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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:

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)

HologramsA standard representation for a (monochromatic) optical wave field generally traveling near the direction is, (in complex notation):

where is the (non-negative) amplitude, and is the (arbitrary) phase of the field. (In the standard notation, dependence is left implicit, since it can be deduced

from the and dependence, and the time dependence, , is left off, since it would simply cancel from both sides of the equations.) With this notation in mind, let's define to be the light scattered from an object we wish to make a hologram of, and is a (nearly) plane-wave reference beam. The hologram is made by allowing and to interfere at the site of a detector (a CCD

camera, say). At the detector, then, the total wave field is and the detected intensity is proportional to:

(1)

where indicates the operation of complex conjugation. This expression describes the detected hologram.

Optical Reconstruction of Holographic ImagesIf the reference beam, , is a good approximation of a plane wave, then the amplitude of

and the phase of , and:

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where is a constant which can be made real by choosing the origin such that

. Then, the 4th term in (1) (ie.: ) is proportional to the original object wave, . Given this approximation, we can recover the image of the object by passing a plane wave through a mask (film) with transparency equal to This wave will then be

modulated so that it is proportional to , and the object-wave term 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 digitally. Of course, 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:

This is the background reference illumination; it is what the detector would see if no object wave were present.

This is the diffraction pattern of the object; it is what the detector would see if no

reference wave were present. Generally, .

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.

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 ImagesOne 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,

can be recorded when no objects are in the beam, the first step in improving the

hologram, is to subtract out the background:

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Our first improved hologram, , 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, , by . Unfortunately, we do not know the amplitude

function directly, as we only have access to the detected intensity of the reference

wave, .At this point, we make some reasonable approximations. Remember, by definition,

In general, , since significant variations in the intensity of the background is commonly seen in holograms. Variations in the

phase, of , however, would be directly related to components of propagating in different directions. Since is a well-collimated laser beam that

propagates in (nearly) one direction only, evidently i. This, plus

the fact that (by definition), means that:

.The next improvement in our digital hologram is therefore:

This modified hologram, , is much improved over the original detected hologram, . The background variations are removed, the object wave is distortion-free, and the

(already small) diffraction pattern overlay, , 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, .

Images produced by digitally simulating the propagation of have 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 might be removed, leaving only the object wave. In itself, we have no way to

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distinguish between . However, if the object that comes from is compact (that is, it forms a small image), then when we propagate over distance

to form an image of the object, and (as well as ) become separated

geometrically. Then is the image, which is compact, real, and localized, while the

other two terms are spread over a much larger region. We physically separate 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

complete shadow image?), back-propagate it to the hologram plane (ie.; calculate )

from which we can calculate the complex representation of and the diffraction

pattern ( ). These two non-object wave terms would then be subtracted from to yield a new estimate for , 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 high-precision images are required.

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i φR ( 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.