Download Structure of Optical Vortices

Structure of Optical Vortices
Jennifer E. Curtis and David G. Grier
Dept. of Physics, James Franck Institute and Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637
(Dated: April 1, 2012)
Helical modes of light can be focused into toroidal optical traps known as optical vortices, which
are capable of localizing and applying torques to small volumes of matter. Measurements of optical
vortices created with the dynamic holographic optical tweezer technique reveal an unsuspected
dependence of their structure and angular momentum flux on their helicity. These measurements
also provide evidence for a novel optical ratchet potential in practical optical vortices.
Beams of light with helical wavefronts focus to rings,
rather than points, and also carry orbital angular momentum [1–3] that they can transfer to illuminated objects [3–6]. When focused strongly enough, such helical
modes form toroidal optical traps known as optical vortices [4, 7, 8], whose properties present novel opportunities for scientific research and technological applications.
For example, optical vortices should be ideal actuators for
microelectromechanical systems (MEMS) [9], and arrays
of optical vortices [10] have shown a promising ability
to assemble colloidal particles into mesoscopic pumps for
microfluidic systems. All such applications will require
a comprehensive understanding of the intensity distribution and angular momentum flux within optical vortices.
This Letter describes measurements of the structure of
optical vortices created with the dynamic holographic
optical tweezer technique [10], and of their ability to exert torques on trapped materials. These measurements
reveal qualitative discrepancies with predicted behavior,
which we explain on the basis of scalar diffraction theory.
A helical mode ψ` (~r) is distinguished by a phase factor
proportional to the polar angle θ around the beam’s axis,
ψ` (~r) = u(r, z) e−ikz ei`θ .
The Structure of Optical Vortices
Jennifer E. Curtis and David G. Grier
Dept. of Physics, James Franck Institute and Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637
(Dated: July 31, 2002)
Helical modes of light can be focused into toroidal optical traps known as optical vortices capable
of localizing and applying torques to small volumes of matter. Measurements of optical vortices
created with the dynamic holographic optical tweezer technique reveal a previously unsuspected
dependence of their structure and angular momentum flux on their helicity, and reveal the influence
of a novel static optical ratchet potential resulting from practical optical vortex implementations.
Beams of light with helical wavefronts have interesting and unusual properties. Such helical modes focus
to bright rings, rather than to discrete intensity maxima. They also carry orbital angular momentum [1, 2],
which they can transfer to illuminated objects [3]. When
brought to a sufficiently strong focus, helical modes of
light form toroidal optical traps known as optical vortices [4–6]. Their ability to trap and apply torques to
small volumes of matter presents novel opportunities for
scientific research and technological applications. For
example, optical vortices should be ideal replacements
for electrostatic and hydraulic motors in microelectromechanical systems (MEMS). Arrays of optical vortices [7]
have shown a promising ability to assemble discrete colloidal particles into mesoscopic pumps for microfluidic
systems. Optical vortex arrays may even be useful for
organizing ultracold clouds of atoms into quantum computing architectures [8]. All such applications will require
a comprehensive understanding of the intensity distribution and angular momentum flux within optical vortices.
This Letter describes measurements of the structure of
optical vortices created with the dynamic holographic
optical tweezer technique [7] and of their ability to exert torques on trapped materials. These measurements
reveal qualitative discrepancies with predicted behavior,
which we explain on the basis of scalar diffraction theory.
A helical mode ψ` (~r) is distinguished by an overall
phase factor proportional to the polar angle θ around
the beam’s axis,
ψ` (~r, z) = u(r, z) e−ikz ei`θ .
(1)
Here, k is the wavelength of light propagating in the ẑ
direction, u(r, z) is the field’s radial profile at position z,
and ` is an integral winding number known as the topological charge. All angles, and thus all phases, appear
along the beam’s axis, r = 0, and the resulting destructive interference cancels the axial intensity. Similarly,
each ray in such a beam has an out-of-phase counterpart
with which it destructively interferes when the beam is
brought to a focus. Constructive interference at a radius
R` from the optical axis yields a bright ring whose width
is comparable to λ, the wavelength of light. In the semiclassical approximation, furthermore, each photon in a
helical mode is found to carry `h̄ angular momentum
phase mask
(a)
2π
illuminator
ϕ
optical
vortex
0
objective
helical
TE
telescope
M
dichroic
mirror
00
video
camera
SLM
(b)
(c)
R`
5 µm
FIG. 1: (a) Schematic diagram of dynamic holographic optical
tweezers creating an optical vortex. The SLM imposes the
phase φ = `θ mod 2π on the incident Gaussian beam. The
resulting helical beam is focused into a optical vortex. The
inset phase mask encodes an ` = 40 optical vortex. (b) Image
of an ` = 40 optical vortex obtained by placing a mirror in
the objective’s focal plane. The central spot is the focused
image of the undiffracted input beam. Inset: magnified view
of one quarter of an ` = 20 optical vortex at reduced power.
(c) Time-lapse image of a single colloidal silica sphere 0.65 µm
in diameter on an ` = 40 optical vortex. Images separated by
1/6 sec. The arrow indicates the direction of motion.
with respect to the beam’s axis [2], so that the torque on
an illuminated surface is proportional to the intensity.
Helical modes can be generated from conventional
beams of light using mode converters.
For example, Hermite-Gaussian modes from appropriately aligned
laser cavities can be converted to helical LaguerreGaussian modes with a pair of cylindrical lenses. More
generally, beams with planar wavefronts have been converted to helical beams using micromachined helical
(1)
Here, ~k = kẑ is the beam’s wavevector, u(r, z) is the
field’s radial profile at position z, and ` is an integral
winding number known as the topological charge. All
phases appear along the beam’s axis, r = 0, and the
resulting destructive interference cancels the axial intensity. Similarly, each ray in such a beam has an out-ofphase counterpart with which it destructively interferes
when the beam is brought to a focus. Constructive interference at a radius R` from the optical axis yields a
bright ring whose width is comparable to λ, the wavelength of light. The semi-classical approximation further
suggests that each photon in a helical mode carries `h̄ orbital angular momentum [2], so that the beam can exert
a torque proportional to its intensity.
Conventional beams of light can be converted into helical modes with a variety of mode converters [11]. Most
implementations yield topological charges in the range
1 ≤ ` ≤ 8 [2, 6]. By contrast, dynamic holographic optical tweezers [10] can create helical modes up to ` = 200,
FIG. 1: (a) Schematic diagram of dynamic holographic optical tweezers creating an optical vortex. The SLM imposes
the phase ϕ(~r) = `θ mod 2π on the incident TEM00 beam,
converting it into a helical beam that is focused into a optical vortex. The inset phase mask encodes an ` = 40 optical
vortex. (b) Image of the resulting optical vortex obtained by
placing a mirror in the focal plane. The central spot is the
diffraction-limited focus of an separate coaxial TEM00 beam
and coincides with the optical axis. (c) Time-lapse image of
a single colloidal sphere traveling around the optical vortex.
and so are ideal for studying how optical vortices’ properties vary with `.
Our system, depicted in Fig. 1, uses a Hamamatsu
X7550 parallel-aligned nematic liquid crystal spatial light
modulator (SLM) [12] to imprint computer-generated
patterns of phase shifts onto the wavefront of a linearly
polarized TEM00 beam at λ = 532 nm from a frequencydoubled Nd:YVO4 laser (Coherent Verdi). The modu-
2
where L`p (x) is a generalized Laguerre polynomial with
radial index p, and w is the beam’s radius [1]. An
LG mode with p = 0 appears as a ring of light whose
p
radius depends on topological charge as R` = w `/2
1
λ
I(r) [Arb.]
12
0
10
R` [µm]
lated wavefront is transferred by a telescope to the back
aperture of a 100 × NA 1.4 oil immersion objective lens
mounted in a Zeiss Axiovert S100TV inverted optical microscope. The objective lens focuses the light into optical
traps, in this case a single optical vortex. The same lens
also forms images of trapped particles that are relayed to
an attached video camera through a dichroic mirror.
The SLM can shift the light’s phase to any of 150 distinct levels in the range 0 ≤ ϕ ≤ 2π radians at each
40 µm wide pixel in a 480 × 480 square array. Imprinting a discrete approximation to the phase modulation
ϕ(~r) = `θ mod 2π onto the incident beam, yields a helical mode with 70 percent efficiency, independent of laser
power over the range studied. Conversion efficiency is
reduced for ` > 100 by the SLM’s limited spatial resolution. Fig. 1(b) is a digital image of an ` = 40 optical
vortex reflected by a mirror placed in the objective’s focal plane. The unmodified portion of the TEM00 beam
travels along the optical axis and comes to a focus in the
center of the field of view. This conventional beam does
not overlap with the optical vortex in the focal plane
and so does not affect our observations. Fig. 1(c) shows
a time-lapse multiple exposure of a single 800 nm diameter colloidal polystyrene sphere trapped on the optical
vortex’s circumference in an 85 µm thick layer of water
between a coverslip and a microscope slide. Angular momentum transferred from the optical vortex drives the
sphere once around the circumference in a little under
2 sec at an applied power of 500 mW. The image shows
11 stages in its transit at 1/6 sec intervals. We studied the same particle’s motions at different topological
charges and applied powers to establish how helicity influences optical vortices’ intensity distribution and local
angular momentum flux.
Observing that a single particle translates around the
circumference of a linearly polarized optical vortex distinguishes the angular momentum carried by a helical
beam of light from that carried by circularly polarized
light. The latter causes an absorbing particle to spin on
its own axis. Observing that the particle instead translates around the optical axis demonstrates that the angular momentum density in a helical beam results from
a transverse component of the linear momentum density,
as predicted [2, 6].
Most observed characteristics of optical vortices have
been interpreted in terms of the properties of LaguerreGaussian (LG) eigenmodes of the paraxial Helmholtz
equation [5, 6]. These have a radial dependence
√ !`
2
2r
2r
r2
`
p
L`p
exp
−
, (2)
up (r, z) = (−1)
w
w2
w2
0
r [µm]
8
10
6
4
2
0
0
50
100
150
`
FIG. 2: Dependence of optical vortex’s radius R` on topological charge `. The dashed line is predicted by Eq. (5) with
no free parameters. Inset: Azimuthally averaged intensity at
` = 40 from the image in Fig. 1(b).
[13]. Practical optical vortices, including the example in
Fig. 1(b), also appear as rings of light and so might be expected to scale in the same way [6]. However, the data in
Fig. 2 reveal qualitatively different behavior. We obtain
R` from digitized images such as Fig. 1(b) by averaging
over angles and locating the radius of peak intensity. Projecting different values of ` reveals that R√` scales linearly
with the topological charge, and not as `.
This substantial discrepancy can be explained by
considering how the phase-modulated beam propagates
through the optical train. The field in the focal plane
of a lens of focal length f is related in scalar diffraction
theory to the field at the input aperture (and thus at
the face of the SLM) through a Fourier transform [14].
Transforming the helical beam first over angles yields
Z Σ
krr0
u` (r, 0) =
r0 u(r0 , −f ) J`
dr0 ,
(3)
f
0
where J` (x) is the `-th order Bessel function of the first
kind, and Σ is the radius of the optical train’s effective aperture. For our system, Σ = 1.7 mm. Setting
u(r, −f ) = u0 for a uniform illumination yields
`
∞
X
ψ` (r, θ, 0)
(−1)(n+ 2 ) [Σkr/(2f )]2(n+`)
i`θ
=
e
. (4)
πΣ2 u0
1 + 2` + n (` + n)! n!
n=0
2
The radius R` of the principal maximum in |ψ` (r, θ, 0)|
is approximated very well by
λf
`
1+
,
(5)
R` = a
πΣ
`0
with a = 2.585 and `0 = 9.80. This also agrees quantitatively with the data in Fig. 2.
3
Even if the mode converter created a pure p = 0 LG
mode, the optical trapping system’s limited aperture, Σ,
still would yield a superposition of radial eigenmodes at
the focal plane [2], and a comparable linear dependence
of R` on `. The superposition of higher-p modes in our
system is evident in the hierarchy of diffraction fringes
surrounding the principal maximum in Fig. 1(b).
This linear dependence on ` leads to scaling predictions for optical vortices’ optomechanical properties with
which we can probe the nature of the angular momentum
carried by helical modes. In particular, a wavelengthscale particle trapped on the circumference of an optical
vortex is illuminated with an intensity I` ∝ P/(2πλR` ),
where P is the power of the input beam, assumingthat
the photon flux is spread uniformly around the vortex’s
circumference in a band roughly λ thick (see Fig. 2). If
we assume that each scattered photon transfers an angular momentum proportional to `h̄, then the particle’s
tangential speed should be proportional to `P/R`2 , and
the time required to make one circuit of the optical vortex
should scale as
The data in Fig. 3(a) show that T` (P ) does indeed
scale according to Eqs. (5) and (6) for larger values of `.
For ` < 40, however, the period is systematically larger
than predicted. Similarly, T` (P ) scales with P as predicted for lower powers, but increases as P increases. In
other words, the particle moves slower the harder it is
pushed. This unexpected effect can be ascribed to the detailed structure of optical vortices created with pixellated
diffractive optical elements; the mechanism presents new
opportunities for studying Brownian transport in modulated potentials.
When projected onto our objective lens’ input pupil,
each of our SLM’s effective phase pixels spans roughly
10λ. Numerically transforming such an apodized beam
reveals a pattern of 2` intensity corrugations, as shown
in Fig. 3(a). These establish a nearly sinusoidal potential
through which the particle is driven by the local angular
momentum flux. We model the intensity’s dependence
on arclength s around the ring as
T` (P ) ∝ R`3 /(`P ).
(6)
√
If the radius had scaled as R` ∝ `, then the particle’s
speed would have been √
independent of `, and the period
would have scaled as `/P . Instead, for ` > `0 , we
expect T` (P ) ∝ `2 /P .
where α is the depth of the modulation, and q = 2`/R`
is its wavenumber. For ` > `0 ≈ 10, q is approximately
independent of `.
This modulated intensity exerts two tangential forces
on the trapped sphere. One is due to the transferred
angular momentum,
I` (s) =
P
(1 + α cos qs) ,
2πλR`
F` (s) = A0
(a)
T`(P) [sec]
8
(b)
β V (s)
10
theory
2
0
−2
−4
6
expt.
0 1 2 3
qs/(2π )
(8)
where we assume a local angular momentum flux of `h̄
per photon. The prefactor A0 includes such geometric
factors as the particle’s scattering cross-section. The
other is an optical gradient force due to the polarizable
particle’s response to local intensity gradients:
Fg (s) = −A0
4
P
(1 + α cos qs) ,
R`
(7)
2πλ ∂I` (s)
P
= A0
α sin qs,
q
∂s
R`
(9)
where sets the relative strength of the gradient force.
Combining Eqs. (8) and (9) yields the tangential force
2
0
0 20 40 60 80 100 0
`
0.5
1.0
1.5 2.0
P [W]
FIG. 3: Time required for colloidal sphere to complete one
circuit of an optical vortex. Dashed curves indicate scaling
predicted by Eq. (6); Solid curves result from fits to Eqs. (12)
and (13). (a) Dependence of T` (P ) on topological charge for
P = 500 mW. Inset: corrugated intensity distribution around
one quarter of the circumference of an ` = 20 optical vortex
measured at reduced intensity, compared with calculated pattern at ` = 40 at the same scale. (b) Dependence of T` (P )
on applied power for ` = 19. The dotted curve includes the
influence of a localized hot spot through Eq. (14). Inset: Potential energy landscape calculated from fits to data in (a)
and (b) for ` = 19 and P = 500 mW.
F (s) = A0
P
(1 − η cos qs) ,
R`
(10)
where we have omitted an irrelevant phase angle, and
1
where η = α(1 + 2 ) 2 . Even if α is much smaller than
unity, both and η can be much larger. In that case,
reducing ` at fixed power increases the depth of the modulation relative to the thermal energy scale kB T , and the
particle can become hung up in the local potential minima. The modulated potential thus increases the effective
drag.
More formally, a particle’s motion along an inclined
sinusoidal potential with strong viscous damping is described by the Langevin equation
γ
ds
= F (s) + Γ(t),
dt
(11)
4
where γ is the viscous drag coefficient and Γ(t) is a zeromean random thermal force. The associated mobility µ
may be expressed as [15]
)
(
1 2
4 η
,
(12)
γµ = 1 + 2 Im
1 2
η`
4 η
`T + i + η`
2 `T + i + . . .
where `T = A0 P η/(4πkB T ) is the topological charge at
which the modulation reaches kB T at power P . Given
this result, the transit time for one cycle should be
T` (P ) = T1
P1 `2
,
P γµ
(13)
where T1 = 2πγR12 /(P1 A0 ) is the expected period for
` = 1 at P = P1 in the absence of modulation. The solid
curve in Fig. 3(a) is a fit to Eqs. (12) and (13) for T1 , `T
and η. The results, T1 P1 /P = 1 msec, η`T = 32, and η =
19, are consistent with the strongly modulated potential
shown in the inset to Fig. 3(b). Rather than smoothly
processing around the optical vortex, the particle instead
makes thermally activated hops between potential wells
in a direction biased by the optical vortex’s torque.
Replacing `/`T with PT /P in Eq. (12) yields an
analogous result for the period’s dependence on applied power for fixed `, as shown in Fig. 3(b). Here,
PT = 4π`ηkB T /A0 is the power at which the modulation
reaches kB T . Using η and P1 T1 obtained from Fig. 3(a),
we find that the sphere’s motions above P = 1.5 W are
slower even than our model predicts. The period’s divergence at high power is due to a localized “hot spot” on
the ` = 19 optical vortex resulting from aberrations in
our optical train. Such hot spots have confounded previous attempts to study single-particle dynamics in helical
beams [3]. Because hot spots also deepen with increasing
power, they retain particles with exponentially increasing
residence times. The total transit time becomes [15]
T (P ) = T` (P ) + TH exp(P/PH ).
(14)
The data in Fig. 3(b) are consistent with TH = 5 msec
and PH = 270 mW. Localization in hot spots becomes
comparable to corrugation-induced drag only for powers above P = 1 W and so does not affect the data in
Fig. 3(a). Consequently, these data offer insights into the
nature of the helical beam’s angular momentum density.
In particular, the simple scaling relation, Eq. (6) is remarkably successful at describing a single particle’s motions around an optical vortex over a wide range of topological charges. This success strongly supports the contention that each photon contributes `h̄ to the local angular momentum flux of a helical beam of light, and not
only to the beam’s overall angular momentum density. It
hinges on our observation that the radius of a practical
optical vortex scales linearly with its topological charge.
The corrugations in apodized optical vortices broadens
this system’s interest. Not only do they provide a realization of the impossible staircase in M. C. Escher’s
lithograph “Ascending and Descending”, but they also
offer a unique opportunity to study overdamped transport on tilted sinusoidal potentials. As a practical Brownian ratchet, this system promises insights germane to
such related phenomena as transport by molecular motors, voltage noise in Josephson junction arrays, and flux
flow in type-II superconductors. Preliminary observations of multiple particles on an optical vortex also suggest opportunities to study transitions from jamming to
cooperativity with increasing occupation.
This work was supported by a grant from Arryx, Inc.,
and in part by the MRSEC program of the National Science Foundation through Grant DMR-9880595 Equipment was purchased with funds from the W. M. Keck
Foundation, and the spatial light modulator was made
available by Hamamatsu Corp.
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