Angular spectrum method: Difference between revisions

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{{Short description|Technique for modeling the propagation of a wave field}}

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 wave]]s. Its mathematical origins lie in the field of [[Fourier optics]]<ref>''Digital Picture Processing'', 2nd edition 1982, Azriel Rosenfeld, Avinash C. Kak, {{ISBN|0-12-597302-0}}, Academic Press, Inc.</ref><ref>''Linear Systems, Fourier Transforms, and Optics'' (Wiley Series in Pure and Applied Optics) Jack D. Gaskill</ref><ref>''Introduction to Fourier Optics'', Joseph W. Goodman.</ref> 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.<ref>[http://www.egr.msu.edu/~fultras-web/background/asa.php ''Angular Spectrum Approach'', Robert J. McGough]</ref>

{{distinguish|Angular spectrum expansion}}

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 wave]]s of the same frequency and different directions. Its mathematical origins lie in the field of [[Fourier optics]]<ref>''Digital Picture Processing'', 2nd edition 1982, Azriel Rosenfeld, Avinash C. Kak, {{ISBN|0-12-597302-0}}, Academic Press, Inc.</ref><ref>''Linear Systems, Fourier Transforms, and Optics'' (Wiley Series in Pure and Applied Optics) Jack D. Gaskill</ref><ref>''Introduction to Fourier Optics'', Joseph W. Goodman.</ref> 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.<ref>[http://www.egr.msu.edu/~fultras-web/background/asa.php ''Angular Spectrum Approach'', Robert J. McGough]</ref>

Modeling the diffraction of a CW (continuous wave), monochromatic (single frequency) field involves the following steps:

# Sampling the complex (real and imaginary) components of a pressure field over a grid of points lying in a cross-sectional plane within the field.

# Taking the 2D-[[Fast Fourier transform|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.

# 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.

# 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,<ref>''Cross-Sectional{{cite Measurementsjournal and Extrapolations of Ultrasonic Fields'',| last1=Waag, | first1=R.C. | last2=Campbell, | first2=J.A. | last3=Ridder, | first3=J. | last4=Mesdag, | first4=P.R., Sonics| title=Cross-Sectional Measurements and Ultrasonics,Extrapolations of Ultrasonic Fields | journal=IEEE Transactions on, JanSonics 1985,and Volume:Ultrasonics 32,| publisher=Institute Issue:of Electrical and Electronics Engineers (IEEE) | volume=32 | issue=1, pp.| 26year=1985 | issn=0018-9537 35| doi=10.1109/t-su.1985.31566 | pages=26–35| bibcode=1985ITSU...32...26W }}</ref><ref>''{{cite journal | last1=Stepanishen | first1=Peter R. | last2=Benjamin | first2=Kim C. | title=Forward and backward projection of acoustic fields using FFT methods'', Stepanishen,| P.journal=The R.;Journal Benjamin,of K.the C.,Acoustical Society of America | publisher=Acoustical Society of America, Journal,(ASA) vol.| volume=71, Apr.| issue=4 | year=1982, p.| 803issn=0001-8124966 | doi=10.1121/1.387606 | pages=803–812| bibcode=1982ASAJ...71..803S }}</ref> 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).<ref>''{{cite journal | last1=Vecchio | first1=Christopher J. | last2=Lewin | first2=Peter A. | title=Finite amplitude acoustic propagation modeling using the extended angular spectrum method'', Chris| Vecchiojournal=The andJournal Peterof Lewin,the JournalAcoustical Society of theAmerica | publisher=Acoustical Society of America, JASA(ASA) | volume=95 (| issue=5), pp.| 2399year=1994 | issn=0001-2408,4966 1994| doi=10.1121/1.409849 | pages=2399–2408| bibcode=1994ASAJ...95.2399V }}</ref><ref>''Acoustic{{cite Propagationconference Modeling|conference=14th UsingAnnual International Conference of the ExtendedIEEE AngularEngineering Spectrumin Method'',Medicine Chrisand Biology Society| last1=Vecchio and| Peterfirst1=Chris | last2=Lewin, | Proceedingsfirst2=Peter ofA. | title=Acoustic propagation modeling using the 14thextended Annualangular Internationalspectrum Conferencemethod | publisher=IEEE Engineering| inyear=1992 Medicine| andisbn=0-7803-0785-2 Biology Society,| doi=10.1109/iembs.1992.5762211 }}</ref><ref>''New{{cite approachesjournal to| nonlinearlast1=Christopher diffractive field propagation'',| first1=P. Ted Christopher| andlast2=Parker | first2=Kevin J. Parker,| title=New approaches to nonlinear diffractive field propagation | journal=The Journal of the Acoustical Society of America, July| 1991,publisher=Acoustical VolumeSociety of America (ASA) | volume=90, Issue| issue=1, pp.| 488year=1991 | issn=0001-4994966 | doi=10.1121/1.401274 | pages=488–499| pmid=1880298 | bibcode=1991ASAJ...90..488C }}</ref><ref>Modeling{{cite ofjournal nonlinear| ultrasoundlast1=Zemp propagation in tissue from array transducers,| first1=Roger J. Zemp,| last2=Tavakkoli | first2=Jahangir Tavakkoli,| andlast3=Cobbold | first3=Richard S. C. Cobbold,| title=Modeling of nonlinear ultrasound propagation in tissue from array transducers | journal=The Journal of the Acoustical Society of America, January| 2003,publisher=Acoustical Society of America (ASA) Volume| volume=113, Issue| issue=1, pp.| 139year=2003 | issn=0001-1524966 | doi=10.1121/1.1528926 | pages=139–152| pmid=12558254 | bibcode=2003ASAJ..113..139Z }}</ref><ref>[http://adsabs.harvard.edu/abs/{{cite thesis|bibcode=1992PhDT........59V |title= Finite Amplitude Acoustic Propagation Modeling Using the Extended Angular Spectrum Method]|last=Vecchio|first=Christopher John|year=1992|type=PhD|publisher=Dissertation Abstracts International}}</ref>

Backward propagation predictions can be used to analyze the surface vibration patterns of acoustic radiators such as [[ultrasonic transducer]]s.<ref>''Transducer{{cite Characterizationjournal using| thelast1=Schafer Angular| Spectrumfirst1=Mark Method'', M.E. Schafer| andlast2=Lewin | first2=Peter P.A. Lewin,| title=Transducer J.characterization Acoust.using Soc.the Am.angular spectrum method | journal=The Journal of the Acoustical Society of America | publisher=Acoustical Society of America (ASA) | volume=85: | issue=5, 2202-2214,| year=1989 | issn=0001-4966 | doi=10.1121/1.397869 | pages=2202–2214| bibcode=1989ASAJ...85.2202S | doi-access=free }}</ref> Forward propagation can be used to predict the influence of inhomogeneous, nonlinear media on acoustic transducer performance.<ref>''{{cite journal | last1=Vecchio | first1=Christopher J. | last2=Schafer | first2=Mark E. | last3=Lewin | first3=Peter A. | title=Prediction of ultrasonic field propagation through layered media using the extended angular spectrum method'', Chris| Vecchio,journal=Ultrasound Markin Schafer,Medicine Peter& Lewin,Biology Ultrasound| Medpublisher=Elsevier Biol.BV 1994;| volume=20( | issue=7):611 | year=1994 | issn=0301-225629 | doi=10.1016/0301-5629(94)90109-0 | pages=611–622| pmid=7810021 }}</ref>

==See also==

* [[WeylWave expansionfield synthesis]]

==References==

{{reflist}}