(This is a general scheme for using two intensity holograms to recover the phase
of the initial signal. It works for optical frequencies as well. For example, if two
holograms, as described, as taken of a microscope image, the wave field can be
recreated and re-imaged at any desired depth, without the artifacts inherent in
optical hologram reconstruction.)
Microwave Holography
General Scheme:
Far-field antenna patterns are easily determined by near-field scanning. When
amplitude and phase are both recorded in the near field, the far field pattern can
be determined by simply propagating the recorded field. In millimeter
wavebands, however, it is difficult to measure phase accurately. (At optical
frequencies, it is impossible) Intensities are easily measured, and the intensity of
the antenna signal mixed with a reference signal is, in fact, a hologram. Opticalstyle holography with radiated reference waves and standard reconstruction
suffers from the same types of spurious images and background noise that
plague optical holography.
With microwave technology it is easy to replace the reference wave with a
constant signal mixed with the output of the probe antenna. This is a method of
reconstructing the fields of the antenna under test (AUT) from near-field
holographic recordings that eliminates spurious images by making use of the
flexibility of this method of recording the hologram.
Symbolism:
The reference signal is
R  Er ( x, y )ei
r
( x, y )
 Er ei
 Er cos( r )  iEr sin( r )
r
 Err  iEri ,
and the AUT field is
A  Ea ( x , y )ei
a
( x, y )
 Ea ei
 Ea cos( a )  iEa sin( a )
a
 Ear  iEai .
The recorded hologram is
H  R A
2
 R  A  RA  R A.
2
2
(1)
Direct solution for the AUT Field
This technique makes use of the ability to shift the phase ( r ) of the reference
wave by an arbitrary angle,  . Let two holograms be recorded, identical except
for a phase difference of  in the reference waves; i.e.:
R(1)  Er ei
r
 Err  iEri
(1)
R( 2 )  Er ei (
(1)
r
 )
 Err  iEri
(2)
(2)
and the holograms are
H(1)  R(1)  A
2
2
H( 2 )  R( 2 )  A .
The solution proceeds as follows:
2
1. Measure A when the reference signal is turned off.
2
2. Measure R when the AUT is turned off. In the case of an injected
2
reference signal, R is simply a constant scalar.
2
2
3. Subtract A and R from the two measured holograms. Referring to
equation (1), we see that:
H  H  R  A
2
2
 RA  R  A
 2( Ear Err  Eai Eri ).
4. From the expressions for the two holograms,
H(1)  2( Ear Err  Eai Eri )
(1)
(1)
H(2 )  2( Ear Err  Eai Eri ),
(2)
solve for the AUT field components:
(2)
Ear
i
i
1 H(1) Er  H(2 ) Er

2 Eri Err  Err Eri
(2)
(2)
(1)
(1)
(2)
(1)
and
 Eai
r
r
1 H(1) Er  H(2 ) Er
.

2 Eri Err  Err Eri
(2)
(2)
(1)
(1)
(2)
(1)
Thus, the AUT field is recovered, artifact free except for measurement errors,
from the measured holograms and the measured (or otherwise known) reference
source.
The denominator in the above expressions, in terms of the reference amplitude,
E r , and phase shift between the two holograms,  , is  Er  sin( ) . Thus,
2
choosing    / 2 allows the denominator to be replaced with R , measured
as described in step (2) above.
2