Abstract

By consideration of the formulae for the reflection and transmission of up to three films, it has been possible to form a generalization for an arbitrary number of films. This generalization is stated as a simple rule whereby the expressions for the reflected and transmitted amplitudes can be readily formed in terms of the Fresnel coefficients of the various boundaries. The generalization is proved by induction. It is also shown that the solutions of the problem by the consideration of the electromagnetic disturbance in each medium, and by the summation of multiple beams, lead to the same result.

© 1948 Optical Society of America

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References

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  1. Rayleigh, Scientific Papers (Cambridge University Press, Teddington, England, 1902), Vol. 3, p. 63.
  2. P. Drude, Theory of Optics (Longmans Green and Company, New York, 1902), p. 302.
  3. T. C. Fry, J. Opt. Soc. Am. 16, 1 (1928).
    [CrossRef]
  4. J. B. Nathanson and C. L. Bartberger, J. Opt. Soc. Am. 29, 417 (1939).
    [CrossRef]
  5. C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).
    [CrossRef]
  6. K. B. Blodgett, Phys. Rev. 57, 921 (1940).
    [CrossRef]
  7. J. A. Stratton, Electro Magnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 511.
  8. K. M. Greenland, Nature 152, 290 (1943).
    [CrossRef]
  9. A. B. Winterbottom, Trans. Faraday Soc. 42, 478 (1946).
    [CrossRef]
  10. A. Vašiček, J. Opt. Soc. Am. 37, 145 (1947).
    [CrossRef]
  11. P. King and L. B. Lockhart, J. Opt. Soc. Am. 36, 513 (1946).
    [CrossRef]
  12. R. Messner, Optik 2, 228 (1947).
  13. L. B. Lockhart and P. King, J. Opt. Soc. Am. 36, 689 (1947).
    [CrossRef]
  14. L. N. Hadley and D. M. Dennison, J. Opt. Soc. Am. 37, 451 (1947).
    [CrossRef]
  15. R. L. Mooney, J. Opt. Soc. Am. 35, 574 (1945).
    [CrossRef]
  16. D. L. Caballero, J. Opt. Soc. Am. 37, 176 (1947).
    [CrossRef] [PubMed]
  17. W. Weinstein, J. Opt. Soc. Am. 37, 576 (1947).
    [CrossRef] [PubMed]
  18. A. Herpin, Comptes Rendus 225, 182 (1947).
  19. A. Vašiček, J. Opt. Soc. Am. 37, 623 (1947).
    [CrossRef]
  20. See reference 2, p. 296.
  21. R. W. Wood, Physical Optics (MacMillan Company, Ltd., London, 1939), third edition, p. 416.
  22. W. König, Handbuch der Physik 20, 229 (1928).
  23. See reference 1, p. 67.

1947 (8)

1946 (2)

P. King and L. B. Lockhart, J. Opt. Soc. Am. 36, 513 (1946).
[CrossRef]

A. B. Winterbottom, Trans. Faraday Soc. 42, 478 (1946).
[CrossRef]

1945 (1)

1943 (1)

K. M. Greenland, Nature 152, 290 (1943).
[CrossRef]

1940 (1)

K. B. Blodgett, Phys. Rev. 57, 921 (1940).
[CrossRef]

1939 (2)

1928 (2)

T. C. Fry, J. Opt. Soc. Am. 16, 1 (1928).
[CrossRef]

W. König, Handbuch der Physik 20, 229 (1928).

Bartberger, C. L.

Blodgett, K. B.

K. B. Blodgett, Phys. Rev. 57, 921 (1940).
[CrossRef]

Caballero, D. L.

Dennison, D. M.

Drude, P.

P. Drude, Theory of Optics (Longmans Green and Company, New York, 1902), p. 302.

Fry, T. C.

Greenland, K. M.

K. M. Greenland, Nature 152, 290 (1943).
[CrossRef]

Hadley, L. N.

Herpin, A.

A. Herpin, Comptes Rendus 225, 182 (1947).

King, P.

L. B. Lockhart and P. King, J. Opt. Soc. Am. 36, 689 (1947).
[CrossRef]

P. King and L. B. Lockhart, J. Opt. Soc. Am. 36, 513 (1946).
[CrossRef]

König, W.

W. König, Handbuch der Physik 20, 229 (1928).

Leberknight, C. E.

Lockhart, L. B.

L. B. Lockhart and P. King, J. Opt. Soc. Am. 36, 689 (1947).
[CrossRef]

P. King and L. B. Lockhart, J. Opt. Soc. Am. 36, 513 (1946).
[CrossRef]

Lustman, B.

Messner, R.

R. Messner, Optik 2, 228 (1947).

Mooney, R. L.

Nathanson, J. B.

Rayleigh,

Rayleigh, Scientific Papers (Cambridge University Press, Teddington, England, 1902), Vol. 3, p. 63.

Stratton, J. A.

J. A. Stratton, Electro Magnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 511.

Vašicek, A.

Weinstein, W.

Winterbottom, A. B.

A. B. Winterbottom, Trans. Faraday Soc. 42, 478 (1946).
[CrossRef]

Wood, R. W.

R. W. Wood, Physical Optics (MacMillan Company, Ltd., London, 1939), third edition, p. 416.

Comptes Rendus (1)

A. Herpin, Comptes Rendus 225, 182 (1947).

Handbuch der Physik (1)

W. König, Handbuch der Physik 20, 229 (1928).

J. Opt. Soc. Am. (11)

Nature (1)

K. M. Greenland, Nature 152, 290 (1943).
[CrossRef]

Optik (1)

R. Messner, Optik 2, 228 (1947).

Phys. Rev. (1)

K. B. Blodgett, Phys. Rev. 57, 921 (1940).
[CrossRef]

Trans. Faraday Soc. (1)

A. B. Winterbottom, Trans. Faraday Soc. 42, 478 (1946).
[CrossRef]

Other (6)

Rayleigh, Scientific Papers (Cambridge University Press, Teddington, England, 1902), Vol. 3, p. 63.

P. Drude, Theory of Optics (Longmans Green and Company, New York, 1902), p. 302.

J. A. Stratton, Electro Magnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 511.

See reference 2, p. 296.

R. W. Wood, Physical Optics (MacMillan Company, Ltd., London, 1939), third edition, p. 416.

See reference 1, p. 67.

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Figures (6)

Fig. 1
Fig. 1

The electromagnetic disturbance in a system of n films sectioned in a plane y=const., on a plane x=const.

Fig. 2
Fig. 2

Reflection and transmission at a single surface.

Fig. 3
Fig. 3

The electromagnetic disturbance in a system of n+1 films.

Fig. 4
Fig. 4

The schemes for n and n+1 films.

Fig. 5
Fig. 5

The beams to be summed in calculating the reflected wave.

Fig. 6
Fig. 6

The beams to be summed in calculating the transmitted wave.

Equations (89)

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ξ = ξ 0 exp i ω [ t - μ k ( a k x - c k z ) / v ]
ψ = ψ 0 exp i ω [ t - μ k ( a k x + c k z ) / v ]
μ 1 a 1 = μ 2 a 2 = = μ k a k = = μ n + 2 a n + 2 ,
c k = + ( 1 - a k 2 ) 1 2 = + { 1 - ( μ 1 a 1 / μ k ) 2 } 1 2 ,
R k = P k exp - i ϕ k ,
S k = Q k exp + i ϕ k ,
ϕ k = - ω μ k c k d k / v = - 2 π μ k c k d k / λ 0 .
c k = - i [ ( μ 1 a 1 / μ k ) 2 - 1 ] 1 2 .
Q 1 = P 1 r 12 ,             R 2 = P 1 t 12 ,
r 12 = - r 21 ,
r 12 2 + t 12 t 21 = 1.
r k k + 1 = [ μ k c k - μ k + 1 c k + 1 ] / [ μ k c k + μ k + 1 c k + 1 ] ,
t k k + 1 = 2 μ k c k / [ μ k c k + μ k + 1 c k + 1 ] .
r k k + 1 = [ μ k + 1 c k - μ k c k + 1 ] / [ μ k + 1 c k + μ k c k + 1 ] ,
t k k + 1 = 2 μ k c k / [ μ k + 1 c k + μ k c k + 1 ] .
c k = cos θ k ,             c k + 1 = cos θ k + 1 ,
μ k + 1 / μ k = sin θ k / sin θ k + 1 ,
R k = P k - 1 t k - 1 k + S k r k k - 1 , Q k - 1 = P k - 1 r k - 1 k + S k t k k - 1 ;
P k - 1 = [ R k + S k r k - 1 k ] / t k - 1 k ,
Q k - 1 = [ R k r k - 1 k + S k ] / t k - 1 k .
Q 1 / P 1 = α 13 = [ r 12 + r 23 exp i 2 ϕ 2 ] / [ 1 + r 12 r 23 exp i 2 ϕ 2 ] .
R 3 / P 1 = β 13 = t 12 t 23 exp i ϕ 2 / D 13 .
Q 1 / P 1 = α 14 = [ r 12 + r 23 exp i ( 2 ϕ 2 ) + r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) + r 12 r 23 r 34 exp i ( 2 ϕ 3 ) 1 + r 12 r 23 exp i ( 2 ϕ 2 ) + r 12 r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) + r 23 r 34 exp i ( 2 ϕ 3 ) ] ,
R 4 / P 1 = β 14 = t 12 t 23 t 34 exp i ( ϕ 2 + ϕ 3 ) / D 14 .
α 15 = [ r 12 + r 23 exp i ( 2 ϕ 2 ) + r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) + r 45 exp i ( 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) + r 12 r 23 r 34 exp i ( 2 ϕ 3 ) + r 12 r 34 r 45 exp i ( 2 ϕ 4 ) + r 12 r 23 r 45 exp i ( 2 ϕ 3 + 2 ϕ 4 ) + r 23 r 34 r 45 exp i ( 2 ϕ 2 + 2 ϕ 4 ) 1 + r 12 r 23 exp i ( 2 ϕ 2 ) + r 12 r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) + r 12 r 45 exp i ( 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) + r 23 r 34 exp i ( 2 ϕ 3 ) + r 34 r 45 exp i ( 2 ϕ 4 ) + r 23 r 45 exp i ( 2 ϕ 3 + 2 ϕ 4 ) + r 12 r 23 r 34 r 45 exp i ( 2 ϕ 2 + 2 ϕ 4 ) ] ,
β 15 = t 12 t 23 t 34 t 45 exp i ( ϕ 2 + ϕ 3 + ϕ 4 ) / D 15 ,
r 12 r 23 exp i ( 2 ϕ 2 ) r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) r k k + 1 exp i ( 2 ϕ 2 + 2 ϕ 3 + + 2 ϕ k ) r n n + 1 exp i ( 2 ϕ 2 + 2 ϕ 3 + + 2 ϕ n ) r n + 1 n + 2 exp i ( 2 ϕ 2 + 2 ϕ 3 + + 2 ϕ n + 2 ϕ n + 1 ) .
r l l + 1 r k k + 1 r p p + 1 exp i { + ( 2 ϕ 2 + + 2 ϕ l ) - ( 2 ϕ 2 + + 2 ϕ k ) + ( 2 ϕ 2 + + 2 ϕ p ) } ,
r 12 + r 23 exp i ( 2 ϕ 2 ) + + r n + 1 n + 2 exp i ( 2 ϕ 2 + 2 ϕ 3 + + 2 ϕ n + 1 ) plus
plus
plus
1 plus
plus
plus
β 1 n + 2 = t 12 t 23 t n + 1 n + 2 × exp i ( ϕ 2 + ϕ 3 + + ϕ n + 1 ) / D 1 n + 2 .
r 12 r 23 exp i ( 2 ϕ 2 ) r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) r 45 exp i ( 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) .
r 12 + r 23 exp i ( 2 ϕ 2 ) + r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) + r 45 exp i ( 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) plus r 12 r 23 r 34 exp i ( - 2 ϕ 2 + 2 ϕ 2 + 2 ϕ 3 ) + r 12 r 34 r 45 exp i ( - 2 ϕ 2 - 2 ϕ 3 + 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) + r 12 r 23 r 45 exp i ( - 2 ϕ 2 + 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) + r 23 r 34 r 45 exp i ( + 2 ϕ 2 - 2 ϕ 2 - 2 ϕ 3 + 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) ,
1 plus r 12 r 23 exp i ( 2 ϕ 2 ) + r 12 r 34 exp i ( 2 ϕ 2 + 2 ϕ 3 ) + r 12 r 45 exp i ( 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) + r 23 r 34 exp i ( - 2 ϕ 2 + 2 ϕ 2 + 2 ϕ 3 ) + r 34 r 45 exp i ( - 2 ϕ 2 - 2 ϕ 3 + 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) + r 23 r 45 exp i ( - 2 ϕ 2 + 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) plus r 12 r 23 r 34 r 45 exp i ( + 2 ϕ 2 - 2 ϕ 2 - 2 ϕ 3 + 2 ϕ 2 + 2 ϕ 3 + 2 ϕ 4 ) .
P 1 = [ P 2 exp - i ϕ 2 + r 12 Q 2 exp i ϕ 2 ] / t 12 ,
Q 1 = [ r 12 P 2 exp - i ϕ 2 + Q 2 exp i ϕ 2 ] / t 12 .
Q 1 / P 1 = [ r 12 + ( Q 2 / P 2 ) exp i 2 ϕ 2 ] / [ 1 + r 12 ( Q 2 / P 2 ) exp i 2 ϕ 2 ] .
Q 2 / P 2             as             N 2 n + 3 / D 2 n + 3 , Q 1 / P 1 = [ r 12 D 2 n + 3 + N 2 n + 3 exp i 2 ϕ 2 ] / [ D 2 n + 3 + r 12 N 2 n + 3 exp i 2 ϕ 2 ] .
r 12 D 2 n + 3 + N 2 n + 3 exp i ( 2 ϕ 2 ) = N 1 n + 3 .
D 2 n + 3 + r 12 N 2 n + 3 exp i ( 2 ϕ 2 ) = D 1 n + 3 .
R n + 3 / P 1 = t 12 t 23 t n + 2 n + 3 exp i ( ϕ 2 + + ϕ n + 2 ) / { D 2 n + 3 + r 12 N 2 n + 3 exp i ( 2 ϕ 2 ) } ,
R n + 3 / P 1 = β 1 n + 3 = t 12 t 23 t n + 2 n + 3 × exp i ( ϕ 2 + + ϕ n + 2 ) / D 1 n + 3 .
C n + 1 1 + C n + 1 3 + C n + 1 5 + ,
C n + 1 0 + C n + 1 2 + C n + 1 4 + ,
α 1 n + 2 = [ X + Y r 12 ] / [ X r 12 + Y ] ,
plus
plus
Q 1 = P 1 [ r 12 + t 12 α 2 M t 21 exp i ( 2 ϕ 2 ) + t 12 α 2 M r 21 α 2 M t 21 exp i ( 4 ϕ 2 ) + ] = P 1 [ r 12 + ( t 12 α 2 M t 21 exp i 2 ϕ 2 ) / ( 1 - α 2 M r 21 exp i 2 ϕ 2 ) ] .
α 1 M = Q 1 / P 1 = [ r 12 + α 2 M exp i ( 2 ϕ 2 ) ] / [ 1 + r 12 α 2 M exp i ( 2 ϕ 2 ) ] .
α 1 n + 3 = [ r 12 D 2 n + 3 + N 2 n + 3 exp i ( 2 ϕ 2 ) ] / [ D 2 n + 3 + r 12 N 2 n + 3 exp i ( 2 ϕ 2 ) ] ,
P 2 = P 1 [ t 12 exp i ( ϕ 2 ) + t 12 α 2 M r 21 exp i ( 3 ϕ 2 ) + t 12 α 2 M r 21 α 2 M r 21 exp i ( 5 ϕ 2 ) + ] .
P 2 = P 1 t 12 exp i ( ϕ 2 ) / [ 1 + r 12 α 2 M exp i ( 2 ϕ 2 ) ] ,
R M = P 2 β 2 M = P 1 β 2 M t 12 exp i ( ϕ 2 ) / [ 1 + r 12 α 2 M exp i ( 2 ϕ 2 ) ] ,
β 1 M = R M / P 1 = β 2 M D 2 M t 12 exp i ( ϕ 2 ) / [ D 2 M + r 12 N 2 M exp i ( 2 ϕ 2 ) ] .
β 1 n + 3 = t 12 t 23 t n + 2 n + 3 × exp i ( ϕ 2 + ϕ 3 + + ϕ n + 2 ) / D 1 n + 3 ,
exp i ϕ k + 1 = [ exp - i ω ν k + 1 c k + 1 d k + 1 / v ] × [ exp - ω ν k + 1 κ k + 1 c k + 1 d k + 1 / v ] = 0.
r 12 r k - 1 k exp i ( 2 ϕ 2 + + 2 ϕ k - 1 ) r k k + 1 exp i ( 2 ϕ 2 + + 2 ϕ k - 1 + 2 ϕ k ) r k + 1 k + 2 exp i ( 2 ϕ 2 + + 2 ϕ k - 1 + 2 ϕ k + 2 ϕ k + 1 ) r n n + 1 exp i ( 2 ϕ 2 + + 2 ϕ n ) r n + 1 n + 2 exp i ( 2 ϕ 2 + + 2 ϕ n + 2 ϕ n + 1 ) .
r 12 exp - i ( 2 ϕ 2 + + 2 ϕ k - 1 + 2 ϕ k ) r k - 1 k exp - i ( 2 ϕ k ) r k k + 1 r k + 1 k + 2 exp i ( 2 ϕ k + 1 ) r n n + 1 exp i ( 2 ϕ k + 1 + + 2 ϕ n ) r n + 1 n + 2 exp i ( 2 ϕ k + 1 + + 2 ϕ n + 2 ϕ n + 1 ) .
N 1 n + 2 = N 1 n + 2 exp - i × ( 2 ϕ 2 + + 2 ϕ k - 1 + 2 ϕ k )
D 1 n + 2 = D 1 n + 2 ,
x 1 exp i θ 1 + x 2 exp i θ 2 + + x m exp i θ m
X ¯ 1 Y 2 + Y 1 X 2 ,
r k k + 1 ( X ¯ 1 X 2 + Y 1 Y 2 ) .
N 1 n + 2 = [ ( X ¯ 1 Y 2 + Y 1 X 2 ) + r k k + 1 ( X ¯ 1 X 2 + Y 1 Y 2 ) ] .
D 1 n + 2 = [ ( X 1 X 2 + Y ¯ 1 Y 2 ) + r k k + 1 ( X 1 Y 2 + Y ¯ 1 X 2 ) ] .
N 1 n + 2 / D 1 n + 2 2 = N 1 n + 2 / D 1 n + 2 2 = α 1 n + 2 2 .
α 1 n + 2 2 = [ ( X ¯ 1 Y 2 + Y 2 X 2 ) + r k k + 1 ( X ¯ 1 X 2 + Y 1 Y 2 ) ] [ ( X 1 Y ¯ 2 + Y ¯ 1 X ¯ 2 ) + r ¯ k k + 1 ( X 1 X ¯ 2 + Y ¯ 1 Y ¯ 2 ) ] [ ( X 1 X 2 + Y ¯ 1 Y 2 ) + r k k + 1 ( X 1 Y 2 + Y ¯ 1 X 2 ) ] [ ( X ¯ 1 X ¯ 2 + Y 1 Y ¯ 2 ) + r ¯ k k + 1 ( X ¯ 1 Y ¯ 2 + Y 1 X ¯ 2 ) ] ,
α 1 n + 2 2 = [ A + B r k k + 1 + B ¯ r ¯ k k + 1 ] / [ A + C r k k + 1 + C ¯ r ¯ k k + 1 ] ,
α 1 n + 2 2 = { [ Y 1 + r k k + 1 X ¯ 1 ] [ Y ¯ 1 + r ¯ k k + 1 X 1 ] } / { [ X 1 + r k k + 1 Y ¯ 1 ] [ X ¯ 1 + r ¯ k k + 1 Y 1 ] } .
X 2 = X r k + 1 k + 2 + Y
Y 2 = X + Y r k + 1 k + 2
r exp i δ = [ r + r exp - i X 1 + r exp - i ( X 1 + X 2 ) + r r r exp - i ( 2 X 1 + X 2 ) 1 + r r exp - i X 1 + r r exp - i ( X 1 + X 2 ) + r r exp - i ( 2 X 1 + X 2 ) ] ,
r exp i δ = α 14 ,             r = r 12 ,             r = r 23 ,             r = r 34 ,             - X 1 = 2 ϕ 2 ,             - X 2 = 2 ϕ 3 .
[ text illegible on the printed page ] = [ 1 + r { exp i ( X 1 + X 2 ) + exp - i ( X 1 + X 2 ) } + r { exp i ( X 2 ) + exp - i ( X 2 ) } + r 2 + r 2 + 2 r r { exp i ( X 1 ) + exp - i ( X 1 ) } + r 2 r { exp i ( 2 X 1 + X 2 ) + exp - i ( 2 X 1 + X 2 ) } + r r 2 { exp i ( X 1 + X 2 ) + exp - i ( X 1 + X 2 ) } + r 2 r 2 1 + r { exp i ( X 1 + X 2 ) + exp - i ( X 1 + X 2 ) } + r { exp i ( 2 X 1 + X 2 ) + exp - i ( 2 X 1 + X 2 ) } + r 2 + r 2 + 2 r r { exp i ( X 1 ) + exp - i ( X 1 ) } + r 2 r { exp i ( X 2 ) + exp - i ( X 2 ) } + r r 2 { exp i ( X 1 + X 2 ) + exp - i ( X 1 + X 2 ) } + r 2 r 2 ] .
N 1 n + 2 exp - i ( 2 ϕ 2 + + 2 ϕ n + 1 )
N ¯ n + 2 1 / D ¯ n + 2 1 = - N 1 n + 2 × exp - i ( 2 ϕ 2 + + 2 ϕ n + 1 ) / D ¯ 1 n + 2 ,
N n + 2 1 / D n + 2 1 = - N ¯ 1 n + 2 × exp i ( 2 ϕ 2 + + 2 ϕ n + 1 ) / D 1 n + 2 .
α 1 n + 2 2 = α n + 2 1 2 .
N 1 n + 2 = 1 2 [ ( 1 + r 12 ) ( 1 + r 23 ) ( 1 + r n + 1 n + 2 ) - ( 1 - r 12 ) ( 1 - r 23 ) ( 1 - r n + 1 n + 2 ) ]
D 1 n + 2 = 1 2 [ ( 1 + r 12 ) ( 1 + r 23 ) ( 1 + r n + 1 n + 2 ) + ( 1 - r 12 ) ( 1 - r 23 ) ( 1 - r n + 1 n + 2 ) ] .
α 1 n + 2 = [ ( μ 1 c 1 ) ( μ 2 c 2 ) ( μ n + 1 c n + 1 ) - ( μ 2 c 2 ) ( μ 3 c 3 ) ( μ n + 2 c n + 2 ) ] [ ( μ 1 c 1 ) ( μ 2 c 2 ) ( μ n + 1 c n + 1 ) + ( μ 2 c 2 ) ( μ 3 c 3 ) ( μ n + 2 c n + 2 ) ] .
α 1 n + 2 = [ μ 1 c 1 - μ n + 2 c n + 2 ] / [ μ 1 c 1 + μ n + 2 c n + 2 ] = r 1 n + 2 .
α 1 n + 2 = [ μ n + 2 c 1 - μ 1 c n + 2 ] / [ μ n + 2 c 1 + μ 1 c n + 2 ] = r 1 n + 2 .
β 1 n + 2 = t 12 t 23 t n + 1 n + 2 / 1 2 [ ( 1 + r 12 ) ( 1 + r 23 ) ( 1 + r n + 1 n + 2 ) + ( 1 - r 12 ) ( 1 - r 23 ) ( 1 - r n + 1 n + 2 ) ] .
β 1 n + 2 = 2 μ 1 c 1 / [ μ 1 c 1 + μ n + 2 c n + 2 ] = t 1 n + 2 ,