of 9
PHYSICS
Reduced volume and reflection for bright optical
tweezers with radial Laguerre–Gauss beams
J.-B. B
́
eguin
a
, J. Laurat
b
, X. Luan
a
, A. P. Burgers
a,1
, Z. Qin
a,c
, and H. J. Kimble
a,2
a
Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, CA 91125;
b
Laboratoire Kastler Brossel, Sorbonne Universit
́
e, CNRS,
ENS-Universit
́
e PSL, Coll
`
ege de France, 75005 Paris, France; and
c
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of
Opto-Electronics, Shanxi University, Taiyuan 030006, China
Contributed by H. J. Kimble, August 17, 2020 (sent for review July 6, 2020; reviewed by Takao Aoki and Mark Saffman)
Spatially structured light has opened a wide range of opportu-
nities for enhanced imaging as well as optical manipulation and
particle confinement. Here, we show that phase-coherent illumi-
nation 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
can be exploited for phase-contrast strategies at the wavelength
scale. One example developed here is the suppression of interfer-
ence fringes from reflection near nanodielectric surfaces, with the
promise of improved cold-atom delivery and manipulation.
optical tweezer
|
Gouy phase
|
cold atoms
|
Laguerre–Gauss beams
S
tructuring of light has provided advanced capabilities in a
variety of research fields and technologies, ranging from
microscopy to particle manipulation (1–4). Coherent control of
the amplitude, phase, and polarization degrees of freedom for
light enables the creation of engineered intensity patterns and
tailored optical forces. In this context, Laguerre–Gauss (LG)
beams have been extensively studied. Among other realizations,
tight focusing with subwavelength features was obtained with
radially polarized beams (5, 6), as well as with opposite orbital
angular momentum for copropagating fields (7). LG beams
have also attracted interest for designing novel optical tweezers
(8–10). Following the initial demonstration of an LG-based
trap for neutral atoms (11), various configurations have been
explored, including three-dimensional (3D) geometries with
“dark” internal volumes (12–15) for atom trapping with blue-
detuned light (16, 17).
For these and other applications of structured light, high spa-
tial resolution is of paramount importance. However, in most
schemes, resolution transverse to the optic axis largely exceeds
that along the optic axis. For example, a typical bright opti-
cal tweezer formed from a Gaussian beam with wavelength
λ
= 1
μ
m focused in vacuum to waist
w
0
= 1
μ
m has transverse
confinement
w
0
roughly three times smaller than its longitudinal
confinement set by the Rayleigh range
z
R
=
π
w
2
0
. One way to
obtain enhanced axial resolution is known as 4
π
microscopy (18,
19), for which counterpropagating 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 copropagating beams each
with distinct Gouy phases (20–22), which was proposed and real-
ized mostly in the context of dark (i.e., blue-detuned) optical
traps, either with two Gaussian beams of different waists or off-
set foci (23, 24) or with LG modes of different orders (14, 25).
However, for bright (i.e., red-detuned) trap configurations, a
comparable strategy has remained elusive.
In this article, we show that superpositions of purely radial LG
modes can lead to reduced volume for bright optical traps. We
also provide a scheme for implementation by way of a spatial
light modulator (SLM) for beam shaping extended beyond the
paraxial approximation into a regime of wavelength-scale traps.
Significantly, apart from reduced trap volume, our study high-
lights differential Gouy phase shifts at the wavelength scale as a
tool for imaging. An application is the strong suppression of inter-
ference fringes from reflections of optical tweezers near surfaces
of nanophotonic structures, thereby providing a tool to integrate
cold-atom transport and nanoscale quantum optics, a timely topic
of paramount importance for the development of the waveguide
quantum electrodynamics (QED) research field (26).
LG Superpositions in the Paraxial Limit
To gain an intuitive understanding, we first consider superposi-
tions of LG modes within the familiar paraxial approximation.
The positive frequency components of the electric field are
denoted by
~
E
p
,
i
=
~
x u
p
(
x
,
y
,
z
;
w
i
)
e
ikz
with
x
-oriented linear
polarization and propagation directed toward negative
z
values
with longitudinal wave vector
k
>
0
. The cylindrically symmetric
complex scalar amplitude
u
p
for LG beams is as in refs. 27
and 28 and given explicitly in
SI Appendix
. The parameter
w
i
denotes the waist (i.e.,
1
/
e
2
intensity radius at
z
= 0
for a
p
= 0
Gaussian beam). 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 (20–22)] is given by
Ψ
p
(
z
) = arg(
~
E
p
,
i
·
~
x e
ikz
) =
(2
p
+ 1) arctan(
z
/
z
R
,
i
)
,
with
the
Rayleigh
range
z
R
,
i
=
π
w
2
i
.
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
due to its improve-
ment in atom delivery. For example, the coherent superposition
Significance
Nanoscale dielectric devices are capable of mediating long-
range atom–atom interactions using guided mode photons,
as well as long-range photon–photon interactions mediated
by lattices of atoms. Such systems have the potential to pro-
vide new tools for quantum science. However, an outstanding
challenge is the delivery of single atoms to optical traps near
nanophotonic devices. We introduce a capability that could
increase fidelity for atom-lattice assembly near nanophotonic
structures. Other applications include imaging strategies.
Author contributions: J.-B.B., J.L., X.L., A.P.B., Z.Q., and H.J.K. designed research; J.-B.B.,
J.L., X.L., A.P.B., Z.Q., and H.J.K. performed research; J.-B.B., J.L., and X.L. contributed
new analytic tools; J.-B.B., J.L., and X.L. analyzed data; and J.-B.B., J.L., X.L., A.P.B., Z.Q.,
and H.J.K. wrote the paper.y
Reviewers: T.A., Waseda University; and M.S., University of Wisconsin–Madison.y
The authors declare no competing interest.y
This open access article is distributed under
Creative Commons Attribution-NonCommercial-
NoDerivatives License 4.0 (CC BY-NC-ND)
.y
1
Present address: Department of Electrical Engineering, Princeton University, Princeton,
NJ 08540.y
2
To whom correspondence may be addressed. Email: hjkimble@caltech.edu.y
This article contains supporting information online at
https://www.pnas.org/lookup/suppl/
doi:10.1073/pnas.2014017117/-/DCSupplemental
.y
First published October 2, 2020.
www.pnas.org/cgi/doi/10.1073/pnas.2014017117
PNAS
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|
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|
no. 42
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~
E
0
+
~
E
6
gives a narrower axial focal width as compared with that
of
~
E
0
+
~
E
2
+
~
E
4
. However, this comes at the price of strong axial
side lobes, which are a hindrance for the presented atom delivery
scheme due to significant revivals of reflection fringing. For the
sole purpose of free-space trapping, note that the phase modula-
tion strategy illustrated in this work is deterministic. Cold atoms
initially loaded into a conventional
p
= 0 optical tweezer can be
transfered to an LG beam tweezer by progressively turning on
other
p
-mode components.
Fig. 1
A
and
B
provides the calculated intensity distributions
for the fundamental Gaussian mode
~
E
0
(blue) and for the super-
position (orange) along the
x
axis in the focal plane and along the
z
propagation axis, respectively. As shown by the line cuts and
Insets
in Fig. 1
A
and
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 Fig. 1
A
and
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
SI Appendix
.
Recall that the rms radial size
σ
p
of the beam intensity
increases as
σ
p
=
w
i
2
p
+ 1
(29), associated with the LG basis
scale parameter
w
i
(i.e., fixed Rayleigh range). This leads to a
larger divergence angle for higher radial number
p
. Therefore,
transverse clipping and the impact of diffraction effects due to
the constraints of finite aperture stop sizes in any realistic imag-
ing lens system (e.g., finite lens numerical aperture (NA), pupil
radius) need to be included and are thus analyzed further below.
A
CD
B
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 intensity 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 (gray), with amplitude reflection coefficient
r
=
0
.
8
and focus at the surface
z
=
0.
z
i
indicate successive maxima. All intensities
are normalized to their peak values in
A
,
B
, and
D
.
Also relevant is that individual
u
p
modes have identical spa-
tial profiles
|
u
p
(0, 0,
z
)
|
along
z
. The reduced spatial scale for
the superposition
~
E
Σ
results 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. 1
C
. The Gouy phase for
~
E
Σ
also leads to suppressed inter-
ference fringes in regions near dielectric boundaries as shown in
Fig. 1
D
.
Beyond volume, a second metric for confinement in an opti-
cal tweezer is the set of oscillation frequencies for atoms trapped
in the tweezer’s optical potential. Trap frequencies for cesium
(Cs) atoms localized within tweezers formed from
~
E
0
and
~
E
Σ
as in Fig. 1
A
and
B
are presented in
SI Appendix
, with signifi-
cant increases for
~
E
Σ
as compared with
~
E
0
. The values for trap
volume and frequency are provided later with the full model.
Field Superpositions Generated with an SLM
Various methods have been investigated to produce LG beams
with high purity (30). A relatively simple technique consists of
spatial phase modulation of a readily available Gaussian source
beam with a series of concentric circular two-level phase steps to
replicate the phase distribution of the targeted field
~
E
p
target
with
p
target
>
0
(31). The maximum purity for this technique is
0
.
8
,
with the deficit of
0
.
2
due to the creation of
p
components
other than the single
p
target
. Moreover, it is desirable to gener-
ate 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 simultane-
ously both the phase and the amplitude spatial distribution of
the desired complex electric field (and in principle, the polariza-
tion distribution for propagation phenomena beyond the scalar
field approximation).
Fig. 2 shows numerical results for a Gaussian source field
~
E
s
input to an 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.
Fig. 2
A
illustrates our technique for the case of the target field
~
E
Σ
. Amplitude information for the sum of complex fields com-
prising
~
E
Σ
is encoded in a phase mask by contouring the phase
modulation depth of a superimposed blazed grating as devel-
oped in refs. 32 and 33. 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 identi-
cal to that for
~
E
0
at
1
/
3
of the invested trap light power, which
helps to mitigate losses associated with the blazed grating. We
remark that it is not crucial to convert from
~
E
0
with simultane-
ous amplitude and phase modulation strategies (e.g., consider
flat-top beams) (30).
The resulting intensity distributions in the focal plane are plot-
ted in Fig. 2
B
and
C
(red solid) for comparison with the ideal
~
E
i
=
~
E
0
(gray solid) and ideal
~
E
i
=
~
E
Σ
(black dashed). These
results are encouraging for our efforts to experimentally gen-
erate tightly focused radial LG superpositions (
SI Appendix
has
initial laboratory results).
Vector Theory of LG Superpositions
Figs. 1 and 2 provide a readily accessible understanding of
focused LG mode superpositions within the paraxial approxima-
tion. To obtain a more accurate description for tight focusing
on a wavelength scale, we next consider a vector theory. Using
the vectorial Debye approximation (34, 35) and an input field
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́
eguin et al.
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PHYSICS
ABC
Fig. 2.
(
A
,
Left
) Calculated transverse phase profile
φ
applied to the SLM to generate the field
~
E
Σ
. (
A
,
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.
~
E
0
with waist
w
0

λ
and polarization aligned along the
x
axis,
we calculate the field distribution at the output of an aplanatic
objective with fixed
NA = sin
θ
max
.
By convention, the ratio of input waist
w
0,
in
to the pupil radius
R
p
is called the filling factor
F
0
w
0,
in
R
p
, where
R
p
=
f
×
NA
for
focal length
f
.
F
0
is an important parameter for focusing LG
beams at finite aperture, and different filling factors may have
very different beam shapes. The curves in Fig. 3 are calcu-
lated numerically using the Debye–Wolf vector theory for filling
factors
F
0
= 0
.
35
and
F
0
= 0
.
45
, each with fixed numerical aper-
ture
NA = 0
.
7
. These parameters provide
1
/
e
2
intensity radii
w
e
2
=
{
1
.
3, 1
.
0
}
μ
m in the focal plane with the input
~
E
0
for
F
0
=
{
0
.
35, 0
.
45
}
, respectively.
For
p
= 0
and
F
0
= 0
.
35
, the intensity profiles in both radial
and axial directions in the focal plane (violet curves in Fig. 3
A
and
B
) are quite similar to those in Fig. 1
A
and
B
. Likewise, for
input of the “0 + 2 + 4” superposition at the same filling factor
F
0
= 0
.
35
(brown curves in Fig. 3
A
and
B
), the intensity pro-
files are again similar to Fig. 1
A
and
B
and evidence reductions
in both transverse and longitudinal widths relative to the
p
= 0
input even in the vector theory with wavelength-scale focusing.
More quantitatively, with
F
0
= 0
.
35
the full width at half max-
imums (FWHMs) for the
~
E
Σ
input are
x
Σ
= 0
.
84
μ
m,
y
Σ
=
0
.
72
μ
m, and
z
Σ
= 2
.
78
μ
m, corresponding to a focal volume
V
Σ
= 1
.
7
μ
m
3
for the central peak. For the
p
= 0
input with
F
0
= 0
.
35
, the FWHMs of the central peaks for each direction
are
x
0
= 1
.
55
μ
m,
y
0
= 1
.
51
μ
m, and
z
0
= 10
.
3
μ
m, corre-
sponding to a focal volume
V
0
= 24
μ
m
3
. The latter reduces to
V
0
= 5
.
62
μ
m
3
under transverse clipping with
F
0
= 1
(
Optimal
Filling Factors
). The ratio of focal volumes defined via FWHMs
for inputs with
p
= 0
and the
~
E
Σ
superposition is then
V
0
/
V
Σ
'
14
. Moreover, for red-detuned optical traps associated with the
line cuts in Fig. 3
A
and
B
, we find trap frequencies for input
~
E
Σ
to be
ω
Σ
x
= 2
π
×
124
kHz and
ω
Σ
z
= 2
π
×
33
kHz.
However, increasing of the filling factor beyond
F
0
= 0
.
35
for
the 0 + 2 + 4 superposition input does not lead to increases in
trap frequencies nor further reductions of the focal volume. As
shown by the brown curves in Fig. 3
C
and
D
for filling factor
F
0
= 0
.
45
, the central width of the focus is not reduced; rather,
the peak of two side lopes increases. This is not the case for the
p
= 0
input (violet curves in Fig. 3
C
and
D
), for which the fitted
waist
w
0
'
1
μ
m for
F
0
= 0
.
45
as compared with
w
0
'
1
.
3
μ
m for
F
0
= 0
.
35
. The existence of an “optimal” filling factor for super-
*
In the paraxial limit with
l
=
0, LG modes are completely specified by wavelength,
beam waist, and mode order
p
, including beam divergence and rms intensity radius,
as applied to the source and input fields. However, for wavelength-scale focusing
and finite apertures, the full vector theory is required with more complex parametric
dependencies.
positions of LG beams is related to the truncation of the highest
order (i.e.,
p
value) in the superposition, which is discussed in
ref. 36.
Filling Factor Dependence for Trap Frequencies and Dimensions.
An
important operational issue for bright tweezer trapping of atoms
and molecules is confinement near the intensity maxima shown
in Fig. 3. From various metrics, here we choose to quantify local-
ization by way of trap vibrational frequencies near the bottom of
the trapping potential (i.e., the central intensity maximum for a
red-detuned trap), which are modified by pupil apodization and
diffraction effects for focused radial LG beams according to their
radial mode number
p
(37).
A
B
CD
Fig. 3.
Focused intensity distributions calculated within the vectorial Debye
approximation for
x
-polarized inputs
~
E
0
(violet) and
~
E
Σ
(brown). The NA
is NA
=
0
.
7, and two filling factor values are compared. (
A
) The
x
-line cut
transverse intensity profiles for
F
0
=
0
.
35.
Left
(right)
Inset
provides the
x
-
y
intensity distribution in the focal plane
z
=
0 for
~
E
0
(
~
E
Σ
). (
B
) The
z
-line
cut axial intensity profiles for
F
0
=
0
.
35.
Left
(right)
Inset
provides the
x
-
z
intensity distribution in the plane
y
=
0 for
~
E
0
(
~
E
Σ
). (
C
) Same as
A
for
F
0
=
0
.
45. (
D
) Same as
B
for
F
0
=
0
.
45. Plotted intensities are normalized to
their maximum values.
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́
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Fig. 4.
Radial and axial trap frequencies as a function of filling factor
F
0
for
the
p
=
0 input (dashed) and the 0 + 2 + 4 superposition input (solid;
A
) trap
frequencies in
x
(blue) and
y
(red) directions and (
B
) trap frequencies in the
z
direction (black). The gray shaded area represents regions where the trap
center becomes a saddle point with a local intensity minimum. All frequen-
cies are evaluated from the trap origin (
x
=
y
=
z
=
0) with normalized trap
depth at
U
/
k
B
=
1 mK.
As an example, we show in Fig. 4 the variation of trap frequency
near the trap minimum (intensity maximum) around
x
,
y
,
z
= 0
for the
x
-polarized input field distributions
~
E
0
and
~
E
Σ
as a func-
tion of the objective lens filling factor
F
0
within the vectorial
Debye propagation model (34, 35). The angular trap frequencies
ω
x
,
ω
y
, and
ω
z
are obtained by fitting the trap minimum at
z
= 0
to
a harmonic potential and extracting the frequency. Fig. 4
A
corre-
sponds to the transverse trap frequencies
ω
x
and
ω
y
, while Fig. 4
B
is for the axial frequency
ω
z
with corresponding intensity distri-
butions for
F
0
= 0
.
35
,
0
.
45
shown in Fig. 3. The gray region in
Fig. 4
B
arises when the curvature at
z
= 0
becomes antitrapping
for
1
.
0
.
F
0
.
1
.
26
with the trap minimum located away from the
origin. Plots showing the evolution of the trap around these filling
factors can be found in
SI Appendix
.
Note that a choice around the local extremum
F
0
0
.
35
not
only alleviates practical requirements of the objective lens (e.g.,
focal length and working distance) but also permits focused fields
not dominated by diffraction losses. The reduction of both trans-
verse and longitudinal intensity widths for input
~
E
Σ
relative to
~
E
0
displayed in Fig. 3 is now evident in Fig. 4 for trapping frequen-
cies even in the vector theory with wavelength-scale focusing. For
well-chosen filling factors
F
0
,
ω
z
for
~
E
Σ
can be larger than any
possible value for
ω
z
achieved with the
~
E
0
beam (no matter the
value of
F
0
).
Polarization Ellipticity for Tight Focusing.
Necessarily, tight focus-
ing of optical fields is accompanied by a longitudinal polarization
component, which requires a description beyond the atomic
scalar polarizability and which results in spatially dependent
elliptical polarization and dephasing mechanisms for atom trap-
ping (38–41). Given the local polarization vector
ˆ

, one can
define the vector
C
= Im(ˆ

×
ˆ

?
)
, which measures the direction
and degree of ellipticity.
|
C
|
= 0
corresponds to linear polar-
ization, while
|
C
|
= 1
for circular polarization. Fig. 5
A
provides
C
y
in the focal plane for the
~
E
Σ
superposition input. Due to
tighter confinement, the polarization gradient reaches
dC
y
/
dx
=
1
.
6
/
μ
m for
~
E
Σ
superposition input, to be compared with
0
.
4
/
μ
m
for the
p
= 0
input
~
E
0
.
We can further quantify the impact of this ellipticity for trap-
ping atoms by the light shifts (scalar, vector, and tensor shifts) of
the 0 + 2 + 4 superposition for trapping the Cs atom, as shown in
Fig. 5
B
and
C
. Here, we choose the wavelength at a magic wave-
length of Cs (
λ
= 935
.
7
nm) with a given trap depth
U
/
k
B
= 1
mK (for
NA = 0
.
7
and
F
0
= 0
.
35
). Vector light shifts are clearly
observed in the transverse direction in Fig. 5
B
. The trap centers
for different
m
F
levels in
6
S
1
/
2
,
F
= 4
ground state are shifted
away from
x
= 0
by
δ
x
30
nm. As the vector light shift is equiv-
alent to a magnetic field gradient along the
x
direction, it can be
suppressed in experiment by an opposite magnetic gradient as
demonstrated in ref. 39.
Optimal Filling Factors
As already shown in Fig. 4 and discussed in the previous section,
the truncation of LG beams in finite apertures will lead to opti-
mal filling factors for the superposed LG beam input such as the
0 + 2 + 4 superposition. This section focuses on better under-
standing of this optimization, beginning with Fig. 6
A
(36). Here,
we plot the electric field amplitudes for
p
= 0
,
p
= 4
and the
0 + 2 + 4 superposition for filling factor
F
0
= 0
.
35
. For
F
0
=
0
.
35
, the
p
= 4
electric field amplitude (blue curve) is already
partially truncated by the aperture (gray area). Further increase
of the filling factor will misrepresent the
p
= 4
LG beam on the
input pupil, and as a result, the foundation of spatial reduction
due to Gouy phase superposition will have to be reconsidered.
The pupil apodization effects will modify the spatial properties
of the focused radial LG beams according to their radial mode
number
p
(37). In fact, larger filling factor truncates the LG
beams and can generate completely different field profiles (even
bottle beams for a single LG
p
= 1
mode input).
Beyond the intuitive picture of truncation of high-order LG
beams at larger filling factor, we further developed a simple
model based on the analysis of Gouy phase to predict the optimal
filling factor (36). For focusing an LG beam with waist
w
0,
in
by
a lens with focal length
f
(assuming the input waist is at the lens
position), the ABCD matrix from Gaussian optics predicts the
input waist and output waist (
w
0
) are related by
w
0
=
f
λ/π
w
0,
in
.
This leads to a Gouy phase as
d
ψ
G
dz
2
p
+ 1
z
R
=
(2
p
+ 1)
π
λ
w
2
0,
in
f
2
=
(2
p
+ 1)
π
λ
F
2
0
NA
2
.
[1]
In the last step, we use the fact that
F
0
=
w
0,
in
/
f
NA. This sug-
gests for a NA
= 1
system, the phase gradient increases quadrat-
ically with
F
0
(or input waist
w
0,
in
). However, this phase gradient
cannot be arbitrarily high for a finite aperture objective. As
shown in ref. 36, the maximum phase gradient for an objective
with fixed NA is given as
AB
C
Fig. 5.
Polarization ellipticity and vector light shift for the 0 + 2 + 4 input.
(
A
) Polarization ellipticity
C
y
in the focal plane for the 0 + 2 + 4 input with
NA
=
0
.
7 and
f
0
=
0
.
35. (
B
and
C
) The light shifts for a Cs atom at magic
wavelength 935.7 nm with trap depth
U
/
k
B
=
1 mK for transverse (
B
) and
axial directions (
C
). The dashed lines indicate the
m
F
levels in 6
S
1
/
2
,
F
=
4
ground state (red dashed) and in 6
P
3
/
2
,
F
=
4 excited state (blue dashed).
In
B
, we can see the ground-state trap is shifted away for the center by
δ
x
30 nm for the
m
F
=
4 sublevel.
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PHYSICS
2
(1 −
1−NA
2
)
0
=0.35
AB
Fig. 6.
Interpretation of optimal filling factor
F
0
for focusing LG beams with finite aperture objective (36). (
A
) Electric field amplitude of
p
=
0 (red curve),
p
=
4 (blue curve), and 0 + 2 + 4 superposition (green curve) at filling factor
F
0
=
0
.
35. The gray shaded region represents the physical cutoff from the
entrance pupil of the objective with NA
=
0
.
7. (
B
) Phase gradient of the focused field for different LG beam inputs assuming NA
=
1. The horizontal dashed
line indicates the maximum available phase gradient from a finite objective with NA
=
0
.
7. The crossing of the horizontal dashed line and phase gradient
for
p
=
4 LG beam (green curve) corresponds to a filling factor
F
0
'
0
.
35.
(
d
ψ
G
dz
)
max
=
k
(1
1
NA
2
)
.
[2]
By setting equal the result from Eq.
1
to this maximum phase
gradient, we can solve for the optimal filling factor as
F
0,
opt
=
1
NA
(
2
2
p
+ 1
)
1
2
(
1
1
NA
2
)
1
2
.
[3]
For NA
= 0
.
7
,
p
= 4
, this equation predicts an optimal
F
0,
opt
'
0
.
36
. In Fig. 6
B
, we show the plot of phase gradient for
p
= 0
to
p
= 8
based on Eq.
1
. The maximum phase gradient for NA
= 0
.
7
is also indicated with the horizontal dashed line. The crossing of
NA
= 1
phase gradient (colored curves) with the maximum phase
gradient for finite aperture predicts the maximum filling factor
for each
p
mode to preserve its property.
Filling Factors and Trap Volumes.
We have investigated more glob-
ally parameter sets that could provide optimal values for the
filling factor
F
0
, where optimal would be formulated specific to
the particular application, such as imaging or reduced scattering
in the focal volume as investigated in the next section. In apply-
ing optical tweezers for atom trapping, an optimal filling factor
might correspond to the highest trap frequency for a given trap
depth. It is indeed possible to derive a relation between trap fre-
quency
ω
and filling factor
F
0
, at least within the Debye–Wolf
formalism, as described in more detail in ref. 36.
Alternatively, for imaging applications, “optimality” might be
defined by the value of
F
0
that achieves the smallest focal vol-
ume for a given NA. Clearly, the focal volumes for both imaging
and trapping at the wavelength scale are significantly impacted
by diffraction and clipping losses of the input field distributions.
To investigate this question, Fig. 7
A
displays volumes
V
0
and
V
Σ
calculated for
x
-polarized inputs of the fields
~
E
0
and
~
E
Σ
,
with details of our operational definition of “volume” given in
SI Appendix
.
Beginning with
F
0

1
in Fig. 7
A
, we note that
V
0
approaches
a lower limit that corresponds to the well-known diffraction-
limited point spread function for a uniformly filled objective of
NA
= 0
.
7
, which is indeed smaller than
V
Σ
for the field
~
E
Σ
in
the same limit
F
0

1
. However, for more modest values
F
0
'
0
.
3
0
.
7
, the volume
V
0
achieved by
~
E
0
is significantly larger
than
V
Σ
for
~
E
Σ
if one matches the input waist at the same objec-
tive lens entrance for both fields (diagrams of the input fields at
the objective entrance for different filling factors can be found
in Fig. 7
B
and
SI Appendix
). Moreover, the volume
V
Σ
achieved
for
F
0
= 0
.
38
is below even the diffraction limit
V
0
for
F
0

1
.
Importantly, the reduced trap volume for
V
Σ
from
~
E
Σ
derives
from improved axial localization along
z
beyond that achievable
with
~
E
0
for any value of
F
0
(
SI Appendix
).
Beyond this general discussion of volume, the behavior of the
underlying intensities in the focal volume is complicated for both
~
E
0
and
~
E
Σ
, with the former well documented in textbooks and
research literature and the latter much less so. Hence, Fig. 7
B
and
C
panels display intensity distributions for
~
E
Σ
(Fig. 7
B
)
across the source aperture and (Fig. 7
C
) in the focal plane that
F
0
=0.39
F
0
=0.74
F
0
=1.0
F
0
=3.0
i)
ii)
iii)
iv)
|E|
2
-6-4-20246
-6-4-20246
2
0
-2
2
0
-2
x
(
μ
m)
x
(
μ
m)
z
(
μ
m)
z
(
μ
m)
0.8
0.6
0.2
0.4
0
C
B
F
0
=1.0
F
0
=0.39
F
0
=0.74
F
0
=3.0
E
Σ
,inc
0.8
0.6
0.2
0.4
0
A
Fig. 7.
Trap volumes and focal-plane intensity profiles for a range of filling
factors
F
0
, all with NA
=
0
.
7 (36). (
A
) Trap volume as a function of filling
factor for the
p
=
0 input (orange) and the 0 + 2 + 4 superposition input
(blue). Here, the trap volume is defined as
V
= ∆
x
y
z
, with
x
,
y
,
z
the full widths at half maxima for the intensity distributions along
x
,
y
,
z
.
The smallest trap volume we find is
V
Σ
=
1
.
7
μ
m
3
at
F
0
=
0
.
35. (
B
) The input
electric field profile for the 0 + 2 + 4 superposition at filling factors
F
0
=
0
.
39,
F
0
=
0
.
74,
F
0
=
1, and
F
0
=
3
.
0. (
C
) Intensity profiles near the focus at
filling factors
F
0
=
0
.
39,
F
0
=
0
.
74,
F
0
=
1, and
F
0
=
3. Note that the airy rings
observed in
i–iii
are still present in
iv
, although not revealed due to their
small size and the limited contrast.
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are much more complicated than those for
E
0
and which exhibit
structure in regions well outside the central maxima (including
strong side lobes and extended axial variations) (
SI Appendix
).
Such side lobes can introduce atom heating for transport of cold
atoms in moving tweezers, thereby reducing atom delivery effi-
ciency from free space to dielectric surfaces. Two other filling
factors
F
0
= 1
and
F
0
= 3
are also presented in Fig. 7
B
and
C
.
Note that
F
0
= 1
in Fig. 7
C
,
iii
corresponds to a flattened trap
intensity in the axial direction with properties similar to Bessel
beams (42), with plots of this effect and discussion found in
SI
Appendix
. Finally,
F
0
= 3
in Fig. 7
C
,
iv
approaches the limit of a
uniform input with a well-known diffraction-limited spot size.
While we have considered here two examples of optimization
by way of trapping frequencies and volumes in tweezer traps,
similar analyses can be formulated to optimize other metrics
(36). Indeed, the systematic search for optimal values of
F
0
briefly described in this section immediately found the peaks
shown in Fig. 4 for atom trapping, which we first identified by
a considerably more painful random search. Moreover, because
trap volumes for focused red traps scale as
V
trap
(
ω
z
ω
x
ω
y
)
1
,
the expressions for trap frequencies in the axial and transverse
directions can be combined to find optimal filling factors
F
0
to
minimize trap volume around the center of a trap for a given
input profile
E
in
for comparison with more global measures of
volume (e.g., FWHM) (
SI Appendix
).
LG Beams Reflected from Dielectric Nanostructures
Excepting Fig. 1
D
, we have thus far directed attention to free-
space optical tweezers for atoms and molecules. However, there
are important settings for both particle trapping and imaging
in which the focal region is not homogeneous but instead, con-
tains significant spatial variations of the dielectric constant over
a wide range of length scales from nanometers to micrometers.
Important examples in atomic, molecular, and optical (AMO)
physics include recent efforts to trap atoms near nanophotonic
structures such as dielectric optical cavities and photonic crys-
tal waveguides (PCWs) (26, 39, 43–46). These efforts have been
hampered by large modification of the trapping potential of an
optical tweezer in the vicinity of a nanophotonic structure, prin-
cipally associated with specular reflection that produces high-
contrast interference fringes extending well beyond the volume
of the tweezer.
The magnitude of the problem is already made clear in the
paraxial limit by the blue curve in Fig. 1
D
. The otherwise
smoothly varying tweezer intensities in free space, shown in Fig. 1
A
and
B
, become strongly modulated in Fig. 1
D
by the reflec-
tion of the tweezer field from the dielectric surface. Given that
the goal for the integration of cold atoms and nanophotonics
is to achieve one-dimensional (
1
D) and two-dimensional (
2
D)
atomic lattices trapped at distances
z
.
λ/
10
from surfaces and
that one interference fringe in Fig. 1 spans
z
=
λ/
2
, it is clear
that free-space tweezer traps cannot be readily employed for
direct transport of atoms along a linear trajectory in
z
to the
near fields of nanoscale dielectrics without implementing more
complicated trajectories. These trajectories not only require the
tweezer spot to traverse along
z
but
x
or
y
as well (47). Further
insight for direct transport along
z
is provided by the animations
in
SI Appendix
for the evolution of the intensity of a conventional
optical tweezer as the focal spot is moved from an initial distance
z
i

λ
to a final distance
z
0
= 0
at the dielectric surface. Placing
atoms at distances
z
.
λ/
10
from dielectric surfaces is possible by
combining LG beam optical tweezers and utilizing guided modes
(GMs) of the dielectric structure. These GMs can be configured
in such a way to attract the atoms via the dipole force to sta-
ble trapping regions
z
.
λ/
10
from the dielectric. Such trapping
configurations are discussed in ref. 45.
That said, Fig. 1
D
investigates a strategy to mitigate this diffi-
culty by exploiting the rapid spatial variation of the Gouy phase
Ψ
Σ
for the field
~
E
Σ
as compared with
Ψ
0
for the field
~
E
0
. As
shown by the orange curve in Fig. 1
D
, the contrast and spatial
extent of near-field interference are greatly reduced for
~
E
Σ
due
to rapid spatial dephasing between input and reflected fields.
To transition this idea into the regime of nanophotonic struc-
tures with tightly focused tweezer fields on the wavelength scale,
we start with a free-space LG beam in the paraxial limit with
waist much larger than the optical wavelength,
w
0

λ
. The opti-
cal field for this initial LG beam is first “sculpted” with the
SLM and then tightly focused as in Fig. 2 with fields in the
free-space focal volume calculated from the Debye–Wolf for-
malism and serving as a background field without scattering. We
then solve for the scattered field in the presence of a dielectric
nanostructure in the focal volume.
An example to validate directly the possibility of reduced
reflection and “fringe” fields for wavelength-scale optical
ABC
Fig. 8.
Simulation of aligning a tightly focused LG beam to reflect and scatter from an APCW directed out of the page (indicated by the gray rectangles)
with focal position aligned to the geometric center of the APCW (
A
) for the input field distribution of a
p
=
0 Gaussian beam with initial polarization
along
x
, (
B
) for the input field distribution of the 0 + 2 + 4 superposed LG beam with polarization along
x
, and (
C
) for the input field distribution of the
0 + 2 + 4 superposed LG beam with polarization along y. All three plots are calculated with the background field derived from the Debye–Wolf integral with
NA
=
0
.
7 and
F
0
=
0
.
35 (36).
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PHYSICS
AC
B
Fig. 9.
(
A
) Optical tweezer focused on a 2D photonic crystal slab. (
B
) Fitting of the Comsol simulated field (dashed line) with the Debye–Wolf evaluated
field reflecting from a planar surface (red solid line). (
C
) The Comsol simulated field of the LG 0 + 2 + 4 superposition
E
Σ
reflected from a 2D PCW that is
composed of a dielectric slab with a 2D square lattice of holes as in ref. 49.
tweezers near nanophotonic devices is presented in Fig. 8, which
displays intensity distributions calculated for (Fig. 8
A
) a focused
p
= 0
Gaussian beam input and (Fig. 8
B
and
C
) a focused
0 + 2 + 4 superposed LG beam aligned to an alligator pho-
tonic crystal waveguide (APCW) (48) for NA
= 0
.
7
and
F
0
=
0
.
35
. This result confirms the spatial reduction of fringe fields
from the superposition of LG beams near complex dielectric
nanostructures.
We stress that our methods for finding the reflected and scat-
tered fields for nanophotonic devices illuminated by coherent
sums of LG fields can be readily extended from
1
D to
2
D slab
PCWs (49, 50). One such result for a
2
D square lattice (49) has
been calculated with the vector theory and is displayed in Fig. 9,
again with reduced reflected fields brought by interference from
the range of Gouy phases.
Atom Transport to a Photonic Crystal
To investigate the efficiency for atom transport from free-space
optical tweezers to reflective traps near dielectric surfaces, we
have performed Monte Carlo simulations of atom trajectories by
Fig. 10.
Single frame from an animation of atom delivery to the APCW (in red) by way of a moving optical tweezer. (
Left
) Gaussian beam
E
0
. (
Right
)
Coherent superposition of LG beams
~
E
Σ
. In the displayed frame, atoms are absent to better highlight the intensity distributions of the two optical tweezers.
The white arrows indicate the direction of motion of the tweezer focus as implemented in refs. 51 and 52, and the grids are 10
×
10
μ
m. The full animation
depicts noninteracting atoms as white “dots” as might have been initially loaded and cooled into the tweezers far from the APCW (20 atoms for each
tweezer). A movie can be found at the following link:
https://data.caltech.edu/records/1446.
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