Abstract

Stationary TE nonlinear surface waves at the interface separating two nonlinear diffusive Kerr-like media are computed for differing diffusion mechanisms and diffusion lengths. The stability of the surface waves on each branch of stationary solutions is determined by launching the appropriate wave and following its evolution, using the beam-propagation method. The stability criteria follow those determined earlier [ J. Opt. Soc. Am. B 5, 559 ( 1988)] for non-diffusive Kerr media. The critical power necessary for observing surface waves is determined and shown to depend on the nature of the diffusion process. Analysis supports the observed numerical behavior of the diffusion coefficient at the interface as a function of increasing diffusion length.

© 1990 Optical Society of America

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References

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  1. M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).
  2. W. J. Tomlinson, Opt. Lett. 5, 323 (1980);W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, Appl. Opt. 21, 2041 (1982).
    [Crossref] [PubMed]
  3. N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, Pis’ma Zh. Eksp. Teor. Fiz. 88, 107 (1985) [Sov. Phys JETP 61, 62 1985].
  4. A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988);Phys Lett. A 129, 231 (1988);Phys. Rev. A 39, 1809 (1989).
    [Crossref]
  5. P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E. M. Wright, Opt. Lett. 13, 690 (1988).
    [Crossref]
  6. D. R. Andersen, Phys. Rev. A 37, 189 (1988).
    [Crossref] [PubMed]
  7. D. A. B. Miller, S. D. Smith, and B. S. Wherrett, Opt. Commun. 35, 221 (1980).
    [Crossref]
  8. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
    [Crossref]
  9. H. B. De Vore, Phys. Rev. 102, 86 (1956).
    [Crossref]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 16.
  11. P. Varatharajah, A. B. Aceves, A. C. Newell, and J. V. Moloney, “Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” submitted to Phys. Rev. A.
  12. I. Stakgold, Green’s Functions and Boundary Value Problems (Wiley, New York, 1979).

1988 (3)

1985 (1)

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, Pis’ma Zh. Eksp. Teor. Fiz. 88, 107 (1985) [Sov. Phys JETP 61, 62 1985].

1980 (2)

1979 (1)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

1973 (1)

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

1956 (1)

H. B. De Vore, Phys. Rev. 102, 86 (1956).
[Crossref]

Aceves, A.

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988);Phys Lett. A 129, 231 (1988);Phys. Rev. A 39, 1809 (1989).
[Crossref]

P. Varatharajah, A. B. Aceves, A. C. Newell, and J. V. Moloney, “Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” submitted to Phys. Rev. A.

Akhmediev, N. N.

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, Pis’ma Zh. Eksp. Teor. Fiz. 88, 107 (1985) [Sov. Phys JETP 61, 62 1985].

Andersen, D. R.

D. R. Andersen, Phys. Rev. A 37, 189 (1988).
[Crossref] [PubMed]

De Vore, H. B.

H. B. De Vore, Phys. Rev. 102, 86 (1956).
[Crossref]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 16.

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

Gossard, A. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

Heatley, D. R.

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, Pis’ma Zh. Eksp. Teor. Fiz. 88, 107 (1985) [Sov. Phys JETP 61, 62 1985].

Kuzmenko, Y. V.

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, Pis’ma Zh. Eksp. Teor. Fiz. 88, 107 (1985) [Sov. Phys JETP 61, 62 1985].

McCall, S. L.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

Miller, D. A. B.

D. A. B. Miller, S. D. Smith, and B. S. Wherrett, Opt. Commun. 35, 221 (1980).
[Crossref]

Miyagi, M.

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988);Phys Lett. A 129, 231 (1988);Phys. Rev. A 39, 1809 (1989).
[Crossref]

P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E. M. Wright, Opt. Lett. 13, 690 (1988).
[Crossref]

P. Varatharajah, A. B. Aceves, A. C. Newell, and J. V. Moloney, “Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” submitted to Phys. Rev. A.

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988);Phys Lett. A 129, 231 (1988);Phys. Rev. A 39, 1809 (1989).
[Crossref]

P. Varatharajah, A. B. Aceves, A. C. Newell, and J. V. Moloney, “Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” submitted to Phys. Rev. A.

Nishida, S.

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

Passner, A.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 16.

Smith, S. D.

D. A. B. Miller, S. D. Smith, and B. S. Wherrett, Opt. Commun. 35, 221 (1980).
[Crossref]

Stakgold, I.

I. Stakgold, Green’s Functions and Boundary Value Problems (Wiley, New York, 1979).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 16.

Tomlinson, W. J.

Varatharajah, P.

P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E. M. Wright, Opt. Lett. 13, 690 (1988).
[Crossref]

P. Varatharajah, A. B. Aceves, A. C. Newell, and J. V. Moloney, “Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” submitted to Phys. Rev. A.

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 16.

Weigmann, W.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

Wherrett, B. S.

D. A. B. Miller, S. D. Smith, and B. S. Wherrett, Opt. Commun. 35, 221 (1980).
[Crossref]

Wright, E. M.

Appl. Phys. Lett. (1)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Weigmann, Appl. Phys. Lett. 35, 451 (1979).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

D. A. B. Miller, S. D. Smith, and B. S. Wherrett, Opt. Commun. 35, 221 (1980).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

H. B. De Vore, Phys. Rev. 102, 86 (1956).
[Crossref]

Phys. Rev. A (1)

D. R. Andersen, Phys. Rev. A 37, 189 (1988).
[Crossref] [PubMed]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

N. N. Akhmediev, V. I. Korneev, and Y. V. Kuzmenko, Pis’ma Zh. Eksp. Teor. Fiz. 88, 107 (1985) [Sov. Phys JETP 61, 62 1985].

Sci. Rep. Res. Inst. Tohoku Univ. Ser. B (1)

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

Other (3)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 16.

P. Varatharajah, A. B. Aceves, A. C. Newell, and J. V. Moloney, “Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media,” submitted to Phys. Rev. A.

I. Stakgold, Green’s Functions and Boundary Value Problems (Wiley, New York, 1979).

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Figures (9)

Fig. 1
Fig. 1

Plots of solutions F and N for Eqs. (3) and (4) versus x for r0 = 0.25, r1 = 0.5, and β = 0.7. Here the solutions from two different methods coincide.

Fig. 2
Fig. 2

Plots of NSW and excitation density (N) versus x for varying diffusion lengths r1 ≡ 0.125, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0, β = 0.7, and α = 0.25. (a) F+ surface waves, (b) F surface waves. Here the left-hand medium is nondiffusive (zero flux to the left across the interface).

Fig. 3
Fig. 3

Plots of NSW and excitation density (N) versus x for varying diffusion lengths r0 = 0.125, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0, β = 0.4, and α = 0.25. (a) F+ surface waves, (b) F− surface waves. Here the right-hand medium is nondiffusive (zero flux to the right across the interface).

Fig. 4
Fig. 4

Plots of NSW and excitation density (N) versus x for varying diffusion lengths r ¯ = r 0 = r 1 = 0.125 , 0.25 , 0.5 , 1.0 , 2.0 , 4.0 and α = 0.25. (a) F+ surface waves for β = 0.7, (b) F surface waves for β = 0.5. Here both media are diffusive and the diffusion is uniform.

Fig. 5
Fig. 5

Graphs of effective wave number β versus power P for varying diffusion lengths r1 = 0.5, 1.0, 2.0 and for (a) α = 0.25 and (b) α = 0.5. The diffusionless (r0 = r1 = 0) curve is included for comparison. This represents case (1), in which only the right-hand medium is diffusive (zero flux to the left across the interface).

Fig. 6
Fig. 6

Graphs of effective wave number β versus power P for varying diffusion lengths r0 = 0.5, 1.0, 2.0 and for (a) α = 0.25 and (b) α = 0.5. The diffusionless (r0 = r1 = 0) curve is included for comparison. This corresponds to case (2), in which only the left-hand medium is diffusive (zero flux to the right across the interface).

Fig. 7
Fig. 7

Graphs of effective wave number β versus power P for varying ratios of the diffusion lengths r = r0/r1 = 0.25/2.0, 0.5/2.0, 1.0/2.0, 2.0/2.0, 2.0/1.0, 2.0/0.5 and for (a) α = 0.25 and (b) α = 0.5. This represents case (3), in which both media are diffusive.

Fig. 8
Fig. 8

Plots of threshold power versus ratios of the diffusion lengths r for (a) α = 0.25, (b) α = 0.5. Here to compute r from 1 to 0 we fix r1 = 2.0 and let r0 vary from 0.0 to 2.0, and to compute r from 1 to ∞ we fix r0 = 2.0 and let r1 vary from 2.0 to 0.0. We do it this way to avoid stiffness by allowing large values of either r0 or r1. This shows the sharp switch in threshold power at r = 1.0.

Fig. 9
Fig. 9

Plots of NSW shapes at the threshold power for values of r = 0.125, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0 (here the ratio r is calculated from the same values of r0 and r1 that are given in Fig. 7) and for (a) α = 0.25 and (b) α = 0.5. The switch in direction of the peak at r = 1.0 is evident in both cases.

Equations (55)

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2 i β z F ( x , z ) + x x F ( x , z ) [ β 2 n 2 ( x | F ( x , z ) | 2 ) ] F ( x , z ) = 0 ,
N r i 2 d 2 N d x 2 = α i | F | 2 ,
d 2 F d x 2 q 0 2 F + N F = 0 , N r 0 2 d 2 N d x 2 = α 0 F 2 , x 0
d 2 F d x 2 q 1 2 F + N F = 0 , N r 1 2 d 2 N d x 2 = α 1 F 2 , x 0 ,
d N d x = 0 , x = 0
N = d N d x = 0 , at x = + ;
r 0 2 d N d x | x = 0 = r 1 2 d N d x | x = 0 + , N = d N d x = 0 at x = ± .
d H / d x = 0 ,
H = { α 0 ( d F d x ) 2 + 1 2 r 0 2 ( d N d x ) 2 α 0 q 0 2 F 2 1 2 N 2 + α 0 N F 2 , x < 0 α 1 ( d F d x ) 2 + 1 2 r 1 2 ( d N d x ) 2 α 1 q 1 2 F 2 ½ N 2 + α 1 N F 2 , x > 0 .
α 0 ( d F d x ) 2 + 1 2 r 0 2 ( d N d x ) 2 α 0 q 0 2 F 2 1 2 N 2 + α 0 N F 2 = 0 , x < 0
α 1 ( d F d x ) 2 + 1 2 r 1 2 ( d N d x ) 2 α 1 q 1 2 F 2 1 2 N 2 + α 1 N F 2 = 0 , x > 0
F + ( x ) = ( 2 α 0 ) 1 / 2 q 0 sech [ q 0 ( x x 0 ) ] , x < 0
F ( x ) = ( 2 α 0 ) 1 / 2 q 0 sech [ q 0 ( x + x 0 ) ] , x < 0 ,
F + ( x ) = ( 2 α 1 ) 1 / 2 q 1 sech [ q 1 ( x x 1 ) ] , x > 0
F ( x ) = ( 2 α 1 ) 1 / 2 q 1 sech [ q 1 ( x + x 1 ) ] , x > 0 ,
τ ( x ) = r 1 + r 0 2 + r 1 r 0 2 tanh ( x d ) .
d 2 F d x 2 q i 2 F + N F = 0 , N d d x ( τ 2 d N d x ) = α i F 2 .
d y j d x = g j ( x , y j ) , j = 1 , , 4 ,
y 1 = N , y 2 = d N d x , y 3 = F , y 4 = d F d x , g 1 ( x , y j ) = y 2 , g 2 ( x , y j ) = 1 τ 2 ( y 1 d τ 2 d x y 2 α i y 3 2 ) , g 3 ( x , y j ) = y 4 , g 4 ( x , y j ) = y 3 ( q i 2 y 1 ) .
E 1 , k = y 1 , k y 1 , k 1 h 2 ( y 2 , k + y 2 , k 1 ) , E 2 , k = y 2 , k y 2 , k 1 h 2 a k [ 2 ( y 1 , k + y 1 , k 1 ) b k c k ] , E 3 , k = y 3 , k y 3 , k 1 h 2 ( y 4 , k + y 4 , k 1 ) , E 4 , k = y 4 , k y 4 , k 1 h 2 ( y 3 , k + y 3 , k 1 ) d k , k = 2 , , m ,
h = ( x m x 1 ) ( m 1 ) , a k = 1 τ 2 ( k ) + τ 2 ( k 1 ) , b k = ( y 2 , k + y 2 , k 1 ) [ ( d τ 2 d x ) k + ( d τ 2 d x ) k 1 ] , c k = α i ( y 3 , k + y 3 , k 1 ) 2 , d k = q i 2 ½ ( y 1 , k + y 1 , k 1 ) .
E 3 , 1 = y 2 , 1 , E 4 , 1 = A q 0 y 3 , 1 y 4 , 1
E 3 , 1 = y 1 , 1 r 0 y 2 , 1 , E 4 , 1 = q 0 y 3 , 1 y 4 , 1
E 1 , m + 1 = y 2 , m , E 2 , m + 1 = B q 1 y 3 , m y 4 , m
E 1 , m + 1 = y 1 , m + r 1 y 2 , m , E 2 , m + 1 = q 1 y 3 , m + y 4 , m
n = 1 4 M j , n Δ y n , k 1 + n = 5 8 M j , n Δ y n 4 , k = E j , k , j = 1 , , 4
n = 1 4 M j , n Δ y n , 1 = E j , 1 , j = 3 , 4
n = 1 4 M j , n Δ y n , m = E j , m + 1 , j = 1 , 2
M k = [ 1 e k 0 0 1 e k 0 0 f k 1 + g k u k 0 f k 1 + g k u k 0 0 0 1 e k 0 0 1 e k υ k 0 w k 1 υ k 0 w k 1 ] , k = 2 , , m .
M 1 = [ 1 τ ( x 1 ) 0 0 0 0 q 0 1 ]
M 1 = [ 0 1 0 0 0 0 s 0 1 ]
M m + 1 = [ 0 1 0 0 0 0 s 1 1 ]
M m + 1 = [ 1 τ ( x m ) 0 0 0 0 q 1 1 ]
e k = h / 2 , f k = h τ 2 ( k ) + τ 2 ( k 1 ) , g k = 1 2 f k [ ( d τ 2 d x ) k + ( d τ 2 d x ) k 1 ] , u k = α i f k ( y 3 , k + y 3 , k 1 ) , υ k = h 4 ( y 3 , k + y 3 , k 1 ) , w k = h 2 [ q i 1 2 ( y 1 , k + y 1 , k 1 ) ] , s 0 = q 0 [ A + ( d A d y 3 , 1 ) y 3 , 1 ] ,
s 1 = q 1 [ B + ( d B d y 3 , m ) y 3 , m ] .
N ( x ) = α 1 0 G r 1 ( x , y ) | F ( y ) | 2 d y , x > 0
N ( x ) = α 0 0 G r 0 ( x , y ) | F ( y ) | 2 d y , x < 0
N ( x ) = α 0 0 G r 0 r 1 ( x , y ) | F ( y ) | 2 d y + α 1 0 G r 0 r 1 ( x , y ) | F ( y ) | 2 d y , < x <
G r 1 ( x , y ) = { 1 r 1 exp ( y / r 1 ) cosh ( x / r 1 ) , x < y 1 r 1 exp ( x / r 1 ) cosh ( y / r 1 ) , x > y
G r 0 ( x , y ) = { 1 r 0 exp ( x / r 0 ) cosh ( y / r 0 ) , x < y 1 r 0 exp ( y / r 0 ) cosh ( x / r 0 ) , x > y
G r 0 r 1 ( x , y ) = { 1 2 r 0 exp ( x y r 0 ) + ( r 0 r 1 2 r 0 2 ) exp ( x + y r 0 ) , x < y 1 2 r 0 exp ( y x r 0 ) + ( r 0 r 1 2 r 0 2 ) exp ( x + y r 0 ) , x > y , x < 0 1 2 r 0 exp ( y / r 0 ) exp ( x / r 1 ) , x > 0
G r 0 r 1 ( x , y ) = { 1 2 r 1 exp ( y / r 1 ) exp ( x / r 0 ) , x < 0 1 2 r 1 exp ( x y r 1 ) + ( r 1 r 0 2 r 1 2 ) exp ( x + y r 1 ) , x < y 1 2 r 1 exp ( y x r 1 ) + ( r 1 r 0 2 r 1 2 ) exp ( x + y r 1 ) , x > y , x > 0
d 2 N d x 2 = α 1 r 1 3 [ exp ( x / r 1 ) 0 x cosh ( y / r 1 ) | F ( y ) | 2 d y + cosh ( x / r 1 ) x exp ( y / r 1 ) | F ( y ) | 2 d y ] α 1 r 1 2 | F ( x ) | 2 , x > 0 .
d 2 N d x 2 ( x = 0 + ) = α 1 r 1 2 0 exp ( u ) [ | F ( r 1 u ) | 2 | F ( 0 ) | 2 ] d u .
d 2 N d x 2 ( x = 0 + ) α 1 r 1 d | F | 2 d x ( x = 0 ) 0 u exp ( u ) d u .
d 2 N d x 2 ( x = 0 + ) α 1 r 1 2 [ | F ( ) | 2 F ( 0 ) | 2 ] 0 exp ( u ) d u ,
d 2 N d x 2 = α 0 r 0 [ cosh ( x / r 0 ) x exp ( y / r 0 ) | F ( y ) | 2 d y + exp ( x / r 0 ) x 0 cosh ( y / r 0 ) | F ( y ) | 2 d y ] α 0 r 0 2 | F ( x ) | 2 , x < 0 .
d 2 N d x 2 ( x = 0 ) = α 0 r 0 2 0 exp ( u ) [ | F ( r 0 u ) | 2 | F ( 0 ) | 2 ] d u .
d 2 N d x 2 ( x = 0 ) α 0 r 0 d | F | 2 d x ( x = 0 ) 0 u exp ( u ) d u .
d 2 N d x 2 ( x = 0 ) α 0 r 0 2 [ | F ( ) | 2 | F ( 0 ) | 2 ] 0 exp ( u ) d u ,
d N d x = { α 0 r 1 2 r 0 3 0 exp ( y / r 0 ) | F ( y ) | 2 d y + α 1 2 r 0 r 1 0 exp ( y / r 1 ) | F ( y ) | 2 d y , x = 0 α 0 2 r 0 r 1 0 exp ( y / r 0 ) | F ( y ) | 2 d y + α 1 r 0 2 r 1 3 0 exp ( y / r 1 ) | F ( y ) | 2 d y , x = 0 + .
d N d x = { α 0 r 1 2 r 0 2 0 exp ( u ) | F ( r 0 u ) | 2 d u + α 1 2 r 0 0 exp ( υ ) | F ( r 1 υ ) | 2 d υ , x = 0 α 0 2 r 1 0 exp ( u ) | F ( r 0 u ) | 2 d u + α 1 r 0 2 r 1 2 0 exp ( υ ) | F ( r 1 υ ) | 2 d υ , x = 0 + .
d N d x { 1 2 r 0 2 ( r 0 α 1 r 1 α 0 ) | F ( 0 ) | 2 + r 1 2 r 0 ( α 1 + α 0 ) d | F | 2 d x ( x = 0 ) , x = 0 1 2 r 1 2 ( r 0 α 1 r 1 α 0 ) | F ( 0 ) | 2 + r 0 2 r 1 ( α 1 + α 0 ) d | F | 2 d x ( x = 0 ) , x = 0 + .
d N d x ( x = 0 ) = α 0 2 r ¯ 0 exp ( u ) | F ( r ¯ u ) | 2 d u + α 1 2 r ¯ 0 exp ( υ ) | F ( r ¯ υ ) | 2 d υ
d N d x 2 ( x = 0 ) 1 2 r ¯ ( α 1 α 0 ) | F ( 0 ) | 2 + 1 2 ( α 1 + α 0 ) × d | F | 2 d x ( x = 0 ) .

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