Field of the Invention

The present invention generally relates to optimisation of electromagnetic designs. In some embodiments, the present invention relates more particularly to the areas of antenna engineering and microwave engineering and application of a cross-entropy method for the optimisation of antenna, electromagnetic component and microwave component design.

Background of the Invention

Progress in evolutionary optimisation has led to a change in modern design approaches in the field of electromagnetics (EM). A few examples where optimisation methods have advanced our knowledge of electromagnetic design problems and produced novel configurations include microstrip antennas, antenna arrays of arbitrary shapes and electromagnetic bandgap (EBG) structures and EBG-based antennas. Manual design methods became unfeasible due to the need to control several design variables, the large time required for simulation of each design, as well as in some instances the need to improve not just one, but multiple performance parameters. This led to the active application of optimisation methods in electromagnetics and microwave engineering.

Summary of the Disclosure

According to a first aspect of the present disclosure, there is provided a method for optimising an electromagnetic device design, the method including:

(a) evaluating a formulated fitness function for each design in an i-th candidate group of designs sampled from a distribution of designs, each design in the distribution of designs including multiple design variables characterised by respective one or more i-th probability distributions represented by an i-th set of distributional parameters, the formulated fitness function being evaluated based on electromagnetic performance criteria of the corresponding design in the i-th candidate group of designs; and (b) determining an (i+1 )-th set of distributional parameters representing one or more (i+1 )-th probability distributions characterising the respective multiple design variables, the (i+1 )-th set of distributional parameters being determined based on a reduction of cross entropy between an empirical distribution characterising a selected i-th elite group of designs from the i-th candidate group of designs based on the evaluated fitness function and the one or more (i+1 )-th probability distributions.

The method may further comprise selecting a respective type of probability distribution to characterise each one of the multiple design variables. The selection of the respective type of probability distribution may be based on matching support of the selected type of probability distribution with feasible values of the respective design variable.

Two or more of the multiple design variables may be constrained by a design condition.

The multiple design variables may include at least a discrete variable and a continuous variable.

Selecting the i-th elite group of designs may include ranking the i-th candidate group of designs according to the corresponding evaluated fitness function and selecting the i-th elite group of designs based on top-ranked designs in the i-th candidate group of designs. The reduction of cross-entropy may include minimising a Kullback-Leibler distance between the empirical distribution characterising the selected i-th elite group of designs and the one or more (i+1 )-th probability distributions.

The determining may further be based on weighted values of the i-th set of distributional parameters. The method may further include iterations of steps (a) to (b) wherein i is initiated at 1 and incremented by 1 at each iteration. The iterations may terminate based on diversity in the i-th candidate or elite group of designs. The diversity may include a maximum variation in the i-th candidate or elite group of designs. The method may further include introducing a perturbation to the (i+1 )-th set of distributional parameters based on a comparison of the evaluated fitness function of a top-ranked design over consecutive iterations.

The sampling may include random sampling. The ratio in the number of the i-th elite group of designs to the i-th candidate group of designs may be between 15% and 25%.

According to a second aspect of the present disclosure, there is provided an electromagnetic device design optimised by the method of the first aspect.

The antenna may be a resonant cavity antenna including a transverse permittivity gradient over a plurality of concentric segments having one or more respective continuous variables and one or more respective discrete variables. The one or more respective continuous variables may be continuously variable widths. The one or more respective discrete variables may be discretely variable permittivity values, and/or discretely variable thickness values. The continuously variable widths may be constrained to a fixed sum.

According to a third aspect of the present disclosure, there is provided a resonant cavity antenna having a design optimised by the method of the first aspect, the resonant cavity antenna including a transverse permittivity gradient over a plurality of concentric segments having respective continuously variable widths and respective discretely variable permittivity values.

The probability distribution for each of the continuously variable widths may be the Dirichlet distribution.

The plurality of concentric segments may have decreasing permittivity values from the antenna's centre towards the antenna's edges. The formulated fitness function may be indicative of directivity-bandwidth product of the antenna. Alternatively the formulated fitness function may be indicative of side-lobe levels of the antenna. Still alternatively the formulated fitness function is indicative of both directivity-bandwidth product and side-lobe levels of the antenna. According to a fourth aspect of the present disclosure, there is provided a non-transitory computer-readable medium including instructions which, when executed by one or more computers, cause the one or more computers to execute the method of the first aspect.

As used herein, except where the context requires otherwise, the term "comprise" and variations of the term, such as "comprising", "comprises" and "comprised", are not intended to exclude further additives, components, integers or steps.

Further aspects of the present invention and further embodiments of the aspects described in the preceding paragraphs will become apparent from the following description, given by way of example and with reference to the accompanying drawings. Brief Description of the Drawings

Figure 1 illustrates an implementation of a cross-entropy method (CEM) based optimisation.

Figure 2 illustrates contour plots of two test fitness functions of known optimised solutions, plotted together with candidate solutions at different iterations of the cross- entropy method (CEM) based optimisation illustrated in Fig. 1 .

Figure 3 illustrates a schematic plan view and a schematic top view of an example resonant cavity antenna (RCA) for optimisation.

Figure 4a illustrates a plot of an evaluated fitness function versus the number of function evaluations in the RCA optimisation example of Fig. 3. Figure 4b illustrates a plot of population diversity versus the number of iterations in the RCA optimisation of example of Fig. 3.

Figure 5 illustrates a prototype of an optimised RCA with a transverse permittivity gradient (TPG) superstrate.

Figure 6 illustrates the simulated and measured boresight directivity versus frequency of the optimised RCA prototype of Fig. 5.

Figure 7 illustrates the simulated and measured radiation patterns of the optimised RCA prototype of Fig. 5 in the E- and H-planes at various radio frequencies. Figure 8 illustrates an implementation of a method of interfacing MATLAB with CST MS.

Detailed Description

In this specification, "optimisation" and like terms can include but are not limited to mean an improvement in performance (for example, electromagnetic performance). In this specification, an "electromagnetic device" can include but is not limited to an electromagnetic component, system or sub-system that creates an electromagnetic field of any type, including guided waves, radiating waves and leaky waves.

An aspect of the present disclosure relates to a technique for optimisation of

electromagnetic designs based on a cross-entropy method (CEM). The disclosed technique has applications in a number of electromagnetic fields, such as antenna design. A feature of a CEM-based optimisation technique (or CEM-based algorithm for simplicity) is that its iterative procedure is independent of the probability distribution describing the optimisation problem or optimisation goal. It means that the technique can be used for different types of decision variables (including the use of a mix of both continuous and discrete design variables) without any significant changes to the logic of the method.

In some implementations, a CEM-based algorithm in accordance with the present disclosure is used in conjunction with three-dimensional electromagnetic full-wave simulation, at each iteration or at least one of the iterations, to evaluate performance of an electromagnetic device under optimisation. The simulation is based on solving Maxwell's equations. The simulation may be carried out by an appropriate numerical method, such as one using finite differences or finite elements. In one implementation, the CEM-based algorithm in accordance with the present disclosure is used in the context of an electromagnetic device, whether it is radiating or non-radiating. The electromagnetic device may be operated at a selected frequency or frequency range within a part of the electromagnetic spectrum (e.g. 1 kHz - 100 THz). For example, the operating frequency may be at radio frequencies. As another example, the operating frequency may be at optical frequencies. A skilled person would appreciate that the description herein is, with minor modifications, also applicable to other means than three-dimensional electromagnetic full-wave simulation, such as 2D simulation or circuit analysis,

THE CROSS ENTROPY METHOD

The CEM is a stochastic optimisation technique based on reducing (e.g. minimizing) the cross-entropy between probability distributions. In the present disclosure, while cross- entropy is determined by the Kullback-Leibler divergence, a skilled person would appreciate that cross-entropy is used to measure dissimilarity. The idea of the CEM optimiser evolved from a method to estimate the probability of rare events. This method refocuses the search in more promising terrain, improving the prospects of locating a better solution. For this to occur, at each iteration, two tasks are generally performed:

1 ) create a sample by sampling candidate designs from a current distribution of designs characterised by distributional parameters;

2) update the distributional parameters to characterise the distribution of designs for the next iteration by reducing (e.g. minimising) its cross entropy with the empirical distribution describing the current best designs.

In statistics, the probability distribution of a random variable is a description of the probability of each possible value, or range of values, that X could take. Probability distributions are used in the CEM to characterize the current sample. The current sample is then used to create a new distribution. In any specific implementation of the CEM, only a finite-dimensional family of probability distributions is used. The

distributions within the family are identified by shape parameters; for example, the normal (Gaussian) distribution of a random variable X E R is characterised by its mean μ and variance σ ² , the Dirichlet distribution of a random variable Y E R is characterised by its concentration parameter or vector a, whereas a discrete distribution of a random variable Z is characterised by its parameter p. The CEM-based algorithm aims to produce a sequence of distributions, each characterising a current sample, that are increasingly concentrated around the optimal design.

The disclosed method for optimising an electromagnetic design is an iterative process. While employing multiple iterations is common, the method does not preclude employing only one iteration. In a general form, the disclosed method includes (a) sampling an i-th candidate group of designs from a distribution of designs, (b) evaluating a formulated fitness function for each design in the i-th candidate group of designs, (c) selecting an i-th elite group of designs from the i-th candidate group of designs based on the evaluated fitness function, and (d) determining an (i+1 )-th set of distributional parameters representing one or more (i+1 )-th probability distributions characterising the respective multiple design variables. In a multiple-iteration

implementation, i may be initiated at 1 , incremented by 1 at each iteration and terminated once a termination criteria is determined to be reached. Each design in the distribution of designs includes multiple design variables (e.g. one or more dielectric widths and one or more permittivity values) characterised by the respective one or more i-th probability distributions (e.g. Dirichlet distribution for dielectric widths and general discrete distribution for permittivity values) represented by the i-th set of distributional parameters (e.g. a for Dirichlet-distributed dielectric widths, and p for discretely distributed permittivity values). The multiple design variables may be jointly or independently characterised by the corresponding one or more i-th probability distributions. For example, a design variable may be characterised by a single probability distribution. Alternatively or additionally, a subset of the multiple design variables may be characterised by one or more appropriate probability distributions. In one implementation, the multiple design variables include at least a discrete variable (e.g. permittivity values) and a continuous variable (e.g. dielectric widths). In another implementation, the multiple design variables include only discrete variables or only continuous variables.

In some implementations, the multiple design variables are not all independent. In other words, at least one design variable may be constrained by a design condition. In some instances, the design condition can be presented by a mathematical equation or inequality. For example, several design variables may be constrained by a fixed sum (e.g. fixed width to limit the dimension or physical footprint of a device). As another example, the length of each section of a radio-frequency filter used in a mobile phone is a variable but the total length of the filter should be less than or equal to a fixed value (determined by the space budget of the phone design). In other instances, the design condition can be a relationship between variables. For example, the design condition may be a monotonically increasing or decreasing value of over some of the multiple variables (e.g. monotonically decreasing permittivity values from the centre to the edges in an RCA example described below).

These statistical measures are to be used in conjunction with a fitness function formulation. The formulation of the fitness function depends on the goal of the optimisation. The formulated fitness function, which is indicative of a figure of merit of the electromagnetic design under optimisation, is evaluated based on electromagnetic performance criteria of the corresponding design in the i-th candidate group of designs. Formulation of the fitness function for evaluation may depend on the objective of the optimisation and the electromagnetic device under optimisation. For example, if the electromagnetic device is an antenna, the electromagnetic performance criteria may be selected as the directivity-bandwidth product of the antenna, or it may be selected as the side-lobe levels of the antenna, or it may be selected as a weighted or non-weighted combination of both. The (i+1 )-th set of distributional parameters characterise the distribution in the next iteration from which the next sample is drawn. The (i+1 )-th set of distributional parameters is determined based on a reduction of cross entropy between empirical distributions of the i-th elite group of designs and the one or more (i+1 )-th probability distributions. While in one implementation the reduction of the cross entropy is a minimisation of the cross entropy, it does not need to be the case.

The selection of the respective probability distributions depends on the nature of the design variables. For continuous variables, examples to be selected to generate possible solutions include the normal and beta distribution families. Examples of discrete probability distribution families that can be selected include binomial, Poisson, Bernoulli, discrete uniform, and geometric. A basis for the selection is to choose a family of sampling distributions for which cross-entropy optimisations can be efficiently performed. For example, the selection is based on a way being known of determining, exactly or approximately, the member of the sampling distribution family which is most like a given empirical distribution, according to the cross-entropy measure of similarity. An alternative or additional basis for the selection is to choose a family of sampling distributions whose support matches the feasible values for the design variables. For example, the selection is based on the choice of distributional parameters being able to easily be chosen so that the set of designs which might be randomly generated matches the set of designs which meet basic design limits.

An implementation of the CEM-based algorithm 100 is summarised in the flow chart in Fig. 1 . The algorithm 100 starts at step 102 and terminates at step 1 16. The algorithm 100 considers N candidate designs per iteration. The considered designs at each iteration is called a generation. Each candidate design is characterized by the vector of design variables where d is the number of variables or the problem dimensionality. In step 102, the algorithm 100 initialises a distribution of designs. Initialisation of the distribution may include selecting a respective type of probability distribution to characterise each one of the multiple design variables. As mentioned, the selection of the respective probability distributions depends on the nature of the design variables.

The parameter vector represents the initial distribution, which is generally selected to spread its probability fairly evenly across all possible designs. Subsequently, each generation / ^' is drawn from the distribution

In step 104, candidate designs are sampled from the distribution of designs. The sampling of candidate designs may include random ly sampling from the distribution of designs. To avoid doubt, the entire population of designs need not be available or retrieved in order to generate the sample of candidate designs. Since the distribution of designs is well characterised by the distributional parameters (e.g. using mean μ and variance a ² for a normally distributed variable), a sample can be generated (e.g.

randomly generated) using the distributional parameters. The number of candidate designs A/ to sample depends on the time, precision and success rate requirements. From empirical data, the number of candidate designs can range between 3d and 7d, where d is the dimensionality (i.e. the number of degrees of freedom). Of course the number of candidate designs can be outside this range. In a general example described below, the number of candidate designs is 100. In an antenna design example described below, the number of candidate designs is 45. In step 106, a formulated fitness function for each design in the candidate group of designs is evaluated. Depending on the goal of the optimisation, a fitness function is formulated. For example, for a goal of balancing between antenna directivity and antenna bandwidth, the formulation of the fitness function may include at least directivity and bandwidth, such as in the form of a directivity-bandwidth product. If a greater weight is to be given directivity over bandwidth (or vice versa), then a weighted combination of directivity and bandwidth may be formulated as the fitness function. The evaluated fitness function may result in a numerical value, with a larger (or smaller) numerical value indicating a more desired electromagnetic performance and with a smaller (or larger) numerical value indicating a less desired electromagnetic performance. In some implementations, the numerical value may be a weighted combination of different electromagnetic performance criteria. For example, for an antenna, both the directivity- bandwidth product and the side-lobe levels are important electromagnetic performance criteria. The evaluated fitness function in one form can provide a certain weight (e.g. 95%) to the directivity-bandwidth product and a complementary weight (e.g. 5%) to side-lobe levels to combine the two criteria.

In step 108, the candidate group of designs are ranked according to the evaluated fitness function. Depending on the objective of the optimisation, whether it is a maximization or minimization problem, the fitness values are sorted in descending or ascending order, respectively. A subset N _el (e.g. top-ranked) of the candidate group of designs are selected as the elite group of designs. The ratio of elite to candidate designs (i.e. N _el : N) determines the compromise between the speed and the

thoroughness of the search. A large ratio decreases the convergence speed but increases the optimality of the final design, while a small ratio exhibits the opposite behaviour. Through observations, the inventors have found that the ratio N _el : N can range between 15% and 25% for achieving an acceptable balance, though ratios outside this range might prove suitable for different problem types. In a general example described below, each candidate group of designs has 100 designs and each elite group of designs has 20 designs, which represents a 20% ratio. In an antenna design example described below, each candidate group of designs has 45 designs and each elite group of designs has 10 designs, which represents a 22% ratio. In step 1 10, the (i+1 )-th set of distributional parameters representing (i+1 )-th probability distributions characterising the respective multiple design variables is determined. The determination allows the distributions of design at the next iteration to be fitted to or otherwise describe the selected elite group of designs. In one implementation, the determination is based on minimising cross entropy between the empirical distributions characterising the i-th elite group of design and the (i+1 )-th probability distributions In this implementation, the algorithm 100 finds distributional parameters that best-fit or best describe the elite group of designs

according to the Kullback-Leibler distance. In step 1 12, the determined set of distributional parameters may be smoothed over consecutive iterations. In other words, distributional parameters from one or more previous iterations may be used to influence the distributional parameters determined at the current iteration. Smoothing serves to reduce the danger that the distribution will concentrate too quickly around a suboptimal design. In one implementation, a weighted value of the i-th set of distributional parameters may be used to adjust the (i+1 )-th set of distributional parameters. For example, a smoothing parameter can be used to slightly modify the new distributional parameters:

where is a vector of unsmoothed distributional parameters at the current iteration,

and is a vector of smoothed distributional parameters from a previous iteration.

In step 1 14, the diversity of the candidate or elite designs is determined and compared with a diversity threshold. Based on the comparison, the algorithm 100 may be terminated or continued to iterate. In general, the diversity measures the spread of the distribution of designs at the current iteration. One way of quantifying the diversity is to use the maximum variation between the candidate or elite design in any one or more design parameters. The diversity threshold is indicative of the acceptable level of precision in the ultimate design. Responsive to the comparison, if the determined diversity is below the diversity threshold, the algorithm 100 is terminated, otherwise the algorithm 100 is continued to iterate. A skilled person would appreciate a number of steps above are optional steps. For example, steps 1 12 and 1 14 are optional. Another optional step that is intended to improve the global search ability of the CEM is the use of a mutation operator. In one implementation, the algorithm 100 introduces a perturbation to the (i+1 )-th set of distributional parameters based on a comparison of the evaluated fitness function of a top-ranked design over consecutive iterations. For example, if the best fitness value has not been improved for a number of iterations, the algorithm 100 will randomly add small perturbations to the shape parameters to avoid being stuck at a local optimum.

GENERAL EXAMPLE: BENCHMARK PROBLEMS To demonstrate the applicability of the disclosed method, the CEM-based algorithm 100 is applied to benchmark problems with fitness functions having known locations of global and local solutions. The application of the algorithm 100 and its results are as follows.

Two continuous test fitness functions are formulated and used, one unimodal and one multimodal.

The unimodal function is the Axis Parallel Hyper-Ellipsoid, also called the Sum of Squares function, which is described by the equation

with the global minimum

The multimodal test function is the continuous multidimensional Ackley function, formulated as

with the global minimum

Both test fitness functions in this general example are restricted to the domain is the problem dimension, and inverted to create maximization problems. Fig. 2 illustrates the convergence of the CEM to the optimal points for the 2-D Sum of square s. The set of feasible values for each variable is located between , which is a

"compact interval" or a continuous range of values, including both its endpoints. The beta distribution family is selected to draw the sampling distributions from. This selection is based on the beta distribution family also ranging over a compact interval. A linear transformation is used to convert the beta distribution's support, [0, 1 ], to match the required compact interval of [-10,10]. The stopping criterion is the threshold meaning that when the best fitness value reached a point higher than the threshold, the CEM-based algorithm 100 stopped searching and the optimisation is considered successful. If the algorithm 100 is trapped in a suboptimal solution and stopped progressing for 50 iterations, the process was terminated and the optimisation was considered unsuccessful.

The maximum number of dimensions in this general example on test fitness functions is 30, which is a highly multidimensional search space compared to electromagnetic design problems typically handled with global optimisation methods. The mean number of function evaluations (MNFE) and success rate were calculated by averaging data of 9 optimisation runs. The sample size (N) is fixed to 100 candidates with The results are summarised in Table I.

It can be seen that 100% success rate was achieved for up to 15 dimensions with convergence. Even when the threshold was not attained, the solution found by the algorithm 100 was close to the known optimum. To approach 100% success rate for higher-dimensional search spaces, the sample size N should be increased. For example, for Ackley function when the sample size N is set to 200 candidates, the success rate was firmly 100% and MNFE was equal to 13044.

Comparison with Particle Swarm Optimisation (PSO) Technique

The CEM-based algorithm 100 can be compared with other metaheuristic techniques. As a comparative example, the nature-inspired PSO algorithm is applied to the same test fitness functions in the above general example. The PSO algorithm exploits the knowledge of groups (swarms, flocks) and their cooperation to reach the global best location in the fitness function. . It is usually recommended to decrease the influence of the previous velocities of particles over the course of a run from to

However, for low-dimensional cases the speed of convergence can

be further increased by decreasing the inertia weight to 0.01. Therefore, for analyses on test functions with low dimensions (2-D and 5-D), was set to 0.01 , but for higher dimensional cases (10-D to 30-D) it followed the general rules with This ensured the best performance of PSO for the comparison with CEM. Boundary conditions is another factor which has a significant effect on the performance of the PSO. Here, absorbing walls were applied to the particles trying to escape the solution space. When a particle hits the boundary of the solution space in one of the dimensions, the velocity in that dimension is zeroed, and the particle will eventually be pulled back toward the allowed solution space . As it is understood to be the most commonly used boundary condition, it was applied in this comparative example. The same threshold as for the general example using the CEM-based algorithm 100 was used as the stopping criterion for the PSO algorithm and the population size is also fixed to 100 candidates to ensure a proper comparison with the CEM-based algorithm 100. The optimisation result of the Sum of Squares and Ackley functions by PSO is summarized in Table II. A comparison against Table I shows that for both unimodal and multimodal test functions CEM performed more than two times faster than PSO.

In comparison, characteristics of CEM-based algorithms of the disclosed method are: (I) the total number of function evaluations is relatively low compared to PSO and (II) no adjustment of the internal parameters is required. These two properties are beneficial for optimisation of EM design problems where fitness functions are frequently evaluated through simulation and, thus, MNFE defines the overall optimisation time. By using a CEM-based algorithm, the time required to complete the optimisation can be

significantly decreased, and the need to adjust internal parameters of the optimizer can be avoided, which may also decrease the design time. It should also be noted that using a larger number of candidate designs achieve a higher success rate. It can be attributed to knowledge from statistics that when more experiments are conducted, the resulting distribution is more precise.

SPECIFIC EXAMPLE: DESIGN OF RESONANT CAVITY ANTENNA The CEM-based algorithm in accordance with the present disclosure may be used in conjunction with three-dimensional electromagnetic full-wave simulation, at each iteration or at least one of the iterations, that numerically solves Maxwell's equations (e.g. using finite differences or finite elements) to evaluate the performance of an electromagnetic device under optimisation. To illustrate this use, the CEM-based algorithm is applied in the context of a radio-frequency electromagnetic radiating device (e.g. a resonant cavity antenna operating in the frequency range between 10 and 20 GHz). A skilled person would also appreciate that, with minor modifications, the CEM- based algorithm may also be applied to optimise a radio-frequency electromagnetic non-radiating device (e.g. a microwave filter) or an optical device. Resonant cavity antennas (RCAs), also known as Fabry-Perot cavity antennas, EBG resonator antennas, partially reflective surface antennas and 2D leaky-wave antennas, are currently under active research due to their attractive high-gain performance, simple feed structure and planar profile. These properties make them a potential alternative to the conventional antennas, such as reflectors, horns, lenses and arrays, especially for applications at microwave and millimetre wave frequencies. In RCAs, a partially reflective superstrate is placed above a fully reflective ground plane to form a resonant cavity in between, as schematically shown in Fig. 3, which depicts a form of RCA design for optimisation. The fields in the cavity experience multiple reflections, and when appropriate conditions are satisfied, a directive beam is generated at boresight. Here, the CEM-based algorithm 100 is applied to RCA designs to find the optimal combination of parameters for a superstrate with transverse permittivity gradients (TPG) made of concentric dielectric rings. For illustrative purposes, the level of quantization is fixed to three segments, allowing the widths of rings to each be continuous variables. The number of possible combinations of dielectric materials (each having discretely variable permittivity values and/or discretely variable thickness values) with variable widths of the rings in such a superstrate create a multidimensional optimisation problem suitable for the CEM.

Allowing the optimisation to involve both continuous and discrete variables is useful for designs employing off-the-shelf dielectric materials of continuously variable dimension (in this case, width of the concentric rings). As the choice of commercially available dielectrics is limited to some specific permittivity values, the results obtained by continuous optimisation methods are not practical for implementation. The use of discrete variables allows a designer to define a database list of available materials (e.g. permittivity values) to be used in optimisation. Second, due to the use of a specific probability distribution family, it was possible to fix the diameter of the superstrate

the widths of its sections

Model description - Framework of the optimisation problem

The RCA under optimisation and is schematically shown in Fig. 3. The overall diameter of its superstrate and ground plane was set to D = 48 mm, which is

is a free-space wavelength corresponding to the first resonance freq uency of the cavity thickness of the superstrate is t. The number of permittivity

steps in the TPG superstrate was chosen to be equal three, which is understood to be a reasonable trade-off between the fabrication complexity and performance in such a compact footprint. Therefore, the vector of parameters has 6 variables which is a 5-dimensional parameter space due

to the enforced dependency of widths, i.e.

In general, the permittivities and the widths of the rings form the vector of variables and superstrate diameter D is fixed. This optimisation problem can be mathematically written as:

where is a set of discrete permittivities and is the level of quantization in the TPG

superstrate.

To apply the CEM, these design parameters are modelled using two probability distribution families. The feasible values of permittivity of each ring are defined by commercially available microwave dielectric materials. Based on such feasible values of permittivity, the permittivity vector is sampled from a general discrete distribution on the available permittivities. Here, the family of all probability distributions on the finite permittivity set is low-dimensional. There is no need to narrow down the sampling distribution family any further. Thus, at each iteration, the probability distribution for is described by a stochastic vector of length , which is the number of available

permittivity values in a database (defined by a designer according to many factors, such as design requirements or material availability). For convenience, these vectors are stored in a matrix. Initially, this matrix is uniform, which means that every permittivity has the same probability to be in a ring. Then, based on fitness function evaluations, those permittivities of sections that result in a better fitness function earn a higher probability of occurring in the next iteration. The ring widths cannot be sampled independently, due to the constraint that they sum to D/2. In addition, for practical reasons, we enforce that for each ring. Thus, the vector elementwise

nonnegative and sum to 1 . These constraints combine to bound a multidimensional geometric region called a simplex. The Dirichlet distribution family has a simplex as its support, and is therefore selected to provide suitable sampling distributions to match the vector above. The matrix and the vector parameter of the Dirichlet distribution are updated at each iteration based on the elite group of designs. An appropriately formulated fitness function reflects a representative figure of merit, which defines the result of the entire optimisation. In this RCA optimisation example, the goal is set at achieving the maximal directivity-bandwidth product (DBP) in the range of frequencies from 10 to 20 GHz. The DBP is calculated as the product of peak directivity and 3-dB directivity fractional bandwidth BW:

where BW was calculated by and are the lowest and the highest frequencies at the half-power level of

boresight directivity, respectively. Due to the multiplicative nature of this fitness function, the highest value of BW in this RCA optimisation example is restricted to:

It affects the optimisation in the way that when the bandwidth reaches this limit, only designs with higher peak directivity are searched. This technique is applied here to achieve a result with a hi her value of peak directivity as well as wide BW. In

this example, has been set however, this constraint can be changed to

any other value, which would result in a different superstrate profile and performance parameters if optimisation is necessary.

Side-lobe levels are typically higher in wideband RCAs, especially in the E-plane at the end of design frequency range. Their effects are included in the fitness function to ensure low side lobe levels (SLLs) in the entire frequency range. Fitness of SLLs in E- plane was defined as the average difference between the desired and the current value of SLLs over the frequency range from

where is the desired SLL in E-plane and is the current SLL

in the range between

Thus, multiple objectives are taken into account in a single objective by a conventional weighted aggregation:

where / = 20 (found empirically). While the impedance matching is not included in the objective function defined by (10) in this example, a skilled person would appreciate that it may be included with minor modifications.

Since the superstrate is non-periodic and has small lateral dimensions (diameter at the lowest frequency), classic design approaches involving unit-cell

optimisations or transmission line modelling cannot be used for the analysis of such compact RCAs. Instead, the full-wave analysis of a complete antenna should be performed for an accurate prediction of boresight directivity. Thus, RCA design models were simulated in CST Microwave Studio (CST MS), and CEM optimisation was performed using MATLAB. When full-wave EM software is used to evaluate the fitness function, the total optimisation time mostly depends on simulation time, which might be very large for complex models. Therefore, it is desirable to use some measures to reduce the model complexity. Here, the bilateral x- and y-symmetry of the RCA superstrate are employed, thus simulating only its quarter by applying appropriate boundary conditions in two planes.

The details of the RCA optimisation parameters used in this specific design example are given in Table III.

In this antenna design example, the sample size N is set to 45, from which 10 elite designs are selected at each iteration. Further, the smoothing parameter is set to within the interval (0.7, 1 ), however, smaller values may be used to decelerate the convergence. In this antenna design example, was chosen. The time required for evaluation of one generation (45 candidate designs) was observed at approximately 20 minutes. In this antenna design example, the stopping criterion was configured to be based on a measure of elite design diversity, defined as the maximum variation between the N _el elite designs in a current iteration. In other implementations, the diversity may be measured based on the N candidate designs. For both vectors this variation was calculated

using Euclidean distances.

The algorithm 100 stopped when diversity was equal to 0 for permittivities and less than the threshold d = 0.009 mm for widths. To evaluate the fitness function, the boresight directivity and the level of the first side lobes were calculated in the range from 10 GHz to 20 GHz with a step of 0.5 GHz. Fig. 4a shows the convergence of the algorithm for one of the optimization results and Fig. 4b illustrates the decrease in the population diversity over the course of the optimisation. It took approximately 14 hours to complete the full optimisation on an Intel Core i7, 3.6 GHz processor with 32GB of RAM.

The permittivities were chosen from the range of Rogers TMM laminates, i.e. TMM3

TMM10i In general, this list can be continued by any other available

materials.

Separate optimisations were performed for three values of the superstate thickness and the resulting antenna performance were analysed. Seven runs per each thickness were executed to compromise between statistical certainty of performance and total run time. Table IV shows the best solutions found for each superstrate thickness. It can be seen that the peak directivity reaches 19.38 dBi for the superstrate of thickness 7.62 mm, lowest side lobe levels can be achieved with t=6.985 mm, and the widest radiation bandwidth was obtained for t=6.35 mm. Thus, each design has some advantages over others, and the choice in favour of one of the solutions can be made based on design objectives. It is interesting to note that without any enforcement made during

optimisation, permittivity profiles of the optimal solutions have a gradual decrease of permittivity from the centre towards the edges.

PROTOTYPE FABRICATION AND MEASUREMENT RESULTS Another aspect of the present disclosure is the fabrication of an RCA based on the CEM-based algorithm. Among the three optimised models, the one with the least superstrate thickness (i=6.35 mm) is chosen for illustrative purposes. . The resulting prototype is shown in Fig. 5. The fabrication of the resulting prototype includes subtractive manufacturing a disk and two concentric rings (e.g. from TMM10i, TMM10 and TMM4 dielectric slabs, respectively), and inserting one into another. The prototype includes two nylon spacers to support the thick superstrate above the aluminium ground plane. The prototype includes a rectangular slot in a ground plane fed by a coaxial-to- waveguide adapter WR-75. In this prototype, the slot dimensions are set to be

12x7.5mm ² for impedance matching. The overall diameter of the ground plane is equal to the diameter of the superstrate.

Fig. 6 shows the simulated and measured boresight directivity versus frequency of this prototype. The measured radiation characteristics of the prototype were obtained in a spherical near-field measurement system at the Australian Antenna Measurement Facility (AusAMF). A general agreement between simulations and measurements can be seen in Fig. 6. The measured peak directivity is 17.6 dBi, and 3-dB directivity bandwidth extends from 1 1 .3 to 19.1 GHz, which is 51 % fractional bandwidth. The comparison of simulated and measured patterns in H- and E-plane at 6 selected frequencies within the operating band are shown in Fig. 7. The simulated and measured beamwidths at all frequencies have an excellent agreement. Overall, measured SLLs are less than -15 dB in H-plane, and less than -10 dB in E-plane, except for 18 GHz where SLLs are higher.

These results demonstrate that by optimising the TPG superstrate by the CEM-based algorithm an optimal permittivity profile may be achiever, which provides relatively high DBP with a relatively thin superstrate. For i=7.62 mm, the RCA with an optimal superstrate configuration provides a 3 dB increase in peak directivity.

It is noted that a CEM-base algorithm of the disclosed method allows for adjustability, including adjusting the sample size size of elite designs and smoothing

parameter Therefore, optimal values of these parameters (which depend on a particular problem) can be found after a couple of first optimisation runs. It is envisaged that the CEM-based algorithm of the disclosed method may be apply to other antenna designs, such as dielectric resonator antennas and lenses, as well as artificial structures, metasurfaces and cloaks.

INTERFACING MATLAB AND CST MS The disclosed optimisation method may be carried out by computing resources, such as one or more computer systems, each system including one or more computers. In the case of a single computer, disclosed optimisation method may be entirely carried out by the single computer. In the case of multiple computers, the disclosed optimisation method may be carried out in a distributed manner by the multiple computers, with one or more steps carried out by one computer and one or more other steps carried out by another computer or other computers. In one example, the multiple computer systems are collocated or closely located. In another example, the multiple computer systems are separately located and communicatively connected via a communications network, such as a public network (e.g. the Internet), a private network (e.g. a local area network), or a virtual private network (e.g. a virtual local area network).

The disclosed optimisation method may be fully or partially carried out by the computing resources. Where the disclosed optimisation method is partially carried out by the computing resources, one or more steps of the disclosed optimisation method are carried out by the computing resources, while one or more other steps of the disclosed optimisation method are not necessarily carried out by the computing resources. For example, the step of sampling from a distribution of designs may be achieved by retrieving a sample of designs from a database, in which data (e.g. the distribution of designs) may be entered manually or provided by other computing resources.

Fig. 8 illustrates an implementation 800 of a method to interface a CEM-based algorithm according to the disclosed method (using either optimisation toolbox or custom-built code) in MATLAB environment with external function evaluations in CST MS. In this implementation, instead of carrying data of each candidate between the software packages, the data of a plurality of designs in the distribution of designs is loaded in CST MS to perform continuous simulations. Thus, the reason for an increased speed is a minimized number of software launches. Although the independence of candidate designs in a sample allows to run the simulations in parallel using parallel computing, the described technique is beneficial in case of relatively small models for optimisation and unavailable powerful computing resources.

In this implementation of interfacing, the general steps include:

802: In MATLAB environment, candidate designs are generated in accordance with a CEM-based optimisation technique, and design variables of each candidate are written to a text file. MATLAB launches a macro file, which is written in Visual Basic for

Applications (VBA) macro editor.

804: CST MS environment: macro reads the text file with the parameters of each model to be simulated, opens the model and writes every candidate as a separate sequence of a Parameter Sweep. To store the simulation results of each candidate, a user-defined watch is used. It creates another text file in a Result folder and writes the results after each simulation. When all simulations are completed, control retunes to MATLAB.

806: In MATLAB environment, the results are read from the Result folder, the fitness function is evaluated and the stopping condition is checked. 808: The iterations continue if the stopping condition is not satisfied.

810: The iterations terminate if the stopping condition is satisfied.

It will be appreciated that the disclosed optimisation method is applicable to

electromagnetic designs other than an antenna design, such as passive microwave circuits such as filters, active microwave circuits such as amplifiers, millimetre-wave circuits, terahertz devices, optical devices, and radio-frequency devices. It will be understood that the present disclosure extends to all alternative combinations of two or more of the individual features mentioned or evident from the text or drawings. All of these different combinations constitute various alternative aspects of the present disclosure.