Abstract

We investigate theoretically gain-guided modes in unstable resonators with a uniformly reflective mirror in the area of highly efficient steady-state lasers, where the gain saturation is the main efficiency factorly. We achieved self-consistent Hermite–Gaussian modes at significant gain saturation as well as the connection of the mode’s scaling factor and mode amplitude coefficients with the system parameters by using complex paraxial wave optics. A new stabilization mechanism, saturation guiding, works together with gain guiding in unstable resonators. We obtained more actual results for mode generation and selection by integrating the laser rate equation.

© 2001 Optical Society of America

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  1. K. J. Snell, N. McCarthy, M. Piche, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
    [CrossRef]
  2. V. Magni, G. Valentini, S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum. Electron. 23, 1105–1134 (1991).
    [CrossRef]
  3. V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 915–921.
  5. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
    [CrossRef]
  6. F. Salin, J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992).
    [CrossRef] [PubMed]
  7. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
    [CrossRef]
  8. Y. Sun, A. E. Siegman, “Optical mode properties of laterally offset gain and index guiding structures,” IEEE J. Quantum Electron 32, 790–795 (1996).
    [CrossRef]
  9. C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
    [CrossRef]
  10. D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.
  11. J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
    [CrossRef]
  12. J. Whittle, D. R. Skinner, “Transfer efficiency formula for diffusely reflecting laser pumping cavities,” Appl. Opt. 5, 1179–1182 (1966).
    [CrossRef] [PubMed]
  13. Ref. 4, pp. 485–490.
  14. A. Hardy, “Gaussian modes of resonators containing saturable gain medium,” Appl. Opt. 19, 3830–3835 (1980).
    [CrossRef] [PubMed]
  15. G. J. Ernst, W. J. Witteman, “Mode structure of active resonators,” IEEE J. Quantum Electron. QE-9, 911–919 (1973).
    [CrossRef]
  16. Ref. 4, pp. 777–814.
  17. W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [CrossRef]
  18. Y. A. Ananiev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
    [CrossRef]
  19. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]

1999 (1)

C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

1998 (1)

J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
[CrossRef]

1996 (1)

Y. Sun, A. E. Siegman, “Optical mode properties of laterally offset gain and index guiding structures,” IEEE J. Quantum Electron 32, 790–795 (1996).
[CrossRef]

1992 (2)

F. Salin, J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992).
[CrossRef] [PubMed]

V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
[CrossRef]

1991 (1)

V. Magni, G. Valentini, S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum. Electron. 23, 1105–1134 (1991).
[CrossRef]

1990 (1)

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

1988 (1)

K. J. Snell, N. McCarthy, M. Piche, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

1983 (1)

N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

1980 (1)

1973 (2)

G. J. Ernst, W. J. Witteman, “Mode structure of active resonators,” IEEE J. Quantum Electron. QE-9, 911–919 (1973).
[CrossRef]

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[CrossRef]

1971 (1)

Y. A. Ananiev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

1966 (1)

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Ananiev, Y. A.

Y. A. Ananiev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

De Silvestri, S.

V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
[CrossRef]

V. Magni, G. Valentini, S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum. Electron. 23, 1105–1134 (1991).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Ernst, G.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

Ernst, G. J.

G. J. Ernst, W. J. Witteman, “Mode structure of active resonators,” IEEE J. Quantum Electron. QE-9, 911–919 (1973).
[CrossRef]

Golla, D.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

Hardy, A.

Knoke, S.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

Lin, A. P.

J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
[CrossRef]

Magni, V.

V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
[CrossRef]

V. Magni, G. Valentini, S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum. Electron. 23, 1105–1134 (1991).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

McCarthy, N.

K. J. Snell, N. McCarthy, M. Piche, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

Newstein, M.

N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Perry, N.

N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Piche, M.

K. J. Snell, N. McCarthy, M. Piche, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

Qian, Lie-Jia

V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
[CrossRef]

Rabinowitz, P.

N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Salin, F.

Schöne, W.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

Serral, C.

C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Shen, D. Y.

J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
[CrossRef]

Sherstobitov, V. E.

Y. A. Ananiev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

Siegman, A. E.

Y. Sun, A. E. Siegman, “Optical mode properties of laterally offset gain and index guiding structures,” IEEE J. Quantum Electron 32, 790–795 (1996).
[CrossRef]

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 915–921.

Skinner, D. R.

Snell, K. J.

K. J. Snell, N. McCarthy, M. Piche, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

Song, J.

J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
[CrossRef]

Squier, J.

Sun, Y.

Y. Sun, A. E. Siegman, “Optical mode properties of laterally offset gain and index guiding structures,” IEEE J. Quantum Electron 32, 790–795 (1996).
[CrossRef]

Svelto, O.

V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Tünnermann, A.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

Ueda, K.

J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
[CrossRef]

Valentini, G.

V. Magni, G. Valentini, S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum. Electron. 23, 1105–1134 (1991).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

van Druten, N. J.

C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

van Exter, M. P.

C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Welling, H.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

Whittle, J.

Witteman, W. J.

G. J. Ernst, W. J. Witteman, “Mode structure of active resonators,” IEEE J. Quantum Electron. QE-9, 911–919 (1973).
[CrossRef]

Woerdman, J. P.

C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

J. Song, A. P. Lin, D. Y. Shen, K. Ueda, “High optical-to-optical efficiency of LD pumped CW Nd:YAG laser under pumping distribution control,” Appl. Phys. B 66, 539–542 (1998).
[CrossRef]

IEEE J. Quantum Electron (1)

Y. Sun, A. E. Siegman, “Optical mode properties of laterally offset gain and index guiding structures,” IEEE J. Quantum Electron 32, 790–795 (1996).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. Serral, M. P. van Exter, N. J. van Druten, J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

G. J. Ernst, W. J. Witteman, “Mode structure of active resonators,” IEEE J. Quantum Electron. QE-9, 911–919 (1973).
[CrossRef]

J. Appl. Phys. (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

K. J. Snell, N. McCarthy, M. Piche, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

V. Magni, S. De Silvestri, Lie-Jia Qian, O. Svelto, “Rod-imaging supergaussian unstable resonator for high power solid-state lasers,” Opt. Commun. 94, 87–91 (1992).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum. Electron. (1)

V. Magni, G. Valentini, S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum. Electron. 23, 1105–1134 (1991).
[CrossRef]

Phys. Rev. A (1)

N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Sov. J. Quantum Electron. (1)

Y. A. Ananiev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

Other (4)

Ref. 4, pp. 777–814.

Ref. 4, pp. 485–490.

D. Golla, S. Knoke, W. Schöne, G. Ernst, A. Tünnermann, H. Welling, “Design and operation of a 250 W CW, diode laser side-pumped Nd:YAG rod laser,” in Advanced Solid-State Lasers, B. H. T. Chai, S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 207–209.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 915–921.

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Figures (9)

Fig. 1
Fig. 1

Saturation of the gain profile by a Gaussian beam: g 0 = 1 cm-1; α0 = 1 cm-1; ω g b = 3; curve 1, S = 0; curve 2, S = 1; curve 3, S = 5; curve 4, S = 10.

Fig. 2
Fig. 2

(a) Distributed longitudinally uniform Gaussian-like saturable gain medium in an unstable plano–convex resonator with uniform reflectivity mirrors. (b) Left- and right-circulating normalized intensities in a laser oscillator with large output coupling, according to the Rigrod analysis. R 1 and R 2 are the mirrors power reflectances.

Fig. 3
Fig. 3

Resonator magnification function F(M): curve 1, obtained with Eq. (17); curve 2, 1.2 ln M; curve 3, obtained with Eq. (23); curve 4, ln M.

Fig. 4
Fig. 4

Alternative model with a relatively short active medium located in front of the output plane mirror of the unstable plano–convex resonator.

Fig. 5
Fig. 5

Quadratic approach mode attenuation coefficients γ m+ n (M): curves 1 and 2, γ0, γ1 for a three-level system, respectively, with G = 1.2, α0/g 0 = 1.4; curves 3 and 4, the same for a four-level system with G = 1.3; curve 5, 1/M 2; curve 6, 1/M 4.

Fig. 6
Fig. 6

Mode attenuation coefficients γ0 and γ1 dependent on S for a four-level system at M = 1, curves 1 and 2, respectively. Curves 3 and 4 represent the same dependence with mode areas reduced by 10%.

Fig. 7
Fig. 7

Mode attenuation coefficients γ0(S) and γ1(S) for a three-level system with M = 1; α0/g 0 = 2; G = 2, curves 1 and 2, respectively. Curve 3 was obtained for a fourth-order super-Gaussian beam whose amplitude radius was reduced by a factor of 0.9.

Fig. 8
Fig. 8

Relative error from approximation (36) in the TEM00 mode transverse integrand Eq. (28): δ=gsr-gsrIr|r=ωS/gS0I0at r = ω b depending on S.

Fig. 9
Fig. 9

More realistic behavior of four-level system mode attenuation coefficients of γ0(M) and γ1(M) at G = 1.16, curves 1 and 3, respectively, compared with the same results obtained with the quadratic approach, curves 2 and 4, respectively.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

η=IoutIavail=S1+S,
gr=g0+α0exp-2r2ωg2-α0.
Ir, z=I0zexp-2r2/ωb2z,
gSr, z=g0+α0exp-2r2ωg2-α01+Szexp-2r2ωb2z, Sz=I0zIS,
ωb/ωg1
ωb2ωg2  g0g0+α0S1+S.
kz=2πλ0n0-i λ02πgA2z2,
gA2zd2gAr, zdr2r=0.
Mdz=ABCD=1dzn0-n0γ˜2zdz1
n0γ˜2zi λ02π gA2z=i λ0g0π1+Sz×g0+α0g01ωg2-Sz1+Sz1ωb2z.
Mz+dz=MdzMz,
Sz=2PincavzISπωb2z,
M˜l=cosγ˜ln0γ˜-1 sinγ˜l-n0γ˜ sinγ˜lcosγ˜l.
|γ˜2|l2=λg0l2π1+Sg0+α0g01ωg2-S1+S1ωb2=1πg0l1+Sλlωx2,
M˜l=1-θ˜l-3θ˜l1-θ˜; θ˜γ˜2l23; lln0.
ÃB˜C˜D˜=m-4+mθ˜lm+1-2θ˜m-1l-2m+2θ˜lm-4+mθ˜,
3|θ˜|lFM=λ0πωb2,
FM=3M-1M+1+4M2M+13,
ωb2=S1+Sg0g0+α0+1+SFMg0+α0lωg2.
ωb2=S1+S+1+SFMg0lωg2,
ωb2=S1+S+FMln G0ωg2.
ωb2=S1+Sg0g0+α0+FMln G0-ln AS0ωg2, AS0exp-α0l1+S.
FM1.2 ln M.
FM=M2-1M2+1.
FM=M2-12,
ũmx=Hm2x ik02q˜exp-ik0x22q˜,
γ˜mũmx=ik2πB˜1/2- ũmxexp-ik2πB˜Ãx2-2xx+D˜x2dx,
|γm|1Mm+1/2.
dIr, zdz=gSrIr, z.
gSr=g01+S-2g01+Sg0+α0g0·1ωg2-S1+S1ωb2r2,
gSr=g01+S1-2.4 ln Mln G0r2ωb2.
P02lP00G20M1.22, P12lP10G20M1.24,
|γm,n|=1/M2m+n+1.
FMln M.
G20RM2m+n+1G20Rγm+n=1,
γ0= 1M21-exp-1W1+α0g0; γ1=1M41-exp-1W1+α0g0 1+12W1+α0g0,
gSrgSg01+Sexp-2r2ωb2 W,
γ0=1M21+W, γ1=1M4 1+W/21+W2.
δ=gsr-gsrIr|r=ωS/gS0I0

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