Abstract

The use of a TEM01*-mode laser beam has been proposed as a means of focusing an atomic beam to nanometer- scale spot diameters. We have analyzed the classical trajectories of atoms through a TEM01*-mode laser beam, using methods developed for particle optics. The differential equation that describes the properties of the: first- order paraxial lens hi exactly the same form as the bell-shaped magnetic Newtonian lens that was first analyzed by Glaser for the focusing of electrons in an electron-microscope objective. We calculate the first-order properties of the lens, obtaining cardinal elements that are valid over the entire operating range of the lens including the thick and the immersion regimes. Contributions to the spot size are discussed, including four aberrations plus diffraction and atomic-beam-collimation effects. Explicit expressions for spherical chromatic, spontaneous-emission, and dipole-fluctuation aberrations are obtained. Examples are discussed for a sodium atomic beam, showing that subnanometer-diameter spots may be achieved with reasonable laser and atomic- beam parameters. Optimization of the lens is also discussed.

© 1991 Optical Society of America

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References

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  1. J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
    [Crossref]
  2. J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111 (1980).
    [Crossref]
  3. V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
    [Crossref]
  4. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
    [Crossref]
  5. V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
    [Crossref]
  6. G. M. Gallatin and P. L. Gould, J. Opt. Soc. Am. B 8, 502 (1991).
    [Crossref]
  7. W. Glaser, Z. Phys. 117, 285 (1941); see also Ref. 9 below.
    [Crossref]
  8. See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).
  9. P. Grivet, Electron Optics, 2nd ed. (Pergamon, Oxford, 1972).
  10. P. W. Hawkes and E. Kasper, Electron Optics (Academic, London, 1989), Vol. 1.
  11. See also, e.g., A. B. El-Kareh and J. C. J. El-Kareh, Electron Beams, Lenses, and Optics (Academic, New York, 1970), Vol. 2; M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
    [Crossref]
  12. See, e.g., J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, Menlo Park, Calif., 1964), p. 8.
  13. See, e.g., R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972), pp. 456ff.
  14. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
    [Crossref]
  15. M. Scheinfein and M. Isaacson, J. Vac. Sci. Technol. B 4, 326 (1986); M. Scheinfein, “Transmission electron energy loss spectroscopy at 5Å spatial resolution,” Ph.D. dissertation (Cornell University, Ithaca, NY., 1985).
    [Crossref]

1991 (1)

1988 (1)

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
[Crossref]

1987 (1)

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[Crossref]

1986 (1)

M. Scheinfein and M. Isaacson, J. Vac. Sci. Technol. B 4, 326 (1986); M. Scheinfein, “Transmission electron energy loss spectroscopy at 5Å spatial resolution,” Ph.D. dissertation (Cornell University, Ithaca, NY., 1985).
[Crossref]

1985 (1)

1980 (2)

1978 (1)

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
[Crossref]

1941 (1)

W. Glaser, Z. Phys. 117, 285 (1941); see also Ref. 9 below.
[Crossref]

Ashkin, A.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111 (1980).
[Crossref]

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
[Crossref]

Balykin, V. I.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
[Crossref]

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111 (1980).
[Crossref]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
[Crossref]

Cohen-Tannoudji, C.

Dalibard, J.

El-Kareh, A. B.

See also, e.g., A. B. El-Kareh and J. C. J. El-Kareh, Electron Beams, Lenses, and Optics (Academic, New York, 1970), Vol. 2; M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
[Crossref]

El-Kareh, J. C. J.

See also, e.g., A. B. El-Kareh and J. C. J. El-Kareh, Electron Beams, Lenses, and Optics (Academic, New York, 1970), Vol. 2; M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
[Crossref]

Freeman, R. R.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111 (1980).
[Crossref]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
[Crossref]

Gallatin, G. M.

Glaser, W.

W. Glaser, Z. Phys. 117, 285 (1941); see also Ref. 9 below.
[Crossref]

Goldstein, H.

See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).

Gordon, J. P.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

Gould, P. L.

Grivet, P.

P. Grivet, Electron Optics, 2nd ed. (Pergamon, Oxford, 1972).

Hawkes, P. W.

P. W. Hawkes and E. Kasper, Electron Optics (Academic, London, 1989), Vol. 1.

Isaacson, M.

M. Scheinfein and M. Isaacson, J. Vac. Sci. Technol. B 4, 326 (1986); M. Scheinfein, “Transmission electron energy loss spectroscopy at 5Å spatial resolution,” Ph.D. dissertation (Cornell University, Ithaca, NY., 1985).
[Crossref]

Kasper, E.

P. W. Hawkes and E. Kasper, Electron Optics (Academic, London, 1989), Vol. 1.

Letokhov, V. S.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
[Crossref]

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[Crossref]

Mathews, J.

See, e.g., J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, Menlo Park, Calif., 1964), p. 8.

Ovchinnokov, Yu. B.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
[Crossref]

Pathria, R. K.

See, e.g., R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972), pp. 456ff.

Pearson, D. B.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111 (1980).
[Crossref]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
[Crossref]

Scheinfein, M.

M. Scheinfein and M. Isaacson, J. Vac. Sci. Technol. B 4, 326 (1986); M. Scheinfein, “Transmission electron energy loss spectroscopy at 5Å spatial resolution,” Ph.D. dissertation (Cornell University, Ithaca, NY., 1985).
[Crossref]

Sidorov, A. I.

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
[Crossref]

Walker, R. L.

See, e.g., J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, Menlo Park, Calif., 1964), p. 8.

J. Mod. Opt. (1)

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnokov, and A. I. Sidorov, J. Mod. Opt. 35, 17 (1988).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Vac. Sci. Technol. B (1)

M. Scheinfein and M. Isaacson, J. Vac. Sci. Technol. B 4, 326 (1986); M. Scheinfein, “Transmission electron energy loss spectroscopy at 5Å spatial resolution,” Ph.D. dissertation (Cornell University, Ithaca, NY., 1985).
[Crossref]

Opt. Commun. (1)

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
[Crossref]

Phys. Rev. Lett. (1)

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978).
[Crossref]

Z. Phys. (1)

W. Glaser, Z. Phys. 117, 285 (1941); see also Ref. 9 below.
[Crossref]

Other (6)

See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).

P. Grivet, Electron Optics, 2nd ed. (Pergamon, Oxford, 1972).

P. W. Hawkes and E. Kasper, Electron Optics (Academic, London, 1989), Vol. 1.

See also, e.g., A. B. El-Kareh and J. C. J. El-Kareh, Electron Beams, Lenses, and Optics (Academic, New York, 1970), Vol. 2; M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
[Crossref]

See, e.g., J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, Menlo Park, Calif., 1964), p. 8.

See, e.g., R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972), pp. 456ff.

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Figures (5)

Fig. 1
Fig. 1

Laser focusing of atoms in a TEM01* laser beam. Cross- sectional view of the focus of the laser beam, with laser intensity represented by a gray scale. The atomic beam propagates coaxially with the laser beam, being focused by the gradient in the laser intensity.

Fig. 2
Fig. 2

Sample trajectory, described by Eq. (16), of an atom initially traveling parallel to the z axis at a radius of 0.1 μm. The locations of the focal point and the principal plane are shown along with the definitions of the angle ϕ and the focal length f. For this trajectory q = 1.42.

Fig. 3
Fig. 3

Focal length f and principal plane location zp as a function of q. Note that the focal length has a minimum at q = 2, where zp = −f.

Fig. 4
Fig. 4

Trajectory R1(Z) used in determining aberration coefficients for finite object and image distances. The ray crosses the z axis at the object position z0 with slope α0 and again at the image position zi with slope αi. For this particular ray α0 = 0.025, q = 1.15, and L = 5.33 μm.

Fig. 5
Fig. 5

Ray traces of atomic trajectories through a TEM01* laser atomic lens, (a) Cases A–C of Table 1, P0 = 0.1 W, v0 is 1 × 104, 5 × 104, and 1 × 105 cm/s, respectively, (b) Cases D–F of Table 1, P0 = 1.0 W, v0 is 1 × 104, 5 × 104, and 1 × 105 cm/s, respectively, (c) MFL condition with q = 2.

Tables (5)

Tables Icon

Table 1 First-Order Properties of a TEM01* Laser-Atomic Lens for Sodiuma

Tables Icon

Table 2 Laser Power and Detuning Necessary to Achieve MFL Conditions for Three Atomic Velocitiesa

Tables Icon

Table 3 FWHM Spot Diameters (nm) Arising from Each of the Contributions for the Six Cases of Table 1a

Tables Icon

Table 4 Comparison with GG Values of 1/e2 Spot Diameters 2σ0 for a Gaussian Atomic Beama

Tables Icon

Table 5 FWHM Spot Diameters (nm) at the MFL Condition for Sodium Atoms at Three Atomic Velocitiesa

Equations (92)

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r ¨ + 1 m U ( r , z ) r = 0.
d d z [ ( 1 - U ( r , z ) E 0 ) 1 / 2 ( 1 + r 2 ) - 1 / 2 r ] + 1 2 E 0 [ 1 - U ( r , z ) E 0 ] - 1 / 2 ( 1 + r 2 ) 1 / 2 U ( r , z ) r = 0.
U ( r , z ) = Δ 2 ln [ 1 + p ( r , z ) ] ,
p ( r , z ) = I ( r , z ) I s γ 2 γ 2 + 4 Δ 2 .
I ( r , z ) = 8 I 0 w 0 2 r 2 w 4 ( z ) exp ( - 2 L 2 w 0 2 r 2 L 2 + z 2 ) .
r + 1 2 E 0 U ( r , z ) r = 0.
p 0 = 8 γ 2 γ 2 + 4 Δ 2 I 0 I s .
U 2 ( r , z ) = Δ p 0 w 0 2 2 w 4 ( z ) r 2 .
r + p 0 Δ 2 E 0 w 0 2 w 4 ( z ) r = 0.
k 2 = p 0 Δ 2 E 0 L 2 w 0 2 ,
q 2 = k 2 + 1.
R + k 2 ( 1 + Z 2 ) 2 R = 0.
R + 2 cot ϕ R + k 2 R = 0
y + q 2 y = 0.
R ( ϕ ) = ( 1 / sin ϕ ) ( c 1 sin q ϕ + c 2 cos q ϕ ) ,
R ( ϕ ) = r 0 L sin q ϕ q sin ϕ .
z f = L cot ( n π / q ) .
z p = - L cot ( n π / 2 q )             ( n odd ) , z p = L tan ( n π / 2 q )             ( n even ) ,
f = ( - 1 ) n + 1 L sin ( n π / q ) .
M = 1 m = ( - 1 ) n sin ϕ o sin ϕ i ,
ɛ = R 2 ( Z i ) R 1 R 2 - R 1 R 2 Z o Z i R 1 ( Z ) W ( R 1 , R 1 , Z ) d Z ,
ɛ sph = M α o 3 ( C sph o / L ) ,
ɛ chr = M α o ( C chr o / L ) ,
ɛ spont = M α o ( C spont o / L ) ,
ɛ dip = M α o 3 ( C dip o / L ) ,
h ( ϕ ) = sin [ q ( ϕ - ϕ o ) ] q sin ϕ sin ϕ o .
R 2 ( ϕ ) = g ( ϕ ) = sin ϕ o sin ϕ sin [ q ( ϕ - α ) ] sin [ q ( ϕ o - α ) ] ,
ɛ = - M ϕ o ϕ i h ( ϕ ) W ( α o h , α o h , ϕ ) d Z d ϕ d ϕ .
C sph o = | L α o 3 ϕ o ϕ i h ( ϕ ) W sph ( α o h , α o h , ϕ ) d ϕ sin 2 ϕ |
C chr o = | L α o ϕ o ϕ i h ( ϕ ) W chr ( α o h , α o h , ϕ ) d ϕ sin 2 ϕ | .
W sph ( R , R , Z ) = k 2 R 3 [ ( p 0 L 2 w 0 2 - k 2 ) 1 ( 1 + Z 2 ) 4 + 4 L 2 w 0 2 1 ( 1 + Z 2 ) 3 ] - k 2 R 2 R [ 2 Z ( 1 + Z 2 ) 3 ] - k 2 R R 2 [ 1 ( 1 + Z 2 ) 2 ] .
C sph o = | k 2 L q 4 sin 4 ϕ o ( p 0 L 2 w 0 2 + 4 ) × ϕ o ϕ i sin 2 ϕ sin 4 [ q ( ϕ - ϕ o ) ] d ϕ + k 2 L q 4 sin 4 ϕ o ( 4 L 2 w 0 2 - 3 ) × ϕ o ϕ i sin 4 [ q ( ϕ - ϕ o ) ] d ϕ - k 2 L q 2 sin 4 ϕ o × ϕ o ϕ i sin 2 ϕ sin 2 [ q ( ϕ - ϕ o ) ] d ϕ + 4 k 2 L q 3 sin 4 ϕ o × ϕ o ϕ i sin ϕ cos ϕ cos [ q ( ϕ - ϕ o ) ] × sin 3 [ q ( ϕ - ϕ o ) ] d ϕ | .
C sph o = L sin 4 ϕ o 3 π k 2 8 q 5 [ L 2 2 w 0 2 ( p 0 + 8 ) - 5 + 2 k 2 3 ] - L sin 4 ϕ o 1 8 ( 4 k 2 + 3 ) ( 3 p 0 L 2 w 0 2 + 15 - 4 k 2 ) × [ sin ( 2 ϕ o + 2 π q ) - sin ( 2 ϕ o ) ] .
ϕ i = ϕ o + ( n π / q ) ,
C sph o = C sph 0 o + C sph 1 o M + C sph2 o M 2 + C sph3 o M 3 + C sph4 o M 4 .
- cot ϕ o = 1 M sin ( π / q ) + cot ( π / q ) ,
C sph 0 o = C sph 4 o = 3 π k 2 L 8 q 5 sin 4 ( π / q ) [ L 2 2 w 0 2 ( p 0 + 8 ) - 5 + 2 k 2 3 ] - L sin ( 2 π / q ) 8 ( 4 k 2 + 3 ) sin 4 ( π / q ) ( 3 p 0 L 2 w 0 2 + 15 - 4 k 2 ) ,
C sph1 o = C sph3 o = 3 π k 2 L cos ( π / q ) 2 q 5 sin 4 ( π / q ) [ L 2 2 w o 2 ( p 0 + 8 ) - 5 + 2 k 2 3 ] - L [ 3 + cos ( 2 π / q ) ] 4 ( 4 k 2 + 3 ) sin 3 ( π / q ) ( 3 p 0 L 2 w 0 2 + 15 - 4 k 2 ) ,
C sph 2 o = 3 π k 2 L [ 2 + cos ( 2 π / q ) ] 4 q 5 sin 4 ( π / q ) [ L 2 2 w 0 2 ( p 0 + 8 ) - 5 + 2 k 2 3 ] - 3 L cos ( 2 π / q ) 2 ( 4 k 2 + 3 ) sin 3 ( π / q ) [ 3 p 0 L 2 w 0 2 + 15 - 4 k 2 ] .
ɛ sph = α i 3 C sph i L ,
C sph i = M 4 C sph o ,
C sph i ( M = 0 ) = C sph 4 o .
α i = r 0 f = r 0 L sin ( π / q ) .
δ sph = 3 π k 2 r 0 3 16 L 2 q 5 sin ( π / q ) [ L 2 2 w 0 2 ( p 0 + 8 ) - 5 + 2 k 2 3 ] - r 0 3 cos ( π / q ) 8 L 2 ( 4 k 2 + 3 ) ( 3 p 0 L 2 w 0 2 + 15 - 4 k 2 ) .
r + Δ p 0 2 E 0 w 0 2 w 4 ( z ) r = Δ p 0 2 E 0 w 0 2 w 4 ( z ) r .
W chr ( R , R , Z ) = k 2 ( 1 + Z 2 ) 2 R ,
C chr o = π k 2 L 2 q 3 sin 2 ϕ o .
C chr o = C chr 0 o + C chr 1 o M + C chr 2 o M 2 .
C chr 0 o = C chr 2 o = - π k 2 L 2 q 3 1 sin 2 ( π / q ) ,
C chr 1 o = - π k 2 L q 3 cos ( π / q ) sin 2 ( π / q ) .
ɛ chr = α i ( C chr i / L ) ,
C chr i = M 2 C chr o ,
δ chr = π k 2 r 0 2 q 3 sin ( π / q ) Δ E 1 / 2 E 0 ,
r ¨ + 1 m U ( r , z ) r = 1 m F r ( t ) .
r + 1 2 E 0 U ( r , z ) r = 1 2 E 0 F r ( z v 0 ) .
W ( R , R , Z ) = L 2 E 0 F r ( L Z v 0 ) .
ɛ 2 = R 2 ( Z i ) 2 ( R 1 R 2 - R 1 R 2 ) 2 Z o Z i d Z Z o Z i d Z R 1 ( Z ) R 1 ( Z ) × W ( Z ) W ( Z ) .
C spont o = L α o [ ϕ o ϕ i d ϕ sin 2 ϕ ϕ o ϕ i d ϕ sin 2 ϕ h ( ϕ ) h ( ϕ ) × W spont ( ϕ ) W spont ( ϕ ) ] 1 / 2 ,
C dip o = L α o 3 [ ϕ o ϕ i d ϕ sin 2 ϕ ϕ o ϕ i d ϕ sin 2 ϕ h ( ϕ ) h ( ϕ ) × W dip ( ϕ ) W dip ( ϕ ) ] 1 / 2 .
F spont ( t ) F spont ( t ) = 2 3 γ p ( r , z ) 2 ( h λ ) 2 δ ( t - t ) .
p ( r , z ) p 0 L 2 w 0 2 R 2 ( 1 + Z 2 ) 2 = p L 2 w 0 2 R 2 ( ϕ ) sin 4 ϕ .
W spont ( ϕ ) W spont ( ϕ ) = L 3 h 2 v 0 γ p 0 12 w 0 2 E 0 2 λ 2 R 2 ( ϕ ) ( sin 6 ϕ ) δ ( ϕ - ϕ ) ,
C spont o = L 2 h λ w 0 E 0 [ L v 0 γ p 0 12 ϕ o ϕ i h 4 ( ϕ ) ( sin 2 ϕ ) d ϕ ] 1 / 2 .
C spont o = L 2 h λ w 0 E 0 ( L v 0 γ p 0 12 ) 1 / 2 1 q 2 sin 2 ϕ o × [ ϕ o ϕ i sin 4 [ q ( ϕ - ϕ o ) ] sin 2 ϕ ] 1 / 2 .
R 2 = - sin ( π / q ) cos ( q ϕ ) sin ϕ .
ɛ 2 = L 2 r 0 2 sin 2 ( π / q ) 0 π / q d ϕ sin 2 ϕ 0 π / q d ϕ sin 2 ϕ R 1 ( ϕ ) R 1 ( ϕ ) × W ( ϕ ) W ( ϕ ) .
C spont i ( M = 0 ) = L q 2 sin 2 ( π / q ) ( L 3 h 2 v 0 γ p 0 12 w 0 2 E 0 2 λ 2 ) 1 / 2 × ( 0 π / q sin 4 q ϕ sin 2 ϕ d ϕ ) 1 / 2 ,
δ spont = 2 ( 2 ln 2 ) 1 / 2 r 0 q 2 sin ( π / q ) ( L 3 h 2 v 0 γ p 0 12 w 0 2 E 0 2 λ 2 ) 1 / 2 × ( 0 π / q sin 4 q ϕ sin 2 ϕ d ϕ ) 1 / 2 ,
F dip ( t ) F dip ( t ) = 2 Δ 2 4 ( p ) 2 p 2 exp ( - γ t - t ) .
W dip ( Z ) W dip ( Z ) = k 4 p 0 2 L 4 w 0 4 R 6 ( Z ) ( 1 + Z 2 ) 8 exp ( - γ L v 0 Z - Z ) .
exp ( - γ L v 0 Z - Z ) 2 v 0 γ L δ ( Z - Z ) .
W dip ( ϕ ) W dip ( ϕ ) = 2 v 0 γ L k 4 p 0 2 L 4 w 0 4 R 6 ( ϕ ) ( sin 18 ϕ ) δ ( ϕ - ϕ ) .
C dip o = L 3 ω 0 2 ( 2 v 0 γ L ) 1 / 2 k 2 p 0 q 4 sin 4 ϕ o × [ ϕ o ϕ i sin 6 ϕ sin 8 [ q ( ϕ - ϕ o ) ] d ϕ ] 1 / 2 .
C dip i ( M = 0 ) = L 3 w 0 2 ( 2 v 0 γ L ) 1 / 2 k 2 p 0 q 4 sin 4 ( π / q ) × [ 0 π / q sin 6 ϕ ( sin 8 q ϕ ) d ϕ ] 1 / 2
δ dip = 4 ( ln 2 ) 1 / 2 r 0 3 w 0 2 ( v 0 γ L ) 1 / 2 k 2 p 0 q 4 sin ( π / q ) × [ 0 π / q sin 6 ϕ ( sin 8 q ϕ ) d ϕ ] 1 / 2 .
δ diffr = 0.61 λ dB α i .
δ diffr = 0.61 λ dB L r 0 sin ( π / q ) .
δ source = M d o ( f / z o ) d o ,
δ tot = ( δ sph 2 + δ chr 2 + δ spont 2 + δ dip 2 + δ diffr 2 + δ source 2 ) 1 / 2 .
Δ MFL = 3 E 0 λ 2 π 2 w 0 2 ,
P 0 MFL = 18 π 3 E 0 2 2 γ 2 λ 4 w 0 2 I s ,
δ sph 9 π 1024 ( p 0 + 8 ) r 0 3 w 0 2 ,
δ chr = 3 π 16 Δ E 1 / 2 E 0 r 0 ,
δ spont = π 5 / 2 2 ( ln 2 6 ) 1 / 2 ( γ p 0 λ 5 ) 1 / 2 h m w 0 2 r 0 v 0 3 / 2 ,
δ dip = 3 ( 91 ln 2 ) 1 / 2 256 p 0 ( λ γ ) 1 / 2 r 0 3 w 0 3 v 0 1 / 2 ,
δ diffr = 0.61 π h m λ w 0 2 r 0 v 0 ,
δ tot 2 = A r 0 6 w 0 4 + B r 0 2 + C w 0 4 r 0 2 v 0 3 + D r 0 6 v w 0 6 + E w 0 4 r 0 2 v 0 2 ,
A = [ 9 π 1024 ( p 0 + 8 ) ] 2 ,
B = [ 3 π 16 Δ E 1 / 2 E 0 ] 2 ,
C = π 5 ln 2 24 γ p 0 λ 5 h 2 m 2 ,
D = 819 ln 2 65536 λ γ p 0 2 ,
E = 0.372 π 2 h 2 m 2 λ 2 .

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