modelling of a circular fresnel zone plate lens for electromagnetic wave antenna application

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. 2005; 18:429–439Published online 3 October 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jnm.589

Modelling of a circular Fresnel zone plate lens forelectromagnetic wave antenna application

Tae Yong Kim1, Yukio Kagawa2,n,y and Ling Yun Chai2

1Division of Internet Engineering, Dongseo University, San 69-1, Chure-2 Dong, Sasang-ku, Pusan 617 716, Korea2Department of Electronics and Information Systems, Akita Prefectural University, Yurihonjo,

Akita 015 0055, Japan

SUMMARY

The application of Fresnel lenses is often made to the receiving antennas for television and radio by way ofthe satellite communication, for the astronomical observation and so on. Their advantage lies in structuralsimplicity and low cost. A typical Fresnel zone plate lens (FZPL) consists of coaxial conductive ring bandsseparately placed over the surface of a dielectric plate. For the antenna application a receiving horn isprovided at the focal point. In our previous work, the experiment was carried out for this arrangement,where the horn aperture is larger than the wavelength. Comparison was made with the solution of theKirchhoff’s scalar formula with reasonable agreement (Trans EIC Japan 1998; J-81-B-II(8):823–828; TransEIC Japan 1996; J-79-B-II(11):959–963). The present work is to validate the numerical treatment. Themethod of moment and the theory of the physical optics are here considered with three-dimensionalvectorial formulation. The focal characteristic is obtained not only for the normal incidence but also for theslightly oblique incidence. Copyright # 2005 John Wiley & Sons, Ltd.

KEY WORDS: fresnel lens; method of moment; physical optics; Kirchhoff ’s formula

1. INTRODUCTION

Fresnel zone plate lenses (FZPL) have been used as a flat optical lens [1]. Their electromagneticwave equivalence has also widely been used for antenna applications. The use for receiverantennas was reported for the communication satellite, televisions, radios, astronomicalobservation and so on, because of their structural simplicity and low cost [2]. Hoashi et al.reported the evaluation of a FZPL antenna for the focusing characteristics based on theKirchhoff’s scalar formula [3, 4], which was validated by the experimental results. Gou et al.considered the far-field analysis of the offset characteristics for FZPL antenna design [5].

The present paper considers the FZPL as an electromagnetic scattering and diffractionproblem in more exact manner by means of the three-dimensional method of moment and

Received 1 October 2000Revised 1 April 2004Accepted 1 July 2005Copyright # 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: Y. Kagawa, Department of Electrical and Information Systems, Akita Prefectural University,Yurihonjo, Akita 015 0055, Japan.

physical optics modelling. The method of moment (MoM) is a direct boundary integralequation method, suitable for the analysis of electromagnetic scattering and diffractionproblems [6–10], while the physical optics (PO) provides a short-wavelength assumption, inwhich only the illuminated parts of an object are taken into calculation. The MoM reduces theintegral equation to a set of linear algebraic equations. Although the application is direct,however, when the dimension of the object is large relative to the wavelength in operation,enormous computational resources are required. This is overcome here by using a restartconjugate gradient method for the numerical solutions [8–10]. Some numerical simulation aredemonstrated and the comparison with other solutions is presented.

2. FORMULATION

2.1. Moment method using a triangular patch scheme

The operation of the zone plate lens antenna lies in the wave interference as the results of thediffraction and the scattering, and the radiation from the surface currents induced on theconductive body. The integral expression is applied to the surfaces of the conducting body ofarbitrary shape for the electric field [11, 12]. The formulation is therefore the direct applicationof our previous work [8]. The boundary condition is enforced in such a way that the tangentialelectric field vanishes on the perfectly conducting surface,

#n� Einc ¼ #n� ð joAÞ þ #n�rF ¼ 0 on S ð1Þ

where #n denotes the outward unit normal vector on the conducting surface, Einc is an impressedelectric field and o denotes the angular frequency. The magnetic vector potential A and theelectric scalar potential F are expressed as follows:

AðrÞ ¼ mZS0Jðr0ÞGðr; r0Þ dS0 ð2Þ

FðrÞ ¼j

oe

ZS0ðr0 � Jðr0ÞÞGðr; r0Þ dS0 ð3Þ

Gðr; r0Þ ¼ e�jkR=R ð4Þ

where k ¼ oðemÞ1=2 represents the wave number corresponding to the external medium(permittivity e; permeability m) and G is the Greens’ function. R ¼ jr� r0j; where r and r0 are,respectively, the position vectors at a field point and at the source point.

The unknown current distributions Jðr0Þ is expanded in the form

JffiXNn¼1

Jnfn ð5Þ

where Jn is the unknown current coefficient and fn is the known expansion function. As shownin Figure 1, the expansion function fn is defined on a pair of triangular Tþn and T�n sharing acommon edge ln as

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:429–439

T. Y. KIM, Y. KAGAWA AND L. Y. CHAI430

fn ¼ln

2A�nq�n for T�n ð6Þ

where ln is the length of the common edge, A�n is the area of the triangle patch T�n ; and q�n is theposition vector with respect to T�n [6]. Equation (1) is integrated on both sides over the surfaceS; so that the MoM expression is

hfm;Einci ¼ johfm;Ai þ hfm;rFi ð7Þ

where the notation ha; bi denotes a symmetric inner product orRSa � b dS: Here the weighting

function fm is chosen to be the same as the expansion function fn: The linear algebraic equationis obtained so that

XNn¼1

ZmnJn ¼ Vm or ½Zmn�fJng ¼ fVmg; m ¼ 1; 2; . . . ;N ð8Þ

where Zmn is a generalized system matrix and Vm is a driving vector component independent ofexcitation. This is a fully occupied matrix equation, which has to be solved numerically for theunknown current coefficient Jn: The system matrix Zmn is large, full and dense. The iterativeconjugate gradient method is used for numerical solution.

2.2. Method of physical optics

With the physical optics, the field illuminated and the surface currents over the illuminated partsare only considered. The total electric field EðrÞ at the position r exterior to the surface of anobject is evaluated so that

EðrÞ ¼ EincðrÞ þ EscatðrÞ ð9Þ

where EincðrÞ is the incident electric field and EscatðrÞ is the scattered field, which is

EscatðrÞ ¼ikZ4p

ZS

R� R� dPeðr0ÞeikR

R3dS ð10Þ

where k is the wave number, Z (¼ m=e) is the wave impedance, R ð¼ r� r0) is the distance vectorfrom the source point r0 to the point of observation r; and R ¼ jRj:

Triangle T−

n

Triangle T +n

n-th edgenl

ρn+

ρn−

r

O

Figure 1. Triangular surface elements sharing a common edge ln:


MODELLING OF A CIRCULAR FRESNEL ZONE PLATE LENS 431

dPeðr0Þ is the source strength at point r0 on S; which is

dPeðr0Þ � Jsðr0Þ dS ¼ n0 �Hðr0Þ dS ð11Þ

where Jsðr0Þ is the induced electric current density at r0 on S: It is assumed that r is not extremelyclose to r0:

If Js is assumed to be the same current as obtained by the geometrical optics, Jgos

Jsðr0Þ ¼ Jgos ðr0Þ ¼

2n0 �Hincðr0Þ for the illuminated region

0 for the shadow region

(ð12Þ

where

Hincðr0Þ ¼E0

Z0ð�sin finc; cos finc; 0Þe�jkR ð13Þ

where E0 is the source electric field strength, finc is the incident angle ðp� xincÞ:

2.3. Determination of Fresnel zone configuration

A typical FZPL consists of closed conductive opaque annular rings as shown in Figure 2. Theconducting opaque zones are assumed to lie in the x–y plane, centred at ð0; 0; 0Þ in the Cartesianco-ordinates. The focal point must lie at ð0; 0;FÞ for the normal incidence, where F is the focallength. According to the concept of the Fresnel’s principle, the focal point is created in such away that rays via the boundaries of the ring yield the in-phase superposition for the successivepath-length of the one-half of the wavelength difference. The relation of the inner and outerradius of jth zone in the FZPL is thus determined as found in physics text books as

Routj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF þ jlþ l=4Þ2 � F2

qð14Þ

Plane wave

Conducting annular ring

R inj

j-th zone

outR j

FFocal point

Einc

ξ inc

x

y

z

incident angle

Figure 2. Geometry of a circular Fresnel zone-plate lens.



Rinj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF þ ðj � 1=2Þlþ l=4Þ2 � F2

qð15Þ

where l is the wavelength corresponding to the operation frequency in free space.

conducting opaque annuli

aperture or open annuli(a) (b)

Figure 3. Two complement models for FZPL: (a) diffraction model with obstacles; and (b) transmissionmodel through aperture.

0.012

0.0096

0.0072

0.0048

0.0024

0.0

0.006

0.0048

0.0036

0.0024

0.0012

0.0

0.006

0.0048

0.0036

0.0024

0.0012

0.0

0.012

0.0096

0.0072

0.0048

0.0024

0.0

(a)|Jx|-component

(b)

|Jy|-component

|Jx|-component |Jy|-component

x

y

arbitraryunit

arbitraryunit

Figure 4. Examples of the surface current distribution (for F ¼ 5:0 cm; l ¼ 2:5 cm): (a) shows themagnitude distribution for the normal incidence xinc ¼ 08; and (b) the magnitude distribution for the

oblique incidence xinc ¼ 48:



As shown in Figure 3, two complementary formulations are possible for the diffractions withconductive rings and transmission through the apertures. The Kirchhoff formula provides thesolution for the transmission through the aperture within the scalar wave approximation [4].

3. NUMERICAL RESULTS

In this section, the numerical results are presented for some Fresnel zone plate lenses. Theconducting annular rings are discretized into triangular patches. The unknown coefficients of

-25

-20

-15

-10

-5

0

5

10

15

20

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Rel

ativ

e ga

in [d

B]

Distance from zone plate [m]

MoMPhysical Optics

Kirchhoff apprx.(aperture model)

yz

x-polarized wave

x

ξ inc= 0

ο

Figure 5. Relative field intensity distributions (in decibels) along the axis from the FZPL (F ¼ 5:0 cm;l ¼ 2:5 cm; xinc ¼ 08).

-25

-20

-15

-10

-5

0

5

10

15

20

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Rel

ativ

e ga

in [d

B]


MoMPhysical Optics

Kirchhoff apprx.(aperture model)-25

-20

-15

-10

-5

0

5

10

15

20

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Rel

ativ

e ga

in [d

B]


MoMPhysical Optics


(a) (b)

Figure 6. Relative field intensity distributions (in decibels) along the axis from the FZPL for obliqueincidence (F ¼ 5:0 cm; l ¼ 2:5 cm): (a) xinc ¼ 28; and (b) xinc ¼ 48:



the induced current are obtained by solving a system of equations. The scattered electric fieldcan be calculated directly from the induced currents. The scattered field solution from theconducting zone plates can also be obtained from the physical optics (PO) [13], which is, to beexact, limited to electrically large scatterers. For the complementary solution method,Kirchhoff’s approximation can be applied to the apertures.

The geometry of the FZPL are determined based on Equations (14) and (15). The FZPL withfive conduction opaque rings is considered, which is operated at 11.92GHz (l ¼ 2:5 cm). Thefocal length is set to be at F ¼ 5:0 cm ð¼ 2l).

Assume that x-polarized plane wave with incident angle xinc arrives from the rear side of theFZPL as shown in Figure 2. The conductive zone rings are divided into 5352 triangular elementswith 7224 unknowns. Figure 4 shows the examples of the computed surface current distributionsfor two incident angles. Note that the currents are local because of the short wavelength, and

Figure 7. Electric field intensity distributions of the dominant component for three incident angles in focalplane (F ¼ 5:0 cm; l ¼ 2:5 cm): (a) xinc ¼ 08; (b) xinc ¼ 28 and (c) xinc ¼ 48



both x-component and y-component are induced. If the aperture size is wider in terms of thewavelength, the aperture field may be approximately equal to that of the incident field except thefield near the edges for the Kirchhoff’s approximation. In the PO method, the surface currentsare also considered in which the y-component of the induced surface currents does not exist.

Figure 5 shows the field intensity distributions along the axis for the plane wave of normalincidence (xinc ¼ 08). The plus signed curve refers to the solution due to the MoM, which iscompared with those of other solution methods, PO method and Kirchhoff’s scalar waveapproximation. The discrepancy is profound except in the mid-range field including the focal

-15

-10

-5

0

5

10

15

20

0.2 0.25 0.3 0.35 0.4

Rel

ativ

e ga

in [d

B]


MoMPhysical Optics


yz

x-polarized wave

x

ξ inc=0

o

Figure 8. Relative field intensity distributions (in decibels) along the axis from the FZPL (F ¼ 30 cm;l ¼ 2:5 cm; xinc ¼ 08).

-15

-10

-5

0

5

10

15

20

0.2 0.25 0.3 0.35 0.4

Rel

ativ

e ga

in [d

B]


MoMPhysical Optics

Kirchhoff apprx.(aperture model)-15

-10

-5

0

5

10

15

20

0.2 0.25 0.3 0.35 0.4

Rel

ativ

e ga

in [d

B]


MoMPhysical Optics


(a) (b)

Figure 9. Relative field intensity distributions (in decibels) along the axis from the FZPL for obliqueincidence (F ¼ 30 cm; l ¼ 2:5 cm): (a) xinc ¼ 28; and (b) xinc ¼ 48:



point. While the induced currents are ignored in the Kirchhoff’s approximation, the diffractionis not properly considered in the solutions of the PO method. The location of the focal point orthe point of the maximum gain is slightly shorter for the solution with the Kirchhoffapproximation than F ¼ 5:0 cm as designed. The distributions for the oblique incidence areshown in Figures 6(a) and (b). As the shift of the incident angle is small, though the focal pointdoes not lie on the axis, the decrease of the gain is small.

In the experiments [4], a horn with an opening of radius 1.6 cm is provided to receive thewave. Figure 7 shows the field distribution at the focal plane. It should be noted that the sign ofthe phase changes within the horn aperture.

Next, the examination is extended to the case of longer focal length. The FZPL is designed forthree-zone lens with focal length F ¼ 30 cm (12l). The rings are divided into 11 080 triangleelements with 15 876 unknowns. The calculated gains versus the distance from the lens areshown in Figure 8 for normal incidence. The focusing characteristics for the oblique incidenceare shown in Figures 9(a) and (b). Though the PO method gives closer solution to that of the

Figure 10. Electric field distributions of the dominant component for three incident angles in Z ¼ 30 cmplane (F ¼ 30 cm; l ¼ 2:5 cm):(a) xinc ¼ 08, (b) xinc ¼ 28 and (c) xinc ¼ 48



MoM in some cases and in some regions, Kirchhoff approximation is not always too bad.Figure 10 shows the field distribution on the focal plane, in which the phase is not so muchchanged within the horn aperture.

4. CONCLUDING REMARKS

The MoM technique is very general and applicable to the variety of problems. The FZPLs arehere considered as an electromagnetic scattering and diffraction models. The numerical resultsare presented compared with other solutions, to examine the validity of the Krichhoff’s scalarfield model used in our previous work [3, 4]. Our demonstration shows that the numericalsolutions give different results as their modelling depends on the different assumption. Theexperimental results were reported to agree reasonably with the Kirchhoff ’s solutions. Theexperiment was however too rough to justify the solutions, as the output from the horn with arather wide opening was only measured for evaluation. The FZPLs are operated at a relativelyshort wavelength, modelling for which requires a large number of the elements. The resultinglarge system matrix is successfully handled by the iterative conjugate gradient method of arestart version for the numerical solutions.

REFERENCES

1. Jenkins FA, White HE. Fundamentals of Optics. McGraw-Hill: New York, 1981.2. Hristov HD. Fresnel Zones in Wireless Links, Zone Plate Lenses and Antennas. Artech House Inc., Boston: U.S.A,

2000.3. Hoashi T, Onodera T, Kagawa Y. Oblique incidence characteristics of Fresnel-zone plate lens antenna. Transactions

of EIC Japan 1998; J-81-B-II(8):823–828.4. Hoashi T, Onodera T, Hagio F, Kagawa Y. On the evaluation of receiving power of a Fresnel-zone-plate lens

antenna. Transactions of EIC Japan 1996; J-79-B-II(11):959–963.5. Guo YJ, Barton SK. Offset Fresnel zone plate antenna. International Journal of Satellite Communication 1994;

12:381–384.6. Rao SM, Wilton DR, Glisson AW. Electromagnetic scattering by the surfaces of arbitrary shape. IEEE Transactions

1982; AP-30:409–418.7. Kim TY, Kagawa Y. Electromagnetic scattering from inhomogeneous material bodies. JASCOME 1997; 14:19–24.8. Kim TY, Kagawa Y, Hoashi T. Electromagnetic wave scattering and diffraction from arbitrarily shaped structures.

KJJC-AP/EMC/EMT’98, Pusan: Korea, 1998; 49–52.9. Vanderberg PM. Iterative computation techniques in scattering based upon the integrated square error criterion.

IEEE Transactions 1984; AP-32:1063–1071.10. Wang JJH, Dubberley JR. Computational of fields in an arbitrarily shaped heterogeneous dielectric or biological

body by an iterative conjugate gradient method. IEEE Transactions 1989; MTT-37:1119–1125.11. Analoui M, Kagawa Y. Electromagnetic scattering from conductor coated material bodies. IJNM Electronic

Networks, Devices and Fields 1991; 4:287–299.12. Analoui M, Kagawa Y. On the surface-patch and wire-grid modeling for planar antenna mounted on metal housing.

IEICE Transactions of Communications 1993; E76-B(11):1450–1455.13. Bhattacharryya AK. High-frequency Electromagnetic Techniques. Wiley: New York, 1995.

AUTHORS’ BIOGRAPHIES

Tae-Yong Kim did his Bachelor’s and Master’s degree at Pukyon University, Pusan, Korea. After militaryservice, he continued his graduate study at Okayama University, Graduate School of Natural Science andTechnology, from 1997 to 2000, under the Japan’s Ministry of Education, Science and Sports Scholarship



Program. He obtained the PhD (Eng.) in 2001. His interests include the numerical modelling andsimulation of the electromagnetic wave problems. At present Dr Kim is Lecturer, Division of InternetEngineering, Dongseo University, Pusan, Korea.

Yukio Kagawa is Professor at Department of Electronics and Information Systems, Akita PrefecturalUniversity, Yurihonjo, Akita, since 2001. He obtained Dr Eng. from Tohoku University in 1963, andbecame Professor of Electrical communication Engineering, Toyama University in 1970 and he joinedOkayama University in 1990. He is professor emeritus, both Toyama and Okayama University. Hisinterests include the numerical simulation of field problems and inverse problems. Prof. Kagawa is Fellowof IEEE, IOA and Japan Society of Simulation Technology (JSST) in which he was JSST President for theterm 1998–2000.

Lin-Yun Chai completed his undergraduate study at Xi’dian University, Xian, China. He did graduatestudy at Okayama University, Graduate School of Natural Science and Technology, and obtained MSc(Eng.) and PhD (Eng.) in the mathematical electronics science in 2000 and 2004, respectively. Dr Chaijoined Department of Electrics and Information Systems, Akita Prefectural University as Instructor in2004.



modelling of a circular fresnel zone plate lens for electromagnetic wave antenna application

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