DOI: 10.1007/s00340-004-1618-z

Appl. Phys. B 79, 679–682 (2004)

Lasers and OpticsApplied Physics B

m. morinaga Focusing ground-state atomswith an electrostatic fieldInstitute for Laser Science, University of Electro-Communications and CREST,Chofu, Tokyo 182-8585, Japan

Received: 19 April 2004/Revised version: 14 July 2004Published online: 25 August 2004 • © Springer-Verlag 2004

ABSTRACT A general formula for the trajectory of atoms inan arbitrary potential with axial symmetry is derived. We ap-ply this formula to show that an axially symmetric electrostaticimaging lens for ground-state neutral atoms is possible. Becauseof its simple construction and capability of focusing atoms ofany species, such a lens system can be used in a variety ofapplications.

PACS 03.75.Be; 39.25.+k; 32.80.Pj

1 Introduction

A lens system is one of the key components inatom optics, and various types of lenses have been extensivelystudied, including lenses based on magnetic hexapoles [1],microwave fields [2], standing waves of light [3], and holo-graphic manipulation of wavefronts [4]. Among them an elec-trostatic lens has many attractive features such as its simpleconstruction and ability to manipulate atoms of any species.However, it is known that unfortunately an axially symmetricthin lens using a static electromagnetic field is always concavefor ground-state atoms [5]. To avoid this property, electro-static lenses without axial symmetry [6] or magnetic hexapolelenses for low-field-seeking states in a Zeeman sub-level [1]have been examined.

In this paper, first we derive a general formula for the atomtrajectory based on the paraxial expansion of an arbitrary po-tential. Then we apply this formula to an electrostatic lenssystem with axial symmetry to show that such a system is al-ways concave for fast neutral atoms, and always convex forslow neutral atoms. Here, by ‘fast’ (‘slow’) we mean that thetotal energy of the atom is larger (smaller) than the potentialenergy at far distance, where the lens potential is negligible.This means that, in the case of a convex lens system (‘slow’atoms), atoms should always stay inside the lens potential re-gion. To overcome this restriction, we also present a schemethat utilizes the gravity of the Earth in addition to the electro-static field.

� Fax: +81-424/85-8960, E-mail: morinaga@ils.uec.ac.jp

2 Impossibility of making a thin convex lens

In this section we briefly review the proof of theimpossibility of making an electrostatic thin convex lens withaxial symmetry for fast ground-state atoms (see also [5]).

The dipole force acting on an atom in an inhomogeneouselectrostatic field can be expressed as

F = (P ·∇)E = α∑

i

Ei∂i E = α∑

i

Ei∇Ei = α

2∇|E|2 (1)

(∂i Ej = ∂j Ei comes from ∇ × E = 0), where P and α are thedipole moment and the polarizability of the atom, respectively(P = αE and α > 0 for atoms in a stable state). Thus the dipoleforce is derived from a Stark potential

U = −α

2|E|2 (2)

with F = −∇U .Suppose a ‘fast’ atom is incident on a thin electrostatic lens

with rotational symmetry around the z axis (Fig. 1). We as-sume that the initial atom velocity v0 is parallel to the z axiswith an impact parameter r0. The impulse I exerted on theatom during the flight is

I =∫

F(x(t), y(t), z(t))dt . (3)

Because the lens is thin and the atom moves fast, the distancefrom the z axis does not change considerably while the atom

FIGURE 1 An atom is launched into a thin lens parallel to the optical axis.The radial component of the impulse exerted on the atom is approximatelyproportional to the integral over the cylinder surface of the force that acts onthe atom

680 Applied Physics B – Lasers and Optics

passes through the lens, and thus the above integral can bereplaced with

I = 1

v0

∫F(r0x, r0y, z)dz . (4)

The radial component Ir can be calculated as an integral of Frover the cylinder of radius r0:

Ir = 1

2πr0v0

∮cylinder surface

F dS

= 1

2πr0v0

∫cylinder interior

∇ FdV . (5)

Because

∇ F = ∇ ∇ 1

2α|E2| = α

∑i, j

(∂i Ej)2 ≥ 0 , (6)

Ir ≥ 0 and we conclude that the lens is concave.

Is this estimation correct?

If we apply the same argument to a charged-particle lens sys-tem, the result will be that the electrostatic lens is neitherconvex nor concave (Ir = 0 follows from ∇ F = 0), which iswrong: an electrostatic lens for a charged particle is alwaysconvex.

Another contradiction arises if we add a strong uniformbias field E0 to the electrostatic lens field E ′. In this case,the force is dominantly governed by the term linear in E ′, sothat again ∇ F ∼ 0. The same calculation as for the charged-particle optics leads to the conclusion that such a lens shouldbe convex.

These examples show that we have to handle the systemmore carefully.

3 Refined treatment

We follow a procedure similar to that used in elec-tron optics [7] but using a general potential U . This treatmentincludes the effects of acceleration in the longitudinal direc-tion and of the change of radial distance r during the flight,which were neglected in the previous discussion.

Let us consider the motion of a particle under the poten-tial U with rotational symmetry around the z axis: U = U(z, r)with r ≡ √

x2 + y2 (Fig. 2). Defining G ≡ −U/m (m: mass)and using the paraxial expansion

G(z, r) = g0(z)+ 1

2g1(z)r

2 , (7)

equations of motion can be written as{r = ∂r G(z, r) = g1(z)r ,

z = ∂z G(z, r) ∼= ddz g0(z) .

(8)

The second equation can easily be integrated, giving

z = √2g0 +η , (9)

FIGURE 2 An atom passes through a potential with axial symmetry

where η is an integration constant. We redefine g0 +η/2 as g0so that (9) becomes

z = √2g0 . (10)

Using the relation ddt = dz

dtddz = √

2g0ddz and defining R(z) as

R(z) ≡ g1/40 (z)r(z) , (11)

the first equation of (8) leads to a differential equation in z:

d2

dz2R(z) =

[1

2

g1

g0+ 1

4

g′′0

g0− 3

16

g′20

g20

]R(z) . (12)

Note that, from the definition (11), R(z) is proportional to r(z)outside the lens potential where g0 is constant.

[Example] For a charged particle of charge q,

G(z, r) = − q

mφ(z, r) , (13)

where φ is the scalar potential of the electrostatic field. Defin-ing Φ(z) ≡ φ(z, 0) and using the expansion1

φ(z, r) = Φ(z)− 1

4Φ′′(z)r2 , (14){

g0 = − qm Φ ,

g1 = q2m Φ′′ ,

(15)

(12) becomes

d2

dz2R = − 3

16

Φ′2

Φ2R . (16)

The coefficient of R on the right-hand side of the above equa-tion is always negative, so that an electrostatic lens is every-where convex (with respect to R) for a charged particle.For neutral atoms

G = 1

m

(α

2|E|2 +β

). (17)

Here the constant β is equal to the total energy, i.e. kinetic en-ergy + potential energy, of the atom (the offset of the potentialenergy is chosen so that it goes to zero at far distance). Using(14) and (7),{

g0 = 1m ( α

2 Φ′2 +β) ,

g1 = α2m

(−Φ′Φ′′′ + 12Φ′′2) ,

(18)

1 Expansion to higher orders is given byφ(z, r) = ∑∞

ν=0(−1)ν( 1

ν!)2

Φ(2ν)(z)( r

2

)2ν.

MORINAGA Focusing ground-state atoms with an electrostatic field 681

and (12) becomes

d2

dz2R = 3

8αβ

(Φ′′

α2 Φ′2 +β

)2

R . (19)

The sign of the coefficient of R on the right-hand side of (19) isdetermined by the sign of β so that, if the atom is incident from‘outside the lens potential’ (β > 0) the lens is everywhere con-cave, whereas in the case where the atom starts from inside thepotential (‘in-lens beam source’) with such low velocity thatit cannot escape from the potential (β < 0) the lens is every-where convex.

4 Numerical calculations of the atom trajectories

In the following we show an example of atom tra-jectories for a particular lens, which consists of a metal plateof thickness d with a hole of inner radius r0 put in a uniformelectric field E0 (Fig. 3).

For the case d = r0,

Φ(z) = E0z

1 −0.4

r1√z2 + r2

1

(20)

with r1 = 1.2r0 gives a good approximation for the scalarpotential of the electrostatic field on the z axis (i.e. the opti-cal axis). We use parameters for sodium atoms and the valuefor the polarizability αNa/4πε0 = 24.4 ×10−30 m3 is takenfrom [8]. Figure 4 shows the potential curve on the z axis forr1 = 2 mm and E0 = 1 kV/mm in units of mK, and we calcu-late the atom trajectories in this system. Trajectories for atoms

FIGURE 3 The lens system. A metal plate of thickness d with a hole ofradius r0 is sandwiched by a pair of electrodes with V0 = E0 L

FIGURE 4 The potential curve on the z axis for the bias electric field E0 =1 kV/mm

FIGURE 5 Atom trajectories. Solid line: β = −0.024 mK, dashed line:β = 0 mK, and dash-dotted line: β = 0.024 mK

FIGURE 6 Power vs. initial atom velocity v0 for the bias electric field E0 =1 kV/mm

FIGURE 7 The trajectories of atoms with β = −0.024 mK

starting from z = −10 mm with positive, zero, and nega-tive β (β = 0.024 mK, 0 mK, and −0.024 mK) are shown inFig. 5. The corresponding initial velocities are v0 = 0.30 m/s,0.26 m/s, and 0.23 m/s. The lens has a negative power (i.e.the inverse of the focal length f ) for atoms with positive β,zero power for β = 0, and a positive power for negative β, aspredicted. This dependence of the power on the initial atomvelocity v0 is plotted in Fig. 6.

In Fig. 7 trajectories of atoms with β = −0.024 mK start-ing from an off-axis position are also shown.

5 Effects of gravity

In Sect. 3 we have seen that both the beam sourceand the focal plane have to be inside the lens potential in orderto make a convex lens. However, we now show that the latterrestriction can be removed with the help of the gravity of theEarth.

We take the optical axis (z axis) in the vertical direc-tion (+z direction ‘downward’). We consider the case where

682 Applied Physics B – Lasers and Optics

atoms are released from a cold trap, so that gravitational ac-celeration dominates the vertical motion and the scheme pre-sented in Sect. 3 will not give a simple description any more.So here we simply give an example of numerically calculatedtrajectories.

The lens system is shown in Fig. 8. Now a hole of radius r2is drilled in the third electrode of the previous setup to extractatoms to the free space.

For the scalar potential on the z axis we use the followingexpression:

Φ(z) = E0

z − L −

√(z − L)2 + r2

2

2− 0.4r1z√

z2 + r21

(21)

(r1 = 1.2r0 as before). Near the hole of the third electrode(21) gives only a rough approximation of the real scalar po-tential, but is sufficient to describe the behavior of the atomtrajectory qualitatively. In Fig. 9 the potential curve on thez axis is shown for r1 = 0.4 mm, r2 = 1 mm, L = 4 mm, E0 =V0/L = 1 kV/mm, and g = 9.8 m/s2, and in Fig. 10 we havecalculated atom trajectories under these parameters assumingthat the atoms start from z = −3 mm with zero longitudinal

FIGURE 8 The lens system. The third electrode of the previous setup isreplaced with a thin plate with a hole of radius r2

FIGURE 9 The potential curve on the z axis for the bias electric field E0 =1 kV/mm

FIGURE 10 The trajectories of atoms released from z = −3 mm

velocity, demonstrating a focusing of atoms outside the lensregion.

6 Conclusion and remarks

We have shown that an electrostatic focusing lensfor ground-state atoms with axial symmetry is possible. Infree space, both the beam source and the focal plane have tobe inside the lens potential. However, it was shown that, undergravity, we can extract atoms to outside the lens potential.Such a lens system could be used to focus, image, or collimatecold atoms of any species including those that do not possessmagnetic moments.

In this paper, we did not estimate aberrations of the lenssystem. When gravity is present, even a small misalignment ofthe optical axis to the direction of gravity might also cause anadditional aberration. Analysis of such effects is left for futurework.

ACKNOWLEDGEMENTS We thank Hilmar Oberst for his care-ful reading of the manuscript. This work is partly supported by the 21stCentury COE program of the University of Electro-Communications on ‘Co-herent Optical Science’ supported by the Ministry of Education, Culture,Sports, Science and Technology.

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