Abstract

Ellipsometry measurements on several different kinds of rough surfaces were compared with stylus measurements of the surface texture. For steeply sloped periodic surfaces, the ellipsometric angles Δ and ψ varied rapidly as the angle of incidence was varied near a diffraction minimum. This effect is interpreted in terms of the Kirchhoff theory and is ascribed to interference between singly and doubly reflected light waves. For a set of Ni replicas of machined surfaces with random surface profiles, Δ and ψ varied systematically with surface texture. These variations persisted even after the surface composition was changed by evaporating first Al then Au on the surfaces. The systematic effects due to surface roughness are in disagreement with those of a previous experiment and are not readily explainable in terms of the Kirchhoff theory. The possible reasons for this are discussed along with the prospects for using ellipsometry as a tool for measuring surface roughness.

© 1980 Optical Society of America

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References

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  1. For an extensive review of the subject see R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
    [CrossRef]
  3. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
    [CrossRef]
  4. Ref. 1, Chap. 4.
  5. J. Kruger, P. C. S. Hayfield, in Handbook on Corrosion Testing and Evaluation, W. H. Ailor, Ed. (Wiley, New York, 1971), p. 783.
  6. Ref. 1, p. 419.
  7. K. Vedam, Surf. Sci. 56, 221 (1976).
    [CrossRef]
  8. F. Meyer, E. E. Kluizenaar, D. den Engelsen, J. Opt. Soc. Am. 63, 529 (1973).
    [CrossRef]
  9. Y. J. Van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
    [CrossRef]
  10. H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
    [CrossRef]
  11. W. Primak, Surf. Sci. 16, 398 (1969).
    [CrossRef]
  12. K. Vedam, S. S. So, Surf. Sci. 29, 379 (1972).
    [CrossRef]
  13. M. M. Ibrahim, N. M. Bashara, Sur. Sci. 30, 632 (1972).
    [CrossRef]
  14. J. M. Elson, J. M. Bennett, J. Opt. Soc. Am. 69, 31 (1979).
    [CrossRef]
  15. C. A. Fenstermaker, F. L. McCrackin, Surf. Sci. 16, 85 (1969).
    [CrossRef]
  16. I. Ohlidal, F. Lukes, Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
    [CrossRef]
  17. F. L. McCrackin, Natl. Bur. Stand. U.S. Tech. Note 479 (1969).
  18. E. L. Church, J. M. Zavada, J. Opt. Soc. Am. 66, 1136A (1976); Appl. Opt. 14, 1788 (1975).
    [PubMed]
  19. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3.
  20. T. Smith, Surf. Sci. 56, 252(1976).
    [CrossRef]
  21. P. M. Lonardo, Ann. CIRP 23, 189 (1974); P. Chiesorin, P. M. Lonardo, Tech. Pap. Soc. Manuf. Eng. MS77–216 (1977); P. M. Lonardo, Ann. CIRP 27, 531 (1978).
  22. For a discussion of surface roughness parameters see American National Standards Institute B46.1-1978, Surface Texture (American Society of Mechanical Engineers, New York, 1978), p. 27.
  23. T. Smith, G. Lindberg, Surf. Technol. 8, 1 (1979).
    [CrossRef]
  24. R. M. A. Azzam, N. M. Bashara, Phys. Rev. B: 58, 4721 (1972).
    [CrossRef]
  25. A similar system is discussed by detail by E. C. Teague, Natl. Bur. Stand. U.S. Tech. Note 902 (1976).
  26. The present system is discussed briefly by T. V. Vorburger, E. C. Teague, F. E. Scire, Dimensions/NBS 62, 18 (1978).
  27. J. Peklenik, Proc. Inst. Mech. Eng. London Part 3 182, 108 (1967).
    [CrossRef]
  28. See, for example, D. J. Whitehouse, J. F. Archard, Proc. R. Soc. London Ser. A: 316, 97 (1970).
    [CrossRef]
  29. J. Peklenik, Ann. CIRP 15, 381 (1967).
  30. M. C. Hutley, V. M. Bird, Opt. Acta 20, 771 (1973).
    [CrossRef]
  31. V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
    [CrossRef]
  32. A. Hessel, A. A. Oliner, Appl. Opt. 4, 1275 (1965).
    [CrossRef]
  33. Ref. 19, Chap. 5.
  34. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), p. 161.
  35. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 435.
  36. Ref. 34, p. 74.
  37. Ref. 19, Chap. 4.

1979 (2)

1978 (1)

The present system is discussed briefly by T. V. Vorburger, E. C. Teague, F. E. Scire, Dimensions/NBS 62, 18 (1978).

1976 (4)

A similar system is discussed by detail by E. C. Teague, Natl. Bur. Stand. U.S. Tech. Note 902 (1976).

E. L. Church, J. M. Zavada, J. Opt. Soc. Am. 66, 1136A (1976); Appl. Opt. 14, 1788 (1975).
[PubMed]

T. Smith, Surf. Sci. 56, 252(1976).
[CrossRef]

K. Vedam, Surf. Sci. 56, 221 (1976).
[CrossRef]

1974 (2)

Y. J. Van der Meulen, N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
[CrossRef]

P. M. Lonardo, Ann. CIRP 23, 189 (1974); P. Chiesorin, P. M. Lonardo, Tech. Pap. Soc. Manuf. Eng. MS77–216 (1977); P. M. Lonardo, Ann. CIRP 27, 531 (1978).

1973 (2)

1972 (4)

K. Vedam, S. S. So, Surf. Sci. 29, 379 (1972).
[CrossRef]

M. M. Ibrahim, N. M. Bashara, Sur. Sci. 30, 632 (1972).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Phys. Rev. B: 58, 4721 (1972).
[CrossRef]

I. Ohlidal, F. Lukes, Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

1970 (1)

See, for example, D. J. Whitehouse, J. F. Archard, Proc. R. Soc. London Ser. A: 316, 97 (1970).
[CrossRef]

1969 (4)

F. L. McCrackin, Natl. Bur. Stand. U.S. Tech. Note 479 (1969).

C. A. Fenstermaker, F. L. McCrackin, Surf. Sci. 16, 85 (1969).
[CrossRef]

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
[CrossRef]

W. Primak, Surf. Sci. 16, 398 (1969).
[CrossRef]

1967 (2)

J. Peklenik, Ann. CIRP 15, 381 (1967).

J. Peklenik, Proc. Inst. Mech. Eng. London Part 3 182, 108 (1967).
[CrossRef]

1965 (1)

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
[CrossRef]

1962 (2)

Archard, J. F.

See, for example, D. J. Whitehouse, J. F. Archard, Proc. R. Soc. London Ser. A: 316, 97 (1970).
[CrossRef]

Archer, R. J.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Phys. Rev. B: 58, 4721 (1972).
[CrossRef]

For an extensive review of the subject see R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

M. M. Ibrahim, N. M. Bashara, Sur. Sci. 30, 632 (1972).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Phys. Rev. B: 58, 4721 (1972).
[CrossRef]

For an extensive review of the subject see R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3.

Bennett, J. M.

Bird, V. M.

M. C. Hutley, V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

Church, E. L.

E. L. Church, J. M. Zavada, J. Opt. Soc. Am. 66, 1136A (1976); Appl. Opt. 14, 1788 (1975).
[PubMed]

den Engelsen, D.

Elson, J. M.

Fenstermaker, C. A.

C. A. Fenstermaker, F. L. McCrackin, Surf. Sci. 16, 85 (1969).
[CrossRef]

Hayfield, P. C. S.

J. Kruger, P. C. S. Hayfield, in Handbook on Corrosion Testing and Evaluation, W. H. Ailor, Ed. (Wiley, New York, 1971), p. 783.

Hessel, A.

Hien, N. C.

Hutley, M. C.

M. C. Hutley, V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

Ibrahim, M. M.

M. M. Ibrahim, N. M. Bashara, Sur. Sci. 30, 632 (1972).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 435.

Kinosita, K.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
[CrossRef]

Kluizenaar, E. E.

Kruger, J.

J. Kruger, P. C. S. Hayfield, in Handbook on Corrosion Testing and Evaluation, W. H. Ailor, Ed. (Wiley, New York, 1971), p. 783.

Lindberg, G.

T. Smith, G. Lindberg, Surf. Technol. 8, 1 (1979).
[CrossRef]

Lonardo, P. M.

P. M. Lonardo, Ann. CIRP 23, 189 (1974); P. Chiesorin, P. M. Lonardo, Tech. Pap. Soc. Manuf. Eng. MS77–216 (1977); P. M. Lonardo, Ann. CIRP 27, 531 (1978).

Lukes, F.

I. Ohlidal, F. Lukes, Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

McCrackin, F. L.

F. L. McCrackin, Natl. Bur. Stand. U.S. Tech. Note 479 (1969).

C. A. Fenstermaker, F. L. McCrackin, Surf. Sci. 16, 85 (1969).
[CrossRef]

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
[CrossRef]

Meyer, F.

Nishibori, M.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
[CrossRef]

Ohlidal, I.

I. Ohlidal, F. Lukes, Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

Oliner, A. A.

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
[CrossRef]

Peklenik, J.

J. Peklenik, Proc. Inst. Mech. Eng. London Part 3 182, 108 (1967).
[CrossRef]

J. Peklenik, Ann. CIRP 15, 381 (1967).

Primak, W.

W. Primak, Surf. Sci. 16, 398 (1969).
[CrossRef]

Sakata, H.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
[CrossRef]

Scire, F. E.

The present system is discussed briefly by T. V. Vorburger, E. C. Teague, F. E. Scire, Dimensions/NBS 62, 18 (1978).

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), p. 161.

Smith, T.

T. Smith, G. Lindberg, Surf. Technol. 8, 1 (1979).
[CrossRef]

T. Smith, Surf. Sci. 56, 252(1976).
[CrossRef]

So, S. S.

K. Vedam, S. S. So, Surf. Sci. 29, 379 (1972).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3.

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
[CrossRef]

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
[CrossRef]

Teague, E. C.

The present system is discussed briefly by T. V. Vorburger, E. C. Teague, F. E. Scire, Dimensions/NBS 62, 18 (1978).

A similar system is discussed by detail by E. C. Teague, Natl. Bur. Stand. U.S. Tech. Note 902 (1976).

Twersky, V.

Van der Meulen, Y. J.

Vedam, K.

K. Vedam, Surf. Sci. 56, 221 (1976).
[CrossRef]

K. Vedam, S. S. So, Surf. Sci. 29, 379 (1972).
[CrossRef]

Vorburger, T. V.

The present system is discussed briefly by T. V. Vorburger, E. C. Teague, F. E. Scire, Dimensions/NBS 62, 18 (1978).

Whitehouse, D. J.

See, for example, D. J. Whitehouse, J. F. Archard, Proc. R. Soc. London Ser. A: 316, 97 (1970).
[CrossRef]

Yokota, H.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
[CrossRef]

Zavada, J. M.

E. L. Church, J. M. Zavada, J. Opt. Soc. Am. 66, 1136A (1976); Appl. Opt. 14, 1788 (1975).
[PubMed]

Ann. CIRP (2)

P. M. Lonardo, Ann. CIRP 23, 189 (1974); P. Chiesorin, P. M. Lonardo, Tech. Pap. Soc. Manuf. Eng. MS77–216 (1977); P. M. Lonardo, Ann. CIRP 27, 531 (1978).

J. Peklenik, Ann. CIRP 15, 381 (1967).

Appl. Opt. (1)

Dimensions/NBS (1)

The present system is discussed briefly by T. V. Vorburger, E. C. Teague, F. E. Scire, Dimensions/NBS 62, 18 (1978).

J. Opt. Soc. Am. (6)

J. Res. Natl Bur. Stand. Sect. A: (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, J. Res. Natl Bur. Stand. Sect. A: 67, 363 (1963).
[CrossRef]

Natl. Bur. Stand. U.S. Tech. Note (2)

F. L. McCrackin, Natl. Bur. Stand. U.S. Tech. Note 479 (1969).

A similar system is discussed by detail by E. C. Teague, Natl. Bur. Stand. U.S. Tech. Note 902 (1976).

Opt. Acta (2)

M. C. Hutley, V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

I. Ohlidal, F. Lukes, Opt. Acta 19, 817 (1972); I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

Phys. Rev. B (1)

R. M. A. Azzam, N. M. Bashara, Phys. Rev. B: 58, 4721 (1972).
[CrossRef]

Proc. Inst. Mech. Eng. London Part 3 (1)

J. Peklenik, Proc. Inst. Mech. Eng. London Part 3 182, 108 (1967).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

See, for example, D. J. Whitehouse, J. F. Archard, Proc. R. Soc. London Ser. A: 316, 97 (1970).
[CrossRef]

Sur. Sci. (1)

M. M. Ibrahim, N. M. Bashara, Sur. Sci. 30, 632 (1972).
[CrossRef]

Surf. Sci. (6)

K. Vedam, Surf. Sci. 56, 221 (1976).
[CrossRef]

C. A. Fenstermaker, F. L. McCrackin, Surf. Sci. 16, 85 (1969).
[CrossRef]

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, Surf. Sci. 16, 265 (1969).
[CrossRef]

W. Primak, Surf. Sci. 16, 398 (1969).
[CrossRef]

K. Vedam, S. S. So, Surf. Sci. 29, 379 (1972).
[CrossRef]

T. Smith, Surf. Sci. 56, 252(1976).
[CrossRef]

Surf. Technol. (1)

T. Smith, G. Lindberg, Surf. Technol. 8, 1 (1979).
[CrossRef]

Other (11)

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3.

For a discussion of surface roughness parameters see American National Standards Institute B46.1-1978, Surface Texture (American Society of Mechanical Engineers, New York, 1978), p. 27.

Ref. 19, Chap. 5.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), p. 161.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 435.

Ref. 34, p. 74.

Ref. 19, Chap. 4.

Ref. 1, Chap. 4.

J. Kruger, P. C. S. Hayfield, in Handbook on Corrosion Testing and Evaluation, W. H. Ailor, Ed. (Wiley, New York, 1971), p. 783.

Ref. 1, p. 419.

For an extensive review of the subject see R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of the ellipsometer. The surface normal is constructed with respect to the average plane of the surface. The plane of incidence is determined by the surface normal and the direction of the incident light ray. For measurements discussed in this paper, the detector was located in the plane of incidence and the roughness lay direction was either parallel or perpendicular to the plane of incidence. All but a few measurements were made on the specular reflected beam, i.e., θ2 = θ1.

Fig. 2
Fig. 2

Calculations for Δ as a function of θ1 on Si with several values of rms slope Sr and several values of SiO2 film thickness t: (a) Sr = 0, t = 0; (b) Sr = 0, t = 40 Å; (c) Sr = 0, t = 80 Å; (d) Sr = 0.04, t = 0; (e) Sr = 0.10, t = 0.

Fig. 3
Fig. 3

Calculations for ψ as a function of θ1 on Si. (a), (b), (c), and (e) are the same as for Fig. 2.

Fig. 4
Fig. 4

Roughness profiles of the five periodic surfaces studied in this experiment: (a) PRS, Ra = 0.49 μm; (b) PRS, Ra = 3.07 μm; (c) sinusoidal specimen, Ra = 0.3 μm; (d) sinusoidal specimen, Ra = 1.0 μm; (e) sinusoidal specimen, Ra = 3.0 μm. The two roughness patches on the PRS have nominally triangular profiles, but the peaks and valleys are significantly rounded off for the 0.49-μm patch.

Fig. 5
Fig. 5

Amplitude density function (ADF) for the 32G|| surface orientation. The curve is the average of five such curves measured at five distributed positions on the surface.

Fig. 6
Fig. 6

Autocorrelation function (ACF) for the 32G || surface. The curve is the average of five such curves measured at distributed positions on the surface.

Fig. 7
Fig. 7

Δ vs θ1 for the PRS: ★ – patch with Ra = 0.49 μm; ▲ – patch with Ra = 3 μm; ○ – smooth patch.

Fig. 8
Fig. 8

ψ vs θ1 for the PRS; symbols are the same as in Fig. 7.

Fig. 9
Fig. 9

P results vs scattering angle, θ1 + σ2, for the sinusoidal specimens. θ1 was held constant at ~72°, and P and A were measured for various diffraction spots. The specular spot for each surface occurs at θ1 + θ2 144°: ● – surface with 3-μm Ra; ◆ – surface with 1-μm Ra; ▲ – surface with 0.3-μm Ra.

Fig. 10
Fig. 10

A results vs θ1 + θ2 for the sinusoidal specimens: ● - surface with 3-μm Ra; ■ - surface with 1-μm Ra; ▲ - surface with 0.3-μm Ra

Fig. 11
Fig. 11

Δ vs mean square slope S r 2 for various patches of the roughness comparison standard. || (⊥) means that the direction of maximum slopes was oriented parallel (perpendicular) to the plane of incidence. The nine measurements were taken as shown for four different sets of experimental conditions: (a) λ = 5461 Å, θ1 = 65°, Ni surface; (b) λ = 6328 Å, θ1 = 60°, Ni surface; (c) λ = 6328 Å, θ1 = 65°, 200-Å Al evaporated on Ni surface; (d) λ = 6328 Å, θ1 = 65°, 1500-Å Au evaporated on top of 200-Å Al on Ni surface. Curves (b), (c), and (d) have been shifted arbitrarily for clarity. The dotted line shows the calculation based on the theory of Ohlidal and Lukes for the conditions of data set b.

Fig. 12
Fig. 12

ψ vs mean square slope for the roughness comparison standard. Same conditions as in Fig. 11.

Fig. 13
Fig. 13

Schematic diagram showing the relevant quantities for optical scattering from a rough surface, where the scattered vector K2 is in the plane of incidence.

Fig. 14
Fig. 14

Typical incident, intermediate, and reflected wave vectors for scattering from the 0.49-μm PRS: (a) single scattering; (b) double scattering; (c) the doubly scattered ray used in the calculation.

Fig. 15
Fig. 15

Results calculated with Eq. (15) for Δ and ψ as a function of θ1 for the 0.49-μm PRS: (a) Δ vs θ1; (b) ψ vs θ1.

Tables (1)

Tables Icon

Table I Roughness Parameters for the Patches of the Comparison Specimen Studied by Ellipsometrya

Equations (21)

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

R p / R s = tan ψ exp ( i Δ ) ,
tan ψ exp ( i Δ ) = tan ψ 0 exp ( i Δ 0 ) + ½ S r 2 f p 1 + ½ S r 2 f s ,
tan 2 ψ = tan 2 ψ cos ( Δ - 180 ° ) .
Λ a = 2 π R a / S a .
E 2 ( Q ) = [ - i exp ( - i K ρ 0 ) / 4 π ρ 0 ] K 2 × { n L × E 0 - ( μ / ) 1 / 2 [ ( K 2 / K ) × ( n L × H 0 ) ] } exp ( i v · r ) d S 0 .
v = K 1 - K 2 .
H 1 = ( / μ ) 1 / 2 [ ( K 1 / K ) × E 1 ] ,
r c cos θ 1 L λ / 4 π
E 2 s ( Q ) = K E 1 s - l l R s [ ζ ( sin θ 2 - sin θ 1 ) - ( cos θ 2 + cos θ 1 ) ] exp ( iv · r ) ) d x , E 2 p ( Q ) = K E 1 p - l l R p [ ζ ( sin θ 2 - sin θ 1 ) - ( cos θ 2 + cos θ 1 ) ] exp ( iv · r ) d x .
R s = cos θ 1 L - ( η ^ 2 - sin 2 θ 1 L ) 1 / 2 cos θ 1 L + ( η ^ 2 - sin 2 θ 1 L ) 1 / 2 , R p = η ^ 2 cos θ 1 L - ( η ^ 2 - sin 2 θ 1 L ) 1 / 2 η ^ 2 cos θ 1 L + ( η ^ 2 - sin 2 θ 1 L ) 1 / 2 .
E 2 s = - 2 K E 1 s cos θ 1 - l l R s exp [ i v z ζ ( x ) ] d x , E 2 p = - 2 K E 1 p cos θ 1 - l l R p exp [ i v z ζ ( x ) ] d x ,
tan ψ exp ( i Δ ) = E 2 p / E 1 p E 2 s / E 1 s = - l l R p exp ( i v z ζ ) d x - l l R s exp ( i v z ζ ) d x .
tan ψ exp ( i Δ ) = [ R p ( θ 1 + 15 ° ) + R p ( θ 1 - 15 ° ) ] [ 1 - exp ( i α cos θ 1 ) ] [ R s ( θ 1 + 15 ° ) + R s ( θ 1 - 15 ° ) ] [ 1 - exp ( i α cos θ 1 ) ] ,
E 2 j = E 1 2 j + E 2 2 j ,
E 2 2 j = δ R j ( θ 1 + 15 ) R j ( 75 ° ) k = 1 10 exp ( i β k ) ,
tan ψ exp ( i Δ ) = ( E 1 2 p + E 2 2 p ) / E 1 p ( E 1 2 s + E 2 2 s ) / E 1 s = Q 1 p + Q 2 p Q 1 s + Q 2 s ,
Q 1 p = [ R p ( θ 1 + 15 ° ) + R p ( θ 1 - 15 ° ) ] [ 1 - exp ( i α cos θ 1 ) ] , Q 1 s = [ R s ( θ 1 + 15 ° ) + R s ( θ 1 - 15 ° ) ] [ 1 - exp ( i α cos θ 1 ] , Q 2 p = δ R p ( θ 1 + 15 ° ) R p ( 75 ° ) k = 1 10 exp ( i β k ) , Q 2 s = δ R s ( θ 1 + 15 ° ) R s ( 75 ° ) k = 1 10 exp ( i β k ) .
tan ψ exp ( i Δ ) = R p ( θ 1 ) exp ( i v z ζ ) + ½ d 2 R p ( θ 1 ) d ζ 2 ζ 2 exp ( i v z ζ ) R s ( θ 1 ) exp ( i v z ζ ) + ½ d 2 R s ( θ 1 ) d ζ 2 ζ 2 exp ( i v z ζ ) ,
exp ( i v z ζ ) = - l l exp ( i v z ζ ) d x .
ζ 2 exp ( i v z ζ = S r 2 exp ( i v z ζ ) ,
f p = ( R s ) - 1 ( d 2 R p / d ζ 2 ) , f s = ( R s ) - 1 ( d 2 R s / d ζ 2 ) .

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