Abstract

The focusing of an atomic beam using near-resonant laser light is considered. Path-integral techniques are employed to transform the problem into a standard diffraction integral. This approach is general and allows us to deal with thick laser lenses. Starting from the basic form of the potential energy for an atom in a laser beam, we derive the propagation kernel for the atomic wave function for the particular case of a TEM01*, or doughnut, mode laser beam. Both the full three-dimensional propagation kernel and its paraxial approximation are discussed. We show that the paraxial case can be obtained from the three-dimensional case by a stationary-phase approximation of the propagation equation. Numerical results for the focusing of a Gaussian atomic beam are presented. These results show that spot diameters on the order of 20 Å are obtainable for many reasonable choices of laser and atomic beam parameters and that for most of these cases the thin-lens approximation is not valid. The effects of the lowest-order aberrations are also briefly discussed. Spherical aberration is found to contribute significantly to the focal spot diameter, at least for the doughnut mode laser beam considered here.

© 1991 Optical Society of America

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References

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  1. See, for example, the following special issues: “Mechanical Effects of Light,” P. Meystre and S. Stenholm, eds., J. Opt. Soc. Am. B2, No. 11 (1985); “Laser Cooling and Trapping,” S. Chu and C. Weiman, eds., J. Opt. Soc. Am. B6, No. 11 (1989); and the following review articles: W. D. Phillips and H. J. Metcalf, Sci. Am. 256, 50–56 (1987); W. D. Phillips, P. L. Gould, and P. D. Lett, Science 239, 877–883 (1988).
    [CrossRef] [PubMed]
  2. J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
    [CrossRef]
  3. J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111–113 (1980).
    [CrossRef]
  4. V. I. Balykin, V. S. Letokhov, and A. I. Sidorov, JETP Lett. 43, 217–220 (1986).
  5. V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
    [CrossRef]
  6. V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151–156 (1987); Phys. Today,  42(4), 23–28 (1989).
    [CrossRef]
  7. A. Ashkin, Phys. Rev. Lett. 25, 1321–1324 (1970).
    [CrossRef]
  8. A. Ashkin, Phys. Rev. Lett. 40, 729–732 (1978); R. J. Cook, Phys. Rev. Lett. 44, 976–979 (1980); J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707–1720 (1985).
    [CrossRef]
  9. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  10. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1617 (1980).
    [CrossRef]
  11. In the definition of Δ, we include the Doppler shift due to the atomic beam velocity. This is justified for the following reasons. We are considering monochromatic collimated atomic beams. Thus the velocity is the same for all the atoms, and for the focusing configuration considered, it is essentially constant along each trajectory. Hence it enters the potential simply as a constant parameter. To lowest order, and with the laser beam treated as a superposition of paraxial plane waves, the effect of the velocity is to replace Δ by Δ − V· kL, where kL is the wave vector of the (laser) plane wave.10 For many practical cases the spread in kL values in the laser beam is small, and thus the only effect of the atomic motion for the configuration shown in Fig. 1 is to replace Δ by Δ − kLV.
  12. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1092.
  13. H. Kogelnik and T. Li, Appl. Opt. 5, 1550–1567 (1966); A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 6.
    [CrossRef] [PubMed]
  14. As a numerical example of a diffraction-limited atomic beam, consider sodium atoms traveling at a speed of 500 m/sec with an incident spot size (σ0) of 0.1 μ m. The diffraction angle through this aperture is approximately 10−4 rad. Thus the beam has a characteristic spread of transverse velocities that is on the order of 5 cm/sec owing completely to diffraction. A similar angular spread would result from collimation with a 100-μ m-diameter opening positioned 50 cm upstream from the 0.1-μ m beam-defining aperture. Laser cooling of the transverse and longitudinal velocities would serve to further relax the collimation requirements.
  15. This value of Δ was chosen in Ref. 6 to minimize spherical aberration. It should be noted, however, that the coefficient of the term that leads to the lowest order spherical aberration, that is the coefficient of the ρ2/2 term [where ρ2= (x2+ y2)/w02 in the notation of Ref. 6] in square brackets in Eq. (6) there, is actually given by 3 + 5α/8 and not by 1 − α/2, where α= P0γ2/πΔ2ISw02= 2(I0/IS) (γ2/Δ2). Hence this choice of detuning (α= 2) does not eliminate the lowest order spherical aberration. In fact, we find that this term contributes significantly to enlarging the minimum spot size (at least for the doughnut mode laser beam considered here), unless the lens aperture is severely restricted. Since we find it impossible to eliminate the lowest order spherical aberration, we chose this value of detuning because it comes close to optimizing the radial potential well depth of the lens.
  16. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 108.
  17. J. J. McClelland and M. Scheinfein, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication).

1988 (1)

V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
[CrossRef]

1987 (1)

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151–156 (1987); Phys. Today,  42(4), 23–28 (1989).
[CrossRef]

1986 (1)

V. I. Balykin, V. S. Letokhov, and A. I. Sidorov, JETP Lett. 43, 217–220 (1986).

1980 (2)

1978 (2)

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 40, 729–732 (1978); R. J. Cook, Phys. Rev. Lett. 44, 976–979 (1980); J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707–1720 (1985).
[CrossRef]

1970 (1)

A. Ashkin, Phys. Rev. Lett. 25, 1321–1324 (1970).
[CrossRef]

1966 (1)

Ashkin, A.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1617 (1980).
[CrossRef]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111–113 (1980).
[CrossRef]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 40, 729–732 (1978); R. J. Cook, Phys. Rev. Lett. 44, 976–979 (1980); J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707–1720 (1985).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 25, 1321–1324 (1970).
[CrossRef]

Balykin, V. I.

V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
[CrossRef]

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151–156 (1987); Phys. Today,  42(4), 23–28 (1989).
[CrossRef]

V. I. Balykin, V. S. Letokhov, and A. I. Sidorov, JETP Lett. 43, 217–220 (1986).

Bjorkholm, J. E.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111–113 (1980).
[CrossRef]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1092.

Feynman, R. P.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Freeman, R. R.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111–113 (1980).
[CrossRef]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
[CrossRef]

Gordon, J. P.

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1617 (1980).
[CrossRef]

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Kogelnik, H.

Letokhov, V. S.

V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
[CrossRef]

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151–156 (1987); Phys. Today,  42(4), 23–28 (1989).
[CrossRef]

V. I. Balykin, V. S. Letokhov, and A. I. Sidorov, JETP Lett. 43, 217–220 (1986).

Li, T.

McClelland, J. J.

J. J. McClelland and M. Scheinfein, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication).

Meyer-Arendt, J. R.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 108.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1092.

Ovehinnikov, Y. B.

V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
[CrossRef]

Pearson, D. B.

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Opt. Lett. 5, 111–113 (1980).
[CrossRef]

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
[CrossRef]

Scheinfein, M.

J. J. McClelland and M. Scheinfein, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication).

Sidorov, A. I.

V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
[CrossRef]

V. I. Balykin, V. S. Letokhov, and A. I. Sidorov, JETP Lett. 43, 217–220 (1986).

Appl. Opt. (1)

J. Mod. Opt. (1)

V. I. Balykin, V. S. Letokhov, Y. B. Ovehinnikov, and A. I. Sidorov, J. Mod. Opt. 35, 17–34 (1988).
[CrossRef]

JETP Lett. (1)

V. I. Balykin, V. S. Letokhov, and A. I. Sidorov, JETP Lett. 43, 217–220 (1986).

Opt. Commun. (1)

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151–156 (1987); Phys. Today,  42(4), 23–28 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606–1617 (1980).
[CrossRef]

Phys. Rev. Lett. (3)

J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41, 1361–1364 (1978).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 25, 1321–1324 (1970).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 40, 729–732 (1978); R. J. Cook, Phys. Rev. Lett. 44, 976–979 (1980); J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707–1720 (1985).
[CrossRef]

Other (8)

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

In the definition of Δ, we include the Doppler shift due to the atomic beam velocity. This is justified for the following reasons. We are considering monochromatic collimated atomic beams. Thus the velocity is the same for all the atoms, and for the focusing configuration considered, it is essentially constant along each trajectory. Hence it enters the potential simply as a constant parameter. To lowest order, and with the laser beam treated as a superposition of paraxial plane waves, the effect of the velocity is to replace Δ by Δ − V· kL, where kL is the wave vector of the (laser) plane wave.10 For many practical cases the spread in kL values in the laser beam is small, and thus the only effect of the atomic motion for the configuration shown in Fig. 1 is to replace Δ by Δ − kLV.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1092.

As a numerical example of a diffraction-limited atomic beam, consider sodium atoms traveling at a speed of 500 m/sec with an incident spot size (σ0) of 0.1 μ m. The diffraction angle through this aperture is approximately 10−4 rad. Thus the beam has a characteristic spread of transverse velocities that is on the order of 5 cm/sec owing completely to diffraction. A similar angular spread would result from collimation with a 100-μ m-diameter opening positioned 50 cm upstream from the 0.1-μ m beam-defining aperture. Laser cooling of the transverse and longitudinal velocities would serve to further relax the collimation requirements.

This value of Δ was chosen in Ref. 6 to minimize spherical aberration. It should be noted, however, that the coefficient of the term that leads to the lowest order spherical aberration, that is the coefficient of the ρ2/2 term [where ρ2= (x2+ y2)/w02 in the notation of Ref. 6] in square brackets in Eq. (6) there, is actually given by 3 + 5α/8 and not by 1 − α/2, where α= P0γ2/πΔ2ISw02= 2(I0/IS) (γ2/Δ2). Hence this choice of detuning (α= 2) does not eliminate the lowest order spherical aberration. In fact, we find that this term contributes significantly to enlarging the minimum spot size (at least for the doughnut mode laser beam considered here), unless the lens aperture is severely restricted. Since we find it impossible to eliminate the lowest order spherical aberration, we chose this value of detuning because it comes close to optimizing the radial potential well depth of the lens.

J. R. Meyer-Arendt, Introduction to Classical and Modern Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 108.

J. J. McClelland and M. Scheinfein, National Institute of Standards and Technology, Gaithersburg, Md. 20899 (personal communication).

See, for example, the following special issues: “Mechanical Effects of Light,” P. Meystre and S. Stenholm, eds., J. Opt. Soc. Am. B2, No. 11 (1985); “Laser Cooling and Trapping,” S. Chu and C. Weiman, eds., J. Opt. Soc. Am. B6, No. 11 (1989); and the following review articles: W. D. Phillips and H. J. Metcalf, Sci. Am. 256, 50–56 (1987); W. D. Phillips, P. L. Gould, and P. D. Lett, Science 239, 877–883 (1988).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Schematic showing the relationship of the atomic beam to the laser beam.

Fig. 2
Fig. 2

Plot of the intensity for a doughnut mode laser beam as a function of the axial distance z and the radial distance ρ.

Fig. 3
Fig. 3

Plot of the atomic beam width with the laser on, σE(solid curve), and with the laser off, σ (dashed curve). The parameter values for this case are given in the text.

Fig. 4
Fig. 4

Plot of the logarithm of the position of best focus, f, in micrometers, as a function of the logarithm of the laser power, P0, in watts. The solid curve is the thick-lens result, and the dashed curve is the thin-lens result. The parameter values are given in the text.

Fig. 5
Fig. 5

Plot of the logarithm of the spot radius, σE, in angstroms, as a function of the logarithm of the laser power, P0, in watts. The solid curve is the thick-lens result, and the dashed curve is the thin-lens result. The parameter values are given in the text.

Equations (25)

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Ψ ( x b , t b ) = d 3 x a K ( x b , t b , x a , t a ) Ψ ( x a , t a ) ,
K ( b , a ) = x b , t b x a , t a = a b δ x exp [ { i S [ x ( t ) ] } ] .
S [ x ( t ) ] = t a t b d t { m 2 [ t x ( t ) ] 2 - U [ x ( t ) ] } .
U ( x ) = Δ 2 ln ( 1 + I I S γ 2 γ 2 + 4 Δ 2 ) ,
K ( b , a ) N exp [ ( i / ) S c ] ,
I ( x , y , z ) = 8 I 0 w 0 2 ( x 2 + y 2 ) w 4 ( z ) exp ( - 2 L 2 w 0 2 x 2 + y 2 L 2 + z 2 ) ,
U ( x ) = C 2 x 2 + y 2 ( L 2 + z 2 ) 2 ,
C = 4 Δ P 0 L 4 π w 0 4 I S γ 2 γ 2 + 4 Δ 2 .
δ S [ x ( t ) ] δ x ( t ) | x ( t ) = x c ( t ) = - m t 2 x c + U ( x c ) = 0.
x 0 ( t ) = x a + x b - x a t b - t a ( t - t a ) x a + V ( t - t a ) .
K ( b , a ) exp ( i t a t b d t { m 2 V 2 - U [ x 0 ( t ) ] } ) .
K ( b , a ) exp { i [ m 2 T ( Z 2 + ρ b 2 ) + f 1 C T 2 Z 3 ρ b 2 + ( f 2 C T 2 Z 3 - m T ) ρ b · ρ a + ( f 3 C T 2 Z 3 + m 2 T ) ρ a 2 ] } ,
f 1 = arctan ( z b / L ) - arctan ( z a / L ) 2 L ( z a 2 L 2 + 1 ) - 1 2 ( z b z b 2 + L 2 - z a z a 2 + L 2 ) ( z a 2 L 2 - 1 ) - ( 1 z b 2 + L 2 - 1 z a 2 + L 2 ) z a , f 2 = arctan ( z b / L ) - arctan ( z a / L ) L ( z a z b L + 1 ) + ( z b z b 2 + L 2 - z a z a 2 + L 2 ) ( z a z b L 2 - 1 ) + ( 1 z b 2 + L 2 - 1 z a 2 + L 2 ) ( z b + z a ) , f 3 = - arctan ( z b / L ) - arctan ( z a / L ) 2 L ( z b 2 L 2 + 1 ) - 1 2 ( z b z b 2 + L 2 - z a z a 2 + L 2 ) ( z b 2 L 2 - 1 ) - ( 1 z b 2 + L 2 - 1 z a 2 + L 2 ) z b .
Ψ ( x a , t a ) = exp [ - ( i / ) E t a + i k z a ] ϕ ( ρ a , z a ) ,
z a ( 1 S c + k z a ) = 0
z a S c = - k .
z b - z a T = k m ,
ϕ ( ρ b , z b ) d 2 ρ a G ( ρ b , z b , ρ a , z a ) ϕ ( ρ a , z a ) ,
G ( b , a ) = exp { i [ ( f 1 C m 2 Z 2 k + k 2 Z ) ρ b 2 + ( f 2 C m 2 Z 2 k - k Z ) ρ b · ρ a + ( f 3 C m 2 Z 2 k + k 2 Z ) ρ a 2 ] } .
2 i k z ϕ + T 2 ϕ = 0 ,
ϕ ( ρ , z ) = σ 0 σ ( z - z 0 ) exp [ - i arctan ( 2 z - z 0 k σ 0 2 ) - ρ 2 σ 2 ( z - z 0 ) + i k ρ 2 2 R ( z - z 0 ) ] ,
σ ( z ) = σ 0 ( 1 + 4 z 2 / k 2 σ 0 4 ) 1 / 2 ,             R ( z ) = z ( 1 + k 2 σ 0 4 / 4 z 2 ) .
ϕ ( ρ b , z b ) exp ( { i f 1 C m 2 Z 2 2 k + i k 2 Z - ( f 2 C m 2 Z 2 2 k - k Z ) 2 [ 4 σ 2 ( z a - z 0 ) - 2 i k R ( z a - z 0 ) - 2 i f 3 C m Z 2 2 k - 2 i k Z ] } ρ b 2 ) .
σ E ( z b ) = σ ( z a - z 0 ) { 4 σ 4 ( z a - z 0 ) + [ k R ( z a - z 0 ) + f 3 C m Z 2 2 k + k Z ] 2 } 1 / 2 k Z - f 2 C m 2 Z 2 2 k , R E ( z b ) = k 2 { f 1 C m 2 Z 2 2 k + k 2 Z - 2 ( f 2 C m 2 Z 2 2 k - k Z ) 2 ( k R ( z a - z 0 ) + k Z + f 3 C m Z 2 2 k ) - 1 [ 16 σ 4 ( z a - z 0 ) + 4 ( k R ( z a - z 0 ) + k Z + f 3 C m Z 2 2 k ) 2 ] } .
2 σ SPHER ( 3 + 5 4 I 0 I S γ 2 Δ 2 ) 6 σ 0 3 w 0 2 17 6 4 σ 0 3 w 0 2 , 2 σ CHROM 4 σ 0 δ V V , 2 σ SPONT 2 σ 0 ( 2 V 3 λ γ I S I 0 ) 1 / 2 2 σ 0 ( 2 γ V 3 λ Δ 2 ) 1 / 2 , 2 σ DIP 1.44 ( V λ γ ) 1 / 2 ( γ 2 Δ 2 I 0 I S ) ( σ 0 w 0 ) 3 1.44 ( V λ γ ) 1 / 2 ( σ 0 w 0 ) 3 ,

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