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Radar absorbent materials (RAM) have long been used for applications requiring the absorption of electromagnetic waves at RF and microwave frequencies. The work outlined in this brief summary describes numerical optimisation of multi-layer RAM using a genetic algorithm approach. The application for which this solution was developed was the minimisation of target radar cross section.
The present description deals with the simplified case of normal incidence and considers only the minimisation of the reflection coefficient for a given thickness in the objective function for the optimisation process. In practice the objective function would be expanded to incorporate material density to minimise absorber mass.
The first research on absorbers was carried out in the 1930's with the first absorber patent in 1936. Initially designed for reducing the effects of antenna structural supports on performance and then later during the development of radar in WWII for signature reduction and stealth applications.
Early designs such as Salisbury screen were narrow-band absorbers relying on resonance effects. Designers soon moved to broad-band designs using tapered material characteristics to provide an impedance match between free space and reflector. This was effected either geometrically using conical or pyramidal structures, or by varying the material characteristics of a flat absorber as a function of depth. An example of this type of RAM is the Jaumann absorber with alternate plastic and resistive sheets, the resistivity of the dissipative sheets varying with depth.
Current design methods are still based on these principles. There are three goals in the design, minimisation of reflection coefficient, minimisation of thickness and reduction in mass of the absorber. Each may take priority depending on the application. Major enhancements in modern RAM have come about through improved materials.
Traditional design of layered absorbers has relied on approximate closed form expressions , intuitive engineering 'feel', extensive experimentation or relatively simple optimisation methods.
The equations governing material behaviour are cast as a function of the relevant material parameters, these being permittivity and permeability as a function of frequency, conductivity and the physical layer thickness. Using these properties, the solution can be obtained for wave transmission and reflection at the material interfaces giving absorber performance as a function of frequency, incidence angle and polarisation.
The main problem of this methodology is the large available parameter space. This means that intuitive design principles cannot be used for effective optimisation and there may be realisable designs available which significantly outperform those generated in this way.
Deterministic, gradient based, serial search optimisation techniques may be applied, for example downhill simplex, Powell's method and conjugate gradient. Such methods have been found to be successful in a number of electromagnetic optimisation tasks. For the present application, however, they have serious deficiencies.
A recent alternative is genetic algorithm optimisation. This approach has been demonstrated to yield excellent results in the most difficult optimisation problems. The reason for this is that this method relies only on the value of the objective function and not its derivative. No search vector need be defined and the parameter space is covered at random through the search operators of crossover and mutation.
This technique is explored below in the design of multi-layer absorbers. The next section discusses the derivation of the objective function for the multi-layered absorber performance.
For brevity normal incidence is considered, see [1] for a more extensive discussion of this analysis. In addition absorber density has not been included in the optimisation task. The characterisation is made on the basis of the reflection coefficient of the absorber throughout the required bandwidth. It is assumed in the following analysis that the RAM is mounted on a reflective metallic backing.
The problem of transmission of a plane TEM wave through a material interface can be formulated in the context of standard transmission line theory. The material characteristics determine the intrinsic impedance of the medium and the propagation constant for wave transmission, given in Equations (1) and (2) respectively.
(1)
(2)
For an interface between semi-infinite half space of medium 1, impedance Z1, and half space of medium 2, impedance Z2, the reflection coefficient is given by Equation (3) (wave travelling from left to right, from medium 1 to 2).
(3)
Where the material layer of medium 2 is of finite thickness, Equation (3) cannot be applied trivially. In fact the impedance of medium 2 at the first interface is modified by returning waves reflected internally at medium 2's second interface. The modification of the impedance depends on the phase of the returning wave, which in turn depends on the layer thickness, the wavelength in the medium, and also the attenuation in the medium if it is lossy.
The propagation constant describes the loss and phase of the returning wave. Generally the propagation constant is complex, the real part being the attenuation constant and the imaginary part the phase constant. The so called wave impedance at the interface between medium 1 and 2 where medium 2 is of finite thickness d is given by Equation (4).
(4)
Equation (3) can now be used for the reflection coefficient substituting the intrinsic impedance of medium 2, Z2, with the modified wave impedance Z2w. The reflection coefficient of a multi-layered material can be found by an iterative procedure starting with the backing layer of the material, assumed to be semi-infinite in extent, and progressing forward through the layers calculating the wave impedance at each internal boundary, until the exterior boundary is reached. The reflection coefficient of the absorber is then computed using the wave impedance at this outer boundary.
A recursive algorithm based on Equation (3) was implemented as the objective function for the genetic optimisation algorithm. The genetic algorithm (GA) code was implemented as a binary coded GA, able to select material characteristics from which to construct the absorber from a table of available materials using a binary identification encoding.
Solutions were achieved using simple ranking selection, a one point crossover operator with probability of crossover Pcross= 0.8, and probability of mutation Pmut = 0.0005. These values correspond well with those values used by other authors and are within the usually adopted limits.
A number of sophisticated operators have been used by other authors which seek to enhance convergence and efficiency of the search and selection processes within the parameter space. High levels of sophistication in the GA operators were found to be unnecessary in the present application as the objective function is trivial to compute resulting in short calculation times for the optimisation task.