Download Motion Induced by Light: Photokinetic Effects in the Rayleigh Limit

Motion Induced by Light: Photokinetic Effects in the Rayleigh Limit
David B. Ruffner,1, 2 Aaron Yevick,1 and David G. Grier1
1
Department of Physics and Center for Soft Matter Research,
New York University, New York, New York 10003, USA
2
Spheryx, Inc., New York, New York 10016, USA
Structured beams of light can move small objects in surprising ways. Particularly striking examples include observations of polarization-dependent forces acting on optically isotropic objects and
tractor beams that can pull objects opposite to the direction of the light’s propagation. Here we
present a theoretical framework in which these effects vanish at the leading order of light scattering
theory. Exotic optical forces emerge instead from interference between different orders of multipole
scattering. These effects create a rich variety of ways to manipulate small objects with light, socalled photokinetic effects. Applying this formalism to the particular case of Bessel beams offers
useful insights into the nature of tractor beams and the interplay between spin and orbital angular
momentum in vector beams of light, including a manifestation of orbital-to-spin conversion.
Beams of light exert forces and torques on illuminated
objects. The resulting photokinetic effects have attracted
considerable interest because of their role in optical micromanipulation [1]. Gradients in the light’s intensity
give rise to induced-dipole forces that are responsible for
optical trapping [2]. Scattering and absorption give rise
to radiation pressure through transfer of the light’s momentum. The momentum density, in turn, is steered by
gradients in the wavefronts’ phase [3] and also by the
curl of the light’s spin angular momentum density [4–
6]. The repertoire of effects governed by the amplitude,
phase and polarization profiles of a structured beam of
light can be surprising. Circularly polarized beams of
light, for example, influence optically isotropic objects
in ways that linearly polarized light cannot [6, 7]. Appropriately structured beams of light can even transport
small objects upstream against the direction of propagation [8–12], and are known as tractor beams [13].
Here we present a theory for photokinetic effects in
vector beams of light that lends itself to interpretation
of experimental results. It naturally distinguishes strong
and easily observed first-order effects from more subtle
second-order effects. Our formulation focuses primarily on objects that are substantially smaller than the
wavelength of light, the so-called Rayleigh regime, for
which we obtain analytical results. This analysis explains observations of spin-dependent forces acting on
isotropic objects [6], establishes the operating conditions
for propagation-invariant tractor beams [10–12], and predicts as-yet unobserved effects such as orbital-to-spin
conversion in helical Bessel beams. The theory is useful
therefore for developing and optimizing optical micromanipulation tools such as first-order tractor beams [9] and
knotted force fields [14].
DESCRIPTION OF A BEAM OF LIGHT
The vector potential of a monochromatic beam of light
of frequency ω may be decomposed into Cartesian com-
ponents as
A(r, t) =
3
X
aj (r) eiϕj (r) e−iωt êj ,
(1)
j=1
where aj (r) is the real-valued amplitude of the beam
along direction êj at position r and ϕj (r) is the associated phase. Both the amplitude and phase profiles are
readily modified with diffractive optical elements, for example. This factorization is useful therefore because it
expresses A(r, t) in terms of quantities that can be controlled experimentally.
For vector potentials that satisfy the Coulomb gauge
condition, ∇ · A(r) = 0, the electric and magnetic fields
are related to the vector potential by
E(r, t) = −∂t A(r, t) = iω A(r, t)
and
B(r, t) = ∇ × A(r, t).
(2a)
(2b)
The amplitudes of the vector potential’s components contribute to the light’s intensity through
I(r) =
3
ω2 X 2
1
2
|E(r, t)| =
a (r),
2µc
2µc j=1 j
(3)
where µ is the permeability of the medium. With this,
the local polarization may be written as
3
X
ω
ˆ(r) = p
aj (r)eiϕj (r) êj .
2µcI(r) j=1
(4)
This description provides a point of departure for discussing light’s properties and its ability to exert forces.
PROPERTIES OF A BEAM OF LIGHT
The time-averaged linear momentum density carried
by the beam of light is given by Poynting’s theorem,
g(r) =
1
< {E∗ (r, t) × B(r, t)} .
2µc2
(5)
2
where c is the speed of light in the medium. Expressing
g(r) in terms of experimentally controlled quantities,
g(r) =
3
ω X 2
1
a (r) ∇ϕj (r) + ∇ × s(r),
2µc2 j=1 j
2
(6)
p(r, t) = αe E(r, t)
reveals that the light’s momentum is guided both by
phase gradients [3, 6] and also by the curl of the timeaveraged spin angular momentum density [4–6, 15–17]
ω
= {A∗ (r, t) × A(r, t)}
2µc2
i
I(r) ˆ(r) × ˆ∗ (r).
=
ωc
s(r) =
(7a)
(7b)
The first term on the right-hand side of Eq. (6) emerges
from Noether’s theorem and has been identified as the
canonical momentum density of light [17, 18]. The second
describes a spin current [4] and has been described as
a virtual momentum density because its non-divergent
nature limits its ability to couple to matter [17, 18].
A beam of light also can carry orbital angular momentum [4, 15, 19, 20],
3
X
ω
I(r)
j (r) r × ∇∗j (r)
`(r) = i
2µc2
j=1
(8a)
3
ω2 X 2
a (r) [r × ∇ϕj (r), ] ,
=
2µc j=1 j
Rayleigh approximation that the light’s instantaneous
electric field is uniform across the illuminated object’s
volume. The electromagnetic field then induces electric
and magnetic dipole moments in the particle,
(8b)
which is distinct from its spin angular momentum. Equation (8) suggests little connection between `(r) and s(r).
A direct connection has been noted, however, in experimental studies of particles’ circulation in circularly polarized optical traps [6, 7, 21] and has been dubbed spinto-orbital conversion. Quantum-mechanically consistent
definitions of s(r) and `(r) make this connection explicit
[17]. Equations (7) and (8) emphasize the relationship
of these quantities to experimentally controllable amplitudes and phases in the laboratory reference frame. They
consequently offer insight into the converse process by
which the orbital angular momentum content of a beam
imbues the light with spin angular momentum, which has
received less attention.
OPTICAL FORCES IN THE RAYLEIGH REGIME
First-order photokinetic effects
Optical forces arise from the transfer of momentum
from the beam of light to objects that scatter or absorb
the light. They are not simply proportional to g(r), however, but rather arise from more subtle mechanisms. To
illustrate this point, we consider the force exerted by a
beam of light on an object that is smaller than the wavelength of light. Taking this limit permits us to adopt the
m(r, t) = αm B(r, t),
and
(9a)
(9b)
respectively, where αe = αe0 + iαe00 is the particle’s com0
00
plex electric polarizability and αm = αm
+ iαm
is the
corresponding magnetic polarizability.
The induced electric dipole moment responds to the
light’s electromagnetic fields through the time-averaged
Lorentz force [22, 23]
1
< {(p(r, t) · ∇) E∗ (r, t)}
2
1
+ < {∂t p(r, t) × B∗ (r, t)}
2


3


X
1
= < αe
Ej (r, t) ∇ Ej∗ (r, t) .
(10)

2 
Fe (r) =
j=1
Expressed in terms of experimentally controlled parameters, the electric-dipole force has the form [16]
Fe (r) =
3
X
1
µc αe0 ∇I(r) + µc αe00
a2j (r) ∇ϕj (r). (11)
2
j=1
The first term on the right-hand side of Eq. (11) is
the manifestly conservative intensity-gradient force that
is responsible for trapping by single-beam optical traps
[2]. The second describes a non-conservative force [3, 16]
that is proportional to the phase-gradient contribution
to g(r). This is the radiation pressure experienced by a
small particle and is responsible for the transfer of orbital
angular momentum from helical modes of light [24]. Because αe00 > 0 for conventional materials, radiation pressure tends to drive illuminated objects downstream.
Surprisingly, the spin-curl contribution to g(r) has no
counterpart in Fe (r) [23, 25], this missing term having
been canceled by the induced dipole’s coupling to the
magnetic field in Eq. (10) [16]. This means that the radiation pressure experienced by a small particle is not
simply proportional to the Poynting vector [5, 16, 23],
as might reasonably have been assumed. The absence
of any spin-dependent contribution may be appreciated
because s(r) involves products of different components
of the polarization whereas Fe (r) does not [16]. Although spin-dependent forces arise naturally for chiral
objects [26, 27], compelling experimental evidence for
spin-dependent forces acting on isotropic objects [6] cannot be explained as a first-order photokinetic effect [25].
The corresponding result for magnetic dipole scattering has a form [23, 28] analogous to Eq. (11),


3

X
1 
Fm (r) = < αm
Bj (r, t)∇Bj∗ (r, t) .
(12)

2 
j=1
3
This contribution to the total optical force similarly is
comprised of a conservative intensity-gradient term and
a non-conservative phase-gradient term. At the dipole
order of light scattering, the photokinetic forces on optically isotropic scatterers are governed by intensity gradients and phase gradients, and depend only indirectly on
the light’s polarization [29].
Equation (11) also constrains designs for tractor
beams. A beam propagating along ẑ with g(r) · ẑ > 0
would exert a retrograde force if Fe (r) · ẑ < 0. In a
propagation-invariant beam satisfying ∂z I(r) = 0, we exP3
pect j=1 aj (r) ∂z ϕj (r) > 0. The phase-gradient term
in Eq. (11) therefore has a positive axial projection for
conventional materials with αe00 > 0. This means that
propagation-invariant beams cannot act as tractor beams
for induced electric dipoles. A similar line of reasoning yields an equivalent result for Fm (r). The apparent
absence of propagation-invariant tractor beams at the
dipole order of multipole scattering is consistent with
numerical studies [10, 11] that find no pulling forces for
particles much smaller than the wavelength of light.
contributions to optical forces appear at higher orders
of multipole scattering due to interference between the
fields scattered by induced electric and magnetic multipoles. Higher-order effects than those captured by
Eq. (14) may be accentuated in particles larger than the
wavelength of light, particularly through Mie resonances.
Equation (14) also suggests a mechanism by which
propagation-invariant modes can act as tractor beams.
The first two terms on the right-hand side of Eq. (14)
are directed substantially transverse to g(r), and so cannot contribute to retrograde forces. The prefactor of the
third term is negative for conventional materials, and
thus inherently describes a pulling force. The overall axial force can be directed upstream for objects that satisfy
1
µ2
∗
(15)
< {αe αm } > 2 = c αe + αm .
k
c
This condition is not generally met for particles smaller
than the wavelength of light, which is why tractor-beam
action generally is anticipated for larger particles for
which Mie resonances [10, 11, 31] and Fano resonances
[32] favor forward scattering.
Second-order photokinetic effects
APPLICATION TO BESSEL BEAMS
Exotic effects such as polarization-dependent forces
and tractor beam action are restored by the interference
between the scattered fields. For the particular case of
the electric and magnetic dipole fields, such interference
leads to a force of the form [10, 23, 25, 28]
k4
< {p∗ (r, t) × m(r, t)}
12πc0
k3 µ
=
−= {αe∗ αm } ∇I(r)
12π0
Fem (r) = −
(13)
+ 0 ω 2 = {αe∗ αm } < {[A∗ (r, t) · ∇] A(r, t)}
∗
− 2ω < {αe αm } g(r) .
(14)
The first term on the right-hand side of Eq. (14) contributes to the intensity-gradient trapping force. The
second describes a non-conservative force that has been
identified [23], but not widely discussed. It is influenced
by spatial variations in the light’s polarization but is symmetric under exchange of the components’ indexes and
so does not depend on the spin angular momentum density. Together, these terms have been identified with the
imaginary part of the Poynting vector [18, 23] The third
term in Eq. (14) is proportional to the linear momentum
density and therefore includes both phase-gradient and
spin-curl contributions.
Interference between electric and magnetic dipole scattering therefore can account for spin-dependent forces of
the kind observed in optical trapping experiments [6, 7,
21, 30]. It can be shown, moreover, that spin-dependent
As an application of this formalism, we consider the
forces exerted by a monochromatic Bessel beam [33, 34],
whose vector potential may be written in cylindrical coordinates, r = (ρ, θ, z), as
Am,α (r, t) = a0 e−iωt P̂Jm (ρ̃) eimθ+i cos α kz .
(16a)
Here, a0 is the wave’s amplitude, k = ω/c is the
wavenumber of an equivalent plane wave at frequency
ω and ρ̃ = sin α kρ is a dimensionless radial coordinate. Bessel beams are distinguished by the integer winding number, m, and the convergence angle, α, which
are treated as adjustable parameters. The operator P̂
projects the scalar wavefunction into a vector field satisfying the Coulomb gauge condition. The particular projection operator [35],
1
P̂TE
φ = − ∇ × ẑ,
k
(16b)
describes an azimuthally polarized transverse-electric
Bessel beam. Using operator notation not only yields
compact expressions for the vector Bessel beams, but also
clarifies their symmetries.
The convergence angle, α, reduces the axial component of the momentum density relative to that of a plane
wave. An object that scatters light into the forward direction might thereby increase the momentum density
in the beam, and so would have to recoil upstream to
conserve momentum. This is the basis for the proposal
[8, 10, 11] that a Bessel beam can act as a tractor beam.
4
The winding number, m, governs the helical pitch of
the beam’s wavefronts and endows the light with orbital
angular momentum [20, 36] whose axial component,
`(r) · ẑ = m a20
ω 2
J (ρ̃),
2µc2 m
(17)
is proportional to m.
Remarkably, this Bessel beam’s spin angular momentum density,
s(r) = m a20
2
(ρ̃)
1 1 dJm
ẑ,
2µω ρ dρ
(18)
also is proportional to the helical winding number m.
Equation (18) therefore describes orbital-to-spin angular
momentum conversion, an effect that appears not to have
been described previously. Having wavefront helicity control the light’s degree of spin polarization complements
spin-to-orbital conversion [7] in which a beam’s state of
polarization contributes to its wavefronts’ helicity. Like
spin-to-orbital conversion, orbital-to-spin conversion vanishes in the paraxial limit (α → 0).
In considering the forces exerted by Bessel beams, it
is convenient to replace the dipole polarizabilities by
the first-order Mie scattering coefficients, a1 and b1 ,
which are dimensionless and tend to be of the same
order of magnitude for small particles. They are related to the polarizabilities by αe = i6π0 n2m a1 /k 3 and
αm = i6πb1 /(µk 3 ) [10, 37]. With this substitution,
the net optical force due to dipole scattering, F(r) =
Fe (r) + Fm (r) + Fem (r), has an axial component
Distinguishing first- and second-order photokinetic effects is useful for assessing the nature of the forces exerted by beams of light on Rayleigh particles. Trapping
by optical tweezers and the torque exerted by transfer of orbital angular momentum are examples of firstorder effects. Second-order effects include the spin-curl
force, which has been studied experimentally [6] and
observed numerically [38], but has not previously been
explained explicitly [5, 16]. The pulling force exerted
by propagation-invariant tractor beams, including Bessel
beams, similarly turns out to be a second-order effect.
Orbital-to-spin conversion similarly emerges as a property of helical Bessel beams and is the counterpart to
spin-to-orbital conversion, which has been widely discussed [7, 21]. Beyond revealing subtle properties of
vector light waves, this formalism also provides a basis
for developing new modes of optical micromanipulation.
Whereas tractor beams with continuous propagation invariance are inherently limited in their ability to transport small objects, Eq. (11) provides guidance for designing more powerful and longer-range [39] tractor beams
with discrete propagation invariance that can operate at
dipole order. Solenoidal waves [9] appear to be an example of such first-order tractor beams. The theory of
photokinetic effects suggests how to optimize such modes
and how to discover new examples.
This work was supported by the National Science
Foundation through Grant Number DMR-1305875. We
are grateful to Giovanni Milione, Aleksandr Bekshaev
and Konstantin Bliokh for valuable discussions.
2
µFz (r)
= < a1 + cos2 α b1 fm,α
(ρ̃)
2
cos α sin α
3πk 2 a20
2
2
+ < {b1 } sin2 α Jm
(ρ̃) − < {a∗1 b1 } fm,α
(ρ̃), (19)
where
2
fm,α
(ρ̃) =
dJm (ρ̃)
dρ̃
2
+
Jm (ρ̃)
ρ̃
2
.
(20)
The first two terms on the right-hand side of Eq. (19)
arise from electric and magnetic dipole scattering, respectively, and tend to push the scatterer along the +ẑ
direction. The third term, which arises from interference,
generally is negative and describes a pulling force. The
Bessel beam acts as a tractor beam only if this secondorder term dominates the first-order terms. This condition is met by scatterers whose polarizabilities satisfy
Eq. (15). Similar results can be obtained for the radially
polarized Bessel beam. On the basis of these considerations, we conclude that a Bessel beam only acts as a
tractor beam for Rayleigh particles under exceptional circumstances. The formulation of first- and second-order
optical forces reveals that this limitation is a generic feature of propagation-invariant beams of light and is not
specific to Bessel beams.
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