HOLOGRAPHIC PROJECTION TO AN ARBITRARY PLANEFROM SPHERICAL NEAR-FIELD MEASUREMENTS

Allen C. Newell, Bert SchlüperNearfield Systems Inc.,

1330 East 223rd St. Suite 524, Carson, CA 90745

Robert J. DavisThe Mitre Corp.

202 Burlington Road, Bedford, MA 01730-1420

Abstract

Holographic back-projections of planar near-fieldmeasurements to a plane have been available for sometime. It is also straightforward to produce a hologramfrom cylindrical measurements to another cylindricalsurface and from spherical measurements to anotherspherical surface1-7. In many cases the AUT isapproximately a planar structure and it is desirable tocalculate the hologram on a planar surface fromcylindrical or spherical near-field or far-fieldmeasurements. This paper will describe a recentlydeveloped spherical hologram calculation where the far-field pattern can be projected on any plane byspecifying the normal to the plane. The resultinghologram shows details of the radiating antenna as wellas the energy scattered from the supporting structure.Since the hologram is derived from pattern data over acomplete hemisphere, it generally shows more detailthan holograms from planar measurements made at thesame separation distance.

Keywords: Antenna measurements; Spherical near-field;Measurement diagnostics.

1.0 Introduction

For reference, the planar near-field hologram will first bedefined. Let us represent the measured planar near-fielddata on the plane z = d as B(x,y,d) where x,y defines theposition of the probe on the measurement plane. From thenear-field data we first compute the plane wave spectrum ofthe measured data using the Fast Fourier Transform (FFT).If we denote the spectrum of the measured data as

( , )x yD k k , the transmitting vector plane wave spectrum

of the AUT as ( , )x yt k kr

, and the receiving vector plane

wave spectrum of the probe as ( , )x ys k kr, the coupling

equation is

( )2

( , ) ( , ) ( , )

( , , )4

x y

x y x y x y

i di xk yk

D k k t k k s k k

e B x y d e dx dyγ

π

−− +

= •

= ∫

r r

(1)

In the above equation, ( , )x yk k represents the x-ycomponents of the propagation vector on the x-y plane andthe FFT output is automatically given as a function of these

variables. The time convention i te ω− is used throughout.

If the Hologram is computed before probe correction, weuse the calculated spectrum and the inverse Fouriertransform to obtain

( )( , , ) ( , ) x yh i xk yki dh x y x yB x y d D k k e e dk dkγ += ∫ . (2)

The results of this calculation can be interpreted asequivalent to what the probe would measure if the planarmeasurement had been performed at z = dh . Thiscalculation is straightforward and fairly simple since weautomatically have the plane wave spectrum as a functionof the x-y components of the propagation vector.

If we perform the probe correction on the measuredspectrum, but do not multiply by the Cos(θ) to obtain theelectric field, the hologram of the A-component of theprobe-corrected spectrum is,

( )( , , ) ( , ) x yh i xk yki dI h A x y x yB x y d t k k e e dk dkγ += ∫ , (3)

where ( , , )I hB x y d is the new hologram. This result canbe interpreted as equivalent to what an ideal, A polarizedprobe would measure on the plane z = dh . In general we

may use any component of the probe-corrected spectrum in(3), and the results will represent the vector component thatan ideal, perfectly polarized probe would measure on aplane. For instance, the AUT may be nominally linearlypolarized, but if we use circular components in the probecorrection, and use the right hand circular component inEquation (3), the results will represent what a point sourceprobe that is right circularly polarized would measure onthe new plane.

If the electric field has been calculated by multiplying thespectrum by Cos(θ), or if the electric field is obtained fromanother measurement, we must first convert to the planewave spectrum before computing the hologram. If theelectric field is given as a function of the plane wave vectorcomponents, this conversion is given by

( , )( , )

( )

( , )( , )

( )

A x yA x y

E x yE x y

k kt k k

Cos

k kt k k

Cos

θ

θ

Ε=

Ε=

(4)

where the subscripts A and E denote the azimuth andelevation vector components for an azimuth over elevationcoordinate system. In general, any orthogonal vectorcomponents can be used. After the conversion to planewave components using equation (4) we then use Equation(3) to obtain the hologram. Since Cos(90) = 0, we can onlyuse the electric field values over a region slightly less thanthe forward hemisphere where θ < 90 degrees. A test forthis condition must be included in the computer programthat implements Equation (4).

If the electric field is not already given as at equally spacedintervals of the plane wave vector components, we mustfirst interpolate from the given angle coordinates to( , )x yk k coordinates. This is necessary for instance if theelectric field data comes from far-field measurements,spherical near-field or cylindrical near-field measurements.In those cases, the results are generally the electric fieldcomponents as a function of spherical angles such as

( ),A θ φΕ , ( ),A A EΕ or ( ),A α εΕ . The equations thatare used for the interpolation from angular coordinates toplane wave vector coordinates are

sin cos cos sin sin ,

sin sin sin cos sin ,

cos cos cos cos cos .

x

y

z

k E Akk

Ekk E Ak

θ φ α

θ φ α ε

γ θ α ε

= = =

= = =

= = = =

. (5)

and the interpolation is illustrated schematically in Figure 1for θ,φ coordinates . The lines of the rectangular gridrepresent the x-y plane where the plane wave vectorcomponents ( , )x yk k are defined. The hologram is alsocalculated on this plane.

2.0 General Hologram on arbitrary plane.

The procedure for obtaining a general hologram on anarbitrary plane will follow the same general approach asoutlined above. The only difference will be in the specificequations that are used to convert from electric field toplane wave spectrum and to interpolate from anglecoordinates to plane wave components. In this case, theplane used to define the plane wave components will not ingeneral be the x-y plane. It will instead be the plane on

Figure 1 Schematic of interpolation from sphericaltheta, phi coordinates to plane-wave vectors kx , ky.

ky

kx

which the hologram is desired and will be referred to as theholographic plane.

For illustration purposes, let us assume that the far electricfield of the AUT has been determined over a completesphere using either spherical near-field measurements orfar-field measurements. It is assumed that both theamplitude and phase of the far-field are known as a functionof spherical angles (θ,φ). We will denote these results as

( ) ( ), , ,θ φθ φ θ φΕ Ε . Next we define a reference

direction, denoted as mev

, that is specified by the angles

( ),m mθ φ such that the plane on which the hologram is

desired is normal to the unit vector mev

. This referencedirection will usually be the direction of the main beam orthe normal to the AUT aperture. In general, an infinitenumber of coordinate systems are normal to me

v, each one

differing by the rotation of the plane about the unit vector.To make the coordinates specific, we define unit vectors

1ev

and 2ev

such that they are parallel to the θ- and φ field

vectors ( ),m mEθ θ φr

and ( ),m mEφ θ φr

as illustrated byFigure 2.

Mathematical expressions for the unit vectors are obtainedfrom Equation (5) and the equations that define the vectorcomponents for the different coordinate systems. For the

θ,φ coordinate system, the x, y and z components of the unitvectors are

( )1

cos( )cos( ), cos( )sin( )

sin( )

m m

m m m m

m

eθ φ

θ φ θ φθ

= −

v(6)

( )2

sin( )cos( )

0

m

m meφ

φ φ− =

v(7)

( )sin( )cos( )

, sin( )sin( )cos( )

m m

m m m m m

m

eθ φ

θ φ θ φθ

=

v(8)

Calculation of the dot and cross products of these unitvectors will confirm that they form an orthogonal set andtherefore 1e

vand 2e

v define the holographic plane normal to

mev

on which the hologram is to be calculated. Thisholographic plane is illustrated schematically in Figure 3.The solid circle represents the direction ( ),m mθ φ and thebroad lines represent the rectangular coordinate system

Figure 2 Illustration of unit vectors e1 and e2 thatdefine the holographic plane

Figure 3 Schematic of the holographic plane and therectangular coordinates for the plane.

Y

Z

e1

e2

em

Y

Z

normal to this direction which defines the holographicplane. The origin of the new coordinates is coincident withthe origin of the original measurement coordinates.

The x-, y- and z-components of a propagation vector for anarbitrary plane wave as defined in the original measurementcoordinate system are

( )sin( ) cos( )

, sin( )sin( )cos( )

kk

θ φθ φ θ φ

θ

=

r

r . (9)

The components of the same propagation vector in theholographic plane coordinate system are

( ) ( )( ) ( ) ( )

11

cos( ) cos( )sin( )cos( )sin( )sin( ) cos( )sin( )

cos( ) sin( )

cos cos sin( ) cos( )

cos sin sin( )sin( ) sin cos( )

m m

m m

m

m m

m m m

k k ek k

θ φθ φθ φ θ φ

θ θ

θ φ θ φθ φ θ φ θ θ

= • = • −

=

+ −

rv

r r

(10)

( ) ( )

22

sin( )sin( ) cos( )sin( )sin( ) cos( )

cos( ) 0

sin sin( )cos( ) cos sin( )sin( )

m

m

m m

k k ek k

φθ φθ φ φ

θ

φ θ φ φ θ φ

− = • = •

= − +

rv

r r

(11)

( ) ( )( ) ( ) ( )

sin( )cos( )sin( )cos( )sin( )sin( ) sin( )sin( )

cos( ) cos( )

sin cos sin( ) cos( )

sin sin sin( )sin( ) cos cos( )

m mm

m m m

m

m m

m m m

k k ek k

θ φθ φθ φ θ φ

θ θ

θ φ θ φθ φ θ φ θ θ

= • = •

=

+ +

rv

r r

(12)

The notation showing that the vectors are functions of (θ,φ)or (θm, φm) has been deleted for brevity.

To compute the hologram on the newly defined plane wemust first convert from electric field components to planewave components defined in the holographic coordinatesystem using an equation similar to (4). This requiresknowing the angle between the reference direction and thedirection of propagation for each plane wave. We note thatthe angle between the reference direction and thepropagation vector, and denoted by Θ , that corresponds toθ for the original measurement system is

( )

( ) ( )( ) ( )( )

cos

sin( )cos( )sin( )cos( )sin( )sin( ) sin( )sin( )

cos( ) cos( )

sin cos sin( ) cos( )

sin sin sin( )sin( )

cos cos( )

m

m m

m m

m

m m

m m

m

k ek

θ φθ φθ φ θ φ

θ θ

θ φ θ φθ φ θ φθ θ

Θ = •

= •

=

+

−

rv

r

(13)

And therefore the conversion from electric field to planewave spectrum is given by

( ) ( )( ),

,cos

tθ φ

θ φΕ

=Θ

rr

(14)

As in the case of the hologram on the x-y plane, thisequation is only valid for the "hemisphere" region wherethe denominator is non-zero. The boundaries of this regionare defined by the condition that

[ ]cos( )tan( )

sin( ) cos( )cos( ) sin( )sin( )m

bndm m m

θθθ φ φ φ φ

−=+

(15)

To complete the calculation of the hologram, we then useEquations (10) and (11) to interpolate from (θ,φ)coordinates to (k1,k2) coordinates which gives ( )1 2,t k k

r.

Equation (3) is then used to calculate the hologram on theholographic plane for one or more of the vector componentsof plane wave spectrum. The distance dh is the distancefrom the origin of the spherical coordinates to the surface ofthe antenna and may be either positive or negativedepending on the location of the antenna.

3.0 Measurement Results

Three measurements were performed to illustrate thefeatures of the hologram from spherical data. The firstantenna shown in Figure 4 is an X-Band slotted array withmajor dimensions of approximately 10 wavelengths.Conductive tape was placed over two elements to simulatefaults in the antenna. Both spherical and planar near-fieldmeasurements were performed on this antenna with theprobe approximately 22 wavelengths from the antenna. Atthis distance, the far-field from the planar data was onlyvalid to 40 degrees off-axis. As a result, the hologram fromthe planar data showed little detail of the array elements andthe null at the fault was not very sharp as show in Figure 5.The spherical hologram showed much more detail of boththe element locations and the deep, sharp null at thesimulated fault. The greater detail in the sphericalhologram is possible since accurate pattern data is availableover the complete hemisphere normal to the antenna. Thewide-angle portions of the pattern produce the fine detail inthe hologram.

The setup for the second measurement is shown in Figure 6.Even though the antenna is small, the airplane model that itis mounted on has some effect on the pattern as currents areinduced on the airplane, and so measurements must bemade at 0.5-degree intervals. The measurements wereperformed at 15.75 GHz over the complete sphere and thehologram was computed for the plane normal to the Z-axis.One useful feature of the hologram is that it can becalculated for planes with different dimensions. A smallplane centered on the radiating elements will show theamplitude and phase distribution for the actual antenna. Ifthe size of the plane is increased to include the major

Figure 4 Slotted array antenna with conductivetape over an element to simulate a fault.

Figure 5 Comparison of holograms from planar andspherical near-field measurements.

Figure 6 Aircraft model with small antenna onspherical near-field range.

-6 -4 -2 0 2 4 6Y-Position in Wavelengths

-30

-20

-10

0

Rel

ativ

e Am

plitu

de in

dB

Spherical DataPlanar Data

portion of the aircraft and the dynamic range of thehologram is increased to show very low signal levels, thescattering from the aircraft can be clearly seen. The detailand large dynamic range results from a combination of thefine data point spacing, the low noise level in the measureddata and the spherical data over the complete hemispherethat encloses the hologram plane.

Both of the holograms in Figures 5 and 7 have beencalculated for a plane normal to the Z-axis where theinterpolation from angles to k1 and k2 are fairly simple. Asa test of the general hologram, a fan beam antenna wasmounted with the peak of the main beam near theta = 90,phi = 0. This orientation of the antenna is referred to as anequatorial mount. The hologram for this antenna is shownin Figure 8 and demonstrates again the sharp resolution ofthe spherical hologram. The relative amplitude of the

individual elements in the antenna is also apparent and canbe used to make adjustments in element amplitude andphase.

4.0 Conclusions

Spherical near-field measurements can be used to obtainhigh-resolution holographic projections to any arbitraryplane. Measurements have been presented to illustrate thetechnique for planes normal to the Z-axis and normal to theX-axis.

5.0 References[1] Langsford, P.A.; Hayes, M.J.C.; Henderson, R.,“Holographic diagnostics of a phased array antenna fromnear field measurements”. AMTA 1989, p. 10-32.

[2] Repjar, A.; Guerrieri, J.; Kremer, D.; Canales, N.;Wilkes, R.J., “Determining faults on a flat phased arrayantenna using planar near-field techniques”. AMTA 1991,p. 8-11.

[3] Guler, M.G.; Joy, E.B.; Black, D.N.; Wilson, R.E.,“Far-field spherical microwave holography”. AMTA 1992,p. 8-3.

[4] Rochblatt, D.J.; Seidel, B.L., “Microwave antennaholography”, Microwave Theory and Techniques, IEEETransactions on , Volume: 40 Issue: 6 , June 1992, Page(s):1294 –1300.

[5] Isernia, T.; Leone, G.; Pierri, R.; Soldovieri, F.,“Microwave diagnostics by holography and phaseretrieval”. AMTA 1994, p. 244.

[6] Farhat, K.S.; Williams, N., “Microwave holographyapplications in antenna development”, Novel AntennaMeasurement Techniques, IEE Colloquium on , 1994Page(s): 3/1 -3/4.

[7] Guler, M.G.; Joy, E.B., “High resolution sphericalmicrowave holography”, Antennas and Propagation, IEEETransactions on , Volume: 43 Issue: 5 , May 1995, Page(s):464 –472.

Figure 7 Hologram amplitude from spherical near-field data on aircraft model.

Figure 8 Hologram for plane normal to theta = 90degrees for fan beam antenna.

-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0Hologram AmplitudeFrom Spherical Near-Field DataHologram Settings; X=0 in, Y=0 in., Z=5.5 in

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0