Abstract

We studied the photoemission process in films of amorphous Ge deposited on silica substrates. We measured, for both polarizations and for several angles of incidence, with direct and back illumination (through the substrate), the reflectance, the transmittance, and the photoemissive yield of each film without exposure to air. We analyzed the possibility of deducing with minimum error, from the experimental results, the index of refraction, the thickness, and the escape probability of photoexcited electrons and we verified the validity of the model used for computation.

© 1971 Optical Society of America

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  1. S. M. Sze, J. L. Moll, and T. Sugano, Solid-State Electron. 7, 509 (1964).
    [Crossref]
  2. C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
    [Crossref]
  3. G. A. Katrich and O. G. Sarbei, Sov. Phys. Solid State 3, 1181 (1961).
  4. B. Pistoulet and M. Rouzeyre, in Basic Problems in Thin Films Physics(Vandenhoeck and Ruprecht, Göttingen, 1965), p. 648.
  5. H. Kanter and W. A. Feibelman, J. Appl. Phys. 33, 3580 (1962).
    [Crossref]
  6. R. E. Collins and L. W. Davies, Solid-State Electron. 7, 445 (1964).
    [Crossref]
  7. C. A. Mead, Phys. Rev. Letters 8, 56 (1962).
    [Crossref]
  8. G. Niquet, P. Vernier, and P. Hartmann, Compt. Rend. 270, 1234 (1970).
  9. P. Vernier and E. Coquet, in Ref. 4, p. 328.
  10. P. Vernier, E. Coquet, and M. Bigueurre, Compt. Rend. 262, 1141, 1728 (1966).
  11. P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
    [Crossref]
  12. P. M. Grant and W. Paul, Phys. Rev. 37, 3110 (1966).
  13. A. M. Clark, Phys. Rev. 154, 750 (1967).
    [Crossref]
  14. J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
    [Crossref]
  15. P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).
  16. F. Abelès, in Progress in Optics II, edited by E. Wolf(North-Holland, Amsterdam, 1963), p. 251.
  17. P. Rouard and P. Bousquet, in Progress in Optics IV, edited by E. Wolf(North-Holland, Amsterdam, 1965), p. 147.
  18. M. G. Bouchard, in Ref. 4, p. 301.
  19. G. Baldini and L. Rigaldi, J. Opt. Soc. Am. 60, 495 (1970).
    [Crossref]
  20. T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).
  21. P. Bousquet, Opt. Acta 3, 153 (1965); J. Phys. (France) 18, 447 (1957).
    [Crossref]
  22. M. Berman, M. R. Kerchner, and S. Ergun, J. Opt. Soc. Am 60, 647 (1970).
    [Crossref]
  23. M. Pauty and J. P. Goudonnet, Rev. Phys. Appliquée 4, 24 (1969).
    [Crossref]
  24. R. Stuart, F. Wooten, and W. Spicer, Phys. Rev. 135A, 495 (1964).
    [Crossref]
  25. S. V. Pepper, J. Opt. Soc. Am. 60, 805 (1970).
    [Crossref]

1970 (7)

G. Niquet, P. Vernier, and P. Hartmann, Compt. Rend. 270, 1234 (1970).

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
[Crossref]

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).

T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).

M. Berman, M. R. Kerchner, and S. Ergun, J. Opt. Soc. Am 60, 647 (1970).
[Crossref]

G. Baldini and L. Rigaldi, J. Opt. Soc. Am. 60, 495 (1970).
[Crossref]

S. V. Pepper, J. Opt. Soc. Am. 60, 805 (1970).
[Crossref]

1969 (1)

M. Pauty and J. P. Goudonnet, Rev. Phys. Appliquée 4, 24 (1969).
[Crossref]

1967 (1)

A. M. Clark, Phys. Rev. 154, 750 (1967).
[Crossref]

1966 (2)

P. M. Grant and W. Paul, Phys. Rev. 37, 3110 (1966).

P. Vernier, E. Coquet, and M. Bigueurre, Compt. Rend. 262, 1141, 1728 (1966).

1965 (1)

P. Bousquet, Opt. Acta 3, 153 (1965); J. Phys. (France) 18, 447 (1957).
[Crossref]

1964 (3)

R. Stuart, F. Wooten, and W. Spicer, Phys. Rev. 135A, 495 (1964).
[Crossref]

S. M. Sze, J. L. Moll, and T. Sugano, Solid-State Electron. 7, 509 (1964).
[Crossref]

R. E. Collins and L. W. Davies, Solid-State Electron. 7, 445 (1964).
[Crossref]

1963 (1)

J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
[Crossref]

1962 (3)

C. A. Mead, Phys. Rev. Letters 8, 56 (1962).
[Crossref]

C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
[Crossref]

H. Kanter and W. A. Feibelman, J. Appl. Phys. 33, 3580 (1962).
[Crossref]

1961 (1)

G. A. Katrich and O. G. Sarbei, Sov. Phys. Solid State 3, 1181 (1961).

Abelès, F.

F. Abelès, in Progress in Optics II, edited by E. Wolf(North-Holland, Amsterdam, 1963), p. 251.

Ashley, E. J.

T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).

Baldini, G.

Bennett, J. M.

T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).

Berman, M.

M. Berman, M. R. Kerchner, and S. Ergun, J. Opt. Soc. Am 60, 647 (1970).
[Crossref]

Bigueurre, M.

P. Vernier, E. Coquet, and M. Bigueurre, Compt. Rend. 262, 1141, 1728 (1966).

Bouchard, M. G.

M. G. Bouchard, in Ref. 4, p. 301.

Bousquet, P.

P. Bousquet, Opt. Acta 3, 153 (1965); J. Phys. (France) 18, 447 (1957).
[Crossref]

P. Rouard and P. Bousquet, in Progress in Optics IV, edited by E. Wolf(North-Holland, Amsterdam, 1965), p. 147.

Chabrier, G.

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
[Crossref]

Clark, A. M.

A. M. Clark, Phys. Rev. 154, 750 (1967).
[Crossref]

Collins, R. E.

R. E. Collins and L. W. Davies, Solid-State Electron. 7, 445 (1964).
[Crossref]

Coquet, E.

P. Vernier, E. Coquet, and M. Bigueurre, Compt. Rend. 262, 1141, 1728 (1966).

P. Vernier and E. Coquet, in Ref. 4, p. 328.

Cornaz, J.

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
[Crossref]

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).

Crowell, C. R.

C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
[Crossref]

Davey, J.

J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
[Crossref]

Davies, L. W.

R. E. Collins and L. W. Davies, Solid-State Electron. 7, 445 (1964).
[Crossref]

Donovan, T. M.

T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).

Ergun, S.

M. Berman, M. R. Kerchner, and S. Ergun, J. Opt. Soc. Am 60, 647 (1970).
[Crossref]

Feibelman, W. A.

H. Kanter and W. A. Feibelman, J. Appl. Phys. 33, 3580 (1962).
[Crossref]

Goudonnet, J. P.

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
[Crossref]

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).

M. Pauty and J. P. Goudonnet, Rev. Phys. Appliquée 4, 24 (1969).
[Crossref]

Grant, P. M.

P. M. Grant and W. Paul, Phys. Rev. 37, 3110 (1966).

Hartmann, P.

G. Niquet, P. Vernier, and P. Hartmann, Compt. Rend. 270, 1234 (1970).

Howarth, L. E.

C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
[Crossref]

Kanter, H.

H. Kanter and W. A. Feibelman, J. Appl. Phys. 33, 3580 (1962).
[Crossref]

Katrich, G. A.

G. A. Katrich and O. G. Sarbei, Sov. Phys. Solid State 3, 1181 (1961).

Kerchner, M. R.

M. Berman, M. R. Kerchner, and S. Ergun, J. Opt. Soc. Am 60, 647 (1970).
[Crossref]

Labate, E. C.

C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
[Crossref]

Mead, C. A.

C. A. Mead, Phys. Rev. Letters 8, 56 (1962).
[Crossref]

Moll, J. L.

S. M. Sze, J. L. Moll, and T. Sugano, Solid-State Electron. 7, 509 (1964).
[Crossref]

Montgomery, M. D.

J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
[Crossref]

Niquet, G.

G. Niquet, P. Vernier, and P. Hartmann, Compt. Rend. 270, 1234 (1970).

Pankey, T.

J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
[Crossref]

Paul, W.

P. M. Grant and W. Paul, Phys. Rev. 37, 3110 (1966).

Pauty, M.

M. Pauty and J. P. Goudonnet, Rev. Phys. Appliquée 4, 24 (1969).
[Crossref]

Pepper, S. V.

Pistoulet, B.

B. Pistoulet and M. Rouzeyre, in Basic Problems in Thin Films Physics(Vandenhoeck and Ruprecht, Göttingen, 1965), p. 648.

Rigaldi, L.

Rouard, P.

P. Rouard and P. Bousquet, in Progress in Optics IV, edited by E. Wolf(North-Holland, Amsterdam, 1965), p. 147.

Rouzeyre, M.

B. Pistoulet and M. Rouzeyre, in Basic Problems in Thin Films Physics(Vandenhoeck and Ruprecht, Göttingen, 1965), p. 648.

Sarbei, O. G.

G. A. Katrich and O. G. Sarbei, Sov. Phys. Solid State 3, 1181 (1961).

Spicer, W.

R. Stuart, F. Wooten, and W. Spicer, Phys. Rev. 135A, 495 (1964).
[Crossref]

Spicer, W. E.

T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).

Spitzer, W. G.

C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
[Crossref]

Stuart, R.

R. Stuart, F. Wooten, and W. Spicer, Phys. Rev. 135A, 495 (1964).
[Crossref]

Sugano, T.

S. M. Sze, J. L. Moll, and T. Sugano, Solid-State Electron. 7, 509 (1964).
[Crossref]

Sze, S. M.

S. M. Sze, J. L. Moll, and T. Sugano, Solid-State Electron. 7, 509 (1964).
[Crossref]

Tirman, R. J.

J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
[Crossref]

Vernier, P.

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
[Crossref]

G. Niquet, P. Vernier, and P. Hartmann, Compt. Rend. 270, 1234 (1970).

P. Vernier, E. Coquet, and M. Bigueurre, Compt. Rend. 262, 1141, 1728 (1966).

P. Vernier and E. Coquet, in Ref. 4, p. 328.

Wooten, F.

R. Stuart, F. Wooten, and W. Spicer, Phys. Rev. 135A, 495 (1964).
[Crossref]

Compt. Rend. (3)

G. Niquet, P. Vernier, and P. Hartmann, Compt. Rend. 270, 1234 (1970).

P. Vernier, E. Coquet, and M. Bigueurre, Compt. Rend. 262, 1141, 1728 (1966).

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Compt. Rend. 271, 255 (1970).

J. Appl. Phys. (1)

H. Kanter and W. A. Feibelman, J. Appl. Phys. 33, 3580 (1962).
[Crossref]

J. Opt. Soc. Am (1)

M. Berman, M. R. Kerchner, and S. Ergun, J. Opt. Soc. Am 60, 647 (1970).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Acta (1)

P. Bousquet, Opt. Acta 3, 153 (1965); J. Phys. (France) 18, 447 (1957).
[Crossref]

Opt. Commun. (1)

P. Vernier, J. P. Goudonnet, J. Cornaz, and G. Chabrier, Opt. Commun. 1, 391 (1970).
[Crossref]

Phys. Rev. (5)

P. M. Grant and W. Paul, Phys. Rev. 37, 3110 (1966).

A. M. Clark, Phys. Rev. 154, 750 (1967).
[Crossref]

C. R. Crowell, W. G. Spitzer, L. E. Howarth, and E. C. Labate, Phys. Rev. 127, 2006 (1962).
[Crossref]

R. Stuart, F. Wooten, and W. Spicer, Phys. Rev. 135A, 495 (1964).
[Crossref]

T. M. Donovan, W. E. Spicer, J. M. Bennett, and E. J. Ashley, Phys. Rev. 2, 397 (1970).

Phys. Rev. Letters (1)

C. A. Mead, Phys. Rev. Letters 8, 56 (1962).
[Crossref]

Rev. Phys. Appliquée (1)

M. Pauty and J. P. Goudonnet, Rev. Phys. Appliquée 4, 24 (1969).
[Crossref]

Solid-State Electron. (3)

S. M. Sze, J. L. Moll, and T. Sugano, Solid-State Electron. 7, 509 (1964).
[Crossref]

R. E. Collins and L. W. Davies, Solid-State Electron. 7, 445 (1964).
[Crossref]

J. Davey, R. J. Tirman, T. Pankey, and M. D. Montgomery, Solid-State Electron. 6, 205 (1963).
[Crossref]

Sov. Phys. Solid State (1)

G. A. Katrich and O. G. Sarbei, Sov. Phys. Solid State 3, 1181 (1961).

Other (5)

B. Pistoulet and M. Rouzeyre, in Basic Problems in Thin Films Physics(Vandenhoeck and Ruprecht, Göttingen, 1965), p. 648.

P. Vernier and E. Coquet, in Ref. 4, p. 328.

F. Abelès, in Progress in Optics II, edited by E. Wolf(North-Holland, Amsterdam, 1963), p. 251.

P. Rouard and P. Bousquet, in Progress in Optics IV, edited by E. Wolf(North-Holland, Amsterdam, 1965), p. 147.

M. G. Bouchard, in Ref. 4, p. 301.

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Figures (16)

Fig. 1
Fig. 1

Scheme of the experimental device. A, liquid-nitrogen trap (the diffusion pump is not represented). P, Plate supporting the photomultiplier PM1 rotating around the sample. C, Evaporation source. S, Monochromator exit slit. L, Silica lens. D, Diaphragm. G, Glan prism. Q, Silica plate. M, Film being measured. E, Diffusing fluorite screen. PM1, PM2, Photomultipliers 51 UVP. G, Double galvanometer recording PM1 and PM2 signals.

Fig. 2
Fig. 2

Definition of error points. N, Point with coordinates x, y, z equal to the exact values of n, k, and d/λ for the film. ∑1, Surface F1(x,y,z) = M1 + ΔM1. ∑1′, ∑2′, ∑3′, ∑2, and ∑3 have not been represented. N1, Projection of N on ∑1. a1, Unitary vector normal to ∑1 at N1. The error point P1 associated to the measurement M1 is the intersection with the plane z = 1 of a1 extended from the origin.

Fig. 3
Fig. 3

Principle used for the calculation of reflectance, transmittance, and photoemissive yield for direct illumination. Interference is taken in account for the calculation of , T, and Y, and then these quantities are combined to calculate R, T, and r, by merely adding irradiances and photoelectric currents.

Fig. 4
Fig. 4

Error points for determination of the index of refraction and thickness of a Ge film at 2537 Å (n = 1.65, k = 2.70, d = 448 Å). We have represented in the plane z = 1 (Fig. 2) the error points associated with measurements of transmittance with p polarization (+), transmittance with s polarization (×), reflectance with s polarization and direct illumination (○), reflectance with s polarization and back illumination (⊝), reflectance with p polarization and direct illumination (●), reflectance with p polarization and back illumination (filled circle with bar through it). The numbers indicate the angle of incidence in degrees, except for transmittance with s polarization because, then, the error point does not depend on the angle of incidence.

Fig. 5
Fig. 5

Error points for determination of the index of refraction and thickness of a Ge film at 2537 Å (n = 1.65, k = 2.70, d = 224 Å). Code as in Fig. 4.

Fig. 6
Fig. 6

Error points for determination of the index of refraction and thickness of a Ge film at 2537 Å (n = 1.65, k = 2.70, d = 56 Å). Code as in Fig. 4.

Fig. 7
Fig. 7

Error points for determination of the index of refraction and thickness of a Ge film at 2537 Å (n = 1.65, k = 2.70, d = 28 Å). Code as in Fig. 4.

Fig. 8
Fig. 8

Values of xp, yp, zp, found for n, k, d/λ in successive iterations with the method given in Appendix B for a 200-Å-thick Ge film (n = 1.46, k = 2.46).

Fig. 9
Fig. 9

Values of xp, yp, zp, found for n, k, d/λ in successive iterations with the method given in Appendix B for a 30-Å-thick Ge film (n = 1.90, k = 2.41).

Fig. 10
Fig. 10

Estimated values of thickness vs wavelength for a Ge Film.

Fig. 11
Fig. 11

Variation of n and k with thickness for Ge amorphous films.

Fig. 12
Fig. 12

Variation of n and k with wavelength for a Ge amorphous film. +×, for two different 185-Å-thick films. ●, for a 225-Å-thick film. ○, for a 190-Å-thick film deposited on a substrate at 250°C.

Fig. 13
Fig. 13

Variation of the calculated ratio r/r+ of a Ge film (n = 1.68, k = 2.79, d = 143 Å, λ = 2537 Å) with the attenuation length L for different angles of incidence and polarizations. The experimental point for r/r+, reported on the calculated curve, gives the estimation of L.

Fig. 14
Fig. 14

Variation of the calculated ratio r/r+ of an Au film (n = 1.34, k = 1.64, d = 332 Å, λ = 2537 Å) with the attenuation length L for different angles of incidence and polarizations. The experimental point for r/r+, reported on the calculated curve, gives the estimation of L.

Fig. 15
Fig. 15

Variation of attenuation length with film thickness.

Fig. 16
Fig. 16

Variation of attenuation length with light wavelength. +×●, for three different films deposited on cold substrate. ○, for a film deposited on a substrate at 250°C.

Tables (3)

Tables Icon

Table I Measurements used for optical-index and thickness calculations for two Ge films.

Tables Icon

Table II Estimations of the escape probability p0 of an electron excited at the surface of the same film, obtained from the yield r for different angles of incidence and polarization, with direct and back illumination. Estimations are multiplied by an unknown constant c.

Tables Icon

Table III Mean estimation of cp0 for different films.

Equations (56)

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F i ( x , y , z ) = M i .
F i ( x , y , z ) = M i ,
F i ( x , y , z ) - F i ( x , y , z ) < Δ M i .
M i - Δ M i < F i ( x , y , z ) < M i + Δ M i             i = 1 , 2 , 3.
Δ V = 8 Δ M 1 Δ M 2 Δ M 3 / [ a 1 · ( a 2 × a 3 ) ] ,
x = ( F i / x ) / ( F i / z )             y = ( F i / y ) / ( F i / z ) .
x = ( G / x ) / ( G / z )             y = ( G / y ) / ( G / z ) .
G [ x = ( G / x ) / ( G / z ) ,             y = ( G / y ) / ( G / z ) ]
H [ x = ( H / x ) / ( H / z ) ,             y = ( H / y ) / ( H / z ) ] .
R + = + + S T 2 exp ( - 2 ρ / cos i 2 ) 1 - S - exp ( - 2 ρ / cos i 2 )
R - = - ( 1 - S ) 2 exp ( - 2 ρ / cos i 2 ) 1 - S - exp ( - 2 ρ / cos i 2 ) + S
T = ( 1 - S ) T exp ( - ρ / cos i 2 ) 1 - S - exp ( - 2 ρ / cos i 2 ) .
a 1 · ( a 2 × a 3 ) = [ OP 1 · ( OP 2 × OP 3 ) ] / ( O P 1 · O P 2 · O P 3 ) .
Σ p i [ F i ( x , y , z ) - M i ] 2 = E ( x , y , z ) .
p ( z ) = p 0 e - z / L ,
r = 0 d D ( z ) p 0 e - z / L d z .
D ( z ) = 4 π n k λ cos i 0 E x 2 + E y 2 + E z 2 E 0 2 .
r + = p 0 [ Y + + Y - T S exp ( - 2 ρ / cos i 2 ) [ 1 - S - exp ( - 2 ρ / cos i 2 ) ] ( 1 - S ) ]
r - = p 0 Y - exp ( - ρ / cos i 2 ) 1 - S - exp ( - 2 ρ / cos i 2 ) .
sin i 0 = n 1 sin i 1 = n 2 sin i 2 ,
p - j q = n 1 cos i 1             ( p and q real ) ,
ϕ ( p , q , u ) = ( p - u - j q ) / ( p + u + j q ) ,
r 0 s = ϕ ( p , q , cos i 0 ) ,
r 0 p / r 0 s = ϕ ( p , q , sin i 0 t g i 0 ) ,
r 1 s = ϕ ( p , q , n 2 cos i 2 ) ,
r 1 p / r 1 s = ϕ ( p , q , n 2 sin i 2 t g i 2 ) ,
ψ = ( 4 π d n 1 cos i 1 ) / λ ,
= e - j ψ / 2 ,
+ = - r 0 + r 1 2 ) / ( 1 - r 0 r 1 2 ) 2 ,
- = ( - r 1 + r 0 2 ) / ( 1 - r 0 r 1 2 ) 2 ,
T = | ( 1 - r 0 ) ( 1 + r 1 ) 1 - r 0 r 1 2 | 2 × n 2 cos i 2 cos i 0 .
F i ( x p - 1 , y p - 1 , z p - 1 ) + F i x ( x p - 1 , y p - 1 , z p - 1 ) δ x p + F i y ( x p - 1 , y p - 1 , z p - 1 ) δ y p + F i z ( x p - 1 , y p - 1 , z p - 1 ) δ z p .
δ x Σ p i ( F i x ) 2 + δ y Σ p i F i x F i y + δ z Σ p i F i x F i z = Σ p i ( F i - M i ) F i x , δ x Σ p i F i x F i y + δ y Σ p i ( F i y ) 2 + δ z Σ p i F i y F i z = Σ p i ( F i - M i ) F i y , δ x Σ p i F i x F i z + δ y Σ p i F i y F i z + δ z Σ p i ( F i z ) 2 = Σ p i ( F i - M i ) F i z .
r 0 = ρ 0 e - j ϕ 0 ,
t 0 s = 1 + r 0 s ,
t 0 p = ( 1 / n 1 ) ( 1 - r 0 p ) .
r 1 = ρ 1 e - j ϕ 1 ,
t 1 s = 1 - r 1 s ,
t 1 p = ( n 2 / n 1 ) ( 1 + r 1 p ) .
p q = n k ,
p 2 - q 2 = n 2 - k 2 - sin 2 i 0 ,
α = 4 π q / λ ,
β = 1 / L ,
δ = 4 π p / λ ,
sin ϕ 2 = δ / ( β 2 + δ 2 ) 1 2 ,
cos ϕ 2 = β / ( β 2 + δ 2 ) 1 2 ,
f ( i 1 ) = cos i 1 2 - sin i 1 2 cos i 1 2 + sin i 1 2 = p 2 + q 2 - sin 2 i 0 p 2 + q 2 + sin 2 i 0 .
D 2 = 1 + ρ 1 2 ρ 0 2 e - 2 α d + 2 ρ 1 ρ 0 [ cos ( δ d + Φ 1 + Φ 0 ) ] e - α d .
Y s + = 4 π n k λ n 0 cos i 0 t 0 s 2 β + α 1 D s 2 A s + ,
A s + = 1 - [ 1 + ρ 1 s 2 β + α β - α + 2 ρ 1 s cos ( Φ 1 s - Φ 2 ) α + β ( β 2 + δ 2 ) 1 2 ] × e - ( α + β ) d + 2 ρ 1 s β + α ( δ 2 + β 2 ) 1 2 cos ( δ d + Φ 1 s - Φ 2 ) e - α d + ρ 1 s 2 β + α β - α e - 2 α d ,
Y s - = 4 π n k λ n 0 cos i 0 t 2 s 2 t 1 s 2 β - α e - α d D s 2 A s - ,
A s - = 1 + ρ 0 s 2 β - α β + α - 2 ρ 0 s β - α ( β 2 + δ 2 ) 1 2 cos ( Φ 0 s + Φ 2 ) - e - ( β - α ) d [ 1 + ρ 0 s 2 β - α β + α e - 2 α d - 2 ρ 0 s β - α ( β 2 + δ 2 ) 1 2 cos ( δ d + Φ 0 s + Φ 2 ) e - α d ] ,
Y p + = 4 π n k λ n 0 cos i 0 ( cos i 1 2 + sin i 1 2 ) D p 2 A p + ,
A p + = 1 - [ 1 + ρ 1 p 2 β + α β - α + 2 ρ 1 p × cos ( Φ 1 p - Φ 2 ) α + β ( β 2 + δ 2 ) 1 2 f ( i 1 ) ] e - ( β + α ) d + 2 ρ 1 p β + α ( β 2 + δ 2 ) 1 2 f ( i 1 ) cos ( δ d + Φ 1 p - Φ 2 ) e - α d + ρ 1 p 2 β + α β - α e - 2 α d ,
Y p - = 4 π n k λ n 0 cos i 0 t 2 p 2 t 1 p 2 β - α × ( cos i 1 2 + sin i 1 2 ) e - α d D p 2 A p - ,
A p - = 1 + ρ 0 p 2 β - α β + α - 2 ρ 0 p β - α ( β 2 + δ 2 ) 1 2 cos ( Φ 0 p + Φ 2 ) f ( i 1 ) - e - ( β - α ) d [ 1 + ρ 0 p 2 β - α β + α e - 2 α d - 2 ρ 0 p β - α ( β 2 + δ 2 ) 1 2 f ( i 1 ) cos ( δ d + Φ 0 p + Φ 2 ) e - α d ] .