Abstract

The propagation of partially coherent Hermite-cosh-Gaussian (H-ChG) beams through atmospheric turbulence is studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams in turbulence is derived. It is shown that the angular spread of partially coherent H-ChG beams with smaller spatial correlation length σ0, smaller waist width w 0, smaller beam parameter Ω0, and larger beam orders m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w 0, Ω0, and smaller m, n. Under a certain condition partially coherent H-ChG beams may generate the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The angular spread of partially coherent Hermite-Gaussian (H-G), cosh-Gaussian (ChG), Gaussian Schell-model (GSM) beams, and fully coherent H-ChG, H-G, ChG, Gaussian beams is studied and treated as special cases of partially coherent H-ChG beams. The results are interpreted physically.

©2008 Optical Society of America

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References

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  1. E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [Crossref] [PubMed]
  2. E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [Crossref]
  3. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  4. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2008 (1)

2007 (3)

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2900 (2007).
[Crossref]

H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
[Crossref]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
[Crossref]

2006 (3)

2005 (3)

2004 (1)

2003 (2)

2002 (2)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

1998 (2)

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

1980 (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

1979 (3)

1978 (3)

1977 (1)

1972 (1)

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Banakh, V. A.

Baykal, Y.

Cai, Y.

Casperson, L. W.

Chen, X.

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Dogariu, A.

Eyyuboglu, H. T.

Eyyubolu, H. T.

H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
[Crossref]

Farina, J. D.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Gbur, G.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Gori, F.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

He, S.

Ji, X.

Leader, C.

Lü, B.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Mironov, V. L.

Narducci, L. M.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Ouyang, C.

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Plonus, M. A.

Shirai, T.

Tovar, A. A.

Wang, S.

Wolf, E.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Yura, H. T.

Zahid, M.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B: Lasers Opt. (1)

H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (6)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005).
[Crossref]

Opt. Eng. (1)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Other (1)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

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Figures (6)

Fig. 1.
Fig. 1. Angular spread θ sp of a partially coherent H-ChG beam versus the spatial correlation length σ0. The calculation parameters are z=10km, m=n=1, w 0=3cm, Ω0=100m-1.

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Fig. 2.
Fig. 2. Angular spread θ sp of a partially coherent H-ChG beam versus the waist width w 0. The calculation parameters are z=10km, m=n=1, σ0=1.732cm, Ω0=100m-1, C 2 n =10-14m-2/3.

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Fig. 3.
Fig. 3. Angular spread θ sp of a partially coherent H-ChG beam versus the beam parameter Ω0. The calculation parameters are z=10km, m=n=1, w 0=3cm, σ0=4cm, C 2 n =10-15m-2/3.

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Fig. 4.
Fig. 4. Angular spread θ sp of a partially coherent H-ChG beam versus the beam order m, n. The calculation parameters are z=10km, w 0=1cm, σ0=2 cm, Ω0=300m-1, C 2 n =10-14m-2/3.

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Fig. 5.
Fig. 5. Normalized rms width w(z) of the four equivalent partially coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z in free space and in turbulence. a: the corresponding fully coherent Gaussian beam; b: the equivalent GSM beam; c: the equivalent partially coherent H-G beam: d: the equivalent partially coherent H-ChG beam; e: the equivalent partially coherent ChG beam. The calculation parameters are listed in Table 1, and the other parameters are λ=1.06µm, C 2 n =10-14m-2/3.

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Fig. 6.
Fig. 6. Normalized rms width w(z) of the three fully coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z. a: the corresponding fully coherent Gaussian beam; b: the equivalent fully coherent ChG beam; c: the equivalent fully coherent H-ChG beam; d: the equivalent fully coherent H-G beams. The calculation parameters are listed in Table 2, and the other parameters are λ=1.06µm, C 2 n =10-14m-2/3.

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Tables (2)

Tables Icon

Table 1. Beam parameters relating to Fig. 5.

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Tables Icon

Table 2. Beam parameters relating to Fig. 6.

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Equations (82)

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U ( ρ , z = 0 ) = H m ( 2 w 0 ρ x ) H n ( 2 w 0 ρ y ) exp ⁡ ( − ρ x 2 + ρ y 2 w 0 2 ) cosh ⁡ ( Ω 0 ρ x + Ω 0 ρ y ) ,
W ( ρ 1 , ρ 2 , z = 0 ) = H m ( 2 w 0 ρ 1 x ) H n ( 2 w 0 ρ 1 y ) exp ⁡ ( − ρ 1 x 2 + ρ 1 y 2 w 0 2 )
× cosh ( Ω 0 ρ 1 x + Ω 0 ρ 1 y ) exp ⁡ [ − ( ρ 1 x − ρ 2 x ) 2 2 σ 0 2 ]
× H m ( 2 w 0 ρ 2 x ) H n ( 2 w 0 ρ 2 y ) exp ⁡ ( − ρ 2 x 2 + ρ 2 y 2 w 0 2 )
× cosh ( Ω 0 ρ 2 x + Ω 0 ρ 2 y ) exp ⁡ [ − ( ρ 1 y − ρ 2 y ) 2 2 σ 0 2 ] ,
W ( ρ 1 ′ , ρ 2 ′ , z ) = ( k 2 π z ) 2 ∬ d 2 ρ 1 ∬ d 2 ρ 2 W ( ρ 1 , ρ 2 , z = 0 )
× exp ⁡ { i k 2 z [ ( ρ 1 ′ − ρ 1 ) 2 − ( ρ 2 ′ − ρ 2 ) 2 ] } 〈 exp ⁡ [ ψ ( ρ 1 , ρ 1 ′ ) + ψ * ( ρ 2 , ρ 2 ′ ) ] 〉 m ,
〈 exp ⁡ [ ψ ( ρ 1 , ρ 1 ′ ) + ψ * ( ρ 2 , ρ 2 ′ ) ] 〉 m
≈ exp { − 1 ρ 0 2 [ ( ρ 1 − ρ 2 ) 2 + ( ρ 1 − ρ 2 ) · ( ρ 1 ′ − ρ 2 ′ ) + ( ρ 1 ′ − ρ 2 ′ ) 2 ] } ,
ρ 0 = ( 0.545 C n 2 k 2 z ) − 3 5 ,
u = ρ 2 + ρ 1 2 , v = ρ 2 − ρ 1 .
I ( ρ ′ , z ) = W ( ρ ′ , ρ ′ , z )
  = ( k 4 π z ) 2 ∬ d 2 u ∬ d 2 v exp ( − 2 u 2 w 0 2 ) exp ( − v 2 ε 2 ) exp ( − i k z u · v ) exp ( i k z ρ ′ · v )
  × H m [ 2 w 0 ( u x − v x 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y − v y 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× { exp [ 2 Ω 0 ( u x + u y ) ] + exp [ − Ω 0 ( v x + v y ) ]
+ exp [ Ω 0 ( v x + v y ) ] + exp [ − 2 Ω 0 ( u x + u y ) ] }
1 ε 2 = 1 2 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 .
w ( z ) = ∫ ρ ′ 2 I ( ρ ′ , z ) d 2 ρ ′ ∫ I ( ρ ′ , z ) d 2 ρ ′ .
w ( z ) = P + Q k 2 z 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 16 5 ,
P = R 1 R 0 ,
Q = R 2 R 0 ,
R 0 = exp ⁡ ( w 0 2 Ω 0 2 ) L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 1 ,
R 1 = w 0 2 { exp ⁡ ( w 0 2 Ω 0 2 ) ( 1 + w 0 2 Ω 0 2 2 L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 1 + 2 w 0 2 Ω 0 2 2
× [ L n 0 ( − w 0 2 Ω 0 2 ) L m − 1 1 ( − w 0 2 Ω 0 2 ) + L m 0 ( − w 0 2 Ω 0 2 ) L n − 1 1 ( − w 0 2 Ω 0 2 ) ] + w 0 2 Ω 0 2
× [ L n 0 ( − w 0 2 Ω 0 2 ) L m − 2 2 ( − w 0 2 Ω 0 2 ) + L m 0 ( − w 0 2 Ω 0 2 ) L n − 2 2 ( − w 0 2 Ω 0 2 ) ] ) + m + n + 1 2 } ,
R 2 = exp ( w 0 2 Ω 0 2 ) { 2 ( 1 w 0 2 + 1 σ 0 2 ) L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 2 w 0 2 [ L m − 1 1 ( − w 0 2 Ω 0 2 )
× L n 0 ( − w 0 2 Ω 0 2 ) + L m 0 ( − w 0 2 Ω 0 2 ) L n − 1 1 ( − w 0 2 Ω 0 2 ) ] } + 2 ( 1 w 0 2 + 1 σ 0 2 − Ω 0 2 ) + 2 ( m + n ) w 0 2
θ sp ( z ) = w ( z ) z | z → ∞ = Q k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 .
θ sp ( z ) = Q 1 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 1 = 2 ( m + n + 1 w 0 2 + 1 σ 0 2 ) .
θ sp ( z ) = Q 2 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 2 = 2 [ ( 1 w 0 2 + 1 σ 0 2 ) − Ω 0 2 1 + exp ⁡ ( w 0 2 Ω 0 2 ) ] .
θ sp ( z ) = Q 3 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 3 = 2 ( 1 w 0 2 + 1 σ 0 2 ) .
θ sp ( z ) = Q 4 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 4 = exp ⁡ ( w 0 2 Ω 0 2 ) [ 2 w 0 2 L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 2 w 0 2 [ L m − 1 1 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 )
+ L m 0 ( − w 0 2 Ω 0 2 ) L n − 1 1 ( − w 0 2 Ω 0 2 ) ] + 2 ( 1 w 0 2 − Ω 0 2 ) + 2 ( m + n ) w 0 2 .
θ sp ( z ) = Q 5 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 5 = 2 ( m + n + 1 w 0 2 ) .
θ sp ( z ) = Q 6 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 6 = 2 [ ( 1 w 0 2 − Ω 0 2 1 + exp ⁡ ( w 0 2 Ω 0 2 ) ] .
θ sp ( z ) = Q 7 k 2 + 4 ( . 0545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 7 = 2 w 0 2 .
Q = Q 1 = Q 2 = Q 3 = Q 7
Q 4 = Q 5 = Q 6 = Q 7
w ( z ) = F F 0 ,
F 0 = ∫ I ( ρ ′ , z ) d 2 ρ ′ ,
F 0 = ∫ ρ ′ 2 I ( ρ ′ , z ) d 2 ρ ′ .
F 0 = ∬ W ( 0 ) ( ρ , ρ , z = 0 ) d 2 ρ
= 1 2 2 m + n − 1 m ! n ! w 0 2 π [ exp ( w 0 2 Ω 0 2 ) L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 1 ] .
F = F 1 + F 2 + F 3 + F 4 ,
F 1 = ∫ ρ ′ 2 I 1 ( ρ ′ , z ) d 2 ρ ′ ,
F 2 = ∫ ρ ′ 2 I 2 ( ρ ′ , z ) d 2 ρ ′ ,
F 3 = ∫ ρ ′ 2 I 3 ( ρ ′ , z ) d 2 ρ ′ ,
F 4 = ∫ ρ ′ 2 I 4 ( ρ ′ , z ) d 2 ρ ′ ,
I 1 ( ρ ′ , z ) = 1 4 ( k 2 π z ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y - v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
  × exp ( − 2 u 2 w 0 2 ) exp ( − v 2 ε 2 ) exp ( − i k z u · v ) exp ( i k z ρ ′ · v ) exp [ 2 Ω 0 ( u x + u y ) ] ,
I 2 ( ρ ′ , z ) = 1 4 ( k 2 π z ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y - v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ( − 2 u 2 w 0 2 ) exp ( − v 2 ε 2 ) exp ( − i k z u · v ) exp ( i k z ρ ′ · v ) exp [ Ω 0 ( v x + v y ) ] .
∫ x 2 exp ⁡ ( − i 2 π x s ) d x = − 1 ( 2 π ) 2 δ ″ ( s ) ,
F 1 = F 11 + F 12 ,
F 11 = − 1 4 ( z k ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y − v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ⁡ ( − 2 u 2 w 0 2 ) exp ⁡ ( − v 2 ε 2 ) exp ⁡ ( − i k z u · v ) exp ⁡ [ 2 Ω 0 ( u x + u y ) ] δ ″ ( v x ) δ ( v y ) ,
F 12 = − 1 4 ( z k ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y − v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ⁡ ( − 2 u 2 w 0 2 ) exp ⁡ ( − v 2 ε 2 ) exp ⁡ ( − i k z u · v ) exp ⁡ [ 2 Ω 0 ( u x + u y ) ] δ ( v x ) δ ″ ( v y ) ,
∫ f ( x ) δ ( x ) d x = f ( 0 ) ,
∫ exp ⁡ [ − ( x − y ) 2 ] H m ( x ) H n ( x ) d x = 2 n π m ! y n − m L n n − m ( − 2 y 2 ) ,
F 11 = − 1 4 ( z k ) 2 2 n n ! π w 0 2 exp ⁡ ( w 0 2 Ω 0 2 2 ) L n 0 ( − w 0 2 Ω 0 2 ) ∬ d v x d u x
× H m [ 2 w 0 ( u x − v x 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] exp ⁡ ( 2 u x 2 w 0 2 )
× exp ⁡ ( − v x 2 ε 2 ) exp ⁡ ( − i k z u x v x ) exp ⁡ ( 2 Ω 0 u x ) δ ″ ( v x ) .
∫ exp ( − x 2 ) H m ( x + y ) H n ( x + z ) d x = 2 n π m ! y n − m z n − m L n n − m ( − 2 y z ) ,
∫ f ( x ) δ ″ ( x ) d x = f ″ ( 0 ) ,
F 11 = − 1 4 ( z k ) 2 2 m + n − 1 m ! n ! π w 0 2 exp ⁡ ( w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 )
× [ ( − k 2 w 0 2 4 z 2 − 2 ε 2 ) L m 0 ( − w 0 2 Ω 0 2 ) ] − k 2 w 0 4 Ω 0 2 4 z 2 L m 0 ( − w 0 2 Ω 0 2 ) − k 2 w 0 4 Ω 0 2 z 2 L m − 2 2 ( − w 0 2 Ω 0 2 )
− ( 2 w 0 2 + k 2 w 0 2 2 z 0 ) L m − 1 1 ( − w 0 2 Ω 0 2 ) − k 2 w 0 4 Ω 0 2 z 2 L m − 1 1 ( − w 0 2 Ω 0 2 ) ] .
F 2 = F 21 + F 22 ,
F 21 = − 1 4 ( z k ) 2 2 m + n − 1 m ! n ! π w 0 2 [ ( − k 2 w 0 2 4 z 2 − 2 ε 2 ) + Ω 0 2 − m ( 2 w 0 2 + k 2 w 0 2 2 z 2 ) ] .
w ( z ) = P + Q k 2 z 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 16 5 .

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