Abstract

It is demonstrated that ab initio microscopic calculations of the linear optical reflection response of mesoscopic films, e.g., quantum wells, can always be represented by a sheet-model description, even if one goes beyond the so-called electric-dipole approximation. A general expression for the sheet-conductivity tensor is established, and the reflection matrix of the mesoscopic film is studied. In the wake of an investigation of the moment expansion of the sheet conductivity, new jump conditions for the electromagnetic field across a so-called electric-dipole–electric-dipole sheet are established. Finally, the inconsistencies associated with the use of standard (textbook) jump conditions in analysis of the linear optical response of mesoscopic films are identified.

© 1995 Optical Society of America

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References

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  1. J. D. E. McIntyre, in Optical Properties of Solids. New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), p. 555.
  2. M. Yamamoto and T. Namioka, Appl. Opt. 31, 1612 (1992).
    [CrossRef] [PubMed]
  3. N. Raj and D. R. Tilley, in The Dielectric Function of Condensed Systems, L. V. Keldysh, D. A. Kirzhnitz, and A. A. Maradudin, eds., Vol. 24 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1989), p. 459.
    [CrossRef]
  4. V. M. Agranovich, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds., Vol. 1 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1982), p. 187.
  5. F. Forstman and R. R. Gerhardts, Metal Optics Near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  6. O. Keller, A. Liu, and A. Zayats, Opt. Commun. 110, 604 (1994).
    [CrossRef]
  7. O. Keller, Phys. Rev. B 38, 8041 (1988).
    [CrossRef]
  8. O. Keller, in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).
  9. O. Keller, J. Opt. Soc. Am. B 12, 997 (1995).
    [CrossRef]
  10. O. Keller, Phys. Rev. B 37, 10588 (1988).
    [CrossRef]
  11. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  12. P. J. Feibelman, Prog. Surf. Sci. 12, 287 (1982).
    [CrossRef]
  13. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  14. A. Liebsch, G. Hincelin, and T. López-Rios, Phys. Rev. B 41, 10463 (1990).
    [CrossRef]
  15. D. E. Aspnes, in Optical Properties of Solids. New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), p. 799.
  16. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  17. P. J. Feibelman, Phys. Rev. B 12, 1319 (1975).
    [CrossRef]
  18. A. Bagchi, Phys. Rev. B 15, 3060 (1977).
    [CrossRef]
  19. O. Keller and A. Liu, Phys. Rev. B 49, 2072 (1994).
    [CrossRef]
  20. E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
    [CrossRef]
  21. K. Hattori, T. Mori, H. Okamoto, and Y. Hamakawa, Phys. Rev. Lett. 60, 825 (1988).
    [CrossRef] [PubMed]
  22. E. A. Vinogradov, A. V. Zayats, and F. A. Pudonin, Sov. Phys. Solids 33, 197 (1990).
  23. A. V. Zayats, Yu. A. Repeyev, D. N. Nikogosyan, and E. A. Vinogradov, J. Lumin. 52, 335 (1992).
    [CrossRef]

Agranovich, V. M.

V. M. Agranovich, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds., Vol. 1 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1982), p. 187.

Alieva, E. V.

E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, in Optical Properties of Solids. New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), p. 799.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bagchi, A.

A. Bagchi, Phys. Rev. B 15, 3060 (1977).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Feibelman, P. J.

P. J. Feibelman, Phys. Rev. B 12, 1319 (1975).
[CrossRef]

P. J. Feibelman, Prog. Surf. Sci. 12, 287 (1982).
[CrossRef]

Firsov, E. I.

E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
[CrossRef]

Forstman, F.

F. Forstman and R. R. Gerhardts, Metal Optics Near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1986).
[CrossRef]

Gerhardts, R. R.

F. Forstman and R. R. Gerhardts, Metal Optics Near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1986).
[CrossRef]

Hamakawa, Y.

K. Hattori, T. Mori, H. Okamoto, and Y. Hamakawa, Phys. Rev. Lett. 60, 825 (1988).
[CrossRef] [PubMed]

Hattori, K.

K. Hattori, T. Mori, H. Okamoto, and Y. Hamakawa, Phys. Rev. Lett. 60, 825 (1988).
[CrossRef] [PubMed]

Hincelin, G.

A. Liebsch, G. Hincelin, and T. López-Rios, Phys. Rev. B 41, 10463 (1990).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Keller, O.

O. Keller, Phys. Rev. B 38, 8041 (1988).
[CrossRef]

O. Keller, in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).

O. Keller, J. Opt. Soc. Am. B 12, 997 (1995).
[CrossRef]

O. Keller, Phys. Rev. B 37, 10588 (1988).
[CrossRef]

O. Keller, A. Liu, and A. Zayats, Opt. Commun. 110, 604 (1994).
[CrossRef]

O. Keller and A. Liu, Phys. Rev. B 49, 2072 (1994).
[CrossRef]

Kuzik, L. A.

E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
[CrossRef]

Liebsch, A.

A. Liebsch, G. Hincelin, and T. López-Rios, Phys. Rev. B 41, 10463 (1990).
[CrossRef]

Liu, A.

O. Keller, A. Liu, and A. Zayats, Opt. Commun. 110, 604 (1994).
[CrossRef]

O. Keller and A. Liu, Phys. Rev. B 49, 2072 (1994).
[CrossRef]

López-Rios, T.

A. Liebsch, G. Hincelin, and T. López-Rios, Phys. Rev. B 41, 10463 (1990).
[CrossRef]

McIntyre, J. D. E.

J. D. E. McIntyre, in Optical Properties of Solids. New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), p. 555.

Mori, T.

K. Hattori, T. Mori, H. Okamoto, and Y. Hamakawa, Phys. Rev. Lett. 60, 825 (1988).
[CrossRef] [PubMed]

Namioka, T.

M. Yamamoto and T. Namioka, Appl. Opt. 31, 1612 (1992).
[CrossRef] [PubMed]

Nikogosyan, D. N.

A. V. Zayats, Yu. A. Repeyev, D. N. Nikogosyan, and E. A. Vinogradov, J. Lumin. 52, 335 (1992).
[CrossRef]

Okamoto, H.

K. Hattori, T. Mori, H. Okamoto, and Y. Hamakawa, Phys. Rev. Lett. 60, 825 (1988).
[CrossRef] [PubMed]

Pudonin, F.

E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
[CrossRef]

Pudonin, F. A.

E. A. Vinogradov, A. V. Zayats, and F. A. Pudonin, Sov. Phys. Solids 33, 197 (1990).

Raj, N.

N. Raj and D. R. Tilley, in The Dielectric Function of Condensed Systems, L. V. Keldysh, D. A. Kirzhnitz, and A. A. Maradudin, eds., Vol. 24 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1989), p. 459.
[CrossRef]

Repeyev, Yu. A.

A. V. Zayats, Yu. A. Repeyev, D. N. Nikogosyan, and E. A. Vinogradov, J. Lumin. 52, 335 (1992).
[CrossRef]

Tilley, D. R.

N. Raj and D. R. Tilley, in The Dielectric Function of Condensed Systems, L. V. Keldysh, D. A. Kirzhnitz, and A. A. Maradudin, eds., Vol. 24 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1989), p. 459.
[CrossRef]

Vinogradov, E. A.

E. A. Vinogradov, A. V. Zayats, and F. A. Pudonin, Sov. Phys. Solids 33, 197 (1990).

A. V. Zayats, Yu. A. Repeyev, D. N. Nikogosyan, and E. A. Vinogradov, J. Lumin. 52, 335 (1992).
[CrossRef]

Yakovlev, V. A.

E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
[CrossRef]

Yamamoto, M.

M. Yamamoto and T. Namioka, Appl. Opt. 31, 1612 (1992).
[CrossRef] [PubMed]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zayats, A.

O. Keller, A. Liu, and A. Zayats, Opt. Commun. 110, 604 (1994).
[CrossRef]

Zayats, A. V.

E. A. Vinogradov, A. V. Zayats, and F. A. Pudonin, Sov. Phys. Solids 33, 197 (1990).

A. V. Zayats, Yu. A. Repeyev, D. N. Nikogosyan, and E. A. Vinogradov, J. Lumin. 52, 335 (1992).
[CrossRef]

Other (23)

J. D. E. McIntyre, in Optical Properties of Solids. New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), p. 555.

M. Yamamoto and T. Namioka, Appl. Opt. 31, 1612 (1992).
[CrossRef] [PubMed]

N. Raj and D. R. Tilley, in The Dielectric Function of Condensed Systems, L. V. Keldysh, D. A. Kirzhnitz, and A. A. Maradudin, eds., Vol. 24 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1989), p. 459.
[CrossRef]

V. M. Agranovich, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds., Vol. 1 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1982), p. 187.

F. Forstman and R. R. Gerhardts, Metal Optics Near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1986).
[CrossRef]

O. Keller, A. Liu, and A. Zayats, Opt. Commun. 110, 604 (1994).
[CrossRef]

O. Keller, Phys. Rev. B 38, 8041 (1988).
[CrossRef]

O. Keller, in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).

O. Keller, J. Opt. Soc. Am. B 12, 997 (1995).
[CrossRef]

O. Keller, Phys. Rev. B 37, 10588 (1988).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

P. J. Feibelman, Prog. Surf. Sci. 12, 287 (1982).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

A. Liebsch, G. Hincelin, and T. López-Rios, Phys. Rev. B 41, 10463 (1990).
[CrossRef]

D. E. Aspnes, in Optical Properties of Solids. New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), p. 799.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

P. J. Feibelman, Phys. Rev. B 12, 1319 (1975).
[CrossRef]

A. Bagchi, Phys. Rev. B 15, 3060 (1977).
[CrossRef]

O. Keller and A. Liu, Phys. Rev. B 49, 2072 (1994).
[CrossRef]

E. V. Alieva, E. I. Firsov, L. A. Kuzik, V. A. Yakovlev, and F. Pudonin, Phys. Lett. A 152, 89 (1991).
[CrossRef]

K. Hattori, T. Mori, H. Okamoto, and Y. Hamakawa, Phys. Rev. Lett. 60, 825 (1988).
[CrossRef] [PubMed]

E. A. Vinogradov, A. V. Zayats, and F. A. Pudonin, Sov. Phys. Solids 33, 197 (1990).

A. V. Zayats, Yu. A. Repeyev, D. N. Nikogosyan, and E. A. Vinogradov, J. Lumin. 52, 335 (1992).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Schematic illustration showing the sheet-model radiation chanels. The quantum well occupies the region − dz ≤ 0, and the equivalent current-density sheet (gray area) is placed at z = z0. Radiation from a source point located at r′ can reach the observation point at r by following various paths. Thus the contribution from the background field (dashed lines) consists of a direct term plus an indirect one (reaching the observation point on reflection from the substrate). The radiation from the source point located at r0 in the sheet reaches r by direct or indirect propagation, as indicated by the solid lines. As shown by the dotted lines, the source r0 is induced by the primary radiation r′, reaching r0 by the direct and the indirect paths.

Equations (104)

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E i ( r , t ) = E i ( q 0 , ω ) exp ( i q 0 z ) exp [ i ( q · r ω t ) ]
Q ( r , t ) = Q ( z ; q , ω ) exp [ i ( q · r ω t ) ]
E ( z ) = E B ( z ) i μ 0 ω QW G B ( z , z ) · σ B ( z , z ) · E B ( z ) d z d z .
r ( 0 ) = r p ( e z e z e x e x ) + r s e y e y ,
E B ( z ) = [ U exp ( i q 0 z ) + r ( 0 ) exp ( i q 0 z ) ] · E i ( 0 ) ,
G B ( z , z ) · P B ( z , z ) + ( c 0 ω ) 2 δ ( z z ) e z e z ,
J ( z ) = QW σ B ( z , z ) · E B ( z ) d z ,
E ( z ) = E B ( z ) i μ 0 ω QW P B ( z , z ) · σ B ( z , z ) · E B ( z ) d z d z , z d ,
P B ( z , z ) = exp ( q 0 z ) 2 i q 0 [ ( e y e y + e r e r ) exp ( i q 0 z ) + ( r s e y e y + r p e r e i ) exp ( i q 0 z ) ] , z d ,
E B ( z ) = N ( z , z 0 ) · E B ( z 0 ) ,
P B ( z , z ) = P B ( z , z 0 ) · N ( z , z 0 ) ,
N ( z , z 0 ) = [ U exp ( i q 0 z ) + r ( 0 ) exp ( i q 0 z ) ] · [ U exp ( i q 0 z 0 ) + r ( 0 ) exp ( i q 0 z 0 ) ] 1 z = z or z
E ( z ) = E B ( z ) i μ 0 ω P B ( z , z 0 ) · J S ( z 0 ) , z d ,
J S ( z 0 ) = S ( z 0 ) · E B ( z 0 ) ,
S ( z 0 ) = QW N ( z , z 0 ) · σ B ( z , z ) · N ( z , z 0 ) d z d z .
E r ( d ) = r ( d ) · E i ( d ) .
E ( d ) = E B ( d ) i μ 0 ω P B ( d , z 0 ) · S ( z 0 ) · E B ( z 0 ) .
E B ( z ) = { U exp [ i q 0 ( z + d ) ] + r ( 0 ) exp [ i q 0 ( z d ) ] } · E i ( d ) ,
E B ( d ) = [ U + r ( 0 ) exp ( 2 i q 0 d ) ] · E i ( d ) ,
E B ( z 0 ) = { U exp [ i q 0 ( z 0 + d ) ] + r ( 0 ) exp [ i q 0 ( z 0 d ) ] } · E i ( d ) .
E ( d ) = E i ( d ) + r ( d ) · E i ( d ) ,
r ( d ) = r ( 0 ) exp ( 2 i q 0 d ) i μ 0 ω P B ( d , z 0 ) · S ( z 0 ) · { U exp [ i q 0 ( z 0 + d ) ] + r ( 0 ) exp [ i q 0 ( z 0 d ) ] } .
r ( d ) = r ( 0 ) exp ( 2 i q 0 d ) ,
r ( d ) = r ( d ) i μ 0 ω P B ( d , z 0 ) · S ( z 0 ) · { U exp [ i q 0 ( z 0 + d ) ] + r ( d ) exp [ i q 0 ( z 0 + d ) ] } .
N ( z , z 0 ) = N ( 0 , z 0 ) · N ( z , 0 ) ,
N ( z , z 0 ) = N ( z , 0 ) · N ( 0 , z 0 ) ,
S ( z 0 ) N ( 0 , z 0 ) · S ( 0 ) · N ( 0 , z 0 ) ,
P B ( d , z 0 ) = P B ( d , 0 ) · N ( z 0 , 0 ) = exp ( i q 0 d ) P B ( 0 , 0 ) · N ( z 0 , 0 ) .
r ( d ) = exp ( 2 i q 0 d ) r ( 0 )
r ( 0 ) = r ( 0 ) i μ 0 ω P B ( 0 , 0 ) · S ( 0 ) · [ U + r ( 0 ) ] .
σ B = [ σ x x B 0 σ x z B 0 σ y y B 0 σ z x B 0 σ z z B ] ,
S = [ S x x 0 S x z 0 S y y 0 S z x 0 S z z ] .
r = [ r x x 0 r x z 0 r y y 0 r z x 0 r z z ] .
E r ( d ) = [ ( r x x q q 0 r x z ) e x e x + r y y e y e y + ( r z z q 0 q 0 r z x ) e z e z ] · E i ( d ) .
r x x + r z z q q 0 r x z q 0 q r z x = 0 .
r p r z z q 0 q r z x ,
E r ( d ) = r ( d ) · E i ( d ) ,
r ( d ) = r p ( d ) ( e z e z e x e x ) + r s ( d ) e y e y .
r p ( d ) = exp ( 2 i q 0 d ) r p ( 0 ) ,
r s ( d ) = exp ( 2 i q 0 d ) r s ( 0 ) .
r p ( 0 ) = r p ( 0 ) + 1 2 0 ω q 0 { ( q 0 ) 2 [ 1 r p ( 0 ) ] 2 × S x x ( 0 ) q 2 [ 1 + r p ( 0 ) ] 2 S z z ( 0 ) + q q 0 [ 1 r p 2 ( 0 ) ] [ S z x ( 0 ) S x z ( 0 ) ] } ,
r s ( 0 ) = r s ( 0 ) μ 0 ω 2 q 0 [ 1 + r s ( 0 ) ] 2 S y y ( 0 ) ,
E ( z ) = E B ( z ) i μ 0 ω P B ( z , 0 ) · S ( 0 ) · E B ( 0 ) .
E B ( z ) = E i p ( 0 ) [ e i exp ( i q 0 z ) + r p ( 0 ) e r exp ( i q 0 z ) ] + E i s ( 0 ) e y [ exp ( i q 0 z ) + r s ( 0 ) exp ( i q 0 z ) ] ,
P B ( z , 0 ) = exp ( i q 0 z ) 2 i q 0 { [ 1 + r s ( 0 ) ] e y e y + e r [ e r + r p ( 0 ) e i ] } .
E B ( 0 ) = E i p ( 0 ) [ e i + r p e r ] + E i s ( 0 ) e y ( 1 + r s ) .
E ( z ) = E i p ( 0 ) [ e i exp ( i q 0 z ) + r p ( 0 ) e r exp ( i q 0 z ) ] + E i s ( 0 ) e y [ exp ( i q 0 z ) + r s ( 0 ) exp ( i q 0 z ) ] ,
r p ( 0 ) = r p ( 0 ) μ 0 ω 2 q 0 [ e r + r p ( 0 ) e i ] · S ( 0 ) · [ e i + r p ( 0 ) e r ] ,
r s ( 0 ) = r s ( 0 ) μ 0 ω 2 q 0 [ 1 + r s ( 0 ) ] 2 e y · S ( 0 ) · e y .
N ( z , z 0 ) = N ( z 0 , z 0 ) + ( z z 0 ) N ( z , z 0 ) z | z = z 0 + ,
N ( z 0 , z 0 ) U ,
N ( z , z 0 ) z | z = z 0 = i q 0 [ U exp ( i q 0 z 0 ) r ( 0 ) exp ( i q 0 z 0 ) ] · [ U exp ( i q 0 z 0 ) + r ( 0 ) exp ( i q 0 z 0 ) ] 1 .
N ( z , 0 ) z | z = 0 = i q 0 ( 1 + r p 1 r p e x e x + 1 + r s 1 + r s e y e y + 1 r p 1 + r p e z e z ) .
S ( 0 ) = S ED ED + S MD / EQ ED ( 0 ) + S ED MD / EQ ( 0 ) + S MD / EQ MD / EQ ( 0 ) + ,
S ED ED = QW σ B ( z , z ) d z d z ,
S MD / EQ ED ( 0 ) = N ( 0 , 0 ) z · [ QW z σ B ( z , z ) d z d z ] ,
S ED MD / EQ ( 0 ) = N ( 0 , 0 ) z · [ QW z σ B ( z , z ) d z d z ] ,
S MD / EQ MD / EQ ( 0 ) = N ( 0 , 0 ) z · [ QW z , z σ B ( z , z ) d z d z ] . N ( 0 , 0 ) z ,
N ( 0 , 0 ) z N ( z , 0 ) z | z = 0 .
E ( z ) = E B ( z ) i μ 0 ω P B ( z , 0 ) · J S ( 0 ) , z < d .
E ( z ) = E B ( z ) i μ 0 ω P < > B ( z , 0 ) · J S ( 0 ) , z 0 + ,
P < > B ( z , 0 ) = 1 2 i q 0 [ t s e y e y + t p e ( q , κ ) e i ] exp ( i κ z ) ,
t s = 1 + r s ,
t p = 1 / 2 ( ω ) ( 1 + r p ) .
κ = [ ( ω c 0 ) 2 ( ω ) q 2 ] 1 / 2
J S ( 0 ) = S ED ED · E B ( 0 ) .
E ( 0 ) = E B ( 0 ) i μ 0 ω P B ( 0 , 0 ) · J S ( 0 ) ,
E ( 0 + ) = E B ( 0 + ) i μ 0 ω P < > B ( 0 + , 0 ) . J S ( 0 ) ,
P B ( 0 , 0 ) = 1 2 i q 0 [ ( 1 + r s ) e y e y + e r e r + r p e r e i ] ,
P < > B ( 0 + , 0 ) = 1 2 i q 0 ( t s e y e y + t p e e i ) .
e x · [ E ( 0 + ) E ( 0 ) ] = e x · [ E B ( 0 + ) E B ( 0 ) ] i μ 0 ω e x · [ P < > B ( 0 + , 0 ) P B ( 0 , 0 ) ] · J S ( 0 )
r p = q 0 κ q 0 + κ ,
e x · [ P < > B ( 0 + , 0 ) P B ( 0 , 0 ) ] = i q ( c 0 ω ) 2 e z .
e x · [ E ( 0 + ) E ( 0 ) ] = 1 0 q ω e z · J S ( 0 ) .
e y · [ E ( 0 + ) E ( 0 ) ] = 0 ,
e z · [ D ( 0 + ) D ( 0 ) ] 0 e z · [ E ( 0 + ) E ( 0 ) ] .
e z · [ D ( 0 + ) D ( 0 ) ] 0 e z · [ E B ( 0 + ) E B ( 0 ) ] i μ 0 0 ω e z · [ P < > B ( 0 + , 0 ) P B ( 0 , 0 ) ] · J S ( 0 ) .
e z · [ P < > B ( 0 + , 0 ) P B ( 0 , 0 ) ] = i q ( c 0 ω ) 2 e x .
e z · [ D ( 0 + ) D ( 0 ) ] = q ω e x · J S ( 0 )
i ω ρ ( z ) = i q J x ( z ) + d J z ( z ) d z ,
ρ S QW ρ ( z ) d z
ρ S = q ω QW J x ( z ) d z + 1 i ω [ J z ( R ) J z ( L ) ] ,
QW J x ( z ) d z = e x · QW σ B ( z , z ) · E B ( z ) d z d z = e x · [ QW σ B ( z , z ) d z d z ] · E B ( 0 ) = e x · S ED ED · E B ( 0 ) = e x · J s ( 0 )
q ω e x · J S ( 0 ) = ρ S 1 i ω [ J z ( R ) J z ( L ) ] .
B ( z ) = 1 i ω ( i q e x + e z d d z ) × E ( z ) .
B ( z ) = B B ( z ) i μ 0 ( q e x + κ e z ) × P < > B ( z , 0 ) · J S ( 0 ) , z 0 + ,
B ( z ) = B B ( z ) i μ 0 ( q e x q 0 e z ) × P B ( z , 0 ) · J S ( 0 ) , z d .
B ( 0 + ) B ( 0 ) = i μ 0 { q e x × [ P < > B ( 0 + , 0 ) P B ( 0 , 0 ) ] + e z × [ κ [ P < > B ( 0 + , 0 ) + q 0 P B ( 0 , 0 ) ] } · J S ( 0 ) .
e x · [ B ( 0 + ) B ( 0 ) ] = μ 0 e y · J S ( 0 ) ,
e y · [ B ( 0 + ) B ( 0 ) ] = μ 0 e x · J S ( 0 ) ,
e z · [ B ( 0 + ) B ( 0 ) ] = 0 .
E y ( 0 + ) E y ( 0 ) = 0 ,
B x ( 0 + ) B x ( 0 ) = μ 0 J y S ( 0 ) ,
B z ( 0 + ) B z ( 0 ) = 0 ,
E x ( 0 + ) E x ( 0 ) = 1 0 q ω J z S ( 0 ) ,
D z ( 0 + ) D z ( 0 ) = q ω J x S ( 0 ) ,
B y ( 0 + ) B y ( 0 ) = μ 0 J x S ( 0 ) ,
Jump ( E y ) = ω q Jump ( B z ) .
Jump ( B y ) = μ 0 ω q Jump ( D z ) .
J S ( 0 ) = S 0 · E ( 0 ) ,
S 0 = S ED ED · [ U i μ 0 ω P B ( 0 , 0 ) · S ED ED ] 1 .
e x · [ E ( 0 + ) E ( 0 ) ] = 1 0 q ω e z · S 0 · E ( 0 ) ,
e z · [ D ( 0 + ) D ( 0 ) ] = 1 0 q ω e x · S 0 · D ( 0 ) .
e z × [ B ( 0 + ) B ( 0 ) ] = μ 0 [ e x e x + e y e y ] · S 0 · E ( 0 ) .

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