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Zone Plates and Holograms:
In previous notes (“Scalar Diffraction, 1/30/13), we gave an example of using beam
propagation via the plane wave spectrum to show the focusing properties of zone plates.
We will now show how zone plates can be calculated, how their wavefront properties can
be manipulated, and how this relates to digital holography.
A zone plate may be calculated by means of geometry:
ZP
h
D
Z
F
Given the wavelength and desired focal length, F, we can calculate the range of heights,
h1, where D = Z+n, and the heights, h2, where D = Z + (n+1/2), (where n is an integer).
Then we create the zone plate by making the areas “near” heights h1 open and the areas
“near” h2 closed (or vice-versa).
This is equivalent to taking a plane wave from the left, and a spherical wave from the
focal point, F, and comparing them at the plane of the Zone Plate. Where they are in
phase (or nearly so), the zone plate is transparent; Where they are out of phase, the Zone
Plate is opaque. This is, in fact, a digitally-defined Hologram, where the “Image” is
simply a point.
Extended Depth of Field Zone Plates:
Zone plates are interesting for imaging in regions of the E-M spectrum where suitable
materials for lenses don’t exist. Examples would be X-rays or Gamma-rays.
Here is the small central portion of a cubic-phase zone plate designed for Lawrence
Livermore labs’ 2.4 nanometer radiation source. The idea was to create a scanning
microscope by focusing the short wave radiation onto a object which would be scanned
and the transmitted (or, scattered) radiation detected. The problem was that a normal
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zone plate had a Depth of Focus much smaller than any object that could produce
significant attenuation. The solution was to design a zone plate by comparing the phase
of an incoming plane wave to the desired outgoing cubic-distorted spherical wave.
A digital hologram can be designed this way, as long as you have a numeric (or
symbolic) description of the desired outgoing wave.
Digitally-Detected Holograms:
By interfering two coherent waves at a detector, we can record the areas where they are
in phase and out of phase. This constitutes a hologram. An example of an optical system
that does this is:
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Here is a picture of the bench-test optics:
This system was designed (via the detector, magnification, and maximum Numerical
Aperture) to detect objects as small as 2 microns. Here is what the detected hologram of
a 5 micron wire looked like:
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The top images are and in-focus image of the wire, and a hologram taken 1 cm out of
focus. (The DOF of the system was about 100 microns for a 5 micron object.) The
bottom images are reconstructions of the wire’s image by propagating a modified version
of the detected hologram. Note how the background noise has been suppressed.
A typical hologram of cloud drops (simulated by polystyrene beads in water for the
bench) is:
The difference between just propagating the image (as an amplitude-modified plane
wave) and propagating a modified image is shown in the partial resconstruction of two
beads in the upper right of the above image:
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Finally, here is the image, hologram, and reconstructed hologram of a barb from a guinea
feather:
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The algorithm used to decode the above holograms is in Appendix A.
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Appendix A
Improved Algorithm for Reconstructing Digital
Holograms
(From 1995, unpublished)
Holograms
A standard representation for a (monochromatic) optical wave field generally traveling
near the z direction is, (in complex notation):
O  Ao ( x , y )eik o ( x , y )
where Ao ( x , y ) is the (non-negative) amplitude, and  o ( x , y ) is the (arbitrary) phase of
the field. (In the standard notation, z dependence is left implicit, since it can be deduced
i t
from the x and y dependence, and the time dependence, e , is left off, since it would
simply cancel from both sides of the equations.)
With this notation in mind, let's define O to be the light scattered from an object we
wish to make a hologram of, and R is a (nearly) plane-wave reference beam. The
hologram is made by allowing O and R to interfere at the site of a detector (a CCD
camera, say). At the detector, then, the total wave field is   ( R  O) and the
detected intensity is proportional to:
H 
2
  R  O
2
  R  O R  O

(1)
 R  O  RO   R  O,
2
2

where indicates the operation of complex conjugation. This expression describes the
detected hologram.
Optical Reconstruction of Holographic Images
If the reference beam, R , is a good approximation of a plane wave, then the amplitude of
R: AR ( x , y )  constant and the phase of R: R ( x , y )  constant , and:
R  AR ( x , y )e i R ( x , y )
 RC
where RC is a constant which can be made real by choosing the origin such that
R  0

. Then, the 4th term in (1) (ie.: R O ) is proportional to the original object wave, O .
Given this approximation, we can recover the image of the object by passing a plane
wave through a mask (film) with transparency equal to H. This wave will then be
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


modulated so that it is proportional to H , and the object-wave term R O  RC O will
produce an image at the original z-distance away from the hologram. Equivalently, we
can simulate the propagation of a wave field equal to H digitally.
Of course, R is never exactly a uniform plane wave, so distortion and noise are thereby
introduced. The other terms in (1) that are not proportional to the object wave also
introduce noise into the image. In addition, if the wave used in reconstructing the image
optically is not a uniform plane wave, then even more distortion and noise are present in
the image.
The meaning of all of the terms in (1) are:
2
R : This is the background reference illumination; it is what the detector would see if
no object wave were present.
2
O : This is the diffraction pattern of the object; it is what the detector would see if no
reference wave were present. Generally, O  R .
2
2
RO  : This is the conjugate object wave, distorted by any non-uniformity's in the
reference wave. The conjugate object wave is the same as the object wave, except
it focuses in the opposite direction; thus the conjugate wave will be out-of-focus
at the point where the object wave forms an image and vice-versa.
RO: This is the object wave, distorted by non-uniformity's in the reference wave. A
distorted reference wave will distort the final image.
Special Digital Reconstruction of Holographic Images
One of the advantages of having access to a digitized hologram is that we are no longer
bound to the limitations of optical reconstruction. Since the detected reference wave,
2
R , can be recorded when no objects are in the beam, the first step in improving the
hologram, H, is to subtract out the background:
2
H  H  R
 O  RO   R O.
2
Our first improved hologram, H  , has one less noise term than the standard `optical'
hologram. For further improvement, we would like to remove the distorting effect of

multiplying the object wave, O , by R . Unfortunately, we do not know the amplitude

function R directly, as we only have access to the detected intensity of the reference
2
wave, R .
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At this point, we make some reasonable approximations. Remember, by definition,
R  AR ( x, y )e i R ( x , y ) . In general, AR ( x , y )  constant , since significant variations
in the intensity of the background is commonly seen in holograms. Variations in the
phase,  R ( x , y ) of R , however, would be directly related to components of R
propagating in different directions. Since R is a well-collimated laser beam that
propagates in (nearly) one direction only, evidently  R ( x , y )  constant i. This, plus
the fact that AR ( x , y )  0 (by definition), means that:
R  R 
2
R .
The next improvement in our digital hologram is therefore:
H  
H
R
2
O  RO  RO
2

R
2
2
O

 O  O.
R
This modified hologram, H  , is much improved over the original detected hologram,
H . The background variations are removed, the object wave is distortion-free, and the
2
(already small) diffraction pattern overlay, O , is reduced by division with the
(generally larger) background. Virtually the only noise term of note left is the out-of
focus conjugate object wave, O .
Images produced by digitally simulating the propagation of H  have 10  20 times
the Signal/Noise of optically reconstructed hologram images (and are even better than
many in-focus images, made with coherent light). They are a much better target for
automatic image analysis techniques than images from optically reconstructed holograms.
There is a further, iterative, technique whereby even the last two noise terms of H 
might be removed, leaving only the object wave. In H  itself, we have no way to

distinguish between O, O , and.
2
O
R
. However, if the object that O comes from is
compact (that is, it forms a small image), then when we propagate H  over distance

to form an image of the object, O and O (as well as
O
R
z
2
) become separated
geometrically. Then OZ  z is the image, which is compact, real, and localized, while the
other two terms are spread over a much larger region. We physically separate OZ  z
from the other terms by selecting a small area around the image. We can then apply any
apriori knowledge we have about the image to improve it (for example, should it be a
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complete shadow image?), back-propagate it to the hologram plane (ie.; calculate OZ 0 )

from which we can calculate the complex representation of O and the diffraction

pattern ( O  O O ). These two non-object wave terms would then be subtracted from
2
H  to yield a new estimate for O: H  , which could then be propagated forward to
find the next approximation to the image. We have not tried this yet, since the current
algorithm works very well, but it might prove useful for applications where highprecision images are required.
 R x, y  is an actual constant only for the case of in-line or Gabor holograms – for off-axis reference
holograms, perhaps a linear phase function can be used as an approximation, depending on the angle of the
reference beam. This has not been experimentally attempted, due to the difficulty of doing off-axis
reference holography with limited resolution CCD cameras.
i
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