Download Atom dynamics in two counter-propagating evanescent Bessel

Indian J Phys (February 2014) 88(2):197–202
DOI 10.1007/s12648-013-0390-5
ORIGINAL PAPER
Atom dynamics in two counter-propagating evanescent Bessel beams
S Al-Awfi*
Department of Physics, Taibah University, P.O. Box 30002, Medina, Saudi Arabia
Received: 30 June 2013 / Accepted: 29 August 2013 / Published online: 11 September 2013
Abstract: It is shown that total internal reflection of two counter-propagating Bessel beams with a phase singularity can
generate evanescent light that displays a rotational character. This effect retains the phase singularity of input light, as well
reflects associated orbital angular momentum. The product optical system gives rise to optical forces that can trap lateral
in-plane motion of such atoms, thus acting as a trap. We have explained basic properties of this processes and discuss how
they can be used to influence atoms localized in vicinity of the surface.
Keywords:
Bessel mode; Orbital angular momentum; Phase singularity
PACS Nos.: 37.10.De; 37.10.Gh
1. Introduction
It is well established that laser light prepared as a Bessel
beam shows little diffraction, typically exhibiting a number
of concentric high intensity rings separated by dark rings
[1–4]. In particular, the central peak is considered to be
remarkably stable against diffraction and this property is
used in recent applications including atom guides, nonlinear optics and optical atom sorting [5]. We point out here
that non-diffractive feature also makes these beams suitable for generation of surface optical vortices in which
typically a Bessel beam propagating within an optically
dense medium is totally internally reflected at a planar
interface with vacuum. A light field, possessing the rotational features of Bessel beam and tightly bound to the
surface as an evanescent mode, is generated in vacuum
region with its intensity in vacuum depending on Bessel
order and displaying interesting spatial variations while
still decaying exponentially with distance away from the
surface [6, 7]. We present the theory exploring surface
mode due to a totally internally reflected Bessel beam, its
optical angular momentum content, interference of
multiple beams and possible applications that can be
experimentally realized for systems in vicinity of the surface [8–11].
Present work aims to determine behavior associated
with optical angular momentum whose origin is neither
from photon spin nor from surface morphology, the analogous phase properties being primarily conferred by a
helical beam structure. Developing earlier work [12],
present focus is on utilization of surface field for the control of vicinal atoms. We suppose that before Bessel light is
switched on, atoms are trapped vertically by optical
potential created by evanescent light. For an atom at rest in
z-direction, the fields imported by Bessel light are basically
two-dimensional and hence motion occurs only in the
plane, parallel to the surface.
2. Theory of evanescent field of a single beam
Consider first electric field of a Bessel beam traveling
along z-direction in a medium of a constant refractive
index n, characterized by integer l, circular frequency x,
and axial wave vector k ¼ nk0 , where k0 ¼ x=c is the wave
vector in vacuum. For Bessel light plane polarized along y^,
the field vector can be written in cylindrical coordinates as
EIk‘ ðr; q; zÞ ¼ y^F k‘ ðrk ; zÞ exp iðkz xtÞ exp i‘u
*Corresponding author, E-mail: [email protected]
ð1Þ
where F k‘ ðrkp
; zÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi
is the standard envelope function of rk and
z, with rk ¼ x2 þ y2 :
Ó 2013 IACS
198
S Al-Awfi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8p2 kr2 w20 I
n¼
n2 e 0 c
y
pð1 2‘Þ
1
H ¼ ‘ tan
þ
x cos /
4
ð4Þ
ð5Þ
3. Atom–field interaction
Fig. 1 Schematic representation of an atom in a vacuum, trapped
above the point on a surface where there is total internal reflection of
a single Bessel beam
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2I=n2 e0 c
z ‘þ1=2
F k‘ ðrk ; zÞ ¼
ð2pkr w0 Þ1 zmax
z2
pð1 2‘Þ
exp 2
J‘ ðkr rk Þ
exp i
4
zmax
ð2Þ
Here w0 is input beam waist, kr is radial wavevector and
zmax is typical ring spacing. Last term J‘ ðkr rÞ is Bessel
function of the order l. Such a light field can be
arranged, as shown in Fig. 1, to strike the internal planar
surface of a medium, in which it is propagating—that is,
in contact with vacuum. If interface with vacuum
occupies the plane z ¼ 0 and angle of incidence h,
exceeds total internal reflection angle, an evanescent
mode is created in vacuum. Main requirements are
applicability of standard phase-matching condition of
boundary reflection and the condition that electric field
vector component tangential to surface be continuous
across boundary. To be able to define the evanescent
electric field, we must first obtain expressions
appropriate for a beam incident at an angle /. In this
case, the explicit form of evanescent electric field that
displays angular momentum properties, as well as the
mode characteristics, is as follows [12]:
!
x sin / ‘þ1=2
ðx sin /Þ2
evan
Ek‘ ðRÞ ¼ y^n
exp zmax
z2max
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1
x2 þ y2
J‘
exp iH
w0 1 þ kz=pw2 2
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
exp zk0 n2 sin2 ð/Þ 1
expðik0 nx sin /Þ
ð3Þ
where n and H are given by
We now consider an atom in vacuum region with position
vector RðtÞ ¼ ½rk ðtÞ; z where rk ðtÞ ¼ ½xðtÞ; yðtÞ is in-plane
position of the atom. We assume that the atom is trapped
vertically at a fixed position z close to upper surface.
Trapping occurs by a potential well that is significantly
stronger than potential created by evanescent light [9, 10].
For an atom at rest at a fixed vertical position z, the fields
due to Bessel beam are essentially two-dimensional, hence
motion occurs only in x–y plane, parallel to surface.
Electronic properties of the atom are simply cast in form of
a two-level approximation, assuming a ground state
(denoted as level g, of energy Eg) and an excited state
(level e, of energy Ee) and a frequency x0 such that hx0 ¼
Ee Eg : Interaction of the atom with surface plasmon
vortex is taken in dipole approximation, with a
Hamiltonian
Hint ¼ l Eevan
k‘ ðx; y; zÞ
ð6Þ
We adopt the field–dipole orientation picture in which
atomic transition dipole l aligns itself adiabatically along
local electric field direction. Interaction of the atom with
electric field of evanescent Bessel beam leads to two
important dynamical attributes of motion, namely Rabi
frequency Xevan
k‘ ðRÞ that characterizes the interaction of a
neutral atom of electric dipole moment l approaching to
wards outer surface from vacuum region (z [ 0) and so,
interacting with evanescent light. Secondly, evanescent
light phase hevan
k‘ ðrk Þ which corresponds to momentum
imparted by evanescent light to the atom, which are given
by
evan
Xevan
h
k‘ ðRÞ ¼ l Ek‘ ðRÞ=
y
pð2‘ 1Þ
1
ðx;
yÞ
¼
‘
tan
hevan
k‘
4
pffiffiffiffi x cos /
e2 xx sin /
c
ð7Þ
ð8Þ
Substituting for electric field in vacuum region, we secure
the following result for Rabi frequency associated with
evanescent field arising from internal reflection of a single
Bessel beam
Atom dynamics in two counter-propagating
199
Dissipative force Fdiss has been exploited in heating and
cooling atomic motion. As Eq. (15) suggests, to influence
the atom in x y directions, integer l must be greater than
zero, otherwise the atom will only be subject to a force in
x-direction. In this case, the dissipative force reduces to
(
pffiffiffiffi
)
CðRÞX2 ðRÞ e2 x sin /=c
Fdiss ðR; VÞ ¼ 2h
ð16Þ
D2 ðR; VÞ þ 2XðRÞ2 þ C2 ðRÞ
ðx sin /=zmax Þ‘þ1=2
Xevan
k‘ ðRÞ ¼ n 2 1=4
1 þ kx sin /=pw20
!
ðx sin /Þ2
exp z2max
0
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
x cos / þ y
B
C
ffiA
J‘ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
w0 1 þ kx sin /=pw20
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
exp zk0 n2 sin2 / 1
ð9Þ
evan
Rabi frequency Xevan
k‘ ðRÞ and optical phase hk‘ ðrk Þ are
extremely important for determining and manipulating
optical forces, which we now consider.
4. Manipulating optical forces
Total steady state force acting on the centre of mass of an
atom moving near the metallic sheet surface of transition
frequency x0 moving in vacuum region at velocity vector
_ is given by well known expression [13]
VðtÞ ¼ rðtÞ
FðR; VÞ
(
)
CðRÞX2 ðRÞrhðRÞ ð1=2ÞDðR; VÞXðRÞrXðRÞ
¼ 2h
D2 ðR; VÞ þ 2XðRÞ2 þ C2 ðRÞ
¼ Fdiss þ Fdip
ð10Þ
where CðRÞ is decay emission rate of the atom and DðR; VÞ
is dynamic detuning given by
DðR; VÞ ¼ D0 V rh
ð11Þ
where D0 ¼ ðx x0 Þ is static detuning of light from
atomic resonance and rh is gradient of light phase, in view
of Eq. (8), it can be written as
rhðx; yÞ ¼ aðx; yÞ þ bðx; yÞ
ð12Þ
On the other hand, second term in Eq. (10) is dipole force
Fdip , used for trapping process and can be written as
(
)
DðR; VÞXðRÞrXðRÞ
Fdip ðR; VÞ ¼ 2h 2
ð17Þ
D ðR; VÞ þ 2XðRÞ2 þ C2 ðRÞ
This means that atoms also become subject to a light
induced dipole force Fdip which depends on the order of
mode l. Explicit form of dipole potential U for such mode
is such that Fdip ¼ rU, which means that
"
#
hDðR; VÞ
2XðRÞ2
ln 1 þ 2
UðR; VÞ ¼
ð18Þ
2
D ðR; VÞ þ C2 ðRÞ
5. Discussion of two counter-propagating beams
An undesirable feature in case of evanescent field of a
single Bessel beam, as far as interaction with atoms is
concerned, is that there is a plane wave with a wavevector equal to in-plane component of incident light
traveling along the surface. The corresponding term in
phase function gives a dissipative force, as for a plane
wave in free space, but which acts on the atom along
where
aðx; yÞ ¼
pffiffiffiffi
‘y
h
i ð e2 x sin /=cÞ
2
x2 cos / 1 þ ðy=x cos /Þ
ð13Þ
bðx; yÞ ¼
‘
h
i
x cos / 1 þ ðy=x cos /Þ2
ð14Þ
Thus the dissipative force Fdiss can be given as
2haCðRÞX2 ðRÞ
Fdiss ðR; VÞ ¼
x^
D2 ðR; VÞ þ 2X2 ðRÞ þ C2 ðRÞ
2hbCðRÞX2 ðRÞ
y^
þ
D2 ðR; VÞ þ 2X2 ðRÞ þ C2 ðRÞ
ð15Þ
Fig. 2 Schematic representation of an atom in a vacuum, trapped
above the point on a surface where there is total internal reflection of
a counter-propagating Bessel beam
200
S Al-Awfi
x-direction. A possible arrangement in which this undesirable motion can be eliminated is by creation of a situation whereby in-plane waves of two beams counterpropagate as the situation shown in Fig. 2. In this case,
two Bessel beams, labeled I and II, are incident at angles
/1 ¼ /2 ¼ /; both are reflected and have field components within the surface in vacuum region. We assume
that two beams are identical in modal form and spatial
distribution, except for their directions of propagation and
possibility of an associated change in sign of angular
momentum quantum number l. Rabi frequencies and
phases may also differ in the sign of angular momentum
quantum number l. For Rabi frequency of the evanescent
field of beam I we write
‘þ1=2
ðx sin /=zmax Þ
2 1=4
1 þ kx sin /=pw20
!
ðx sin /Þ2
exp z2max
0
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
x cos / þ y
B
C
ffiA
J‘ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
w0 1 þ kx sin /=pw20
(
h
F1þ2
dip ðR; VÞ ¼ 2
1
X‘1/
ðRÞ ¼ n þ
ð19Þ
The corresponding phase gradient can be written as
1
1
1
rh‘1/
ðx; yÞ ¼ a‘1/
ðx; yÞ^
x þ b‘1/
ðx; yÞ^
y
ð20Þ
In consequence, dynamic detuning of evanescent field of
beam I is
1
1
1
1
D‘1/
ðR; VÞ ¼ D0 V rh‘1/
¼ D0 a‘1/
Vx b‘1/
Vy
ð21Þ
where Vx and Vy are in-plane Cartesian velocity
components. Representation of second beam is derived
from the first by rotation about y-axis through an angle /.
Hence we can write
2
2
X‘2ð/Þ
ðRÞ ¼ X‘1ð/Þ
ðRÞ
ð22Þ
2
2
D‘2ð/Þ
ðR; VÞ ¼ D0 V rh‘1ð/Þ
2
2
¼ D0 a‘1ð/Þ
Vx b‘1ð/Þ
Vy
ð23Þ
Consequently, combined effect of two beams is to give rise
to optical forces acting on centre of mass of atom. Total
dissipative force can be written as the vector sum
(
!
X21/ a1/ x^ b1/ y^
1þ2
Fdiss ðR; VÞ ¼ 2hC
D21/ þ X21/ þ C2 ‘
1
1 )
0
2
X1ð/Þ a1ð/Þ x^ b1ð/Þ y^
A
þ@
ð24Þ
D21ð/Þ þ X21ð/Þ þ C2
‘2
Similarly the total dipole force is
Fig. 3 The Optical dipole potential distribution (in unit of
hC0 =2) due to the surface mode with ‘ ¼ 1
U0 ¼ D1/ X21/ rX21/
!
D21/ þ X21/ þ C2
D1ð/Þ X21ð/Þ rX21ð/Þ
D21ð/Þ þ X21ð/Þ þ C2
‘1 )
!
ð25Þ
‘2
where, for convenience, we have indicated orbital angular
momentum quantum number for each beam as a subscript
of expression for corresponding force. Of two average
optical forces, it is easy to see that Fdip w can act as a
repulsive force provided that detuning D0 is positive (blue
detuning). In this case, optical system works as an effective
atomic mirror. On the other hand, Fdip acts as an attractive
force provided that detuning D0 is negative (red detuning)
and optical system works as a successful optical lattice.
Figure 3 shows variations in x y plane of dipole
potential Uðx; yÞ generated by evanescent field, which are
created at upper surface by the two counter-propagating
Bessel beams with negative detuning. The potential is
given in units of convenient scaling parameter U0 ¼ hC0 =2
and distances are measured in units of beam waist w0 .
Parameters of a sodium atom and evanescent beams are
such wavelength k ¼ 589:0 nm; beam-waist w0 ¼ 40k;
irradiance I ¼ 2 108 Wm2 ; angles of incident
/1 ¼ /2 ¼ p=4; e1 ¼ 1, e2 ¼ 2:3; l ¼ 2:6ea0 (e is electron charge, a0 is Bohr radius) and spontaneous rate C is
taken to be the free space value C0 ¼ 6:1 107 s1 (this is
in fact a very good approximation in trajectory region
which is sufficiently far from the surface). The static detuning is taken to be jD0 j ¼ 103 C0 .
It is seen that for D0 \0, dipole potential exhibits a
minimum at points where the intensity is maximum. It is
seen that the atom is confined in annulus-shaped potential
well, spiraling outwards in an elliptically shaped. It can be
deduced from Fig. 3 that from a quantum–mechanical
point of view solutions of two-dimensional Schrödinger
equation with Uðx; yÞ as potential must exist. In ground
state, atomic wave function peaks in vicinity of the central
minimum associated with dipole potential. It can also be
Atom dynamics in two counter-propagating
201
Fig. 4 Atomic trajectories of Na atom in optical system arrangement
shown in Fig. 2 for a negative detuning case with angle of incidence
/1 ¼ /2 ¼ 42:00 and angular momentum quantum numbers ‘1 ¼
‘2 ¼ 1 (vertical axis y=w0 and horizontal axis x=w0 )
seen from Fig. 3 that for parameters assumed above, central well depth is approximately 36:0U0 . This is sufficiently
deep to exhibit many quasi-harmonic trapping (vibrational)
states. Vibrational frequency can be estimated simply using
the parabolic approximation [14].
Fig. 5 Variation of the transverse distances x=w0 and y=w0 with time
for the case in Fig. 4
6. Result of atom dynamics
Dynamics of an atom of mass M approaching a surface for
a given set up is obtainable by solving the equation of
motion
M
d2 R
1þ2
¼ F1þ2
diss ðR; VÞ þ Fdip ðR; VÞ
dt2
ð26Þ
It is thus clear that Eq. (26) constitutes a set of coupled
ordinary differential equations. These, for a given set of
initial conditions, can be solved numerically using standard
routines.
Figure 4 shows typical atomic trajectory on x y plane
based on solution of Eq. (26) for Na atom, assuming D0 \0
with the initial conditions such that the atom starts
from rest i.e. Vð0Þ ¼ ð0; 0Þ, at the point ðxð0Þ; yð0ÞÞ ¼
ð0:5; 0:5Þw0 . This rotational motion is an expected signature of the light orbital angular momentum effects. Radial
features of trajectory immediately suggest that atom may
be trapped in an annulus-shaped quantum well due to light
and this is confirmed by inspecting the distribution of
dipole potential in Fig. 3. It is clear that this radial confinement leads to vibrational motion in a radial direction
and results in an overall zigzag trajectory.
Figure 5 displays the variation of transverse distances
x=w0 and y=w0 with time, while Fig. 6 displays the evolution of velocity components. It is found that axial
velocity Vz grows in magnitude with time while the
transverse velocities Vx and Vy exhibit periodic oscillations. An important feature displayed by results depicted in
Fig. 6 Variation of the transverse velocities components in unit V0
where V0 ¼ kC0 for the case in Fig. 4. Both velocities indicate the
rapid onset of oscillatory motions of same time
Figs. 4 and 5 is that changing the sign of angular
momentum quantum number ‘ from (?1) to (-1) causes
change in rotational motion from clockwise to opposite
(counterclockwise) sense.
7. Conclusions
In summary, this paper has dealt with the basic features
that can arise when an atom interacts with an evanescent
Bessel beams possessing orbital angular momentum. The
main effects of such light on atomic behavior are elucidated for two counter-propagating beams where we find
reciprocity between axial and radial motions and the
existence of a static torque and a characteristic dipole
202
potential. The effects of orbital angular momentum have
been discussed here in connection with evanescent Bessel
modes. As we have shown, the orbital angular momentum
of evanescent Bessel modes is explicit and its influence on
atomic motion is more straightforward to interpret. Finally
it is to be noted that such system with highly significant
enhancements of evanescent light fields can be realized by
introducing a metallic film. In evanescent surface plasmon
modes, the fields can be at least an order of magnitude
larger than reported values [15–17].
References
[1] J Durnin Opt. Soc. Am. A Opt. Image Sci. 4 651 (1987)
[2] J Durnin J Miceli and J Eberly Phys. Rev. Lett. 58 1499 (1987)
[3] D McGloin, V Garcés-Chávez and K Dholakia Opt. Lett. 28 657
(2003)
[4] D McGloin and K Dholakia Contemporary Phys. 40 15 (2005)
S Al-Awfi
[5] D McGloin, G Spalding H Melville W Sibbett and K Dholakia
Opt. Commun. 225 215 (2003)
[6] S Al-Awfi Indian J. Phys. 87 53 (2013)
[7] S Kurilkina, V Belyi and N Kazak Opt. Commun. 283 3860
(2010)
[8] A Raghu, Yogesha and S Ananthamurthy Indian J. Phys. 84
1051 (2010)
[9] S Al-Awfi S Bougouffa and M Babiker Opt. Commun. 283 1022
(2009)
[10] V Lembessis, S Al-Awfi, M Babiker and D Andrews Optics 13
064002 (2011)
[11] J Kirk C Bennett M Babiker and S Al-Awfi Phys. Low. Dim.
Struct. 3/4 127 (2002)
[12] S Al-Awfi Plasmonics 8 529 (2013)
[13] E Hinds Adv. Atom. Mole. 2 1(1993)
[14] L Allen, M Babiker, W Lai and V Lembessis Phys. Rev. A54
4259 (1996)
[15] H Raether Surface Plasmons on Smooth and Rough Surfaces and
on Gratings (Berlin: Springer) (1988)
[16] D Andrews M Babiker V Lembessis and S Al-Awfi J. Phys.
Status Solidi RRL 4 241 (2010)
[17] B Barman and K C Sarma Indian J. Phys. 86 703 (2012)