Abstract

An analytic model based on a single encoding–diffraction sequence is developed to describe the first main region of focusing that occurs when a cw off-resonant monochromatic Gaussian beam enters a medium of two-state atoms. A quantitative definition of the dispersive regime for self-focusing is given, and an explicit form is obtained for the field encoding produced by the initial propagation. The phase profile of this encoded field incorporates the effects of both the medium and diffraction, and we find a simple expression for the fundamental length scale, which determines where diffractive effects become dominant. An analytic expression is obtained for the on-axis field, which results once the encoded field begins to diffract freely, and simple expressions for the position and size of the main enhancement are given. We give a simple interpretation of the focusing and the two parameters required for its characterization. The model gives excellent agreement to the full numeric solution over a wide parameter range and, in particular, explains the rapid on-axis oscillations at the outset of focusing.

© 1988 Optical Society of America

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References

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  1. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  2. See, in particular, Secs. 5 and 8 of Ref. 1 and references therein.
  3. W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
    [CrossRef]
  4. M. LeBerre, E. Ressayre, A. Tallet, F. P. Mattar, “Qausi-trapping of Gaussian beams in two-level systems,” J. Opt. Soc. Am. B 2, 956–967 (1985).
    [CrossRef]
  5. M. LeBerre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604–1618 (1982).
    [CrossRef]
  6. M. LeBerre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
    [CrossRef]
  7. M. G. Boshier, W. J. Sandle, “Self-focusing in a vapour of two-state atoms,” Opt. Commun. 42, 371–376 (1982).
    [CrossRef]
  8. D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous-wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
    [CrossRef]
  9. M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, N. Peyghambarian, “Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement,” J. Opt. Soc. Am. B 1, 591–605 (1984).
    [CrossRef]
  10. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 112.
  11. N. Wright, M. C. Newstein, “Self-focusing of coherent pulses,” Opt. Commun 9, 8–13 (1973).
    [CrossRef]
  12. A. Icsevgi, W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
    [CrossRef]
  13. This is not a simple frequency scan since both I and F are scaled in units of (1 + Δ2).
  14. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 31.
  15. E. T. Copson, Asymptotic Expansions, Vol. 55 of Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge U. Press, Cambridge, 1965), pp. 91–98.
    [CrossRef]
  16. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 720.
  17. R. J. Ballagh, J. Cooper, W. J. Sandle, “Effective two-state behavior of collisionally perturbed Zeeman degenerate atomic transition and its application to optical bistability,” J. Phys. B 14, 3881–3890 (1981).
    [CrossRef]

1986 (1)

1985 (1)

1984 (2)

1982 (2)

M. G. Boshier, W. J. Sandle, “Self-focusing in a vapour of two-state atoms,” Opt. Commun. 42, 371–376 (1982).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604–1618 (1982).
[CrossRef]

1981 (1)

R. J. Ballagh, J. Cooper, W. J. Sandle, “Effective two-state behavior of collisionally perturbed Zeeman degenerate atomic transition and its application to optical bistability,” J. Phys. B 14, 3881–3890 (1981).
[CrossRef]

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

1973 (1)

N. Wright, M. C. Newstein, “Self-focusing of coherent pulses,” Opt. Commun 9, 8–13 (1973).
[CrossRef]

1969 (1)

A. Icsevgi, W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[CrossRef]

1968 (1)

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Ballagh, R. J.

D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous-wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
[CrossRef]

R. J. Ballagh, J. Cooper, W. J. Sandle, “Effective two-state behavior of collisionally perturbed Zeeman degenerate atomic transition and its application to optical bistability,” J. Phys. B 14, 3881–3890 (1981).
[CrossRef]

Boshier, M. G.

M. G. Boshier, W. J. Sandle, “Self-focusing in a vapour of two-state atoms,” Opt. Commun. 42, 371–376 (1982).
[CrossRef]

Cooper, J.

R. J. Ballagh, J. Cooper, W. J. Sandle, “Effective two-state behavior of collisionally perturbed Zeeman degenerate atomic transition and its application to optical bistability,” J. Phys. B 14, 3881–3890 (1981).
[CrossRef]

Copson, E. T.

E. T. Copson, Asymptotic Expansions, Vol. 55 of Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge U. Press, Cambridge, 1965), pp. 91–98.
[CrossRef]

Gibbs, H. M.

Haus, H. A.

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Icsevgi, A.

A. Icsevgi, W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[CrossRef]

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 720.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 720.

Lamb, W. E.

A. Icsevgi, W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[CrossRef]

LeBerre, M.

M. LeBerre, E. Ressayre, A. Tallet, F. P. Mattar, “Qausi-trapping of Gaussian beams in two-level systems,” J. Opt. Soc. Am. B 2, 956–967 (1985).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, N. Peyghambarian, “Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement,” J. Opt. Soc. Am. B 1, 591–605 (1984).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604–1618 (1982).
[CrossRef]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 31.

Mattar, F. P.

McClelland, D. E.

Newstein, M. C.

N. Wright, M. C. Newstein, “Self-focusing of coherent pulses,” Opt. Commun 9, 8–13 (1973).
[CrossRef]

Peyghambarian, N.

Ressayre, E.

M. LeBerre, E. Ressayre, A. Tallet, F. P. Mattar, “Qausi-trapping of Gaussian beams in two-level systems,” J. Opt. Soc. Am. B 2, 956–967 (1985).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, N. Peyghambarian, “Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement,” J. Opt. Soc. Am. B 1, 591–605 (1984).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604–1618 (1982).
[CrossRef]

Rushford, M. C.

Sandle, W. J.

D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous-wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
[CrossRef]

M. G. Boshier, W. J. Sandle, “Self-focusing in a vapour of two-state atoms,” Opt. Commun. 42, 371–376 (1982).
[CrossRef]

R. J. Ballagh, J. Cooper, W. J. Sandle, “Effective two-state behavior of collisionally perturbed Zeeman degenerate atomic transition and its application to optical bistability,” J. Phys. B 14, 3881–3890 (1981).
[CrossRef]

Tai, K.

Tallet, A.

M. LeBerre, E. Ressayre, A. Tallet, F. P. Mattar, “Qausi-trapping of Gaussian beams in two-level systems,” J. Opt. Soc. Am. B 2, 956–967 (1985).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, K. Tai, H. M. Gibbs, M. C. Rushford, N. Peyghambarian, “Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement,” J. Opt. Soc. Am. B 1, 591–605 (1984).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604–1618 (1982).
[CrossRef]

Wagner, W. G.

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Wright, N.

N. Wright, M. C. Newstein, “Self-focusing of coherent pulses,” Opt. Commun 9, 8–13 (1973).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 112.

J. Opt. Soc. Am. B (3)

J. Phys. B (1)

R. J. Ballagh, J. Cooper, W. J. Sandle, “Effective two-state behavior of collisionally perturbed Zeeman degenerate atomic transition and its application to optical bistability,” J. Phys. B 14, 3881–3890 (1981).
[CrossRef]

Opt. Commun (1)

N. Wright, M. C. Newstein, “Self-focusing of coherent pulses,” Opt. Commun 9, 8–13 (1973).
[CrossRef]

Opt. Commun. (1)

M. G. Boshier, W. J. Sandle, “Self-focusing in a vapour of two-state atoms,” Opt. Commun. 42, 371–376 (1982).
[CrossRef]

Phys. Rev. (2)

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

A. Icsevgi, W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[CrossRef]

Phys. Rev. A (2)

M. LeBerre, E. Ressayre, A. Tallet, “Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium,” Phys. Rev. A 25, 1604–1618 (1982).
[CrossRef]

M. LeBerre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

Prog. Quantum Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Other (6)

See, in particular, Secs. 5 and 8 of Ref. 1 and references therein.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 112.

This is not a simple frequency scan since both I and F are scaled in units of (1 + Δ2).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 31.

E. T. Copson, Asymptotic Expansions, Vol. 55 of Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge U. Press, Cambridge, 1965), pp. 91–98.
[CrossRef]

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1968), p. 720.

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Figures (8)

Fig. 1
Fig. 1

Development of the radial profiles of |C| (solid curve) and |M| (dashed curve) for the DF solution with I00 = 20, F = 2, and Δ = 15. The maxima of the |M| profiles move inward as z successively takes the values z = 0, z = 0.04, and z = 0.08.

Fig. 2
Fig. 2

Relative contributions of the phase-gradient term (solid curve), the amplitude curvature term (dotted curve), and the in-phase curvature term (dashed curve) to the curvature of the DF solution at (ρc, zc) for a range of Δ. The on-axis intensity and effective Fresnel number are kept constant at I00 = 20 and F = 2, respectively. The point D defines the boundary of the dispersive regime.

Fig. 3
Fig. 3

Radial profiles at z = z c / 2 0.0527 of (a) the phase ϕDF (long-dashed curve), ϕDM (solid curve), and ϕ1 (short-dashed curve); (b) the derivatives ∂ϕDF/∂ρ (long-dashed curve), ∂ϕdm/∂ρ (solid curve), and ∂ϕ1/∂ρ (short-dashed curve). The initial on-axis intensity I00 = 20, and FΔ = 30.

Fig. 4
Fig. 4

Trajectories of the saddle point and its approximate forms in the u plane for I00 = 20 and FΔ = 30. Each curve is parameterized by the distance ζ given in units of zc. The solid curve is the exact saddle-point trajectory; the dashed curve is u from ζ = 0 to ζ = 0.7115zc and u+ for ζ > 0.7115zc; the dotted curve is ub [see Eq. (5.11)].

Fig. 5
Fig. 5

Intensity profiles of (a) the full numeric solution and (b) the DME/FSD model. Parameters are I00 = 20, F = 2, and Δ = 15.

Fig. 6
Fig. 6

On-axis intensity for the full numeric solution (solid curve), the DME/FSD model (dashed curve), and the saddle-point approximation (dotted curve) for I00 = 20, F = 2, and Δ = 15.

Fig. 7
Fig. 7

On-axis intensity for the full numeric solution (solid curve), the DME/FSD model (dashed curve), and the saddle-point approximation (dotted curve) for I00 = 1000, F = 1, and Δ = 100.

Fig. 8
Fig. 8

On-axis intensity for (a) the full numeric solution and (b) the DME-FSD model (dashed curve) and the saddle-point approximation (dotted curve) for I00 = 1000, F = 10, and Δ = 100. The inset in (b) gives the on-axis phase of the Gaussian contribution (solid curve) and the wing contribution (dotted curve) of the saddle-point solution.

Equations (80)

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( ρ , z ) z = i T 2 ( ρ , z ) 2 F ( 1 + i Δ ) ( ρ , z ) 1 + | ( ρ , z ) | 2 .
Δ ( ω L ω 0 ) / Γ ,
sat = ( / d ) ( γ Γ ) 1 / 2 ,
T 2 2 ρ 2 + 1 ρ ρ .
F α 0 ( 1 + Δ 2 ) z R
( ρ , z = 0 ) = A 00 exp ( ρ 2 ) ,
I 0 I ( ρ , z = 0 ) = A 00 2 exp ( 2 ρ 2 ) ,
I 00 I ( ρ = 0 , z = 0 ) = A 00 2 .
M 2 F ( 1 + i Δ ) 1 + I
C i T 2 ,
DF ( ρ , z ) = ( ρ , z = 0 ) exp [ 1 2 ( 1 + i Δ ) ( 4 F z + I 0 I ) ] .
ϕ DF = Δ 2 ( 4 F z + I I 0 ) .
C = A e i ϕ { i [ T 2 A A ( ϕ ρ ) 2 ] ( T 2 ϕ + 2 A A ρ ϕ ρ ) } .
A z = A ( T 2 ϕ + 2 A A ρ ϕ ρ ) 2 F A 1 + A 2 ,
ϕ z = [ T 2 A A ( ϕ ρ ) 2 ] 2 F Δ 1 + A 2 ,
T 2 ϕ + 2 A A ρ ϕ ρ ,
T 2 A A ( ϕ ρ ) 2 ,
T 2 A DF = 4 A DF 1 + I [ 1 I 0 + 2 ρ 2 I 0 + ρ 2 ( 1 + I 0 1 + I ) 2 ( 1 I ) ] ,
ϕ DF ρ = 2 Δ ρ I I 0 1 + I ,
T 2 ϕ DF = 4 Δ I I 0 ( 1 + I ) 3 [ ( 1 + I ) 2 + 2 ρ 2 ( I I 0 1 ) ] ,
A DF ρ = 2 ρ A DF 1 + I 0 1 + I .
F ( 1 + Δ 2 ) 1 / 2 4 A 00 exp ( 2 ) ,
| Δ e | [ 16 I 00 exp ( 4 ) F 2 1 ] 1 / 2 .
d I d z = 4 F I 1 + I ,
I 0 I = 4 F z I 0 1 + I 0 ,
ϕ DF Δ 2 F z 1 + I 0 ,
ϕ DF ρ 8 F Δ z I 0 ρ ( 1 + I 0 ) 2 ,
| C | A ( ϕ DF ρ ) 2 .
| C | A 00 exp ( ρ 2 ) [ 8 F Δ z I 0 ρ ( 1 + I 0 ) 2 ] 2 .
z c [ 1 + I ( ρ c max , 0 ) ] 2 ( 1 + Δ 2 ) 1 / 4 8 F | Δ | I ( ρ c max , 0 ) 5 / 4 ρ c max .
z c = ( 2 F | Δ | ln I 00 ) 1 / 2 ,
| ( ϕ DF ρ ) 2 | > | 2 A DF A DF ρ ϕ DF ρ + T 2 ϕ DF | ,
| Δ ( I I 0 ) | > | 2 ( 1 + I 0 ) + 2 ( 1 I I 0 ) 1 + I 1 + I ρ 2 | .
Δ > 8 F ln I 00 .
I 0 I + 4 F z c 1 + 2 ( 2 F ) 1 / 2 ( Δ ln I 00 ) 1 / 2 ,
Δ > 1 .
Δ d = 8 F ln I 00 + 1 .
ϕ = ϕ DF + ϕ 1 ,
ϕ z ( ϕ ρ ) 2 2 F Δ 1 + I ,
ϕ 1 z = ( ϕ DF ρ ) 2 2 ϕ DF ρ ϕ 1 ρ ( ϕ 1 ρ ) 2
( ϕ DF ρ ) 2 ,
ϕ 1 [ 8 F Δ I 0 ρ ( 1 + I 0 ) 2 ] 2 z 3 3 ,
ϕ DM = Δ 2 F z 1 + I 0 [ 1 + κ I 0 2 ρ 2 ( 1 + I 0 ) 3 ] ,
κ 16 3 ln I 00 ( z z c ) 2
ϕ 1 ρ = ( z z c ) 2 ϕ DF ρ θ ,
θ 8 I 0 3 ( 1 + I 0 ) 3 ln I 00 [ 1 + I 0 + 4 ρ 2 ( I 0 1 ) ] .
8 I 0 ( I 0 1 ) 3 ln I 00 ( 1 + I 0 ) 3 ,
ϕ DM ρ = ϕ DF ρ + ϕ 1 ρ
DM ( ρ , z ) = A 00 exp ( ρ 2 ) exp { 2 i Δ F z 1 + I 0 [ 1 + κ I 0 2 ρ 2 ( 1 + I 0 ) 3 ] } ,
ϕ DM ρ = ϕ DF ρ [ 1 + ( z z c ) 2 θ ] ;
max ( ϕ DM ρ ) 2 max ( ϕ DF ρ ) 2 z = z c { ( z z c ) 2 [ 1 + ¼ ( z z c ) 2 ] 2 } ,
( ρ , z ) = i 2 ζ 0 DM ( ρ , z enc ) exp [ i ρ 2 / ( 4 ζ ) ] J 0 [ ρ ρ / ( 2 ζ ) ] ρ d ρ ,
ζ ( z z enc ) .
( ρ = 0 , z ) = i 2 ζ 0 DM ( ρ , z enc ) exp [ i ρ 2 / ( 4 ζ ) ] ρ d ρ .
( ρ = 0 , z ) = i A 00 4 ζ 0 exp [ 1 4 ζ w ( x ) ] d x ,
w ( x ) 4 ζ x + i { g 1 + I 0 [ 1 + κ I 0 2 x ( 1 + I 0 ) 3 ] + x }
g 8 Δ F z enc ζ .
1 + g u ( 1 + u ) 5 { 2 ( 1 + u ) 3 + κ u [ 1 + u + 2 ( 1 u ) ln u I 00 ] } = 0 ,
g 8 Δ F z enc ζ 1 + 4 i ζ
u I 0 = I 00 exp ( 2 x ) .
u ± = ( 1 + g ) ± [ ( 1 + g ) 2 1 ] 1 / 2 .
u b = 2 ( 1 + g ) ,
ζ > z enc ln I 00 4 .
( ρ = 0 , z ) = E G + E S ,
E G = { A 00 exp [ i 2 F Δ z enc ( 1 + I 00 ) ] } 1 ( 1 + 16 ζ 2 ) 1 / 2 exp [ i tan 1 ( 4 ζ ) ] .
E S = A 00 { π 2 ζ w } 1 / 2 exp ( w 4 ζ ) ,
w d 2 w d x 2 = 4 igu ( 1 + u ) 3 { 1 u + 2 κ u ( 1 + u ) 3 × [ 1 u 2 + ( 1 3 u + u 2 ) ln ( u I 00 ) ] } .
T 2 ϕ DM ϕ DF ρ [ θ ρ ( z z c ) 2 ] ,
max ( ϕ DF ρ ) ( F Δ ) 1 / 2 z z c
max ( θ ρ ) 4
max ( T 2 ϕ DM ) 4 ( F Δ ) 1 / 2 z z c max ( ϕ DF ρ ) 2 .
z enc = 1.1 z c ,
2 { 1 + 5 F Δ } 1 / 2 | M | max
| 1 u | ζ = 0 .
ζ m = ( F Δ ) 1 / 2 + [ F Δ + ( 9 / 2 ) ln I 00 ] 1 / 2 6 ( 2 ln I 00 ) 1 / 2 ,
ζ m 3 8 z c ln I 00 = 3 8 2 ( ln I 00 F Δ ) 1 / 2
A m [ 4 π 2 3 ( F Δ ln I 00 ) 1 / 2 ] 1 / 2 .
I max ( A 00 + A m ) 2 .
{ 4 z R ζ p } { a [ ( 1 / 2 ) ln I 00 ] 1 / 2 } = { a [ 2 π / ( F Δ ) 1 / 2 ] } λ ,
ζ p ~ 1 4 2 ( ln I 00 F Δ ) 1 / 2 ,

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