Angular Spectrum Approach

The Angular Spectrum Approach (ASA) was first used in optics to propagate fields from a source plane to a destination plane [1]. ASA decomposes wavefields within a single plane into plane wave components via a 2D spatial Fourier transform. Each plane wave component is then propagated through the Fourier domain to a destination plane. The wavefield in the destination plane is then reconstructed via an inverse spatial Fourier transform. When performed on a uniform spatial grid, the forward and inverse Fourier transforms are performed with FFT's, thereby providing a fast and simple computational framework. Because ASA is both efficient and easy to understand, ASA has been previously utilized in therapeutic ultrasound in the design of transducers [2].

However, ASA is subject to several sources of numerical error due to aliasing and grid truncation. These errors have been analyzed [3], and an optimized ASA has been developed for several situations. To mitigate aliasing errors, a low-pass filter that restricts the propagation vectors (angular restriction) is utilized, as well as zero-padding to alleviate grid truncation errors. In addition, the user may select either a spectral propagator, which uses the analytical Fourier transform for propagating plane wave components, or a spatial propagator, which uses a numerical Fourier transform. Attenuation has been incorporated into the FOCUS ASA implementation in order to simulate absorption in soft tissue.

Since ASA requires a source plane as initial data, the Fast Nearfield Method (FNM) is used to initialize ASA. For large volumetric field computations on uniform grids, ASA outperforms all other computational methods [4]. Additional physics, such as refraction through multiple boundaries [5] and nonlinear wave propagation [6] have been incorporated and may be selected by the user. Since ASA decomposes wavefields into plane waves, layered media is easily modeled using the transmission coefficients and Snell's law relationships for plane waves. Nonlinear linear fields are synthesized by solving Burger's equation for a specified number of harmonics. Thus, observed phenomena such as wave steepening and shock wave formation may be included in thermal therapy simulations.


  1. J. W. Goodman, "Introduction to Fourier Optics", 2nd ed., McGraw-Hill, New York, 1996.
  2. G. T. Clement and K Hynynen, "Field Characterization of therapeutic ultrasound phased arrays through forward and backward planar projection", J. Acoust. Soc. Am., 108(1):441-446, 2000.
  3. X. Zeng and R. J. McGough, "Evaluation of the Angular Spectrum Approach for Simulations of Near-Field Pressures", J. Acoust. Soc. Am., 123(1):68-76 ,2008.
  4. X. J. Zeng and R. J. McGough. "Optimal simulations of ultrasonic fields produced by large thermal therapy arrays using the angular spectrum approach." J. Acoust. Soc. Am. 125, 2967-2977 (2009).
  5. P. T. Christopher and K. J. Parker, "New approaches to the linear propagation of acoustic fields", J. Acoust. Soc. Am., 90(1): 507-521, 1991.
  6. P. T. Christopher and K. J. Parker, "New approaches to nonlinear diffractive field propagation", J. Acoust. Soc. Am., 90(1):488-499, 1991.