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 ε and dielectric overlayer ε<sub>0</sub>. The frequency range of interest is such that Re(ε) < 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).
  3. H. Davies, Proc. IEEE 101, 209 (1954).
  4. H. J. Juranek, Z. Phys. 233, 324 (1970).
  5. E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
  6. E. Kretschmann, Z. Phys. 227, 412 (1969).
  7. E. A. Stern, Phys. Rev. Lett. 19, 1321 (1967).
  8. D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
  9. J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
  10. E. Kretschmann and E. Kröger, J. Opt. Soc. Am. 65, 15C (1975).
  11. A. Maradudin and D. Mills, Phys. Rev. B 11, 1392 (1975).
  12. J. M. Elson, Phys. Rev. B 12, 2541 (1975).
  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).
  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, Semiannual Report No. 5, ARPA Order 2175, March 1974-Sept. 1974, p. 4.
  19. D. Gloge, E. Chinnock and H. Earl, Bell Syst. Tech. J. 48, 511 (1969).
  20. See Refs. 3, 5, 6, 8, 10, 11, and also J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970).
  21. J. M. Elson, Michelson Laboratory, in Technical Report No. AFWL-TR-75-210 for Air Force Weapons Laboratory, Kirtland AFB, (Apr., 1976).
  22. J. L. Stanford, J. Opt. Soc. Am. 60, 49 (1970).
  23. T. Tamir and H. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971); J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 538 (1970).
  24. A Braunstein and M. Braunstein, J. Vac. Sci. Technol. 8, 412 (1970).
  25. W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968).
  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 µm.
  27. J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in Techinal Report No. AFWL-TR-74-218 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, Semiannual Report No. 3, 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).
  32. MgF2, LiF, and A12O3 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).
  33. S. Ballard, J. Browder, and J. Ebersole, in American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–12.

Archibald, P. C.

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, Semiannual Report No. 3, 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).

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).

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).

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, Semiannual Report No. 3, ARPA Order 2175, March 1974–Sept. 1974, p. 15.

Bennett, J.

J. Bennett and M. Elson, in High Energy Laser Mirrors and Windows, Semiannual Report No. 5, 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).

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.

T. Tamir and H. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971); J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 538 (1970).

Braunstein, A

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

Braunstein, M.

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

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).

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 µm.

Chinnock, E.

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

Crowell, J.

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

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).

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. 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).

J. M. Elson, Michelson Laboratory, in Technical Report No. AFWL-TR-75-210 for Air Force Weapons Laboratory, Kirtland AFB, (Apr., 1976).

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

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

Elson, M.

J. Bennett and M. Elson, in High Energy Laser Mirrors and Windows, Semiannual Report No. 5, 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).

Harris, J. S.

J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in Techinal Report No. AFWL-TR-74-218 for Air Force Weapons Lab., Kirtland AFB, p. 36 (July, 1974).

J. S. Harris (private communication).

Heitmann, W.

W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968).

Hunderi, O.

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

Juranek, H. J.

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

Kretschmann, E.

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

E. Kretschmann, Z. Phys. 227, 412 (1969).

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

Kröger, E.

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

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

Maradudin, A.

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

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

Mills, D.

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

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

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).

Rice, S. O.

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

Ritchie, R. H.

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

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

Ritter, E.

W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968).

Slomba, A. F.

J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in Techinal Report No. AFWL-TR-74-218 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).

H. E. Bennett, P. C. Archibald, and J. L. Stanford, in High Energy Laser Mirrors and Window, Semiannual Report No. 3, ARPA Order 2175, March 1974–Sept. 1974, p. 15.

Stern, E. A.

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

Stratton, J. A.

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

Tamir, T.

T. Tamir and H. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971); J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 538 (1970).

Wolfe, W. L.

MgF2, LiF, and A12O3 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).

Other (33)

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.

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

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

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

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

E. Kretschmann, Z. Phys. 227, 412 (1969).

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

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

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

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

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

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

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

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

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

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

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, Semiannual Report No. 5, ARPA Order 2175, March 1974-Sept. 1974, p. 4.

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

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

J. M. Elson, Michelson Laboratory, in Technical Report No. AFWL-TR-75-210 for Air Force Weapons Laboratory, Kirtland AFB, (Apr., 1976).

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

T. Tamir and H. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971); J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 538 (1970).

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

W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968).

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 µm.

J. S. Harris and A. F. Slomba, Perlin-Elmer Corp. in Techinal Report No. AFWL-TR-74-218 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, Semiannual Report No. 3, 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).

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).

MgF2, LiF, and A12O3 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).

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

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