# The Spherical Harmonic Family of Beampatterns

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. General Solution

#### 2.2. Special Properties of Sums of Spherical Bessel Functions

#### 2.3. Modifications via Imaginary Shift in Coordinates

#### 2.4. Other Simple Modifications and Properties

_{mod}= 4, this imposed modulation of integer orders matches the natural modulation of the ${j}_{n}$ and ${Y}_{n}^{0}$.

## 3. Methods

_{max}= 200 corresponds to an active high amplitude transducer radius of approximately 10 mm, with a residual decay beyond that radius. We also assume a flat transducer so that the cone angle of Figure 1 is set to $\pi /2$; this models the use of a conventional piston transducer, albeit with active control of the amplitude and phase of the excitation across the transducer.

## 4. Results

_{n}= 1) and the square root weighted sum (a

_{n}= n

^{1/2}). These display a strong central axis amplitude with oscillations in the radial direction as shown in Figure 6.

_{0}[kx], thus resembling the classical Bessel beam described by Durnin as implemented within the framework of the optical axicon [2,3].

_{mod}= 2.5. This modulates the amplitudes of the integer orders within the sum, and thereby serves as a rough equivalent to changing the angle of incidence in conical optical axicon configurations.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Geometry for expansion of spherical waves within a cone of polar angle ${\theta}_{s}$, and with the surface of the cone being an active source capable of generating spatial distributions in the form of spherical Bessel functions and sums of spherical Bessel functions. The azimuthal angle is $\phi $, the radial coordinate ${r}_{s}$ lies on the active source, and z represents the zenith direction.

**Figure 2.**The first four orders of spherical Bessel functions j

_{n}(x) as a function of argument x (abscissa). Note that each initial maximum is successively delayed and diminished, while the asymptotes can create out-of-phase cancellation The peaks from left to right correspond to integer orders 0 (blue line), 1 (yellow line), 2 (green line), and 3 (orange line).

**Figure 3.**First four orders of the spherical harmonic functions ${Y}_{n}^{0}\left(\theta \right)$ as a function of $\theta $ (abscissa). The zero order is flat and successive orders are increasingly higher order in cosine terms. These are also related to the Legendre polynomials. When $\theta =\pi /2$, at the surface of a flat piston transducer, the values of ${Y}_{n}^{0}\left(\pi /2\right)$ reduce to simple oscillations around ±0.318 as a function of increasing n with a period of 4n. The colors correspond to integer orders 0 (blue line), 1 (yellow line), 2 (green line), and 3 (orange line).

**Figure 4.**The sum of the first 200 spherical Bessel functions of argument x, weighted as in Equation (5), from x = {1,250}. This illustrates a general property of the sums resembling a bandlimited interpolation function within a range where x ≤ N.

**Figure 5.**The use of a Gaussian function as f(n) in Equation (6) with a summation up to N = 200 integer orders results in a Gaussian function of x (abscissa). The blue curve is the sum of the Gaussian-weighted spherical Bessel functions, where the Gaussian function of n has a standard deviation of 70, and an amplitude of $\sqrt{\pi /2}$ The yellow curve is the same Gaussian function of x. This illustrates how easily apodization functions can be included into the framework of the sum of spherical Bessel functions.

**Figure 6.**5 MHz ultrasound beampatterns from the sum of the first 200 orders of spherical Bessel Functions with (

**top**) $\sqrt{n}$ weighting and (

**bottom**) equal weighting on a flat, radially symmetric transducer located at the left-hand side of each figure. Linear amplitude plots are shown with lateral range −1.5 to 1.5 mm and axial range from the origin to 20 mm. On the far right is the color bar used for beampattern figures, with the vertical scale indicating the percent of the maximum value.

**Figure 7.**Axial beampattern plots for cases shown in Figure 6, (top and bottom, respectively) 5 MHz beams with distance in mm. Vertical units: arbitrary units of wave amplitude.

**Figure 8.**Transverse wave amplitude (arbitrary units) as a function of lateral distance from the center axis for the two cases shown in Figure 7. These are measured at 8 mm range from the transducer and are closely approximated by a Bessel function of zero order.

**Figure 9.**Beampatterns with the same parameters as Figure 6 (top) with the addition of the z offset by imaginary number q = 1/2 (

**top**) and 1 (

**bottom**). The q factor produces stronger source amplitudes at radii greater than 9 mm (beyond the scale shown), which contribute to the emerging pattern at axial range greater than 10 mm.

**Figure 10.**Beampatterns for 5 MHz ultrasound examples with transducer surface on the left, illustrating (

**top**) the localization property of limiting the range of N, and (

**bottom**) the modulation property of oscillations of a

_{n}. Note that the axial range in Figure 10 (

**bottom**) is extended to 30 mm.

**Figure 11.**5 MHz beampattern produced with only a single spherical Bessel function of integer order n = 199 and with imaginary offset parameter q = 1/2. The transducer is at the left and the axial range extends to 120 mm.

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**MDPI and ACS Style**

Parker, K.J.; Alonso, M.A.
The Spherical Harmonic Family of Beampatterns. *Acoustics* **2022**, *4*, 958-966.
https://doi.org/10.3390/acoustics4040059

**AMA Style**

Parker KJ, Alonso MA.
The Spherical Harmonic Family of Beampatterns. *Acoustics*. 2022; 4(4):958-966.
https://doi.org/10.3390/acoustics4040059

**Chicago/Turabian Style**

Parker, Kevin J., and Miguel A. Alonso.
2022. "The Spherical Harmonic Family of Beampatterns" *Acoustics* 4, no. 4: 958-966.
https://doi.org/10.3390/acoustics4040059