Optimisation of stepped permittivity impedance loaded absorber D.G. Holtby, K.L. Ford and B. Chambers A novel type of pyramidal absorber is presented. This absorber utilises a combination of impedance loading and an exponential stepped permit- tivity proﬁle. The new design offers signiﬁcant improvement in the absorber’s low frequency performance when compared with existing pyramidal absorbers. Optimisation of the design parameters has been carried out to maximise the absorber’s bandwidth at 240 dB reﬂectivity. Introduction: For many years there has been interest in the design of broadband microwave absorbers, for use in anechoic chambers, that are either effective at low frequencies or are physically small in size. In prac- tice, one of these characteristics is usually traded-off for the other, depend- ing on the required application. Previous authors have investigated alternative conﬁgurations of geometric transition (GT) absorbers. These include arrays of multi-layer wedges [1], arrays of doubly periodic curved pyramids [2], arrays of multi-level wedges based on Chebychev polynomials [3] and variation of the geometrical proﬁle in conjunction with the use of permittivity proﬁles [4]. Other work has been carried out on the design of a layered radar absorber using a genetic algorithm (GA) [5]. In addition, there has recently been some success in improving the low frequency performance of pyramidal absorbers by loading them with an impedance layer in the form of a frequency selective surface (FSS) [6]. This Letter introduces a new absorber topology called the stepped permittivity impedance loaded (SPIL) absorber which incorpor- ates an FSS embedded within a pyramidal absorber that has an exponen- tial permittivity proﬁle. The SPIL absorber is then optimised for maximum bandwidth using a particle swarm optimisation (PSO) routine and its performance is then compared with that of a homogeneous unloaded pyramidal absorber and an FSS loaded pyramidal absorber. l d t PEC p FSS t Fig. 1 Side view of layered pyramidal absorber embedded with FSS g w p p Fig. 2 Unit cell of square loop FSS Design: Fig. 1 shows a cross-section of a typical pyramidal absorber that includes an impedance layer in its base at a distance, d, from the perfect electrical conductor (PEC) back plane. The premise of the SPIL absorber design was to divide the pyramid into layers such that a stepped permit- tivity proﬁle existed through the absorber thickness. The FSS provides additional impedance matching, as discussed in [5], when an unloaded absorber’s geometric and permittivity proﬁle alone cannot offer further performance improvement. The reﬂectivity of the SPIL absorber can then be conﬁgured for maximum bandwidth by optimisation of the stepped permittivity proﬁle and the FSS parameters. To carry out the simulations it was assumed that a homogeneous absorber has a bulk per- mittivity as given in [1] and this is denoted throughout this Letter as 1 bulk . The discrete steps in permittivity between adjacent layers of foam were deﬁned by multiplying 1 bulk by a position dependent multiplier given by, Ae 2BxþC where A, B and C are variables and x is the distance from the PEC boundary in millimetres; hence it was assumed that the permit- tivity proﬁle varies exponentially with distance. Other proﬁles could be adopted; however an exponential proﬁle is close to that experienced in practical manufacturing processes. Fig. 2 shows a unit cell of the FSS. While there is a wide range of FSS designs available, a square loop bandstop FSS was chosen as it has reasonable bandwidth and its geometry minimises the metal in the structure which may affect the high frequency performance of the absorber. The square loop FSS can be characterised by an inter-cell spacing, g, and a track width, w. In the present work, the dimensions of the Emerson and Cuming VHP-4 absorber were used to provide continuity with previous studies [5]. The dimensions of the absorber are p ¼ 38 mm, l ¼ 77 mm and t ¼ 25 mm. The absorber thickness was divided into 20 layers, with a thickness d t of 5 mm in all but the last layer, which was 7 mm. A PSO was employed, in the manner described in [7], to optimise the par- ameters d, g, w, A, B and C of the SPIL absorber. PSO is a stochastic optimisation routine that is modelled on the behaviour that a ﬂock of birds or a swarm of insects exhibit when searching for a source of food. In PSO, each one of the design’s parameters is represented by a dimension in the ‘solution space’. A swarm of 20 solutions (or agents) with randomised parameters were created to occupy this space at a certain instant i. The reﬂectivity response of each agent was then simulated using CST Microwave Studio with the appropriate boundary conditions to simulate an inﬁnite periodic array of SPIL pyramids. Using these results, a ﬁtness value was assigned to each agent that was directly proportional to the absorber’s bandwidth below a reﬂectivity of 240 dB. The agents were then free to ‘ﬂy’ within the boundary conditions of the solution space, effectively altering the sol- ution’s design parameters. The speciﬁc boundary conditions used were 1 , A , 20, 0.002 , B , 0.02 and 0 , C , 2 for the permittivity parameters and 0 , d , 25 and 0 , g, w , 19 for the FSS parameters, providing the sum of g and w did not exceed 19 mm. At each time step, the velocity of an agent was expressed by V ði þ 1Þ¼ w  V ðiÞþ c1  randð0; 1Þð pbestðiÞ posðiÞÞ þ c2  randð0; 1ÞðgbestðiÞ posðiÞÞ where rand(0,1) indicated a random number chosen between 0 and 1, c1 and c2 were the cognitive and social rates respectfully, w was the initial weight factor, pos(i) was the current position of the agent, pbest(i) was the best position found by that agent and gbest(i) was the best position found by the entire swarm. Both c1 and c2 were set equal to 2, and w was a time dependant weight factor determined by wðk Þ¼ 0:9  0:5  k N where k was the current iteration number and N was the total number of iterations. Once the velocity of each agent was known, the new position of each agent was calculated from the following equation: posði þ 1Þ¼ V ðiÞþ posðiÞ The optimisation routine was set to run for 100 iterations, which was considered to be sufﬁcient for the agents to congregate around the best possible solution location. Results: The optimised permittivity variables for the SPIL absorber were found to be A ¼ 19.33, B ¼ 0.082 and C ¼ 0.68. The resulting stepped permittivity proﬁle is shown in Fig. 3 and is compared with that of an optimised stepped permittivity absorber which was not loaded with an FSS. The values of the stepped permittivity absorber were A ¼ 4.56, B ¼ 0.167 and C ¼ 1.63. The permittivity at the back layer of the SPIL absorber was multiplied by a factor of almost 14, which dropped to 1 (i.e. 1 bulk ) at about 45 mm from the PEC boundary, and continued to fall gradually to 0.68 times 1 bulk at the tip of the absorber. The optimised FSS parameters were found to be d ¼ 23.34 mm, g ¼ 3.29 mm and w ¼ 0.53 mm. Fig. 4 shows the reﬂectivity response of the SPIL absorber, compared with those of an unmodiﬁed absorber, a homogeneous absorber with an FSS and a stepped permittivity absorber. The latter two were optimised using the discussed PSO. It can be seen that the combination of an FSS and permittivity proﬁling offers major performance improvement over previously discussed designs. The reﬂectivity of the SPIL absorber falls below 240 dB at a frequency of 1.6 GHz, compared to values of 6.26, 3.15 and 2.87 GHz for the unmodiﬁed, FSS loaded and stepped permittivity absorbers, respectively. ELECTRONICS LETTERS 26th March 2009 Vol. 45 No. 7