Reduced volume and reflection for optical tweezers with radial Laguerre-Gauss beams
J.-B. B ́eguin,
1
J. Laurat,
2
X. Luan,
1
A. P. Burgers,
1,
∗
Z. Qin,
1, 3
and H. J. Kimble
1,
†
1
Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Laboratoire Kastler Brossel, Sorbonne Universit ́e, CNRS,
ENS-Universit ́e PSL, Coll`ege de France, 4 Place Jussieu, 75005 Paris, France
3
State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China
(Dated: February 12, 2020)
Spatially structured light has opened a wide range of opportunities for enhanced imaging as well
as optical manipulation and particle confinement. Here, we show that phase-coherent illumination
with superpositions of radial Laguerre-Gauss (LG) beams provides improved localization for bright
optical tweezer traps, with narrowed radial and axial intensity distributions. Further, the Gouy
phase shifts for sums of tightly focused radial LG fields extend the range of imaging methods and
permit novel phase-contrast microscopy strategies at the wavelength scale. One application is the
suppression of interference fringes from reflection near nano-dielectric surfaces, with the promise of
improved cold-atom delivery and manipulation.
Structuring of light has provided advanced capabilities
in a variety of research fields and technologies, ranging
from microscopy to particle manipulation [1–4]. Coher-
ent control of the amplitude, phase, and polarization de-
grees of freedom for light enables the creation of engi-
neered intensity patterns and tailored optical forces. In
this context, Laguerre-Gauss (LG) beams have been ex-
tensively studied. Among other realizations, tight focus-
ing with subwavelength features was obtained with radi-
ally polarized beams [5, 6], as well as with opposite or-
bital angular momentum for copropagating fields [7]. LG
beams have also attracted interest for designing novel op-
tical tweezers [8–10]. Following the initial demonstration
of a LG-based trap for neutral atoms [11], various con-
figurations have been explored, including 3D geometries
with “dark” internal volumes [12–14] for atom trapping
with blue-detuned light [15, 16].
For these and other applications of structured light,
high spatial resolution is of paramount importance. How-
ever in most schemes, resolution transverse to the optic
axis exceeds that along the optic axis. For example, an
optical tweezer formed from a Gaussian beam with wave-
length
λ
= 1
μ
m focused in vacuum to waist
w
0
= 1
μ
m
has transverse confinement
w
0
roughly 3
×
smaller than
its longitudinal confinement set by the Rayleigh range
z
R
=
πw
2
0
/λ
. One way to obtain enhanced axial resolu-
tion is known as 4
π
microscopy [17, 18], for which coun-
terpropagating beams form a standing wave with axial
spatial scale of
λ/
2 over the range of
z
R
. However, 4
π
microscopy requires interferometric stability and delicate
mode matching. Another method relies on copropagat-
ing beams each with distinct Gouy phases [19–21], which
was proposed and realized mostly in the context of dark
optical traps, either with two Gaussian beams of differ-
ent waists or offset foci [22, 23], or with two LG modes
∗
Present address: Department of Electrical Engineering, Prince-
ton University, Princeton, New Jersey 08540, USA
†
hjkimble@caltech.edu
−
3
0
3
x
[
μ
m
]
0
1
I
(
x
,
0
,
0)
(a)
−
20
0
20
z
[
μ
m
]
0
1
I
(0
,
0
,
z
)
(b)
−
10
0
10
z
[
μ
m
]
−
2
π
2
π
Ψ
(0
,
0
,
z
)
[
rad
]
(c)
0
5
10
z
[
μ
m
]
0
1
I
(0
,
0
,
z
)
z
1
z
2
z
3
(d)
0
0
+
2
+
4
0
1
0
1
0
0
+
2
+
4
0
1
0
1
FIG. 1. Comparison between the fundamental Gaussian mode
~
E
0
(blue) and the superposition of radial
p
modes
~
E
Σ
(orange)
with
p
= 0
,
2
,
4. The plots are calculated for the paraxial case,
with
w
0
= 1
μ
m and
λ
= 1
μ
m. (a)
x
-line cut transverse inten-
sity profiles. Insets provide the x-y distribution in the focal
plane. (b)
z
-line cut axial intensity profiles. Insets correspond
to the x-z distribution in the
y
= 0 plane. (c) Gouy phases Ψ
0
(blue) for
~
E
0
and Ψ
Σ
(orange) for
~
E
Σ
along the optical axis
z
. (d) Reflection fringes due to a semi-infinite planar surface
(grey), with amplitude reflection coefficient
r
=
−
0
.
8 and fo-
cus at the surface
z
= 0.
z
i
indicate successive maxima. All
intensities are normalized to their peak values in (a,b,d).
of different orders [14]. However, for bright trap config-
urations, a comparable strategy has remained elusive.
In this Letter, we show that superpositions of purely
radial LG modes can lead to reduced volume for bright
optical traps. We also provide a scheme for implemen-
arXiv:2001.11498v2 [quant-ph] 10 Feb 2020
2
(a)
z
=
0
f
~
E
i
~
E
s
~
E
f
x
y
z
x
s
y
s
0
2
π
φ
[
rad
]
−
3
0
3
x
[
μ
m
]
0
1
I
(
x
,
0
,
0)
(b)
−
20
0
20
z
[
μ
m
]
0
1
I
(0
,
0
,
z
)
(c)
FIG. 2. (a)
left-
Calculated transverse phase profile
φ
applied to the spatial phase modulator (SLM) to generate the field
~
E
Σ
.
right-
An incident Gaussian source field
~
E
s
is incident on the SLM. The first order diffracted field
~
E
i
on the exit plane of the
SLM is then focused by an objective lens with effective focal length
f
to form the field
~
E
f
in the focal plane at
z
= 0. (b) Line
cuts along
x
of
|
~
E
f
|
2
in the focal plane for modulation of the SLM with
φ
(
x
s
,y
s
) calculated to generate
~
E
Σ
(red solid line),
ideal target intensity
|
~
E
Σ
|
2
(black dashed line), and Gaussian intensity
|
~
E
0
|
2
(gray line). (c) As in (b), but for line cuts along
z
with
x
=
y
= 0.
tation by way of a spatial light modulator (SLM) for
beam shaping extended beyond the paraxial approxima-
tion into a regime of wavelength-scale traps. Signifi-
cantly, apart from reduced trap volume, our study high-
lights differential Gouy phase shifts at the wavelength
scale as a novel tool for imaging. An application is the
strong suppression of interference fringes from reflections
of optical tweezers near surfaces of nanophotonic struc-
tures, thereby providing a tool to integrate cold-atom
transport and nanoscale quantum optics.
Within the paraxial approximation, we denote the
positive frequency component of the electric field by
~
E
p,i
=
~xu
p
(
x,y,z
;
w
i
)
e
−
ikz
with
x
-oriented linear po-
larization and propagation directed towards negative
z
values with longitudinal wave-vector
k >
0.
The
cylindrically-symmetric complex scalar amplitude
u
p
for
LG beams is as in [24, 25], and given explicitly in [26].
The parameter
w
i
denotes the waist, i.e., 1
/e
2
Gaus-
sian beam intensity radius for the fundamental Gaus-
sian beam with
p
= 0. The azimuthal mode number
l
is dropped with
l
= 0 throughout (i.e., pure radial
LG beams with radial number
p
). For a given optical
frequency, the phase of the field relative to that of a
plane wave propagating along
−
z
(i.e., the Gouy phase
[19–21]) is given by Ψ
p
(
z
) = arg(
~
E
p,i
·
~xe
ikz
) = (2
p
+
1) arctan(
z/z
R,i
), with the Rayleigh range
z
R,i
=
πw
2
i
/λ
.
Although we have analyzed diverse superpositions of
radial LG modes, for clarity we confine our discussion
here to the particular superposition
~
E
Σ
=
~
E
0
+
~
E
2
+
~
E
4
.
Figure 1(a, b) provide the calculated intensity distribu-
tions for the fundamental Gaussian mode
~
E
0
(blue) and
for the superposition (orange), along the
x
-axis in the fo-
cal plane and along the
z
-propagation axis, respectively.
As shown by the line cuts and insets in Figure 1 (a, b),
there is a large reduction in focal volume
V
Σ
for
|
~
E
Σ
|
2
relative to
V
0
for
|
~
E
0
|
2
. Here,
V
= ∆
x
∆
y
∆
z
, with
∆
x,
∆
y,
∆
z
taken to be the full widths at half maxima
for the intensity distributions along
x,y,z
in Figure 1(a,
b), leading to
V
0
/V
Σ
'
22 where
V
0
= 8
.
6
μ
m
3
and
V
Σ
= 0
.
39
μ
m
3
as detailed in [26]. Recall that individual
u
p
modes have identical spatial profiles
|
u
p
(0
,
0
,z
)
|
along
z
. The reduced spatial scale for the superposition
~
E
Σ
re-
sults from the set of phases
{
Ψ
p
(
z
)
}
for
p
= 0
,
2
,
4, with
Gouy phases for the total fields
~
E
0
and
~
E
Σ
shown in Fig-
ure 1(c). The Gouy phase for
~
E
Σ
also leads to suppressed
interference fringes in regions near dielectric boundaries
as shown in Figure 1(d).
A second metric for confinement in an optical tweezer
is the frequency of oscillation for atoms trapped in the
tweezer’s optical potential. The set of trap frequencies
for Cs atoms localized within tweezers formed from
~
E
0
and
~
E
Σ
as in Figure 1(a,b) are presented in [26]. As
expected the trap frequencies have significant increases
for
~
E
Σ
as compared to
~
E
0
.
Various methods have been investigated to produce LG
beams with high purity [27]. A relatively simple tech-
nique consists of spatial phase modulation of a readily
available Gaussian source beam with a series of concen-
tric circular binary phase steps to replicate the phase
distribution of the targeted field
~
E
p
target
with
p
target
>
0
[28]. The maximum purity for this technique is
∼
0
.
8,
with the deficit of
∼
0
.
2 due to the creation of
p
com-
ponents other than the single
p
target
. Moreover, it is
desirable to generate not only high purity LG beams
for a single
p
target
but also arbitrary coherent sums of
such modes, as for
~
E
Σ
. Rather than generate separately
each component from the set of required radial modes
{
p
}
target
, here we propose a technique with a single SLM
that eliminates the need to coherently combine multiple
beams for the set
{
p
}
target
. Our strategy reproduces si-
multaneously both the phase and the amplitude spatial
distributions of the desired complex electric field (and in
principle, the polarization distribution for propagation
phenomena beyond the scalar field approximation).
3
Figure 2(a) illustrates our technique for the case of
the target field
~
E
Σ
. Amplitude information for the sum
of complex fields comprising
~
E
Σ
is encoded in a phase
mask by contouring the phase-modulation depth of a su-
perimposed blazed grating as developed in [29, 30]. For
atom trapping applications with scalar polarizability, the
tweezer trap depth is proportional to the peak optical
intensity in the focal plane, where for the coherent field
superposition
~
E
Σ
, the peak intensity reaches a value iden-
tical to that for
~
E
0
at only 1
/
9 of the invested trap light
power, which helps to mitigate losses associated with the
blazed grating.
Figure 2 shows numerical results for a Gaussian source
field
~
E
s
input to a SLM to create the field
~
E
i
leaving the
SLM.
~
E
i
is then focused by an ideal thin spherical lens
and propagated to the focal plane at
z
= 0 by way of the
Fresnel-Kirchhoff scalar diffraction integral. The result-
ing intensity distributions in the focal plane are plotted
in Figure 2(b,c) (red solid) for comparison with the ideal
~
E
i
=
~
E
0
(grey solid) and ideal
~
E
i
=
~
E
Σ
(black dashed).
These results are encouraging for our efforts to experi-
mentally generate tightly focused radial LG superposi-
tions.
Figures 1 and 2 provide a readily accessible under-
standing of focused LG mode superpositions within the
paraxial approximation. To obtain a more accurate de-
scription for tight focusing on a wavelength scale, we next
consider a vector theory. Using the vectorial Debye ap-
proximation [31, 32] and an input field
~
E
0
with waist
w
0
λ
and polarization aligned along the
x
-axis, we
calculate the field distribution at the output of an apla-
natic objective with numerical aperture of NA = 0
.
7.
The filling factor
F
0
for the objective with pupil radius
R
p
is defined as
F
0
=
w
0
/R
p
, with
F
0
= 0
.
35 for the plots
in Figure 3. These parameters provide a 1
.
3
μ
m waist in
the focal plane for the input
~
E
0
. The corresponding in-
tensity distributions along the x-axis in the focal plane
and along the propagation axis z are presented in Fig-
ure 3 (a), (b), respectively, and are quite similar to those
in Figure 1. Reductions in both transverse and longi-
tudinal widths for input
~
E
Σ
relative to
~
E
0
are evident
even in the vector theory with wavelength-scale focusing.
Results for tweezer volume and oscillation frequency are
detailed in [26] for Figure 3(a), (b), for which
V
0
/V
Σ
'
14
where
V
0
= 24
μ
m
3
and
V
Σ
= 1
.
7
μ
m
3
.
Tight focusing is accompanied by a longitudinal polar-
ization component, which leads to a spatially-dependent
elliptical polarization and to dephasing mechanisms for
atom trapping [33, 34]. Given the local polarization
vector
~
, one can define the vector
~
C
= Im(
~
×
~
?
),
which measures the direction and degree of ellipticity.
|
C
|
= 0 corresponds to linear polarization while
|
C
|
= 1
for circular polarization. Figure 3 (c) provides
C
y
in
the focal plane for the superposition input
~
E
Σ
. Due
to tighter confinement, the polarization gradient reaches
dC
y
/dx
= 1
.
6
/μm
for input
~
E
Σ
, to be compared to
0
.
4
/μm
for
~
E
0
. Fig. 3(d) plots the total field (input plus
reflected) for an infinite planar surface in this condition
of tight focusing.
Excepting panels (d) in Figures 1 and 3, we have thus
far directed attention to free-space optical tweezers for
atoms and molecules. However, there are important set-
tings for both particle trapping and imaging in which the
focal region is not homogeneous but instead contains sig-
nificant spatial variations of the dielectric constant over a
wide range of length scales from nanometers to microns.
Important examples in AMO Physics include recent ef-
forts to trap atoms near nano-photonic structures such
as dielectric optical cavities and photonic crystal waveg-
uides (PCWs) [34–40]. These efforts have been hampered
by large modification of the trapping potential of an op-
tical tweezer in the vicinity of a nano-photonic structure,
principally associated with specular reflection that pro-
duces high-contrast interference fringes extending well
beyond the volume of the tweezer.
Panels (d) in Figures 1 and 3 investigate a strategy
to mitigate this situation by exploiting the rapid spatial
variation of the Gouy phase Ψ
Σ
for the field
~
E
Σ
as com-
pared to Ψ
0
for the field
~
E
0
. As shown in panels (d), the
−
3
0
3
x
[
μ
m
]
0
1
I
(
x
,
0
,
0)
(a)
−
20
0
20
z
[
μ
m
]
0
1
I
(0
,
0
,
z
)
(b)
−
2
0
2
x
[
μ
m
]
−
2
0
2
y
[
μ
m
]
(c)
0
5
10
z
[
μ
m
]
0
1
I
(0
,
0
,
z
)
z
1
z
2
z
3
(d)
0
0
+
2
+
4
0
0
+
2
+
4
0
1
0
1
0
1
0
1
−
0
.
3
0
.
0
0
.
3
FIG. 3. Focused intensity distributions calculated within the
vectorial Debye approximation for inputs
~
E
0
(violet) and
~
E
Σ
(brown). The numerical aperture is NA = 0
.
7 with filling
factor
F
0
= 0
.
35, leading to a focused waist
w
0
'
1
.
3
μ
m for
input
~
E
0
. (a)
x
-line cut transverse intensity profiles. The
insets provide the x-y intensity distribution in the focal plane
z
= 0. (b)
z
-line cut axial intensity profiles. The insets
correspond to the x-z distribution. (c) Polarization ellipticity
C
y
in the focal plane for input
~
E
Σ
. (d) Reflection fringes
from a semi-infinite planar surface, with amplitude reflection
coefficient
r
=
−
0
.
8. All intensities are normalized to their
maximum values in (a,b,d).