driel, and J. F. Young, Can. J. Phys. 63, 104 (1985).
[Crossref]

J. Chem. Phys. (1)

R. Chance, A. Prock, and R. Silbey, J. Chem. Phys. 60, 2184 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rep. (1)

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[Crossref]

Phys. Rev. (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Phys. Rev. 174, B13 (1968).
[Crossref]

Phys. Rev. B (6)

H. Morawitz and M. R. Philpott, Phys. Rev. B 12, 4869 (1974).

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Phys. Rev. B 21, 4389 (1980).
[Crossref]

J. E. Sipe, Phys. Rev. B 22, 1589 (1980).
[Crossref]

J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, Phys. Rev. B 27, 1141 (1983).
[Crossref]

J. A. Litwin, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 31, 5543 (1985).
[Crossref]

W. Lukosz, Phys. Rev. B 22, 3030 (1980).
[Crossref]

Phys. Rev. Lett. (2)

W. H. Weber and G. W. Ford, Phys. Rev. Lett. 44, 1774 (1980).
[Crossref]

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983).
[Crossref]

Prog. Surf. Sci. (1)

P. Feibelman, Prog. Surf. Sci. 12, No. 4 (1982).
[Crossref]

Surf. Sci (1)

J. E. Sipe, Surf. Sci 105, 489 (1981). I take this opportunity to correct some typographical errors in that paper. Equation (8) should read asF↔(k)=2πω˜2w0-1[s^s^+p^0+p^0+kz-w0-iη-s^s^+p^0-p^0-kz+w0+iη]-4πz^z^;Eq. (11) should read asF↔(κ;Z)=2πiω˜2w0-1(s^s^+p^0+p^0+)θ(Z)exp(iw0Z)+2πiω˜2w0-1(s^s^+p^0-p^0-)θ(-Z)exp(-i-w0Z)-4πz^z^δ(Z); and Eq. (16) should read as g(κ;Z)=2πiw0-1[θ(Z)exp(iw0Z)+θ(-Z)exp(-iw0Z)].
[Crossref]

Surf. Sci. (1)

G. W. Ford and W. H. Weber, Surf. Sci. 109, 451 (1981).
[Crossref]

Other (6)

J. E. Sipe and G. I. Stegeman, in Surface Polaritons, D. L. Mills and V. M. Agranovich, eds. (North-Holland, Amsterdam, 1982), pp. 661–701.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984).

T. F. Heinz, Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1982) (unpublished).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).

Z. Knittl, Optics of Thin Films: An Optical Multilayer Theory (Wiley, New York, 1976).

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Figures (3)

Fig. 1
Fig. 1

(a) The orthogonal triad ( s ^ , κ ^ , z ^). The vectors are all real. (b) The orthogonal triads ( s ^ , ν ^ + , p ^ +) and ( s ^ , ν ^ - , p ^ -) drawn for κ < ω ˜n and n real. (c) The vector ν + = κ κ ^ + w z ^ for κ < ω ˜n and n real. Here κ = ω ˜n sin θ, w = ω ˜n cos θ.

Fig. 2
Fig. 2

A dipole layer with dipole moment per unit area p = σd in a background dielectric constant leads to a field between the charged sheets of −4πσ/. The integral of this over the distance between the sheets is −4πσd/ = −4πp/.

Fig. 3
Fig. 3

(a) A general layered geometry. The dielectric constant is 1 for z > 0 and 2 for z < −D; in between, the optical properties are unspecified. (b) An interface at z = 0. (c) A three-layer geometry, with dielectric constants as indicated.

Equations (85)

Equations on this page are rendered with MathJax. Learn more.

· D = 0 , c × H - D ˙ = 0 , · B = 0 , c × E + B ˙ = 0 ,
D = E + 4 π P t , H = B - 4 π M t .
f ( r , t ) = f ( r ) e - i ω t + c . c . = 2 Re [ f ( r ) e - i ω t ]
P t ( r ) = χ e E ( r ) + P ( r ) , M t ( r ) = χ b B ( r ) + M ( r ) .
· E ( r ) = 0 , × B ( r ) + i ω ˜ μ E ( r ) = 0 , · B ( r ) = 0 , × E ( r ) - i ω ˜ B ( r ) = 0 ,
1 + 4 π χ e , μ - 1 1 - 4 π χ b
( 2 + ω ˜ 2 n 2 ) E ( r ) = 0 , ( 2 + ω ˜ 2 n 2 ) B ( r ) = 0 ,
n μ
κ = u x ^ + v y ^ κ κ ^ ,
E ( r ) = E exp ( i ν · r ) , B ( r ) = B exp ( i ν · r ) ,
ν + κ + w z ^ = u x ^ + v y ^ + w z ^ , ν - κ - w z ^ = u x ^ + v y ^ - w z ^ ,
w ( ω ˜ 2 μ - κ 2 ) 1 / 2 ,
s ^ κ ^ × z ^ .
U = x ^ x ^ + y ^ y ^ + z ^ z ^
U = s ^ s ^ + κ ^ κ ^ + z ^ z ^ .
ν · E = 0
p ^ ± ν - 1 ( κ z ^ w κ ^ ) .
ν ^ ± ν - 1 ( κ κ ^ ± w z ^ ) ,
( s ^ · s ^ ) = ( p ^ + · p ^ + ) = ( p ^ - · p ^ - ) = ( ν + · ν ^ + ) = ( ν ^ - · ν ^ - ) = 1 ,
p ^ + p ^ + + ν ^ + ν ^ + = p ^ - p ^ - + ν ^ - ν ^ - = κ ^ κ ^ + z ^ z ^ ,
U = s ^ s ^ + p ^ + p ^ + + ν ^ + ν ^ + = s ^ s ^ + p ^ - p ^ - + ν ^ - ν ^ - .
s ^ × ν ^ ± = p ^ ± , ν ^ ± × p ^ ± = s ^ , p ^ ± × s ^ = ν ^ ± ,
E + ( r ) = ( E s + s ^ + E p + p ^ + ) exp ( i ν + · r ) , B + ( r ) = n ( E p + s ^ - E s + p ^ + ) exp ( i ν + · r ) .
E - ( r ) = ( E s - s ^ + E p - p ^ - ) exp ( i ν - · r ) , B - ( r ) = n ( E p - s ^ - E s - p ^ - ) exp ( i ν - · r ) ,
· E ( r ) = - 4 π - 1 · P ( r ) , × B ( r ) + i ω ˜ μ E ( r ) = - 4 π i ω ˜ μ P ( r ) + 4 π μ × M ( r ) , · B ( r ) = 0 , × E ( r ) - i ω ˜ B ( r ) = 0 ,
P ( r ) = P δ ( z - z 0 ) exp ( i κ · R ) , M ( r ) = M δ ( z - z 0 ) exp ( i κ · R ) ,
E ( r ) = E + ( r ) exp ( - i ω z 0 ) θ ( z - z 0 ) + E - ( r ) exp ( i ω z 0 ) θ ( z 0 - z ) + δ ( z - z 0 ) exp ( i κ · R ) , B ( r ) = B + ( r ) exp ( - i ω z 0 ) θ ( z - z 0 ) + B - ( r ) exp ( i ω z 0 ) θ ( z 0 - z ) + δ ( z - z 0 ) exp ( i κ · R ) ,
θ ( z ) = 1 z > 0 = 0 , z < 0
E = E s s ^ + E κ κ ^ + E z z ^ , B = B s s ^ + B κ κ ^ + B z z ^ .
θ ( z - z 0 ) = z ^ δ ( z - z 0 ) , δ ( z - z 0 ) = z ^ δ ( z - z 0 ) ,
δ ( z - z 0 ) z ^ × [ ( E s + s ^ + E p + p ^ + ) - ( E s - s ^ + E p - p ^ - ) ] + i κ × E δ ( z - z 0 ) + z ^ × E δ ( z - z 0 ) - i ω ˜ B δ ( z - z 0 ) = 0.
E s = E κ = 0 ,
E p + + E p - + i κ ν w - 1 E z - i ω ˜ ν w - 1 B s = 0 , E s + - E s - - i ω ˜ B κ = 0 , B z = 0.
n δ ( z - z 0 ) z ^ × [ ( E p + s ^ - E s + p ^ + ) - ( E p - s ^ - E s - p ^ - ) ] + i κ × B δ ( z - z 0 ) + z ^ × B δ ( z - z 0 ) + i ω ˜ μ E δ ( z - z 0 ) = - 4 π i ω ˜ μ P δ ( z - z 0 ) + 4 π i μ κ × M δ ( z - z 0 ) + 4 π μ z ^ × M δ ( z - z 0 ) .
B s = 4 π μ M s , B κ = 4 π μ M κ ,
E s + + E s - = 4 π i ω ˜ 2 μ w - 1 P s - 4 π i κ ω ˜ μ w - 1 M z , E p + - E p - = - 4 π i ω ˜ μ n - 1 P κ , E z = - 4 π - 1 P z ,
E s + = 2 π i ω ˜ 2 μ w - 1 s ^ · P - 2 π i ω ˜ ν μ w - 1 p ^ + · M , E s - = 2 π i ω ˜ 2 μ w - 1 s ^ · P - 2 π i ω ˜ ν μ w - 1 p ^ - · M , E p + = 2 π i ω ˜ 2 μ w - 1 p ^ + · P + 2 π i ω ˜ ν μ w - 1 s ^ · M , E p - = 2 π i ω ˜ 2 μ w - 1 p ^ - · P + 2 π i ω ˜ ν μ w - 1 s ^ · M , E s = E κ = 0 , E z = - 4 π - 1 P z , B s = 4 π μ M s , B κ = 4 π μ M κ B z = 0.
P ( κ ; z ) exp ( - i κ · R ) P ( r ) d R ,
P ( r ) = d κ ( 2 π ) 2 P ( κ ; z ) exp ( i κ · R ) = d κ ( 2 π ) 2 d z 0 [ δ ( z - z 0 ) P ( κ ; z 0 ) exp ( i κ · R ) ] ,
P ( r ) = d κ ( 2 π ) 2 P ( κ ; z ) exp ( i κ · R ) M ( r ) = d κ ( 2 π ) 2 M ( κ ; z ) exp ( i κ · R ) ,
E ( r ) = d κ ( 2 π ) 2 E ( κ ; z ) exp ( i κ · R ) B ( r ) = d κ ( 2 π ) 2 B ( κ ; z ) exp ( i κ · R ) ,
E ( κ ; z ) = G E P ( κ ; z - z ) · P ( κ ; z ) d z + G E M ( κ ; z - z ) · M ( κ ; z ) d z B ( κ ; z ) = G B P ( κ ; z - z ) · P ( κ ; z ) d z + G B M ( κ ; z - z ) · M ( κ ; z ) d z ,
G E P ( κ ; z ) = 2 π i ω ˜ 2 μ w - 1 ( s ^ s ^ + p ^ + p ^ + ) e i ω z θ ( z ) 2 + 2 π i ω ˜ 2 μ w - 1 ( s ^ s ^ + p ^ - p ^ - ) e - i ω z θ ( - z ) - 4 π - 1 z ^ z ^ δ ( z ) , G E M ( κ ; z ) = 2 π i ω ˜ ν μ w - 1 ( p ^ + s ^ - s ^ p ^ + ) e i ω z θ ( z ) + 2 π i ω ˜ ν μ w - 1 ( p ^ - s ^ - s ^ p ^ - ) e - i ω z θ ( - z ) , G B P ( κ ; z ) = 2 π i ω ˜ ν μ w - 1 ( s ^ p ^ + - p ^ + s ^ ) e i ω z θ ( z ) + 2 π i ω ˜ ν μ w - 1 ( s ^ p ^ - - p ^ - s ^ ) e - i ω z θ ( - z ) , G B M ( κ ; z ) = 2 π i ν 2 μ w - 1 ( s ^ s ^ + p ^ + p ^ + ) e i ω z θ ( z ) + 2 π i ν 2 μ w - 1 ( s ^ s ^ + p ^ - p ^ - ) e - i ω z θ ( - z ) + 4 π μ ( s ^ s ^ + κ ^ κ ^ ) δ ( z ) .
· E ( r ) = - 4 π - 1 · P ( r ) , × B ( r ) + i ω ˜ E ( r ) = - 4 π i ω ˜ P ( r ) , · B ( r ) = 0 , × E - i ω ˜ B ( r ) = 0
P ( r ) = d κ ( 2 π ) 2 P ( κ ; z ) exp ( i κ ; R ) ,
E ( r ) = d κ ( 2 π ) 2 E ( κ ; z ) exp ( i κ · R ) ,
E ( κ ; z ) = G ( κ ; z - z ) · P ( κ ; z ) d z
G ( κ ; z ) = 2 π i ω ˜ 2 w - 1 ( s ^ s ^ + p ^ + p ^ + ) θ ( z ) e i ω z + 2 π i ω ˜ 2 w - 1 ( s ^ s ^ + p ^ - p ^ - ) θ ( - z ) e - i ω z - 4 π - 1 z ^ z ^ δ ( z ) .
E ( r ) = E 1 + p ^ 1 + exp ( i ν 1 + · r ) + E 1 - p ^ 1 exp ( i ν 1 - · r ) , E ( r ) = E 2 + p ^ 2 + exp ( i ν 2 + · r ) + E 2 - p ^ 2 - exp ( i ν 2 - · r ) ,
e i ( z ) = [ E i + exp ( i w i z ) E i - exp ( - i w i z ) ] .
e 1 ( 0 ) = M 12 P e 2 ( - D ) ,
M i j p = 1 T i j p [ T i j p T i j p - R i j p R j i p R i j p - R i j p 1 ] .
M 12 p = 1 t 12 p [ 1 r 12 p r 12 p 1 ]
r i j p = w i j - w j i w i j + w j i ,             t i j p = 2 n i n j w i w i j + w j i
r j i = - r i j , t i j t j i - r i j r j i = 1 ,
r i j s = w i - w j w i + w j ,             t i j s = 2 w i w i + w j .
e i ( z a ) = M i ( z a - z b ) e i ( z b ) ,
M i ( z ) = [ exp ( i w i z ) 0 0 exp ( - i w i z ) ] ,
e 1 ( 0 ) = M 13 e 3 ( 0 ) , e 3 ( 0 ) = M 3 ( D ) e 3 ( - D ) , e 3 ( - D ) = M 32 e 2 ( - D ) ,
M 12 = M 13 M 3 ( D ) M 32 .
T 12 = t 13 t 32 exp ( i w 3 D ) 1 - r 31 r 32 exp ( 2 i w 3 D ) , R 12 = r 13 + t 13 r 32 t 31 exp ( 2 i w 3 D ) 1 - r 31 r 32 exp ( 2 i w 3 D ) ,
P ( r ) = P ( z ) exp ( i κ · R ) .
E = ( s ^ t 21 s s ^ + p ^ 1 + t 21 p p ^ 2 + ) · [ 2 π i ω ˜ 2 w 2 - 1 ( s ^ s ^ + p ^ 2 + p ^ 2 + ) · - 0 exp ( - i w 2 z ) P ( z ) d z ] = 2 π i ω ˜ 2 w 2 - 1 ( s ^ t 21 s s ^ + p ^ 1 + t 21 p p ^ 2 + ) · - 0 exp ( - i w 2 z ) P ( z ) d z
E ( z ) = - 4 π 2 - 1 z ^ z ^ · P ( z ) + 2 π i ω ˜ 2 w 2 - 1 ( s ^ s ^ + p ^ 2 + p ^ 2 + ) · - z exp [ i w 2 ( z - z ) ] P ( z ) d z + 2 π i ω ˜ 2 w 2 - 1 ( s ^ r 21 s s ^ + p ^ 2 - r 21 p p ^ 2 + ) · - 0 exp [ - i w 2 ( z + z ) ] P ( z ) d z + 2 π i ω ˜ 2 w 2 - 1 ( s ^ s ^ + p ^ 2 - p ^ 2 - ) · z 0 exp [ - i w 2 ( z - z ) ] P ( z ) d z .
P ( r ) = P δ ( z - 0 + ) exp ( i κ · R ) .
E = 2 π i ω ˜ 2 w 1 - 1 ( s ^ s ^ + p ^ 1 + p ^ 1 + ) · P + 2 π i ω ˜ 2 w 1 - 1 ( s ^ r 12 s s + p ^ 1 + r 12 p p ^ - ) · P = 2 π i ω ˜ 2 w 1 - 1 [ s ^ ( 1 + r 12 s ) s ^ + p ^ 1 + ( p ^ 1 + + r 12 p p ^ 1 - ) ] · P .
E = 2 π i ω ˜ 2 w 1 - 1 [ s ^ ( 1 + R 12 s ) s ^ + p ^ 1 + ( p ^ 1 + + R 12 p p ^ 1 - ) ] · P ,
E = 2 π i ω ˜ 2 w 2 - 1 ( s ^ T 21 s s ^ + p ^ 1 + T 21 p p ^ 2 + ) · - - D exp [ i w 2 ( - D - z ) ] P ( z ) d z .
P ( r ) = P δ ( z - z 0 ) exp ( i κ · R ) ,
E ( z ) = 2 π i ω ˜ 2 w 3 - 1 ( s ^ s ^ + p ^ 3 + p ^ 3 + ) · P θ ( z - z 0 ) exp [ i w 3 ( z - z 0 ) ] + 2 π i ω ˜ 2 w 3 - 1 ( s ^ s ^ + p ^ 3 - p ^ 3 - ) · P θ ( z 0 - z ) exp [ - w 3 ( z - z 0 ) ] - 4 π 3 - 1 z ^ z ^ · P δ ( z - z 0 ) .
e 3 ( z 0 + ) = v + e 3 ( z 0 - ) ,
v = [ v + - v - ]
v ± = 2 π i ω ˜ 2 w 3 - 1 q ^ 3 ± · P ,
e 1 ( 0 ) = M 13 M 3 ( - z 0 ) e 3 ( z 0 + ) , e 3 ( z 0 - ) = M 3 ( z 0 + D ) M 32 e 2 ( - D ) ,
e 1 ( 0 ) = M 13 M 3 ( - z 0 ) v + M 13 M 3 ( D ) M 32 e 2 ( - D ) .
M i j M j i = M i ( z ) M i ( - z ) = I ,
e 2 ( - D ) = - M 23 M 3 ( - D - z 0 ) v + M 23 M 3 ( - D ) M 31 e 1 ( 0 ) .
e 1 ( 0 ) = [ E 1 + 0 ] ,             e 2 ( - D ) = [ 0 E 2 - exp ( i w 2 D ) ] .
E 1 + = t 31 exp ( - e 3 z 0 ) v + + r 32 exp ( 2 i w 3 D ) exp ( i w 3 z 0 ) v - 1 - r 32 r 31 exp ( 2 i w 3 D ) .
E 1 + = t 31 1 - r 32 r 31 exp ( 2 i w 3 D ) [ - D 0 exp ( - i w 3 z ) v + ( z ) d z + r 32 exp ( 2 i w 3 D ) - D 0 exp ( i w 3 z ) v - ( z ) d z ] ,
v ± ( z ) = 2 π i ω ˜ 2 w 3 - 1 q ˜ 3 ± · P ( z ) ,
e 1 ( 0 ) = M 13 M 3 ( - z ) e 3 ( z ) , e 3 ( z ) = M 3 ( z + D ) M 32 e 2 ( - D )
F(k)=2πω˜2w0-1[s^s^+p^0+p^0+kz-w0-iη-s^s^+p^0-p^0-kz+w0+iη]-4πz^z^;
F(κ;Z)=2πiω˜2w0-1(s^s^+p^0+p^0+)θ(Z)exp(iw0Z)+2πiω˜2w0-1(s^s^+p^0-p^0-)θ(-Z)exp(-i-w0Z)-4πz^z^δ(Z);
g(κ;Z)=2πiw0-1[θ(Z)exp(iw0Z)+θ(-Z)exp(-iw0Z)].

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