Issue 
Acta Acust.
Volume 8, 2024
Special Issue: Numerical, computational and theoretical acoustics



Article Number  26  
Number of page(s)  13  
DOI  https://doi.org/10.1051/aacus/2024022  
Published online  13 August 2024 
Scientific Article
A localized approximation approach for the calculation of beam shape coefficients of acoustic and ultrasonic Bessel beams
^{1}
Department of Electrical and Computer Engineering, São Carlos School of Engineering, University of São Paulo, 400 Trabalhador sãocarlense Ave. 13566590, São Carlos, SP, Brazil
^{2}
CORIAUMR 6614 – Normandie Université, CNRSUniversité et INSA de Rouen, Campus Universitaire du Madrillet, 76800 SaintEtienne du Rouvray, France
^{*} Corresponding author: leo@sc.usp.br
Received:
31
December
2023
Accepted:
5
June
2024
The description of acoustical waves can be achieved using an expansion over basic functions with weighting coefficients which may be called beam shape coefficients (BSCs). There is a strong analogy between the scalar formalism of acoustical waves and the vectorial electromagnetic formalism, known as generalized Lorenz–Mie theory (GLMT), describing the interaction between a homogeneous sphere and an arbitrary illuminating beam. In particular, BSCs have been introduced as well in GLMT and the mathematical arsenal to evaluate them, developed since several decades, can be used mutatis mutandis to evaluate BSCs in acoustics. In particular, the present paper introduces a method named localized approximation to the evaluation of the acoustical BSCs, similar to the localized approximation used to evaluate electromagnetic BSCs, in the case of Bessel beams. Such a formalism akin to the electromagnetic GLMT may be viewed as an acoustical GLMT and should allow a renewal of the calculation of various properties of acoustical wave scattering, in particular to the design of acoustical tweezers similar to optical tweezers.
Key words: Acoustic scattering / Beam shape coefficients / Bessel beams / Generalized Lorenz–Mie theory / Localized approximation
© The Author(s), Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In some laser light scattering theories, like in generalized Lorenz–Mie theories (GLMTs), e.g. [1, 2], or in Extended Boundary Condition Method (EBCM) for structured beams, e.g. [3–5], the illuminating laser beam may be encoded using beam shape coefficients (BSCs) which intervene when expanding the fields over vector wave functions. There exists an arsenal to evaluate these BSCs, including quadratures and localized approximations (LAs), e.g. Gouesbet et al. for a review [6].
It however happens that the structure of scalar scattering theories exhibits many analogies with the one of vectorial scattering discussed above, so that the arsenal developed in this framework, may be transferred, mutatis mutandis, to the case of acoustical scattering. The present paper is then devoted to a discussion of quadratures and localized approximation methods to the case of acoustical Bessel beams.
When no analytical, rigorous solutions can be found for the BSCs, it is usually advantageous to have recourse to alternative, approximate schemes such as the LAs [7–9]. Although widely known and explored in optics, these methods remained largely overlooked in the field of acoustic scattering.
To the best of the authors’ knowledge, the first paper that introduced the concept of localized approximations for explicit BSC calculation in acoustic scattering appeared in 2022 and is due to Li et al. in the evaluation of radiation forces on elastic spheres [10]. In this work, LA BSCs are calculated for Gaussianlike beams under both on and offaxis configurations, the BSCs for the latter configuration being found from the former through the use of a translational addition theorem of spherical wave functions. Then, in 2023, Li and Zhang, again using LA schemes, evaluated acoustic forces over multilayered, eukaryotic spherical cells arbitrarily positioned in space with respect to a Gaussian beam [11]. In both works, LA schemes as borrowed from optics were shown to provide good results, with the proviso that the confinement parameter remains small. Finally, and also in the field of scattering by acoustic beams, the van de Hulst principle of localization has been introduced by Marston in 2007 in the interpretation of the scattering dependence on the axicon angle of Bessel beams [12] and in the analysis of scattering of focused beams by spheres considering quasiGaussian Bessel beam superpositions [13].
The paper is organized as follows. Section 2 deals with generalities for arbitrary shaped acoustical waves satisfying the Helmholtz equation, and introduces acoustical BSCs using a quadrature approach. Section 3 deals with the LAprocedure using an empirical approach, introduces a variant named Integral Localized Approximation (ILA), and applies it to the case of on and offaxis Bessel beams. Section 4 provides simulation results and discussions, which validate the localization procedures for beams having small axicon angles and in the absence of topological charges.
These observations are similar to the ones obtained in optics where it has been observed that the quality of the procedures deteriorates when the axicon angle increases [14–19], and/or when the topological charge increases [20–23]. These observations concerning the LAs reinforce the parallel between vectorial and scalar scatterings. Section 5 is a conclusion.
2 Generalities
Let us start by introducing the usual set of two Cartesian coordinate systems (x, y, z) and (u, v, w) in acoustic and ultrasonic scattering, with origins at O_{P} and O_{B}, respectively, see Figure 1. The first system is assumed to be attached to a homogeneous spherical particle and the second to the incident beam. Here, +u, +v and +w are parallel to +x, +y and +z, respectively. Even though we shall not be interested in the geometrical and physical properties of the scatterer, the relative position (x_{0}, y_{0}, z_{0}) of the beam with respect to the center of the sphere (at which the origin O_{P} is located) is of importance. For x_{0} = y_{0} =0 and ±z propagation, we have the simplified onaxis configuration, while for arbitrary x_{0}, y_{0} ≠0, the beam is under an offaxis configuration. Spherical (r, θ, ϕ) and cylindrical (ρ, ϕ, z) coordinates are attached to (x, y, z) for convenience and further calculations. In cylindrical coordinates, the relative position of O_{B} is designated as (ρ_{0}, ϕ_{0}, z_{0}).
Figure 1 Coordinate systems for acousticmatter analysis. A Cartesian coordinate system (x, y, z) is introduced supposing a hypothetical spherical scatterer centered at its origin O_{P}. A second Cartesian coordinate system (u, v, w) is attached to the impinging beam, with origin O_{B}. The position of O_{B} with respect to O_{P} is here denoted as (x_{0}, y_{0}, z_{0}). Attached to the xyz system is a cylindrical coordinate system (ρ, ϕ, z) such that the relative position of O_{B} can also be written as (ρ_{0}, ϕ_{0}, z_{0}). 
Let us assume a +zpropagating incident acoustical beam with a time harmonic factor exp(+iωt), where ω is the operating frequency, written in terms of a complex scalar potential ψ_{i}(r, θ, ϕ) expressed as a partial wave expansion using spherical wave functions [24–26]:
where is the field strength, (“pw” stands for “plane wave”), j_{n}(.) are spherical Bessel functions of the first kind and of integer order n and are associated Legendre functions, which follows Robin’s convention [27]. This choice for reflects the preference of the authors and incorporate a factor of (−1)^{m} which might not be present in other works. Finally, k = 2π/λ is the wave number and λ is the wavelength. Assuming the absence of nonlinear effects and propagation of the acoustical waves in a lossless medium [28–32], equation (1) is a rigorous solution to the scalar Helmholtz equation which can be used to describe acoustic radiation pressure fields in inviscid fluids.
The expansion coefficients – known as the Beam Shape Coefficients (BSCs) – can be isolated in equation (1) using orthogonality relations for and exp(imϕ), see e.g. Refs. [2, 24, 33]:
For beams which exactly satisfy the scalar Helmholtz equation ∇^{2}ψ_{i} + k^{2}ψ_{i} = 0, the rdependent factor 1/j_{n}(kr) is cancelled after integration. This is the case of Bessel beams, see Gong et al. [34]. The BSCs calculated from equation (2) are called the quadrature BSCs.
3 The localized approximation
According to the localization principle of van de Hulst [35], one can associate to each partial wave in equation (1) (or to each term of order n in the plane wave amplitude functions [2]) an acoustical ray or bundle of rays parallel to the w axis and travelling at a transverse distance (n + 1/2)/k from O_{uvw}. The formal justification for the use of a LA approach in optics relies on the asymptotic behavior of Bessel functions of order (n + 1/2) in the expressions for the Mie scattering coefficients [2, 35]. Since similar functions also appear for spherical scatterers in acoustics, see equation (5) in Mitri and Silva [36] for the case of a rigid sphere, Mitri [37] (Appendix B) for the general case of an elastic sphere in an ideal fluid and equation (4) in Hasegawa [38] for both fluid and solid isotropic spheres, it is plausible to conjecture that a LA scheme do indeed exist in acoustics.
Let R = kr. Slightly modifying a procedure justified in optics [39], we then empirically propose the following procedure for finding the acoustic LA BSCs as follows:
Decompose the original, intended field ψ_{i} into azimuthal waves:

Express the azimuthal modes under a form that emphasizes a “plane wave” factor which is multiplied by the remaining part of , viz., :
where is the field strength [e.g., of equation (1)].
Then, the localized approximation to is given by:
where L ≡ L(n) is a function of n and is still to be found, see below.
It is seen from equations (3)–(5) that the azimuthal waves carry a plane wave factor which, together with the ϕdependent exponential exp(imϕ), is removed before application of a localization operator, say , which sets θ → π/2 and R = kr → L^{1/2}. In compact form, therefore, the LA BSCs can be written as:
with
For a plane wave of the form , one sees that , where δ_{ij} is the Kronecker delta. Therefore, , ∀n, as expected [40, 41].
An important point concerns the choice of L in the LA scheme and, consequently, of . In Refs. [10, 11], no explicit mention to a choice of L is made, since for onaxis Gaussian beams, the only nonzero LA BSCs are those for which m = 0 and, therefore, regardless of the choice of L, see equation (5) of Li et al. [10]. For offaxis beams, the use of addition theorem for evaluating obscures the explicit appearance of , so that a general formalism cannot be devised from such an approach.
But in the optical domain, have different expressions for m = 0 and m ≠ 0 [9, 42], and it could be the case that L^{1/2} should assume different values for different m as well. That such is not the case is inspired by the rigorous procedure presented by Gouesbet [39] in optics and, after an empirical slight modification, we propose:
valid for both on and offaxis beams. In other words, since now L^{1/2} = n + 1/2, the operator will set θ → π/2 and R = kr → n + 1/2.
The quality of the empirical procedure is now checked by modeling the case of Bessel beams.
3.1 LA BSCs for acoustical Bessel beams
An acoustic or ultrasonic Bessel beam is an ideal solution to the scalar homogeneous Helmholtz equation in cylindrical coordinates. It carries resistance to diffraction and, consequently, selfhealing properties, so that their transverse field profile is, except for phase variations, invariant under propagation [43, 44].
Let a vorder Bessel beam with axicon angle α have transverse and radial wave numbers k_{ρ} = ksinα and k_{z} = kcosα, respectively. For onaxis beams (x_{0} = y_{0} = 0) with z_{0} = 0, we can express it as:
For offaxis beams, the field of equation (9) is displaced by (x_{0}, y_{0}, z_{0}). Instead of equation (9), one then finds an expression in terms of cylindrical coordinates ρ, ϕ and z:
where
In order to expand equation (10) into azimuthal modes, we proceed as follows. First, we introduce Neumann’s addition theorem for Bessel functions [45]:
with
From equations (12) and (13), equation (10) can be rewritten as:
Now, let m = p + v, or p = m − v. Then, from equation (14),
which, for highly paraxial beams, cosα ≈ 1, and exp(−ik_{z}z) = exp(−ikcosαz) ≈ exp(−ikz) = exp(−ikrcosθ). This means that, under this restrict case, a plane wave factor of the form can be put into evidence in equation (15). Using the fact that R = kr, equation (15) can then be recast under the form:
Notice that the condition of highly paraxial beams is necessary to ensure a quasiplane wave factor under the brackets of equation (16), but that we have not applied the paraxial approximation cosα ≈ 1 and sinα ≈ α to the remaining part of equation (15) because, for all practical purposes regarding the determination of an approximate expression for the BSCs, the paraxial approximation does not need to be forced to hold for all factors in equation (16). As observed in optics using a variant known as the integral localized approximation, the LA is not able to accurately remodel a Bessel beam for high axicon angles [16].
Comparison between equations (4) and (16) allows us to identify . Using the fact that kρ = krsinθ, one then finds:
Applying the localization operator on equation (17) leads to
Substituting equation (18) in equation (6) and making use of equation (8) then leads to the final expression for the BSCs of a general offaxis arbitraryorder Bessel beam:
with Z_{0} given by equation (13).
Equation (19) is the most important formula of the present work. It can be readily compared with the exact BSCs calculated from quadratures [34] which, using the time harmonic convention exp(+iωt), can be recast under the form:
Since (−1)^{m} = (−1)^{m} and i^{−m} = −i)^{m}, can be put in a form best suited for comparison with :
For onaxis beams (Z_{0} = 0), equations (19) and (21) simplify, respectively, to:
and
It is seen that, for onaxis Bessel beams, the only nonzero BSCs are those with m = v, as expected [34].
3.2 Asymptotic correspondence between the LA and exact BSCs
Besides the Nbeam method used in optics, and under study in acoustics, an alternative way to at least partially justify the choice of for acoustical fields is by looking at the asymptotic behavior of the associated Legendre polynomials for large n (n → ∞) and small axicon angles.
To see it clearly, let [33]
so that, from equation (21), for m ≥ 0 can be put under the form
where
Similarly, for m < 0, it is readily seen from equations (21) and (26) that
In a similar fashion, from equation (19), it is inferred that:
and, remembering that J_{−m}(x) = (_{−}1)^{m}J_{m}(x),
Next, we introduce the MacDonald expansion for associated Legendre polynomials as their degree n → ∞, see equation (1), Section 5.72 in Watson [45]:
with x = 2(n + 1/2)sin(α/2).
For highly paraxial beams, a first approximation can be set with cosα/2 ≈ 1 and sin(α/2) ≈ α/2, such that x ≈ (n + 1/2)α ≈ (n + 1/2)sinα and, retaining only the first term in equation (30),
It is then seen from equations (25), (27)–(29) and (31) that, for sufficiently small axicon angles, large n and for all m, . Under these conditions, the choice R = L^{1/2} = (n + 1/2) is fully appropriate. Since L^{1/2} must be independent of the choice of the acoustical field (otherwise, one would have to deal with a different LA scheme for each incident beam, therefore leading to an infinite number of LAs), it is admissible at the present time to take this form of L^{1/2} to be best suited for achieving accurate results using the LA approach.
3.3 The acoustical integral localized approximation
In 1998, Ren et al. proposed a variation of the LA scheme whose main advantage was to avoid, in the process of obtaining the BSCs of optical wave fields, the decomposition into azimuthal modes, which corresponds in the acoustical or ultrasonic case to equation (3) above.
The main idea behind such proposal was to gain a certain level of flexibility in dealing with arbitraryshaped beams, since such a decomposition could, in principle, be tedious.
Such an approach can be readily translated to acoustic and ultrasonic fields. Since the modes in equation (3) are proportional to exp(imϕ) [for Bessel beams, this is seen from equation (15)] and using simple orthogonality relations for exponential functions, step (i) in the LA scheme can be avoided if we evaluate according to:
In view of equations (6) and (32), the BSCs can now be found from:
Because equation (33) now involves an integration process, this new LA scheme has been given the name integral localized approximation (ILA) [9].
The fact is that, for offaxis Bessel beams, the extraction of the BSCs from equation (33) still demands the expansion of ψ_{i}(r, θ, ϕ) into azimuthal modes before performing the ϕintegration. This has been shown to hold for both zeroorder and higherorder optical Bessel beams, where azimuthal modes appears naturally when using Neumann’s addition theorem for Bessel functions [42, 46].
Starting from equation (10), imposing the localization operator , expanding the resulting formula according to Neumann’s addition theorem and performing the integration over ϕ, the BSCs extracted from equation (33) using the ILA can be shown to be the same as those derived from the LA itself and given in equation (19), as expected.
3.4 A general relation between exact and LA BSCs
An interesting and general relation between the exact and LA BSCs for acoustic/ultrasonic fields can be extracted from equations (1) and (33).
To see it clearly, notice that the localization operator, once applied on equation (1), produces
where we have explicitly written the r.h.s. in terms of the exact BSCs.
Now, from equation (33),
Equation (35) establishes an intrinsic relation between and the exact BSCs of arbitraryshaped beams. Since [27]
where (.)!! is the double factorial, in general only a set of exact BSCs [either with (p − m) even or (p − m) odd] will contribute to the value of . In addition, because for p < m, equation (35) can be written in the following final form:
The importance of equation (37) relies on the fact that, whenever a comparison between exact and LA BSCs is to be performed, or when one needs to infer the limit of applicability of the LA for a particular beam whose exact BSCs are known, there is no need to actually go through the analytical process of finding the LA BSCs presented in previous sections. The disadvantage, of course, resides on the infinite summation, which not only can increase computational burden but also forces some truncation criterion to be imposed based on prespecified tolerated error.
4 Simulations and results
In this section we illustrate the behavior of equations (19) and (21), and their simplified form for onaxis Bessel beams, equations (22) and (23), respectively, together with field reconstructions for both onaxis and offaxis beams with different axicon angles. To do so, in all simulations we have set λ = 50 μm. Algorithms have been developed based on the equations of the previous section using the commercial software Wolfram Mathematica 12.1 Student Edition. They are available upon reasonable request. Simulations were then run on a personal laptop [Intel(R) Core(TM) i73630QM CPU @ 2.40 GHz, 16.0 GB]. Arbitrary precision (infiniteprecision arithmetic) has been chosen for the calculations.
The infinite sum in equation (1) is truncated in accordance with Wiscombe’s convergence criterion [47], which is wellestablished in the field of optics and has also been recently introduced in acoustics and ultrasonics [24]. It states that there is a maximum n, say n_{max}, above which the partial waves will have negligible contribution to the total field. The value of n_{max} can be calculated in terms of the product kr, such that [47]:
As a first example, let us assume the simple onaxis configuration, for which x_{0} = y_{0} = 0. For this case, the only nonzero beam shape coefficients (BSCs) are , as can be seen from equations (22) and (23).
Figure 2 shows exact and localized approximation (LA) BSCs for a zeroorder Bessel beam (v = 0), for six different axicon angles (α = 1°, 5°, 10°, 20°, 35° and 50°), along with the logarithmic percent error . A 1% error therefore corresponds to the horizontal line ln[error(%)] = 0. Without loss of generality, we have set z_{0} = 0.
Figure 2 (a) Exact (red, solid, with circular marks) and LA (blue, dashed, with triangular marks) BSCs for an onaxis zero order Bessel beam as a function of n for α = 1°. (b) The logarithmic percent error corresponding to (a). (c) and (d) α = 5°; (e) and (f) α = 10°; (g) and (h) α = 20°; (i) and (j) α = 35°; (k) and (l) α = 50°. In all cases, λ = 50 μm. 
Several important features are observed. First, the smaller the value of α, the better is the agreement between the exact and LA BSCs. However, notice that, for a fixed α, the logarithmic error slightly increases as n increases. The highest errors are seen for values of n coinciding with those points for which tends to zero. The most critical scenario is, of course, that of α = 50°. In this particular case, although both the exact and the LA BSCs show a similar oscillatory pattern (Fig. 2k), high errors are observed for all n (Fig. 2l), with the oscillations observed for the LA BSCs occurring for higher n when compared with those observed for the exact BSCs.
Such a delay clearly increases as α increases and reflects itself in the transverse locations of the bright annular disks of the Bessel beam. This is illustrated in Figure 3, where the exact and LA transverse field intensities, and , together with the logarithmic error, are plotted for α = 1° (Figs. 3a–3d), 10° (Figs. 3e–3h) and 50° (Figs. 3i–3l). It must be stressed that, even though subtle differences between the original (exact) and LAremodeled beams can be noticed, the latter is a genuine acoustical beam on its own, that is, the remodeled beam is still an exact solution to the homogeneous scalar Helmholtz equation.
Figure 3 Exact and LAremodeled field intensities and , and logarithmic percent errors for the zeroorder Bessel beams of Figure 2 with α = 1°, 10° and 50°. (a)–(d) α = 1°. (e)–(h) α = 10°. (i)–(l) α = 50°. The corresponding BSCs are shown in Figure 2(a), (e) and (k), respectively. 
The effect of a nonzero topological charge can be appreciated in Figure 4 for α = 50°, which shows the exact and LAremodeled field intensities for v = 0, 3 and 7. One can clearly see that, as v increases, errors become higher at the positions of global maxima of ψ_{i}. This has important consequences on the application of localized schemes for predicting acoustic and ultrasonic pressure forces. Better results, at least at transverse positions close to the most intense bright annular disk, are expected for smaller values of v. In addition, for very small particles, only those partial waves with low values of n enter into the calculation of acoustical forces, so that only the first few BSCs need to be taken into account. This means that, regardless of the value of α, the LA scheme might still produce good physical predictions. This is clearly seen for the zeroorder Bessel beams of Figure 2 for all values of α, a feature also noticed in the optical domain [16].
Figure 4 (a) Exact (black, dotdashed) and LAremodeled (red, dashed) field intensities and (in logarithmic scale), and logarithmic percent errors for an onaxis Bessel beams with α = 50°, for v = 0. (b) and (c) Same as (a), but now for v = 3 and 7, respectively. 
For offaxis configuration, it can be seen from equations (19) and (21) that:
It is seen from equation (39) that the percent error for a given n and m is actually independent of the relative lateral position or displacement (x_{0}, y_{0}) of the beam with respect to the origin O_{P}. Therefore, even though the number of BSCs demanded to reproduce ψ_{i}(r, θ, ϕ) will increase for ρ_{0} ≠ 0 [this statement is readily confirmed from either equation (19) or (21) by setting Z_{0} ≠ 0], just like for onaxis beams, discrepancies between the exact and LA BSCs will only depend on n, m and α for offaxis Bessel beams. This means that the analysis of BSCs for the offaxis case resembles that previously performed for onaxis beams. This fact contrasts significantly with that observed for optical Bessel beams, for which the LA scheme could be highly dependent on x_{0} and y_{0} [16]. Finally, notice that the ratio is also independent of the beam order v.
As an example, Figure 5 shows how the ratio behaves as a function of n for a zeroorder Bessel beam with α = 1° (Figs. 5a–5c), 10° (Figs. 5d–5f) and 30° (Figs. 5g–5i). Specific lateral displacements of ρ_{0} = 50λ, 5λ and 4λ have been chosen with the sole purpose of having nonzero BSCs for m ≠ 0, since equation (39) does not depend on ρ_{0}. The deleterious effect of an increasing axicon angle is readily seen. Such an effect becomes more pronounced at values of n corresponding to minima of .
Figure 5 (a)–(c) Ratio (black solid lines, with circular marks) for a Bessel beam with v = 0 for α = 1° and m = 0, 2 and 5, respectively. (d)–(f) Same as before, now for α = 10°. (g)–(i) Same as (a)–(c), now for α = 30°. For reference purposes, the exact BSCs are also shown (blue dashed lines, with triangular marks), with multiplicative factors being introduced for better visualization. The choice of ρ_{0} ≠ 0 allows for nonzero BSCs for m ≠ 0 but has not other implications in the comparison between the plots, since equation (39) does not depend on ρ_{0} (had we chosen the same ρ_{0}, the results would remain unchanged). The beam is assumed to be displaced along the x axis, that is, ϕ_{0} = 0 or, equivalently, ρ_{0} = x_{0}. 
An interesting feature of equation (39) is its independence with respect to the relative position (ρ_{0}, ϕ_{0}, z_{0}) of the Bessel beam. This means that, for a given axicon angle, the ratio between the LA and exact BSCs depends only on n and m. But it must be stressed that, since for fixed n and m the value of changes as we move from an onaxis to an offaxis configuration, and also because changes as ρ_{0} ≠ 0 changes, the reconstructed fields still depend on the relative position of the beam, as it should be.An example of field reconstruction for an offaxis beam and from both exact and LA BSCs is illustrated in Figure 6 for v = 0 and α = 30°, assuming ρ_{0} ≡ x_{0} = 4λ = 200 μm. In acoustic scattering by spherical particles, force calculations assume that the particle is centered at O_{P}, that is, at x = y = 0. Since errors are high (Fig. 6c) the LA might fail to provide accurate force predictions for offaxis beams for such a high value of the axicon angle. It can be checked that smaller values of α improves the agreement between the original field and the one calculated through the LA.
Figure 6 (a) Original transverse field intensity for an off axis zeroorder Bessel beam with α = 30° and ρ_{0} = x_{0} = 4λ(y_{0} = z_{0} = 0). (b) Same as (a), but now for the remodeled LA field using the BSCs given in equation (19). (c) Logarithmic percent error for the field intensities of (a) and (b). 
As expected from the results for onaxis beams, notice that in Figure 6 there is a noticeable loss of symmetry in the Bessel pattern with respect to the optical axis of the beam when using localization schemes (Fig. 6b). This comes from the fact that, as x increases, the number of BSCs demanded to calculate increases in accordance with the localization principle of van de Hulst (the same being valid for the field along y). But, as n increases, discrepancies between the exact and LA BSCs also increases, as revealed in Figure 2 for an onaxis Bessel beam.
The choice of Wiscombe’s criterion for truncating the sum in equation (1) has been qualitatively justified for Bessel beams by Ambrosio and Gouesbet in Ref. [24] (see Fig. 4 of this reference). In this work, the authors derived finite series expressions for the BSCs and compared them with those analytically extracted from quadratures. For field reconstruction using the LA method, a similar verification can be performed. Consider, for instance, adding or subtracting a value n_{add} from n_{max}, see equation (38). Figure 7 shows how and behave as n_{add} goes from −20 to 20, for the offaxis Bessel beam of Figure 6. As is clearly seen, as n_{add} approaches zero, that is, when truncation occurs at n_{max}, the calculated exact and LA field intensities have converged to their final values of 1 and 0.76129978, respectively, with excellent agreement.
Figure 7 Verification of Wiscombe’s criterion for the offaxis Bessel beam of Figure 6, at a fixed position (x, y, z) = (4λ, y, z). Here, the value of n_{add} is added to that evaluated from Wiscombe’s criterion [47], see equation (38) in order to truncate the sum in equation (1). 
Although ideally the LA method seeks to establish a complete identification , such an identity is spoiled in view of equation (37). In fact, the LA BSCs are weighting sums of the exact coefficients with the same m. Because of equation (36), only those terms with (p − m) even in the r.h.s. of equation (37) will contribute to the final value of , which means that, for even m, only the exact BSCs with p even will contribute to the LA BSCs, while for m odd, only the exact BSCs with p odd will contribute to the LA BSCs. This is true not only for Bessel beams, but for any arbitraryshaped beam.
As an example of application of equation (37), consider an onaxis Bessel beam (z_{0} = 0) with α = 1° and v = 0. Table 1 shows for specific values of n, as calculated from both equation (19) and equation (37), the latter being truncated at p_{max} = 20, 100, and 400. As p_{max} increases, there is greater agreement between the LA BSCs as computed from equations (19) and (37). This is also observed in Table 2 for α = 50°, keeping all previous parameters unchanged.
5 Conclusions
In this work we have introduced the acoustical localized approximation for arbitraryorder Bessel beams. The beam shape coefficients have been compared with those computed from quadrature schemes and field remodeling has been shown for both on and offaxis configurations.
In contrast with previous works which dealt with Gaussian beams, here we have justified the presence of prefactors in the expressions for the localized approximation beam shape coefficients by having recourse to asymptotic behavior of associated Legendre functions of very large degrees. A rigorous justification of the localized approximation for arbitraryshaped acoustic/ultrasonic beams is still open to investigation.
A variant of the localized approximation, which attempts at avoiding the decomposition of the original field into azimuthal modes, viz., the integral localized approximation, has also been presented. The beam shape coefficients for Bessel beams calculated in this manner are expectedly the same as those calculated from the original localization scheme. A general relationship between the localized and exact beam shape coefficients has been also provided, therefore eliminating the need to actually derive the former coefficients whenever the latter (exact) coefficients are available for any acoustic/ultrasonic beam.
The results shown in this paper reveals that the localized approximation can be used to describe acoustical and ultrasonic Bessel beams with (very) low axicon angles. For zeroorder beams and small axicon angles, errors at the acoustical axis are usually small or negligible. However, as the order of the beam increases, the corresponding errors at the locations of maximum of the transverse field pattern (the most intense annular ring) increase as well, thus indicating that physical predictions involving, for instance, acoustic pressure force calculations and extracted from the localized beam shape coefficients might not be accurate enough in comparison with those computed from exact beam shape coefficients derived from quadrature schemes.
Obviously, the use of a localized approximation is superfluous when exact beam shape coefficients are available. However, this work has provided a path for a systematic and general justification of localized schemes for arbitraryshaped beams. In the optical realm, justification has been given in terms of what is called the Nbeam method [39]. Besides, it was also shown that the localized approximation cannot be rigorously justified for Bessel beams or, more in general, beams carrying a propagation factor of the form exp(−ik_{z}z) instead of exp(−ikz), e.g., Ref. [14]. We hope that the present work paves the way for equivalent deeper investigations in acoustics and ultrasonics.
Funding
This work was partially supported by the National Council for Scientific and Technological Development (CNPq) (406949/20212, 309201/20217) and by The São Paulo Research Foundation (FAPESP) (2021/061210).
Conflict of interest
The authors declare no conflict of interest.
Data availability statement
The data are available from the corresponding author on request.
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Cite this article as: Ambrosio LA & Gouesbet G. 2024. A localized approximation approach for the calculation of beam shape coefficients of acoustic and ultrasonic Bessel beams. Acta Acustica, 8, 26.
All Tables
All Figures
Figure 1 Coordinate systems for acousticmatter analysis. A Cartesian coordinate system (x, y, z) is introduced supposing a hypothetical spherical scatterer centered at its origin O_{P}. A second Cartesian coordinate system (u, v, w) is attached to the impinging beam, with origin O_{B}. The position of O_{B} with respect to O_{P} is here denoted as (x_{0}, y_{0}, z_{0}). Attached to the xyz system is a cylindrical coordinate system (ρ, ϕ, z) such that the relative position of O_{B} can also be written as (ρ_{0}, ϕ_{0}, z_{0}). 

In the text 
Figure 2 (a) Exact (red, solid, with circular marks) and LA (blue, dashed, with triangular marks) BSCs for an onaxis zero order Bessel beam as a function of n for α = 1°. (b) The logarithmic percent error corresponding to (a). (c) and (d) α = 5°; (e) and (f) α = 10°; (g) and (h) α = 20°; (i) and (j) α = 35°; (k) and (l) α = 50°. In all cases, λ = 50 μm. 

In the text 
Figure 3 Exact and LAremodeled field intensities and , and logarithmic percent errors for the zeroorder Bessel beams of Figure 2 with α = 1°, 10° and 50°. (a)–(d) α = 1°. (e)–(h) α = 10°. (i)–(l) α = 50°. The corresponding BSCs are shown in Figure 2(a), (e) and (k), respectively. 

In the text 
Figure 4 (a) Exact (black, dotdashed) and LAremodeled (red, dashed) field intensities and (in logarithmic scale), and logarithmic percent errors for an onaxis Bessel beams with α = 50°, for v = 0. (b) and (c) Same as (a), but now for v = 3 and 7, respectively. 

In the text 
Figure 5 (a)–(c) Ratio (black solid lines, with circular marks) for a Bessel beam with v = 0 for α = 1° and m = 0, 2 and 5, respectively. (d)–(f) Same as before, now for α = 10°. (g)–(i) Same as (a)–(c), now for α = 30°. For reference purposes, the exact BSCs are also shown (blue dashed lines, with triangular marks), with multiplicative factors being introduced for better visualization. The choice of ρ_{0} ≠ 0 allows for nonzero BSCs for m ≠ 0 but has not other implications in the comparison between the plots, since equation (39) does not depend on ρ_{0} (had we chosen the same ρ_{0}, the results would remain unchanged). The beam is assumed to be displaced along the x axis, that is, ϕ_{0} = 0 or, equivalently, ρ_{0} = x_{0}. 

In the text 
Figure 6 (a) Original transverse field intensity for an off axis zeroorder Bessel beam with α = 30° and ρ_{0} = x_{0} = 4λ(y_{0} = z_{0} = 0). (b) Same as (a), but now for the remodeled LA field using the BSCs given in equation (19). (c) Logarithmic percent error for the field intensities of (a) and (b). 

In the text 
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