# Angular spectrum method

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[1][2][3] but it has been applied extensively in the field of ultrasound. 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.[4]

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

In addition to predicting the effects of diffraction,[5][6] 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).[7][8][9][10][11]

Backward propagation predictions can be used to analyze the surface vibration patterns of acoustic radiators such as ultrasonic transducers.[12] Forward propagation can be used to predict the influence of inhomogeneous, nonlinear media on acoustic transducer performance.[13]

## References

1. ^ Digital Picture Processing, 2nd edition 1982, Azriel Rosenfeld, Avinash C. Kak, ISBN 0-12-597302-0, Academic Press, Inc.
2. ^ Linear Systems, Fourier Transforms, and Optics (Wiley Series in Pure and Applied Optics) Jack D. Gaskill
3. ^ Introduction to Fourier Optics, Joseph W. Goodman.
4. ^ Angular Spectrum Approach, Robert J. McGough [1]
5. ^ Cross-Sectional Measurements and Extrapolations of Ultrasonic Fields, Waag, R.C. Campbell, J.A. Ridder, J. Mesdag, P.R., Sonics and Ultrasonics, IEEE Transactions on, Jan 1985, Volume: 32, Issue: 1, pp. 26- 35.
6. ^ Forward and backward projection of acoustic fields using FFT methods, Stepanishen, P. R.; Benjamin, K. C., Acoustical Society of America, Journal, vol. 71, Apr. 1982, p. 803-812.
7. ^ Finite amplitude acoustic propagation modeling using the extended angular spectrum method, Chris Vecchio and Peter Lewin, Journal of the Acoustical Society of America, JASA 95 (5), pp. 2399-2408, 1994.
8. ^ Acoustic Propagation Modeling Using the Extended Angular Spectrum Method, Chris Vecchio and Peter Lewin, Proceedings of the 14th Annual International Conference IEEE Engineering in Medicine and Biology Society, 1992.
9. ^ New approaches to nonlinear diffractive field propagation, P. Ted Christopher and Kevin J. Parker, Journal of the Acoustical Society of America, July 1991, Volume 90, Issue 1, pp. 488-499.
10. ^ Modeling of nonlinear ultrasound propagation in tissue from array transducers, Roger J. Zemp, Jahangir Tavakkoli, and Richard S. C. Cobbold, Journal of the Acoustical Society of America, January 2003, Volume 113, Issue 1, pp. 139-152.
11. ^ Finite Amplitude Acoustic Propagation Modeling Using the Extended Angular Spectrum Method
12. ^ Transducer Characterization using the Angular Spectrum Method, M.E. Schafer and P.A. Lewin, J. Acoust. Soc. Am. 85:5, 2202-2214, 1989.
13. ^ Prediction of ultrasonic field propagation through layered media using the extended angular spectrum method, Chris Vecchio, Mark Schafer, Peter Lewin, Ultrasound Med Biol. 1994;20(7):611-22.