Angular spectrum method
Encyclopedia
The angular spectrum method is a technique for modeling the propagation of a wave field. This technique involves expanding a complex wave field into a summation of infinite number of plane waves. Its mathematical origins lie in the field of Fourier Optics
Fourier optics
Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...

but it has been applied extensively in the field of ultrasound
Ultrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...

. The technique can predict an acoustic pressure field distribution over a plane, based upon knowledge of the pressure field distribution at a parallel plane. Predictions in both the forward and backward propagation directions are possible.

Modeling the diffraction of a CW (continuous wave), monochromatic (single frequency) field involves the following steps:
1. Sampling the complex (real and imaginary components of a) pressure field over a grid of points lying in cross-sectional plane within the field.
2. Taking the 2D-FFT (two dimensional Fourier Transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

) of the pressure field contour - this will decompose the field into a 2D "angular spectrum" of component plane waves each traveling in a unique direction.
3. Multiplying each point in the 2D-FFT by a propagation term which accounts for the phase change that each plane wave will undergo on its journey to the prediction plane.
4. Taking the 2D-IFFT (two dimensional inverse Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

) of the resulting data set to yield the field contour over the prediction plane.

In addition to predicting the effects of diffraction, the model has been extended to apply to non-monochromatic cases (acoustic pulses) and to include the effects of attenuation, refraction, and dispersion. Several researchers have also extended the model to include the nonlinear effects of finite amplitude acoustic propagation (propagation in cases where sound speed is not constant but is dependent upon the instantaneous acoustic pressure).

Backward propagation predictions can be used to analyze the surface vibration patterns of acoustic radiators such as ultrasonic transducers. Forward propagation can be used to predict the influence of inhomogeneous, nonlinear media on acoustic transducer performance.