Abstract

The effects on reflectivity of a statistical variation in the thickness of layers in a multilayered Bragg reflector are studied. Analytic expressions are obtained for 〈ρ〉 and 〈ρρ*〉, the expected value of the reflection and reflectivity coefficients as a function of σ, the standard deviation in layer thickness. These expressions are then compared with values obtained using a computer routine which “builds” a reflector with the desired parameters and σ value, and then calculates the reflection. The results of the computer experiment are presented in the form of p(ρρ*), the probability distribution function of a statistical Bragg reflector. Finally, simple phenomenological expressions are presented for the reflectivity probability distribution.

© 1978 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (MacMillan, New York, 1964).
  2. P. Yeh, A. Yariv, and C. Hong, J. Opt. Soc. Am. 67, 423 (1977).
    [Crossref]
  3. A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).
  4. N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
    [Crossref]
  5. C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
    [Crossref]
  6. S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955) [Sov. Phys.-JETP 2, 466 (1956)].
  7. J. P. Vander Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. (to be published).
  8. Private communication with Wess Icenogle of Spectra Physics, Mountain View, Calif.
  9. Private communication with Steve Silver of OCLI, Santa Rosa, Calif.
  10. K. O. Hill, Appl. Opt. 13, 1853 (1974).
    [Crossref] [PubMed]
  11. A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
    [Crossref]
  12. C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
    [Crossref]
  13. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1971), p. 85.
  14. Petr Beckmann, The Scattering of Electromagnetic Waves From Rough Surfaces (MacMillan, New York, 1963), p. 81.
  15. H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
    [Crossref]
  16. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [Crossref]
  17. W. Streifer, D. R. Scifres, and R. D. Burnham, J. Opt. Soc. Am. 66, 1359 (1976).
    [Crossref]
  18. H. W. Harman, Principles of the Statistical Theory of Communication, (McGraw-Hill, New York, 1963), p. 75.

1977 (3)

P. Yeh, A. Yariv, and C. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[Crossref]

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
[Crossref]

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

1976 (2)

1974 (1)

1973 (2)

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[Crossref]

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

1970 (1)

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

1955 (1)

S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955) [Sov. Phys.-JETP 2, 466 (1956)].

Ashkin, A.

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).

Beckmann, Petr

Petr Beckmann, The Scattering of Electromagnetic Waves From Rough Surfaces (MacMillan, New York, 1963), p. 81.

Bey, P. P.

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (MacMillan, New York, 1964).

Burnham, R. D.

Harman, H. W.

H. W. Harman, Principles of the Statistical Theory of Communication, (McGraw-Hill, New York, 1963), p. 75.

Hill, K. O.

Hong, C.

P. Yeh, A. Yariv, and C. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[Crossref]

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

Icenogle, Wess

Private communication with Wess Icenogle of Spectra Physics, Mountain View, Calif.

Illegems, M.

J. P. Vander Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. (to be published).

Katzir, A.

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
[Crossref]

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1971), p. 85.

Kogelnik, H.

H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
[Crossref]

Livanos, A. C.

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
[Crossref]

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

Mikulyak, R. M.

J. P. Vander Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. (to be published).

Rytov, S. M.

S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955) [Sov. Phys.-JETP 2, 466 (1956)].

Scifres, D. R.

Shellan, J. B.

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
[Crossref]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

Silver, Steve

Private communication with Steve Silver of OCLI, Santa Rosa, Calif.

Streifer, W.

Tang, C. L.

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

Vander Ziel, J. P.

J. P. Vander Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. (to be published).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (MacMillan, New York, 1964).

Yariv, A.

P. Yeh, A. Yariv, and C. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[Crossref]

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
[Crossref]

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[Crossref]

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).

Yeh, P.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, Appl. Phys. Lett. 31, 276 (1977).
[Crossref]

Bell. Syst. Tech. J. (1)

H. Kogelnik, Bell. Syst. Tech. J. 55, 109–126 (1976).
[Crossref]

IEEE J. Quantum Electron. (3)

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[Crossref]

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IEEE J. Quantum Electron. QE-13, 296 (1977).
[Crossref]

J. Opt. Soc. Am. (2)

Zh. Eksp. Teor. Fiz. (1)

S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955) [Sov. Phys.-JETP 2, 466 (1956)].

Other (8)

J. P. Vander Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. (to be published).

Private communication with Wess Icenogle of Spectra Physics, Mountain View, Calif.

Private communication with Steve Silver of OCLI, Santa Rosa, Calif.

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).

H. W. Harman, Principles of the Statistical Theory of Communication, (McGraw-Hill, New York, 1963), p. 75.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1971), p. 85.

Petr Beckmann, The Scattering of Electromagnetic Waves From Rough Surfaces (MacMillan, New York, 1963), p. 81.

M. Born and E. Wolf, Principles of Optics (MacMillan, New York, 1964).

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Figures (13)

FIG. 1
FIG. 1

Geometry of reflection with n cells used in low-reflectivity case.

FIG. 2
FIG. 2

Average reflectance as a function of layer standard deviation and shift from center of the band gap, indicating the broadening and lowering of the response curve, for the case of 25 cells.

FIG. 3
FIG. 3

Average amplitude reflectance in low-reflectance limit as a function of cell standard deviation and the number of cells. Note that the parameters of each structure have been chosen to give a 10% reflectance for a perfect reflector.

FIG. 4
FIG. 4

Average intensity reflectivity in low-reflectance limit as a function of cell standard deviation and the number of cells. Note how, as becomes large the asymptotic reflectance becomes 0.01/(2N + 1).

FIG. 5
FIG. 5

Plot of the function G(κL).

FIG. 6
FIG. 6

Plot of the data given in Table I (σr = 2%).

FIG. 7
FIG. 7

Comparison of computer experimental results and theory for a 10-cell structure with various values of σr.

FIG. 8
FIG. 8

Experimental distribution of P(R) (circles) as compared to theoretical prediction (solid line) for σr = 2% for structures with 10 cells. The average reflectivity 〈R〉 and the reflectivity of a perfect structure Rp are indicated by the thin vertical lines.

FIG. 9
FIG. 9

Same as Fig. 8, but for structure with 25 cells.

FIG. 10
FIG. 10

Same as Fig. 8, but for structure with 50 cells.

FIG. 11
FIG. 11

Experimental distribution of P(R) as compared to theoretical prediction for structures containing 25 cells and have σr = 5%.

FIG. 12
FIG. 12

Same as Fig. 11, but for σr = 7.5%.

FIG. 13
FIG. 13

Same as Fig. 11, but for σr = 10%. Note the expected broadening of the curve with increasing σr.

Tables (1)

Tables Icon

TABLE I Table comparing results of the computer experiment with the two analytic expressions. Note the close agreement between the second order ϕ′ expression and the experiment. Rp = ρρ* for perfect structure; 〈R〉 = 〈ρρ*〉; σr = 0.02; n1 = 3.6; n2 = 3.4.

Equations (121)

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ρ = r 1 ( 1 + exp [ 2 i ( k 1 x a 1 + k 2 x b 1 ) ] + exp { 2 i [ k 1 x ( a 1 + a 2 ) + k 2 x ( b 1 + b 2 ) ] } + + exp { 2 i [ k 1 x ( a 1 + a 2 + + a N ) + k 2 x ( b 1 + b 2 + - r 1 exp ( 2 i k 2 x b 1 ) ( 1 + exp [ 2 i ( k 1 x a 1 + k 2 x b 2 ) ] + + exp { 2 i [ k 1 x ( a 1 + + a N - 1 ) + k 2 x ( b 2 + + b N ) ] } ) ,
k i x = ( ω / c ) n i cos θ i ,             i = 1 , 2 ,             r 1 = k 1 x - k 2 x k 1 x + k 2 x ,
a p = a ( 0 ) + u p ,             p = 1 , 2 , N b p = b ( 0 ) + v p , a ( 0 ) , b ( 0 ) = ideal thickness of layers ,
exp { 2 i [ k 1 x ( a 1 + + a p ) + k 2 x ( b 1 + + b p ) ] } = exp { 2 i p [ k 1 x a ( 0 ) + k 2 x b ( 0 ) ] } × exp { 2 i [ k 1 x ( u 1 + + u p ) + k 2 x ( v 1 + + v p ) ] } .
e 2 i p [ k 1 x a ( 0 ) + k 2 x b ( 0 ) ] e - 2 [ k 1 x 2 p σ a 2 + k 2 x 2 p σ b 2 ] = e 2 i p k Λ e - 2 p k 2 σ 2 ,
k 2 σ 2 k 1 x 2 σ a 2 + k 2 x 2 σ b 2 , k Λ k 1 x a ( 0 ) + k 2 x b ( 0 ) .
ρ = [ r 1 / ( 1 - e 2 i k Λ e - 2 k 2 σ 2 ) ] × [ 1 - e 2 i ( N + 1 ) k Λ e - 2 ( N + 1 ) k 2 σ 2 - e 2 i k 2 x b ( 0 ) × e - 2 k 2 x 2 σ b 2 ( 1 - e 2 i N k Λ e - 2 N k 2 σ 2 ) ] .
ρ = ( 2 N + 1 ) r 1 ( 1 - N k 2 σ 2 ) = r 0 ( 1 - N k 2 σ 2 ) .
ρ = r 0 e - G 2 σ 2 / 6 r 0 [ 1 - ( G 2 / 6 ) σ 2 ] ,
ρ = ½ r 0 [ e - ( π / 2 ) σ r a 2 - ( π / 2 ) σ r b 2 ] r 0 [ 1 - ( π / 4 ) ( σ r a 2 + σ r b 2 ) ] ,
ρ ρ * = r 1 2 [ ( N + 1 + 2 e - 2 N k 2 σ 2 ( 1 - e 2 k 2 σ 2 ) 2 [ 1 - ( N + 1 ) e 2 k 2 σ 2 N + N e 2 ( N + 1 ) k 2 σ 2 ] ) + ( N + 2 e - 2 ( N - 1 ) k 2 σ 2 ( 1 - e 2 k 2 σ 2 ) 2 × [ 1 - N e 2 k 2 σ 2 ( N - 1 ) + ( N - 1 ) e 2 N k 2 σ 2 ] ) - ( 2 cos ( 2 k 2 x b 1 ( 0 ) ) ( e - 2 k 2 x 2 σ b 2 + e - 2 k 1 x 2 σ a 2 ) × e - 2 ( N - 1 ) k 2 σ 2 ( 1 - e 2 k 2 σ 2 ) 2 [ 1 - N e 2 ( N - 1 ) k 2 σ 2 + ( N - 1 ) e 2 N k 2 σ 2 ] + N ) ]
ρ ρ * = r 1 2 { [ 2 N 2 + 2 N + 1 - 2 ( N 2 + N ) cos ( 2 k 2 x b ( 0 ) ) ] - k 2 σ 2 [ ( 4 3 N 3 + 2 N 2 + 2 3 N ) - 2 ( 2 3 N 3 + N 2 + N 3 ) cos ( 2 k 2 x b ( 0 ) ) ] } .
R = i κ e i ϕ S ,
S = - i κ e - i ϕ R ,
κ = | d S / d z R | = | a . r . / u . l . a . i . | = r 1 Λ 0 / 2 = 2 N r 1 L
R ( - ½ L ) = 1 ,
S ( ½ L ) = 0.
ρ ( - 1 2 ) = S ( - L 2 ) / R ( - L 2 ) = S ( - L 2 ) .
R - i ϕ R - κ 2 R = 0.
R = R 0 ( z ) + R 1 ( z ) + 2 R 2 ( z ) + .
R 0 - κ 2 R 0 = 0 ,
R 1 - κ 2 R 1 = i ϕ R 0 ,
R 2 - κ 2 R 2 = i ϕ R 1 ,
R n - κ 2 R n = i ϕ R n - 1 ,             n 1.
R 0 = cosh [ κ ( ½ L - z ) ] cosh κ L ,
R n = 1 κ - L / 2 z i ϕ ( ξ ) R n - 1 ( ξ ) sinh [ κ ( z - ξ ) ] d ξ - i κ C 1 sinh [ κ ( ½ L + z ) ] cosh κ L × - L / 2 L / 2 ϕ ( ξ ) R n - 1 ( ξ ) cosh [ κ ( ½ L - ξ ) ] d ξ             n = 1 , 2.
ρ ( - L 2 ) = - i κ e - i ϕ ( - L / 2 ) R ( - L 2 ) .
R ϕ ( z 0 ) ϕ ( z ) ϕ ( z + z 0 ) lim W 1 2 W z = - W W ϕ ( z ) ϕ ( z + z 0 ) d z
= Σ 2 ( 1 - z 0 l )             for z 0 l = 0             for z 0 l .
ρ ( - L 2 ) = i e - i ϕ ( - L / 2 ) [ tanh κ L - 2 Σ 2 l 8 κ C 1 3 × ( 2 C 1 5 - 1 4 S 1 S 4 - C 1 C 2 - C 1 + κ L S 1 ) ] ,
ρ ( - L 2 ) ρ * ( - L 2 ) = tanh 2 κ L - 2 Σ 2 l 2 κ C 1 4 [ κ L 4 - 1 16 S 4 - S 1 2 ( C 1 C 2 + C 1 - 2 C 1 5 - κ L S 1 + 1 4 S 1 S 4 ) ] .
ρ ( - L 2 ) = ie - i ϕ ( - L / 2 ) κ L ( 1 - 2 Σ 2 L l 4 ) ,
ρ ( - L 2 ) ρ * ( - L 2 ) = κ 2 L 2 ( 1 - 2 Σ 2 L l 6 ) .
ρ ( - L 2 ) = i e - i ϕ ( - L / 2 ) ( tanh κ L - 2 Σ 2 l 8 κ ) ,
ρ ( - L 2 ) ρ * ( - L 2 ) = tanh 2 κ L - 2 Σ 2 l 2 κ e - 2 κ L ( 2 κ L - 1 ) 1 - e - 2 κ L [ 4 + 2 Σ 2 l κ ( κ L - 1 2 ) ] .
z 2 π Λ ( z ) d z z ( 2 π Λ 0 - 2 π Λ 0 2 δ Λ ) z ) ) d z = 2 π z Λ 0 + ϕ ( z ) ,
ϕ ( z ) = - 2 π Λ 0 2 δ Λ ( z ) ,
R δ Λ ( z 0 ) = δ Λ ( z + z 0 ) δ Λ ( z ) = S 2 ( 1 - z 0 l )             z 0 l = 0             otherwise .
2 ϕ ( z ) ϕ ( z + z 0 ) = 4 π 2 Λ 0 4 δ Λ ( z ) δ Λ ( z + z 0 ) , 2 Σ 2 ( 1 - z 0 l ) = 4 π 2 Λ 0 4 S 2 ( 1 - z 0 l ) .
2 Σ 2 = ( 4 π 2 / Λ 0 4 ) S 2 ,
l = l .
Δ t ( z ) Δ t ( z + z 0 ) = σ ¯ 2 [ 1 - ( 2 z 0 / Λ 0 ) ] , σ ¯ 2 = ½ ( σ a 2 + σ b 2 ) ,             Λ 0 / 2 a ( 0 ) b ( 0 ) .
S 2 = ( 2 σ ¯ ) 2 ,
2 Σ 2 = ( 4 π 2 / Λ 0 4 ) ( 2 σ ¯ ) 2 .
ρ ( - L 2 ) = i e - i ϕ ( - L / 2 ) [ tanh κ L - σ ¯ 2 π 2 Λ 0 3 κ C 1 3 × ( 2 C 1 5 - 1 4 S 1 S 4 - C 1 C 2 - C 1 + κ L S 1 ) ] ,
ρ ( - L 2 ) ρ * ( - L 2 ) = tanh 2 κ L - 4 σ ¯ 2 π 2 Λ 0 3 κ C 1 4 [ κ L 4 - 1 16 × S 4 - S 1 2 ( C 1 C 2 + C 1 - 2 C 1 5 - κ L S 1 + 1 4 S 1 S 4 ) ] = tanh 2 x - 2 π 2 σ ¯ 2 L Λ 0 3 G ( x )
ρ [ - ( L / 2 ) ] ρ * [ - L / 2 ) ] - tanh 2 x tanh 2 x = - 2 π 2 σ ¯ 2 L Λ 0 3 G ( x ) tanh 2 x , x κ L , G ( x ) - 1 x [ 1 2 C 1 4 ( 1 4 S 4 - x ) + S 1 C 1 4 ( C 1 C 2 + C 1 - 2 C 1 5 - x S 1 + 1 4 S 1 S 4 ) ] .
R = i κ e i ϕ S ,
S = - i κ e - i ϕ R ,
R = i κ S + i κ ( e i ϕ - 1 ) S ,
S = - i κ R - i κ ( e - i ϕ - 1 ) R ,
ϕ ( z ) = ω c n 1 i = 1 z / Λ u i + ω c n 2 i = 1 z / Λ v i
R = R 0 + R 1 + R 2 + ,
S = S 0 + S 1 + S 2 + ,
R 0 = i κ S 0 ,
S 0 = - i κ R 0 ,
R n = i κ S n + χ S n - 1 ,             n 1
S n = - i κ R n + χ * R n - 1 ,             n 1.
ρ S 0 ( 0 ) + S 1 ( 0 ) R 0 ( 0 ) = S 0 ( 0 ) + S 1 ( 0 ) .
R 0 ( z ) = cosh κ ( L - z ) C 1 ,
S 0 ( z ) , + i sinh κ ( L - z ) C 1 .
S 1 - κ 2 S 1 = - i κ χ S 0 + χ * R 0 + χ * R 0 .
S 1 ( 0 ) = i κ C 1 2 0 L [ ( e - i ϕ - 1 ) cosh 2 κ ( L - z ) - ( e i ϕ - 1 ) sinh 2 κ ( L - z ) ] d z ,
ρ ρ * = S 0 ( 0 ) + S 1 ( 0 ) 2 S 0 2 + S 1 S 0 * + S 1 * S 0 ,
Δ r 2 ρ ρ * - S 0 2 = S 0 ( S 1 * - S 1 ) .
Δ r 2 = - κ tanh κ L cosh 2 κ L 0 L ( 2 - e i ϕ ( z ) - e - i ϕ ( z ) ) d z .
2 ϕ 2 ( z ) = ω 2 c 2 z Λ ( n 1 2 σ a 2 + n 2 2 σ b 2 ) = 2 z Ψ Λ ,
Ψ ( ω 2 / 2 c 2 ) ( n 1 2 σ a 2 + n 2 2 σ b 2 )
Δ r 2 = - 2 κ S 1 C 1 3 0 L ( 1 - e - Ψ ( z / Λ ) ) d z = - 2 κ S 1 Λ C 1 3 ( N Ψ + e - N Ψ - 1 Ψ ) .
Δ r 2 r 0 2 = - 2 κ Λ tanh κ L cosh 2 κ L ( N Ψ + e - N Ψ - 1 Ψ ) .
p ( r ) = C q - 1 Γ ( q - 1 ) e c / ( r p - r ) ( r p - r ) q ,
q = 3 + ( r p - r ) 2 r 2 - r 2 ,
C = ( r p - r ) [ ( r p - r ) 2 + ( r 2 - r 2 ) ] r 2 - r 2 .
r ρ ρ * = r p - ( κ S 1 / C 1 3 ) f ,
f 0 L ( 2 - e i ϕ ( z ) - e - i ϕ ( z ) ) d z , f = 2 Λ N Ψ + e - N Ψ - 1 Ψ .
r 2 - r 2 = ( κ 2 S 1 2 / C 1 6 ) ( f 2 - f 2 ) .
( r 2 - r 2 ) 1 / 2 = κ S 1 C 1 3 2 3 N L ( π 2 ) 2 ( σ ¯ Λ / 2 ) 2 .
r p - r = κ S 1 C 1 3 N L ( π 2 ) 2 ( σ ¯ Λ / 2 ) 2 ,
( r 2 - r 2 ) 1 / 2 r p - r = 2 3 3 1.15
q = 15 4 ,
C = 7 4 κ S 1 C 1 3 N L ( π 2 ) 2 ( σ ¯ Λ / 2 ) 2 ,
p ( r ) = C 11 / 4 Γ ( 11 / 4 ) e C / ( r p - r ) ( r p - r ) 15 / 4 , Γ ( 11 4 ) = ( 7 4 ) ! 1.608.
r peak = r p - C q = r p - 7 15 κ S 1 C 1 3 N L ( π 2 ) 2 ( σ ¯ Λ / 2 ) 2 ,
r p - r peak r p - r = 7 15 .
R 0 - κ 2 R 0 = 0
R n - κ 2 R n = i ϕ R n - 1 ,             n = 1 , 2
R ( - ½ L ) = 1
S ( ½ L ) = 0 ,
R = R 0 + R 1 + 2 R 2
S = - i e - i ϕ ( R / κ ) .
R 0 ( z ) = cosh { κ [ ( L / 2 ) - z ] } cosh ( κ L ) .
R 1 ( - L 2 ) = R 2 ( - L 2 ) = R 1 ( L 2 ) = R 2 ( L 2 ) = 0.
B n sinh { κ [ ( L / 2 ) + z ] } ,
1 κ - L / 2 z i ϕ ( η ) R n - 1 ( η ) sinh [ κ ( z - η ) ] d η .
R n ( z ) = B n sinh [ κ ( L 2 + z ) ] + 1 κ - L / 2 z i ϕ ( η ) R n - 1 ( η ) sinh κ ( z - η ) ] d η .
R ( - L 2 ) = R 0 ( - L 2 ) + R 1 ( - L 2 ) + 2 R 2 ( - L 2 ) .
R 1 ( - L 2 ) = - i C 1 - L / 2 L / 2 ϕ ( η ) R 0 ( η ) cosh [ κ ( L 2 - η ) ] d η = 0             since ϕ ( η ) = 0
R 2 ( - L 2 ) = - i C 1 - L / 2 L / 2 ϕ ( ξ ) R 1 ( ξ ) cosh [ κ ( L 2 - ξ ) ] d ξ .
R 2 ( - L 2 ) = - κ C 1 2 ξ = - L / 2 L / 2 η = - L / 2 ξ ϕ ( ξ ) ϕ ( η ) cosh [ κ ( L 2 - ξ ) ] cosh [ κ ( ξ - η ) ] sinh [ κ ( L 2 - η ) ] d ξ d η + κ C 1 3 ξ = - L / 2 L / 2 η = - L / 2 L / 2 ϕ ( ξ ) ϕ ( η ) cosh [ κ ( L 2 - ξ ) ] cosh [ κ ( L 2 + ξ ) ] cosh [ κ ( L 2 - η ) ] sinh [ κ ( L 2 - η ) ] d ξ d η .
R ϕ ( ξ - η ) l Σ 2 δ ( ξ - η ) ,
a b δ ( x ) f ( x ) d x = f ( 0 )             for             a < 0 < b
0 b δ ( x ) f ( x ) d x = ½ f ( 0 )             for             b > 0 ,
R 2 ( - L 2 ) = Σ 2 l 8 C 1 3 ( - C 1 C 2 - C 1 + 2 C 1 5 + κ L S 1 - 1 4 S 1 S 4 ) .
ρ ( - L 2 ) = - i κ e - i ϕ ( - L / 2 ) [ R 0 ( - 1 2 L ) + R 1 ( - 1 2 L ) + 2 R 2 ( - 1 2 L ) ]
ρ ( - 1 2 L ) ρ * ( - 1 2 L ) = 1 κ 2 ( R 0 R 0 * + 2 R 1 R 1 * + 2 R 0 R 2 * + R 0 * R 2 ) z = - L / 2 .
R 0 R 2 * + R 0 * R 2 = R 0 R 2 * + R 0 * R 2 .
R 1 ( - L 2 ) R 1 * ( - L 2 ) = κ 2 C 1 4 - L / 2 L / 2 - L / 2 L / 2 ϕ ( η ) ϕ ( ξ ) × sinh [ κ ( L 2 - η ) ] sinh [ κ ( L 2 - ξ ) ] × cosh [ κ ( L 2 - η ) ] cosh [ κ ( L 2 - ξ ) ] d η d ξ .
R 1 ( - L 2 ) R 1 * ( - L 2 ) = κ Σ 2 l C 1 4 ( 1 32 S 4 - 1 8 κ ) .
R t ( z 0 ) = Δ t ( z ) Δ t + ( z + z 0 ) = lim W 1 2 W - W W Δ t ( z ) Δ t ( z + z 0 ) d z .
Δ t ( z ) Δ t ( z + z 0 ) = σ a 2 P n 1 ( z 0 ) + σ b 2 P n 2 ( z 0 ) ,
P n 1 = a 0 Λ 0 ( 1 - z 0 a 0 )             for             z 0 < a 0 , = 0             otherwise , P n 2 = b 0 Λ 0 ( 1 - z 0 b 0 )             for             z 0 < b 0 , = 0             otherwise ,
Δ t ( z ) Δ t ( z + z 0 ) = σ ¯ 2 ( 1 - z 0 Λ 0 / 2 )             z 0 < Λ 0 / 2 = 0             otherwise .
f 0 L ( 2 - e i ϕ ( z ) - e - i ϕ ( z ) ) d z ,
ϕ ( z ) = ω c n 1 i = 1 z / Λ u i + ω c n 2 i = 1 z / Λ v i .
f 2 = 0 L 0 L ( 2 - e i ϕ ( z ) - e - i ϕ ( z ) ) ( 2 - e i ϕ ( z ) - e - i ϕ ( z ) ) d z d z = 4 L 2 - 8 L 0 L e - i ϕ ( z ) d z + 2 0 L 0 L e i [ ϕ ( z ) + ϕ ( z ) ] d z d z + 2 0 L 0 L e i [ ϕ ( z ) - ϕ ( z ) ] d z d z .
- 8 L 0 L e - i ϕ ( z ) d z = - 8 L 0 L e ( 2 / 2 ) ϕ 2 ( z ) d z = - 8 L 0 L e ( Ψ / Λ ) z d z = 8 L 2 N Ψ ( e - N Ψ - 1 ) .
2 0 L 0 L e i [ ϕ ( z ) + ϕ ( z ) ] d z d z .
e i [ ϕ ( z ) + ϕ ( z ) ] = e 2 i ϕ ( z ) e i Ω ( z - z ) = e - 2 2 ϕ 2 ( z ) e - ( 2 / 2 ) Ω 2 ( z - z ) = e - 4 z Ψ / Λ e - ( Ψ / Λ ) ( z - z ) ,
2 0 L 0 L e i [ ϕ ( z ) + ϕ ( z ) ] d z d z = 4 3 L 2 N 2 Ψ 2 ( 3 4 - e - N Ψ + 1 4 e - 4 N Ψ ) .
2 0 L 0 L e i [ ϕ ( z ) - ϕ ( z ) ] d z d z = 4 z = 0 L z = 0 z e - ( ψ / Λ ) ( z - z ) d z d z = 4 L 2 N Ψ ( 1 + 1 N Ψ ( e - N Ψ - 1 ) ) .
f 2 - f 2 = 4 3 ( N L Ψ ) 2 4 3 N 2 L 2 ( π 2 ) 4 ( σ ¯ Λ / 2 ) 4 .