Offset Fresnel lens antenna

Y.J. Guo PhD I.H. Sassi, MSc S.K. Barton. FlEE

Indexing terms: Antennas, Fresnel lens, Kirchofldifiaction, Offset

~

Abstract: An analysis of the offset Fresnel lens antenna is presented using the vectorial Kirchhoff diffraction theory. Equations for predicting the copolarised and crosspolarised radiation patterns are given. Two approaches to evaluating the far-field integration are introduced. Antenna performance such as the sidelobe and the cross- polarisation levels is investigated. An experimental offset Fresnel lens is reported. Measured results agree well with the theoretical prediction.

1 Introduction

In recent years, there has been a growing interest i various types of Fresnel zone antennas [l-71. Two main advantages of the Fresnel zone antenna are the low cost and conformability. By employing an offset configu- ration, a lens or reflector conformal to a building or vehicle surface can be produced at very low cost. These advantages are especially attractive in such applications as direct broadcasting by satellite (DBS) reception and very small aperture terminals (VSAT).

To date, research on Fresnel zone antennas has been mainly focused on either the circular zone plate reflector [l, 3-51, or the circular inhomogeneous dielectric lens [2]. Although the simple Fresnel lens antenna suffers from low effciency of about 8% [6,8], it serves as a very attractive indoor candidate when a large window or an electrically transparent wall is available. In the applica- tion of DBS, for example, an offset Fresnel lens can be produced by simply painting a zonal pattern on a window glass or a blind with conducting (absorbing) material. The satellite signal passes through the trans- parent zones and is focused at an indoor feed.

The concept of the Fresnel lens was developed by Lord Rayleigh and R.W. Wood before the turn of the century and it remained mainly as an optical and millimeter-wave device for a long time [SI. In 1961, Buskirk and Hendrix reported a circular Fresnel lens antenna for X-band operation [lo]. Recently, Leyten and Herben published the first vectorial analysis of the circu- lar Fresnel lens and compared it with the parabolic reflector [a]. In this paper, a vectorial analysis of, and microwave experiments on, offset Fresnel lens antennas are fist presented. Based on the vectorial Kirchhoff dif-

0 IEE, 1994 Paper 1310H (Ell), first received 11th October 1993 and in revised form 5th April 1994 The authors are with the Department of Electronic & Electrical Engin- eering, University of Bradford, Richmond Road, Bradford BD7 IDP, United Kingdom

IEE hoc.-Microw. Antennas hopag., Vol. 141, No. 6, December 1994

fraction theory, equations for predicting the far-field pat- terns are derived in Section 2. Owing to an oblique factor, the far-field integration for the offset Fresnel lens is different from that for the standard aperture antenna. Section 3 introduces two approaches to evaluating the integration efficiently. Such radiation performance as sidelobe and crosspolarisation levels is investigated numerically and an experimental offset Fresnel lens is reported in Section 4. Measured results agree well with the theoretical prediction.

2 Formulation of the radiation field

The Fresnel lens consists of a number of alternately transparent and opaque half-wave zones. The opaque zones are covered with conducting or absorbing material. In this paper, we assume that the Fresnel lens consists of a series of zonal openings in a finite conducting plate (see Fig. la). For the offset configuration, the zone boundaries are described by a set of elliptical equations [7]. Project- ing the elliptical boundaries in the direction of the maximum radiation yields a set of eccentric circles (see Fig. lb). We choose such a Cartesian co-ordinate system that the direction of the maximum radiation coincides with the z axis and the lens is symmetric about the yz plane (see Fig. 2). Then, the plane where the Fresnel lens is placed is described by

z = y tan a (1) where a is the offset angle [7]. The projection of the Fresnel zone boundaries on the xy plane is given by

(y - b)’ + x’ = a’

a = [nfl + (nl/2)’(1 + tanZ a)]”’

b = n l tan a/2 (n = 1, 2, ...)

(2)

( 3 4

(34

where the parameters a and b are given by

withfas the focal length. When illuminated by a feedhorn placed at the focal

point F (see Fig. 2), the radiation of the Fresnel lens is mainly attributed to the aperture field over the transpar- ent zones. As a good approximation, we extend the con- ductor around the Fresnel lens to infinity. Then, by virtue of Love’s field-equivalence principle [ 1 11, the transparent zones can be represented by an equivalent magnetic current J, on an infinite conducting plane without open- ings. With E, as the tangential electrical field on the lens

This work was supported by the UK Science & Engineering Research Council (SERC) under research grant GR/H44851.

517

sufface, J, is given by

J,,, = -n x E, (over transparent zones) (4) where n, the unit vector normal to the lens surface, is given by

n = cos az - sin ay

J,,, vanishes over the region covered with conductor.

0

b

Fig. 1 maximum radiation (I Fresnel lens b Projection

An offset Fresnel lens and its projection in the direction of

Fig. 2 Co-ordinates used in the theoretical model

As the Fresnel lens is usually electrically large and the width of half-wave zones are normally greater or compa- rable with the operating wavelength [12], the following Kirchhoff approximation can be employed:

n x E, = n x E, (over transparent zones) (5)

518

where E/ is the feed radiation field. Without loss of gen- erality, we assume the far field of the feed is of the follow- ing form:

(6) E, = exp (-jkR)/R (BE, sin cp + 4E+ cos cp),

Substituting eqns. 5 and 6 into eqn. 4 yields

J , = XJ,, + yJ,, + ZJ,, = exp (-jkR)/R{x[(cos a cos 0 sin

x sin +Eo + cos a cos' $E+]

+ y cos a sin p cos $[-cos BE, + E,J

+ z sin a sin cp cos +[-cos BEB + E o ] } (7) where the spherical coordinates (R, 8, p) are related with the Cartesian co-ordinates (x, y, z) through

- sin a sin e)

R = [xz + yz + (f+ y tan (84 e = tan-'[(x2 + y2)l/'/(f+ y tan a)] (8b)

p = tan-'(y/x) ( 8 4

J,, = tan aJ,, (9)

Eqn. 7 shows that

By virtue of a magnetic-type vector potential [6, 111, the far field of the magnetic current is given by

E = jk/(2nr cos a) exp (-jkr)

x [{Jmx(c sin + + $ cos 5 cos $1

- J,[c cos $ + +[tan a sin 5 - cos sin $)I} x exp {jk[x sin cos $ + y(sin sin $

+ tan a cos e)]} dx dy (10) where (r, <, $) are the conventional spherical co-ordinates corresponding to the Cartesian co-ordinates (x, y, 2).

According to Ludwig's definition three [13], the co- polarised and the crosspolarised unit vectors are given respectively by

e, = (C sin $ + $ cos (I) e, = (5 cos $ - $ sin +).

( 1 1 4 ( 1 1 4

The scalar product of eqn. 10 with e, and e 2 , respec- tively, yields the CO- and cross polarised fields of the Fresnel lens antenna:

E,, = jk/(2nr cos a) exp (-jkr)

x J{J,,,dsinz + + cos i cos2 $1

- J , cos $[sin $(I - cos [) + tan a sin a} x exp {jk[x sin 5 cos $ + y(sin sin $

s

+ tan a cos [)I} dx dy ( I 2 4 E,,,, = jk/(2nr cos a) exp (-jkr)

x

- J,,[(cosz$ + cos 5 sin2$)

- tan a sin < sin $1) exp {jk[x sin 5 cos $

+ y(sin 5 sin $ + tan a cos [)I} dx dy

cos + sin $( I - cos e)

(12b)

Eqn. 12 shows that the x-component of the magnetic current, which corresponds to the y-component of the

I E E hoc.-Microw. Antennas Propag., Vol. 141, No. 6 , December 1994

aperture field, has no contribution to the crosspolarised field in the E- and H-planes, whereas the y-component of the magnetic current, which corresponds to the x- component of the aperture field, has no contribution to the copolarised field in the E-plane. When the aperture field has a pure linear polarisation in the y-direction, we have

J, = xf(8, $)(cos a cos e - sin a sin 8 sin cp)

x exp ( - jkR) /R (13) and eqn. 12 simplifies to

E,,, = jk/(2nr cos a) exp ( -jkr)(sin’ ((I + cos 5 cos’ ((I)

x I J m x exp { jk [x sin cos ((I

+ y(sin [ sin ((I + tan a cos e)]} dx dy (14a) E,,, = jk/(2xr cos a) exp ( - jkr) sin $ cos ((I(1 - cos [)

x 1.l- exp { j k [ x sin cos ((I

+ Asin 5 sin ((I + tan a cos 01) dx d y (14b) Eqn. 14 represents a far field without crosspolarisation in the E- and H-planes. According to eqn. 7, it can be pro- duced with a feed radiation pattern of the following type:

E, = f ( & $1 (154 E+ = f ( e , 4) COS e (15b)

Owing to the flatness of the Fresnel lens, its optimal feed radiation pattern given by eqn. 15 is different from that for the conventional parabolic reflector by a factor of cos 0 [ll]. It should be pointed out that this conclusion is based on the Kirchoff approximation given by eqn. 5. For a practical Fresnel lens antenna, the scattering from the edges of the conducting zones inevitably produces an aperture field with crosspolarisation, thus giving rise to a crosspolarised radiation field.

3 Evaluation of the integration

Section 2 shows that, to obtain the radiation pattern of the offset Fresnel lens, it is required to evaluate the following two dimensional integration :

I(L $1 = JndX, Y ) exp CA@, ((Ib + V(L((I)y)l dx dy

(16) where Jmi represents the x-component or the y- component of the magnetic current, and the spatial spectra U((, ((I) and v(c, ((I) are given by

I

u(c, ((I) = k sin cos ((I (174 u(c, ((I) = k(sin sin ((I + tan a cos e) ( 17b)

It is seen that the offset configuration introduces an oblique factor exp ( j k tan a cos e), which makes the relationship between the spatial spectra and the observa- tion angle different from that for the symmetrical con- figuration [6, 81. If the x y plane is chosen as the plane of the lens, the oblique factor will disappear, but the direc- tion of the maximum radiation will form an angle a with the z-axis, and the definitions of the CO- and cross- polarisations, as well as the expression of the feed field, will become more complicated.

IEE Proc.-Microw. Antennas Propag., Vol. 141, No. 6, December I994

Although the integration in eqn. 15 appears as a Fourier type and can be efficiently evaluated by using the fast Fourier transform technique (FFT) [14], it is not straightforward to obtain plane-cut patterns except for the E-plane one. With ((I = 0 for the H-plane, for example, eqn. 16 describes a curve instead of a straight line in the spectral plane, while the FFT gives only a dis- crete set of spectra on the rectangular grid. To tackle the problem, two approaches are introduced as follows.

The first approach is the one used for analysing the offset parabolic reflector antennas [14, 1.51. For conve- nience, we rearrange eqn. 16 as

K ((I) = exp (-.ivyo) J,h Y )

(18)

(19)

with N as the number of transparent and opaque half- wave zones in the Fresnel lens. The domain of integration in eqn. 18 is a circle centred at (0, 0, 0) with radius R , given by

s, x exp [j(ux + uy)] dx dy

where

yo = N I tan a/2

R, = [ N f I + (N1/2)*(1 + tan’ a)]”’ (20)

Then, we extend the domain of integration into a square with side length D and expand the magnetic current J,,(x, y ) as the following finite Fourier series:

where the upper and lower limits are chosen differently for convenience of implementing the FFT and

w = 2x10 (224

C q p = Jmi(x> Y ) ~ X P [ j o ( q x + PY)I dx dy (226) I Substituting eqn. 21 into eqn. 18 produces

(23) sin [(U - qw)D/2] sin [(U - po)D/2] [(U - qw)D/21 [(U - P ~ ) D / ~ I

Eqn. 23 can be calculated at any given point (c, $), and the coefficients C,, can be evaluated efficiently by using the FFT [lS]. When only the gain of the antenna is concerned, we have

I ( 0 , O ) = exp ( - jky , tan a)Coo (24) The second approach is to obtain a discrete set of spectra by evaluating the integration in eqn. 22b with the FFT algorithm first. Then, the discrete spectra are used to produce a continuous spectral function by using the sub- domain interpolations such as the cubic-spline inter- polation and the bivariate interpolation. Mathematically, this gives

(25) where ~ ( u - pw, U - qw) is the subdomain basis function centred at (PO, qw). As the phase of C , changes rapidly with p and q, however, it was found that eqn. 25 failed to

519

give stable results. Therefore, an interpolation of the magnitude is employed instead as follows:

When eqn. 26 is used, the second approach produces antenna patterns very close to the ones obtained with eqn. 23.

Comparing eqn. 23 with eqn. 26, it is seen that the first approach is equivalent to approximating a continuous spectrum with all-domain basis functions. For a given observation angle, a double summation which involves a large number of sin (x) /x operations is required. This is very time-consuming for electrically large lenses. As a subdomain interpolation technique, the second approach requires only a few adjacent discrete spectra for a given observation angle. Therefore, the CPU time used by the second approach is much less than that by the first one. It is suggested, therefore, that the first approach be used when the phase pattern is of concern, and the second be used when the amplitude pattern only is required. In the following Section, the second approach with bivariate basis functions is employed to produce the numerical results.

4 Numerical and experimental results

To investigate the radition performance of the offset Fresnel lens antenna numerically, a pyramidal horn was used as the feed. The aperture field of the feedhorn was assumed to be an expanded waveguide mode with a quadratic phase error [16, 171. Using both the equivalent electric current and the equivalent magnetic current [ll], the radiation field is obtained as

u e , 4) = q e , 4) = jk(1 + cos e)Py(e, (p)/(4n) (27)

where

P , = E , I"" r2 cos (nx/A) exp { j [ - (n / l ) - 4 2 -B/2

x (x2/Ri + y2/R2) + k k + Y Y ) ~ ) dx d~ (28)

(294

(296)

with

p = k sin 0 cos 4 y = k sin 0 sin 4

In eqn. 28, A and B are the width and height of the horn aperture, and R, and R, are, respectively, the distances from the aperture to the two virtual apex lines [16, 171. Varying A and B changes the edge illumination level of the offset FZP lens.

As shown in Section 2, the projection of the offset Fresnel lens boundary on the xy plane produces a circle with diameter equal to the lens width 2R,. To investigate the performance of the offset configuration, it is appropri- ate to have the lens width fixed. Fig. 3 shows the normal- ised E-plane and H-plane patterns of an offset Fresnel lens with varying offset angle. The lens comprises four transparent half-wave zones with R , fixed at 0.3 m, and operates at 10.39 GHz. The feed parameters are for a pyramidal horn with an aperture of 4.1 cm by 2.8 an, made by the Mid-Century Microwavegear Ltd. As the Kirchhoff approximation is suitable mainly for near-in patterns, we restrict the observation angle in a range of (-SO", 50"). For the E-plane patterns, it is Seen that when the offset angle is increased from 0" to 20" and a", the

520

3 dB beamwidths remain almost constant, the skirts of the main beams become wider, and the sidelobe level is increased from - 15.9 dB, to - 15.4 dB and - 14.8 dB,

O r A

I I I I -50 -2 5 0 25 50

observat ion a n g l e 5 , deg

I I '

-40 ' I I -50 -25 0 25 50

Copolarised patterns of a Fresnel lens with fixed width and observation a n g l e 5, deg

Fig. 3 varying ofset angle (I E-plane b H-plane - a=o"

OL = 20" - _ ==a" ~ ~ _ _

respectively. Less change is observed in the H-plane pat- terns. The antenna directivity is actually reduced by 0.1 dB and 0.7 dB, respectively. Fig. 4 shows the cross- polarised patterns of the same lens antenna in the JI = 45" plane. It is observed that increasing the offset

- 50 -25 0 25 50 -100

observation angle 5, deg

Fig. 4 __ m = O "

Crosspolarised patterns of the Fresnel lens

e = 209 Or=40"

~ _ _ -

IEE Proc.-Microw. Antennas Propag., Vol. 141, No. 6, December 1994

angle hardly changes the crosspolarisation level. Increas- ing the offset angle further leads to further reduction of the antenna directivity and further increase of the side- lobe level.

For conventional lenses and reflectors, decreasing the edge illumination level effectively reduces the sidelobe level. However, this is not true for the Fresnel lens. Numerical results show that increasing the dimension of the feed does not change the sidelobe level significantly. This is because in a Fresnel lens antenna spherical phase errors in an range of f tr/2 exist over all the transparent half-wave zones. These severe phase errors dominate the sidelobe level of the antenna. Besides, as the scattering from the edges of the conducting zones has not been con- sidered in the theoretical model given in Section 2, it is expected that the sidelobe level of a practical Fresnel lens will be higher than predicted. The simple (not phase- corrected) Fresnel lens should, therefore, only be used when the interference through sidelobes is not a major concern.

As an experimental prototype, an offset Fresnel lens comprising three transparent half-wave zones with f= 0.14m, offset angle a = 20” and operating at 10.39 GHz was designed. The conducting pattern was printed on a 2mm dielectric substrate with relative dielectric constant 6, = 2.1. The same pyramidal horn used in the numerical analysis was employed as the feed. Fig. 5a shows the E-plane patterns of the antenna. For

-1 0 m T3

.- F -20

I I I I - 2 5 0 25 50

observation angle 6, deg

b

-10 - m V

-50 -2 5 0 25 50 observation angle 5 . deg

Fig. 5 numerical predictions a E-plane b H-plane __ Theoretical (CO) ~~~~ Theoretical (CO)

Experimental patterns of an offset Fresnel lens compared with

Experimental (cross)

IEE Proc.-Microw. Antennas Propag., Vol. 141, No. 6, December 1994

the copolarisation, the theoretical pattern has a wider main beam, but lower sidelobes. It follows the trend of the experimental pattern as far as 50” away from the direction of the maximum radiation. Theoretically, the crosspolarisation in the E-plane is below -60 dB, whereas it is shown in Fig. 5a that the measured cross- polarisation was about -29 dB. Fig. 5b shows the corre- sponding results in the H-plane. For the copolarisation, it is seen that the experimental pattern has about the same sidelobe level as that of the predicted one, but slightly wider main beam. The two far-out patterns have similar profiles, but are less close than the E-plane patterns. The predicted crosspolarisation in the H-plane is below -45 dB, whereas it is shown in Fig. 56 that the measured crosspolarisation was about -22 dB. One reason for the discrepancy between the theoretical and experimental results is that the lens width is only 12.4 wavelengths and the focal length is only 4.8 wavelengths, so there is a strong multiple scattering between different conducting zones, and the lens and feed, which is not considered in the theoretical model. The other reason is the feed align- ment accuracy. Better agreement may be expected with a larger Fresnel lens and better alignment mechanism. Fur- thermore, the higher crosspolarisation level in the H-plane is attributed to the stronger edge scattering from conducting zones in this plane.

5 Conclusions

Theory for analysing offset Fresnel lens antennas has been presented, and the radiation performance, including sidelobe and crosspolarisation levels, has been investi- gated. Two approaches to evaluating the radiation inte- gral efficiently have been introduced. It has been shown that with a fixed lens width (or a fixed projected area in the direction of the maximum radiation), the H-plane and crosspolarisation patterns remain almost constant with offset angle. There is no significant deterioration of gain or E-plane sidelobe level if the offset angle is not very large (say < 40“). As the sidelobe level of the Fresnel lens is not sensitive to the edge-illumination level, its feed should be designed to produce the maximum efficiency. The theoretical prediction was compared with one partic- ular experimental pattern of a small Fresnel lens with 20” offset angle, and fairly good agreement was obtained.

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