Optical theorem for acoustic non-diffracting beams and application to radiation force and torque

Likun Zhang; Philip L. Marston

doi:10.1364/BOE.4.001610

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.

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 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,

ψ_{i} = ψ_{i 0} (x, y; β) exp (i κ z - i ω t),

and scattered by an object centered at the origin (refer to Fig. 1), resulting in a scattered far field with a complex amplitude A_s(θ, ϕ),

ψ_{s} = ψ_{0} A_{s} (n) exp (i k r - i ω t) / r,

an extended optical theorem for the extinction cross section, denoted by σ_ext, was derived in the context of acoustics by [14] as an azimuthal angle integration

σ_{ext} = \frac{4 π}{k} Im [\frac{1}{2 π} \int_{0}^{2 π} g^{*} (ϕ) A_{s} (n (β, ϕ)) d ϕ],

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 reduced Whittaker integral

ψ_{i 0} (x, y; β) = \frac{ψ_{0}}{2 π} \int_{0}^{2 π} g (ϕ) exp [i μ (x cos ϕ + y sin ϕ)] d ϕ, with μ = k sin β .

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) = k n (β, ϕ),

have a tilted conical angle β relative to the beam’s axis (refer to Fig. 1). The scattering in the theorem (3), A_s(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 A_s(n(θ, ϕ)).

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The extinction results from both scattering and absorption. One can write σ_ext = σ_sca + σ_abs, with σ_sca and σ_abs denoting the cross sections of scattering and absorption, respectively. The scattering cross section, σ_sca, is given by the integral of the scattering coefficient over the whole solid angle element dΩ = sinθdθdϕ as,

σ_{sca} = \int {| A_{s} (n) |}^{2} d Ω .

On taking the limits, β = 0 and g(ϕ) = 1, our extended theorem (3) reduces to the well-known theorem for a plane wave [28–30],

σ_{ext} = \frac{4 π}{k} Im [A_{s} (0, 0)],

where A_s(0, 0) is the forward scattering of the incident plane wave ψ_i = 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,

ψ_{i} = ψ_{0} i^{m} J_{m} (μ ρ) exp (i κ z + i m ϕ),

where

ρ = \sqrt{x^{2} + y^{2}}

, 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 (i m ϕ) .

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

A_{s} (n (θ, ϕ)) = \frac{exp (i m ϕ)}{i k} \sum_{n = | m |}^{\infty} \frac{(s_{n} - 1)}{2} (2 n + 1) \frac{(n - m)!}{(n + m)!} P_{n}^{m} (cos β) P_{n}^{m} (cos θ),

where the functions

P_{n}^{m}

are associated Legendre functions (see also [31]). The functions s_n in the partial wave coefficients (s_n − 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 |s_n| ≤ 1 and only for non-absorptive scattering are all coefficients |s_n| equal to unity.

By using (9) and (10), and letting θ = β, it immediately follows from the optical theorem (3) that

σ_{ext} = \frac{π}{k^{2}} \sum_{n = | m |}^{\infty} (2 n + 1) {[P_{n}^{m} (cos β)]}^{2} Re [2 (1 - s_{n})],

where Re denotes the real part. Given that |s_n| ≤ 1, one always has Re[2(1 − s_n)] ≥ 0, and hence σ_ext ≥ 0 as required; only in the limit of no scattering, s_n = 1, does no extinction σ_ext = 0 occur. The cross section has the dimension of area. For an object being a sphere of radius a, the dimensionless efficiency factor becomes,

Q_{ext} \equiv σ_{ext} / (π a^{2}) = \frac{1}{{(k a)}^{2}} \sum_{n = | m |}^{\infty} (2 n + 1) \frac{(n - m)!}{(n + m)!} {[P_{n}^{m} (cos β)]}^{2} Re [2 (1 - s_{n})],

which recovers the result in [10] derived from an analytical integration of energy flux (where the scattering coefficients were written as (s_n − 1)/2 = α_n + iβ_n, with α_n and β_n 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 − s_n)] as

Re [2 (1 - s_{n})] = ({| s_{n} - 1 |}^{2}) + (1 - {| s_{n} |}^{2}) .

The first part, |s_n − 1|², associated with the modula of scattering coefficients, corresponds to the contribution from the scattering. The second part, 1 − |s_n|², associated with the difference of |s_n|² from unity, corresponds to the contribution from the absorption. Hence it follows from (12) and (13) that

Q_{sca} \equiv σ_{sca} / (π a^{2}) = \frac{1}{{(k a)}^{2}} \sum_{n = | m |}^{\infty} (2 n + 1) {[P_{n}^{m} (cos β)]}^{2} ({| s_{n} - 1 |}^{2}),

Q_{abs} \equiv σ_{abs} / (π a^{2}) = \frac{1}{{(k a)}^{2}} \sum_{n = | m |}^{\infty} (2 n + 1) {[P_{n}^{m} (cos β)]}^{2} (1 - {| s_{n} |}^{2}) .

which again recovers the results in [10] derived from an analytical integration of energy flux of corresponding fields.

In the special case of non-absorptive scattering (|s_n| = 1), Q_abs = σ_abs/(πa²) = 0. In this case using the phase shifts δ_n with

s_{n} = exp (i 2 δ_{n}),

both the extinction in (12) and the scattering in (14) relate to the phase shifts as

Q_{ext, sca} = σ_{ext, sca} / (π a^{2}) = \frac{4}{{(k a)}^{2}} \sum_{n = | m |}^{\infty} (2 n + 1) {[P_{n}^{m} (cos β)]}^{2} {sin}^{2} (δ_{n}) .

Lastly, the corresponding powers are given as

P_{ext, sca, abs} = I_{0} σ_{ext, sca, abs} = I_{0} (π a^{2}) Q_{ext, sca, abs},

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 = (ρ_oc_o/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

F_{z} = c_{0}^{- 1} (P_{ext} cos β - P_{sca} {〈 cos θ 〉}_{s})

or in terms of a dimensionless force function Y_p as

F_{z} = π a^{2} c_{0}^{- 1} I_{0} Y_{p}, Y_{p} = Q_{ext} cos β - Q_{sca} {〈 cos θ 〉}_{s},

where the factors, cos β and 〈cos θ〉_s, correspond to the axial projections of extracted momentum and scattered momentum [10, 11, 14], respectively, and

{〈 cos θ 〉}_{s} = \frac{\int cos θ {| A_{s} (n) |}^{2} d Ω}{\int {| A_{s} (n) |}^{2} d Ω} .

The factor 〈cos θ〉_s is an asymmetry parameter of the scattering: 〈cos θ〉_s 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 Q_ext = Q_sca + Q_abs, an implicit term in (20) associated with absorption, Q_abs cos β, degrades the negative force [10].

The cross section and efficiency factor associated with the scattering asymmetry factor 〈cos θ〉_s in (21) for a sphere of radius a are

σ_{asym . sca} \equiv \int cos θ {| A_{s} (n) |}^{2} d Ω, Q_{asym . sca} \equiv σ_{asym . sca} / (π a^{2}) .

When a sphere is on the axis of the Bessel beam (8), the efficiency of the scattering asymmetry, Q_asym.sca, can be read from [10] (Q_asym.sca = −Y₁ therein) as

Q_{asym . sca} = \frac{σ_{asym . sca}}{π a^{2}} = {(\frac{2}{k a})}^{2} \sum_{n = | m |}^{\infty} [2 (α_{n} α_{n + 1} + β_{n} β_{n + 1})] \frac{(n - m + 1)!}{(n + m)!} P_{n}^{m} (b) P_{n + 1}^{m} (b) .

Together the extinction (12) from the optical theorem and the asymmetry (23) give the dimensionless force function in (20) as

\begin{array}{l} Y_{p} & = & \frac{cos β}{(k a^{2})} \sum_{n = | m |}^{\infty} (- 4 α_{n}) (2 n + 1) \frac{(n - m)!}{(n + m)!} {[P_{n}^{m} (cos β)]}^{2} \\ - {(\frac{2}{k a})}^{2} \sum_{n = | m |}^{\infty} [2 (α_{n} α_{n + 1} + β_{n} β_{n + 1})] \frac{(n - m + 1)!}{(n + m)!} P_{n}^{m} (cos β) P_{n + 1}^{m} (cos β), \end{array}

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

T_{z} = \frac{m}{ω} P_{abs},

or in terms of a dimensionless torque efficiency factor Q_T for a sphere of a radius a as,

T_{z} = π a^{2} I_{0} Q_{T} / ω, Q_{T} = m Q_{abs} .

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

Q_{T} = m Q_{abs} = \frac{m}{{(k a)}^{2}} \sum_{n = | m |}^{\infty} (2 n + 1) \frac{(n - m)!}{(n + m)!} {[P_{n}^{m} (cos β)]}^{2} (1 - {| s_{n} |}^{2}),

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].

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].

Acknowledgments

Zhang acknowledges the support from NASA and ONR, and Marston acknowledges support from ONR.

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Optical theorem for acoustic non-diffracting beams and application to radiation force and torque

Abstract

Corrections

1. Introduction

2. Optical theorem of a non-diffracting beam: Review

3. Application to extinction by a sphere

4. Application to the axial radiation force on a sphere

5. Application to the radiation torque on a sphere

6. Conclusions and discussion

Acknowledgments

References and links

Cited By

Figures (1)

Equations (27)

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