Abstract

First-order perturbation theory and a nonorthogonal coordinate transformation are used to calculate light scattering from rough metallic surfaces having a dielectric overlayer. The theory considers arbitrary polarization, incident intensity profile, and angle of incidence, and it is valid for complex values of the dielectric constants of the metal and dielectric overlayer 0. The frequency range of interest is such that Re() < 0. The results are applied to cases of periodic and random roughness where the dielectric overlayer replicates the substrate profile. Comparison to experiment is made in the case of periodic roughness, and numerical results are given in both cases.

© 1976 Optical Society of America

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  1. In the case of periodic roughness the rectangular profile is considered. Even though the slope of a rectangular profile is infinite at the steps the large majority of the surface has zero slope. In fact, it may be shown that the average slope of the surface is zero.
  2. S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
    [Crossref]
  3. H. Davies, Proc. IEEE 101, 209 (1954).
  4. H. J. Juranek, Z. Phys. 233, 324 (1970).
    [Crossref]
  5. E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
    [Crossref]
  6. E. Kretschmann, Z. Phys. 227, 412 (1969).
    [Crossref]
  7. E. A. Stern, Phys. Rev. Lett. 19, 1321 (1967).
    [Crossref]
  8. D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
    [Crossref]
  9. J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
    [Crossref]
  10. E. Kretschmann and E. Kröger, J. Opt. Soc. Am. 65, 15C (1975).
    [Crossref]
  11. A. Maradudin and D. Mills, Phys. Rev. B 11, 1392 (1975).
    [Crossref]
  12. J. M. Elson, Phys. Rev. B 12, 2541 (1975).
    [Crossref]
  13. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, New York, 1963).
  14. L. Deryugin, Radiotekhn. 15, 9 (1960).
  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 38.
  16. D. Mills, A. Maradudin, and E. Burstein, Ann. Phys. (N.Y.) 56, 504 (1970).
    [Crossref]
  17. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1969), p. 100.
  18. J. Bennett and M. Elson, in High Energy Laser Mirrors and Windows, , ARPA Order 2175, March 1974–Sept. 1974, p. 4.
  19. D. Gloge, E. Chinnock, and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
    [Crossref]
  20. See Refs. 3, 5, 6, 8, 10, 11, and also J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970).
    [Crossref]
  21. J. M. Elson, Michelson Laboratory, in for Air Force Weapons Laboratory, Kirtland AFB, (Apr., 1976).
  22. J. L. Stanford, J. Opt. Soc. Am. 60, 49 (1970).
    [Crossref]
  23. T. Tamir and H. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971); J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 538 (1970).
    [Crossref]
  24. A. Braunstein and M. Braunstein, J. Vac. Sci. Technol. 8, 412 (1970).
    [Crossref]
  25. W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968).
    [Crossref] [PubMed]
  26. G. Charlton (private communication), Air Force Weapons Laboratory, Kirtland AFB, New Mexico. Detailed examination ThF4 overcoated rectangular groove gratings yielded good profile definition in the vicinity of the steps. The overlayer thickness was 2.7 μ m.
  27. J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in for Air Force Weapons Lab., Kirtland AFB, p. 36 (July, 1974).
  28. J. S. Harris (private communication).
  29. H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, , ARPA Order 2175, March 1974–Sept. 1974, p. 15.
  30. J. M. Bennett and J. M. Elson, High Energy Laser Mirrors and Windows, Semi-Annual Report No. 7, Mar 1975–Sep 1975, ARPA Order 2175 (1975b) (to be published).
  31. CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
    [Crossref]
  32. MgF2, LiF, and Al2O3 are standard optical coating materials. See W. L. Wolfe in Handbook of Military Infrared Technology, Naval Research Laboratory, p. 355 (1965) and J. L. Stanford and H. E. Bennett, Appl. Opt. 8, 2556 (1969); H. E. Bennett, J. M. Bennett, and E. J. Ashely, ibid.2, 156 (1963).
    [Crossref] [PubMed]
  33. S. Ballard, J. Browder, and J. Ebersole, in American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–12.

1975 (3)

E. Kretschmann and E. Kröger, J. Opt. Soc. Am. 65, 15C (1975).
[Crossref]

A. Maradudin and D. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

J. M. Elson, Phys. Rev. B 12, 2541 (1975).
[Crossref]

1974 (1)

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[Crossref]

1971 (1)

1970 (7)

A. Braunstein and M. Braunstein, J. Vac. Sci. Technol. 8, 412 (1970).
[Crossref]

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[Crossref]

H. J. Juranek, Z. Phys. 233, 324 (1970).
[Crossref]

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

D. Mills, A. Maradudin, and E. Burstein, Ann. Phys. (N.Y.) 56, 504 (1970).
[Crossref]

See Refs. 3, 5, 6, 8, 10, 11, and also J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970).
[Crossref]

J. L. Stanford, J. Opt. Soc. Am. 60, 49 (1970).
[Crossref]

1969 (2)

D. Gloge, E. Chinnock, and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
[Crossref]

E. Kretschmann, Z. Phys. 227, 412 (1969).
[Crossref]

1968 (2)

W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968).
[Crossref] [PubMed]

CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

1967 (1)

E. A. Stern, Phys. Rev. Lett. 19, 1321 (1967).
[Crossref]

1960 (1)

L. Deryugin, Radiotekhn. 15, 9 (1960).

1954 (1)

H. Davies, Proc. IEEE 101, 209 (1954).

1951 (1)

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Archibald, P. C.

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, , ARPA Order 2175, March 1974–Sept. 1974, p. 15.

Ashely, E. J.

CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

Ballard, S.

S. Ballard, J. Browder, and J. Ebersole, in American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–12.

Beaglehole, D.

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[Crossref]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, New York, 1963).

Bennett, H. E.

CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, , ARPA Order 2175, March 1974–Sept. 1974, p. 15.

Bennett, J.

J. Bennett and M. Elson, in High Energy Laser Mirrors and Windows, , ARPA Order 2175, March 1974–Sept. 1974, p. 4.

Bennett, J. M.

CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

J. M. Bennett and J. M. Elson, High Energy Laser Mirrors and Windows, Semi-Annual Report No. 7, Mar 1975–Sep 1975, ARPA Order 2175 (1975b) (to be published).

Bertoni, H.

Braunstein, A.

A. Braunstein and M. Braunstein, J. Vac. Sci. Technol. 8, 412 (1970).
[Crossref]

Braunstein, M.

A. Braunstein and M. Braunstein, J. Vac. Sci. Technol. 8, 412 (1970).
[Crossref]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1969), p. 100.

Browder, J.

S. Ballard, J. Browder, and J. Ebersole, in American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–12.

Burstein, E.

D. Mills, A. Maradudin, and E. Burstein, Ann. Phys. (N.Y.) 56, 504 (1970).
[Crossref]

Charlton, G.

G. Charlton (private communication), Air Force Weapons Laboratory, Kirtland AFB, New Mexico. Detailed examination ThF4 overcoated rectangular groove gratings yielded good profile definition in the vicinity of the steps. The overlayer thickness was 2.7 μ m.

Chinnock, E.

D. Gloge, E. Chinnock, and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
[Crossref]

Crowell, J.

Davies, H.

H. Davies, Proc. IEEE 101, 209 (1954).

Deryugin, L.

L. Deryugin, Radiotekhn. 15, 9 (1960).

Earl, H.

D. Gloge, E. Chinnock, and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
[Crossref]

Ebersole, J.

S. Ballard, J. Browder, and J. Ebersole, in American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–12.

Elson, J. M.

J. M. Elson, Phys. Rev. B 12, 2541 (1975).
[Crossref]

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[Crossref]

J. M. Elson, Michelson Laboratory, in for Air Force Weapons Laboratory, Kirtland AFB, (Apr., 1976).

J. M. Bennett and J. M. Elson, High Energy Laser Mirrors and Windows, Semi-Annual Report No. 7, Mar 1975–Sep 1975, ARPA Order 2175 (1975b) (to be published).

Elson, M.

J. Bennett and M. Elson, in High Energy Laser Mirrors and Windows, , ARPA Order 2175, March 1974–Sept. 1974, p. 4.

Gloge, D.

D. Gloge, E. Chinnock, and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
[Crossref]

Harris, J. S.

J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in for Air Force Weapons Lab., Kirtland AFB, p. 36 (July, 1974).

J. S. Harris (private communication).

Heitmann, W.

Hunderi, O.

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[Crossref]

Juranek, H. J.

H. J. Juranek, Z. Phys. 233, 324 (1970).
[Crossref]

Kretschmann, E.

E. Kretschmann and E. Kröger, J. Opt. Soc. Am. 65, 15C (1975).
[Crossref]

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

E. Kretschmann, Z. Phys. 227, 412 (1969).
[Crossref]

Kröger, E.

E. Kretschmann and E. Kröger, J. Opt. Soc. Am. 65, 15C (1975).
[Crossref]

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

Maradudin, A.

A. Maradudin and D. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

D. Mills, A. Maradudin, and E. Burstein, Ann. Phys. (N.Y.) 56, 504 (1970).
[Crossref]

Mills, D.

A. Maradudin and D. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

D. Mills, A. Maradudin, and E. Burstein, Ann. Phys. (N.Y.) 56, 504 (1970).
[Crossref]

Motyka, R. J.

CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

Rice, S. O.

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Ritchie, R. H.

Ritter, E.

Slomba, A. F.

J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in for Air Force Weapons Lab., Kirtland AFB, p. 36 (July, 1974).

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, New York, 1963).

Stanford, J. L.

J. L. Stanford, J. Opt. Soc. Am. 60, 49 (1970).
[Crossref]

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, , ARPA Order 2175, March 1974–Sept. 1974, p. 15.

Stern, E. A.

E. A. Stern, Phys. Rev. Lett. 19, 1321 (1967).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 38.

Tamir, T.

Wolfe, W. L.

MgF2, LiF, and Al2O3 are standard optical coating materials. See W. L. Wolfe in Handbook of Military Infrared Technology, Naval Research Laboratory, p. 355 (1965) and J. L. Stanford and H. E. Bennett, Appl. Opt. 8, 2556 (1969); H. E. Bennett, J. M. Bennett, and E. J. Ashely, ibid.2, 156 (1963).
[Crossref] [PubMed]

Ann. Phys. (N.Y.) (1)

D. Mills, A. Maradudin, and E. Burstein, Ann. Phys. (N.Y.) 56, 504 (1970).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Gloge, E. Chinnock, and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
[Crossref]

Commun. Pure Appl. Math. (1)

S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

J. Opt. Soc. Am. (4)

J. Vac. Sci. Technol. (1)

A. Braunstein and M. Braunstein, J. Vac. Sci. Technol. 8, 412 (1970).
[Crossref]

Phys. Rev. (1)

CaF2 is often used to enhance surface roughness as in Ref. 22. See also H. E. Bennett, J. M. Bennett, E. J. Ashely, and R. J. Motyka, Phys. Rev. 165, 755 (1968).
[Crossref]

Phys. Rev. B (3)

A. Maradudin and D. Mills, Phys. Rev. B 11, 1392 (1975).
[Crossref]

J. M. Elson, Phys. Rev. B 12, 2541 (1975).
[Crossref]

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[Crossref]

Phys. Rev. Lett. (1)

E. A. Stern, Phys. Rev. Lett. 19, 1321 (1967).
[Crossref]

Phys. Status Solidi B (1)

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[Crossref]

Proc. IEEE (1)

H. Davies, Proc. IEEE 101, 209 (1954).

Radiotekhn. (1)

L. Deryugin, Radiotekhn. 15, 9 (1960).

Z. Phys. (3)

H. J. Juranek, Z. Phys. 233, 324 (1970).
[Crossref]

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
[Crossref]

E. Kretschmann, Z. Phys. 227, 412 (1969).
[Crossref]

Other (13)

In the case of periodic roughness the rectangular profile is considered. Even though the slope of a rectangular profile is infinite at the steps the large majority of the surface has zero slope. In fact, it may be shown that the average slope of the surface is zero.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 38.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1969), p. 100.

J. Bennett and M. Elson, in High Energy Laser Mirrors and Windows, , ARPA Order 2175, March 1974–Sept. 1974, p. 4.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, New York, 1963).

MgF2, LiF, and Al2O3 are standard optical coating materials. See W. L. Wolfe in Handbook of Military Infrared Technology, Naval Research Laboratory, p. 355 (1965) and J. L. Stanford and H. E. Bennett, Appl. Opt. 8, 2556 (1969); H. E. Bennett, J. M. Bennett, and E. J. Ashely, ibid.2, 156 (1963).
[Crossref] [PubMed]

S. Ballard, J. Browder, and J. Ebersole, in American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–12.

J. M. Elson, Michelson Laboratory, in for Air Force Weapons Laboratory, Kirtland AFB, (Apr., 1976).

G. Charlton (private communication), Air Force Weapons Laboratory, Kirtland AFB, New Mexico. Detailed examination ThF4 overcoated rectangular groove gratings yielded good profile definition in the vicinity of the steps. The overlayer thickness was 2.7 μ m.

J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in for Air Force Weapons Lab., Kirtland AFB, p. 36 (July, 1974).

J. S. Harris (private communication).

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, , ARPA Order 2175, March 1974–Sept. 1974, p. 15.

J. M. Bennett and J. M. Elson, High Energy Laser Mirrors and Windows, Semi-Annual Report No. 7, Mar 1975–Sep 1975, ARPA Order 2175 (1975b) (to be published).

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Figures (6)

FIG. 1
FIG. 1

Schematic representation of some symbols used in this work. The angle of incidence in this example is θ0 = 5° and the angles of scattering are θN. In this example the grating spacing d = 1.1λ and the θ−1 = −55° and θ+1 = 85° diffraction angles are shown. The θ+1 (forward scattering) is defined to be positive and the θ−1 (back scattering) is defined as negative.

FIG. 2
FIG. 2

Schematic representation of a grating having a rectangular profile with a replicating dielectric layer. The substrate has dielectric constant , whereas the overlayer has physical thickness τ and is characterized by dielectric constant 0. The ratio of groove width to grating spacing in this example is A = 0.67. Also, the groove depth is denoted by H (not shown).

FIG. 3
FIG. 3

Predicted variation of the R(−1) ratio (p-polarized to s-polarized light scattered in the N = −1 order) versus the normalized grating spacing d/λ. The theoretical plot is given for a bare surface and the circles represent data taken by Harris and Slomba (Ref. 27) from ion etched rectangular groove substrates with gold overcoats. The incident angle and wavelength are 40° and 10.6 μm.

FIG. 4
FIG. 4

Predicted variation of the p-polarized scattering efficiency P p ( - 1 ), the s-polarized scattering efficiency P s ( - 1 ), and the polarization ratio R(−1) (all in the N = −1 order) vs the angle of incidence in degrees. The factor H/λ is the normalized groove depth where λ = 10.6 μm. The gratings have a ThF4 overcoat of normalized physical thickness τ/λ = 0.02358. Three values of d/λ are given where the d/λ = 0.799 grating is a triangular groove ruled in gold whereas the d/λ = 0.958 and 1.597 gratings are rectangular groove ion etched substrates with gold overcoats. Comparison of experiment and theory for R(−1) is made. The experimental data are obtained from Harris and Slomba (Ref. 27).

FIG. 5
FIG. 5

Predicted variation of polarization ratio versus energy is shown. The angle of incidence is θ0 = 45° and the scattering surface is uncoated Al in the bare surface plots and Al2O3/Al in the oxide surface plots. The Al2O3 physical thickness is τ = 30 Å. The lower set of curves represents the polarization ratio in the specular beam whereas the upper set of curves represents the polarization ratio in the θ = 30° backscattering direction.

FIG. 6
FIG. 6

Predicted variation of the scattering probability into solid angle dΩ per unit scattering factor (dP/dΩ)/δ2g (kk0) vs the angle of scattering θ, in deg, is shown. Various values of physical thicknesses τ are assumed and the angle of incidence is 45°. One set of curves corresponds to scattering from an Al substrate with a MgF2 dielectric layer while the other set is for a Ag substrate and LiF dielectric layer. The incident wavelength for the MgF2/Al and LiF/Ag curves are 5461 and 6328 Å, respectively. The planes of incidence and scattering coincide. Note that the surface scattering factor g(kk0) does not enter into these plots.

Equations (73)

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

ê 1 = x ˆ + ζ 1 z ˆ ,
ê 2 = y ˆ + ζ 2 z ˆ ,
ê 3 = z ˆ ,
× × A ( r , t ) - [ z - ζ ( x , y ) ] ( ω / c ) 2 A ( r , t ) = 0 ,
× × - [ z - ζ ( x , y ) ] ( ω / c ) 2 = L ,
( L ( 0 ) + L ( 1 ) ) A = 0 ,
L ( 0 ) = { [ - ( 2 u 2 2 + 2 u 3 2 ) - ( u 3 ) ω 2 c 2 ] [ 2 u 2 u 3 ] [ 2 u 3 u 1 ] [ 2 u 1 u 2 ] [ - ( 2 u 1 2 + 2 u 3 2 ) - ( u 3 ) ω 2 c 2 ] [ 2 u 3 u 2 ] [ 2 u 1 u 3 ] [ 2 u 2 u 3 ] [ - ( 2 u 1 2 + 2 u 2 2 ) - ( u 3 ) ω 2 c 2 ] } .
L 11 ( 1 ) = ζ 1 2 u 3 u 1 + 2 ζ 2 2 u 3 u 2 + ( ζ 1 u 1 + ζ 2 u 2 ) u 3 ,
L 12 ( 1 ) = ζ 1 2 u 3 u 2 ,
L 13 ( 1 ) = - ζ 1 2 u 3 2 ,
L 21 ( 1 ) = - ζ 2 2 u 3 u 1 ,
L 22 ( 1 ) = ζ 2 2 u 3 u 2 + 2 ζ 1 2 u 3 u 1 + ( ζ 1 u 1 + ζ 2 u 2 ) u 3 ,
L 23 ( 1 ) = - ζ 2 2 u 3 2 ,
L 31 ( 1 ) = - ( 2 ζ 1 u 1 2 + 2 ζ 1 u 2 2 ) - 2 ( ζ 1 u 1 u 1 + ζ 1 u 2 u 2 ) - ( ζ 1 2 u 1 2 + ζ 2 2 u 2 u 1 ) ,
L 32 ( 1 ) = - ( 2 ζ 2 u 1 2 2 ζ 2 u 2 2 ) - 2 ( ζ 2 u 2 u 2 + ζ 2 u 1 u 1 ) - ( ζ 1 2 u 1 u 2 + ζ 2 2 u 2 2 ) ,
L 33 ( 1 ) = ( ζ 1 u 1 + ζ 2 u 2 ) u 3 + ζ 1 2 u 3 u 1 + ζ 2 2 u 3 u 2 .
G ( u , t ; u , t ) = ( 2 π ) - 3 d 2 k d ω g ( u 3 , u 3 ) × e i [ k · ( ρ - ρ ) - ω ( t - t ) ] ,
g 11 ( m , v ) = - 2 ν η η 0 0 e η ( τ - u 3 ) e ν u 3 ( ω / c ) 2 ϕ ( τ ) ,
g 31 ( m , v ) = - 2 i k ν η 0 0 e η ( τ - u 3 ) e ν u 3 ( ω / c ) 2 ϕ ( τ ) ,
g 33 ( m , v ) = 2 η 0 k 2 0 e η ( τ - u 3 ) e ν u 3 ( ω / c ) 2 ϕ ( τ ) ,
g 13 ( m , v ) = - 2 i k η η 0 0 e η ( τ - u 3 ) e ν u 3 ( ω / c ) 2 ϕ ( τ ) ,
g 22 ( m , ν ) = 2 η 0 e η ( τ - u 3 ) e ν u 3 β ( τ ) ;
g 11 ( d , v ) = - η η 0 Γ + ( u 3 ) e η ( τ - u 3 ) ( ω / c ) 2 ϕ ( τ ) ,
g 31 ( d , v ) = - i k η 0 Γ + ( u 3 ) e η ( τ - u 3 ) ( ω / c ) 2 ϕ ( τ ) ,
g 33 ( d , v ) = k 2 Γ - ( u 3 ) e η ( τ - u 3 ) ( ω / c ) 2 ϕ ( τ ) ,
g 13 ( d , v ) = - i k η Γ - ( u 3 ) e η ( τ - u 3 ) ( ω / c ) 2 ϕ ( τ ) ,
g 22 ( d , v ) = K - ( u 3 ) e η ( τ - u 3 ) β ( τ ) ;
g 11 ( v , v ) = - η 2 ( ω / c ) 2 ( e - η u 3 - u 3 + ψ ( τ ) ϕ ( τ ) e 2 η τ e - η ( u 3 + u 3 ) ) ,
g 31 ( v , v ) = i k 2 ( ω / c ) 2 ( ( u 3 - u 3 ) u 3 - u 3 e - η u 3 - u 3 + ψ ( τ ) ϕ ( τ ) e 2 η τ e - η ( u 3 + u 3 ) ) ,
g 33 ( v , v ) = k 2 2 η ( ω / c ) 2 ( e - η u 3 - u 3 - ψ ( τ ) ϕ ( τ ) e 2 η τ e - η ( u 3 + u 3 ) ) ,
g 13 ( v , v ) = - i k 2 ( ω / c ) 2 ( ( u 3 - u 3 ) u 3 - u 3 e - η u 3 - u 3 - ψ ( τ ) ϕ ( τ ) e 2 η τ e - η ( u 3 + u 3 ) ) ,
g 22 ( u , v ) = 1 2 η ( e - η u 3 - u 3 - α ( τ ) β ( τ ) e 2 η τ e - η ( u 3 + u 3 ) ) ;
Γ ± ( τ ) = ( ν 0 + η 0 ) e η 0 τ ± ( ν 0 - η 0 ) e - η 0 τ ,
K ± ( τ ) = ( ν + η 0 ) e η 0 τ ± ( ν - η 0 ) e - η 0 τ ,
ϕ ( τ ) = ( ν 0 + η 0 ) ( η 0 + η 0 ) e η 0 τ + ( ν 0 - η 0 ) ( η 0 - η 0 ) e - η 0 τ ,
ψ ( τ ) = ( ν 0 + η 0 ) ( η 0 - η 0 ) e η 0 τ + ( ν 0 - η 0 ) ( η 0 + η 0 ) e - η 0 τ ,
β ( τ ) = ( η + η 0 ) ( ν + η 0 ) e η 0 τ + ( η 0 - η ) ( ν - η 0 ) e - η 0 τ ,
α ( τ ) = ( η 0 + η ) ( ν + η 0 ) e η 0 τ + ( η + η 0 ) ( ν - η 0 ) e - η 0 τ .
H 1 ( 1 ) = A 3 ( 1 ) u 2 - A 2 ( 1 ) u 3 + u 2 ( ζ 1 A 1 ( 0 ) + ζ 2 A 2 ( 0 ) ) - ζ 2 A 3 ( 0 ) u 3 ,
H 2 ( 1 ) = A 1 ( 1 ) u 3 - A 3 ( 1 ) u 1 - u 1 ( ζ 1 A 1 ( 0 ) + ζ 2 A 2 ( 0 ) ) + ζ 1 A 3 ( 0 ) u 3 ,
H 3 ( 1 ) = A 2 ( 1 ) u 1 - A 1 ( 1 ) u 2 + ζ 2 A 3 ( 0 ) u 1 - ζ 1 A 3 ( 0 ) u 2 .
A ( 1 ) ( u , t ) = - d 3 u d t G ( u , t ; u , t ) L ( 1 ) · A ( 0 ) ( u , t ) .
[ 2 u 1 2 + 2 u 2 2 + 2 u 3 3 + ( u 3 ) ( ω c ) 2 ] A ( 0 ) = 0.
A I ( 0 ) ( u , t ) = ( 2 π ) - 3 d 2 k d ω a k , ω ( 0 ) ( u 3 ) e i ( k · ρ - ω t ) ,
A v ( u , t ) = I e - i q τ ( ( k ˆ 0 cos θ 0 + ê 3 sin θ 0 ) e i q ( τ - u 3 ) + ( k ˆ 0 cos θ 0 - ê 3 sin θ 0 ) e - i q ( τ - u 3 ) ψ 0 ( τ ) ϕ 0 ( τ ) ) e i ( k 0 · ρ - ω 0 t ) ,
A d ( u , t ) = 2 I cos θ 0 e - i q τ ϕ 0 ( τ ) { k ˆ 0 q 0 Γ 0 + ( u 3 ) + ê 3 k 0 Γ 0 - ( u 3 ) } × ê i ( k 0 · ρ - ω 0 t ) ,
A m ( u , t ) = 4 I 0 q 0 e - i q τ ϕ 0 ( τ ) cos θ 0 ( k ˆ 0 ν 0 - i k 0 ê 3 ) e i ( k 0 · ρ - ω 0 t ) e ν 0 u 3 ,
ϕ 0 ( τ ) = ( ν 0 0 - i q 0 ) ( q 0 + q 0 ) e - i q 0 τ + ( ν 0 0 + i q 0 ) ( q 0 - q 0 ) e i q 0 τ ,
ψ 0 ( τ ) = ( ν 0 0 - i q 0 ) ( q 0 - q 0 ) e - i q 0 τ + ( ν 0 0 + i q 0 ) ( q 0 + q 0 ) e i q 0 τ ,
Γ 0 ± ( u 3 ) = ( ν 0 0 - i q 0 ) e - i q 0 u 3 ± ( ν 0 0 + i q 0 ) e i q 0 u 3 .
A v ( u , t ) = I ê 2 e - i q τ ( e i q ( τ - u 3 ) - α 0 ( τ ) β 0 ( τ ) e - i q ( τ - u 3 ) ) e i ( k 0 · ρ - ω 0 t ) ,
A d ( u , t ) = 2 e 2 I q e - i q τ β 0 ( τ ) K 0 - ( u 3 ) e i ( k 0 · ρ - ω 0 t ) ,
A m ( u , t ) = - 4 ê 2 i I q q 0 e - i q τ β 0 ( τ ) e ν 0 u 3 e i ( k 0 · ρ - ω 0 t ) ,
β 0 ( τ ) = ( q + q 0 ) ( ν 0 - i q 0 ) e - i q 0 τ + ( q 0 - q ) ( ν 0 + i q 0 ) e i q 0 τ ,
α 0 ( τ ) = ( q 0 - q ) ( ν 0 - i q 0 ) e - i q 0 τ + ( q + q 0 ) ( ν 0 + i q 0 ) e - i q 0 τ ,
K 0 ± ( u 3 ) = ( ν 0 - i q 0 ) e - i q 0 u 3 ± ( ν 0 + i q 0 ) e i q 0 u 3 .
ζ k - k 0 = ζ ( u 1 , u 2 ) e - i ( k 0 - k ) · ρ d 2 ρ ,
ζ ( u 1 ) = N = - 1 2 C N e i 2 π N u 1 / d ,
ζ k - k 0 2 L 2 = π 2 N = - C N 2 δ ( k 2 ) δ ( k 1 - k 0 - 2 π N d ) ,
sin θ N = sin θ 0 + N / D .
g ( k - k 0 ) = - L 1 L 1 d τ 1 - L 2 L 2 d τ 2 G ( τ ) cos ( k - k 0 ) · τ .
S ¯ T = ( c / 8 π ) Re ( ( 0 ) × H ( 0 ) * + ( 1 ) × H ( 1 ) * ) ,
A ϕ ( 1 ) = 2 I ( 2 π ) - 2 e - i ω 0 t ( ω 0 c ) cos θ 0 d 2 k e i k · ρ e - i q τ e η ( τ - u 3 ) ζ k - k 0 β ( τ ) × ( B θ sin ϕ cos ϕ ϕ 0 ( τ ) + i B ϕ cos ϕ sin ϕ β 0 ( τ ) )
A θ ( 1 ) = 2 I ( 2 π ) - 2 e - i ω 0 t cos θ 0 × d 2 k e i k · ρ e - i q τ e η ( τ - u 3 ) ζ k - k 0 ( η cos θ - i k sin θ ) ϕ ( τ ) × ( P θ ϕ 0 ( τ ) cos ϕ + i P ϕ sin ϕ sin ϕ β 0 ( τ ) ) ,
B θ = q 0 ( 4 η 0 ν 0 0 ( 0 - ) + ( 1 - 0 ) Γ 0 + K - ) ( ω 0 / c ) ,
B ϕ = ( ω 0 / c ) 2 [ 4 η 0 q 0 ( 0 - ) + i K - K 0 ( 1 - 0 ) ] ,
P θ = 4 η 0 q 0 0 ( 0 - ) ( k k 0 + ν ν 0 0 cos ϕ ) + ( i k k 0 0 Γ - Γ 0 - + η 0 q 0 Γ + Γ 0 + cos ϕ ) ( 1 - 0 ) ,
P ϕ = - η 0 ( ω 0 / c ) [ 4 q 0 0 ν ( 0 - ) + i ( 1 - 0 ) Γ + K 0 - ] .
d P d Ω = cos θ 0 ( ω 0 / c ) 4 cos 2 θ π 2 ζ k - k 0 2 L 2 × ( B θ sin ϕ cos ϕ / ϕ 0 ( τ ) + i B ϕ cos ϕ sin ϕ / β 0 ( τ ) 2 β ( τ ) 2 + P θ cos ϕ / ϕ 0 ( τ ) + i P ϕ sin ϕ sin ϕ / β 0 ( τ ) 2 ϕ ( τ ) 2 ) ,
S ¯ ( 1 ) = ( c / 8 π ) Re ( g ( 1 ) × H g * + r ( 1 ) × H r ( 1 ) * + g ( 1 ) × H r ( 1 ) * + r ( 1 ) × H g ( 1 ) * ) .
P ϕ ( N ) = ( ω 0 c ) 2 cos θ 0 cos θ N C N 2 × ( B ϕ 2 sin 2 ϕ β 0 ( τ ) 2 β ( τ ) 2 + P θ N 2 cos 2 ϕ ϕ 0 ( τ ) 2 ϕ ( τ ) 2 ) ,
R ( N ) = P θ N 2 β 0 ( τ ) 2 β ( τ ) 2 B ϕ 2 ϕ 0 ( τ ) 2 ϕ ( τ ) 2 .
C N 2 = { [ 2 H sin ( N A π ) ] / π N } 2 ,