Abstract

We describe a two-level model for the mesospheric sodium resonance fluorescence excited by a train of pulses. For pulse durations that are short compared with the 16-ns natural lifetime, population-rate equations are inadequate and must be replaced by density-matrix (Bloch) equations. We briefly contrast these two approaches and discuss several issues associated with pulse-train excitation. Analytical approximations to averages over atomic velocities and the transverse spatial profile of the laser are described. Estimates of return photon numbers for trains of pulses, ranging in duration from tens of picoseconds to half a nanosecond, are made and these estimates are compared with those of more conventional long-pulse lidar schemes. Roughly similar photon numbers are predicted for both long and short pulses whenever their average intensities are comparable. For average intensities, less than ~10 W/cm2, trains of ≈0.5-ns pulses yield greater photon returns than trains of ≈30-ps pulses. For larger intensities the reverse can be true.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
    [CrossRef]
  2. G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
    [CrossRef]
  3. C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
    [CrossRef]
  4. R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).
  5. L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Maui Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
    [CrossRef]
  6. C. W. Allen, Astrophysical Quantities (Athlone, University of London, London, 1955), p. 113.
  7. See, for instance, L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York1975).
  8. See, for instance, P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 8.
  9. M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
    [CrossRef]
  10. H. M. Gibbs, R. E. Slusher, “Sharp-line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
    [CrossRef]
  11. See, for instance, P. W. Milonni, “Saturation of anomalous dispersion in cw HF lasers,” Appl. Opt. 20, 1571–1578 (1981).
    [CrossRef] [PubMed]
  12. Except for the factor of 3, relation (56) reduces to Eq. (12) used in Section 2.1 under the assumption that I ≪ Isat. The factor of 3 accounts approximately for the fact that with circularly polarized light the Na atom will be optically pumped into a two-level system with Δm = ±1, with |m| having its largest allowed value, and that the dipole matrix elements are largest in this case. This is the same factor of 3 that appears in Eq. (30) and gives a saturation intensity consistent with observations of power broadening in (optically pumped) Na (see Ref. 9).
  13. L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York, 1975), pp. 57–58.
  14. See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
    [CrossRef]
  15. The effect of recoil and the redistribution of atomic velocities has been investigated by L. C. Bradley, MIT Lincoln Laboratory, Lexington, Mass. 02173-0073. (personal communication, 1990).

1989 (1)

See C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

1987 (1)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Maui Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

1986 (1)

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

1985 (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

1981 (1)

1980 (1)

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

1978 (1)

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
[CrossRef]

1977 (1)

M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
[CrossRef]

1972 (1)

H. M. Gibbs, R. E. Slusher, “Sharp-line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

Allen, C. W.

C. W. Allen, Astrophysical Quantities (Athlone, University of London, London, 1955), p. 113.

Allen, L.

See, for instance, L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York1975).

L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York, 1975), pp. 57–58.

Blamont, J. E.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
[CrossRef]

Bos, F.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
[CrossRef]

Bradley, L. C.

The effect of recoil and the redistribution of atomic velocities has been investigated by L. C. Bradley, MIT Lincoln Laboratory, Lexington, Mass. 02173-0073. (personal communication, 1990).

Chanin, M. L.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
[CrossRef]

Citron, M. L.

M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
[CrossRef]

Eberly, J. H.

See, for instance, L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York1975).

L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York, 1975), pp. 57–58.

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Gabel, C. W.

M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
[CrossRef]

Gardner, C. S.

See C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Maui Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, R. E. Slusher, “Sharp-line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

Gray, H. R.

M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
[CrossRef]

Knight, P. L.

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Megie, G.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
[CrossRef]

Milonni, P. W.

See, for instance, P. W. Milonni, “Saturation of anomalous dispersion in cw HF lasers,” Appl. Opt. 20, 1571–1578 (1981).
[CrossRef] [PubMed]

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Sechrist, C. F.

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Segal, A. C.

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Slusher, R. E.

H. M. Gibbs, R. E. Slusher, “Sharp-line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

Stroud, C. R.

M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
[CrossRef]

Thompson, L. A.

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Maui Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

Voelz, D. G.

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

J. Geophys. Res. (1)

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois, 1. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Nature (London) (1)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Maui Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

Phys. Rep. (1)

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Phys. Rev. A (2)

M. L. Citron, H. R. Gray, C. W. Gabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. A 16, 1507–1512 (1977).
[CrossRef]

H. M. Gibbs, R. E. Slusher, “Sharp-line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

Planet. Space Sci. (1)

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, “Simultaneous nighttime lidar measurements of atmospheric sodium and potassium,” Planet. Space Sci. 26(1), 27–35 (1978).
[CrossRef]

Proc. IEEE (1)

See C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

Other (6)

C. W. Allen, Astrophysical Quantities (Athlone, University of London, London, 1955), p. 113.

See, for instance, L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York1975).

See, for instance, P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 8.

The effect of recoil and the redistribution of atomic velocities has been investigated by L. C. Bradley, MIT Lincoln Laboratory, Lexington, Mass. 02173-0073. (personal communication, 1990).

Except for the factor of 3, relation (56) reduces to Eq. (12) used in Section 2.1 under the assumption that I ≪ Isat. The factor of 3 accounts approximately for the fact that with circularly polarized light the Na atom will be optically pumped into a two-level system with Δm = ±1, with |m| having its largest allowed value, and that the dipole matrix elements are largest in this case. This is the same factor of 3 that appears in Eq. (30) and gives a saturation intensity consistent with observations of power broadening in (optically pumped) Na (see Ref. 9).

L. Allen, J. H. Eberly, Optical Resonance and Two-Leυel Atoms (Wiley, New York, 1975), pp. 57–58.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (19)

Fig. 1
Fig. 1

Upper-level probability p2 versus pulse duration predicted by Bloch equations (solid curve) and rate equations (dotted curve) after a single pulse of intensities (a) 2.67 mW/cm2. (b) 2.67 W/cm2, and (c) 2.67 kW/cm2.

Fig. 2
Fig. 2

As in Fig. 1 for I = 2.67 W/cm2 and longer pulses, showing the increasing accuracy of the rate-equation approximation with increasing pulse duration.

Fig. 3
Fig. 3

Upper-level probability p2 (solid curve) for different scaled (dimensionless) atomic velocities for the case of a 60-ps π pulse of peak intensity 2.67 kW/cm. The N = 1 velocity group corresponds to an atom with zero velocity along the direction of propagation of the pump pulse; since the velocity distribution is symmetric about ξ = 0, only results for positive velocities are shown. For comparison we show the velocity distribution (dashed curve) assumed throughout this work, scaled here so as to be unity for the N = 1 (ξ = 0) velocity group.

Fig. 4
Fig. 4

As in Fig. 3 but for a 0.5-ns π pulse of peak intensity 38.4 W/cm2.

Fig. 5
Fig. 5

Upper-level probability p2 versus peak pulse intensity after a 10-ps pulse. The dashed curve shows the corresponding result when spatial averaging over a transverse Gaussian profile of the pump pulse is performed. The results shown are based on relation (51).

Fig. 6
Fig. 6

As in Fig. 5 but for a 30-ps pulse.

Fig. 7
Fig. 7

As in Fig. 5 but for a 50-ps pulse.

Fig. 8
Fig. 8

As in Fig. 5 but for a 70-ps pulse.

Fig. 9
Fig. 9

As in Fig. 5 but for a 0.5-ns pulse. The results shown are based on relation (52).

Fig. 10
Fig. 10

Upper-level probability p ¯ 2 ( 0 ) after each pulse for a train of 60-ps pulses of peak intensity 30 W/cm2, assuming a pulse separation of 200 ns. Doppler averaging is included. The dashed curve is Eq. (61) without any Dopper averaging.

Fig. 11
Fig. 11

As in Fig. 10 but with the pulse separation reduced to 30 ns.

Fig. 12
Fig. 12

Return photon number N versus IAVG for a train of 10-ps pulses separated by 10 ns, under the assumptions made in this paper for the Na layer abundance and the illuminated and receiver areas.

Fig. 13
Fig. 13

As in Fig. 12 but for 30-ps micropulses separated by 20 ns.

Fig. 14
Fig. 14

As in Fig. 12 but for 30-ps pulses separated by 8 ns.

Fig. 15
Fig. 15

As in Fig. 14 but with the pulse separation reduced to 2 ns.

Fig. 16
Fig. 16

As in Fig. 12 but for 50-ps pulses separated by 20 ns.

Fig. 17
Fig. 17

As in Fig. 16 but with the pulse separation reduced to 5 ns.

Fig. 18
Fig. 18

Comparison of results for return photon number for 25-ps pulses separated by 5 ns (solid curve) with those for 0.5-ns pulses separated by 5 ns (dashed curve).

Fig. 19
Fig. 19

Comparison of results for return photon number for 50-ps pulses separated by 5 ns (solid curve) with those for 0.5-ns pulses separated by 5 ns.

Tables (6)

Tables Icon

Table I Comparison of Exact Numerical Results with Relation (51) for a 60-ps Pulsea

Tables Icon

Table II Comparison of Exact Numerical Results with Approximation (52) for a 0.5-ns pulse. The Peak Intensity 38.4 W/cm2 Corresponds to a π Pulse for a Velocity Group at Resonance with the Pulse.

Tables Icon

Table III Ratio NLP/NSP for a Long Pulse of Duration IAVG and a Train of 50-ps Pulses Separated by 20 ns, Each of Peak Intensity I, Having the Same Average Intensity IAVG as the Long Pulse

Tables Icon

Table IV As in Table III, but with a 10-ns Pulse Separation

Tables Icon

Table V As in Table III, but with a 5-ns Pulse Separation

Tables Icon

Table VI As in Table IV, but for 0.5-ns Pulses Separated by 10 ns

Equations (66)

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

δ ν D = 2 λ ( 2 R T a log 2 M ) 1 / 2 1 GHz
1 τ s ρ s ( S L ) 0 T d t p ¯ 2 ( t ) = 1 τ s C s S 0 T d t p ¯ 2 ( t ) ,
N = ( T 0 A a 4 π z 2 ) 1 τ s C s S 0 T d t p ¯ 2 ( t ) ,
w ( z ) = w 0 ( 1 + z 2 / z 0 2 ) 1 / 2 ,
N = [ T 0 A a w 2 ( z ) C s 4 z 2 ] 1 τ s 0 T d t p ¯ 2 ( t ) .
1 τ s 0 T d t p ¯ 2 ( t ) N p τ s 0 τ sep d t p ¯ 2 ( 0 ) exp ( t / τ s ) τ macro τ sep p ¯ 2 ( 0 ) [ 1 exp ( τ sep / τ s ) ] ,
N ( T 0 A a C s S 4 π z 2 ) τ macro τ sep p ¯ 2 ( 0 ) [ 1 exp ( τ sep / τ s ) ] .
N max ( T 0 A a C s S 4 π z 2 ) τ macro τ sep [ 1 e τ sep / τ s ] .
N = R max τ macro p ¯ 2 ( 0 ) ,
R max = ( T 0 A a C s S 4 π z 2 ) 1 τ sep [ 1 exp ( τ sep τ s ) ]
α = λ 2 8 πτ s ρ ( 4 log 2 π ) 1 / 2 1 δ ν D = 2 . 7 × 10 8 cm 1
p ¯ 2 ( t ) = σ I τ s h ν ,
N = T 0 A a 4 π z 2 λ h c ρ s S L σ I τ p .
N = T 0 2 λ J h c A a 4 π z 2 ρ s σ L = T 0 2 λ J h c A a 4 π z 2 α L ,
N R = η T 0 2 λ J h c A a 4 π z 2 Δ z α s ρ ( z ) ρ ( 0 ) .
A = g 1 g 2 ( 8 π 2 e 2 m c λ 2 ) f 6 . 2 × 10 7 s 1
δ ν 0 = A / 2 π 10 MHz
I sat = π 3 h ν A λ 2 6 mW / cm 2 .
u ˙ = Δ υ β u ,
υ ˙ = Δ u β υ + κ E 0 w ,
w ˙ = γ ( w + 1 ) κ E 0 υ ,
w = p 2 p 1 , p 1 + p 2 = 1 ,
u Δ β υ ,
υ κ E 0 β Δ 2 + β 2 w ,
w ˙ γ ( w + 1 ) κ 2 E 0 2 β Δ 2 + β 2 w = γ ( w + 1 ) 8 πκ 2 c I β Δ 2 + β 2 w ,
8 πκ 2 c I β Δ 2 + β 2 = 3 ћ ω 0 ( λ 2 A 4 π ) I S ( ν ) 2 R ,
p ˙ 2 = γ p 2 R ( p 2 p 1 ) ,
p ¯ 2 = R γ + 2 R = 1 2 I / I sat 1 + I / I sat ,
I sat = 4 πγ ћ ω 0 3 λ 2 A S ( ν ) .
I sat = 4 π 2 γ ћ ω 0 δ ν 0 3 λ 2 A = 4 π 2 ћ ω 0 3 λ 2 ( A 4 π ) = π h ν A 3 λ 2 ,
w ( ) 1 + 1 2 κ 2 E 0 2 | 0 d t e exp [ i t ( Δ i τ p 1 ) t ] | 2 = 1 2 κ 2 E 0 2 1 Δ 2 + τ p 2 ,
w ( ) 1 + 1 2 κ 2 E 0 2 τ p 2 .
w ( ) 1 + 1 2 β κ 2 E 0 2 τ p .
p 2 , rate equations p 2 , Bloch equations = 1 βτ p ,
N short-pulse train N single long pulse = τ s τ sep [ 1 e τ sep / τ s ] p ¯ 2 ( short pulse ) p ¯ ( long pulse ) ,
p ¯ 2 ( short pulse ) p ¯ 2 ( long pulse ) = 1 2 κ 2 E 0 2 τ p 2 [ I / I sat 1 + I / I sat ] 1 = 9 4 π 2 λ 4 ( h ν ) 2 I sat ( I + I sat ) τ p 2 .
υ ˙ κ E 0 w ,
w ˙ κ E 0 υ ,
υ sin θ,
w cos θ
θ κ d t E 0 ( t )
θ = κ E 0 τ p = ( 3 λ 3 A I 4 ћ c π 2 ) 1 / 2 τ p = π / 2 ,
I τ p 2 = ( ћ c π 4 3 λ 3 A ) 8 × 10 18 ,
u ˙ = Δ ¯ υ β ¯ u ,
υ ˙ = Δ ¯ u β ¯ υ + Ω ¯ w n = 0 exp [ ( τ n τ sep ) 2 ] ,
w ˙ = γ ¯ ( w + 1 ) Ω ¯ υ n = 0 exp [ ( τ n τ sep ) 2 ] ,
θ = Ω ¯ d τ exp ( τ 2 ) = Ω ¯ π = π ( βτ p ) 2 I I sat .
p ¯ 2 ( t ) = d ξ W D ( ξ ) p 2 ( t ; ξ ) ,
W D ( ξ ) = Q π e Q 2 ξ 2 ,
Q log 2 πτ p δ ν D .
p ¯ 2 ( 0 ) 1 2 ( 1 + w ) = sin 2 θ 2 = sin 2 [ 1 2 π βτ p ( 2 I / I sat ) 1 / 2 ] .
p ¯ 2 ( 0 ) ( 1 πτ p δ ν D ) sin 2 [ 1 2 π βτ p ( 2 I / I sat ) 1 / 2 ] .
p 2 = I / 2 I sat 1 + Δ 2 / β 2 + I / I sat
p ¯ 2 = π 2 ( 4 log 2 π ) 1 / 2 δ ν 0 δ ν D I / I sat ( 1 + I / I sat ) 1 / 2 exp ( b 2 ) erfc ( b ) ,
b 4 log 2 δ ν 0 δ ν D ( 1 + I / I sat ) 1 / 2 .
p ¯ 2 π 2 ( 4 log 2 π ) 1 / 2 δ ν 0 δ ν D I / I sat ( 1 + I / I sat ) 1 / 2 = 3 α h ν ρ s I τ s ( 1 + I / I sat ) 1 / 2 ,
p ¯ 2 I / 2 I sat 1 + I / I sat .
N LP N SP = 3 αλ h c ρ s I τ LP p ¯ 2 ( 0 ) τ sep / τ macro [ 1 e τ sep / τ s ] ( 1 + I / I sat ) 1 / 2 .
N LP N SP = 1 . 3 p ¯ 2 ( 0 ) I ( 1 + I / I sat ) 1 / 2 τ sep τ s [ 1 exp ( τ sep / τ s ) ] 1 .
p ¯ 2 ( 0 ) spatial = 2 π 0 d r r exp ( r 2 / w 2 ) × sin 2 [ 1 2 π βτ p ( 2 I / I sat ) 1 / 2 exp ( r 2 / w s 2 ) ] = 0 1 d u sin 2 [ 1 2 π βτ p ( 2 I / I sat ) 1 / 2 u ] = 1 2 sin [ π βτ p ( 2 I / I sat ) ] 2 π βτ p ( 2 I / I sat ) 1 / 2 .
p ¯ 2 ( 0 ) = 1 2 [ 1 + w ¯ ( 0 ) ] ,
w ¯ ( 0 ) = [ ( 1 exp ( τ sep / τ s ) ] [ 1 exp ( τ sep / 2 τ s ) cos θ ] 1 exp ( 3 τ sep / 4 τ s ) [ exp ( τ sep / τ s ) exp ( τ sep / 2 τ s ) ] cos θ ,
T 0 A a C s S 4 π z 2 = 0 . 83 ,
N = 0 . 83 τ macro τ sep p ¯ 2 ( 0 ) [ 1 exp ( τ sep / τ s ) ] .
N = 8 . 3 × 10 4 p ¯ 2 ( 0 ) 1 τ sep [ 1 exp ( τ sep / 16 ) ] ,
2 π λδ ν D N p Δ V = τ macro τ sep 2 π h m λ 2 δ ν D .

Metrics