Abstract

The analytical treatment of linear intracavity phase aberrations and their compensation by an appropriately configured intracavity adaptive optic element is presented in this paper. The approximate geometrical treatment of the effects of intracavity phase aberrations is first treated, followed by a complete exact diffraction integral formulation of the adaptive compensation of linear phase-tilt and phase-curvature aberrations. This new analysis yields the exact intracavity phase compensation for phase tilt, curvature-of-field, and astigmatism-phase aberrations in positive branch, confocal unstable resonator cavities.

© 1981 Optical Society of America

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Corrections

Kurt Edmund Oughstun, "Intracavity adaptive optic compensation of phase aberrations. I: Analysis: erratum," J. Opt. Soc. Am. 71, 1409-1409 (1981)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-71-11-1409

References

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  1. A. E. Siegman, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-3, 156–163 (1967).
    [CrossRef]
  2. D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. 12, 997–1010 (1973).
    [CrossRef] [PubMed]
  3. A. N. Chester, “Three-dimensional diffraction calculations of laser resonator modes,” Appl. Opt. 12, 2353–2366 (1973).
    [CrossRef] [PubMed]
  4. D. B. Rensch, “Three-dimensional unstable resonator calculations with laser medium,” Appl. Opt. 13, 2546–2561 (1974).
    [CrossRef] [PubMed]
  5. A. E. Siegman and E. A. Sziklas, “Mode calculations in unstable resonators with flowing saturable gain. 1. Hermite–Gaussian Expansion,” Appl. Opt. 13, 2775 (1974).
    [CrossRef] [PubMed]
  6. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2. Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  7. L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. QE-9, 1102–1113 (1973).
    [CrossRef]
  8. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [CrossRef]
  9. R. R. Butts and P. V. Avizonis, “Asymptotic analysis of unstable laser resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1072–1078 (1978).
    [CrossRef]
  10. S. H. Cho, S. Y. Shin, and L. B. Felsen, “Ray optical analysis of unstable resonators,” J. Opt. Soc. Am. 69, 563–574 (1979).
    [CrossRef]
  11. Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
    [CrossRef]
  12. A. Gerrand and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).
  13. K. E. Oughstun, “Intracavity adaptive optics study. Part I. Linear intracavity phase perturbations and their compensation by intracavity adaptive optics. A. Analysis,” (United Technologies Research Center, East Hartford, Conn., 1980).
  14. A. E. Siegman, “A canonical formulation for analyzing multi-element unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–39 (1976).
    [CrossRef]

1979 (1)

1978 (1)

1976 (1)

A. E. Siegman, “A canonical formulation for analyzing multi-element unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–39 (1976).
[CrossRef]

1975 (1)

1974 (2)

1973 (4)

1972 (1)

Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
[CrossRef]

1967 (1)

A. E. Siegman, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-3, 156–163 (1967).
[CrossRef]

Anan’ev, Yu. A.

Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
[CrossRef]

Avizonis, P. V.

Burch, J. M.

A. Gerrand and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Butts, R. R.

Chen, L. W.

L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. QE-9, 1102–1113 (1973).
[CrossRef]

Chester, A. N.

Cho, S. H.

Felsen, L. B.

S. H. Cho, S. Y. Shin, and L. B. Felsen, “Ray optical analysis of unstable resonators,” J. Opt. Soc. Am. 69, 563–574 (1979).
[CrossRef]

L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. QE-9, 1102–1113 (1973).
[CrossRef]

Gerrand, A.

A. Gerrand and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Horwitz, P.

Oughstun, K. E.

K. E. Oughstun, “Intracavity adaptive optics study. Part I. Linear intracavity phase perturbations and their compensation by intracavity adaptive optics. A. Analysis,” (United Technologies Research Center, East Hartford, Conn., 1980).

Rensch, D. B.

Shin, S. Y.

Siegman, A. E.

A. E. Siegman, “A canonical formulation for analyzing multi-element unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–39 (1976).
[CrossRef]

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2. Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

A. E. Siegman and E. A. Sziklas, “Mode calculations in unstable resonators with flowing saturable gain. 1. Hermite–Gaussian Expansion,” Appl. Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

A. E. Siegman, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-3, 156–163 (1967).
[CrossRef]

Sziklas, E. A.

Appl. Opt. (5)

IEEE J. Quantum Electron. (3)

A. E. Siegman, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. QE-3, 156–163 (1967).
[CrossRef]

L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. QE-9, 1102–1113 (1973).
[CrossRef]

A. E. Siegman, “A canonical formulation for analyzing multi-element unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–39 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

Sov. J. Quantum Electron. (1)

Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
[CrossRef]

Other (2)

A. Gerrand and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

K. E. Oughstun, “Intracavity adaptive optics study. Part I. Linear intracavity phase perturbations and their compensation by intracavity adaptive optics. A. Analysis,” (United Technologies Research Center, East Hartford, Conn., 1980).

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Figures (10)

Fig. 1
Fig. 1

Depiction of the optical path traversed by a typical ray associated with the dominant plane-wave geometrical eigenmode of the unaberrated ideal unstable cavity with geometrical magnification M. The transverse distance of the ray from the optic axis for the final iterations in the cavity is indicated at the left of the figure.

Fig. 2
Fig. 2

Segmentation of an extended inhomogeneous linear phase medium of refractive index n(x, z) into discrete phase planes located at zj.

Fig. 3
Fig. 3

Linear positive branch, confocal unstable resonator cavity with a segmented inhomogeneous phase medium and intracavity optical elements.

Fig. 4
Fig. 4

Positional dependence of the geometrical phase-weighting factor.

Fig. 5
Fig. 5

Behavior of the geometrical phase-weighting factor as a function of the cavity magnification M for a uniformly distributed phase medium in a positive branch, confocal unstable resonator cavity.

Fig. 6
Fig. 6

Unstable resonator cavity configuration with an intracavity adaptive deformable mirror.

Fig. 7
Fig. 7

Propagation of the phase front (indicated by the solid curves) in a single round-trip iteration through the aberration-corrected resonator cavity for a uniform phase distribution reflected off the convex feedback mirror.

Fig. 8
Fig. 8

The perturbed and corrected relative collimated Fresnel number as a function of the applied phase-curvature aberration strength for a linear positive branch, confocal unstable resonator cavity with z1 = z2 = z3 = 503.145 cm, M = 2, R = 3018.868 cm, and Nc = 0.8333.

Fig. 9
Fig. 9

The phase-curvature correction weighting coefficient −β2/δ2 as a function of the applied phase-curvature aberration strength for a linear positive branch, confocal unstable resonator cavity with z1 = z2 = z3 = 503.145 cm, M = 2, R = 3018.868 cm, and Nc = 0.8333.

Fig. 10
Fig. 10

(a) Geometrical mode phase front for the intracavity perturbed and corrected cavity configurations under the influence of a negative phase-curvature aberration δ2. The exiting perturbed cavity mode phase front is divergent, whereas that of the intracavity corrected cavity is collimated, but with an altered magnification. The dotted lines indicate the boundary ray trajectories followed by the unperturbed geometrical mode phase front. (b) Geometrical mode phase front for the intracavity perturbed and corrected cavity configurations under the influence of a positive phase-curvature aberration δ2. The exiting perturbed cavity mode phase front is convergent, whereas that of the intracavity corrected cavity is collimated, but with an altered magnification. The dotted lines indicate the boundary ray trajectories followed by the unperturbed geometrical mode phase front.

Equations (52)

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Δ ζ TOT ( x ) = l = 0 Δ ζ ( x M 2 ) ,
ϕ j ( x ) = z j - Δ z / 2 z j + Δ z / 2 n ( x , z ) d z - Δ z = z j - Δ z / 2 z j + Δ z / 2 [ n ( x , z ) - 1 ] d z .
ϕ j ( x ) = k = 1 δ j k x k ,
Δ ζ ( x ) = z T j = 0 N + 1 { k = 1 δ j k x k + k = 1 δ j k x k M k × [ 1 + ( M - 1 ) j N + 1 ] k } = z T j = 0 N + 1 k = 1 δ j k x k { 1 + 1 M k [ 1 + ( M - 1 ) j N + 1 ] k } .
Δ ζ TOT ( x ) = z T j = 0 N + 1 k = 1 δ j k x k × { 1 + 1 M k [ 1 + ( M - 1 ) j N + 1 ] k } l = 0 1 ( M k ) l = z T j = 0 N + 1 k = 1 δ j k x k 1 M k - 1 { M k + [ 1 + ( M - 1 ) j N + 1 ] k } = z T j = 0 N + 1 k = 1 δ j k α ¯ j k x k .
α ¯ j k = 1 M k - 1 { M k + [ 1 + ( M - 1 ) j N + 1 ] k } .
Δ ζ TOT ( x ) = z T j = 0 N + 1 k = 1 δ k ( z j ) α ¯ k ( z j ) x k ,
α ¯ k ( z j ) = 1 M k - 1 { M k + [ 1 + ( M - 1 ) z j z T ] k } .
α ¯ k ( z T ) = 1 + α ¯ k ( 0 ) ;
δ k ( z n ) = α ¯ k ( z m ) α ¯ k ( z n ) δ k ( z m ) .
α ¯ k ( z m ) α ¯ k ( z n ) = M k + [ 1 + ( M - 1 ) z m z T ] k M k + [ 1 + ( M - 1 ) z n z T ] k ,
δ k ( z n ) = - α ¯ k ( z m ) α ¯ k ( z n ) δ k ( z m ) ,
Δ ζ hom ( x ) = z T k = 1 n k α ¯ k x k ,
α ¯ k = 1 M k - 1 [ M k + M k + 1 - 1 ( k + 1 ) ( M - 1 ) ] ,             k = 1 , 2 , 3 , ,
δ k ( z l ) = - 1 α ¯ k ( z l ) × [ j = 0 j l N + 1 δ k ( z j ) α ¯ k ( z j ) + 1 z T z m z n n k ( z ) α ¯ k ( z ) d z ] .
ϕ ( x ) = δ 1 x ,
γ ˜ u ( x 2 , y 2 ) = - i 2 λ z T e i 2 k z T × exp { - i π λ 2 δ 1 M x [ 1 - 1 M x R x ( z 2 + z 3 ) ] M x + 1 x 2 } × A u ( x 1 , y 1 ) exp { - i π λ 2 δ 1 M x [ 1 - 1 M x R x ( z 2 + z 3 ) ] M x + 1 x 1 } × exp { i π λ z T [ ( x 2 - M x x 1 ) 2 M x + 1 + ( y 2 - M y y 1 ) 2 M y + 1 ] } d x 1 d y 1 .
1 - 1 M x R x ( z 2 + z 3 ) = 1 M x ( 1 + z 1 + z T R x ) ,
β 1 = - δ 1 M x ( 2 z 1 + z 2 + z 3 ) + z 2 + z 3 M x ( 2 z 1 + 2 z 2 + z 3 ) + z 3 ,
ϕ ( x ) = δ 2 x 2 ,
R p = 1 2 δ 2 .
( A B C D ) = [ M x + Ω z T ( 1 + 1 M x ) + Δ Γ 1 M x + τ ] ,
Ω 8 δ 2 2 z 1 ( z 2 + z 3 ) ( 1 + 2 R x z 1 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] - 4 δ 2 { ( 1 + 2 R x z 1 ) [ z T - 1 M x R x ( z 2 + z 3 ) ( z T + z 1 ) ] + 2 R x z 1 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] } ,
Δ 8 δ 2 2 z 1 2 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] - 4 δ 2 z 1 [ z T + z 2 + z 3 - 2 M x R x z T ( z 2 + z 3 ) ] ,
Γ 8 δ 2 2 ( z 2 + z 3 ) ( 1 + 2 R x z 1 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] - 4 δ 2 { ( 1 + 2 R x z 1 ) [ 1 - 2 M x R x ( z 2 + z 3 ) ] + 2 R x ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] } ,
τ 8 δ 2 2 z 1 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] - 4 δ 2 [ z T - 1 M x R x ( z 2 + z 3 ) ( z T + z 1 ) ] .
γ ˜ u ( x 2 , y 2 ) = - i 2 λ z T e i 2 k z T A u ( x 1 , y 1 ) × exp { i k 2 [ ( M x + Ω x ) x 1 2 + ( 1 M x + τ x ) x 2 2 - 2 x 1 x 2 z T ( 1 + 1 M x ) + Δ x + ( M y + Ω y ) y 1 2 + ( 1 M y + τ y ) y 2 2 - 2 y 1 y 2 z T ( 1 + 1 M y ) + Δ y ] } d x 1 d y 1 ,
γ ˜ u ( x 2 ) = ( - i 2 λ z T ) 1 / 2 e i 2 k z T - a a u ( x 1 ) × exp [ i π λ ( M x + Ω x ) x 1 2 + ( 1 M x + τ x ) x 2 2 - 2 x 1 x 2 z T ( 1 + 1 M x ) + Δ x ] d x 1 ,
γ ˜ u ( M x a ξ ) = ( - i 2 N T ) 1 / 2 e i 2 k z T × exp [ i π N c ( 1 + τ x M x + 1 1 + Ω x / M x ) ξ 2 ] × - 1 1 u ( a ζ ) exp [ i π N c ( 1 + Ω x M x ) ( ζ - ξ 1 + Ω x / M x ) 2 ] d ζ .
γ ˜ u [ M x a ( 1 + Ω x M x ) ρ ] = ( - i 2 N T ) 1 / 2 e i 2 k z T × exp [ i π N c ( 1 + Ω x M x ) 2 ( 1 + τ x M x + 1 1 + Ω x / M x ) ρ 2 ] × - 1 1 u ( a ζ ) exp [ i π N c ( 1 + Ω x M x ) ( ζ - ρ ) 2 ] d ζ .
N T a 2 λ z T , N c M 2 a 2 λ z T ( M + 1 ) .
N c P = N c ( 1 + Ω x M x ) ,
( A B C D ) = ( M x + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 z T ( 1 + 1 M x ) + Δ 0 + Δ 1 β 2 + Δ 2 β 2 2 Γ 0 + Γ 1 β 2 + Γ 2 β 2 2 1 M x + τ 0 + τ 1 β 2 + τ 2 β 2 2 ) ,
Ω 0 = - 2 δ 2 ( ( 1 + 2 R x z 1 ) [ z T + ( z 2 + z 3 ) ( 1 - 2 M x R x z T ) ] + z 1 { ( 1 + 2 R x z T ) [ 1 - 2 M x R x ( z 2 + z 3 ) ] + 2 R x ( z 2 + z 3 ) } ) + 8 δ 2 2 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] z 1 ( 1 + 2 R x z 1 ) ,
Ω 1 = - 2 { [ 1 + 2 R x ( z 1 + z 2 ) ] [ z T + z 3 ( 1 - 2 M x R x z T ) ] + ( z 1 + z 2 ) [ ( 1 - 2 M x R x z 3 ) ( 1 + 2 R x z T ) + 2 R x z 3 ] } + 4 δ 2 ( z 2 ( 1 + 2 R x z 1 ) [ z T + z 3 ( 1 - 2 M x R x z T ) ] + z 1 [ 1 + 2 R x ( z 1 + z 2 ) ] { z 2 + z 3 + z 3 × [ 1 - 2 M x R x ( z 2 + z 3 ) ] } + ( z 1 + z 2 ) ( 1 + 2 R x z 1 ) [ z 3 + ( z 2 + z 3 ) ( 1 - 2 M x R x z 3 ) ] + z 2 z 1 [ ( 1 - 2 M x R x z 3 ) ( 1 + 2 R x z T ) + 2 R x z 3 ] ) - 16 δ 2 2 z 2 z 1 ( 1 + 2 R x z 1 ) × { z 2 + 2 z 3 [ 1 - 1 M x R x ( z 2 + z 3 ) ] } ,
Ω 2 = 8 z 3 ( 1 - 1 M x R x z 3 ) ( ( z 1 + z 2 ) [ 1 + 2 R x ( z 1 + z 2 ) ] - 2 δ 2 z 2 { ( z 1 + z 2 ) ( 1 + 2 R x z 1 ) + z 1 [ 1 + 2 R x ( z 1 + z 2 ) ] } + 4 δ 2 2 z 2 2 z 1 ( 1 + 2 R x z 1 ) ) ,
Δ 0 = - 2 δ 2 ( z 1 [ z T + ( z 2 + z 3 ) ( 1 - 2 M x R x z T ) ] + z 1 { z 2 + z 3 + z T [ 1 - 2 M x R x ( z 2 + z 3 ) ] } ) + 8 δ 2 2 z 1 2 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] ,
Δ 1 = - 2 { ( z 1 + z 2 ) [ z T + z 3 ( 1 - 2 M x R x z T ) + z 3 + z T ( 1 - 2 M x R x z 3 ) ] } + 4 δ 2 z 1 ( z 2 [ z T + z 3 ( 1 - 2 M x R x z T ) ] + ( z 1 + z 2 ) { z 2 + z 3 + z 3 [ 1 - 2 M x R x ( z 2 + z 3 ) ] } + ( z 1 + z 2 ) [ z 3 + ( z 2 + z 3 ) ( 1 - 2 M x R x z 3 ) ] + z 2 [ z 3 + z T ( 1 - 2 M x R x z 3 ) ] ) - 16 δ 2 2 z 1 2 z 2 { z 2 + 2 z 3 [ 1 - 1 M x R x ( z 2 + z 3 ) ] } ,
Δ 2 = 8 z 3 ( 1 - 1 M x R x z 3 ) × [ ( z 1 + z 2 ) ( z 1 + z 2 - 4 δ 2 z 1 z 2 ) + 4 δ 2 2 z 2 2 z 1 2 ] ,
Γ 0 = - 2 δ 2 { ( 1 + 2 R x z 1 ) [ 1 - 2 M x R x ( z 2 + z 3 ) ] + ( 1 + 2 R x z T ) [ 1 - 2 M x R x ( z 2 + z 3 ) ] + 2 R x ( z 2 + z 3 ) } + 8 δ 2 2 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] ( 1 + 2 R x z 1 ) ,
Γ 1 = - 2 { [ 1 + 2 R x ( z 1 + z 2 ) ] ( 1 - 2 M x R x z 3 ) + ( 1 - 2 M x R x z 3 ) ( 1 + 2 R x z T ) + 2 R x z 3 } + 4 δ 2 ( z 2 ( 1 + 2 R x z 1 ) ( 1 - 2 M x R x z 3 ) + [ 1 + 2 R x ( z 1 + z 2 ) ] { z 2 + z 3 + z 3 [ 1 - 2 M x R x ( z 2 + z 3 ) ] } + ( 1 + 2 R x z 1 ) [ z 3 + ( z 2 + z 3 ) ( 1 - 2 M x R x z 3 ) ] + z 2 [ ( 1 - 2 M x R x z 3 ) ( 1 + 2 R x z T ) + 2 R x z 3 ] ) + 8 δ 2 2 z 2 { z 2 + 2 z 3 [ 1 - 1 M x R x ( z 2 + z 3 ) ] } ( 1 + 2 R x z 1 ) ,
Γ 2 = 8 z 3 ( 1 - 1 M x R x z 3 ) { 1 + 2 R x ( z 1 + z 2 ) - 4 δ 2 z 2 [ 1 + 1 R x ( 2 z 1 + z 2 ) ] + 4 δ 2 2 z 2 2 ( 1 + 2 R x z 1 ) } ,
τ 0 = - 2 δ 2 { z 1 [ 1 - 2 M x R x ( z 2 + z 3 ) ] + z 2 + z 3 + z T [ 1 - 2 M x R x ( z 2 + z 3 ) ] } + 8 δ 2 2 z 1 ( z 2 + z 3 ) [ 1 - 1 M x R x ( z 2 + z 3 ) ] ,
τ 1 = - 2 [ ( z 1 + z 2 ) ( 1 - 2 M x R x z 3 ) + z 3 + z T ( 1 - 2 M x R x z 3 ) ] + 4 δ 2 ( z 1 z 2 ( 1 - 2 M x R x z 3 ) + ( z 1 + z 2 ) × { z 2 + z 3 + z 3 [ 1 - 2 M x R x ( z 2 + z 3 ) ] } + z 1 [ z 3 + ( z 2 + z 3 ) ( 1 - 2 M x R x z 3 ) ] + z 2 [ z 3 + z T ( 1 - 2 M x R x z 3 ) ] ) + 8 δ 2 2 z 1 z 2 { z 2 + 2 z 3 [ 1 - 1 M x R x ( z 2 + z 3 ) ] } ,
τ 2 = 8 z 3 ( 1 - 1 M x R x z 3 ) × [ z 1 + z 2 - 2 δ 2 z 2 ( 2 z 1 + z 2 ) + 4 δ 2 2 z 2 2 z 1 ] .
( x 2 0 ) = [ M x + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 z T ( 1 + 1 M x ) + Δ 0 + Δ 1 β 2 + Δ 2 β 2 2 Γ 0 + Γ 1 β 2 + Γ 2 β 2 2 1 M x + τ 0 + τ 1 β 2 + τ 2 β 2 2 ] ( x 1 0 ) ,
β 2 = - Γ 1 - ( Γ 1 2 - 4 Γ 0 Γ 2 ) 1 / 2 2 Γ 2 ,
β 2 ( z 1 + z 2 ) = - α ¯ 2 ( z 1 ) α ¯ 2 ( z 1 + z 2 ) δ 2 ( z 1 ) ,
α ¯ 2 ( z 1 ) α ¯ 2 ( z 1 + z 2 ) = M 2 + [ 1 + ( M - 1 ) z 1 z T ] 2 M 2 + [ 1 + ( M - 1 ) z 1 + z 2 z T ] 2 .
γ ˜ u [ M x a ( 1 + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 M x ) ρ ] = - i 2 N T e i 2 k z T exp { i π N c ( 1 + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 M x ) 2 × [ 1 + ( τ 0 + τ 1 β 2 + τ 2 β 2 2 ) M x + M x M x + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 ] ρ 2 } × - 1 1 u ( a ζ ) exp [ i π N c ( 1 + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 M x ) ( ζ - ρ ) 2 ] d ζ ,
N c c = N c ( 1 + Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 M x ) ,
Ω 0 + Ω 1 β 2 + Ω 2 β 2 2 = 0.