Fresnel zone plate lens (FZPL) has widely been used in electromagnetic antenna applications. Most analysis method based on the potential (scalar) wave approximation has
been applied to a few very limited and simplified cases. The present paper analyzes the FZPL in more general form including the diffraction and transmission using the method
of moments (MoM). The focusing gain characteristics in the oblique incidence as well as in the normal incidence are considered. The MoM solution using the three-dimensional
vectorial formulation requires a large memory space for the FZPL as it is operated at a short wavelength. This is simply overcome by using an iterative conjugate gradient
method for the numerical evaluation. The MoM solutions are compared with the other solutions

Kagawa, Yukio

Kim, Tae Yong

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering, Okayama University, Vol.35, Nos.l&2, pp.53-61, March 2001Focusing Characteristic Analysisof Circular Fresnel Zone Plate LensTae Yong KIM! and Yukio KAGAWA2(Received December 19. 2000)Fresnel zone plate lens (FZPL) has widely been used in electromagnetic antenna appli-cations. Most analysis method based on the potential (scalar) wave approximation hasbeen applied to a few very limited and simplified cases. The present paper analyzes theFZPL in more general form including the diffraction and transmission using the methodof moments (MoM). The focusing gain characteristics in the oblique incidence as well asin the normal incidence are considered. The MoM solution using the three-dimensionalvectorial formulation requires a large memory space for the FZPL as it is operated ata short wavelength. This is simply overcome by using an iterative conjugate gradientmethod for the numerical evaluation. The MoM solutions are compared with the othersolutions.1 INTRODUCTIONFresnel zone plate lenses have been used as a flat optical lens [1]. Their equivalence has also widely been used inelectromagnetic antenna applications. The use for receiver antennas of the communication satellite televisions,radio astronomical observation and so on, is reported because of their structural simplicity and low cost [2].Hoashi et al. reported the evaluation of a FZPL antenna based on Kirchhoff's scalar formula (3]-(4]. Guo et al.considered the far-field analysis for the offset FZPL antenna design [5].In the present paper, the FZPL is considered as the electromagnetic scattering and diffraction problemsin the exact manner, for which numerical techniques are attempted to solve the fundamental field equations,subject to the boundary conditions and the geometry constraints. The formulation is presented and the MoM isapplied as a solution method. The MoM is a direct boundary integral equation method, suitable for the analysisof electromagnetic scattering and diffraction problems [6]-[10]. The MoM reduces the integral equation to a setof linear algebraic equation. Although the application is direct, however, when the dimension of the object islarge relative to the wavelength to be considered, enormous computational resources are required. This is simplyovercome by using a restart conjugate gradient method for the numerical solutions [8]-[10]. Some numericalsimulation results and the comparison with other solutions are presented.2 MOMENT METHOD FORMULATIONThe operation of the zone plate lens antenna is based on the wave interference as the results the diffraction,scattering and the radiation from the induced surface currents. The integral expression is applied to 3-D surfaceIThe Graduate School of Natural Science and Technology2Departrnent of Electrical and Electronic Engineering5354 Tae Yang KIM and Yukia KAGAWAof the conducting bodies of arbitrary shape for the electric field. The boundary condition is enforced in such away that the tangential electric field vanishes on the perfectly conducting surfacen x E inc = n x (jwA) + n x \7<1> on S (1)where n denotes the outward unit normal vector on the conducting surface S, E inc is an impressed electric fieldand w denotes the angular frequency. The magnetic vector potential A and the electric scalar potential <1> arederived as the following:A(r) J-l! J(r')G(r, r')dS' (2)Sf<1>(r) L 1(\7'. J(r'))G(r,r')dS' (3)WE: SfG(r,r')e-jkR (4)4nRwhere k = W(E:J-l)1/2 represents the wave number corresponding to the external medium (permittivity E:, per-meability J-l) and R = Ir - r'l. rand r' are the position vectors at a field point and at the source point,respectively.Figure 1 Triangular surface elements sharing a common edge In.The unknown current distributions J(r') is expanded to be solved in the form:(5)where I n is the unknown current coefficient and f n is the known expansion function. As shown in Figure 1, theexpansion function f n is definded on a pair of triangular patches T: and Tn- sharing a common edge in as:. in ± ±f n = A± Pn for Tn2 n (6)where in is the length of the common edge, A; is the area of the triangle patch T!, and P; is the positionvector with respect to T! [6J. Both sides of Equation (1) are integrated over the surface S, so that the MoMexpression is:< f m' E inc >= jw < f m' A > + < f m' \7<1> > (7)where the notation < a, b > denotes a symmetric inner product or Is a . bdS. Here the weighting function f mis chosen to be the same as the expansion function f n' The linear algebraic equation is thus obtainedor {Vm }, m = 1,2,· .. , N (8)Focusing Characteristic Analysis of Circular Fresnel Zone Plate Lenswhere Zmn is a generalized system matrix and Vm is a driving vector depended on an excitation. This fullyoccupied equation system has to be solved numerically for the unknown current coefficient In' Especially for alarge-scale electromagnetic problems, the system matrix Zmn is large, full and dense. The iterative conjugategradient method is applied to solve the discretized linear system for the numerical solution.3 DETERMINATION OF A FRESNEL ZONE CONFIGURATIONA typical FZPL consists of the closed conductive opaque annuli rings as shown in Figure 2. The conductingopaque zones are assumed to lie in the x-y plane, centered at (0,0,0) in the Cartesian coordinates. The focalpoint must lie at (0,0, F), where F is the focal length. According to the concept of the Fresnel's principle,the focal point is created in such a way that rays via the boundaries of the rings yield successively path-lengthdifference of one-half of the wavelength. The relation of the inner and outer radius of j-th zone in the FZPL isthus determined based on the optical ray theory as:55J(F + j>'o + >'0/4)2 - F2J(F + (j - 1/2)>'0 + >'0/4)2 - F2(9)(10)where >'0 is the wavelength corresponding to the operation frequency in free space [4]. As shown in Figure 3, thetwo complementary formulations are possible for diffraction with conductive rings and transmission through theapertures. Kirchhoff scalar wave approximation can be applied to solve the transmission through the aperture.xE incPlane waveyConducting annular ring-41_~-r-j-th zoneR l!utR ~n JJ...--'--........-----+-----1~ ZFFocal pointFigure 2 Geometry of a circular Fresnel zone-plate lens.4 NUMERICAL RESULTSIn this section, the numerical results a·re presented for some example of Fresnel zone plate lenses. The geometryfor the FZPL is determined based on Equation (9) and (10). The conducting annuli rings are discretized totriangular patches. Since the unknown coefficients of the induced current are obtained by solving the system ofequations. The scattered electric field can be calculated directly from the induced currents. The scattered nearfield solution from the conducting zone plates can also be obtained from the physical optics (PO) [11], which56 Tae Yang KIM and Yukia KAGAWA• conducting opaque annuli• aperture or open annuli(a) Diffraction model with obstacles (b) Transmission through apertureFigure 3 Two complementary models for the FZPL.0.0120.00960.00720.00480.0024o.0.0120.00960.00720.00480.0024o.IJxl-component(b)0.0060.00480.00360.00240.0012o.0.0060 .. 00480.00360.00240.0012o.Figure 4 Computed surface current distributions(F=5 em, Ao=2.5 em): (a) is the magnitude patterns for thecase ~inc = 0° and (b) is the magnitude patterns for the case ~inc = 4°.Focusing Characteristic Analysis of Circular Fresnel Zone Plate Lensis, to be exact, limited to an electrically large scatterer. For the complementary solution method, traditionalKirchhoff's approximation can be applied on a short wavelength wave such as the aperture problem, as shownin Figure 3(b).57Magnitude of Ex-0.Magnitude of ExMagnitude of Ex32.521.510.5°0.0 5"0.025't direction [m](a) ~inc =0 0 case0.025(c) ~i'C =4 0 caseOv-direction[m]0.025a., dirEtCtion[m]~ direction (m]Figure 5 Electric field distributions of the dominant component for various incident angles in z=5 em plane(F=5 em, >'0=2.5 em).The FZPL for the five conducting opaque rings is considered and operated at 1l.92GHz (>'0 = 2.5 em). Thefocal length is supposed to be F = 5.0 em (=2>'0)' Assume that x-polarized plane wave with incident angle ~incarrives from the left side of the FZPL as shown in Figure 2. The conductive zone rings are divided into 5352triangular elements with 7224 unknowns.Figure 4 shows the magnitude patterns for the computed surface current distributions of each componentfor various incident angles. Note that y-component as well as x-component for the induced currents on con-ducting annuli rings also exists. In fact, if the aperture size is large in terms of the wavelength, the aperturefield may be approximately equal to those of the incident field except the field near the edges for Kirchhoff'sapproximation. In the PO method, the surface currents are also considered to be the incident electric field sothat y-component of the induced surface currents is not exist. Consequently, the solutions computed from thePO method and Kirchhoff's approximation are not included these y-component surface current for the nextaxial focal characteristic calculations_ The electric field distributions of a dominant component in the receivingplane, z=5 em plane (X - Y plane), for the various incident angle are shown in Figure 5. Although the incidentangle is obliquely various, the patterns of the field distributions in the receiving plane are fixed.5820Tae Yang KIM and Yukia KAGAWA0 .III-5. 1~mc =00 --B-+ • • z-10 IYx-polarized wave :-15 I.-20-250.01MoM~Physical Optics ----)f---Kirchhoff apprx.(aperture model) .0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Distance from zone plate [m]Figure 6 Relative field intensity distributions (in decibels) along the axis from the FZPL (F=5 em, Ao=2.5em, Einc = 0°).201510ffi' 5:£c: 0'CijOlQ)-5>~Q)-10a:-15-20-250.01MoM -+----Physical Optics ----)f---Kirchhoff apprx.(aperture model) .MoM--e····Physical Optics --.•--.Kirchhoff apprx.(aperture model) -·-·e--0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Distance from zone plate [m]Figure 7 Relative field intensity distributions (in decibels) along the axis from the FZPL for oblique incidentcase (F=5 em, >'0=2.5 em).Focusing Characteristic Analysis of Circular Fresnel Zone Plate Lens 59Magnitude of Ex1.'1.210.80.60.'0.2oQ,-direction[m](a) ~in'~o caseMagnitude of Ex1.'1.210.80.60.'0.2o-0.Magnitude of Ex1.61.'1.21gg0.'02o<\- direction 1m](b) ~in< =2 0 casePhase of Ex [rad]l\r direction (rnJ(c) ~in< =4 0 case0.025~direction[mlFigure 8 Electric field distributions of the dominant component for various incident angles in z=30 em plane(F=30 em, Ao=2.5 em)Figure 6 shows the field intensity distributions along the axis for the plane wave of the normal incidence({inc = 0°). The plus signed line refers to the solutions due to the MoM, which is compared with other solutionmethods, PO method and Kirchhoff's scalar wave approximation. The discrepancy for the magnitude of eachsolution is happened. It is occurred that y-component of the induced currents is ignored in the solutions of thePO method and Kirchhoff's approximation. The location of the focal point or the point of the maximum gain isslightly shorter for the solution with Kirchhoff approximation than F=5.0 em. The field intensity distributionsfor oblique incidence are then investigated which is shown in Figure 7.Next, the examination is extended to the case of a longer focal length. The FZPL is designed for three-zonelens and F=30 em (12Ao) at the same operation frequency. The rings are divided into 11080 triangle elementswith 15876 unknowns. The electric field distributions of a dominant component in z=30 em plane (X - Yplane) for the various incident angle are depicted in Figure 8.The calculated gains versus the distance from the zone plate lens are shown in Figure 9 for the normalincidence. The focusing characteristics for the oblique incidence are shown in Figure 10. The PO methodgives closer solutions to the MoM. In some cases and in some regions, Kirchhoff approximation is not too bad.For incident angle {inc = 4°, the focal point may not lie on the axis so that the magnitude of the solutions isdecreased.60 Tae Yang KIM and Yukia KAGAWA201510CO:!2.c: 5'iiiOJ~ 0~Q)C(·5,-10 ,~-150.2xtII. I~mc=Oo~,y .. zx-polarized wave:MoM --+--Physical Optics ----,.---Kirchhoff apprx.(aperture model) ----JlE----0.25 0.3 0.35Distance from zone plate [m]0.4Figure 9 Relative field intensity distributions (in decibels) along the axis from the FZPL (F=30 em, >'0=2.5em, ~inc = 00 ).0.4--+------)E---.._---JE ••••···..···8..····.._._..-._.-·-·e·-·-~~-. _ ifJ -~.~~~""'iIi{i~'''_''''~-'•••~ MoMysical OpticsKirchhoff a perture model)• MoMPhysical OpticsKirchhoff apprx.(aperture model)0.25 0.3 0.35Distance from zone plate [m]201510CO:!2.c: 5-iiiOJQ)> 0~Q)C(·5-10·150.2Figure 10 Relative field intensity distributions (in decibels) along the axis from the FZPL for oblique incidentcase(F=30 em, >'0=2.5 em).Focusing Characteristic Analysis of Circular Fresnel Zone Plate Lens5 CONCLUSIONSThe MoM technique is very general and applicable to the variety of problems. The FZPL is considered as anelectromagnetic scattering and diffraction model to which the MoM is applied. The FZPL is operated at arelatively short wavelength. The MoM solution leads to a large system matrix, which could be handled by theiterative conjugate gradient method of a restart version for the numerical solutions. Numerical results havebeen presented including the oblique incidence, which are compared with other solutions.References1. F.A. Jenkins and H.E. White, Fundamentals of optics, McGraw-hill Book Company, 1981.2. Hristo D. Hristov, Fresnel Zones in wireless links, zone plate lenses and antennas, Artech House Publishers,2000.3. T. Hoashi, T. Onodera and Y. Kagawa, Oblique incidence characteristics of Fresnel-zone plate lens antenna,Trans. EIC Japan J-81-B-II(8), pp. 823-828, 1998.4. T. Hoashi, T. Onodera, F. Hagio and Y. Kagawa, On the evaluation ofreceiving power of a Fresnel-zone-platelens antenna, Trans. EIC Japan J-79-B-II(11), pp. 959-963, 1996.5. Y. J. Guo and S. K. Barton, Offset Fresnel zone plate antenna, Internation J. Satellite Communication 12,pp. 381-384, 1994.6. S. M. Rao, D. R. Wilton and A. W. Glisson, Electromagnetic scattering by the surfaces of srbitrary shape,IEEE Trans. AP-30, pp. 409-418, 1982.7. T. Y. Kim and Y. Kagawa, Electromagnetic scattering from inhomogeneous material bodies, JASCOMEVol. 14, pp. 19-24, 1997.8. T. Y. Kim, Y. Kagawa and T. Hoashi, Electromagnetic wave scattering and diffraction from arbitrarilyshaped structures, KJJC-AP/EMC/EMT'9S, pp. 49-52, 1998.9. Peter M. VAN DEN BERG, Iterative computational techniques in scattering based upon the integratedsquare error criterion, IEEE Trans. AP-32, pp. 1063-1071, 1984.10. Johnson J.H. Wang and John R. Dubberley, Computational of fields in an arbitrarily shaped heterogeneousdielectric ot biological body by an iterative conjugate gradient method, IEEE Trans. MTT-37, pp. 1119-1125, 1989.11. Asoke K. Bhattacharyya, High-frequency electromagnetic techniques, John Wiley & Sons, 1995.61

Focusing Characteristic Analysis of Circular Fresnel Zone Plate Lens

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