In this paper, we present analytical theories of the first and second moments of the time-reversed pulse which are obtained by the second- and the fourth-order moments of stochastic Green's functions in random media. The theories are based on the use of parabolic approximations, two-frequency mutual coherence function, and the circular complex Gaussian assumptions. Several effects are discussed including the superresolution, the coherence length, backscattering enhancement, the dependence on the optical depth, the shower curtain effect, and the bandwidth.
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 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.  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.
 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.  discusses a study on the effects of changing media on time reversal, and metrics for time reversal are discussed by Oestges et al. . 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.
 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.
 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.
2. Formulation of the Problem
 Consider the problem shown in Figure 1. A point source at t (L, 0) emits a Gaussian modulated pulse given by
Its spectrum is given by
where Δω = 2/To is the bandwidth and ωo is the carrier frequency. The Green function G1 is the wave emitted at (L, 0) and observed at (, 0). This field G1 is then time-reversed, which is the same as the conjugate G*1 in frequency domain. This time-reversed field is sent into the same medium. The spectrum of the time-reversed field is the complex conjugate of the spectrum of the original field. Thus this is called the “time reversal mirror” or the “conjugate mirror”.
 The field at s is the sum of all contributions from 2M + 1 transducers and given by the spectrum
The average field is therefore given by
where Γ is the mutual coherent function of the field at s and t with the source at m.
It is therefore necessary to calculate the mutual coherence function in order to find the first moment or the average field. This will be discussed in section 3. Next we consider the second moment.
 The field at s is G*1G2, and therefore the second moment is 〈(G*1G2)(G*3G4)*〉, where (G*1G2) is with ω1 and (G*3G4) is with ω2. We then get the second moment:
where Γ(ω1, ω2) = 〈G*1G2G3G*4〉, G1 = G(, t, ω1), G2 = G(s, , ω1), G3 = G(, t, ω2), G4 = G(s, , ω2), = mho, = nho, ho is the element spacing and we use ho = λo/2 to avoid grating lobes. Note that the second moment requires a study of the fourth-order moments. This will be discussed further in section 4.
 In the next sections we examine the above general formula to obtain analytical expressions for 〈ψ〉 and 〈∣ψ∣2〉.
3. Time-Reversed Coherent Field 〈ψ〉, the First Moment
 The coherent field (4) can be expressed in analytical form under parabolic approximation for Green's function. This has been studied by Borcea et al.  and others. However, here we give detailed analytical expression including the shower curtain effects [Ishimaru et al., 2004]. We let Δz = 0 and examine the transverse resolution and time dependence at z = L.
 In order to study the coherent field ψ〉, we need to examine the mutual coherence function Γm in (5). For the problem shown in Figure 1 we have
where the array is along the x axis and the observation is made along the x axis ( = mho and s = ρs). This is the mutual coherence function between the waves at (s, L) and (0, L) emitted by a point source at and is obtained by noting c = and d = s in equation (20–68) of [Ishimaru, 1997].
 The function H(s) represents the effect of random medium, and for a random distribution of particles it is given by [Ishimaru et al., 2004]:
where P = , a(z′), and b(z′) are the absorption and scattering coefficient, respectively, p(s) is the phase function of the scattering pattern of a single scatterer, and s = 2sin , θ = scattering angle.
 It is often convenient to separate the random medium effect into the coherent and the incoherent components. The phase function p(s) is assumed to be Gaussian.
where τo = (a + b)dz′ is the optical depth (OD) and Fs = 1 − exp [−∫dz′b],
where τa is absorption depth, τs is scattering depth, and ρo is coherence length given by [Ishimaru et al., 2004]
Since H(ρs) is independent of = mho, we can sum up Γm and obtain
The half-power spot size Wo in free space is approximately given by
It can be seen from (7) and (11) that the beam spot size W for time-reversed pulse is approximately given by
In free space, ρo is infinite and W is equal to Wo, but in random medium, as the coherence length shortened, the beam spot size W decreases, showing the superresolution. As seen from (12), the coherence length and the spot size depend on the scattering depth, the phase function, and the location of the random medium. This location dependence is called the “shower curtain effect” as the coherence length and the image resolution depend on whether the random medium is close to or further from objects.
4. Time-Reversed Intensity 〈∣ψ∣2〉
 Let us now consider the intensity (6) when t1 = t2 = t. We have
We write ω1 and ω2 using the center frequency ωc and the difference frequency ωd, and assuming that Γ is a slowly varying function of ωc, we integrate with respect to ωc and obtain
This involves the fourth-order moment. This can be expressed by the second-order moments if we assume that Green's functions are the circular complex Gaussian random functions [Goodman, 1985]. We then get
We can also express the mutual coherence function by
where Gio and Gjo are the free space Green's functions and Hij represents the effects of random medium.
 The free space Green's functions in parabolic approximations at z = L are given by
where = mho, = mho, and s = ρs. G10 is Green's function for a wave emitted at (L, 0) and observed at (, 0), and the distance (L2 + p2) is approximated by L + . Similarly, we obtain G20, G30, G40. Note that G10 and G20 are for ω1 and G30 and G40 are for ω2. With (20) we write
The two-frequency mutual coherence function given in (19) has been studied [Ishimaru et al., 2006] and Hij can be expressed as
Summarizing the above, the final expression for the time-reversed intensity is given by
5. Time Reversal and Backscattering Enhancement
 If we do not time-reverse, the field at s is given by (17) with 〈G1G2G*3G*4〉. Using the circular complex Gaussian assumption, we get
The first term represents the ladder term. The second term represents the cyclic term, which is the cross correlation between forward and backward waves, and this give the enhanced backscattering. This can be expressed in a diagram of Figure 2.
 With the time reversal we have a similar but different diagram in Figure 3. Note that the correlation terms are X21 and X34 in (25), and the ladder terms are X31 and X24 in (25). For the correlation terms the beam spot size becomes small as the coherence length ρo becomes small as shown in (15). For the ladder term, however, the spot size becomes larger as the coherence length ρo becomes smaller than the array size (2M + 1)ho. Note that in conventional diagram the lower level represents the complex conjugate as shown in Figure 2. However, in time reversal a part of the upper level G*1 and a part of the lower level G*4 are complex conjugate and they are shown with dashed lines in Figure 3.
6. Numerical Examples
 As an example, we use L = 50λ, d1 = 25λ, d2 = 25λ, αp = 44.58, N = M = 10, ho = λ/2, and Δω/ωo = 0.05.
 1. Average field and intensity: Figure 4 shows the average field and the intensity as functions of the transverse position ρs and time t. Note that at OD = 1 the field has a typical array pattern in (13), but at OD = 10 it is attenuated in magnitude, and the spot size is reduced for OD = 10 showing superresolution.
 2. Superresolution: Figure 5 shows the superresolution. Note that the beam spot sizes become much smaller for OD = 10. It also shows the effects of the phase function. αp = 44.58 gives the half-power angle of 14.3°, while αp = 106 gives 9.2°. The intensity for αp = 106 gives higher magnitude as the power is focused more than for αp = 44.58.
 3. Dependence on OD (optical depth): Figure 6 shows the effect of αp and the shower curtain effects on spot size as functions of OD. Note that the beam spot size is smaller for αp = 44 than for 106 because the coherence length is smaller. Also, the spot size is smaller for d1 = 25λ because the coherence length is smaller. This is the shower curtain effect.
 4. Effects of bandwidth, time spread, and time delay: Figure 7 shows the spread in time. Note the narrower pulse spread for BW = 0.1. The bandwidth effect has been discussed by Lerosey et al. .
 The paper presents an analytical study of the time reversal in random media. Unlike time reversal in free space, multiple scattering causes superresolution as the transverse spot size is reduced because of the coherence length, which can be smaller than the free space spot size. The analysis also includes the shower curtain effect, which is caused by the change of coherence length due to the location of the random medium. The analysis also includes the effects of the phase function of particles and the backscattering enhancement. Finally, this analysis shows the time spread and the time delay of the time-reversed pulse due to multiple scattering.
 This work was supported by the office of Naval Research Code 321, grant number N00014-05-1-0843, grant number N00014-04-1-0074, and National Science Foundation ECS0601394.