A111D2 REFERENCE

NBS

PUBLICATIONS

NAn INST OF STANDARDS iTECHFITX.

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Richard L/An efficient and accura QC100 U56 NO 8S-3036 V1986 C.2 NBS-PUB 36

AN EFFICIENT AND ACCURATE METHOD

FOR CALCULATING AND

REPRESENTING POWER DENSITY IN

THE NEAR-ZONE OF MICROWAVE

ANTENNAS

Richard L. Lewis

Allen C. Newell

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

December 1985

NBS RESEARCH INFORMATION CENTER

NBSIR 85-3036

AN EFFICIENT AND ACCURATE METHOD

FOR CALCULATING AND

REPRESENTING POWER DENSITY IN

THE NEAR-ZONE OF MICROWAVE

ANTENNAS

Richard L. Lewis

Allen C. Newell

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

December 1985

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

I. Introduction 1

II. A Quick Method for Estimating Near-Zone Field Intensities

Radiated by Circular Aperture Antennas 2

III. Computation of the Electric Field Intensity in the Near-Zone

of Aperture Antennas by Numerical Integration of the PlaneWave Spectrum 17

Appendix 25

References * 27

i i i

An Efficient and Accurate Method for Calculating and Representing

Power Density in the Near-Zone of Microwave Antennas

Richard L. Lewis & Allen C. Newell

An algorithm is presented for calculating near-zone and Fresnelregion fields in front of microwave antennas from discrete numerical

values of the radiated plane-wave spectrum (complex far-field pattern). That is, the near fields are calculated by numerically integrating the plane-wave spectrum representation of the field. The crux of the analysis consists of handing a numerical instability

which arises from integrating discrete data. A criterion is developed

for limiting the integration domain in order to exclude highly oscillatory regions of the integrand. In turn, this leads to restricting

the applicable output range over which the field can be computed.

With the numerical instability problem thus resolved, fast Fourier

transform techniques are used to assure efficient numerical integration over a large (but restricted) output range. The results are conveniently presented as relative power-density contours in planes

formed by the longitudinal coordinate axis and one transverse coor- dinate axis. The algorithm is capable of extremely high accuracy,

which is demonstrated by comparing predicted and measured near fields

for two distinct antennas, along with a comparison against an exact

theoretical model. In the case of circularly symmetric excitation

models for electrically large antenna apertures, the predicted relative near-zone power-density contour plots turn out to be a function

of just the relative aperture distribution. Nomographs for obtaining

absolute near-zone power densities are presented for a few typical

aperture-di stribution functi ons.

Key words: antennas; antenna apertures; near fields; Fresnel

region; plane-wave spectrum.

I. Introduction

There are a number of situations related to safety and interference where

it is desirable to have a reliable and concise method of estimating the power

density levels radiated by microwave antennas. We present a new technique for

predicting near-zone field intensities using plane wave scattering matrix

theory to produce accurate calculations along with a highly efficient method

of graphically representing the results. Using this technique, comparisons of

predicted and measured near fields were carried out for selected antennas and

found to be in excellent agreement.

The input to the calculations is the far-field pattern (plane-wave spectrum) of the antenna obtained from either measured near-field data, design

specifications, or theoretical analysis. From this input, programs based on the plane-wave theory were developed to calculate the field along the x and y axes on a series of planes parallel to the aperture. Then the power-density

values in the x-z and y-z planes are obtained and graphically represented on two contour plots. The ordinate of these graphs covers the range ±4D in units

of D and the abcissa extends from just in front of the antenna out to D

2 /\ in

units of D

2 /A, where D is the antenna diameter and A is the wavelength. This

1
choice of scaling means that a single graph will apply without approximation

to all antennas with the same D/A. ratio and relative aperture distribution.

Different input power levels can be accounted for by scaling.

Theoretical results are presented in section II of this report ignoring

small structure effects (which frequently do not scale with D/A), producing

graphical presentations which just depend on the relative aperture distribution regardless of the D/A ratio. The computational method is presented in

section III, which concludes with specific examples using measured data to

demonstrate the accuracy of this technique.

II. A Quick Method for Estimating Near-Zone Field Intensities

Radiated by Circular Aperture Antennas

There are many instances where an estimate is needed of the near-zone

field intensity radiated by a parabolic dish. One possible need is to evaluate hazards to personnel, explosive detonators, or sensitive electronics. A

goal of the U.S. Environmental Protection Agency (EPA) is to characterize

field intensity levels in the near field of typical aperture or reflector

antennas. On behalf of the EPA, the U.S. National Bureau of Standards (NBS)

has developed techniques for predicting antenna near fields. A few of these

predictions are compared against near-field measurements for specific antennas

and found to be in excellent agreement. Here, we wish to characterize near- zone field intensity levels for a specific class of antennas using these techniques.

Our near-field-prediction procedure is numerical integration of the planewave-spectrum integral representation for the electric field. The plane-wave

spectrum data used by this procedure can be generated either from measured

data or analytical models. In this study, we consider relative aperture distribution functions which are amenable to analytical integration to obtain

theoretical plane-wave spectra.

We have chosen for study a circular aperture with a rotational ly symmetric

tapered aperture illumination. We wish to plot relative near-zone power densities as a function of (normalized) y and z coordinates, where z is measured

along an axis perpendicular to the antenna aperture while y is measured in a plane transverse to the z axis. The expected antenna pattern is a function of

^sin6, where D is the antenna diameter, A is the wavelength, and 0 is the

angular position of the observation point with respect to antenna boresight.

2
Consequently, upon normalizing boresight distance z with respect to D

2 /A and

transverse distance y with respect to D, we find that a single set of curves becomes sufficient to describe the relative near-field antenna pattern for a given D/A ratio and a given relative aperture distribution. Moreover, if the

D

2 maximum transverse distance y is small compared to t— (we assume D > 30A

'max A ~

and y = 4D), then the near-field patterns become invariant with respect to

-’max ’ D/A. When D < 30A a separate near-field pattern for each D/A value is

required. Fortunately, the patterns change slowly so one can interpolate

between just a few patterns.

One relative aperture distribution considered is [1 - (p/a) 2

] 2

, where p

is the radial -di stance variable in the aperture, 0 ^ p ^ a, and a = D/2.

Figure 1 shows this distribution's relative power-density contours in the y-z

plane for the case D > 30A, where -4D ^ y ^ 4D and 0 ^ z ^ D

2 /A. Each contour

corresponds to increments of 2.5 dB below the peak field intensity; the darker

contours correspond to increments of 5 dB below the peak field intensity.

With a rotational ly symmetric aperture distribution the contours at

positive and negative y values will be mirror images. Thus, we only need the

range 0 ^ y ^ 4D. Also, using this range doubles the accuracy of specifying

the graph ordinate y. The result using the same case as figure 1 is shown in

figure 2. Overlaying the contours in figure 2 are dashed lines corresponding

to constant y values, with incremental spacing of ^ between adjacent lines.

In figure 3 the relative field intensity along each of these lines is plotted

as a set of parametric curves. The uppermost curve in figure 3 corresponds to

y I 0.

The preceding figures describe the case D > 30A. To indicate how the

results change for smaller ^ ratios, in figures 4, 5, and 6 we show relative

^ D y-z plane power-density contours for T values equal to 24, 16, and 10, respecA

tively. Relatively little difference exists between figures 2 and 4, whereas

considerable difference exists between off-boresight contours in going from

figure 2 to figure 6. For further comparison, in figures 7 and 8 we show

relative field intensity curves, along equally-spaced parallel lines analogous

to figure 3, for ^ values of 16 and 10, respectively.

The absolute power density in dBm per square centimeter at any point in

front of the antenna, for one watt of input power at the antenna aperture, is

readily arrived at by combining the relative dB reading obtained from the preceding figures with the quantity, 38.57 - 20 x Log 10 D [dBm/cm2 ], where D is

in centimeters.

3
We next present results for a slightly modified relative aperture distribution, of the form a + [1 - (^

)

2

] 2

, where a is a small constant. Thus, in

3

figures 9 and 10 we present relative power density contours and relative fieldintensity curves for a = 0.1 (a pedestal 20 dB down), while in figures 11 and

12 we present results for a = 0.316 (a pedestal 10 dB down). In the first

case the absolute power density in dBm/cm2 for one watt of input power at the

antenna aperture is arrived at by combining the relative dB reading obtained

from figure 9 or 10 with 38.17 - 20 x Log 10 D, while for the second case the

absolute power density is obtained by combining the relative dB reading from

figure 11 or 12 with 38.14 - 20 x Log 10 D. Again, D is in centimeters.

To illustrate the use of these graphs, assume that the antenna is a 11.3 m (37 ft) diameter parabolic reflector operating at 6.5 GHz with 56 kW of input

power, and that we wish to estimate the power density 823 m (2700 ft) out from

the antenna face and 15.2 m (50 ft) off axis. Thus, the y distance is 1.35D

D

2 and the z distance is 0.3^-. From figure 2, we see that at these coordinates

there is a relative power density of -45 dB relative to the peak, an amount

also confirmed by figure 3. The correspondi ng absolute power density per watt

of input power is -67.5 dBm/cm2

, so the actual power density is about 0.01

mW/cm2 at this location. The effect here of a pedestal in the relative

aperture distribution is destructive interference according to figures 9 and

11, so the above estimate may be considered as worst case.

In conclusion then, we have presented a nomograph and associated algorithm

for obtaining the absolute power density in the near field of a circular aperture with a [1 - (£) 2

] 2 relative aperture distribution, along with results for

this distribution on a 10 dB pedestal and on a 20 dB pedestal. It should be

noted, in extrapolating these results to actual antennas, that aperture blockage and struts will produce additional effects that have not been accounted

for here.

4
axis

distance

divided

by

D

0.25 0.5 0.75

Z-axis distance divided by D

2 /A

2 2 Figure 1. Relative Power Density Contours in the Y-Z plane for a [1 - (^) ] 3

aperture distribution: D > 30\; -4D to +4D range on Y.

5
axis

distance

divided

by

D

Figure 2. Relative Power Density Contours in the Y-Z plane for a [1 - (^) 2

] 2 3

aperture distribution: D > 30A.

6
Relative

Field

Strength

in

dB

Z-axis distance divided by D

2 /A

Figure 3. Relative Field Intensity along equally spaced parallel lines for a [1 “ (§) 2

] 2 aperture distribution: D > 30A.

a

7
4

Figure 4. Relative Power Density Contours in the Y-Z plane for a [1 - (§) 2

] 2 a aperture distribution: D = 24A.

8
Figure 5. Relative Power Density Contours in the Y-Z plane for a [1 - (£) 2

] 2 a aperture distribution: D = 16A.

9
-axis

distance

divided

by

D

Figure 6. Relative Power Density Contours in the Y-Z plane for a [1 - (^) 2

] 2

aperture distribution: D = 10A.

10
Relative

Field

Strength

in

dB

0.00 .10 .20 .30 .40 . 50 . 60 . 70 . 80 . 90 !

. '

'

Z-axis distance divided by D

2 /A

Figure 7. Relative Field Intensity along equally spaced parallel lines for a [1 - C^) 2

] 2 aperture distribution: D = 16A.

3

li
Relative

Field

Strength

in

dB

Z-axis distance divided by D

2

/A.

Figure 8. Relative Field Intensity along equally spaced parallel lines for a

[1 - (£) 2

] 2 aperture distribution: D = 10A.

3.

12
0.25 0.5

Z-axis distance divided by D

2 /A

Figure 9. Relative Power Density Contours in the Y-Z plane for a [1 - (^) 2

]

2 3

aperture distribution on a pedestal 20 dB down: D > 30A.

13
Relative

Field

Strength

in

dB

0.0

- 15.0

- 20.0

- 25.0

- 30.0

- 35.0

- 40.0

- 45.0

Z-axis distance divided by D

2 /\

Figure 10. Relative Field Intensity along equally spaced parallel lines for a

[1 - (^) 2

] 2 aperture distribution on a pedestal 20 dB down: D > 30A.

cl

14
4

Figure 11. Relative Power Density Contours in the Y-Z plane for a [1 - (^) 2

] 2 3

aperture distribution on a pedestal 10 dB down: D > 30A.

15
Relative

Field

Strength

in

dB

0.0

- 5.0

- 10.0

- 15.0

- 20.0

- 25.0

- 30.0

- 35.0

- 40.0

- 45.0

Z-axis distance divided by D

2 /A

Figure 12. Relative Field Intensity along equally spaced parallel lines for a

[1 - (£) 2

] 2 aperture distribution on a pedestal 10 dB down: D > 30A.

3

16
III. Computation of the Electric Field Intensity in the Near-Zone of

Aperture Antennas by Numerical Integration

of the Plane-Wave Spectrum

Considerable attention has been given to the problem of calculating the

far-field pattern of an antenna from measured near-field data [1] - [11]. Of

these techniques, planar near-field scanning (plane-wave spectrum deconvolution) has seen the most development [12] - [16]. Here, we wish to consider

the inverse problem of determining near-zone and Fresnel region fields from

numerical values of the complex, vector, far-field pattern. That is, we are interested in numerical evaluation of the plane-wave spectrum integral representing the radiated electric field. We assume that numerical plane-wave spectrum data has previously been generated either from planar near-field measurements or from theoretical far-field patterns.

An earlier algorithm was developed by Yaghjian [17] for carrying out such

a numerical integration. In Yaghjian 1 s analysis, the integration domain was restricted in order to achieve computational efficiency; this also served to

constrain the oscillatory nature of the integrand and so eliminate a catastrophic numerical instability which otherwise occurs at large z-axis distances with discrete data points. Use of the fast Fourier transform (FFT)

provides efficient numerical integration over a large transverse output range.

Extending this earlier work to measured data with fixed data-point spacing

D

2 and to z-axis distances greater than 0.5 —, where D is the antenna diameter

and A is the wavelength, we found that numerical instabilities appeared at the

extremities of the integration range. Attempts to reduce these instabilities

by uniformly decreasing the already restricted integration domain drove the

calculated field to zero at the extremities, an effect that had also been found

by Yaghjian [17]. In order to compensate for these computational difficulties,

a new limiting criterion was developed.

In Yaghjian 1 s analysis, a geometrical (sheaf of angles) criterion was

used to limit the integration domain. In this work, we develop an improved

formula for limiting the integration domain, using a sampling-theorem [18]

criterion, producing a numerically more-stable result over a greater

cartesian-coordi nate output range. Also, we establish upper bounds on the

z-axis distance from the antenna and on the data-point spacing interval (A) of

the plane-wave spectrum in order to ensure reliable computations.

Evaluating the plane-wave spectrum integral, then, we compute near-zone

and Fresnel-region fields at a sequence of fixed z-axis distances as a

17
function of the lateral coordinates x and y. We reduce computational effort

by alternately setting one lateral coordinate equal to zero. By computing

along just the x and y axes we obtain sufficient data to generate near-zone field-intensity contours in the x-z and y-z planes. The maximum lateral coor- dinate range is R

q = However, we empirically determined that an effective

lateral coordinate range about half this size is needed to avoid numerical

instabilities. Thus, this effective lateral coordinate range and the upper

bound on the z-axis distance together define a truncated cylindrical region

bounding the near-field computation domain.

Yaghjian [17] describes how a coordinate-system rotation of the plane- wave spectrum data allows the cylindrical region's axis to subtend an arbitrary angle with respect to the main beam of the antenna; however, this also

results in havi:.g to interpolate when using discrete plane-wave spectrum data.

Here, by aligning the cylindrical region's axis with the perpendicular to the

near-field measurement plane, interpolation becomes unnecessary and so we avoid interpolation error. To be definite, we describe the case where the plane-wave spectrum data

are obtained from planar near-field measurements. The antenna measurement

setup is depicted in figure 13, showing the antenna under test, the near-field

measurement plane with the probe antenna just in front of the test antenna,

and a hypothetical y-z plane cut through the cylindrical region of interest.

Graphical output is represented in the figure by field-strength contours in

the y-z plane. At equally spaced intervals along the z-axis, vertical dashed

lines are drawn to represent the series of y-axis cuts along which the near field is computed by the planar near-field program. For smooth contours, of

course, many more plane-cut computation points would be required along the

z-axis than are indicated by the figure.

As background, and to introduce notation, we give an expression for the

y-component of a transmitting antenna's plane-wave spectrum in terms of mea- sured probe data taken over a planar surface in front of the antenna. It is

assumed that multiple reflections between the transmitting antenna and the

probe are negligible, that both antennas are polarization matched in the same

(y) direction, that the input reflection coefficients at the waveguide leads

to the antennas are negligible, and that the probe is far enough removed from

the transmitting antenna so that the significant probe receiving pattern is

independent of spatial frequency variations, i.e., the incident radiation at

the probe has a relatively narrow angle of arrival. Let B (P) be the signal

received by the probe at position P on the planar scan surface normalized to

18
19

intensity

Contours

are

Plotted.
unity at position P

1 (here, P = xa + ya , where a , a , a are unit cartesian

x y x y z vectors), and let A

q be the insertion loss between the transmitting antenna

and the probe when it is located at position P

1 . Then the plane-wave spectrum's y-component can be expressed [12,13] as

b(K) 1 f

2P

o \ V^ nAYG(o) o o r

B (P)e"

1 -*- dP,

o — — ( 1 )

where P q

is the input power to the antenna, Y

q

is the characteristic admittance of free space, G^Co) is the gain of the probe, d is the z-axis distance

from the planar scan surface to the transmitting antenna, dP = dxdy, A. is the

wavelength, K = k a + k a , and y = ,jk2 - K

2 where k = 2n/\ and K

2 = K-K.

The function B q( P

)

is virtually representable (i.e., b ( K) approaches zero

for K sufficiently greater than k, depending upon d) as the Fourier transform

of a bandlimited function. Consequently, the sampling theorem applies, so the

double integral (1) is completely determined by the sampling of data at discrete equal-spaced lattice points over the scan plane [12]. If we assign band

limits ±k and ±k to k and k , respectively, then (1) becomes [ibid]

xm ymx y

2P

b(K) = i

nA Y G (o)

o o r

v ' -iyd n*

k k £ xm ym r=-N! s=-N 2

n / n \ " 1 K * P B (P ) e rs

o -rs

y

where P-rs

rn

k -x

xm

S 7T

1 a

k -y

ym y To be mathematically exact, N 2 and N 2 should be

infinite; in practice, N x and N 2 are limited by the size of the scan area. As

given above, the FFT would limit to 2Ni x 2N2 the number of points at which

values of b ( K) would be calculated. However, we increase the output spectral

density (number of data points) by zero filling around the scan area boundary.

Thus, setting B Q

(P

rs

) = 0 for |r| > N x or for |s| > N 2 , we have

b(K ) -mn

2P

ttA Y G (o)

o o r

iy3 d mn ^xm^ym

N—x-1

2

1

r

2

N

s=-^

B (P )e ^nnn -rs

o -rs'

( 2 )

N N N N where - ^ m ^ - 1, * ^ n ^ • 1. Here, Kmn

= mA

x

a

x + nA

y

ay

2 k 2 k

y = Vk2 “ K

2

, where A = Kl x -, A = -jr— and N > 2Nj , N > 2N 2

. J mn N mn’ xN’yN x y

and

20
We now return to the development of the near-field computation algorithm.

The electric field can be obtained at any point r = R + za

z

, where

R = xa + ya , from the plane-wave spectrum integration

x y

E(r) = ~-f

f

b(K)e'-"-dK

-OO (3)

where k = K + ya and dK = dk dk . Here we assume that the vector spectraldensity function b(K) is known. If both the x and y components of b(K) have

been determined, then the z component is obtained from b^ ( K) =

- ^

K*b(K). In

the far-field of the antenna (3) becomes

, i kr

E(r) ~ -ik cos0 b(£ R)

where cos0 = z/r and r = y[r*r . In the Fresnel region evanescent modes will not be significant, so the

integration limits in (3) change from ±°° to ±k. The minimum applicable z distance is at least a few wavelengths to ensure that evanescent modes have attenuated. In our technique, the integration is carried out using the FFT for

a single fixed value of z at a time. However, stability problems can become

critical in carrying out a numerical integration. In particular, when z is

moderately large, the factor e

1 ^ 2 in the integrand can oscillate rapidly

between plus and minus one while the rest of the integrand changes very

little. As a result, these terms cancel out analytically, but with

discrete data on a computer they can add numerically to produce a significant

integration error. This can occur when z/D is as small as 2 or 3, where D is

the antenna diameter. One can compensate for this by limiting the domain of

integration even further, say to -k ^ k ^ k ,-k ^ k ^ k , where k < 3 J xo x xo’ yo y yo’ xo

k, ky

Q

< k. That is, we have to evaluate the near-field expression

E(r) _1

2tt dk

x

e

1 ^ 2 b(K) en *V^ dk^ (4)

21
We obtain integration-domain limits for (4) by using sampling theorem

criteria; namely, the maximum change in yz between two adjacent sample points

shall be less than 7T. After some ado, this leads to the result

*S<o * *Vo _

R

o (5)

k ^/fs+z2

where R s n/A E n/A . Here, B = 1 + sin 2d> where d> = tan "k /k . The

o y x

’ M Y

o Y

o yo xo derivation of (5) is presented in the appendix, where we establish that the

R

2 maximum z distance is limited by z <

Using the FFT to evaluate (4), it is expedient to integrate over the

extended domain |k|^k ,|k|^k , where k = k ^ k, and utilize a x xm ’ y ym ’ xm ym ’ window function to limit the effective integration a domain to k < k , xo xm’

kyo < ^ym' ^hus, for a fixed value of z, we can approximate (4) by

Here, E.

0 is the Fresnel-region electric field in the x-y plane corresponding

to a fixed value of z with x = j6 , y J = £6 , where 6 = zt/k and 6 = n/k

x’ J y’ x xm y ym

correspond to measurement-plane data-point spacing increments. Also, F(k

x ,ky) = b(K)u(K)e 1 ^ z

, where u(K) is a window function for limiting the oscillatory

behavior of e

1 ^ 2

. The extended integration domain limits are given by kxm

= 1/2 N A , k = 1/2 N A . In (6), -N /2^j<N /2, -N /2^£<N /2, so the maximum xx’ ym yy

v ’ x

J x ’ y y

’ lateral coordinate range is |x|^l/2 N 6 = n/A , |y|^l/2 N 6 = n/A . Howxx x y y y

ever, good numerical stability of (6) at moderately large z values limits the

effective lateral coordinate range to about | x

| < 1/4 N

x

<5 x

, | y | < 1/4 N 6 . Equation (5) was compared against an alternate expression obtained by

Yaghjian [17]. Although (5) yielded the best results, it lacked the latter's

ease of implementation. However, with a hybrid formula, we obtained an ex- pression that yielded good results along with ease of computation. Accordingly, the window function u(K) is set equal to zero under the condition

22
I k

xl ^ N >

R o

+ D

k

y o The integration cutoff imposed by u(K) can lead to ringing at the extremities

of the computed near field when the amplitude of b(K) at the cutoff point is

large. This, in fact, is the reason for limiting the effective range on x and

y to about half the normal FFT output range. Note that this range limitation

quadruples the number of data points needed for the plane-wave-spectrum

i ntegration.

We now write (6) in a form more suitable for FFT processing. The result

i s

N -1 N -1

E

j ‘ * 1 ' 6

J ] m=o Jo

{ - (m '

- >y

3e^B) ") exp + £L) } , c

X y

where the ranges 0^j^N

x ~l, 0S£^Ny-l apply. As a result, our formulation agrees

with standard FFT practice. It is assumed that N x/2 and N^/2 are both even

i ntegers.

In case we are interested in the field along the y-axis where x = 0, the

preceeding expression becomes

E N = -0,* - ^

N -1

y

E

n=0

V1

E

m=0

aa in - x in -

1

Ax\ - i J2.7X inn f ^ r r

r

e x u , y^ # i M 2-i e

\ Lj " 2

~)Ax,(n - 7^)Ay] Je

2nn£/N

(7)

With this formulation, we simply sum along the k coordinate followed by a one-dimensional FFT on the coordinate. A similar "one-dimensional collapse" formulation is used to compute the near field along the x-axis.

Near-field computations using this algorithm were tested against both

theoretical calculations and experimental data. Thus, in figures 14 and 15 we compare results produced by the planar near-field program against an exact

theoretical expression obtained by Rudduck and Chen [19] for the field along

the z-axis of a uniform circular aperture. The absolute power densities are obtained by each method and compared in the figures for both a 10A diameter

23
circular aperture and a 20A diameter circular aperture at an operating

frequency of 4 GHz. Computed results are plotted as solid lines while

theoretical results are plotted as dashed lines. The on-axis electric-field

strength for a uniformly excited circular aperture is given by

where a = D/2 is the aperture radius. The close correspondence between com- puted and theoretical results in these figures is extremely gratifying.

Next, in figures 16 through 21, we show comparisons between computed

near-zone results and actual measured data taken at corresponding z-axis

distances. The antenna characterized in figures 16 through 18 (antenna #1)

has D/A ratio of 16.2, while the antenna characterized in figures 19 thru 21

(antenna #2) has a D/A ratio of 77.6. For each antenna, we compare measured

data against predicted near fields at the measurement plane and at z-axis

distances of 100 cm and 300 cm beyond that. Both antennas are linearly

polarized in the y-di recti on; also, the reconstructed near-fields and the

comparison measurements are taken parallel to the y-axis. As seen in the

figures, there are fewer reconstructed near-field points than measurements at

the shortest distance, due to using a smaller scan area for generating the

plane-wave spectrum than for taking comparison measurements. Measured results

are plotted as dashed lines while the computed results are plotted as solid

lines. Both measured values and computed results are normalized with respect

to the corresponding peak field strength in the measurement plane. The

agreement shown between measured and computed results is remarkable in view of

the very complex pattern structure of these near-zone fields.

Finally, in figures 22 and 23 we show computed near-zone power-density

contours in the y-z plane for these same two antennas, while in figure 24 we show theoretical near-zone power-density contours for a 10A diameter

uniformly excited circular aperture at 4 GHz. These contours give absolute

power density in dBm/cm2 corresponding to one Watt of input power at the

antenna aperture.

In conclusion, we have presented an improved algorithm for calculating

near-zone fields in the vicinity of radiating aperture or dish antennas and we

have demonstrated that our algorithm is capable of extremely high accuracy.

We have also presented absolute power-density contours for some of the

antennas studied. Such absolute power-density contours are a convenient and

informative method for exhibiting near-zone computation results.

24
Appendix

Here, we wish to obtain integration limits k

xQ , k^

o to bound the oscillatory behaviour of e

] ^ z in the integrand of (4). We do this by making use of

the sampling theorem, which requires that there be at least two sample points

per cycle at the greatest resolvable "z" spatial frequency. That is, we will

only retain those z-component spatial frequencies within the integrand satisfying this criterion, so the maximum allowable incremental change in y will be

71 /z. Since y = Vk2 - K

2

, where K

2 = k£ + k2

, the applicable criterion for

determining the integration domain limits is

L = V 1 - (U* - b) - V 1 - U

2

, ( 8 )

where U

= -r— = V k

2 + k

2 /k, and where b represents a small incremental

k ^ xo yo r change in U

2 corresponding to a change from the sample point k

xQ , k^

Q

to the

point k - A , k - A . For consistency we require that k + k > r xo x’ yo y

J ^ xo yo

A + A . x y

The assumption that evanescent modes are negligible in (3) implies that

j U

2 U < 1. Accordingly, we can apply the binomial expansion, 1 - U

2 - 1 - r - U

4 l

g- - to (8). After cancelling terms and rearranging series expansions we

obtai

n

7i _ b /-j , 1 2 U

8 5U 10

kz 8 y

1

1 - U

2 1 - U2 /2 32 64 (9)

Neglecting terms of order b

2 or smaller, we can express b as the ratio of a fourth order polynomial to a power series in U

2

. Carrying out the indicated

power-series division we obtain

b

IJ2 2 U±

8

U

6

_ 5U 8_ 7U 10

16 " 128 " 256 ( 10 )

25
Now let A e A = A . Then

x y

k

2 b = K2 - (k - A) 2 - (k - A) 2 = 2A (k + k - A)

o xo yo xo yo

or kb = 2A(aU - ==) , where a = cos <j) + sin 4> ; 4> = tan

-i i\ l yo

O

7 7 0 xo

We have (see text following eq. 2) A = n/R , where R = 6 is the maximum

o o2yy

lateral-coordinate range. Consequently, b

2 in (9) is negligible compared to b

provided that R

q >> A. Now we can express (10) as

U

2 +

2 jp U

o

Az _ 0 R? ~ 2 ' °

0

( 11 )

We solve (11) as a simple quadratic equation

A

. U

4 U6 5U 8

where o = -j + -g

+ -p +

in U. Then, for z > R q and neglecting terms smaller than , we expand the

resulting radical as a power series in ^

including the small correction term

ct in the expansion. Iteratively substituting into the expression for a we

/ R \ 13

obtain a corrected series expansion for U, which to an accuracy of — ) is az equivalent to

U =

A

f + 1 + h*

R 1R

2 o ' o

2aR, ( 12 )

Equation (12) 's greatest source of discrepancy comes from neglecting the b

2 AR

terms in (9), so for z large the discrepancy in U is on the order of r—2

y / P / x \2

Thus, for p- >(_o^ 2 '' 1

o V A

see that our consistancyJ condition, k + k > 2A, is satisfied provided ’ xo yo

’ K

, the discrepancy is less than( p- )

. From (12) we readily

„ 2R2 z < o A

Finally, neglecting terms of order A/R , (12) becomes

U = r'o

26
References

[1] Brown, J. , Theoretical analysis of some errors in aerial measurements,

Proc. IEE, 105C , 343-351, Feb. 1958.

[2] Brown, J. and Jull , E.V., The prediction of aerial radiation patterns

from near-field measurements, Proc. IEE, 108B, 635-644, Nov. 1961.

[3] Kerns, D.M. , Correction of near-field antenna measurements made with an arbitrary but known measuring antenna, Electronic Letters 6, 346-347, May

1970.

[4] Newell, A.C. and Kerns, D.M., Determination of both polarization and

power gain of antennas by a generalized 3-antenna measurement method,

Electronic Letters 7, 68-70, Feb. 1971.

[5] Kerns, D.M. , Plane-wave scattering-matrix theory of antennas and

antenna-antenna interactions: formulation and applications, J. Res.

NBS-B, 80B , 5-51, Jan. -March 1976.

[6] Jensen, F. , Electromagnetic near-field far-field correlations, PhD

dissertation, Tech. Univ. Denmark, Lyngby, 1970.

[7] Ludwig, A.C., Near-field far-field transformations using spherical-wave

expansions, IEEE Trans. Antennas and Propagation, AP-19, 214-220, 1971.

[8] Johnson, R.C., Ecker, H.A. , and Hollis, J.S., Determination of far-field

antenna patterns from near-field measurements, IEEE Proceedings 61,

1668-1694, Dec. 1973.

[9] Leach, W.M. Jr. and Paris, D.T., Probe compensated near-field

measurements on a cylinder, IEEE Trans. Antennas and Propagation, AP-21,

435-445, July 1973.

[10] Kummer, W.H. and Gillespie, E.S., Antenna Measurements - 1978, IEEE

Proceedings 66, 483-507, April 1978.

27
[11] Newell, A.C., Baird, R.C., and Wacker, P.F., Accurate measurement of

antenna gain and polarization at reduced distances by an extrapolation

technique, IEEE Trans. Antennas and Propagation, AP-21, 418-431, July

1973.

[12] Kerns, D.M. , Plane-wave scattering-matrix theory of antennas and

antenna-antenna interactions, Nat. Bur. Stand. Monograph 162, June 1981.

[13] Newell, A.C. and Crawford, M. L. , Planar near-field measurements on high

performance array antennas, Nat. Bur. Stand. Report NBSIR-74-380 , July

1974.

[14] Newell, A.C., Planar near-field measurement techniques, Nat. Bur. Stand.

Short Course on Antenna Parameter Measurement by Near-Field techniques

lecture notes, July 1975.

[15] Yaghjian, A.D., Upper-bound errors in far-field antenna parameters

determined from planar near-field measurements, Part 1: analysis, Nat.

Bur. Stand. Tech. Note 667, Oct. 1975.

[16] Newell, A.C., Upper-bound errors in far-field antenna parameters

determined from planar near-field measurements, Part 2: analysis and

computer simulation, Nat. Bur. Stand. Short Course on Antenna Parameter

Measurement by Near-Field Techniques lecture notes, July 1975.

[17] Yaghjian, A.D., Efficient computation of antenna coupling and fields

within the near-field region, IEEE Trans. Antennas and Propagation,

AP-30, 113-128, Jan. 1982.

[18] Landu, H.J., Sampling, data transmission, and the Nyquist rate, IEEE

Proceedings, 55, 1701-1706, Oct. 1967.

[19] Rudduck, R.C.; Chen, C.L.J., New plane-wave spectrum formulations for the near field of circular and strip apertures, IEEE Trans. Antennas and

Propagation, AP-24, 438-449, July 1976.

28
Absolute

Power

Density

in

dB

below

one

mi11i

-Watt/cm

0.0

oo

- 3.0

- 6.0

- 9.0

- 12.0

- 15.0

- 18.0

-21 .0

- 24.0

- 27.0

- 30.0

y m

|

i

0 - 0° -15 -30 .45 .60 .75 .90 1.05 1.20 1.35 1.50

Z-Axis Distance divided by DV*

Figure 14. Comparison of results, along the z-axis, as produced by the planar

program (solid line) and as calculated using an exact theoretical

expression (dashed line) for a 1 0 — A diameter uniformly excited

circular aperture at 4 GHz showing power density in dB below one milliwatt/cm 2

, assuming one watt input power to the aperature.

29
Absolute

Power

Density

in

dB

below

one

mi1

1i

-Watt/cm

0.00 .15 .30 .45 .60 .75 .90 1.05 1.20 1.35 1.50

Z-Axis Distance divided by D^/A

Figure 15. Comparison of results, along the z-axis, as produced by the planar

program (solid line) and as calculated using an exact theoretical

expression (dashed line) for a 20-A diameter uniformly excited

circular aperture at 4 GHz showing power density in dB below one milliwatt/cm 2

, assuming one watt input power to the aperature.

30
Relative

Field

Strength

In

dB

Y-Axis Distance divided by D

Figure 16. Comparison of Measured (dashed line) and Computed (solid line)

Relative E-Plane-Cut Field Intensity Curves for Antenna #1 at

z = 65.3 cm.

31
Relative

Field

Strength

In

dB

0.0

- 5.0

- 10.0

- 15.0

- 20.0

- 25.0

- 30.0

- 35.0

- 40.0

- 45.0

- 50.0

- 2.00 - 1.60

Figure 17. Comparison of Measured (dashed line) and Computed (solid line)

Relative E-Plane-Cut Field Intensity Curves for Antenna #1 at

z = 165.3 cm.

32
Relative

Field

Strength

in

dB

0.0

- 5.0

- 10.0

- 15.0

- 20.0

- 25.0

- 30.0

- 35.0

- 40.0

- 45.0

- 50.0

-2

Y-Axis Distance divided by D

Figure 18. Comparison of Measured (dashed line) and Computed (solid line)

Relative E-Plane-Cut Field Intensity Curves for Antenna #1 at

z = 365.3 cm.

33
Relative

Field

Strength

in

dB

Y-Axis Distance divided by D

Figure 19. Comparison of Measured (dashed line) and Computed (solid line)

Relative E-Plane-Cut Field Intensity Curves for Antenna #2 at

z = 42 cm.

34
Relative

Field

Strength

in

dB

. - t " —

—m —u* ** i -

- 2.00 - 1.60 - 1.20 -.80 -.40 0.00 .40 .80 1.20 1.60 2. 00

Y-Axis Distance divided by D

Figure 20. Comparison of Measured (dashed line) and Computed (solid line)

Relative E-Plane-Cut Field Intensity Curves for Antenna #2 at

z = 142 cm.

35
Relative

Field

Strength

in

dB

0.0

Y-Axis Distance divided by D

Figure 21. Comparison of Measured (dashed line) and Computed (solid line)

Relative E-Plane-Cut Field Intensity Curves for Antenna #2 at

z = 342 cm.

36
2

Z-Axis Distance divided by D

2 /A

Figure 22. Near-Zone Power Density Contours for Antenna #1 in dB below one

mil 1 i-Watt/crrr for one Watt input power to the antenna (2.5 dB

separation between adjacent contours).

37
Y-Axis

Distance

divided

by

0.5 0.75

Z-Axis Distance divided by D

2 /A

Figure 23. Near-Zone Power Density Contours for Antenna #2 in dB below one

mi 1 1 i-Watt/cnr for one Watt input power to the antenna (2.5 dB

separation between adjacent contours).

38
-Axis

Distance

divided

by

0.25 0.5 0.75

Z-Axis Distance divided by D

2 /*

Figure 24. Near-Zone Power Density Contours for a 10-A Diameter Uniformly

Excited Circular Aperture in dB below one mi 11 i -Watt/cm at 4 GHz

for one Watt input power at the aperture (2.5 dB separation betweer

adjacent contours.

)

39
NBS-114A (REV. 2-8 U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBSIR 85-3036

2. Performing Organ. Report No. 3. Publ ication Date

December 1985

4. TITLE AND SUBTITLE

An Efficient and Accurate Method For Calculating and Representing

Power Density in the Near-Zone of Microwave Antennas

5. AUTHOR(S)

Richard L. Lewis & Allen C. Newell

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP)

United States Environmental Protection Agency

Office of Radiation Programs

Analysis and Support Division (ANR-461)

Crystal Mall 2, Washington DC 20460

10.

SUPPLEMENTARY NOTES

J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here) An algorithm is presented for calculating near-zone and Fresnel -region fields in

front of microwave antennas from discrete numerical values of the radiated plane-wave

spectrum (complex far-field pattern). That is, the near fields are calculated by numer- ically integrating the plane-wave spectrum representation of the field. The crux of the

analysis consists of handing a numerical instability which arises from integrating

discrete data. A criterion is developed for limiting the integration domain in order to

exclude highly oscillatory regions of the integrand. In turn, this leads to restricting

the applicable output range over which the field can be computed. With the numerical

instability problem thus resolved, fast Fourier transform techniques are used to assure efficient numerical integration over a large (but restricted) output range. The results

are conveniently presented as relative power-density contours in planes formed by the

longitudinal coordinate axis and one transverse coordinate axis. The algorithm is capab

of extremely high accuracy, which is demonstrated by comparing predicted and measured

near fields for two distinct antennas, along with a comparison against an exact theoretical model. In the case of circularly' symmetric excitation models for electrically

large antenna apertures, the predicted relative near-zone power-density contour plots

turn out to be a function of just the relative aperture distribution. Nomographs for

obtaining absolute near-zone power densities are presented for a few typical aperturedistribution functions.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s)

antennas; antenna apertures; near fields; Fresnel region; plane-wave spectrum.

13.

AVAILABILITY

[J] Unlimited

| | For Official Distribution. Do Not Release to NTIS

PH Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

| XI Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

44

15. Price

USCOMM-DC 6043-P 80

U. S. GOVERNMENT PRINTING OFFICE: 1986—677-405 45.088 REGION NO. 8