Abstract

The reflection of light by a system consisting of a nonabsorbing isotropic film and nonabsorbing isotropic substrate with both boundaries (air–film and film–substrate) rough is considered. The scalar theory of light scattering on such a system has been developed. The formulas characterizing both the coherent and the incoherent components of the reflected-light flux have been derived for the case of the identical film (both boundaries are rough, the air–film boundary is a copy of the film–substrate boundary) as well as for the general film (both boundaries are rough and different). The numerical results of the calculation performed for a system of SiO2–Si are presented and the experimental results for the case of art identical film of SiO2 on a Si single crystal are given. The agreement between the theory and the experiment is fairly good. The correctness of the film thickness and its index of refraction depends to a fairly high degree on the roughness of the air–film and film–substrate boundaries.

© 1971 Optical Society of America

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References

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  1. H. Davies, Proc. Inst. Elec. Engrs. (London) 101, 209 (1954).
  2. H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
    [Crossref]
  3. H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [Crossref]
  4. H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
    [Crossref]
  5. J. O. Porteus, J. Opt. Soc. Am. 53, 1394 (1963).
    [Crossref]
  6. P. Bousquet, J. Phys. (Paris) 25, 50 (1964).
    [Crossref]
  7. P. Bousquet, Rev. Opt. 41, 277 (1962).
  8. D. Fabre, J. Romand, and B. Vodar, J. Phys. (Paris) 25, 55 (1964).
    [Crossref]
  9. K. Nagata and J. Nishiwaki, Japan J. Appl. Phys. 6, 251 (1967).
    [Crossref]
  10. K. Nagata, Japan J. Appl. Phys. 6, 1198 (1967).
    [Crossref]
  11. P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
    [Crossref]
  12. E. Kröger and E. Kretchmann, Z. Physik 237, 1 (1970).
    [Crossref]
  13. I. Ohlídal, K. Navrátil, and F. Lukeš, Opt. Commun. 3, 40 (1971).
    [Crossref]
  14. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  15. The validity of the Kirchhoff conditions cannot be assumed while c→ 0.
  16. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  17. The functions X2(vz, − vz) and X1(vz) are the two-dimensional and one-dimensional characteristics functions, respectively, of a given distribution.
  18. Relation (15) is valid assuming the angle α0 to be sufficiently small.
  19. Equation (18) holds, provided that the reference level of the rough boundary is identical with the plane z= 0 (but not, e.g., with a plane z= 〈ξ〉).
  20. In this case, it is also necessary to replace in Eqs. (26) and (27) the mean thickness d¯ by the quantity dm=d¯-〈ξ1〉+〈ξ2〉=d¯-(π/2)12(σ1-σ2).
  21. Landolt-Börnstein Zahlenwerte und Funktionen, 8. Teil Optische Konstanten (Springer, Berlin, 1962), p. 427.
  22. H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
    [Crossref]
  23. F. Lukeš and E. Schmidt, J. Phys. Chem. Solids 26, 1353 (1965.)
    [Crossref]
  24. F. Lukeš, E. Schmidt, and M. Růžička, Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis VIII,(5), 19 (1967) (in Czech).
  25. H. E. Bennett and J. M. Bennett, in Physics of Thin Films 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), P. 1.
  26. T≈ 2 μ m.
  27. A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  28. I. Ohlídal, RNDr. dissertation, Purkyně University, Brno, 1970.

1971 (1)

I. Ohlídal, K. Navrátil, and F. Lukeš, Opt. Commun. 3, 40 (1971).
[Crossref]

1970 (2)

P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
[Crossref]

E. Kröger and E. Kretchmann, Z. Physik 237, 1 (1970).
[Crossref]

1967 (3)

K. Nagata and J. Nishiwaki, Japan J. Appl. Phys. 6, 251 (1967).
[Crossref]

K. Nagata, Japan J. Appl. Phys. 6, 1198 (1967).
[Crossref]

F. Lukeš, E. Schmidt, and M. Růžička, Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis VIII,(5), 19 (1967) (in Czech).

1965 (1)

F. Lukeš and E. Schmidt, J. Phys. Chem. Solids 26, 1353 (1965.)
[Crossref]

1964 (2)

D. Fabre, J. Romand, and B. Vodar, J. Phys. (Paris) 25, 55 (1964).
[Crossref]

P. Bousquet, J. Phys. (Paris) 25, 50 (1964).
[Crossref]

1963 (2)

1962 (1)

P. Bousquet, Rev. Opt. 41, 277 (1962).

1961 (1)

1960 (1)

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

1956 (1)

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

1954 (1)

H. Davies, Proc. Inst. Elec. Engrs. (London) 101, 209 (1954).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Bennett, H. E.

H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
[Crossref]

H. E. Bennett and J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
[Crossref]

H. E. Bennett and J. M. Bennett, in Physics of Thin Films 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), P. 1.

Bennett, J. M.

H. E. Bennett and J. M. Bennett, in Physics of Thin Films 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), P. 1.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Bousquet, P.

P. Bousquet, J. Phys. (Paris) 25, 50 (1964).
[Crossref]

P. Bousquet, Rev. Opt. 41, 277 (1962).

Croce, P.

P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
[Crossref]

Davies, H.

H. Davies, Proc. Inst. Elec. Engrs. (London) 101, 209 (1954).

Devant, G.

P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
[Crossref]

Fabre, D.

D. Fabre, J. Romand, and B. Vodar, J. Phys. (Paris) 25, 55 (1964).
[Crossref]

Hasunuma, H.

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

Kretchmann, E.

E. Kröger and E. Kretchmann, Z. Physik 237, 1 (1970).
[Crossref]

Kröger, E.

E. Kröger and E. Kretchmann, Z. Physik 237, 1 (1970).
[Crossref]

Lukeš, F.

I. Ohlídal, K. Navrátil, and F. Lukeš, Opt. Commun. 3, 40 (1971).
[Crossref]

F. Lukeš, E. Schmidt, and M. Růžička, Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis VIII,(5), 19 (1967) (in Czech).

F. Lukeš and E. Schmidt, J. Phys. Chem. Solids 26, 1353 (1965.)
[Crossref]

Nagata, K.

K. Nagata, Japan J. Appl. Phys. 6, 1198 (1967).
[Crossref]

K. Nagata and J. Nishiwaki, Japan J. Appl. Phys. 6, 251 (1967).
[Crossref]

Nara, J.

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

Navrátil, K.

I. Ohlídal, K. Navrátil, and F. Lukeš, Opt. Commun. 3, 40 (1971).
[Crossref]

Nishiwaki, J.

K. Nagata and J. Nishiwaki, Japan J. Appl. Phys. 6, 251 (1967).
[Crossref]

Ohlídal, I.

I. Ohlídal, K. Navrátil, and F. Lukeš, Opt. Commun. 3, 40 (1971).
[Crossref]

I. Ohlídal, RNDr. dissertation, Purkyně University, Brno, 1970.

Philipp, H. R.

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

Porteus, J. O.

Romand, J.

D. Fabre, J. Romand, and B. Vodar, J. Phys. (Paris) 25, 55 (1964).
[Crossref]

Ružicka, M.

F. Lukeš, E. Schmidt, and M. Růžička, Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis VIII,(5), 19 (1967) (in Czech).

Schmidt, E.

F. Lukeš, E. Schmidt, and M. Růžička, Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis VIII,(5), 19 (1967) (in Czech).

F. Lukeš and E. Schmidt, J. Phys. Chem. Solids 26, 1353 (1965.)
[Crossref]

Séré, M.

P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
[Crossref]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Taft, E. A.

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

Vašícek, A.

A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Verhaeghe, F.

P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
[Crossref]

Vodar, B.

D. Fabre, J. Romand, and B. Vodar, J. Phys. (Paris) 25, 55 (1964).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis (1)

F. Lukeš, E. Schmidt, and M. Růžička, Folia Facultatis Scientiarum Naturalium Univ. Purkynianae Brunensis VIII,(5), 19 (1967) (in Czech).

J. Opt. Soc. Am. (3)

J. Phys. (Paris) (2)

P. Bousquet, J. Phys. (Paris) 25, 50 (1964).
[Crossref]

D. Fabre, J. Romand, and B. Vodar, J. Phys. (Paris) 25, 55 (1964).
[Crossref]

J. Phys. Chem. Solids (1)

F. Lukeš and E. Schmidt, J. Phys. Chem. Solids 26, 1353 (1965.)
[Crossref]

J. Phys. Soc. Japan (1)

H. Hasunuma and J. Nara, J. Phys. Soc. Japan 11, 69 (1956).
[Crossref]

Japan J. Appl. Phys. (2)

K. Nagata and J. Nishiwaki, Japan J. Appl. Phys. 6, 251 (1967).
[Crossref]

K. Nagata, Japan J. Appl. Phys. 6, 1198 (1967).
[Crossref]

Opt. Commun. (1)

I. Ohlídal, K. Navrátil, and F. Lukeš, Opt. Commun. 3, 40 (1971).
[Crossref]

Phys. Rev. (1)

H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960).
[Crossref]

Proc. Inst. Elec. Engrs. (London) (1)

H. Davies, Proc. Inst. Elec. Engrs. (London) 101, 209 (1954).

Rev. Opt. (1)

P. Bousquet, Rev. Opt. 41, 277 (1962).

Surface Sci. (1)

P. Croce, G. Devant, M. Séré, and F. Verhaeghe, Surface Sci. 22, 173 (1970).
[Crossref]

Z. Physik (1)

E. Kröger and E. Kretchmann, Z. Physik 237, 1 (1970).
[Crossref]

Other (12)

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

The validity of the Kirchhoff conditions cannot be assumed while c→ 0.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

The functions X2(vz, − vz) and X1(vz) are the two-dimensional and one-dimensional characteristics functions, respectively, of a given distribution.

Relation (15) is valid assuming the angle α0 to be sufficiently small.

Equation (18) holds, provided that the reference level of the rough boundary is identical with the plane z= 0 (but not, e.g., with a plane z= 〈ξ〉).

In this case, it is also necessary to replace in Eqs. (26) and (27) the mean thickness d¯ by the quantity dm=d¯-〈ξ1〉+〈ξ2〉=d¯-(π/2)12(σ1-σ2).

Landolt-Börnstein Zahlenwerte und Funktionen, 8. Teil Optische Konstanten (Springer, Berlin, 1962), p. 427.

H. E. Bennett and J. M. Bennett, in Physics of Thin Films 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), P. 1.

T≈ 2 μ m.

A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

I. Ohlídal, RNDr. dissertation, Purkyně University, Brno, 1970.

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Figures (10)

Fig. 1
Fig. 1

Geometric representation of-the interaction of a plane monochromatic wave with a rough boundary. Distance OA = kr/k.

Fig. 2
Fig. 2

Illustrations of multiple reflections inside a thin wedge-type film. W1, W2, …, Wp are virtual wave fronts of the reflected waves.

Fig. 3
Fig. 3

Spectral dependence of the reflectance of thin identical films characterized by different σ values (normal distribution). 1, σ = 0; 2, σ = 250 Å; 3, σ = 400 Å; 4, σ = 600 Å.

Fig. 4
Fig. 4

Spectral dependence of the reflectance of a general thin film with different σ2 values; σ1 is constant (σ1 = 100 Å; normal distribution). 1, ideal film; 2, σ2 = 100 Å; 3, σ2 = 300 Å; 4, σ2 = 500 Å.

Fig. 5
Fig. 5

Spectral dependence of the reflectance of general thin films with different σ1 values; σ2 is constant (σ2 =300 Å; normal distribution). 1, σ1 = 0; 2, σ1 = 300 Å; 3, σ1 =500 Å; 4, perfect Si surface.

Fig. 6
Fig. 6

Spectral dependence of the reflectance of an identical and a general thin film (normal distribution). 1, ideal film; 2, identical film with σ = 500 Å; 3, σ1 =500 Å, σ2 = 0; 4, σ1 = 0, σ2 = 500 Å; 5, σ1 = σ2 = 500 Å.

Fig. 7
Fig. 7

Spectral dependence of the reflectance of an identical and general thin film (normal distribution). 1, ideal film; 2, identical film with σ = 300 Å; 3, σ1 = 300 Å, σ2 = 0; 4, σ1 = 0, σ2 = 300 Å; 5, σ1 = σ2 = 300 Å; 6. rough Si surface (σ = 300 Å).

Fig. 8
Fig. 8

Spectral dependence of the reflectance of general thin films with different indexes of refraction n (normal distribution). σ1 = 200 Å, σ2 = 300 Å, n0 = 1; 1, n = 1.3; 2, n = 1.5; 3, n = 1.7; 4, n = 2.0; 5, n = 2.5.

Fig. 9
Fig. 9

Spectral dependence of the pseudo-σ of a Si surface ground with a SiC abrasive. 1, grain diameter 22 μm; 2, grain diameter 7 μm.

Fig. 10
Fig. 10

Spectral dependence of the reflectance of a rough Si surface coated with an identical film. 1, perfect Si surface; 2, SiO2 film on a rough Si surface (ground with a SiC abrasive of 7-μm diameter); 3, Si surface after dissolution of the SiO2 film; 4, SiO2 film on a rough Si surface (ground with a SiC abrasive of 22-μm diameter); open circles: experimental values, full circles: values calculated according to Eq. (14); 5, Si surface after dissolution of the SiO2 film.

Tables (1)

Tables Icon

Table I Comparison of the values of absolute reflectance of a thin film with rough boundaries at the reflectance minimum, the index of refraction of the film and its thickness, before the correction and after the correction according to Eq. (14).

Equations (39)

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4 π r c cos ϑ λ
E P = 1 4 π S 1 [ E ψ n - ψ E n ] d S 1 ,
ψ = [ exp ( i k 2 R ) ] / R ,
( k 2 = k 2 = k 1 = k 1 = k = 2 π / λ ) .
E = ( 1 + R ) E 1 E / n = i ( 1 - R ) k 1 n E 1 ,
R = ( r 1 + r 2 e i x ) / ( 1 + r 1 r 2 e i x ) ,
x = ( 4 π / λ ) n d ¯ cos β cos φ .
ψ = exp ( i k 2 R 0 ) [ exp ( - i k 2 r ) ] / R 0 ,
E P = - i 2 exp ( i k 2 R 0 ) R 0 λ A ( b + a v x + c v y v z ) × - X x - Y Y exp ( i vr ) d x d y ,
a = sin Θ 2 cos Θ 3 ( 1 + R ) v x = - ( 2 π / λ ) sin Θ 2 cos Θ 3 b = cos Θ 2 ( 1 + R ) - ( 1 - R ) v y = - ( 2 π / λ ) sin Θ 2 sin Θ 3 c = sin Θ 2 sin Θ 3 ( 1 + R ) v z = - ( 2 π / λ ) ( 1 + cos Θ 2 ) .
E P E P * = 4 A 2 R 0 2 λ 2 ( b + a v x + c v y v z ) ( b * + a * v x + c * v y v z ) × ( sin v x X sin v y Y v x v y ) 2 exp [ i v z ξ ( x , y ) ] × exp [ - i v z ξ ( x , y ) ] ,
D { E P } = A 2 4 R 0 2 λ 2 ( b + a v x + c v y v z ) ( b * + a * v x + c * v y v z ) × - X X - X X - Y Y - Y Y exp { i [ v x ( x - x ) + v y ( y - y ) ] } × [ exp [ i v z ( ξ - ξ ) ] - exp ( i v z ξ ) exp ( - i v z ξ ) ] × d x d x d y d y .
X 2 ( v z , - v z ) = exp [ i v z ( ξ - ξ ) ] = - - exp [ i v z ( ξ - ξ ) ] w ( ξ , ξ ) d ξ d ξ X 1 ( v z ) = exp ( i v z ξ ) = - exp ( i v z ξ ) w ( ξ ) d ξ .
w ( ξ , ξ ) = w ( ξ ) δ ( ξ - ξ ) c ( τ ) + w ( ξ ) w ( ξ ) [ 1 - c ( τ ) ] ,
w ( ξ ) = ( 1 / σ 2 π ) exp ( - ξ 2 / 2 σ 2 ) ,
c ( τ ) = exp ( - τ 2 / T 2 ) ,
R c = r 1 2 + r 2 2 + 2 r 1 r 2 cos [ ( 4 π / λ ) n d ¯ ] 1 + r 1 2 r 2 2 + 2 r 1 r 2 cos [ ( 4 π / λ ) n d ¯ ] × exp ( - 16 π 2 σ 2 / λ 2 ) = R 0 exp ( - 16 π 2 σ 2 / λ 2 ) .
R i = R 0 [ 1 - exp ( - 16 π 2 σ 2 / λ 2 ) ] × [ 1 - exp ( - π 2 α 0 2 T 2 / λ 2 ) ] ,
R T = R c + R i .
w ( ξ ) = ( ξ / σ 2 ) exp ( - ξ 2 / 2 σ 2 ) ,
X 1 ( v z ) = 1 - σ v z 2 exp ( - σ 2 v z 2 / 2 ) k = 0 ( σ v z / 2 ) 2 k + 1 k ! ( 2 k + 1 ) + i σ v z ( π / 2 ) 1 2 exp ( - σ 2 v z 2 / 2 ) ,
X 1 ( v z ) 2 = [ Re X 1 ( v z ) ] 2 + [ Im X 1 ( v z ) ] 2 .
R T = R 0 [ 1 - ( 16 π 2 σ rms 2 / λ 2 ) ] .
n 0 sin Θ ¯ p = n sin [ Θ 1 + 2 ( p - 1 ) α ] n 0 sin Θ ¯ 1 = n sin Θ 1 h = ρ tan α .
X P = 2 π λ ( B N p - B N 1 ) = 2 π λ n ρ { sin [ Θ 1 + 2 ( p - 1 ) α ] - sin Θ 1 } = ( p - 1 ) 4 π λ n h cos Θ 1 - ( p - 1 ) 2 α 4 π λ n h sin Θ 1 ( p - 1 ) ( 2 p 2 - 4 p + 3 ) α 2 3 4 π n h cos Θ 1 λ .
X P = ( p - 1 ) ( 4 π / λ ) n h ( p - 1 ) ( 4 π / λ ) n d L ,
d L = d ¯ + ξ 1 - ξ 2 .
E = { 1 + r 1 + ( 1 - r 1 2 ) m = 0 N ( - 1 ) m r 1 m r 2 m + 1 exp [ i ( m + 1 ) ( 4 π / λ ) n d L ] } exp ( i k 1 r ) ,
E n = i ( 1 - r 1 ) k 1 n exp ( i k 1 r ) - i k 1 n exp ( i k 1 r ) m = 0 N ( - 1 ) m r 1 m r 2 m + 1 ( 1 - r 1 2 ) exp [ i ( m + 1 ) ( 4 π / λ ) n d L ] + i exp ( i k 1 r ) m = 0 N ( - 1 ) m r 1 m r 2 m + 1 ( 1 - r 1 2 ) ( m + 1 ) ( 4 π / λ ) n [ n x ( ξ 1 x - ξ 2 x ) + n y ( ξ 1 y - ξ 2 y ) ] × exp [ i ( m + 1 ) ( 4 π / λ ) n d L ] ,
E P = - i exp ( i k 2 R 0 ) R 0 λ - X X - Y Y { r 1 + m = 0 N ( - 1 ) m r 1 m r 2 m + 1 ( 1 - r 1 2 ) exp [ i ( m + 1 ) ( 4 π / λ ) n d L ] } exp ( i vr ) d x d y .
R c = r 1 2 X 1 ( 1 ) ( v z ) 2 + m = 0 N r 1 2 m r 2 2 ( m + 1 ) ( 1 - r 1 2 ) 2 X 1 ( 1 ) ( B m ) 2 X 1 ( 2 ) ( M m ) 2 + 2 m , l = 0 m < l N ( - 1 ) m + l r 1 m + l r 2 m + l + 2 ( 1 - r 1 2 ) 2 X 1 ( 1 ) ( B m ) X 1 ( 2 ) ( M m ) X 1 ( 1 ) ( B l ) X 1 ( 2 ) ( M l ) × cos [ 4 π λ n ( m - l ) d ¯ + φ 1 ( B m ) - φ 1 ( B l ) + φ 2 ( M m ) - φ 1 ( M l ) ] + 2 r 1 X 1 ( 1 ) ( v z ) m = 0 N ( - 1 ) m r 1 m r 2 m + 1 ( 1 - r 1 2 ) X 1 ( 1 ) ( B m ) X 2 ( 2 ) ( M m ) × cos [ 4 π λ ( m + 1 ) m d ¯ + φ 1 ( B m ) + φ 2 ( M m ) - φ 1 ( v z ) ] ,
B m = v z + ( m - 1 ) n ( 4 π / λ ) ,             M m = - ( m + 1 ) n ( 4 π / λ ) ,             v z = - ( 4 π / λ ) .
φ j ( q ) = arctg [ Im X 1 ( j ) ( q ) / Re X 1 ( j ) ( q ) ] ,
R i = r 1 2 ( 1 - X 1 ( 1 ) ( v z ) 2 ) [ 1 - exp ( - π 2 α 0 2 T 1 2 / λ 2 ) ] + m = 0 N r 1 2 m r 2 2 ( m + 1 ) ( 1 - r 1 2 ) 2 × { X 1 ( 1 ) ( B m ) 2 [ 1 - X 1 ( 2 ) ( M m ) 2 ] [ 1 - exp ( - π 2 α 0 2 T 2 2 / λ 2 ) ] + X 1 ( 2 ) ( M m ) 2 × [ 1 - X 1 ( 1 ) ( B m ) 2 ] [ 1 - exp ( - π 2 α 0 2 T 1 2 / λ 2 ) ] + [ 1 - X 1 ( 1 ) ( B m ) 2 ] [ 1 - X 1 ( 2 ) ( M m ) 2 ] × [ 1 - exp ( - π 2 α 0 2 λ 2 T 1 2 T 2 2 T 1 2 + T 2 2 ) ] } + 2 m , l = 0 m < l N ( - 1 ) m + l r 1 m + l r 2 m + l + 2 ( 1 - r 1 2 ) 2 { [ 1 - exp ( - π 2 α 0 2 T 2 2 / λ 2 ) ] × [ X 1 ( 1 ) ( B m ) X 1 ( 1 ) ( B l ) cos Δ m l - X 1 ( 1 ) ( B m ) X 1 ( 1 ) ( B l ) X 1 ( 2 ) ( M m ) X 1 ( 2 ) ( M l ) cos Δ 1 m l ] + [ 1 - exp ( - π 2 α 0 2 T 1 2 / λ 2 ) ] [ X 1 ( 2 ) ( M m ) X 1 ( 2 ) ( M l ) cos Δ m l - X 1 ( 1 ) ( B m ) X 1 ( 1 ) ( B l ) X 1 ( 2 ) ( M m ) × X 1 ( 2 ) ( M l ) cos Δ 1 m l ] + [ 1 - exp ( - π 2 α 0 2 λ 2 T 1 2 T 2 2 T 1 2 + T 2 2 ) ] [ 1 - X 1 ( 1 ) ( B m ) X 1 ( 1 ) ( B l ) cos Δ m l - X 1 ( 2 ) ( M m ) X 1 ( 2 ) ( M l ) cos Δ m l + X 1 ( 1 ) ( B m ) X 1 ( 1 ) ( B l ) X 1 ( 2 ) ( M m ) X 1 ( 2 ) ( M l ) cos Δ 1 m l ] } + 2 m = 0 N ( - 1 ) m r 1 m + 1 r 2 m + 1 ( 1 - r 1 2 ) [ 1 - exp ( - π 2 α 0 2 T 1 2 / λ 2 ) ] × [ X 1 ( 2 ) ( M m ) cos δ m - X 1 ( 1 ) ( B m ) X 1 ( 1 ) ( v z ) X 1 ( 2 ) ( M m ) cos Δ m ] ,
Δ m l = ( m - l ) ( 4 π / λ ) n d ¯ + φ 1 ( B m ) - φ 1 ( B l ) Δ m l = ( m - l ) ( 4 π / λ ) n d ¯ + φ 2 ( M m ) - φ 2 ( M l ) Δ 1 m l = ( m - l ) ( 4 π / λ ) n d ¯ + φ 1 ( B m ) - φ 1 ( B l ) + φ 2 ( M m ) - φ 2 ( M l ) Δ m = ( m + 1 ) ( 4 π / λ ) n d ¯ + φ 1 ( B m ) + φ 2 ( M m ) - φ 1 ( v z ) δ m = ( m + 1 ) ( 4 π / λ ) n d ¯ + φ 2 ( M m ) .
X 1 ( 1 ) ( v z ) = X 1 ( 1 ) ( v z ) = exp ( - 8 π 2 σ 1 2 / λ 2 ) X 1 ( 1 ) ( B m ) = X 1 ( 1 ) ( B m ) = exp { - 8 π 2 [ ( m + 1 ) n - 1 ] 2 σ 1 2 / λ 2 }
X 1 ( 2 ) ( M m ) = X 1 ( 2 ) ( M m ) = exp [ - 8 π 2 ( m + 1 ) 2 n 2 σ 2 2 / λ 2 ] φ 1 ( v z ) = φ 1 ( B m ) = φ 2 ( M m ) = 0.
σ rms σ p p / 2 2 ,
dm=d¯-ξ1+ξ2=d¯-(π/2)12(σ1-σ2).