2. Theory and results

Our analysis starts with the perturbation solutions of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation.^[18–25] Assume that a sound source f_n(ξ′)e^inϕ′ with an arbitrary radial distribution oscillates harmonically at angular frequency ω, and is located in the (r,ϕ) plane normal to the z axis, which is the direction of the beam propagation. According to previous results,^[24–29] the fundamental (primary) sound field is given by

and the n-th-order harmonic beam is given as

where

The subscripts ℓ, j, and n denote the orders of harmonics, and it is always true that n = ℓ + j. The product n × m is the order number of the Bessel function. The terms e^imϕ and e^inmϕ are omitted. Note that the lower order (|ℓ − j|) harmonics caused by nonlinear interaction of the harmonics of the order numbers ℓ and j are assumed to be much smaller than the n-th-order harmonic from these harmonics. This is of course valid only under the condition of relatively weak nonlinearity. In Eqs. (1) and (2), ξ = r/a and η = 2z/ka² are the radial and axial dimensionless coordinates, respectively; f_m(ξ′) represents the radial distribution function at the source plane η = 0 or z = 0. Equations (1) and (2) may be considered as the complex-valued (pressure) amplitudes in dimensionless form.

We now consider the high-order (n) harmonic generation of a general m-th-order Bessel beam. Suppose that on the source, the m-th-order Bessel beam with radial form is given as

Here a scaling parameter α is the dimensionless radial wave number of this beam. Assuming that the aperture of the source extends infinitely, hence the well-known Bessel beam field at fundamental frequency, from Eq. (1), is written as

for a limited-diffraction Bessel beam. The second-harmonic generation of the m-th-order Bessel beam is given as

under the approximation that the propagation distance satisfies an asymptotic condition of α²η/2 > 2π. When 0 ≤ α²η/2 < π/2, the second harmonic beam distribution is

If the distance η is varied, there is an apparent difference between the second harmonic beam profile terms in the above two equations. Beam profile is approximated by in the near field, where 0 ≤ α²η/2 < π/2, and by J_2m(2αξ) in the far field, where α²η/2 > 2π. Between the near-field and far-field regions, depending on propagation distance η, the profile evolves from the near-field term to the far-field term J_m(2αξ). The profile difference of the second harmonic Bessel beam in different regions can be physically explained as follows.^[25] Evolution of the second harmonic beam profile from approximately in the near field to J₀(2αξ) in the far field is due to the combination effects of nonlinearity and diffraction. The profile of the primary beam is J₀(αξ) throughout the entire field, and because the square of this field generates the second harmonic, it behaves as a volume distribution of virtual sources with shading proportional to . Radiation from a source plane with a amplitude profile diffracts as it propagates. Diffraction causes a wave radiated from a given virtual source plane to eventually diverge, decrease in amplitude, and shift slightly in phase with respect to the primary beam. It is the superposition of the diffracting radiation from virtual sources far from the observation point, and the second-harmonic components generated close to the observation point and therefore having profiles proportional to and producing the J₀(2αξ) beam profile in the far field. In contrast, a spherically spreading sound in a lossless fluid always generates a second harmonic with a far directivity proportional to the square of the primary-beam directivity, because the far-field contribution always dominates the near-field contribution at sufficiently large distance. What makes Bessel beams different is that the collimation of the field establishes a balance between these near-field and far-field contributions at the observation point.^[25] Thus, the explanation is also suitable to the circumstance of the higher-order harmonics of general Bessel beams.

For each harmonic of the order n, two points are defined: γ_1n and γ_2n. These two points mark the boundaries between the near field and the far field. According to the value of α²/η, we divide the region of beam propagation into three parts for a general n-th harmonic of the Bessel beam. The region satisfying the following condition

is called the near field (or near diffractionless region). In this region, the radial beam profile is almost independent of the propagation distance. The transition region is defined by

The far field (or far diffractionless region) is given by

where the radial profile is also nearly independent of the distance. The values of γ_1n and γ_2n each are generally a function of the order of harmonics and other factors (such as the absorption coefficients of media). In the present analysis, the absorption is ignored.

In what follows, we will derive a more general result, namely that the n-th-order harmonic component in the m-th-order Bessel beam has a J_n×m(nαξ) function distribution in the far field and has a function distribution in the near field. We first consider the asymptotic case of a sufficiently large α²η ≫ γ_2n. This condition can be relaxed to α²η/2 > γ_2n as will be seen later.

From Eqs. (4) and (5) for n = 1 and 2, we assume in general that ℓ-th-order and j-th-order harmonic components may be represented by

and

From Eq. (2b), we have

Applying Eq. (A4) from Appendix A to Eq. (8), a triple integral is obtained as

where λ = (ℓ² + j² − 2ℓ jcost′)^1/2. Note that q̂_mℓj differs from q̄_mℓj by certain constants. By a procedure similar to that of Ref. [24], integration over ξ′ yields

Furthermore, integrating with respect to η′, we find

with b = iα²(ℓj/n)cos²t and t = t′/2.

As shown in Appendix A, when the propagating beam is in the asymptotical far-field region of α²η/2 > γ_2n, the n-th-order harmonic of the J_m Bessel beam is well approximated by

and

We next treat the other limiting case where α²η/2 < γ_1n. In this near field, the second harmonic of the J_m Bessel beam is described by Eq. (5b). From Eqs. (4) and (5b), we assume that the ℓ-th-order harmonic of the m-th-order Bessel beam is

and that the j-th-order harmonic of the m-th-order Bessel beam is

Inserting Eqs. (13) and (14) into Eq. (2b) gives

From Eq. (2a), we obtain

and

This derivation accounts for beam distribution if one excludes beam diffraction. The detailed derivation of A_n is presented in Appendix A.

3. Discussion

Now we come to examine the validity of our asymptotic approximation. We point out that the above asymptotic conditions of α²η/2 ≫ γ_2n and α²η/2 ≪ γ_1n may be relaxed to α²η/2 > γ_2n and α²η/2 < γ_1n, respectively. From the result of Cunningham and Hamilton^[25] (taking α²η/2 to correspond to σ in Ref. [25]), it is easy to show that γ_1n ≈ 1. Then the amplitude of the on-axis beam in the near field 0 ≤ α²η/2 < γ_1n ≈ 1 is approximated by

and in the far field α²η/2 > γ_2n is approximated by

These two approximations are not valid in the transition region. To find the location of a transition point, let the above two amplitudes be equal and let γ_2n be determined roughly. It is found that the transition point is As a result, where C is almost not affected by any of the factors such as the order number n of the harmonic, α, aperture size, etc. To obtain C, we consider the cases of the fundamental and second harmonic amplitudes. For the fundamental amplitude (n=1, where γ_2n = 0 in the entire region), no transition point exists. For the second harmonic amplitude (n = 2), γ_2n ≈ 2π. It is reasonable to deduce that and for a large n.

It is noted that the present analysis is based on the ideal case under which the aperture of the sound source, i.e., the Bessel beam function, is infinite. For a real Bessel beam, we should take into account the finite dimension of its aperture, which limits the field depth and introduces diffractive effects. Also we must clarify the validities of Eqs. (12) and (16), because they are derived by the so-called quasilinear (or successive) approximation method. Neither of these two equations is a uniformly accurate expression for the n-th-order harmonic component of the Bessel beam. Finally, our analysis ignores the absorption of sound in medium. As pointed out by Cunningham and Hamilton, for sufficiently large values of α²η/2, the absorption has a strong influence on the radial distribution of sound beams; this leads to deviation from the beam profile J_m×n(nαξ) in the far field.

Diffraction, strong nonlinearity, and absorption make the problem much more complicated and are beyond the scope of this work. A full analysis may be based on the numerical solution method of the KZK equation developed by Hamilton et al.^[25,27,28]