Atomic Fresnel images and possible applications in atom lithography

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Atomic Fresnel images and possible applications in atom lithography

U. Janicke, M. Wilkens

To cite this version:

U. Janicke, M. Wilkens. Atomic Fresnel images and possible applications in atom lithography. Journal

de Physique II, EDP Sciences, 1994, 4 (11), pp.1975-1980. �10.1051/jp2:1994243�. �jpa-00248099�

J.

Phys.

II trance 4

(t994)

t975-1980 NOVEMBER1994, PAGE 1975

Classification

Physics

Abstracts

32.90 42.20 42.50

Atomic Fresnel images and possible applications in _atom

lithography

U. Janicke and M. Wilkens

Fakultit fur

Physik,

Universitit Konstanz, 78434 Konstanz,

Germany

(Received

24

May1994,

re~ised 6

July 1994, accepted

lo

August 1994)

Abstract. In the _near field

regime

of diffractive atom

optics,

amplitude

corrugations

of the de

Broglie

_wave front _are

important

and _can lead to

interesting

effects. One class of

near field

phenomena

is the formation of Fresnel

images.

We

study

this effect and possible

applications

in atom

lithography using

_wave

packet

simulations.

Diffraction of atoms most

clearly

demonstrates the ultimate _wave character of the atomic center-of-mass

motion, opening

_a broad class of

potential applications,

in

particular

in atom in-

terferometry.

Various

diffracting objects

like microfabricated transmittion

gratings

and double

slits, standing

_wave

light fields,

and

magneto-optical

fields have been

investigated extensively

both

experimentally

and

theoretically

in the past

[1-6].

In _a

typical diffraction experiment

_one considers _a beam of atoms,

traveling predominantly

in the

z-direction,

which _{traverses a}

diffracting object, aligned

in the

x-direction, (see Fig. I).

Diffraction of the atoms occurs if the transverse coherence

length (in

the

x-direction)

is of the

same order _or

larger

than the

spatial period

of the

diffracting object. Assuming

that the kinetic energy in the z-direction of the

incoming

atoms is much

larger

than the interaction energy and that the

diffracting object

is uniform in the

y-direction,

the diffraction is described

by

_an

effectively

one-dimensional model of the quantum mechanical motion

along

the x-direction.

Utilizing

a

paraxial approximation,

the

spatial

evolution of the atomic wavefunction

16(x, z) along

the z-direction is described

by

the

Schr6dinger equation

ih~)~bix, ^z)

=

[ [

+

Uix, z) _~bix, z)

,

ii)

where _~ is the center-of-mass

velocity

of the

incoming

atoms in the

z-direction,

M the _mass of the

atoms,

and

U(x, z)

is the interaction operator ^of the interaction between the _atoms and

the

diffracting object.

It is convenient to

distinguish

between _a

preparation region along

the

z-direction,

where

U(x, z)

is nonzero, and the

subsequent

free evolution for

z > zo, where

U(x,

_{z >}

zo)

^vanishes.

t976 JOUIINAL DE

PHYSIQUE

II N°11

t

~

~

z~~

~

_fl

~

~

~

~

~

~

Fig.

I. The geometry ^of a

typical

diffraction

experiment.

A beam of atoms _moves in the z-direction and traverses _a

diffracting object (e.g,

_a

standing

_wave

light field)

which is

aligned

in the «-direction.

Behind the

diffracting object

_one

distinguishes

between _{a near} field

(nf)

and _a far field

(ff).

For _an

arbitrarily prepared

wavefunction

16(x,zo),

the state at distance _{z zo} downstream is

given by

^{the Fresnel}

integral

16(x, z)

_c~

dx'

_exp I ~~

(x x'

_)~

lb(x',

_zo

(2)

/

21t(z

_zo

In

analogy

to classical

optics

_one

distinguishes

between _a far field

(z

_zo »

a~ /ldB (Fraun-

hofer

limit, region

denoted

by

"ff' in

Fig. I)

and _{a near} field

(z zo)

_m

a~/ldB (Fresnel limit, region

denoted

by

"nf' in

Fig. I).

Here _a is the illuminated size of the

diffracting object,

and

ldB

_"

h/(M~)

is the atomic de

Broglie wavelength.

The transition from _near field to far field behavior is shown in

figure

2 which

depicts

the free evolution of _a coherent _array of atomic _wave

packets. Up

_to

date,

most of the

experiments

have been discussed in the Fraunhofer limit _{z - oo.} In this limit the

spatial

atom distribution which is observed far behind the interaction

region

is

essentially given by

the momentum distribution of the atoms after the interaction. The dimension of the diffraction

pattern

^scales

linearly

with the distance from the

preparation plane.

In contrast in the _near

field,

no

simple scaling

_can be observed.

Here, corrugations

^of the _wave front _are

important

and interferences

between different

spatial parts

of the wavefunction lead to a

complicated

evolution of the atomic wavefunction in the z-direction.

To discuss the _near field effects which _occur in atom diffraction in _more

detail,

_{we turn} to the concrete

example

of diffraction of atoms from _an off-resonant

standing

wave laser field

(scalar Kapitza-Dirac effect). Figure

3 shows _{a wave}

packet

simulation of this _process. An atomic beam of

ground

state two level atoms, modelled

by

_a broad Gaussian _wave

packet spanning

several

optical

_wave

lengths,

moves

through

_a

standing

_wave laser field with

wavelength I,

located at

z = 0. The width of the _wave

packet corresponds

here to the transverse coherence

length

of the atomic beam. The

light

field has _a Gaussian

envelope exp(-z~ /w~

and is

sufficiently

detuned from the atomic transition

frequency,

_so that the

population

of the excited state

during

the

interaction is

negligible. Figure

3 shows the

intensity

distribution of the

ground

state atoms as a function of _x and z. The atoms are focussed

by

the antinodes of the

standing

_wave

light

field

[7,

8] which act _as an array of thin lenses

(phase grating).

In the focal

plane

of these lenses at z =

f

_an array of _narrow

peaks

^with

period

^d

=1/2

is formed

(first

bold line behind the

light

field in

Fig.

3 and thin solid line in

Fig. 4).

N°11 ATOMIC FRESNEL IMAGES ATOM LITHOGRAPHY 1977

z=0

j

l

@

~

x

Fig.

2. Natural

spreading

of _{an array} of Gaussian _wave

packets.

In the _near field the formation of Fresnel

images

is visible. The far field reveals the Fourier transform of the initial _array, which scales

linearly

with the distance.

The focal

length f

is

approximately given by

~ lt~T ~B

~

~

~~

~

^~~~

~~~

Here fl is the Rabi

frequency

of the

light field,

b is the

detuning

of the

light

field from the atomic transition

frequency,

and _T

=

@ w/~

is the interaction time

(note

that fl and b _are

angular frequencies).

In the

following

it is convenient to use recoil units

fl/wrec

_-

fl, b/wr~c

_-

b,

_Twr~c - T,

ldBk

_-

ldB,

where _wr~c

=

ltk~/(2m)

is the recoil

frequency.

The

parameters

used in the simulation _were fl

=

280,

b

=

-200,

and _T

= 0.044. As

T <

(fl)~~/2,

the interaction time is within the Raman-Nath

regime [6].

^{The focal}

spot

^size Ax

(left-to-right

minimum

distance)

_can be estimated

by Iii

Ax =

21dB~ (4)

This

gives Ax/d

= 0.354 in

good agreement

with the value which _can be read off from

figure

4.

The

focussing

of _an atomic beam

by

the antinodes of _a

standing

_wave

light

field is not restricted to the situation in which the

incoming

beam has _a

large

transverse coherence

length.

Even _a

completely

incoherent

beam,

which in this context may be

regarded

to _{as an} ensemble of

1978 JOURNAL DE

PHYSIQUE

II N°11

j

.,

atomic

beam ',

Standing

_WaVB

_' ,-.~

jjght

field

t>'<l't

f+U3

~ f+U2

L

f+L

~

X

Fig.

3. A coherent atomic

beam,

modelled

by

_a broad Gaussian, traverses _a red detuned

standing

wave

light

field. The atoms _are focussed at the antinodes of the

light

field which _acts

as an array of thin lenses with focal length f. At _z

= f _{an array} of

narrow peaks with period d

=

~/2

is formed

(first

bold

line).

Natural

spreading

of the atom distribution leads to the formation of Fresnel

images

with

periods d/3

and

d/2

at _z

=

f

+

L/3

and _z

=

L/2, respectively (second

and third bold

line).

At

z = f + L the initial distribution from _z

=

f

is reconstructed

(fourth

bold

line).

j~

l$

c

l£

©

x-direction

Fig.

4. The initial atom distribution at _{z =} f, the successive Fresnel

images,

and the reconstruction image ^denoted

by

the four bold lines in

figure

3. Thin solid line: initial distribution at _z

= f ^with

period

d, bold solid line: Fresnel

image

with

period d/3,

bold dotted line: Fresnel

image

with

period

d/2,

thin dotted line: reconstruction

image

with

period

d,

phase

shifted

by

_~.

N°11 ATOMIC FRESNEL IMAGES ATOM LITHOGRAPHY ₁₉₇₉

classical

point-like particles,

is focussed _as the antinodes of the

light

field form

quasi-harmonic potentials. However,

the

subsequent

free evolution in the z-direction is very different in the

two _cases. For _z >

f

the

peaks

in the distribution

spread

out and

overlap.

For _a coherent

beam the

peaks

have _a well-defined

phase

relation and

spatial

interferences between the

peaks

lead to

fringes

in the

intensity

distribution. The

period

of these

fringes

which form the Fresnel

images

discussed below _can be much smaller than d. For _an incoherent beam _no smaller

periods

appear as the

outspreading peaks

have _no fixed

phase

relation. Their intensities

simply

add and the initial

periodic

modulation is smoothed _out.

The free evolution in the z-direction of _a coherent distribution with _a

phase

_or

amplitude

modulation of

period

d leads after the _so called Talbot

length

L

=

d2/ldB

to _a reconstruction of the initial

distribution,

_an effect first observed with

light by

Talbot

[9].

^At intermediate

distances

L/m,

_m

=

2, 3,...,

distributions with smaller

periods d/m

_appear. The

images

with smaller

periods

_are referred to _as Fresnel

images,

the reconstruction

images

_are often called Fourier

images [10, iii.

The locations of Fresnel

images

_were first found in

[12]

^and

[13].

The

general

ID Talbot

problem, including

both finite width and infinite width

complex-valued

diffraction

gratings,

has been

analysed

in detail

by

Clauser and Reinsch

[15].

^{For matter} waves,

experimental

observations of Fourier

images

_were

reported

in

[16, Iii,

observations of Fresnel

images

_were

reported

in

[18].

In

figure 3,

the

amplitude

modulation at _{z =}

f

with

period

d leads to the formation of Fresnel

images

with

periods d/3

at _z

=

f

+

L/3 (second

bold

line)

and

d/2

at _{z =}

f

+

L/2 (third

bold

line).

At _z

=

f

+L the distribution from _z

=

f, phase

shifted

by

_ir, is reconstructed

(fourth

bold

line).

At _z

= 0 the

incoming

_wave

acquires

_a

periodic phase

modulation from the

light

field which acts _{as a}

phase grating leaving

the

amplitude

of the _wave

basically unchanged.

This leads to the formation of _a

nearly fringe-free

reconstruction

image

at _z

= L. The

plane

at z =

f

+ L _can be viewed _as well _as the focal

plane

of this reconstruction

image.

The four distributions

depicted by

the bold lines _are also shown in

figure

4 for better

comparison.

The ratio of Talbot

length

to focal

length

in this simulation _was

L/ f

= S-G-

Fresnel

images

have

potent1al applications

in atom

lithography making

it

possible

to write very small

periodic

structures on a substrate

placed

in the

plane

of _a Fresnel

image.

The

visibility

of the Fresnel

images depends strongly

_on the ratio between the width of the

peaks

and their

separation

in the

original

_array: ^the smaller the width

compared

to the

separation,

the

higher

is the maximal order of

periods.

Note that

standing

_wave diffraction allows _{a more} flexible

preparation

of the initial atomic array than the _use of transmittion

gratings (as

used e-g- in

[16-18]),

^{because both the}

periodicity

and the width of the diffraction

peaks

can be

changed

at will

by varying

the

period

and the

intensity

of the

diffracting light

field.

A favorable candidate for atom

lithography

is chromium with the transition

~S3 -~P(

_at

1 = 425.6 _nm and M

= 52 _u [8]. ^{With these values the recoil}

frequency

is _wr

=

132,

^kHz and

the

parameters

used in the simulation become fl

= 37

MHz,

b

= -26

MHz,

and _T

= 330 _ns.

For _a

typical longitudinal velocity

of _~

= 1000

m/s

the Talbot

length

is L

= 6 _{mm, so} that

the Fresnel

images

with

periods d/2

and

d/3

in

figure

3 would _{appear at} distances behind the focal

plane

of 3 _mm and 2 _mm,

respectively.

These distances and the

parameters

^{for the}

light

field _seem to be well in

experimental

reach.

A restriction for the

quality

of the Fresnel

images

in _a continuous beam

experiment

is the

longitudinal velocity

distribution

A~/~

of the _atoms which leads to different interaction times and to _a

smearing

out of the Fresnel

images. By taking

into account a

longitudinal velocity

distribution with _a

rectangular profile

in _our

simulations,

_we found that _up to

A~/~

₌ 0.25 the Fresnel

images

^of

figure

3 _are still observable. This

velocity

distribution is _narrower than the _one for _a thermal

beam,

but with

velocity selecting

devices it should be

experimentally

accessible.

1980 JOURNAL DE

PHYSIQUE

II N°11

Summarizing,

it has been demonstrated that _wave

packet

simulations allow for _a

qualitative

and

quantitative study

of the formation of atomic Fresnel

images.

We showed that Fresnel

images

_can be created

by focussing

_an coherent atomic beam at the antinodes of _a

standing light

field. Possible

applications

in atom

lithography

_were

investigated

for the

potential

^candidate

chromium. The _necessary

parameters

^found are well in

experimental

reach.

Acknowledgments.

Fruitful discussions with C-R-

Ekstrom,

U.

Drodofsky,

and M. Drewsen _are

gratefully

_ac-

knowledged.

The authors

acknowledge support by

the Deutsche

Forschungsgemeinschaft

and

hospitality

in the _group of Professor J.

Mlynek.

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