No abstract

No abstract

No abstract

Ann Arbor, Michigan 48109-21 22. AbstractThe problem of wave propagation in Jorest is ivuisited. In particular, the effect of the non-planar interjace between the air und the canopy on lateral waves is ezamined. An analytical formulation is obtained Jor the mcan field when both the transmitter nnd receiver air within the folingc. This Jormulntion is based on distortcd Born approzinaalion and is shown that compared to a planar interface, the field of a dipole in a canopy uiith rough interJace is significantly reduced. Analytical FormulationGeometry of the diffraction problem is shown in Fig.1 where the dipole located at heights h and h' inside a canopy with effective dielectric constant €1. The envelope of the canopy-air interface is denoted by d ( z , y ) The permittivity of the upper incdium(air) is denoted by €2. We modify the problcm by extcnding the canop) t_o z = 0 plane and assume that there exists a volumctric polarization current J' = ik~l'~(ezwhert E* = + J? t &. Here E is the incident field, J? is the rcflected wave from the planar interface and & i: the scattered field generated by J itself. To the first order in (z2 -€1) the Born approsimation can be used tc the scattered field where6 is the dyadic Green's function ol tbc half spacc diclectric nicdium 12, 31.In (1) d ( t , y) is a two dimensional random process describing the intcrface between the canopy and air and i: assumed lo be Gaussian with a mean value of m(a positive number) and standard deviation of U. Distortec Born approximation providcs a more acciiratr solution for E3. In this approxiination, a phasc corrrctioii lcriii i 0-7803-5639-W99/$10.00 01999 IEEE.

INTRDDUCTIONTime reversal methods OK-B unique opportunity for solving the problem of e l s t r e magnetic (EM) wave propagation and focusing in a spatially varying (inhomogeneous) medium. While the concept of time reversal or applying time reversal mirrors to focus waves in spatially varying media is new to the field of EM wave propagation, it has been applied in the area of acoustics and ultrasonics for several years [I]. The basic premi~e is quite simple. Let an impulse source (in time and space) he transmitted into some general inhomogeneous medium, and consider the tangential surface fields determined on a closed surface surrounding the impulse. It can be shown mathematically that if these surface fidds are conjugated and re-radiated in time reversed sequence, that the incoming wave, generated by the surf-fields is identical to the outgoing wwe, and the returning energy focuses on the original source point. In the frequency domain, this is equivalent to simply taking the complex conjugate of the received field, and reradiating it. This can he shown to he an application of reciprocity, which can be viewed 68 a BpatiDtemporal matched filter, similar to temporal matched filters commonly applied in isdar and mmmmications.In practical applications of TRM, a source transmits through an inhomogeneous medium, to an a finite array. In the frequency domsln, it can be shown through reciprocity that the following relatiomhip holds:where EL, Ez are electric fields caused by current sources J,, J2, iespectively. Let E, represents the fields caused hy source J1, at the location of an array element, and E2 reprerent the fields caused hy reradiation of this element, at the location of the original source J1. If the reradiated fields from the array element are caused hy current source Jz, and Jz = E;. then the right side, and therefore the left side of (1) becomes purely real, and the received fields *om the array element a t the location of the original source thus are also purely real. Applying this to all elements in the array, the coherent sum of their fields at the original source point will add constructively, and focus the received energy at this point.In any b i t e size array that occupies B limited spatial area the system is diffraction limited. however in sousties, it has been shown that in m inhomogeneous medium a time reversal &may is not always diffraction limited and can achieve super-resolution 121. Basically xatterers near the transmitting army and/or focal point act as an extension of the array in the focusing process. In this paper the same phenomenon is show" to exist for EM waves J 1 . E x = . l . E l l