1.85–5.70 GHz, which is approximately 115.5%, for the VSWR �2.0. The measured bandwidth (115.5%) is wider than the simula-tion one (112.6%). The proposed antenna showed triresonancecharacteristics due to the wide bandwidth.

Figure 8 presents the experimental pattern in the E-plane at f �3.33 GHz. After completing the calibration using a horn antenna,we measured the radiation pattern of the far field. The beamwidthwas measured at approximately 70°. Figure 9 presents the exper-imental pattern in the H-plane at the same frequency. Figure 10shows a comparison of the calculated and measured gain for theproposed antenna. The experimental results are in relatively goodagreement with the calculated results. The antenna gain was ap-proximately 3 dBi over the usable entire band.

4. CONCLUSION

We have presented the characteristics of a cross-shaped micros-tripline-fed printed slot antenna. The proposed antenna with rela-tive permittivity 2.2 and thickness 1.575 mm is analyzed using theFDTD method. The dependence of the design parameters Ws, lu,and offset upon the bandwidth characteristics has been investi-gated. The calculated bandwidth (112.6%) of RT-5880 is widerthan the simulated one (90.6%) of the FR-4 substrate. We alsocompared the optimized offset/Ws ratio and the calculated band-width of RT-5880 substrate for slot width Ws. When Ws is 32 mm,the optimized offset/Ws ratio is 0.344, and the calculated bandwidth is 3.744 MHz (112.6%). The experimental bandwidth of thisantenna is from 1.85 to 5.70 GHz, which is approximately 115.5%,for VSWR � 2.0. The proposed antenna showed triresonancecharacteristics due to the wide bandwidth. This antenna may beuseful as a powerful broadband-array antenna.

REFERENCES

1. H.G. Booker, Slot aerials and their relation to complementary wireaerials, J IEE (London) IIIA 93 (1946), 620–626.

2. R.E. Collin, Antenna and radiowave propagation, McGraw-Hill, NewYork, 1985.

3. R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip designhandbook, Artech House, Boston, 2001, chap. 7.

4. Y. Yoshimura, A microstrip slot antenna, IEEE Trans MicrowaveTheory Tech 20 (1972), 760–762.

5. D.M. Pozar, Reciprocity method of analysis for printed slot andslot-coupled microstrip antennas, IEEE Trans Antennas Propagat 34(1986), 1439–1446.

6. A. Axelrod, M. Kisliuk, and J. Mhoz, Broadband microstrip-fed slotradiator, Microwave J 32 (1989), 81–92.

7. M. Kahrizi, T.K. Sarkar, and Z.H. Maricevic, Analysis of a wideradiating slot in the ground plane of a microstrip line, IEEE TransMicrowave Theory Tech 41 (1991), 29–37.

8. S.M. Shum, K.F. Tong, X. Zhang, and K.M. Luk, FDTD modeling ofmicrostrip-line-fed wide-slot antenna, Microwave Opt Technol Lett 10(1995), 118–120.

9. Y.W. Jang, Broadband T and shunt-stub shaped microstrip-fed slotantenna backed by a ground plane, Microwave Opt Technol Lett 32(2002), 278–280.

10. Y.W. Jang, Experimental study of large bandwidth three-offset mi-crostripline-fed slot antenna, IEEE Microwave Wireless Compon Lett11 (2001), 425–427.

11. K.S. Yee, Numerical solution of initial boundary-value problems in-volving Maxwell’s equations in isotropic media, IEEE Trans AntennasPropagat 14 (1966), 302–307.

12. G. Mur, Absorbing boundary conditions for the finite-difference ap-proximation of the time-domain electromagnetic-field equation, IEEETrans Electromag Compat 23 (1981), 377–382.

13. D.M. Sheen, S.M. Ali, M.D. Abouzahra, and J.A. Kong, Applicationof three-dimensional finite-difference time domain method to the anal-ysis of planar microstrip circuits, IEEE Trans Microwave Theory Tech38 (1990), 849–857.

14. A. Taflove and M.E. Brodwin, Numerical solution of steady-stateelectromagnetic scattering problems using the time-dependent Max-well’s equations, IEEE Trans Microwave Theory Tech 23 (1975),623–630.

© 2004 Wiley Periodicals, Inc.

QUARTER-WAVE PHASE-COMPENSATING MULTIDIELECTRICLENS DESIGN USING GENETICALGORITHMS

Lei Sun, Evor L. Hines, Christos Mias, Roger Green, andDaciana UdreaDivision of Electrical and ElectronicsSchool of EngineeringWarwick UniversityCoventry, CV4 7AL, United Kingdom

Received 2 July 2004

ABSTRACT: This paper illustrates a microwave dielectric lens designusing genetic algorithms (GAs). A quarter-wave phase-compensatingmultidielectric lens, using four types of materials and a zoned structure,is designed using GAs. The dielectric lens shape was designed using aGA with a random initial shape in order to compensate for the phaseerror of the multidielectric lens and achieve the required performance.The simulation result, which is performed using high-frequency simula-tion software (HFSS) from Ansoft, shows that the gain of the side lobesdecreases 21-dB more when using our GAs designed multidielectriclens, as compared with the simulation result of the conical-horn antennawithout multidielectric lens. The performance from �150° to 150° an-gles achieved improvement, which proves that GAs may be an effectivemethod for lens design. Further work will be done in this area. © 2004Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 165–169, 2005;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.20577

Key words: genetic algorithms; Fresnel lens; HFSS

1. INTRODUCTION

This paper describes a “virtual lens” design for a conical hornantenna, by applying genetic algorithm (GA) techniques. Empha-

Figure 10 Antenna gain

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 2, January 20 2005 165

sis is placed on the application of the GA as an artificial intelli-gence method in the telecommunications area. The GA, a “global”numerical optimization method based on genetic recombinationand evolution in nature [1], is an exploratory procedure that isoften able to locate near-optimal solutions to complex problems.To do this, it maintains a set of trial solutions (often calledindividuals), and forces them to “evolve” towards an acceptablesolution [2].

GAs have already been used in the telecommunications area.For example, Michielssen et al. used a GA to synthesize a multi-layer radar-absorbing coating that maximizes the absorption of anelectromagnetic wave over a desired range of frequencies andincident angles [3, 4]. GAs were applied to the problem of thinninglinear and planar arrays in order to obtain the lowest maximum-relative-sidelobe level over a specified bandwidth and scan angle[3, 5]. GAs were also applied to optimize the length, flare angle,and first slot depth of each section of a multisectional corrugatedconical-horn antenna, in order to match the admittance, reduce thevoltage standing wave ratio (VSWR) of the horn, and generate thedesired pattern [6]. Therefore, the application GAs in the telecom-munications field is described in quite a number of papers. Inaddition, GAs have been applied to the design of an optical lens[7]; however, GAs have not been effectively applied to micro-wave-lens design.

The purpose of a microwave lens in antenna design is todecrease or increase the wave speed, so that a plane wave can begenerated after it passes through the microwave lens. To theauthors’ best knowledge, no research has focused on microwave-lens design using a GA. This paper presents the design of amicrowave lens using a GA and simulates the optimized lens witha conical-horn antenna as a source. The lens is designed to operateat 20 GHz. The simulation is performed by using HFSS [8]. Theoptimized results are also described in this paper.

Microwave lens design using GAs is described in section 2, GAresults and discussions are presented in section 3, the simulationanalysis of the GA-designed lens using HFSS is given in section 4,section 5 is the conclusion.

2. MICROWAVE LENS DESIGN USING GENETICALGORITHMS

The multidielectric lens design presented here is based on aFrensel lens. One of the main advantages of a Frensel lens is thatit can be produced at low cost; thus, we can try to balance therelationship between the cost and the size of the antenna. One ofthe main disadvantages of this lens is that it does not attain the

same aperture efficiency as a shaped lens, since it corrects thephase of the feed antenna only at discrete locations over itsaperture, whereas a shaped lens corrects the phase of the feedantenna at every location [9]. The solution to the problem ofdegradation in gain, due to phase quantisation errors over theFresnel lens aperture, is to design the lens to correct the phase atan increased number of locations [10]. One method to accomplishthis is to use additional grooves in the dielectric material [11].Another is to use different dielectric constants for the variouszones in the lens [12], which is called a Fresnel-zone plate lens(FZPL). The cost can be reduced by using the first method, but amore complex fabrication process will result and the lens will bethicker. On the other hand, with the second method, a thinner lenscan be produced at higher cost. Hence, by combining the twomethods, it may be possible to balance the cost and fabricationaspects. Therefore, we attempt to design such a multidielectric lensusing GAs in this paper.

The FZPL, a focusing and imaging device invented and studiedby Fresnel more than 150 years ago [12], can be used for focusingand imaging electromagnetic waves, and a plate lens accomplishesthese functions through diffraction and interference, rather thanrefraction [11]. In principle, the FZPL does not transform theincident spherical wave from the feed into an outgoing plane wavesmoothly [12]. Figure 1 shows the lens structure to be shaped usinga GA.

In Figure 1, F is the focal distance of the lens (the feed sourceis set to the focus point); b1, b2, b3, and b4 are the radii of fourlayers, with permittivity �1, �2, �3, and �4, respectively.

Figure 2 illustrates the top view of the lens. The lens uses fourmaterials with different permittivities. The required values for thedielectric constants depend on the thickness w of the lens base, andthe number P of different dielectric constants as follows [9]:

w ��0

P���i � ��i�1�, i � 2, 3, 4, . . . , P, (1)

where �0 � 15 mm (design frequency � 20 GHz) is the wave-length in free space. The amount of phase correction is expressedin terms of P and is given by 360�/P. The lens designed in thispaper is a 90° phase-correction lens, so p � 4 and w � �0/ 2.According to Eq. (1), the permittivity can be calculated if �1 hasFigure 1 Representation of a quarter-wave multilayer lens

Figure 2 Top view of the lens

166 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 2, January 20 2005

been selected. In theory, to reduce manufacturing cost, �1 shouldbe the lowest value, which is 1. However, to adjust the wave phasewhich passes through the first layer, �1 cannot be equal to theenvironmental permittivity. As a result, �1 is selected as 2 and theother permittivities �2, �3, and �4 are 3.66, 5.83, and 8.49, respec-tively.

The radius of each zone is determined using the followingequation [9, 12]:

bm � �2mq�0�F �w

2� � �m�0

2 � 2

, (2)

where m is the zone number and q is the phase-correction factor(q � 1 for the classical FZPL, q � 0.5 for the half-wave FZPL,and q � 0.25 for the quarter-wave FZPL, which is suitable for thelens in this paper according to [9]).

In 2D design, several points are selected along the vertical andhorizontal coordinates to determine the 2D shape of the lens. TheGA has been used to optimize each points’ horizontal coordinate.Figure 3 is a pictorial representation of how we use the GA todetermine the lens shapes on top of the planar layer. For simplicity,a single-layer design shape is shown. An X–Z coordinate is usedhere. Our system has been implemented using Matlab [13] and in

it, several points can be selected along the Z coordinate in order todivide the lens layer into several parts. To simplify the simulationstructure for using HFSS [8], four points were selected in theprocess of the lens design. In total, 17 points (layer 1 with per-mittivity �1 has five points with one more point in the middle of thelens) were selected, since there are four layers.

The parameter of the objective function for the GA is length x.When the lens has four layers, the chromosome is written as

Chromosome

� ��x10x11x12x13x14�, �x21x22x23x24�, �x31x32x33x34�, �x41x42x43x44��.

(3)

The initial value of xij (i � 1, 2, 3, 4; j � 0, 1, 2, 3, 4) isrepresented in binary by using eight bits, for example, x12 �01101000, so that the real value for xij is given by

xij_real � Q��m�1

8

xij�m� � 2m�1�/28, (4)

Figure 3 Pictorial lens representation for GA lens design

Figure 4 Schematic diagram of the GA-lens design process. [Colorfigure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

TABLE 1 Genetic Algorithm Results for Selected Generations

Gen � 2 Gen � 5 Gen � 10 Gen � 30 Gen � 60 Gen � 100

x1 10.03 9.44 10.11 9.78 9.70 9.70x2 8.85 8.77 8.94 9.02 8.68 9.44x3 7.17 6.75 7.08 6.66 6.83 6.66x4 3.63 4.13 3.29 3.63 3.71 3.71x5 0.08 0.34 0.00 0.00 0.00 0.00x6 3.79 3.96 4.05 3.96 3.96 4.05x7 3.04 3.04 2.61 2.61 2.61 2.70x8 1.94 0.42 1.43 1.35 1.35 1.35x9 0.17 1.60 0.67 0.00 0.00 0.00x10 2.70 2.02 2.53 2.45 2.45 2.53x11 1.10 0.67 2.02 1.60 1.69 1.69x12 0.25 0.42 0.59 0.84 0.84 1.01x13 1.43 1.01 0.42 0.08 0.00 0.00x14 1.86 1.69 2.02 1.77 1.86 1.86x15 1.35 1.26 1.18 1.35 1.35 1.18x16 0.17 0.34 0.67 0.59 0.42 0.59x17 0.00 0.00 0.00 0.00 0.00 0.00

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 2, January 20 2005 167

where xij_real is the real value of xij, Q is the largest quantizationlevel which is half of the focus length, and m is the position of eachbinary value in xij.

3. RESULTS AND DISCUSSIONS

To perform an effective search for better and better structures, GAsonly require “payoff values” (objective-function values) associatedwith individual strings. GAs use random choice as a tool to guidea search toward regions of the search space with likely improve-ments [14]. There are four main functions in a typical GA: repro-duction, crossover, mutation, and fitness. Figure 4 schematicallyillustrates how the GA has been applied in our lens design.

According to [14–16], the GA may or may not be truly doneoptimising a design at 70-to-90 generations, the population sizepresented is set as 30 and the maximum generation iteration is setas 100. To simplify the program structure and reduce calculationtime, one crossover point has been applied in our GA process. Inaddition, the mutation and reproduction probabilities, which areimportant keys to determine the speed of the GA process, are setas 0.01 (the probability of mutation should be 5% or less [16]) and

0.1, respectively. According to Eq. (3), each chromosome involves17 variables and each variable consists of eight genes, whichdetermine the variables’ values.

The fitness function can be set so as to amend phase error,reduce lens weight, increase the main lobe gain, and so forth. Here,for the multidielectric lens, the most important aspect is to amendphase error, so that the fitness function is set to achieve the bestvalue to construct a multidielectric lens with the ability to amendphase error. This is determined in [17, 18].

The fitness function for amending the phase error is given by

Fitness � �Pref � Pcurrent�,

where Pref is the phase at which waves travel in air and Pcurrent isthe wave phase of the current GA design point.

Table 1 shows the GA simulation results for some differentgenerations. The results, which are the horizontal coordinates ofeach selected point, do not improve after 30 generations and theconvergence speed is slower than that before 30 generations.

The results also suggest that after 60 generations, no significantimprovements occur; this fits the theory in [16]. So, 100 genera-tions appear to give a good compromise in terms of the computa-tional efficiency and “accuracy” of results. Also, it may makesense to attack a problem with several runs of a small-population

TABLE 2 Genetic Algorithm Z-Value Results for the 10 Simulations

TimesZ (mm) 1 2 3 4 5 6 7 8 9 10 Average (mm)

Z (mm)0 9.78 9.78 9.34 10.79 9.70 9.70 9.61 9.34 9.78 9.70 9.764.79 8.94 8.94 8.94 9.44 9.61 9.02 8.94 8.94 9.11 8.94 9.089.58 6.83 6.83 6.83 6.83 6.91 6.83 6.66 6.83 6.83 6.83 6.82

14.37 3.71 3.71 3.71 3.63 3.62 3.63 3.71 3.71 3.63 3.71 3.6819.16 0 0 0 0 0 0 0.67 0 0 0 0.0719.26 3.96 3.96 3.96 3.96 4.05 3.96 3.96 4.05 4.05 4.05 4.0022.05 2.70 2.70 2.70 2.61 2.53 2.61 2.70 2.70 2.61 2.70 2.6624.83 1.26 1.26 1.26 1.26 1.35 1.35 1.35 1.35 1.35 1.26 1.3127.62 0 0 0 0 0 0 0 0 0 0 027.72 2.45 2.45 2.70 2.45 2.45 2.70 2.53 2.70 2.70 2.70 2.5829.96 1.69 1.60 1.60 1.69 1.69 1.69 1.69 1.69 1.26 1.69 1.6332.20 1.01 1.01 0.84 0.84 0.84 0.59 1.01 0.84 1.35 0.84 0.9234.44 0 0 0 0 0 0 0 0 0 0 034.54 1.86 1.77 1.77 1.77 2.02 1.86 1.77 1.77 1.77 1.77 1.8136.52 1.18 1.35 1.35 1.18 1.35 1.35 1.18 1.18 1.35 1.18 1.2638.49 0.67 0.59 0.67 0.59 0.59 0.59 0.59 0.67 0.67 0.59 0.6240.47 0 0 0 0 0 0 0 0 0 0 0

Figure 5 Side view of the GA-designed lens shapeFigure 6 3D plot of the GA-designed lens. [Color figure can be viewedin the online issue, which is available at www.interscience.wiley.com.]

168 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 2, January 20 2005

GA rather than with just one run of a large-population GA, as bothwill often cost the same in terms of simulation time [16]. Hence, toachieve the best results, the same program has been run 10 times andthe data for simulation by HFSS is the average of these 10 results,based on the population of 100 generations. The results are listed inTable 2. The table illustrates that the lens consists of four layers withdifferent zone heights and radii. Here, the Z coordinate is fixed and thedistance between the two points in each layer is set equally.

Figure 5 shows a plot of the wide view of the lens designed byour GA. It shows that the lens has four sections with differentmaterial permittivity. Figure 6 shows the 3D plots of the lens. Wecan see that the lens has four layers; the center layer has the largestheight with the lowest permittivity and the other layers’ heightsdecrease gradually from the inner layer to outer layer.

4. SIMULATION ANALYSIS OF THE GA-DESIGNED LENSUSING HFSS

The lens was fed with a conical horn which has a gain of 13 dBiat 20 GHz. Figure 7 describes the HFSS simulation results. Thepeak gain of the horn feed is 13 dBi with a 15-dB decrease in thesidelobe levels, from �150° to 150° angles. The peak gain of themultidielectric lens designed by the GA is 11 dBi with 1 36-dBdecrease in the sidelobe levels in the same range of angles. Theresults show that the multidielectric lens offers a somewhat betterperformance than that of the conical horn feed without a lens. Thedegradation between the peak gain and the sidelobe level of themultidielectric lens (36 dB) is shaper than that of the conical hornfeed (15 dB) from �150° to 150° angles.

5. CONCLUSION

This paper discusses the use of a GA to design a multidielectriclens. The result of the simulation shows that the GA-lens designsexhibit improved performance. However, the lens would be moreattractive if the peak gain of the multidielectric lens could beincreased, and the lens shape be improved, due to the complexfabrication process.

Further research work will attempt to improve the performanceof the antenna lens by applying other advanced GAs, such as thenondominated sorting GA (NSGA), messy GA (MGA), domaindecomposition GAs (DDGAs), and so on. Another important fu-ture work will be to amend the parameters’ values in GA appli-cation, including values of population size, mutation probability,crossover probability, and gene location in each chromosome.

In addition, the fitness function presently focuses on amendingphase error. In future GA applications in lens design, other aspectssuch as weight, frequency band, directivity, and so forth can alsobe considered in tandem in order to achieve the ideal lens for otherapplications.

REFERENCES

1. R.L. Haupt, Comparison between genetic and gradient-based optimi-zation algorithms for solving electromagnetics problems, IEEE TransMagn 31 (1995).

2. J.F. Frenzel, Genetic algorithms a new breed of optimization, IEEEPotentials (1993), 21–24.

3. R.L. Haupt, An introduction to genetic algorithms for electromagnet-ics, IEEE Antennas Propagat Mag 37 (1995).

4. E. Michielssen, J.M. Sajer, S. Ranjtthan, and R. Mittra, Design oflightweight, broad-band microwave absorbers using genetic algorithm,IEEE Trans Microwave Theory Tech MTT-41 (1993), 1024–1031.

5. R.L. Haupt, Thinned arrays using genetic algorithms, IEEE TransAntennas Propagat AP-42 (1994), 993–999.

6. D. Yang, Y. Chung, and R. Haupt, Genetic algorithm optimization ofa multisectinal corrugated conical horn antenna, Microwave Opt Tech-nol Lett 38 (2003), 352–356.

7. X. Chen and N. Yamamoto, Genetic algorithm and its application inlens design, Proc SPIE 2863 (1996), 216–221.

8. HFSS, http://www.ansoft.com/products/hf/hfss.9. H.D. Hristov, Fresnel zones in wireless links, zone plate lenses and

antennas, Artech House, Boston, 2000.10. A. Petosa and A. Lttipiboon, Design and performance of a perforated

dielectric Fresnel lens, IEE Proc Microwave Antennas Propagat 150(2003).

11. D.N. Black and J.C. Witse, Millimeter-wave characteristics of phase-correcting Fresnel zone plates, IEEE Trans Microwave Theory TechMTT-35 (1987), 1122–1129.

12. H.D. Hristov and M.H.A.J. Herben, Millimeter-wave Fresnel-zoneplate lens and antenna, IEEE Trans Microwave Theory Tech 43(1995), 2779–2785.

13. Matlab, http://www.mathworks.com.14. D.E. Goldberg, Genetic algorithms in search, optimization, and ma-

chine learning, Addison-Wesley, Reading, MA, 1989.15. K.F. Man, K.S. Tang, and S. Kwong, Genetic algorithms, Springer-

Verlag, London, 1999.16. Y. Rahmat-Samii and E. Michielssen, Electromagnetic optimization

by genetic algorithms, Wiley, New York, 1999.17. C.A. Balanis, Antenna theory analysis and design, 2nd ed., Wiley, New

York, 1997.18. J.D. Kraus, Antennas for all applications, 3rd ed., McGraw-Hill, New

York, 2002.

© 2004 Wiley Periodicals, Inc.

WIDEBAND MICROSTRIP ANTENNAUSING HOOK-SHAPED FEED

B. Lethakumary, Sreedevi K. Menon, Priya Francis,C. K. Aanandan, K. Vasudevan, and P. MohananCentre for Research in Electromagnetics and AntennasDepartment of ElectronicsCochin University of Science and TechnologyCochin 682 022, India

Received 26 June 2004

ABSTRACT: In this paper, we introduce a novel feeding technique forbandwidth enhancement of a rectangular microstrip antenna. This an-tenna offers an impedance bandwidth of 22% without degrading the effi-ciency. The effect of the feed parameters upon patch characteristicssuch as resonant frequency, impedance bandwidth, and radiation pattern

Figure 7 Simulation results using HFSS

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 2, January 20 2005 169