Mitigating the effect of non-uniform loss on time reversal mirrors

Time reversal mirrors work perfectly only for lossless wave propagation. Here, the performance of time-reversal mirrors is quantitatively defined, and the adverse effect of dissipation on their performance is investigated. An application of the technique of exponential amplification is proposed to overcome the effect of dissipation in the case of uniform loss distributions, and, to some extent, in the case of non-uniform loss distributions. A numerical model of a star graph was employed to test the applicability of this technique on realizations with various random spatial distributions of loss. A subset of the numerical results are also verified by experimental results from an electromagnetic time-reversal mirror. The exponential amplification technique should improve the performance of emerging technologies based on time-reversed wave propagation such as directed communication and wireless power transfer.


I. INTRODUCTION
Time reversal (TR) mirrors can, under ideal circumstances, precisely reconstruct a wave disturbance which happened at an earlier time, at any given later time. They have found numerous applications since the earliest experimental demonstrations of TR using acoustic waves [1]. More recently, TR mirrors have been realized using electromagnetic waves [2,3] expanding their range of applicability. They have applications in communications [4,5], imaging [6,7], source localization [8], non-destructive evaluation [9,10], and sensing [3,11,12].
TR mirrors are used to focus waves both in space and time. An ideal TR mirror consists of a wave source located inside a lossless medium that is completely enclosed by a surface of transceivers. A TR mirror operates in two steps. In step one, the transceivers record and absorb the signal broadcast by the source. In step two, the transceivers rebroadcast time reversed versions of the recorded waves. The waves broadcast in step two eventually focus on the location of the source and reconstruct a time reversed version of the original signal, which was broadcast in step one. This is possible because of the TR invariance property of the lossless wave equation. This ideal TR mirror uses the so called closed TR cavity [13]. However, it is not generally practical, for example, to build a closed TR cavity, whose interior is completely covered with transceivers. Practical TR mirrors have several limitations, which result in an imperfect reconstruction of the original signal. These limitations include i) limited spatial coverage by the transceivers, and ii) dissipation during the wave propagation (which breaks TR invariance of the wave equation) [11,14].
The first limitation of TR mirrors can be overcome by the use of a reflecting wave chaotic cavity with partial spatial coverage and transceivers that have a long recording time [11,14]. However, the limitation due to dissipation persists, and leads to increasing loss of information as the recording time increases.
The technique of iterative TR was proposed to improve the performance of TR mirrors which suffer from the limitations of dissipation and incomplete spatial coverage of transceivers [15]. However, the iterative technique may not converge fast enough, despite a recent result showing the tunability of its convergence [16]. A faster alternative to the iterative technique is the inverse filter technique [15,17]. But, the inverse filter technique is computationally costly and potentially unstable [15].
On the other hand, the technique of exponential amplification was used to overcome the adverse effect of dissipation on TR based sensing techniques [11,12]. The spatial range of these techniques is limited due to dissipation. The exponential amplification technique was used to extend the spatial range of the TR based sensing techniques.
In this paper, another application of the exponential amplification technique is proposed. Unlike the work in Ref. [12], the application involves improving the performance of TR mirrors which suffer from dissipation. The performance of TR mirrors is defined as the normalized correlation between the original pulse and the time reversed version of the reconstructed pulse (after some post-processing). This paper quantitatively demonstrates that the exponential amplification technique mitigates the adverse effects of dissipation on the performance of TR mirrors, for different spatial loss distributions. The technique of exponential amplification can compete with the iterative TR technique in improving this performance of TR mirrors, to some extent. The advantage of the exponential amplification technique is that it is faster to implement, as it does not rely on iterative steps. It is also cheaper computationally.
In Sec. II, the theory supporting the exponential amplification technique, first discussed in Ref. [12], is summarized. The performance of TR mirrors, which was not discussed in Ref. [12], is also defined in Sec. II. In Sec. III, the exponential amplification technique is used to improve the newly defined performance of an electromagnetic TR mirror. In particular, it is experimentally shown that the technique can improve the performance of a TR mirror for the case of spatially uniform distributed loss. Sec. IV investigates the limits of applicability of the exponential amplification in systems with inhomogeneous loss distributions. Sections V and VI present arXiv:1208.5431v1 [nlin.CD] 27 Aug 2012 further discussion and conclusions.

II. THEORY
The theory supporting the exponential amplification technique was originally presented in Ref. [12]. In this section, the theory is summarized and the performance of TR mirrors is defined. To summarize the theory, a cavity with transfer function s(ω) is considered. When a brief pulse, a(t), is broadcast into the cavity, the response is b(t). At a later time, a time reversed version of b(t) denoted b(−t) is broadcast back into the cavity producing reconstructed pulse denoted c(−t) at the original broadcasting antenna or transducer.
Ref. [12] provides a derivation showing that, if the cavity is lossless, a(t) is the same as a time reversed version of c(−t) denoted here c(t). Thus, the TR mirror works perfectly in a lossless cavity.
Even if the cavity is lossy, it is shown that the TR mirror can still recover the original pulse in the special case of uniform loss. Suppose that the uniform loss is represented by a 1/e amplitude decay time τ . In this case, the signal b(t) is exponentially amplified by e 2t/τ , time reversed, and broadcast back into the cavity.b(−t) denotes the signal obtained after exponentially amplifying b(t), and time reversing it. Similarly,c(−t) denotes the reconstructed pulse obtained whenb(−t) is broadcast back into the cavity. Ref. [12] analytically predicts thatc(t)e −2t/τ is equal to a(t) for a uniform loss that is characterized by τ .
Here, we define a new signalĉ(t) which is equal tõ c(t)e −2t/τ .ĉ(t) is called the corrected reconstructed pulse. In this paper, the performance (η) of the TR mirror is defined based on the normalized correlation between the corrected reconstructed pulse (ĉ(t)), and the original pulse (a(t)).
Each of the signals a(t) andĉ(t) have a duration of T . In principle, η can range from −1 to 1. However, since the two signals are aligned to maximize η, η ranges from 0 (poor performance) to 1 (perfect reconstruction). Based on the result from Ref. [12], a(t) =ĉ(t) if the system is uniformly lossy with a 1/e voltage decay time, τ ; therefore, the performance of the TR mirror is expected to be perfect (i.e. η = 1). However, the analysis does not model the effect of noise and limited dynamic range on the exponential amplification technique. If the exponential amplification is not applied to b(t), the performance, η, can still be calculated using the signals a(t) and c(t), instead of using a(t) andĉ(t). But, if the system is lossy, and exponential amplification is not applied, then it is not expected that η = 1.
In Sec. III, this theory, which assumes uniform loss distribution, is successfully tested in an experimental system that approximates the case of uniform loss distribution. However, it is also expected that the technique might be useful in the case of a moderately non-uniform loss distribution. The applicability of the exponential amplification technique to mitigate the effect of non-uniform loss is studied in Sec. IV.

III. OVERCOMING THE EFFECT OF SPATIALLY UNIFORM LOSS: EXPERIMENTAL TEST
The electromagnetic TR mirror operates as shown in Fig. 1. The TR mirror involves an empty one cubic meter aluminum box that has scatterers inside it, and two ports in its walls. The aluminum walls of the cavity are constructed of a single material, and can be assumed to have a uniform spatial distribution of loss. The original pulse was broadcast into the cavity using port 1, and the response signal, which we call "sona", was collected using port 2. The original pulse had a carrier frequency of 7GHz and a Gaussian envelope with a standard deviation of 1ns. The sona was recorded for about 6.5µs with a signal to noise ratio (SNR) greater than 1. During the second step of the TR mirror operation, the sona signal was time reversed and broadcast back into the cavity using port 1; this made use of the spatial reciprocity property of the wave equation. Finally, a reconstructed pulse, which approximates a time reversed version of the original pulse was collected at port 2. The reconstructed pulse (and its temporal sidelobes) was collected over a 10µs duration. The first 6.5µs included a direct recording of the time reversed sona being injected into the cavity from port 1. The last 3.5µs was a recording of the postreconstructed pulse emerging from port 2, reverberating throughout the cavity.
Based on Eq. 1, η can be calculated as the normalized correlation between the following two aligned signals: i) the time reversed version of the reconstructed pulse with its temporal sidelobes (which was a 10µs long recording), and ii) the original pulse broadcast (by considering the 1ns long Gaussian envelope to be located at t = 3.5µs of a 10µs long signal of mostly zero voltage values). Note that these two signals are equivalent to the aligned a(t) and c(t) signals introduced in Sec. II. If the TR mirror were perfect, η would be 1. However, as can be seen in part 4 of Fig. 1, the reconstructed pulse has temporal sidelobes which result in η < 1. The sidelobes arise due to the limitations of practical TR mirrors discussed in Sec. I, which include limited spatial coverage of transceivers and dissipation.
Generally, the sona signal shows an exponential decay which is caused by dissipation and coupling losses. The experimental system discussed here is strongly undercoupled [18], which means that the effect of dissipation dominates the effect of coupling. Therefore, the 1/e voltage decay constant (τ ) which is related to the unloaded quality factor can be readily determined by a line fit to the log of the sona energy as a function of time. The exponential amplification can then be applied to the sona signal before it is time reversed and broadcast into the cavity. The exponential amplification is carried out by multiplying the sona signal by the time dependent amplifying function where t is time in seconds, and F is an adjustable factor. Assuming a precise determination of τ , using F = 1 compensates only for the effect of dissipation on the sona during step 1 of the TR mirror operation; whereas, using F = 2 compensates for the effect of dissipation on the sona during both step 1 and step 2 of the TR mirror operation [12]. The theoretical discussion in Sec. II predicts that F = 2 to maximize η. The exponential amplification is applied to the part of the sona whose signal to noise ratio is, at the very least, greater than 1 (i.e. the 6.5µs long sona in this experiment). In addition, the function A(t, F ) is typically terminated by a smooth ramp-down function whose time span is at least as wide as the time duration of the original pulse, which is 1ns; this prevents additional frequency components from entering into the sona.
Suppose that the sona is exponentially amplified using A(t, F ), before it is time reversed and broadcast into the electromagnetic cavity. The reconstructed pulse obtained with F > 0 will have more sidelobes than the reconstructed pulse obtained with F = 0. However, these sidelobes can be numerically suppressed by manipulating the reconstructed waveform as follows. Fig. 2 shows the reconstructed waveform obtained using sona that is amplified with A(t, F = 2) (in blue); the reconstructed waveform has unwanted temporal sidelobes which are a result of the exponential amplification applied to the sona. These sidelobes were corrected as follows. The 10µs long reconstructed waveform was time reversed so that the 1ns long Gaussian pulse is located at about t = 3.5µs; this waveform was then multiplied by A(t − 3.5µs, F = −2) to get the corrected reconstructed waveform (i.e.ĉ(t)) introduced in Sec. II. A(t, F ) = 1 for t < 0 by convention. In Fig. 2, the corrected reconstructed waveform is time reversed and plotted (in red) to highlight its relationship with the reconstructed waveform. The corrected reconstructed waveform is expected to have a better normalized correlation with the original pulse because of the exponential amplification with F = 2, as predicted in Sec. II.
Several values of F were used to carry out the operation of the TR mirror assisted by exponential amplification. For each F value, the performance of the electromagnetic TR mirror, η, is shown in Fig. 3. The performance of the TR mirror is enhanced the most by using exponential amplification with F ≈ 2. In partic- ular, η increases from ≈ 63% at F = 0 to ≈ 73% at F = 2. This result agrees with the theoretical prediction in Sec. II. The fact that the η < 1 even for F ≈ 2 can be explained by the fact that the TR mirror has only one recording channel, whereas the theory assumes that all the waves are captured. Thus, the exponential amplification technique only overcomes the limitation of the TR mirror associated with dissipation, and not the problem of insufficient spatial coverage of recorders. Noise, which is not considered in the theory, also affects the performance of TR mirrors as seen in Ref. [16].
As shown in Fig. 3, η ≈ 63% if the exponential amplification technique is not applied. The exponential amplification with the optimum F = 2 parameter achieves η ≈ 73% for the experimental set up described. This compares with η ≈ 78% which is achieved by the tunable iterative technique on the same experimental set up [16]. Therefore, the iterative technique performs better than the exponential amplification technique when both of them use their respective optimum parameters. Nonetheless, the exponential amplification has an advantage because of its speed, and computational simplicity.
The result in this section proves that the exponential amplification improves the performance, η, of TR mirrors with uniform spatial loss.

IV. OVERCOMING THE EFFECT OF SPATIALLY NON-UNIFORM LOSS: NUMERICAL TEST
The exponential amplification technique is derived assuming uniform loss distribution. The case of non- uniform loss distribution is better handled with techniques such as the inverse filter, and the iterative TR [15,17], which do not assume uniformity of loss. However, it is expected that the exponential amplification can be applied in cavities with moderately inhomogeneous loss distributions. The applicability of the exponential amplification technique was studied as the inhomogeneity of the spatial distribution of loss was increased. The simulations in this section were designed to supplement the experimental results in Sec.III.

A. Simulation Setup
The numerical model is an implementation of a star graph [16,19]. The star graph is a quasi-1D chaotic system. It is modeled using a driving transmission line that feeds a number of other transmission lines, which are all connected to each other in parallel (see Fig. 4). Each transmission line is terminated with a reflection load. The graph is a one port system. Thus, the original pulse is injected through the driving transmission line, and the sona is also collected from the same line. The original pulse has the same characteristics as the one used in the experiment described in Sec. III.
There are 500 transmission lines in the star graph set up, in addition to the driving line. Their length is given by L i = √ i m for i ranging from 1 to 500. The driving line has a length of 0 m. The characteristic admittances of the 500 lines are Y ci = 1S. The driving line has a characteristic admittance of Y cd = i=500 i=1 Y ci = 500S; this set up eliminates prompt reflection of signals injected through the driving line. The terminal reflection coefficients, Γ i , of the 500 lines are all set to 1. The dissipation is introduced through the frequency dependent propagation constant of the lines, γ i (ω) = ı ω c + α i , where c is the speed of light and α i is the loss constant of line i. α i specifies that the voltage waves decay on line i as e −αiz , where z is distance measured along line i in meters. The spatial inhomogeneity of loss on the star graph is modeled as follows. The α i of each of the 500 lines is randomly chosen from the probability density function (pdf), The pdf has two parameters, n and λ, which define the mean (µ α = λ + λn) and standard deviation (σ α = λ(λ + λn)) of α values. This particular pdf is chosen for the following two reasons. First, the coefficient of variation (R = σ α /µ α ) can be easily varied while keeping µ α constant. Keeping the average spatial loss constant simplifies the problem, and helps us focus on the effect of increasing spatial loss inhomogeneity. The main motivation for using the exponential amplification in the case of non-uniform loss is this: if the 1/e decay constants (i.e. τ ) of the different modes are not extremely different, we can use a single average τ value. Thus, it is not interesting to vary the average loss (effectively τ ) here. In this set up µ α is chosen as (cτ ) −1 , where τ = 1.5µs. The second reason to use this particular pdf is that its support is the set of positive numbers, which should be the case as α should always be positive on the passive transmission lines. Fig. 5 shows plots of the pdf for different values of R, where µ α ≈ 0.002.
The numerical values of the parameters of the star graph were chosen to achieve the following objective. The experimental result in Sec. III was based on an undercoupled cavity. An under-coupled cavity is characterized by the domination of energy loss due to dissipation over energy loss due to coupling [18]. If the cavity is not under-coupled, the advantage of the exponential amplification technique cannot be easily seen because there are no strong internal dissipation effects to be compensated in an over-coupled cavity. To accomplish under-coupling, 500 lines were used in the star graph to increase the backreflection coefficient of the waves that are trying to leave the star graph. This forced the waves to reverberate inside the star graph longer, which decreased the coupling loss by two orders of magnitude compared to the dissipation loss.

B. Simulation Results
A TR mirror was implemented on the star graph by broadcasting the original pulse, collecting a sona, exponentially amplifying the sona with A(t, F ), time reversing the amplified sona, and broadcasting it back into the star graph. The reconstructed pulse was collected, and it was time reversed and multiplied by A(t − t p , −F ). Here, t p is the time when the 1ns Gaussian pulse was expected within the time reversed version of the reconstructed waveform. The signals were collected over a span of 10µs without noise.
The performance (η) of the TR mirror is plotted versus F as shown in Fig. 6. This entire process was repeated FIG. 7: The performance of the TR mirror in a star graph model is plotted as a function of the loss inhomogeneity, R, for the cases of: i) no exponential amplification (shown in black), and ii) optimal exponential amplification, which is always F = 2 (shown in red). It is clear that exponential amplification is advantageous for uniform and moderately non-uniform spatial loss distributions.
for star graphs with different degree of loss inhomogeneity characterized by R. R = 0 represents uniform spatial loss distribution. As R increases, the loss inhomogeneity increases. For each R value, 25 realizations of the star graph were generated; hence, there are error bars included on the η vs F plots for each R value, as shown in Fig. 6.
For each R value shown in Fig. 6, the maximum η was taken over all F values attempted; it turns out that the maximum η is always achieved for F = 2. The maximum η achieved is plotted as a function of the loss inhomogeneity (R) in Fig. 7 (shown in red). The η achieved without using exponential amplification (i.e. F = 0) is also plotted as a function of R in Fig. 7 (shown in black). Fig. 7 demonstrates that the exponential amplification technique improves η significantly, for small values of R, which are close to uniform loss distributions. The exponential amplification is still applicable in cavities with more inhomogeneous loss distributions. However, its effectiveness declines with increasing loss inhomogeneity as expected.
From Fig. 7, we can see that η is not affected significantly by R if exponential amplification is not applied. Here, it is worth repeating the fact that the average spatial loss in the star graph is constant for all the R values considered. On the other hand, the fact that η is not restored to 1 even for R = 0 and F = 2 can be explained by the fact that the star graph is designed to be undercoupled, to match the nature of the experimental cavity in Sec. III. The waves in the star graph will continue to reverberate for longer than the 10µs sona recording time. Thus, the problem is not just dissipation which can be compensated by the exponential amplification technique, but the problem here is also the lack of sufficient temporal coverage, which has previously been shown to affect the quality of TR focusing [20].

V. DISCUSSION
The tunable iterative time-reversal technique overcomes the adverse effect of dissipation on the real-time spatiotemporal wave focusing of time reversal mirrors [16]. The focusing was quantified using the ratio of pulse to temporal sidelobe energy. In this paper, we are less interested in achieving a real-time spatiotemporal wave focusing using time reversal mirrors. Instead we are mainly interested in digitizing the reconstructed time reversed pulse, and numerically processing it to suppress the temporal sidelobes of the pulse. Suppressing the temporal sidelobes improves the fidelity of the reconstructed pulse with the time reversed version of the original pulse (i.e. a(−t)). The temporal sidelobes of the reconstructed pulse can be significant, and they are created as a byproduct of the exponential amplification of the sona. Fig. 2 shows a digitized reconstructed pulse before and after its temporal sidelobes are numerically suppressed. The capability to systematically suppress these temporal sidelobes after the digitization of the reconstructed pulse can be useful. For instance, if the shape of the reconstructed pulse is encoding information in a communication application, it is important to be able to improve the fidelity of the reconstructed pulse (which is measured by η) by suppressing its temporal sidelobes.

VI. CONCLUSION
Exponential amplification improves the performance of time reversal mirrors best if the loss is uniformly distributed in space. However, even under conditions in which the loss is not uniformly distributed, the exponential amplification mitigates the adverse effect of nonuniform loss on the performance of time reversal mirrors. The exponential amplification works best when the theoretically predicted optimum parameter of F = 2 is used. This technique may be advantageous to use in time reversal mirror based communication applications because even if the reconstructed pulse will have significant temporal sidelobes during the recording, the processed reconstructed pulse will closely replicate the shape of the original pulse. This work is supported by ONR MURI grant N000140710734, AFOSR grant FA95500710049, ONR AppEl Center, Task A2, grant N000140911190, and the Maryland Center for Nanophysics and Advanced Materials.