Biomedical Engineering Reference
In-Depth Information
2
2
ψ
=
uI
=
u o
+
ug or
*
+
ug
ro
*
+
u r
.
(9.3)
o r
,
o r
,
In the interesting case where the hologram is illuminated by a replica of the reference wave used for
its recording (
u
=
r
), the hologram would diffract part of that light in a way that mimics the object wave.
This is illustrated in Figure 9.2a. By replacing
u
with
r
in Equation 9.3, one sees that the second term of
ψ produces a replica of the object wave, weighted by the product of mutual coherence function with the
intensity of the reference wave. Consequently, an observer looking at the hologram along the direction
indicated by the viewpoint arrow would see an image with the apparent depth and parallax properties
identical to that of the real object. This image is called the virtual image, since placing a detector (or
screen) at the position of the image would not reveal it.
Another interesting case is that for which the hologram is illuminated by a wave replicating the con-
jugate of the reference wave used for recording (
u
=
r
*). In this case, the hologram would diffract part of
the light in the direction of the object, as illustrated in Figure 9.2b. That diffracted light would mimic the
complex conjugate of the object wave, as indicates the third term of Equation 9.3, in which
u
would be
replaced by
r
*. This means that light diffracted from the hologram would form a real image of the object,
exactly at the position it was located for the recording of the hologram.
Sometimes, the two cross-terms forming the images we have just described are called terms of ±1
order of diffraction. With such nomenclature, the first and last terms of Equation 9.3, that do not carry
holographic image information, are consequently referred to as the zero-order terms.
9.2.5 elimination of Zero-order terms and twin image
From a holographic imaging point of view, the zero-order terms are a nuisance. In the worst possible
case, where
o
and
r
are collinear, the zero-order terms are completely superimposed on both imaging
terms and thus deteriorate the image quality.
A very simple solution to diminish the influence of the zero-order terms is to record the hologram
using a plane reference wave of intensity much stronger than that of the object wave. This way,
oo
* is
negligible compared to the imaging terms, and
rr
*, because it is uniform, much less disturbs the image.
But even then, the two imaging terms will overlap, one being in sharp focus, and its twin image
being out of focus. This problem was identified long ago, and many methods were developed to get
rid of the noise introduced in the images by the zero-order and twin image terms. The most widely
used method simply consists in using a reference wave subtending a nonzero angle with the object
wave, as in Leith and Upatnieks (1962). In this so-called off-axis configuration, the angle between
the two waves modulates the two imaging terms in such a way that illuminating the hologram with
a reconstruction wave
u
will see the two imaging terms diffract in different directions. For large
off-axis angles and/or for large object-to-hologram distances (vs. the object dimensions), complete
spatial separation of the imaging and zero-order terms can be achieved. Exact retrieval of the object
wave, that is, unaltered by the presence of the twin image nor the zero-order terms, then becomes
possible.
However, this method requires highly coherent light sources, such as lasers. To understand why it is
so, let us consider a plane object wave propagating along the optical axis, perpendicular to the detec-
tor's surface, and a plane off-axis reference wave, as illustrated in Figure 9.1. At different points on the
detector where
I
is measured, the reference wave will have traveled a different optical path length. In
opposition, the object wave would have traveled the same optical path length, regardless of where it hits
the detector. Therefore, for interference to occur over the entire detector, the coherence length of the
source should be larger than the optical path length difference of the reference wave at all points on the
detector. If not, then interference will appear only a region smaller than the size of the detector. This
point will be further discussed in Section 9.4.2, for the case of digital holography.
