Chaotic Cavities


Time-Reversal Chaotic Cavities for Medical Ultrasound

Kevin J. Haworth, J. Brian Fowlkes, Paul L. Carson, Oliver D. Kripfgans

Department of Radiology, University of Michigan, Ann Arbor, MI 48109-0553 USA


Time-reversal acoustics is an approach to focusing sound through a complicated medium.  In time-reversal acoustics, a signal is recorded after propagating through the medium.  The recorded signal is reversed in time and retransmitted (first in becomes last out).  Taking advantage of the time-invariance of the lossless wave-equation, the retransmitted wave will propagate back along the paths it came from and refocus through the medium. 

Many times the medium through which the sound travels through is unknown and uncontrollable.  However for time-reversal chaotic cavities (TRCC), one chooses a reverberant cavity as the medium through which the sound propagates.  The cavity is chosen so that the reverberations are chaotic (and ergodic).  The result is that a pulse transmitted from outside the cavity and received inside of it is a unique signature (figures 1 and 2).  If the source is moved to a different location, the acoustic signature recorded will be completely uncorrelated from the acoustic signature recorded at the original location.


                                         Figure 1.                                                                            Figure 2.


                                        Figure 3.                                                                           Figure 4.

Several different approaches have been discussed in the literature for correcting aberrations.  One approach of particular interest is time-reversal acoustics.  This approach entails recording the ultrasonic echo of a point-reflector (Figure 2) and then retransmitting the recorded echos in a time-reversed fashion as demonstrated (Figure 3).

One can think of time-reversal acoustics with chaotic cavities as being a pulse-compression scheme.  Thus TRCCs can be used in situations where other pulse-compression schemes are used.  One area of particular interest is the ability to obtain very large amplitude pulses from pulses that are initially small.


Our group has worked with a variety of TRCCs and demonstrated many different results.  This includes using cavities to focus through a human skull (figures 5 and 6).



                                    Figure 5.                                                                             Figure 6.

We are also interested in predicating how well TRCCs can focus.  As can be seen in figures 4 and 6, there are non-negligible side-lobes.  Using the initial work of Derode et al. (1999, 2000) we have developed a statistical model to predict the expected amplitude of the mainlobe (eqn. 1) and sidelobes (eqn 2). 


Where g(t) describes the function initially transmitted into the cavity and W(t) is the windowing applied to the time-reversed waveform (figure 3). The statistical nature of the cavity is described by σ(t) (the decay shape of the impulse response of the cavity) and  ρ(t) (the autocorrelation function of the impulse response).


Equations 1 and 2 allow us to investigate through simulations, how cavities respond to a variety of parameter choices.  We are currently studying the impact of the parameter choices and will use the results in designing future cavities for specific uses.


Haworth KJ, Fowlkes JB, Carson PL, Kripfgans OD. Generalized Shot Moise Model for Time-Reversal Acoustics in Multiple Scattering Media.  J Acoust Soc Am. to be submitted

Haworth KJ, Fowlkes JB, Carson PL, Kripfgans OD. Modeling time-reversal

focusing in a multiple scattering medium. Acoustics ‘08: Acoustical Society of America and European Acoustics Association joint meeting. 31 June - 4 July 2008, Paris, France (ASA2008Haworth.pdf)

Haworth KJ, Fowlkes JB, Carson PL, Kripfgans OD. Pulse Length Dependence

for Acoustic Time Reversal. Radiological Society of North America. 29 November 2006, Chicago, Illinois, USA, Scientific Assembly Program: pp.469

Derode A, Tourin A, Fink M. Limits of time-reversal focusing through multiple scattering: long-range correlation. J Acoust Soc Am. 2000 Jun;107(6):2987-98.

Derode A, Tourin A, Fink M. Ultrasonic pulse compression with one-bit time reversal through multiple scattering.  J. Appl. Phys. 1999; 85: 6343

Acknowledgement:  This work is supported in part by NIH Grant 1R21CA116043.