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Supplementary Information for
Reduced volume and reflection for bright optical tweezers with radial Laguerre-Gauss beams
J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble
Corresponding author: H. J. Kimble.
E-mail: hjkimble@caltech.edu
This PDF file includes:
Supplementary text
Figs. S1 to S8
SI References
J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble
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12
www.pnas.org/cgi/doi/10.1073/pnas.2014017117
Supporting Information Text
1. Field expression
For convenience, the expression for the complex scalar amplitude function for conventional Laguerre-Gauss beams is reproduced
here from (1, 2)
u
p
(
r,z
) =
2
π
w
0
w
(
z
)
exp
(
r
2
w
(
z
)
2
)
exp
(
ik
r
2
2
R
(
z
)
)
×
L
0
p
(
2
r
2
w
(
z
)
2
)
exp (
i
Ψ
p
(
z
))
,
[1]
with
w
0
the waist,
z
R
=
πw
2
0
the Rayleigh range,
R
(
z
) =
z
(1 +
z
2
R
/z
2
)
the radius of curvature, and
w
(
z
) =
w
0
1 +
z
2
/z
2
R
the
waist at position
z
. The Gouy phase is given by
Ψ
p
(
z
) = (2
p
+ 1)
arctan
(
z/z
R
)
and
L
0
p
correponds to the associated Laguerre
polynomial.
2. Trap frequencies and dimensions
A. The paraxial limit.
The full widths at half maxima for the intensity distributions along
x,y,z
in Fig. 1 of the main text
are
x
0
= ∆
y
0
=
1
.
17
μ
m
and
z
0
=
6
.
28
μ
m
for
~
E
0
, and
x
Σ
= ∆
y
Σ
=
0
.
51
μ
m
and
z
Σ
=
1
.
5
μ
m
for
~
E
Σ
. One metric
for confinement of an atom of mass
M
in an optical tweezer is the frequency of oscillation near the bottom of the tweezer’s
optical potential. Relative to the trap formed by
~
E
0
in Fig. 1 of the main text, transverse and longitudinal trap frequencies
in the paraxial limit for
~
E
Σ
are increased as
ω
Σ
x
=
ω
0
x
5
and
ω
Σ
z
=
ω
0
z
35
/
3
, with trap depth
U
0
for both
~
E
0
and
~
E
Σ
. For
Cs atoms with
U
0
= 1
mK, wavelength
λ
=
1
μ
m
, and waist
w
0
=
1
μ
m
,
ω
Σ
x
= 2
π
×
178 kHz
and
ω
Σ
z
= 2
π
×
61 kHz
. Here,
ω
0
x
=
4
U
0
/Mw
2
and
ω
0
z
=
2
U
0
/Mz
2
R
are the transverse and longitudinal angular trap frequencies for an ideal Gaussian
mode
~
E
0
in the paraxial limit, with an accuracy better than
1%
as compared to the ground state trap frequencies obtained by
numerical solution of the spatial Schrödinger equation.
B. The vector theory.
With reference to Figure S1(a, b), we investigate trap volumes and frequencies beyond the paraxial using
the vector theory with
F
0
= 0
.
35
, where the FWHMs for the ‘0+2+4’ superposition inputs are
x
Σ
=
0
.
84
μ
m
,
y
Σ
=
0
.
72
μ
m
and
z
Σ
=
2
.
78
μ
m
. These parameters lead 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
,
corresponding to a focal volume
V
0
=
24
μ
m
3
. The ratio of focal volumes defined via FWHMs for inputs with
p
= 0
and the
‘0+2+4’ superposition is then
V
0
/V
Σ
'
14
. Moreover, for red-detuned optical traps associated with the line cuts in Figure S1(a,
b) [Fig. 3(a, b) of the main text], 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
.
The angular trap frequencies
ω
x
,
ω
y
and
ω
z
are obtained by numerically solving the spatial time-independent Schrödinger’s
equation for a single Cesium atom and, for simplicity here, a scalar optical trap potential with depth of
1 mK
for the
6
S
1
/
2
ground state of Cs. Trap frequencies as a function of the filling factor are shown in Fig. S2 (a-b) for both
E
0
(purple curve)
and
E
Σ
(brown curve). Fig. S2(c) shows how the trap evolves for changing filling factor and in particular explains the grey
region in Fig. 4(b) of the main text.
Trap volume comparisons between the
p
= 0
input field case and the ‘0+2+4’ superposition input can be found in Fig. S3.
Here we show the reduction of trap volume of the
E
Σ
input beam over the traditional
E
0
Gaussian input which is evident for
filling factors
0
.
3
< F
0
<
0
.
84
in Fig. S3 (b) and (c). The reader will note that the abrupt changes in the
E
Σ
(blue curve)
confinement and volume, which we have labeled ‘A’ and ‘B’, arise from our use of the FWHM to calculate volume. Using
U/
2
as the depth where we measure the trap width creates this change as the barrier between the intensity region at
z
= 0
and the
side-lobe at
z
'
4
μ
m falls below
U/
2
and makes the FWHM point shift to point ‘A’. As a reference, Fig. S4 shows the field at
the input aperture for increasing
F
0
for both
E
0
and
E
Σ
. Figure S4 shows the effect of increasing the filling factor
F
0
for a
finite size entrance aperture.
3. Atom trajectory simulation
In the Monte Carlo simulation of atom transport from free space tweezers to near surface traps, the atom sample is initialized
from a sample of temperature
100
μ
K
in
1 mK
trap depth with position
'
600
μ
m
away from the surface. The tweezer focus is
first accelerated with acceleration
a
=
1 m s
2
towards the surface for
20 ms
and then moves at constant velocity for
10 ms
before decelerating with acceleration
a
=
1 m s
2
to stop at the surface (
z
=
0
μ
m
) as shown in Fig. S5. To demonstrate the
robustness of our scheme, we also simulated the atom transport with reflection coefficient
r
=
0
.
3
and
r
=
0
.
8
shown in the
Main Text. Animations of typical atom trajectories are available in the accompanying supplementary files.
4. Media for 1D simulation
Here we provide a description of the four movies associated with Fig. S6, which are available at
http://dx.doi.org/10.22002/D1.1343
.
All movies are generated under the paraxial approximation with
w
0
= 1
μ
m
and normalized to a trap depth of
U/k
B
=
1 mK
in absence of the reflecting surface. The black dots represent individual, noninteracting atoms. The motion profile for the
optical tweezer is given in Fig. S5.
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J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble
5. Media for 3D simulation
Beyond simulations in 1D, we have also investigated atom transport in 3D for the moving tweezer potential
U
(
x
(
t
)
,y
(
t
)
,z
(
t
))
,
as shown in a movie at the following link
http://dx.doi.org/10.22002/D1.1346
. This animation shows the intensity of an
~
E
Σ
tweezer with focus moving towards
z
= 0
. The black dots represent individual noninteracting atoms
{
i
}
whose trajectories
(i.e., positions
x
i
(
t
)
,y
i
(
t
)
,z
i
(
t
)
) are driven by forces from
U
(
x
i
(
t
)
,y
i
(
t
)
,z
i
(
t
))
. The parameters are as in Fig. S6, again in the
paraxial approximation with
w
0
= 1
μ
m
and normalized to a trap depth of
U/k
B
=
1 mK
in absence of the reflecting surface.
The 3D results for trajectories are rendered into 2D for the animation by an orthographic projection into the
x,z
plane.
6. Preliminary results for generation of LG superpositions with an SLM
In Fig. S7 and Fig. S8 we show preliminary data for generation of LG superposition beams in the lab as in Fig. 2 of the main
text. The SLM used here is the PLUTO-2-NIR-080 from Holoeye (
https://holoeye.com/
).
J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble
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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)
3
0
3
x
[
μ
m
]
0
1
I
(
x
,
0
,
0)
(c)
20
0
20
z
[
μ
m
]
0
1
I
(0
,
0
,
z
)
(d)
0
0
+
2
+
4
0
0
+
2
+
4
0
1
0
1
0
1
0
1
0
0
+
2
+
4
0
0
+
2
+
4
0
1
0
1
0
1
0
1
Fig. S1.
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
and
two filling factor values are compared: For
F
0
= 0
.
35
(as in main article) (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. For
F
0
= 0
.
45
(c,d). Plotted intensities for inputs
~
E
0
(violet) and
~
E
Σ
(brown)
are normalized to their maximum values. Reproduced from Fig. 3 in the main text for convenience.
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J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble
a)
b)
c)
Fig. S2.
Dependence of radial (x-cut, a) and axial (z-cut, b) trap frequencies (
ω
x
and
ω
z
at the bottom of the trap) as functions of the objective lens filling factor
F
0
, all for fixed
numerical aperture
NA
= 0
.
7
. (Violet) the input field distribution is
~
E
0
; (Brown) the input field distribution is
~
E
Σ
. (c) Shows the evolution of the trap in the axial direction as
the filling fraction increases. The reader will note that as
F
0
changes from 1.0 to 1.26 the trap minimum lies away from
z
= 0
and the trap frequency becomes ill-defined.
J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble
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Δ
x
Σ
Δ
y
Σ
Δ
y
0
Δ
x
0
a)
A
B
b)
c)
d)
A
B
푈푈
/2
A
B
Fig. S3.
Trap volume and dimensions as a function of the filling factor
F
0
. (a) Shows the dimensions in the
x
and
y
directions for a
p
= 0
input field and our ‘0+2+4’ input
beam denoted with the
Σ
subscript. (b) shows the
z
(axial) direction character of the trap for different
F
0
for
p
= 0
(orange curve) and our
E
Σ
beam (blue curve). Here we
see a distinct decrease in trap size for
0
.
3
< F
0
<
0
.
84
and note that this lies below the lower limit of axial confinement for input
p
= 0
beams, which approach the dotted
line for large values of
F
0
. (c) The trap volume for
p
= 0
(orange curve) and our
E
Σ
beam (blue curve). Note the discontinuity in the blue curves in (b) and (c) from points A
to B is a consequence of the merging center peak and side-lobes for
0
.
88
.
F
0
.
1
which leads to a jump in the position of the FWHM along
z
. (d) illustrates how this
aforementioned jump occurs, as
F
0
increases the barrier between the main trap minimum at
z
= 0
μ
m and the side-lobe minimum at
z
'
4
.
2
μ
m drops below
U/
2
abruptly
changing where the FWHM is defined from point A to point B.
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J.-B. Béguin, J. Laurat, X. Luan, A. P. Burgers, Z. Qin and H. J. Kimble