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Optical theorem for acoustic
non-diffracting beams and application to
radiation force and torque
Likun Zhang1,∗ and Philip L. Marston2
1 Department
of Physics and Center for Nonlinear Dynamics, University of Texas at Austin,
Austin, TX 78712 USA
2 Department of Physics and Astronomy, Washington State University, Pullman, Washington
99164-2814 USA
∗ [email protected]
Abstract: Acoustical and optical non-diffracting beams are potentially
useful for manipulating particles and larger objects. An extended optical
theorem for a non-diffracting beam was given recently in the context of
acoustics. The theorem relates the extinction by an object to the scattering
at the forward direction of the beam’s plane wave components. Here we use
this theorem to examine the extinction cross section of a sphere centered
on the axis of the beam, with a non-diffracting Bessel beam as an example.
The results are applied to recover the axial radiation force and torque on the
sphere by the Bessel beam.
© 2013 Optical Society of America
OCIS codes: (350.4855) Optical tweezers or optical manipulation; (290.2200) Extinction;
(290.4020) Mie theory; (290.5850) Scattering, particles.
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Received 17 Jun 2013; revised 1 Aug 2013; accepted 2 Aug 2013; published 9 Aug 2013
1 September 2013 | Vol. 4, No. 9 | DOI:10.1364/BOE.4.001610 | BIOMEDICAL OPTICS EXPRESS 1610
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1.
Introduction
An idealized non-diffracting (optical or acoustic) beam is a beam whose transverse intensity
pattern has the feature of propagation-invariance [1–5]. Beams which are locally approximately
non-diffracting are potentially useful for particle manipulation.
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One application is the possibility of generating pulling forces. Situations giving pulling
forces for spheres in non-diffracting Bessel beams have been computed in acoustics [6–10]
and in optics [11–13]. It was noticed early [6–8] that acoustical situations predicting to give
negative forces corresponded to a significant reduction of the far-field scattering into the backward hemisphere relative to the forward hemisphere. An analysis of momentum projection and
conservation associated with optical far-field scattering [11] motivated an analogous analysis
in the acoustical case [10, 14] which shows the relationship between the asymmetry in the scattering and the direction of the radiation force. There have been related theoretical discussions
of momentum projection in optics [15] followed by a demonstration of negative optical forces
in a beam closely resembling intersecting plane waves [16].
There has been significant recent interest in broader applications of acoustical radiation
forces and torques for the manipulation of objects of various sizes [17–22]. Acoustic [23, 24]
and optical [25] beams with an extra azimuthal phase dependence exp(imφ ), called vortex
beams, have a helicoidal wavefront and carry orbital angular momentum. This feature of angular momentum transport allows the beam to exert a torque to rotate an object. Acoustic vortex
beams were analyzed beyond the paraxial approximation in [24] to clarify an analogy with optical vortex beams; the radiation torque on a symmetric object centered on the beam’s axis was
related to the absorption of power.
It has long been beneficial to consider possible areas of overlap between some related issues arising in acoustical and optical fields of research [17, 23–27]. In this paper we illustrate
the application of an extended optical theorem on acoustic radiation forces and torques associated with a non-diffracting beam. The optical theorem for an incident plane wave is known
as one of the central theorems in scattering theory [28–30]; the theorem relates the extinction
section to the complex scattering amplitude at the forward direction. An extended theorem for
a non-diffracting beam was given recently in the context of acoustics [14]. Here we use this
extended optical theorem to examine the extinction cross section on a sphere centered on the
axis of a non-diffracting beam, in particular, to examine the extinction for a sphere centered on
a Bessel beam as an example. The results, together with a prior result of an asymmetry factor
of scattering, are then applied to recover the axial radiation force and torque given in [10, 24].
2.
Optical theorem of a non-diffracting beam: Review
For a non-diffracting beam (with a speed c0 in the medium, a frequency ω, and a wavenumber
k = ω/c0 ) propagating along the z axis with an axial wavenumber κ = k cos β ,
ψi = ψi0 (x, y; β ) exp(iκz − iωt),
(1)
and scattered by an object centered at the origin (refer to Fig. 1), resulting in a scattered far
field with a complex amplitude As (θ , φ ),
ψs = ψ0 As (n) exp(ikr − iωt)/r,
(2)
an extended optical theorem for the extinction cross section, denoted by σext , was derived in
the context of acoustics by [14] as an azimuthal angle integration
Z 2π
1
4π
Im
g∗ (φ )As (n(β , φ ))dφ ,
(3)
σext =
k
2π 0
where Im denotes the imaginary part, and the asterisk denotes the complex conjugate. This
theorem (3) connects the extinction to the scattering at the direction n(β , φ ) by using an angular
function g(φ ) of the beam. This angular function g(φ ) relates to the beam’s profile through a
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1 September 2013 | Vol. 4, No. 9 | DOI:10.1364/BOE.4.001610 | BIOMEDICAL OPTICS EXPRESS 1612
reduced Whittaker integral
ψi0 (x, y; β ) =
ψ0
2π
Z 2π
g(φ ) exp[iµ(x cos φ + y sin φ )]dφ , with µ = k sin β .
(4)
0
This integral (4) represents the beam (1) as a superposition of plane wave components, whose
relative amplitude and phase are given by the function g(φ ), and whose wave-vectors
k(µ cos φ , µ sin φ , κ) = k(sin β cos φ , sin β sin φ , cos β ) = kn(β , φ ),
(5)
have a tilted conical angle β relative to the beam’s axis (refer to Fig. 1). The scattering in the
theorem (3), As (n(β , φ )), is the scattering at the forward direction of the beam’s plane wave
component at the azimuthal angle φ .
Fig. 1. The radiation force and/or torque on an object centered on the axis of an idealized non-diffracting beam relates to the extinction by the object via scattering and/or absorption. The beam, propagating along the z axis, is characterized by an angular function
g(φ ) (refer to text) and by a conical angle β determining the direction of wave vectors
k(β , φ ) = kn(β , φ ) of the beam’s plane wave components. The scattering and/or absorption are characterized by a far-field scattering complex amplitude As (n(θ , φ )).
The extinction results from both scattering and absorption. One can write σext = σsca + σabs ,
with σsca and σabs denoting the cross sections of scattering and absorption, respectively. The
scattering cross section, σsca , is given by the integral of the scattering coefficient over the whole
solid angle element dΩ = sin θ dθ dφ as,
Z
σsca =
|As (n)|2 dΩ.
(6)
On taking the limits, β = 0 and g(φ ) = 1, our extended theorem (3) reduces to the wellknown theorem for a plane wave [28–30],
σext =
4π
Im [As (0, 0)] ,
k
(7)
where As (0, 0) is the forward scattering of the incident plane wave ψi = ψ0 exp(ikz − iωt).
3.
Application to extinction by a sphere
Here we use the extended theorem (3) to examine the extinction cross section of a sphere
centered on an idealized Bessel beam of arbitrary order m,
ψi = ψ0 im Jm (µρ) exp(iκz + imφ ),
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(8)
Received 17 Jun 2013; revised 1 Aug 2013; accepted 2 Aug 2013; published 9 Aug 2013
1 September 2013 | Vol. 4, No. 9 | DOI:10.1364/BOE.4.001610 | BIOMEDICAL OPTICS EXPRESS 1613
p
where ρ = x2 + y2 , and m is an integer with m = 0 as a special case giving the ordinary Bessel
beam. The angular function of this beam is [10, 14]
g(φ ) = exp(imφ ).
(9)
For a sphere of radius a centered on the axis of the beam (8), the far-field scattering was
given in [10] in terms of a partial wave expansion as
As (n(θ , φ )) =
(n − m)!
exp(imφ ) ∞ (sn − 1)
∑ 2 (2n + 1) (n + m)! Pnm (cos β )Pnm (cos θ ),
ik
n=|m|
(10)
where the functions Pnm are associated Legendre functions (see also [31]). The functions sn
in the partial wave coefficients (sn − 1)/2 are the same as that for a plane wave scattering
[2]. These coefficients are known for different types of objects, as determined by boundary
conditions. Notice that |sn | ≤ 1 and only for non-absorptive scattering are all coefficients |sn |
equal to unity.
By using (9) and (10), and letting θ = β , it immediately follows from the optical theorem
(3) that
π ∞
(11)
σext = 2 ∑ (2n + 1)[Pnm (cos β )]2 Re[2(1 − sn )],
k n=|m|
where Re denotes the real part. Given that |sn | ≤ 1, one always has Re[2(1 − sn )] ≥ 0, and
hence σext ≥ 0 as required; only in the limit of no scattering, sn = 1, does no extinction σext = 0
occur. The cross section has the dimension of area. For an object being a sphere of radius a, the
dimensionless efficiency factor becomes,
Qext ≡ σext /(πa2 ) =
∞
1
(n − m)! m
(2n + 1)
[P (cos β )]2 Re[2(1 − sn )],
∑
2
(ka) n=|m|
(n + m)! n
(12)
which recovers the result in [10] derived from an analytical integration of energy flux (where
the scattering coefficients were written as (sn − 1)/2 = αn + iβn , with αn and βn being the real
and imaginary parts).
The separation of the extinction (12) into the scattering and the absorption is straightforward.
One may rewrite the coefficients Re[2(1 − sn )] as
Re[2(1 − sn )] = (|sn − 1|2 ) + (1 − |sn |2 ).
(13)
The first part, |sn − 1|2 , associated with the modula of scattering coefficients, corresponds to
the contribution from the scattering. The second part, 1 − |sn |2 , associated with the difference
of |sn |2 from unity, corresponds to the contribution from the absorption. Hence it follows from
(12) and (13) that
Qsca ≡ σsca /(πa2 ) =
∞
1
(2n + 1)[Pnm (cos β )]2 (|sn − 1|2 ),
∑
(ka)2 n=|m|
(14)
Qabs ≡ σabs /(πa2 ) =
∞
1
(2n + 1)[Pnm (cos β )]2 (1 − |sn |2 ).
∑
(ka)2 n=|m|
(15)
which again recovers the results in [10] derived from an analytical integration of energy flux of
corresponding fields.
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In the special case of non-absorptive scattering (|sn | = 1), Qabs = σabs /(πa2 ) = 0. In this case
using the phase shifts δn with
sn = exp(i2δn ),
(16)
both the extinction in (12) and the scattering in (14) relate to the phase shifts as
Qext,sca = σext,sca /(πa2 ) =
∞
4
(2n + 1)[Pnm (cos β )]2 sin2 (δn ).
∑
2
(ka) n=|m|
(17)
Lastly, the corresponding powers are given as
Pext,sca,abs = I0 σext,sca,abs = I0 (πa2 )Qext,sca,abs ,
(18)
where I0 , a function of the amplitude ψ0 in (8), characterizes the intensity of the incident beam.
In the context of acoustics, if ψ represents the velocity potential field, one has the acoustic
intensity I0 = (ρo co /2)(kψ0 )2 .
4.
Application to the axial radiation force on a sphere
By the axial projection of momentum, the axial radiation force exerted by the general nondiffracting beam (4) on an object of arbitrary shape and location was given in [14] as
Fz = c−1
0 (Pext cos β − Psca hcos θ is )
(19)
or in terms of a dimensionless force function Yp as
Fz = πa2 c−1
0 I0Yp , Yp = Qext cos β − Qsca hcos θ is ,
(20)
where the factors, cos β and hcos θ is , correspond to the axial projections of extracted momentum and scattered momentum [10, 11, 14], respectively, and
R
hcos θ is =
cos θ |As (n)|2 dΩ
R
.
|As (n)|2 dΩ
(21)
The factor hcos θ is is an asymmetry parameter of the scattering: hcos θ is is positive or negative when the scattering at the forward hemisphere is stronger or weaker relative to the scattering at the backward hemisphere. An inspection of (20) shows why even in the idealized case
of negligible absorption, conditions to achieve negative forces usually require that the conic
angle β be large: from the form of (20) the asymmetry must lie between −1 and 1. For a Bessel
beam with m = 0 the plane wave limit is recovered by taking β = 0. It follows from the form
of (20) that the force must be non-negative in that limit with or without absorption. By taking
Qext = Qsca + Qabs , an implicit term in (20) associated with absorption, Qabs cos β , degrades the
negative force [10].
The cross section and efficiency factor associated with the scattering asymmetry factor
hcos θ is in (21) for a sphere of radius a are
σasym.sca ≡
Z
cos θ |As (n)|2 dΩ, Qasym.sca ≡ σasym.sca /(πa2 ).
(22)
When a sphere is on the axis of the Bessel beam (8), the efficiency of the scattering asymmetry,
Qasym.sca , can be read from [10] (Qasym.sca = −Y1 therein) as
Qasym.sca =
σasym.sca
=
πa2
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2
ka
2
∞
∑ [2(αn αn+1 + βn βn+1 )]
n=|m|
(n − m + 1)! m
m
P (b)Pn+1
(b). (23)
(n + m)! n
Received 17 Jun 2013; revised 1 Aug 2013; accepted 2 Aug 2013; published 9 Aug 2013
1 September 2013 | Vol. 4, No. 9 | DOI:10.1364/BOE.4.001610 | BIOMEDICAL OPTICS EXPRESS 1615
Together the extinction (12) from the optical theorem and the asymmetry (23) give the dimensionless force function in (20) as
Yp
=
cos β ∞
(n − m)!
∑ (−4αn )(2n + 1) (n + m)! [Pnm (cos β )]2
(ka)2 n=|m|
2 ∞
2
(n − m + 1)!
m
−
(cos β ), (24)
∑ [2(αn αn+1 + βn βn+1 )] (n + m)! Pnm (cos β )Pn+1
ka n=|m|
which recovers the result in [10] derived from the analytical integration of the axial projection of
stress. Equation (24) was derived to associate with the momentum projection [10]. Its analytical
equivalence with prior results given in [6–9] was noted in [10].
5.
Application to the radiation torque on a sphere
In an axisymmetric field (traveling or standing waves) with an azimuthal phase dependence
exp(imφ ), where the integer m is the topological charge, the axial radiation torque on an axisymmetric object centered on the axis of the field was revealed in [24] to be associated with
the absorption of energy as
Tz =
m
Pabs ,
ω
(25)
or in terms of a dimensionless torque efficiency factor QT for a sphere of a radius a as,
Tz = πa2 I0 QT /ω, QT = mQabs .
(26)
When a sphere is placed on the axis of the vortex Bessel beam (8) with a non-zero integer m,
using the absorption (15) from the optical theorem, it has
QT = mQabs =
∞
(n − m)! m
m
(2n + 1)
[P (cos β )]2 (1 − |sn |2 ),
∑
2
(ka) n=|m|
(n + m)! n
(27)
which were published in [10, 24] (and presented even earlier in [32, 33]), where the absorption
was derived from an analytical integration of energy flux.
In [24], Eq. (25) was generalized to the torque on any axisymmetric object centered in any
vortex wave field (traveling or standing waves; where the derivation started from the conservation of of angular momentum [34]). Experiments in [27] confirm (25) for a disk shaped object
in a vortex wave field having an adjustable topological charge m. For most cases series expansions giving the absorption, like our (15), are not currently available. While some related work
has appeared [35, 36], the reader is cautioned the left side of Eq. (18) in [35] is incorrect. Other
corrections to [35] were given in an erratum published by Mitri et al. late in 2012. Notice also
that since the analysis in [24] included the standing wave case, (25) applies to the standing wave
case examined in [35].
6.
Conclusions and discussion
The applications of the extended acoustic extinction theorem (3) illustrated here concern a
sphere placed on the axis of an idealized Bessel beam of arbitrary integer order m. In that case
the resulting predictions for the extinction cross section and associated efficiency factor (12)
agree with results derived a different way in [10]. Also the normalized radiation force (24)
agrees with the result from analytical integration of the axial radiation stress projection in [10]
(equivalent to prior results in [6–9]). The torque and absorption efficiencies in (27) agree with
the results from analytical integration of angular momentum and energy flux in [10, 24].
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The extinction theorem (3) and the associated axial stress projection relation (19) involving
the scattering asymmetry (21) neither require the object to be a sphere nor to be centered on
the axis [14]. In that case, however, more complicated approaches are needed for evaluating
the scattering amplitude. In the case of a sphere some computational approaches have been
described by other researchers for various choices of beam types [20, 22, 37].
There is an additional complication in the case of small spheres in thermal viscous fluids in
that highly intense sound waves can establish a steady flow pattern commonly referred to as
acoustic streaming [38] not allowed for in the present analysis since all of the flow induced by
the acoustic wave is assumed to oscillate at the frequency ω according to (1) and (2). The magnitude and flow pattern of acoustic streaming tends to be somewhat dependent on the transducer
geometry and apparatus boundary conditions [38]. Methods have been introduced to reduce the
effects of streaming yielding agreement between measured and computed forces for objects in
traveling waves [39, 40].
An additional complication resulting from thermal-viscous dissipation in the surrounding
fluid are the contributions to the effective absorption efficiency (15) and correspondingly the
torque efficiency (27) from certain of the partial waves. For example, in the case of a small
solid sphere when ka . 0.5 the viscous correction to the dipole scattering can become important
when the average density of the sphere differs from that of the surrounding fluid. In the usual
case in which the thickness of the oscillating viscous boundary layer is small relative to the
radius of the sphere, a simple approximation is available for estimating the viscous correction
to s1 [41]. For solid spheres having small intrinsic absorption, this correction tends to decrease
in significance the larger the radius of the sphere [41].
A situation where absorption (either by the sphere or in the adjacent boundary layer) can be
beneficial concerns the induced rotation of a sphere on the axis of an acoustic vortex Bessel
beam. That is because the radiation torque applied to the sphere is proportional to the absorbed
power [23,24] as in the case of a sphere made of a lossy dielectric placed in circularly polarized
light [26]. If it is desirable to activate the absorption associated with the aforementioned viscous
correction to the dipole term s1 , it is necessary to use a beam having m = 1 [41]. When the effect
of the absorption of angular momentum by the surrounding fluid is negligible, the rotation
rate for the sphere may be estimated by balancing the radiation torque with the viscous drag
torque [41] as in the electromagnetic case [26].
Acknowledgment
Zhang acknowledges the support from NASA and ONR, and Marston acknowledges support
from ONR.
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