^{1,a)}and Robert J. McGough

^{1,b)}

### Abstract

The implementation of the angular spectrum approach based on the two-dimensional fast Fourier transform is evaluated for near-field pressure simulations of square ultrasound transducers, where the three-dimensional pressure field is calculated from the normal velocity distribution on the transducer surface. The pressure field is propagated in the spatial frequency domain with the spatial propagator or the spectral propagator. The spatial propagator yields accurate results in the central portion of the computational grid while significant errors are produced near the edge due to the finite extent of the window applied to the spatial propagator. Likewise, the spectral propagator is inherently undersampled in the spatial frequency domain, and this causes high frequency errors in the computed pressure field. This aliasing problem is alleviated with angular restriction. The results show that, in nonattenuating media, the spatial propagator achieves smaller errors than the spectral propagator after the region of interest is truncated to exclude the windowing error. For pressure calculations in attenuating media or with apodized pistons as sources, the spatial and spectral propagator achieve similar accuracies. In all simulations, the angular spectrum calculations with the spatial propagator take more time than calculations with the spectral propagator.

This work was supported in part by NIH R01CA093669, NIH R21CA121235, and NSF Theoretical Foundations Grant No. 0634786.

I. INTRODUCTION

II. THEORY

A. The Rayleigh–Sommerfeld integral

B. The fast near-field method

C. The angular spectrum approach

D. Angular restriction

E. Attenuation calculations

F. Error metric

III. NUMERICAL RESULTS

A. Pressure calculations in nonattenuating media for a square piston with uniform normal particle velocity

B. Pressure calculations in attenuating media for a square piston with uniform normal particle velocity

C. Pressure calculations in nonattenuating media for a square piston with apodized normal particle velocity

IV. DISCUSSION

A. Reference pressure calculations

B. Error and time trade-offs

C. Error sources for the spectral propagator

D. Error sources for the spatial propagator

V. CONCLUSION

### Key Topics

- Spatial analysis
- 35.0
- Wave attenuation
- 12.0
- Fourier transforms
- 9.0
- Particle velocity
- 9.0
- Spatial filtering
- 8.0

## Figures

(Color online) The source plane consisting of a nonzero normal particle velocity distribution in a square area on the piston surface. The remaining area is filled with zeros.

(Color online) The source plane consisting of a nonzero normal particle velocity distribution in a square area on the piston surface. The remaining area is filled with zeros.

A two-dimensional cross section of the three-dimensional reference pressure generated by a square piston in nonattenuating media. The excitation frequency is , and the normal particle velocity distribution is uniform across the piston surface. The reference pressure is computed with the fast near-field method. The result is normalized to the maximum pressure amplitude computed in the three-dimensional volume.

A two-dimensional cross section of the three-dimensional reference pressure generated by a square piston in nonattenuating media. The excitation frequency is , and the normal particle velocity distribution is uniform across the piston surface. The reference pressure is computed with the fast near-field method. The result is normalized to the maximum pressure amplitude computed in the three-dimensional volume.

Simulated pressure generated by a square piston in nonattenuating media computed by the angular spectrum approach using (a) the spectral propagator without angular restriction, (b) the spectral propagator with angular restriction, and (c) the spatial propagator. The excitation frequency is . All fields are calculated in successive transverse planes , so the circular convolution errors generated by the spatial propagator are included.

Simulated pressure generated by a square piston in nonattenuating media computed by the angular spectrum approach using (a) the spectral propagator without angular restriction, (b) the spectral propagator with angular restriction, and (c) the spatial propagator. The excitation frequency is . All fields are calculated in successive transverse planes , so the circular convolution errors generated by the spatial propagator are included.

Axial plots of the absolute value of the simulated complex pressure. Results are shown for the reference, the angular spectrum approach using the spectral propagator without and with angular restriction, and the spatial propagator.

Axial plots of the absolute value of the simulated complex pressure. Results are shown for the reference, the angular spectrum approach using the spectral propagator without and with angular restriction, and the spatial propagator.

Normalized root mean squared errors for the pressure generated by a uniform normal particle velocity distribution in nonattenuating media. The errors are evaluated in three-dimensional volumes where the lateral dimensions are (a) and (b) with . The markers on the curves indicate the corresponding results shown in Fig. 3.

Normalized root mean squared errors for the pressure generated by a uniform normal particle velocity distribution in nonattenuating media. The errors are evaluated in three-dimensional volumes where the lateral dimensions are (a) and (b) with . The markers on the curves indicate the corresponding results shown in Fig. 3.

The absolute value of the complex reference pressure generated by a square piston in attenuating media . The excitation frequency is and the normal velocity distribution is uniform across the piston surface. The reference pressure is computed with the fast near-field method.

The absolute value of the complex reference pressure generated by a square piston in attenuating media . The excitation frequency is and the normal velocity distribution is uniform across the piston surface. The reference pressure is computed with the fast near-field method.

The absolute value of the simulated complex pressure in attenuating media computed with the angular spectrum approach using (a) the spectral propagator without angular restriction, (b) the spectral propagator with angular restriction, and (c) the spatial propagator. All fields are calculated in successive transverse planes and truncated to , where .

The absolute value of the simulated complex pressure in attenuating media computed with the angular spectrum approach using (a) the spectral propagator without angular restriction, (b) the spectral propagator with angular restriction, and (c) the spatial propagator. All fields are calculated in successive transverse planes and truncated to , where .

Normalized root mean squared errors for the pressure generated in attenuating media by a uniformly excited square piston. The errors are evaluated in three-dimensional volumes where the lateral dimensions are with . The markers on the curves indicate the corresponding results shown in Fig. 7.

Normalized root mean squared errors for the pressure generated in attenuating media by a uniformly excited square piston. The errors are evaluated in three-dimensional volumes where the lateral dimensions are with . The markers on the curves indicate the corresponding results shown in Fig. 7.

The absolute value of the simulated complex pressure in nonattenuating media generated by a square piston with apodized normal particle velocity distribution. (a) The reference pressure and (b) the pressure computed by the spectral propagator without angular restriction. The excitation frequency is .

The absolute value of the simulated complex pressure in nonattenuating media generated by a square piston with apodized normal particle velocity distribution. (a) The reference pressure and (b) the pressure computed by the spectral propagator without angular restriction. The excitation frequency is .

Normalized root mean squared errors for the pressure generated in nonattenuating media by a square piston with an apodized normal particle velocity distribution. The errors are evaluated in three-dimensional volumes where the lateral dimensions are with . The marker on the dotted line indicates the result in Fig. 9(b).

Normalized root mean squared errors for the pressure generated in nonattenuating media by a square piston with an apodized normal particle velocity distribution. The errors are evaluated in three-dimensional volumes where the lateral dimensions are with . The marker on the dotted line indicates the result in Fig. 9(b).

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