## 1. Introduction

[2] Time reversal has attracted considerable attention in recent years, particularly because of its potential for communication and imaging through a complex environment. When a wave is emitted by a point source and is received by an array of receivers, and time-reversed and back-propagated in the same medium, the wave is refocused near the original source. The time reversal array is also called the “time-reversed mirror” or the “conjugate mirror”. Interesting superresolution phenomena and application to detection and medical imaging are included in comprehensive reviews given by *Fink et al.* [2000] and applications of time reversal technique for communications have been discussed [*Lerosey et al.*, 2005; *Yun and Iskander*, 2006; *Jian et al.*, 2006]. This paper is focused on a particular problem of analytical study of time reversal effects on shower curtain effects, superresolution, and backscattering enhancement effects in random medium.

[3] If the medium is free space, it is clear that the time-reversed pulse wave is refocused with the resolution determined by the aperture size of the array. It has been known that if the medium is random causing multiple scattering, the wave is refocused with the resolution better than that in free space, contrary to our intuition. This is called the “superresolution” which has been studied experimentally and numerically, and some theoretical explanations have been offered [*Fink et al.*, 2000; *Kuperman et al.*, 1998; *Derode et al.*, 2001a, 2001b; *Blomgren et al.*, 2002; *Lerosey et al.*, 2004; *Clouet and Fouque*, 1997; *Borcea et al.*, 2002]. *Liu et al.* [2007] discusses a study on the effects of changing media on time reversal, and metrics for time reversal are discussed by *Oestges et al.* [2005]. This paper presents a detailed analytical study of time reversal in random media. It includes several features. The relationship between the superresolution and the coherence length has been pointed out in the past. The theory in this paper gives an analytical study based on the circular complex Gaussian assumption which shows the shower curtain effects and the backscattering enhancement in time reversal.

[4] The formulation is based on our previous studies on stochastic Green's functions [*Ishimaru*, 1997; *Ishimaru et al.*, 2004, 2006, 2007]. First we consider the first moment of the refocused field, making use of the mutual coherence function and the Gaussian phase function for the random medium. The point source emits a Gaussian modulated pulse. The first moment consists of two terms. One is the coherent field which is attenuated because of the optical depth and the other is the diffuse component. The coherent image is substantially the same as that in free space, except for attenuations. However, the diffuse component, which is dominant for large optical depth, has a much smaller spot size than that in free space. This superresolution is due to the coherence length which is smaller than the free space spot size. As the multiple scattering increases, the transverse coherence length decreases in proportion to the inverse of the square root of the scattering depth, resulting in a smaller spot size and superresolution. The longitudinal spot size along the propagation direction is substantially the same as the original pulse because this is the first moment. This formulation also gives the shower curtain effect giving higher resolution when the random medium is closer to the source.

[5] Next we consider the second moment. Because of the time reversal back-propagations, this second moment requires the fourth-order Green's functions. We employ the circular complex Gaussian assumption to reduce the fourth moment to the second moment [*Goodman*, 1985]. Since we deal with the time-space Green's function, we used two-frequency mutual coherence functions based on the extended Huygens-Fresnel formulations [*Andrew and Phillips*, 1998; *Ishimaru et al.*, 2006]. The second moment has been studied analytically [*Derode et al.*, 2001a, 2001b] using diffusion approximation in an infinite medium. However, this paper employs the mutual coherence function which gives the shower curtain effects and the backscattering enhancement. Numerical examples are given to illustrate these random media effects on time reversal.