Abstract

Thermal blooming compensation instabilities are examined. The linearized system of thermal blooming compensation (TBC) equations is studied to develop parameters that characterize the stability of phase-only and full-wave (amplitude and phase) compensation for the effects of thermal blooming. The stabilizing effects of microscale wind shear are included in the analysis to provide a mechanism to stabilize the TBC equations. Stability is equated to existence of bounded solutions of the linear TBC equations, and appropriate dimensionless parameters are developed that ensure existence and uniqueness of bounded solutions to the TBC equations. Parameters characterizing stability are expressed in forms analogous to conventional scaling laws.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2006 (1)

P. Sprangle, J. Penano, and B. Hafizi, “Optimum wavelength and power for efficient laser propagation in various atmospheric environments,” J. Directed Energy 2, 71-95 (2006).

2002 (6)

2001 (2)

1998 (3)

1997 (1)

1992 (2)

D. L. Fried, “Branch cuts in the phase function,” Appl. Opt. 31, 144-146 (1992).
[CrossRef]

D. G. Fouche, C. Higgs, and C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” Lincoln Lab. J. 5, 273-292 (1992).

1991 (3)

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862-6876 (1991).
[CrossRef] [PubMed]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

1989 (1)

1974 (1)

1971 (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971). 1971 OSA Spring Meeting Abstract: http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-61-5-648.

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Barchers, J. D.

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Bradley, L. C.

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Duzy, C.

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862-6876 (1991).
[CrossRef] [PubMed]

Ellerbroek, B. L.

Fouche, D. G.

D. G. Fouche, C. Higgs, and C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” Lincoln Lab. J. 5, 273-292 (1992).

Fried, D. L.

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Gonsalver, R. A.

Hafizi, B.

P. Sprangle, J. Penano, and B. Hafizi, “Optimum wavelength and power for efficient laser propagation in various atmospheric environments,” J. Directed Energy 2, 71-95 (2006).

Herrmann, J.

Higgs, C.

D. G. Fouche, C. Higgs, and C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” Lincoln Lab. J. 5, 273-292 (1992).

Holmes, R.

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862-6876 (1991).
[CrossRef] [PubMed]

Karr, T. J.

Lee, D. J.

Link, D. J.

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

Myers, R.

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862-6876 (1991).
[CrossRef] [PubMed]

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

Pearson, C. F.

D. G. Fouche, C. Higgs, and C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” Lincoln Lab. J. 5, 273-292 (1992).

Penano, J.

P. Sprangle, J. Penano, and B. Hafizi, “Optimum wavelength and power for efficient laser propagation in various atmospheric environments,” J. Directed Energy 2, 71-95 (2006).

Platt, B. C.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971). 1971 OSA Spring Meeting Abstract: http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-61-5-648.

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

Rhoadarmer, T. A.

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Roggemann, M. C.

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).
[CrossRef]

Shack, R. V.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971). 1971 OSA Spring Meeting Abstract: http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-61-5-648.

Sprangle, P.

P. Sprangle, J. Penano, and B. Hafizi, “Optimum wavelength and power for efficient laser propagation in various atmospheric environments,” J. Directed Energy 2, 71-95 (2006).

Szeto, R. K.-H.

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

Appl. Opt. (6)

J. Directed Energy (1)

P. Sprangle, J. Penano, and B. Hafizi, “Optimum wavelength and power for efficient laser propagation in various atmospheric environments,” J. Directed Energy 2, 71-95 (2006).

J. Opt. Soc. Am. (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971). 1971 OSA Spring Meeting Abstract: http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-61-5-648.

J. Opt. Soc. Am. A (7)

Lincoln Lab. J. (1)

D. G. Fouche, C. Higgs, and C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” Lincoln Lab. J. 5, 273-292 (1992).

Nature (London) (2)

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141-143 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144-146 (1991).
[CrossRef]

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43-72 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862-6876 (1991).
[CrossRef] [PubMed]

Other (1)

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Maximum value of the API parameter η 0 as a function of normalized spatial frequency κ κ 0 to ensure stability for phase compensation. The figure provides the asymptotic bounds as well as the peak bound (marked with diamonds). The peak frequency is illustrated as well (marked with diamonds).

Fig. 2
Fig. 2

Calculation of the API parameter η 0 as a function of normalized spatial frequency κ κ 0 to ensure stability for phase and full-wave compensation. The figure provides asymptotic bounds for the phase compensation case. The figure provides the result based both on exact calculation of θ FW and on approximate calculation of θ FW . The agreement of the two methods is very good up to the minimum value of η 0 , which is all that is practically required

Equations (153)

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2 E ( r , z , t ) + 2 { E ( r , z , t ) ln [ n ( r , z , t ) ] } = ( 1 c 2 ) t t [ n 2 ( r , z , t ) E ( r , z , t ) ] ,
n ( r , z , t ) = 1 + n T ( r , z , t ) + n H ( r , z , t ) + n 2 E ( r , z , t ) 2 ,
t n H ( r , z , t ) + v ( z ) n H ( r , z , t ) = α ( z ) β ( z ) E ( r , z , t ) 2 ,
β ( z ) = T 0 n T C p ρ 0 T ( z ) T ( z ) ,
t n H ( r , z , t ) + v ( z ) n H ( r , z , t ) = α ( z ) E ( r , z , t ) 2 .
E ( r , z , t ) = exp ( i k 0 z i ω t ) E ( r , z , t ) ,
z z E ( r , z , t ) + i 2 k 0 z E ( r , z , t ) + 2 E ( r , z , t ) ( 1 c 2 ) t t [ n 2 ( r , z , t ) E ( r , z , t ) ] + 2 i ( k 0 c ) t [ n 2 ( r , z , t ) E ( r , z , t ) ] + 2 [ E ( r , z , t ) ln n ( r , z , t ) ] + k 0 2 ( n 2 1 ) ( r , z , t ) E ( r , z , t ) = 0
z = ζ L H c t 0 τ , r = ρ ¯ l f = ρ ¯ L H k 0 , t = t 0 τ , v
( z ) = v 0 v ̃ ( ζ ) , α ( z ) = α 0 α ̃ ( ζ ) , E ( r , z , t ) = E 0 e ( ρ ¯ , ζ , τ ) ,
n H ( r , z , t ) = N H n ̃ H ( ρ ¯ , ζ , τ ) , n T ( r , z , t ) = N H n ̃ T ( ρ ¯ , ζ , τ ) ,
n 2 = N H n ̃ 2 .
1 v 0 = 1 Z 0 Z d z α ̃ ( z L H ) v ( z ) .
L H = ( v 0 α 0 E 0 2 1 k 0 ) 2 3 .
t 0 = 1 v 0 L H k 0 = ( 1 α 0 v 0 2 E 0 2 k 0 2 ) 1 3 .
N H = ( α 0 E 0 2 v 0 k 0 ) 2 3 .
N D k 0 E 0 2 D 0 Z d z α ( z ) v ( z ) .
N H 3 2 = E 0 2 k 0 0 Z d z α ( z ) v ( z ) ,
0 = [ N H ζ ζ v 0 c N H 1 2 ζ τ + i 2 ζ + 2 ] e ( ρ ¯ , ζ , τ ) + 2 [ ( v 0 c ) 2 N H τ τ + 2 i v 0 c N H 1 2 τ ] × { [ n ̃ T ( ρ ¯ , ζ , τ ) + n ̃ H ( ρ ¯ , ζ , τ ) + N H n 2 E 0 2 e ( ρ ¯ , ζ , τ ) 2 ] e ( ρ ¯ , ζ , τ ) } + 2 N H { e ( ρ ¯ , ζ , τ ) [ n ̃ T ( ρ ¯ , ζ , τ ) + n ̃ H ( ρ ¯ , ζ , τ ) + N H n 2 E 0 2 e ( ρ ¯ , ζ , τ ) 2 ] } + 2 [ n ̃ T ( ρ ¯ , ζ , τ ) + n ̃ H ( ρ ¯ , ζ , τ ) + N H n 2 E 0 2 e ( ρ ¯ , ζ , τ ) 2 ] e ( ρ ¯ , ζ , τ )
0 = τ n ̃ H ( ρ ¯ , ζ , τ ) + v ̃ ( ζ ) n ̃ H ( ρ ¯ , ζ , τ ) + α ̃ ( ζ ) e ( ρ ¯ , ζ , τ ) 2 .
[ ζ i 2 2 i n ̃ T ( ρ ¯ , ζ , τ ) i n ̃ H ( ρ ¯ , ζ , τ ) ] u ( ρ ¯ , ζ , τ ) = 0 ,
τ n ̃ H ( ρ ¯ , ζ , τ ) + v ̃ ( ζ ) n ̃ H ( ρ ¯ , ζ , τ ) + α ̃ ( ζ ) u ( ρ ¯ , ζ , τ ) 2 = 0 .
[ ζ L H c t 0 τ i 2 2 i n ̃ T ( ρ ¯ , ζ , τ ) i n ̃ H ( ρ ¯ , ζ , τ ) ] u H ( ρ ¯ , ζ , τ ) = 0 ,
[ ζ + L H c t 0 τ i 2 2 i n ̃ T ( ρ ¯ , ζ , τ ) i n ̃ H ( ρ ¯ , ζ , τ ) ] u b ( ρ ¯ , ζ , τ ) = 0 ,
τ n ̃ H ( ρ ¯ , ζ , τ ) + v ̃ ( ζ ) n ̃ H ( ρ ¯ , ζ , τ ) + α ̃ ( ζ ) u H ( ρ ¯ , ζ , τ ) 2 = 0 ,
u b ( ρ ¯ , ζ = Z L H , τ ) = u b , 0 ( ρ ¯ , τ ) , τ > 0 ,
u H ( ρ ¯ , ζ = 0 , τ ) = K [ u b ( ρ ¯ , ζ = 0 , τ ) ] , τ > L c t 0 ,
n ̃ H ( ρ ¯ , ζ , τ = 0 ) = 0 ,
[ ζ i 2 2 i n ̃ T ( ρ ¯ , ζ , τ ) i n ̃ H ( ρ ¯ , ζ , τ ) ] u H ( ρ ¯ , ζ , τ ) = 0 ,
τ n ̃ H ( ρ ¯ , ζ , τ ) + v ̃ ( ζ ) n ̃ H ( ρ ¯ , ζ , τ ) + α ̃ ( ζ ) u H ( ρ ¯ , ζ , τ ) 2 = 0 ,
0 = ζ Ψ H ( ρ ¯ , ζ , τ ) ( L H c t 0 ) τ Ψ H ( ρ ¯ , ζ , τ ) ( i 2 ) [ 2 Ψ H ( ρ ¯ , ζ , τ ) + Ψ H ( ρ ¯ , ζ , τ ) Ψ H ( ρ ¯ , ζ , τ ) ] i n ̃ T ( ρ ¯ , ζ , τ ) i n ̃ H ( ρ ¯ , ζ , τ ) .
Ψ H ( ρ ¯ , ζ , τ ) = Ψ H , 0 ( ρ ¯ , ζ , τ ) + δ Ψ H , 1 ( ρ ¯ , ζ , τ ) + δ 2 Ψ H , 2 ( ρ ¯ , ζ , τ ) + .
0 = ζ Ψ H , 0 ( ρ ¯ , ζ , τ ) ( L H c t 0 ) τ Ψ H , 0 ( ρ ¯ , ζ , τ ) ( i 2 ) [ 2 Ψ H , 0 ( ρ ¯ , ζ , τ ) + Ψ H , 0 ( ρ ¯ , ζ , τ ) Ψ H , 0 ( ρ ¯ , ζ , τ ) ] ,
0 = ζ Ψ H , 1 ( ρ ¯ , ζ , τ ) ( L H c t 0 ) τ Ψ H , 1 ( ρ ¯ , ζ , τ ) ( i 2 ) [ 2 Ψ H , 1 ( ρ ¯ , ζ , τ ) + Ψ H , 1 ( ρ ¯ , ζ , τ ) Ψ H , 0 ( ρ ¯ , ζ , τ ) ] i n ̃ T ( ρ ¯ , ζ , τ ) i n ̃ H ( ρ ¯ , ζ , τ ) .
0 = ζ u H , 0 ( ρ ¯ , ζ , τ ) ( L H c t 0 ) τ u H , 0 ( ρ ¯ , ζ , τ ) ( i 2 ) 2 u H , 0 ( ρ ¯ , ζ , τ ) ,
0 = ζ W H , 1 ( ρ ¯ , ζ , τ ) ( L H c t 0 ) τ W H , 1 ( ρ ¯ , ζ , τ ) ( i 2 ) 2 W H , 1 ( ρ ¯ , ζ , τ ) i u H , 0 ( ρ ¯ , ζ , τ ) [ n ̃ T ( ρ ¯ , ζ , τ ) + n ̃ H ( ρ ¯ , ζ , τ ) ] ,
0 = ζ χ H ( ρ ¯ , ζ , τ ) L H c t 0 τ χ H ( ρ ¯ , ζ , τ ) + 1 2 2 ϕ H ( ρ ¯ , ζ , τ ) ,
0 = ζ ϕ H ( ρ ¯ , ζ , τ ) L H c t 0 τ ϕ H ( ρ ¯ , ζ , τ ) 1 2 2 χ H ( ρ ¯ , ζ , τ ) n ̃ T ( ρ ¯ , ζ , τ ) n ̃ H ( ρ ¯ , ζ , τ ) ,
0 = ζ χ b ( ρ ¯ , ζ , τ ) + L H c t 0 τ χ b ( ρ ¯ , ζ , τ ) + 1 2 2 ϕ b ( ρ ¯ , ζ , τ ) ,
0 = ζ ϕ b ( ρ ¯ , ζ , τ ) + L H c t 0 τ ϕ b ( ρ ¯ , ζ , τ ) 1 2 2 χ b ( ρ ¯ , ζ , τ ) n ̃ T ( ρ ¯ , ζ , τ ) n ̃ H ( ρ ¯ , ζ , τ ) ,
0 = τ n ̃ H ( ρ ¯ , ζ , τ ) + v ̃ ( ζ ) n ̃ H ( ρ ¯ , ζ , τ ) + 2 α ̃ ( ζ ) χ H ( ρ ¯ , ζ , τ ) ,
χ b ( ρ ¯ , ζ = Z L H , τ ) = 0 , τ > 0 ,
ϕ b ( ρ ¯ , ζ = Z L H , τ ) = 0 , τ > 0 ,
χ H ( ρ ¯ , ζ = 0 , τ ) = L H c t 0 τ d τ a ( τ τ ) χ b ( ρ ¯ , ζ = 0 , τ ) , τ > L H c t 0 ,
ϕ H ( ρ ¯ , ζ = 0 , τ ) = L H c t 0 τ d τ p ( τ τ ) ϕ b ( ρ ¯ , ζ = 0 , τ ) ,
τ > L H c t 0 ,
n ̃ H ( ρ ¯ , ζ , τ = 0 ) = 0 ,
a ( τ ) = ω a exp ( ω a τ ) ,
p ( τ ) = ω p exp ( ω p τ ) .
0 = ζ χ H ( κ ¯ , ζ , s ) L H c t 0 s χ H ( κ ¯ , ζ , s ) κ 2 2 ϕ H ( κ ¯ , ζ , s ) ,
0 = ζ ϕ H ( κ ¯ , ζ , s ) L H c t 0 s ϕ H ( κ ¯ , ζ , s ) + κ 2 2 χ H ( κ ¯ , ζ , s ) n ̃ T ( κ ¯ , ζ , s ) n ̃ H ( κ ¯ , ζ , s ) ,
0 = ζ χ b ( κ ¯ , ζ , s ) + L H c t 0 s χ b ( κ ¯ , ζ , s ) κ 2 2 ϕ b ( κ ¯ , ζ , s ) ,
0 = ζ ϕ b ( κ ¯ , ζ , s ) + L H c t 0 s ϕ b ( κ ¯ , ζ , s ) + κ 2 2 χ b ( κ ¯ , ζ , s ) n ̃ T ( κ ¯ , ζ , s ) n ̃ H ( κ ¯ , ζ , s ) ,
0 = s n ̃ H ( κ ¯ , ζ , s ) i κ ¯ v ̃ ( ζ ) n ̃ H ( κ ¯ , ζ , s ) + 2 α ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) .
n ̃ H ( κ ¯ , ζ , s ) = 2 α ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) s i κ ¯ v ̃ ( ζ ) .
χ b ( κ ¯ , ζ , s ) = ζ Z L H d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] [ n ̃ H ( κ ¯ , ζ , s ) + n ̃ T ( κ ¯ , ζ , s ) ] ,
ϕ b ( κ ¯ , ζ , s ) = ζ Z L H d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] [ n ̃ H ( κ ¯ , ζ , s ) + n ̃ T ( κ ¯ , ζ , s ) ] .
χ b ( κ ¯ , ζ , s ) = 2 ζ Z L H d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + ζ Z L H d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] n ̃ T ( κ ¯ , ζ , s ) ,
ϕ b ( κ ¯ , ζ , s ) = 2 ζ Z L H d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + ζ Z L H d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] n ̃ T ( κ ¯ , ζ , s ) .
χ H ( κ ¯ , ζ , s ) = e L H c t 0 s ζ cos ( κ 2 2 ζ ) χ H ( κ , ζ = 0 , s ) + e L H c t 0 s ζ sin ( κ 2 2 ζ ) ϕ H ( κ , ζ = 0 , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) sin [ ( κ 2 2 ) ( ζ ζ ) ] [ n ̃ H ( κ ¯ , ζ , s ) + n ̃ T ( κ ¯ , ζ , s ) ] .
ϕ H ( κ ¯ , ζ , s ) = e L H c t 0 s ζ cos ( κ 2 2 ζ ) ϕ H ( κ , ζ = 0 , s ) e L H c t 0 s ζ sin ( κ 2 2 ζ ) χ H ( κ , ζ = 0 , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] [ n ̃ H ( κ ¯ , ζ , s ) + n ̃ T ( κ ¯ , ζ , s ) ] .
χ H ( κ ¯ , ζ , s ) = e L H c t 0 s ζ cos ( κ 2 2 ζ ) A ( s ) χ b ( κ , ζ = 0 , s ) + e L H c t 0 s ζ sin ( κ 2 2 ζ ) P ( s ) ϕ b ( κ , ζ = 0 , s ) 2 0 ζ d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] n ̃ T ( κ ¯ , ζ , s ) ,
ϕ H ( κ ¯ , ζ , s ) = e L H c t 0 s ζ cos ( κ 2 2 ζ ) P ( s ) ϕ b ( κ , ζ = 0 , s ) e L H c t 0 s ζ sin ( κ 2 2 ζ ) A ( s ) χ b ( κ , ζ = 0 , s ) 2 0 ζ d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] n ̃ T ( κ ¯ , ζ , s ) .
χ H ( κ ¯ , ζ , s ) = 2 A ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) 2 P ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) 2 0 ζ d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + A ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) n ̃ T ( κ ¯ , ζ , s ) + P ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) n ̃ T ( κ ¯ , ζ , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] n ̃ T ( κ ¯ , ζ , s ) ,
ϕ H ( κ ¯ , ζ , s ) = 2 P ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) cos ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + 2 A ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) sin ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) 2 0 ζ d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] α ̃ ( ζ ) s i κ ¯ v ̃ ( ζ ) χ H ( κ ¯ , ζ , s ) + P ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) cos ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) n ̃ T ( κ ¯ , ζ , s ) A ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) sin ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) n ̃ T ( κ ¯ , ζ , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) cos [ κ 2 2 ( ζ ζ ) ] n ̃ T ( κ ¯ , ζ , s ) .
χ H ( κ ¯ , ζ , τ ) = χ H ( κ ¯ , ζ , τ ) e i κ ¯ v ̃ ( ζ ) τ .
χ H ( κ ¯ , ζ , τ ) M [ χ H ( κ ¯ , ζ , τ ) ] = χ ̃ H ( κ ¯ , ζ , τ ) ,
χ ̃ H ( κ ¯ , ζ , τ ) = e i κ ¯ v ̃ ( ζ ) τ L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) ,
M [ f ( κ ¯ , ζ , τ ) ] = 2 0 Z L H d ζ α ̃ ( ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) 0 τ d τ a ( τ τ ) 0 τ L H c t 0 ( ζ + ζ ) d τ e i κ ¯ v ̃ ( ζ ) τ i κ ¯ v ̃ ( ζ ) τ i κ ¯ v ̃ ( ζ ) L H c t 0 ( ζ + ζ ) f ( κ ¯ , ζ , τ ) 2 0 Z L H d ζ α ̃ ( ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) 0 τ d τ p ( τ τ ) 0 τ L H c t 0 ( ζ + ζ ) d τ e i κ ¯ v ̃ ( ζ ) τ i κ ¯ v ̃ ( ζ ) τ i κ ¯ v ̃ ( ζ ) L H c t 0 ( ζ + ζ ) f ( κ ¯ , ζ , τ ) 2 0 ζ d ζ α ̃ ( ζ ) sin [ κ 2 2 ( ζ ζ ) ] 0 τ L H c t 0 ( ζ ζ ) d τ e i κ ¯ [ v ̃ ( ζ ) v ̃ ( ζ ) ] τ i κ ¯ v ̃ ( ζ ) L H c t 0 ( ζ ζ ) f ( κ ¯ , ζ , τ ) ,
M 0 [ f ( κ ¯ , ζ , s ) ] = A ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) f ( κ ¯ , ζ , s ) + P ( s ) 0 Z L H d ζ e L H c t 0 s ( ζ + ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) f ( κ ¯ , ζ , s ) + 0 ζ d ζ e L H c t 0 s ( ζ ζ ) sin [ κ 2 2 ( ζ ζ ) ] f ( κ ¯ , ζ , s ) .
exp { i a κ ¯ v ̃ ( ζ ) } = exp { i a 2 κ ¯ v ̃ 0 } exp { i a κ ¯ v ̃ 1 ( ζ ) } exp { i a κ ¯ δ v ̃ ( ζ ) } .
D δ v ̃ ( ζ 1 , ζ 2 ) = 4 v 0 2 0 d κ z [ 1 cos ( 2 π κ z ζ 1 ζ 2 ) ] Ψ [ κ z L H , ( ζ 1 + ζ 2 ) L H 2 ] ,
Ψ ( κ z , z ) = 0.00969 π 2 Γ ( 5 6 ) Γ ( 11 6 ) C v 2 ( z ) [ κ z 2 + 1 L 0 2 ( z ) ] 5 6 exp [ κ 2 l 0 2 ( z ) 4 π 2 5.91 2 ] ,
V ( ζ , ζ 1 , ζ 2 ) = [ δ v ̃ ( ζ ) 1 ζ 2 ζ 1 ζ 1 ζ 2 d ζ δ v ̃ ( ζ ) ] 2
= 1 ζ 2 ζ 1 ζ 1 ζ 2 d ζ D δ v ̃ ( ζ , ζ ) 1 2 ( ζ 2 ζ 1 ) 2 ζ 1 ζ 2 d ζ ζ 1 ζ 2 d ζ D δ v ̃ ( ζ , ζ ) .
V ¯ ( ζ 1 , ζ 2 ) = 1 ζ 2 ζ 2 ζ 1 ζ 2 d ζ V ( ζ , ζ 1 , ζ 2 ) .
V ̃ ( Z ) = 0 1 d ζ 0 1 d ζ C v ̃ 2 ( ζ + ζ 2 Z L H ) ζ ζ 2 3 1 2 0 1 d ζ 0 1 d ζ C v ̃ 2 ( ζ + ζ 2 Z L H ) ζ ζ 2 3 ,
C V 2 = 1 Z 0 Z d z C v 2 ( z ) ,
χ H ( κ ¯ , ζ , τ ) M [ χ H ( κ ¯ , ζ , τ ) ] = χ ̃ H ( κ ¯ , ζ , τ ) .
M [ f ( κ ¯ , ζ , τ ) ] = 2 0 ζ d ζ α ̃ ( ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) e i κ ¯ [ v ̃ ( ζ ) v ̃ ( ζ ) ] τ × 0 τ d τ f ( κ ¯ , ζ , τ ) + 2 ζ Z L H d ζ α ̃ ( ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) e i κ ¯ [ v ̃ ( ζ ) v ̃ ( ζ ) ] τ 0 τ d τ f ( κ ¯ , ζ , τ ) .
M ¯ [ f ( κ ¯ , ζ , τ ) ] = 2 0 ζ d ζ α ̃ ( ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 × 0 τ d τ f ( κ ¯ , ζ , τ ) + 2 ζ Z L H d ζ α ̃ ( ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 × 0 τ d τ f ( κ ¯ , ζ , τ ) .
χ H ( κ ¯ , ζ , τ ) M ¯ [ χ H ( κ ¯ , ζ , τ ) ] = χ ̃ H ( κ ¯ , ζ , τ )
χ H ( κ ¯ , ζ , τ ) = k = 0 M k { [ χ ̃ H ( κ ¯ , ζ , τ ) ] } .
T P O [ χ H ( κ ¯ , ζ , τ ) ] = M [ χ H ( κ ¯ , ζ , τ ) ] + χ ̃ H ( κ ¯ , ζ , τ )
T P O [ χ H ( κ ¯ , ζ , τ ) ] = χ H ( κ ¯ , ζ , τ ) ,
θ P O ( κ ) sup τ ( 0 , ) ζ [ 0 , Z L H ] 2 τ 0 ζ d ζ α ̃ ( ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 + 2 τ ζ Z L H d ζ α ̃ ( ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 .
ζ 1 ζ 2 d ζ W ( ζ , ζ ) e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 ζ 1 ζ 2 d ζ W ( ζ , ζ ) exp [ w a κ 2 τ 2 V ( ζ , ζ 1 , ζ 2 ) ] ,
θ P O ( κ ) max τ ( 0 , ) 2 τ e w a κ 2 τ 2 V ¯ ( 0 , Z L H ) max ζ [ 0 , Z L H ] I 0 ( ζ , κ )
= 1 κ 2 1 2 e 1 2 w a 1 2 V ¯ ( 0 , Z L H ) max ζ [ 0 , Z L H ] I 0 ( ζ , κ ) ,
I 0 ( ζ , κ ) = 0 ζ d ζ α ̃ ( ζ ) cos ( κ 2 2 ζ ) sin ( κ 2 2 ζ ) + ζ Z L H d ζ α ̃ ( ζ ) sin ( κ 2 2 ζ ) cos ( κ 2 2 ζ ) ,
lim κ 0 I 0 ( ζ , κ ) κ 2 2 [ 0 ζ d ζ α ̃ ( ζ ) ζ + ζ ζ Z L H d ζ α ̃ ( ζ ) ] .
lim κ I 0 ( ζ , κ ) 2 π 0 Z L H d ζ α ̃ ( ζ ) = 2 π Z L H .
lim κ 0 θ P O ( κ ) κ 2 2 1 2 e 1 2 w a 1 2 V ¯ ( 0 , Z L H ) 0 Z L H d ζ α ̃ ( ζ ) ζ ,
lim κ θ P O ( κ ) 1 κ 2 π Z L H 2 1 2 e 1 2 w a 1 2 V ¯ ( 0 , Z L H ) .
η 1 ( κ ) = [ v 0 2 1 2 e 1 2 C V 2 Z 2 3 V ̃ ( Z L H ) ] w 1 1 2 2 1 κ Z 2 L H 2 μ α ̃ , 1 ,
η 1 ( κ ) = [ v 0 2 1 2 e 1 2 C V 2 Z 2 3 V ̃ ( Z L H ) ] w 1 1 2 2 1 1 π 1 κ Z L H ,
μ α ̃ , 1 = 0 1 d ζ α ̃ ( ζ Z L H ) ζ ,
η 0 = [ v 0 2 1 2 e 1 2 C V 2 Z 2 3 V ̃ ( Z L H ) ] w 1 1 4 w 1 1 4 μ α ̃ , 1 1 2 π 1 2 Z 3 2 L H 3 2 ,
κ 0 = 2 L H π Z μ α ̃ , 1 ( w 1 w 1 ) 1 4 .
η N ( κ ) = [ v 0 2 1 2 e 1 2 C V 2 Z 2 3 V ̃ ( Z L H ) ] w 1 1 4 w 1 1 4 μ α ̃ , 1 1 2 π 1 2 Z 3 2 L H 3 2 κ N ( w 1 w 1 ) N 4 [ π Z μ α ̃ , 1 4 L H ] N 2
= η 0 ( κ κ 0 ) N .
η 0 = 2 1 2 e 1 2 w 1 1 4 w 1 1 4 μ α ̃ , 1 1 2 π 1 2 [ v 0 Z 3 2 α 0 E 0 2 k 0 1 2 C V 2 Z 2 3 V ̃ ( Z L H ) ]
= 0.5277 μ α ̃ , 1 1 2 [ v 0 Z 3 2 α 0 E 0 2 k 0 1 2 C V 2 Z 2 3 V ̃ ( Z L H ) ] .
κ P K 4 = 1 μ α ̃ , 1 1 4 + 1 8 μ α ̃ , 1 ( L H Z ) 2 .
η P K = 1 κ P K v 0 2 1 2 e 1 2 w 1 2 ( κ P K ) C V 2 Z 2 3 V ̃ ( Z L H ) Z L H cos ( κ 2 2 Z L H ) μ α ̃ , S ( κ P K 2 2 ) ,
μ α ̃ , S ( a ) = 0 1 d ζ α ̃ ( ζ Z L H ) sin ( a ζ Z L H ) ,
w ( κ ) = 2 1 2 ( 1 + κ 4 κ 0 4 ) .
η WISP = Z 3 2 α 0 E 0 2 k 0 1 2 2 2 π C V 2 Z 2 3 V ̃ ( Z L H ) ( L 0 Z ) 1 2 ,
L 0 = Z 0 1 d ζ α ( ζ ) α ( 0 ) .
η 0 = 2.6457 μ α ̃ , 1 1 2 ( Z L 0 ) 1 2 η WISP .
K ¯ [ f ( κ ¯ , ζ , τ ) ] = 2 ζ Z L H d ζ α ̃ ( ζ ) sin [ κ 2 2 ( ζ ζ ) ] e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 × 0 τ d τ f ( κ ¯ , ζ , τ ) ,
Δ ¯ [ f ( κ ¯ , ζ , τ ) ] = 2 0 ζ d ζ α ̃ ( ζ ) sin [ κ 2 2 ( ζ ζ ) ] e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 × τ + L H c t 0 ( ζ ζ ) τ + L H c t 0 ( ζ + ζ ) d τ f ( κ ¯ , ζ , τ ) .
T FW [ χ H ( κ ¯ , ζ , τ ) ] = K ¯ [ χ H ( κ ¯ , ζ , τ ) ] + Δ ¯ [ χ H ( κ ¯ , ζ , τ ) ] + χ ̃ H ( κ ¯ , ζ , τ ) .
K ¯ ( κ ) sup τ ( 0 , ) ζ [ 0 , Z L H ] 2 τ ζ Z L H d ζ α ̃ ( ζ ) sin [ κ 2 2 ( ζ ζ ) ] e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2 max τ ( 0 , ) 2 τ 0 Z L H d ζ α ̃ ( ζ ) sin [ κ 2 2 ζ ] e 1 2 κ 2 C V 2 L H 2 3 v 0 2 ζ 2 3 τ 2 .
Δ ¯ ( κ ) sup τ ( 0 , ) ζ [ 0 , Z L H ] 4 L H c t 0 0 ζ d ζ ζ α ̃ ( ζ ) sin [ κ 2 2 ( ζ ζ ) ] e 1 2 κ 2 D δ v ̃ ( ζ , ζ ) τ 2
4 L H c t 0 0 Z L H d ζ ζ α ̃ ( ζ ) sin [ κ 2 2 ( Z L H ζ ) ] .
K ¯ ( κ ) τ K 2 Z L H μ α ̃ , S ( κ 2 2 ) τ K 3 κ 2 C V 2 Z 2 3 v 0 2 Z L H μ α ̃ , S , 2 3 ( κ 2 2 ) + τ K 5 1 4 κ 4 ( C V 2 Z 2 3 v 0 2 ) 2 Z L H μ α ̃ , S , 4 3 ( κ 2 2 ) τ K 7 1 24 κ 6 ( C V 2 Z 2 3 v 0 2 ) 3 Z L H μ α ̃ , S , 2 ( κ 2 2 )
μ α ̃ , S , ν ( a ) = 0 1 d ζ ζ ν α ̃ ( ζ Z L H ) sin ( a ζ Z L H ) ,
K ¯ ( κ ) 2 τ K 0 Z L H d ζ α ̃ ( ζ ) sin [ κ 2 2 ζ ] e 1 2 κ 2 D δ v ̃ ( 0 , ζ ) τ K 2 .
T P O [ χ H ( κ ¯ , ζ , τ ) ] = M [ χ H ( κ ¯ , ζ , τ ) ] + χ ̃ H ( κ ¯ , ζ , τ ) ,
T P O [ χ H ( κ ¯ , ζ , τ ) ] = χ H ( κ ¯ , ζ , τ ) ,
M 0 = max ζ [ 0 , Z L H ] , τ [ 0 , ] χ ̃ H ( κ ¯ , ζ , τ ) .
Ω = { ( ζ , τ , u ) ζ [ 0 , Z L H ] , τ [ 0 , T ] , u M 0 1 θ P O } .
Ω 0 = { ( ζ , τ , u ) ζ [ 0 , Z L H ] , τ [ 0 , T ] , u L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) θ P O 1 θ P O M 0 } .
Y = { χ H ( κ ¯ , ζ , τ ) X χ H ( κ ¯ , ζ , τ ) L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) θ P O 1 θ P O M 0 } .
T P O [ χ H ( κ ¯ , ζ , τ ) ] L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) = M [ χ H ( κ ¯ , ζ , τ ) ]
M χ H ( κ ¯ , ζ , τ )
θ P O M 0 1 θ P O ,
T P O [ χ H , 1 ( κ ¯ , ζ , τ ) ] T P O [ χ H , 2 ( κ ¯ , ζ , τ ) ] = M [ χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) ]
M χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ )
θ P O χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ )
< χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) .
T FW [ χ H ( κ ¯ , ζ , τ ) ] = K ¯ [ χ H ( κ ¯ , ζ , τ ) ] + Δ ¯ [ χ H ( κ ¯ , ζ , τ ) ] + χ ̃ H ( κ ¯ , ζ , τ ) .
K ¯ n u K ¯ n n ! u .
k = 0 N K ¯ n [ χ H ( κ ¯ , ζ , τ ) ] = k = 1 N + 1 K ¯ n [ χ H ( κ ¯ , ζ , τ ) ] + k = 0 N K ¯ n Δ ¯ [ χ H ( κ ¯ , ζ , τ ) ] + k = 0 N K ¯ n [ L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) ] ,
χ H ( κ ¯ , ζ , τ ) = K ¯ N + 1 [ χ H ( κ ¯ , ζ , τ ) ] + k = 0 N K ¯ n Δ ¯ [ χ H ( κ ¯ , ζ , τ ) ] + k = 0 N K ¯ n [ L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) ] = S N [ χ H ( κ ¯ , ζ , τ ) ] .
M 1 = max ζ [ 0 , Z L H ] , τ [ 0 , ] k = 0 N K ¯ n [ L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) ] .
Ω = { ( ζ , τ , u ) ζ [ 0 , Z L H ] , τ [ 0 , T ] , u M 1 1 θ FW } .
Ω 1 = { ( ζ , τ , u ) ζ [ 0 , Z L H ] , τ [ 0 , T ] , u k = 0 N K ¯ n L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) θ FW 1 θ FW M 1 } .
Y = { χ H ( κ ¯ , ζ , τ ) X χ H ( κ ¯ , ζ , τ ) k = 0 N K ¯ n L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) θ FW 1 θ FW M 1 } .
χ 0 ( κ ¯ , ζ , τ ) = L 1 ( M 0 { L [ n ̃ T ( κ ¯ , ζ , τ ) ] } ) .
S N [ χ H ( κ ¯ , ζ , τ ) ] k = 0 N K ¯ n [ χ 0 ( κ ¯ , ζ , τ ) ] = K ¯ N + 1 [ χ H ( κ ¯ , ζ , τ ) ] + n = 0 N K ¯ n Δ ¯ [ χ H ( κ ¯ , ζ , τ ) ]
K ¯ N + 1 [ χ H ( κ ¯ , ζ , τ ) ] + Δ ¯ n = 0 N K ¯ n [ χ H ( κ ¯ , ζ , τ ) ]
K ¯ N + 1 ( N + 1 ) ! M 1 1 θ FW + Δ ¯ n = 0 N K ¯ n n ! M 1 1 θ FW
[ ε 2 + Δ ¯ exp ( K ¯ ) ] M 1 1 θ FW
( θ FW ε 2 ) M 1 1 θ FW
< θ FW M 1 1 θ FW ,
S N [ χ H , 1 ( κ ¯ , ζ , τ ) ] S N [ χ H , 2 ( κ ¯ , ζ , τ ) ] = S N [ χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) ]
× K ¯ N + 1 [ χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) ] + n = 0 N K ¯ n Δ ¯ [ χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) ] K ¯ N + 1 [ χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) ]
+ Δ ¯ n = 0 N K ¯ n [ χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) ]
( K ¯ N + 1 ( N + 1 ) ! + Δ ¯ n = 0 N K ¯ n n ! ) χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ )
[ ϵ + Δ ¯ exp ( K ¯ ) ] χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ )
( ϵ + θ FW ) χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ )
< χ H , 1 ( κ ¯ , ζ , τ ) χ H , 2 ( κ ¯ , ζ , τ ) .

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