HOLOGRAPHIC PROJECTION TO AN ARBITRARY PLANE

FROM SPHERICAL NEAR-FIELD MEASUREMENTS

Allen C. Newell, Bert Schlüper
Nearfield Systems Inc.,
1330 East 223rd St. Suite 524, Carson, CA 90745

Robert J. Davis
The Mitre Corp.
202 Burlington Road, Bedford, MA 01730-1420

Abstract r r
D(k x , k y ) = t (k x , k y ) • s (k x , k y )
Holographic back-projections of planar near-field
e− iγ d (1)
measurements to a plane have been available for some
2 ∫
− i ( xk + yk )
time. It is also straightforward to produce a hologram
= B( x, y, d )e x y dx dy
4π
from cylindrical measurements to another cylindrical
surface and from spherical measurements to another
spherical surface1-7. In many cases the AUT is
approximately a planar structure and it is desirable to
calculate the hologram on a planar surface from In the above equation, (k x , k y ) represents the x-y
cylindrical or spherical near-field or far-field components of the propagation vector on the x-y plane and
measurements. This paper will describe a recently the FFT output is automatically given as a function of these
developed spherical hologram calculation where the far-
field pattern can be projected on any plane by
variables. The time convention e −iω t is used throughout.
specifying the normal to the plane. The resulting
hologram shows details of the radiating antenna as well If the Hologram is computed before probe correction, we
as the energy scattered from the supporting structure. use the calculated spectrum and the inverse Fourier
Since the hologram is derived from pattern data over a transform to obtain
complete hemisphere, it generally shows more detail
B ( x, y, d h ) = ∫ D (k x , k y ) eiγ dh e
i ( xk x + yk y )
than holograms from planar measurements made at the dk x dk y . (2)
same separation distance.
The results of this calculation can be interpreted as
Keywords: Antenna measurements; Spherical near-field; equivalent to what the probe would measure if the planar
Measurement diagnostics. measurement had been performed at z = dh . This
calculation is straightforward and fairly simple since we
1.0 Introduction automatically have the plane wave spectrum as a function
of the x-y components of the propagation vector.
For reference, the planar near-field hologram will first be
defined. Let us represent the measured planar near-field If we perform the probe correction on the measured
data on the plane z = d as B(x,y,d) where x,y defines the spectrum, but do not multiply by the Cos(θ) to obtain the
position of the probe on the measurement plane. From the electric field, the hologram of the A-component of the
near-field data we first compute the plane wave spectrum of probe-corrected spectrum is,
the measured data using the Fast Fourier Transform (FFT).
If we denote the spectrum of the measured data as
D(k x , k y ) , the transmitting vector plane wave spectrum BI ( x, y, d h ) = ∫ t A (k x , k y ) eiγ dh e
i ( xk x + yk y )
dk x dk y ,(3)
r
of the AUT as t (k x , k y ) , and the receiving vector plane
r where BI ( x, y , d h ) is the new hologram. This result can
wave spectrum of the probe as s (k x , k y ) , the coupling
be interpreted as equivalent to what an ideal, A polarized
equation is probe would measure on the plane z = dh . In general we
may use any component of the probe-corrected spectrum in kx
(3), and the results will represent the vector component that = sin θ cos φ = cos E sin A = sin α ,
an ideal, perfectly polarized probe would measure on a k
plane. For instance, the AUT may be nominally linearly ky
polarized, but if we use circular components in the probe = sin θ sin φ = sin E = cos α sin ε , . (5)
correction, and use the right hand circular component in k
Equation (3), the results will represent what a point source kz
probe that is right circularly polarized would measure on = γ = cosθ = cos E cos A = cos α cos ε .
k
the new plane.
and the interpolation is illustrated schematically in Figure 1
If the electric field has been calculated by multiplying the
for θ,φ coordinates . The lines of the rectangular grid
spectrum by Cos(θ), or if the electric field is obtained from represent the x-y plane where the plane wave vector
another measurement, we must first convert to the plane
wave spectrum before computing the hologram. If the components ( k x , k y ) are defined. The hologram is also
electric field is given as a function of the plane wave vector calculated on this plane.
components, this conversion is given by
ky

Ε A (k x , k y )
t A (k x , k y ) =
Cos (θ )
(4)
Ε E (k x , k y )
tE (k x , k y ) =
Cos (θ )
where the subscripts A and E denote the azimuth and
elevation vector components for an azimuth over elevation kx
coordinate system. In general, any orthogonal vector
components can be used. After the conversion to plane
wave components using equation (4) we then use Equation
(3) to obtain the hologram. Since Cos(90) = 0, we can only
use the electric field values over a region slightly less than
the forward hemisphere where θ < 90 degrees. A test for
this condition must be included in the computer program
that implements Equation (4).

If the electric field is not already given as at equally spaced

intervals of the plane wave vector components, we must
first interpolate from the given angle coordinates to
(k x , k y ) coordinates. This is necessary for instance if the Figure 1 Schematic of interpolation from spherical
theta, phi coordinates to plane-wave vectors kx , ky.
electric field data comes from far-field measurements,
spherical near-field or cylindrical near-field measurements.
In those cases, the results are generally the electric field
components as a function of spherical angles such as 2.0 General Hologram on arbitrary plane.
Ε A (θ , φ ) , Ε A ( A, E ) or Ε A (α , ε ) . The equations that The procedure for obtaining a general hologram on an
are used for the interpolation from angular coordinates to arbitrary plane will follow the same general approach as
plane wave vector coordinates are outlined above. The only difference will be in the specific
equations that are used to convert from electric field to
plane wave spectrum and to interpolate from angle
coordinates to plane wave components. In this case, the
plane used to define the plane wave components will not in
general be the x-y plane. It will instead be the plane on
which the hologram is desired and will be referred to as the θ,φ coordinate system, the x, y and z components of the unit
holographic plane. vectors are

For illustration purposes, let us assume that the far electric

 cos(θ m ) cos(φm ) 
e1 (θ m , φm ) =  cos(θ m ) sin(φm ) 
v
field of the AUT has been determined over a complete (6)
sphere using either spherical near-field measurements or
far-field measurements. It is assumed that both the
 − sin(θ m ) 
amplitude and phase of the far-field are known as a function
of spherical angles (θ,φ). We will denote these results as  − sin(φm ) 
Εθ (θ , φ ) , Εφ (θ , φ ) . Next we define a reference e2 (φm ) = cos(φm ) 
v
(7)
v
direction, denoted as em , that is specified by the angles  0 
(θ m , φm ) such that the plane on which the hologram is
v sin(θ m ) cos(φm ) 
desired is normal to the unit vector em . This reference
em (θ m , φm ) = sin(θ m )sin(φm ) 
v
direction will usually be the direction of the main beam or (8)
the normal to the AUT aperture. In general, an infinite
v  cos(θ m ) 
number of coordinate systems are normal to em , each one
differing by the rotation of the plane about the unit vector.
Y
To make the coordinates specific, we define unit vectors
v v
e1 and e2 such that they are parallel to the θ- and φ field
r r
vectors Eθ (θ m , φm ) and Eφ (θ m , φ m ) as illustrated by
Figure 2.

e2 Y

Z
em e1

Figure 3 Schematic of the holographic plane and the

rectangular coordinates for the plane.
Calculation of the dot and cross products of these unit
vectors will confirm that they form an orthogonal set and
v v
therefore e1 and e2 define the holographic plane normal to
v
Figure 2 Illustration of unit vectors e1 and e2 that em on which the hologram is to be calculated. This
define the holographic plane holographic plane is illustrated schematically in Figure 3.
Mathematical expressions for the unit vectors are obtained The solid circle represents the direction (θ m , φm ) and the
from Equation (5) and the equations that define the vector broad lines represent the rectangular coordinate system
components for the different coordinate systems. For the
normal to this direction which defines the holographic r sin(θ ) cos(φ )  sin(θ m ) cos(φm ) 
plane. The origin of the new coordinates is coincident with km k v   
the origin of the original measurement coordinates. r = r • em = sin(θ ) sin(φ )  • sin(θ m ) sin(φm ) 
k k
The x-, y- and z-components of a propagation vector for an  cos(θ )   cos(θ m ) 
arbitrary plane wave as defined in the original measurement
coordinate system are
= sin (θ m ) cos (φm ) sin(θ ) cos(φ )
+ sin (θ m ) sin (φm ) sin(θ )sin(φ ) + cos (θ m ) cos(θ )
r sin(θ ) cos(φ )  (12)
k
r (θ , φ ) = sin(θ ) sin(φ )  . (9)
k
 cos(θ ) 
The notation showing that the vectors are functions of (θ,φ)
or (θm, φm) has been deleted for brevity.
The components of the same propagation vector in the
holographic plane coordinate system are To compute the hologram on the newly defined plane we
must first convert from electric field components to plane
wave components defined in the holographic coordinate
system using an equation similar to (4). This requires
knowing the angle between the reference direction and the
direction of propagation for each plane wave. We note that
the angle between the reference direction and the
propagation vector, and denoted by Θ , that corresponds to
θ for the original measurement system is
r
k v
cos ( Θ ) = r • em
k
r sin(θ ) cos(φ )  cos(θ m ) cos(φm )  sin(θ ) cos(φ )  sin(θ m ) cos(φm ) 
k1 k v   
r = r • e1 = sin(θ )sin(φ )  • cos(θ m ) sin(φ m )  = sin(θ ) sin(φ )  • sin(θ m )sin(φm ) 
k k
 cos(θ )   − sin(θ m )   cos(θ )   cos(θ m ) 

= cos (θ m ) cos (φ m ) sin(θ ) cos(φ ) = sin (θ m ) cos (φ m ) sin(θ ) cos(φ )

+ cos (θ m ) sin (φ m ) sin(θ )sin(φ ) − sin (θ m ) cos(θ ) + sin (θ m ) sin (φ m ) sin(θ )sin(φ )
(10)
− cos (θ m ) cos(θ )
(13)
r sin(θ ) cos(φ )   − sin(φm ) 
k2 k v    And therefore the conversion from electric field to plane
r = r • e2 = sin(θ ) sin(φ )  •  cos(φ m )  wave spectrum is given by
k k
 cos(θ )   0 
r
= − sin (φm ) sin(θ ) cos(φ ) + cos (φ m ) sin(θ ) sin(φ )
r Ε (θ , φ )
t (θ , φ ) = (14)
(11) cos ( Θ )

As in the case of the hologram on the x-y plane, this

equation is only valid for the "hemisphere" region where
the denominator is non-zero. The boundaries of this region
are defined by the condition that
 3.0 Measurement Results
− cos(θ m )
tan(θ bnd ) = Three measurements were performed to illustrate the
sin(θ m ) [ cos(φm ) cos(φ ) + sin(φm )sin(φ )]
features of the hologram from spherical data. The first
(15) antenna shown in Figure 4 is an X-Band slotted array with
major dimensions of approximately 10 wavelengths.
To complete the calculation of the hologram, we then use Conductive tape was placed over two elements to simulate
Equations (10) and (11) to interpolate from (θ,φ) faults in the antenna. Both spherical and planar near-field
r
coordinates to (k1,k2) coordinates which gives t ( k1 , k2 ) . measurements were performed on this antenna with the
probe approximately 22 wavelengths from the antenna. At
Equation (3) is then used to calculate the hologram on the this distance, the far-field from the planar data was only
holographic plane for one or more of the vector components valid to 40 degrees off-axis. As a result, the hologram from
of plane wave spectrum. The distance dh is the distance the planar data showed little detail of the array elements and
from the origin of the spherical coordinates to the surface of the null at the fault was not very sharp as show in Figure 5.
the antenna and may be either positive or negative The spherical hologram showed much more detail of both
depending on the location of the antenna. the element locations and the deep, sharp null at the
simulated fault. The greater detail in the spherical
hologram is possible since accurate pattern data is available
over the complete hemisphere normal to the antenna. The
wide-angle portions of the pattern produce the fine detail in
the hologram.

0
Figure 6 Aircraft model with small antenna on
Figure 4 Slotted array antenna with conductive spherical near-field range.
tape over an element to simulate a fault.
The setup for the second measurement is shown in Figure 6.
Even though the antenna is small, the airplane model that it
Relative Amplitude in dB

-10 is mounted on has some effect on the pattern as currents are

induced on the airplane, and so measurements must be
made at 0.5-degree intervals. The measurements were
performed at 15.75 GHz over the complete sphere and the
hologram was computed for the plane normal to the Z-axis.
One useful feature of the hologram is that it can be
-20 calculated for planes with different dimensions. A small
plane centered on the radiating elements will show the
amplitude and phase distribution for the actual antenna. If
Spherical Data the size of the plane is increased to include the major
Planar Data

-30

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

Figure 5 Comparison of holograms from planar and

spherical near-field measurements.
 Hologram Amplitude 0 individual elements in the antenna is also apparent and can
From Spherical Near-Field Data -5 be used to make adjustments in element amplitude and
Hologram Settings; X=0 in, Y=0 in., Z=5.5 in -10
phase.
-15

-20

-25 4.0 Conclusions

-30

-35

-40
Spherical near-field measurements can be used to obtain
-45 high-resolution holographic projections to any arbitrary
-50 plane. Measurements have been presented to illustrate the
-55
technique for planes normal to the Z-axis and normal to the
-60
X-axis.

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

[2] Repjar, A.; Guerrieri, J.; Kremer, D.; Canales, N.;

Figure 7 Hologram amplitude from spherical near- Wilkes, R.J., “Determining faults on a flat phased array
field data on aircraft model. antenna using planar near-field techniques”. AMTA 1991,
portion of the aircraft and the dynamic range of the p. 8-11.
hologram is increased to show very low signal levels, the
scattering from the aircraft can be clearly seen. The detail [3] Guler, M.G.; Joy, E.B.; Black, D.N.; Wilson, R.E.,
and large dynamic range results from a combination of the “Far-field spherical microwave holography”. AMTA 1992,
fine data point spacing, the low noise level in the measured p. 8-3.
data and the spherical data over the complete hemisphere
that encloses the hologram plane. [4] Rochblatt, D.J.; Seidel, B.L., “Microwave antenna
holography”, Microwave Theory and Techniques, IEEE
0 Transactions on , Volume: 40 Issue: 6 , June 1992, Page(s):
-2 1294 –1300.
-4

-6
[5] Isernia, T.; Leone, G.; Pierri, R.; Soldovieri, F.,
“Microwave diagnostics by holography and phase
-8
retrieval”. AMTA 1994, p. 244.
-10

-12
[6] Farhat, K.S.; Williams, N., “Microwave holography
-14 applications in antenna development”, Novel Antenna
-16 Measurement Techniques, IEE Colloquium on , 1994
-18 Page(s): 3/1 -3/4.
-20
[7] Guler, M.G.; Joy, E.B., “High resolution spherical
microwave holography”, Antennas and Propagation, IEEE
Figure 8 Hologram for plane normal to theta = 90 Transactions on , Volume: 43 Issue: 5 , May 1995, Page(s):
degrees for fan beam antenna. 464 –472.

Both of the holograms in Figures 5 and 7 have been

calculated for a plane normal to the Z-axis where the
interpolation from angles to k1 and k2 are fairly simple. As
a test of the general hologram, a fan beam antenna was
mounted with the peak of the main beam near theta = 90,
phi = 0. This orientation of the antenna is referred to as an
equatorial mount. The hologram for this antenna is shown
in Figure 8 and demonstrates again the sharp resolution of
the spherical hologram. The relative amplitude of the