Time Reversal and Its Application to Tomography with Diffracting Sources

Abstract

An exact time-domain method is proposed to time reverse a transient scalar wave using only the field measured on an arbitrary closed surface enclosing the initial source. Under certain conditions, a timereversed field can be approximated by retransmitting the measured signals in a reversed temporal order. Exact reconstruction for three-dimensional broadband diffraction tomography (a linearized inverse scattering problem) is proposed by time-reversing the measured field back to the time when each secondary source is excited. The algorithm is verified by a numerical simulation. Extension to the case using Green's function in a heterogeneous medium is discussed. DOI: 10.1103/PhysRevLett.92.033902 PACS numbers: 42.30.Wb, 42.25.Fx Time reversal of an acoustic or electromagnetic wave is based on the invariance of the wave equation in a lossless medium under the transform t ! ÿt (t represents the time). Time reversal of a wave can be understood as generating the back-propagation field from the measured forward-propagation field and/or its normal derivative after removing the initial sources. The concept of time reversal has been implemented experimentally and applied to a wide range of studies such as inverse scattering [1], wave-front distortion correction [2,3], and multiple scattering phenomena [4].However, no formula is available for computing the time-reversed (TR) field using only the measured field on a closed surface enclosing the initial source. When both the field and its normal gradient on a closed surface are available, there are formulas [3,5] to derive the TR field. However, it is not practical to measure both the field and its normal gradient simultaneously. For example, the output signal from a piezoelectric transducer is generally a complex combination of these two effects. There are two challenges in deriving the TR field using only the field. First, it is not obvious that Green's function, which is widely used to derive the field in space from the field on a closed surface, can be applied here. This is because the TR field on the closed surface includes both diverging and converging components [3,5]. While the converging component of the TR wave is just the measured signals in the forward propagation in a reversed temporal order (RTO), the diverging component has no counterpart in the forward propagation and, consequently, is not available from measurement in general. Second, in a free space, retransmission of the measured signals in RTO from the detection surface does not reproduce the TR field. This is because the waves retransmitted in one position propagate to the other positions on the surface and change the field there, and, consequently, the field on the surface does not equal the field in the forward propagation in RTO.In this Letter, we find that when time reversal is considered in the time domain, an exact time-reversal method that uses only the field on an arbitrary closed surface can be derived for a wide variety of applications such as tomography with diffracting sources, inverse dif...

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