## 4 Fixed Wavelength Versus Fixed Frequency  
Most commercially available GPR emit a pulse by unloading a radiofrequency capacitor through an avalanche transistor. The broadband pulse is then filtered by the dipole antenna fitted on the avalanche transistor emitter: the transfer function of this dipole antenna, located on the ground surface for efficient coupling, is strongly influenced by the ground permittivity. The impact of the environment permittivity and conductivity on a dipole antenna property has been extensively investigated in [40] and will be further addressed below. Since the dipole length is fixed and the permittivity (and thus electromagnetic velocity) varies, the emitted central frequency varies accordingly, as does the impedance at resonance. However, the cooperative target is designed to operate at a fixed frequency range, determined by the spacing between interdigitated electrodes patterned on the piezoelectric substrate. The returning signal is the convolution of the incoming pulse with the impulse response of the acoustic transducer in the time domain, yielding an efficiency easier to grasp in the frequency domain as the product of the pulse spectrum with the acoustic wave transfer function. Any mismatch will prevent a fraction of the incoming electromagnetic pulse power from coupling with the acoustic transducer and yield a loss in measurement range. Similarly, impedance mismatch prevents a fraction of the incoming electromagnetic pulse from being converted to an acoustic wave as it is reflected by the mismatched antenna. Ideal coupling conditions are met when the antenna impedance is the complex conjugate of the SAW impedance [41; 42]. Because the buried antenna impedance widely varies with surrounding permittivity, while the acoustic transducer operates in a relatively narrow frequency band with respect to the ultra-wideband GPR operating range, such a mismatch can become significant. A broadband antenna geometry, as found for example with a bow tie antenna classically used for GPR applications, will reduce the surrounding permittivity issue at the cost of poorer efficiency in the operating frequency band of the acoustic transducer. An initial analytical investigation in which the dipole impedance is modeled as an Resistive-Inductive-Capacitive (RLC) circuit [43] exhibits geometrical and surrounding media permittivity \(\varepsilon\) dependence through the equivalent passive components value. The capacitance value is determined by the dipole half-length \(l\) and diameter \(a\) (or maximum diameter in the case of conical radiating elements) ([44], p. 305) by:  
\[C=\frac{\pi el}{\log(2l/a)-\log(2)} \tag{3}\]  
The inductance value \(L\) on the other hand has no reason to depend on the permittivity and is solely defined by the permeability and geometric properties of the dipole ([44], p. 311) so that the impedance seen at resonance, when the inductive part of the circuit compensates for the capacitive part, becomes [43]:  
\[R=\frac{L}{C\cdot R_{r}} \tag{4}\]  
with \(R_{r}\) the radiation resistance. This impedance at resonance is inversely proportional to the permittivity of the medium surrounding the antenna through the capacitance term, a cause of varying matching condition with the sensor impedance defined solely by its acoustic design characteristics, while the resonance frequency:  
\[f_{R}=\frac{1}{2\pi\cdot\sqrt{LC}} \tag{5}\]  
of the dipole varies as \(1/\sqrt{\varepsilon}\). Such analytical analysis provides a first insight into the parameters affecting matching conditions between the sensor and the antenna. We wish to investigate further and confirm such issues through numerical simulations. We do so through 3D modeling by Finite Difference Time Domain (FDTD) means. FDTD modeling was selected for considering complex media surrounding the radiating element, with varying permittivity and conductivity, as well as layering or geometrical features such as pipeslocated near the sensor.