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Drawing Holograms by Hand
William J. Beaty, Box 351700, University of Washington,
Seattle WA 98195-1700
ABSTRACT
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Accidental abrasion of a specular surface sometimes produces real and
virtual images of the abrading object. Investigation of this phenomenon
in 1994 led to a simple technique which almost sounds like a joke: a
method for creating white light holograms by scribing interference fringes
one at a time onto a plastic plate by hand. The optics of these abrasion
holograms is similar to Rainbow holography, and the similarities reveal
techniques for controlling distortions, for producing images of opaque
objects, as well as for producing animation effects and images that extend
out through the film plane. As with any rainbow hologram, spatially
coherent or point-source illumination is required, both a transmission
mode and a reflection mode exist, and conjugate illumination produces
pseudoscopic images. This abrasion holography highlights the fact that
the zoneplates comprising a Rainbow hologram function independently not
only of illumination frequency but also of fringe spacing.
Size-independent fringes suggests that truly enormous holograms can be
engraved on a wide variety of very crude everyday surfaces.
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1. INTRODUCTION
In inventing holography Dennis Gabor created a method where an object
generates a complex zoneplate lens. This lens can distort a point light
source image and reconstruct a three-dimensional image of the original
object. Benton improved upon the technique with his Rainbow method.
Under this method the monochromatic illumination requirement was removed
and the zoneplate created by each point on the object no longer covered
the entire hologram. Several other non-holographic techniques for
producing freely observed 3D images also exist. These employ arrays of
tiny cylinder lenses or slotted aperture plates, but these "Lenticular"
devices do not involve optical wave interference and they lack most of the
unique behaviors of holograms.
There is another method for 3D image reconstruction that lies halfway
between the diffraction optics of holography and the geometrical optics of
lenticulars. This third method sometimes appears in nature, and so it has
been repeatedly discovered by observant researchers. I call the plates
produced by this third method "scratch holograms", while Liebermann [1] calls them "holographic engravings," and Plummer and
Gardner [2] name them "mechanical holograms." Abramson [3] calls them scratchograms or "Poor man's holograms" and
also refers to the technique as Incoherent Holography. The physics itself
suggests the terms "single-fringe holography" or "giant-fringe
holography." Since the required materials were available to humans ever
since we started using tools, with tongue in cheek I also call them
"prehistoric" holograms. All these terms refer to the same phenomenon and
the same techniques.
In a sunny parking lot its easy to observe strange patterns of highlights
apparently moving across the car hoods. Rows of tiny parallel scratches
in the paint behave as specular line-scatterers. If the angle between the
observer and the surface is changing, these scratches produce patterns of
reflected sunlight that appear to move across the hood. On one occasion I
noticed a particularly striking example. The highlights on a black car
hood were not just highlights, they were large images, and they appeared
deep within the surface. They also had a distinct structure: close-spaced
radial filaments surrounding an array of whitish patches. After some time
in observing these, the nature of the shapes suddenly became clear. They
were images of a lambs wool polishing mitt, complete with matted fibers in
the center, straighter fibers forming a radial halo, and with the glowing
shape of a human hand visible in the matted fibers. Even more intriguing,
most of these patterns appeared to float many centimeters deep within the
car hood. Apparently they were created as the hood was polished by a
gritty polishing-mitt, and the mitt produced a large number of
microscopic, curved, and somewhat parallel scratches in the paint.
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Figure 1
Car hood after careful "polishing" by a dry paper towel
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The phenomenon was easily reproduced by rubbing a very dirty car hood
with a dry paper towel. Figure 1 shows the results. The hand that holds
the towel must not rotate, and its overall trajectory must be a circular
arc of ~90deg with radius < 30cm. The resulting scratch patterns were
observed under sunlight. Images of handprints were clearly being
reconstructed, with the images appearing deep within the painted hood
rather than upon its surface. The depth of each image was somehow
proportional to the radius of the curved scratches and to the angle of
illumination. At some observing angles the images would even appear to
float above the surface rather than below. Some handprint images were
quite sharp, with the surface texture of the paper towel clearly visible
as in Fig. 2 below.
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Figure 2
Close-up photo of a "virtual image" handprint pattern
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Although the greater optics community may not be familiar with the above
phenomenon, it is actually somewhat well known. Plummer and Gardner [2]] analyzed a similar effect in 1992 after noticing 3D
images during inspection of metal mirrors subjected to an automated
lapping process. Artist Gabriel Liebermann harnessed the effect even
earlier in 1981 to produce 3D images in abraded metal, such as his work
entitled World Brain described by Garfield [1].
Apparently Liebermann kept his technique proprietary, although Garfield
mentions that it was based on an NC milling machine. And as Abramson [3] points out, the inventor Hans Weil created a version in
the mid 1930s that encoded stereo pairs of 2D photographic images in the
form of angled parallel abrasions with two different angles, a sort of
"multiplex stereogram" version of the above technique. The present paper
is based on work from 1994.
2. CURVED LINE-SCATTERERS PRODUCE IMAGES
Plummer and Gardner [2] show that the scratches producing
the above images act as curved line-scatterers. (Their paper delves into
the geometry far more rigorously than below, so please refer to ref [2]
for a full treatment.) For simplicity the dipole pattern of light
reflected from a line-scatterer of thickness << wavelength is ignored.
Also assume that these scratches lack a blaze angle and reflect the
incoming light equally in all radial directions, where each incoming ray
is scattered as a hollow cone of rays. In following diagrams I symbolize
the scatterer as a bent rod, although in reality it may have internal
structure.
A curved line-scatterer formed from a segment of a circle can act as a
convex or concave spherical mirror, producing either an astigmatic real or
virtual image of the distant point source. In figure 3, we have a curved
line-scatterer in the XY plane of radius r and center of curvature C.
Light from a distant point source PS shines downwards and reflects from
the scatterer. Each ray from PS scatters into a hollow cone of rays where
the angle of the cone depends on the position of the cone's vertex along
the length of the curved scatterer. The cones' angles depend on the angle
a. The axis of the cone is tangent to the curved scatterer.
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Figure 3
Light from a distant point source scatters off a curved
line-scatterer,
where each ray from the point source produces conical sets
of scattered rays.
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In fig. 3, other parallel rays from PS striking the scatterer at different
locations produce other cones. The rays from all these cones taken
together, if extended through the cone vertices to locate virtual image
points, intersect to form two separate astigmatic virtual image loci. One
virtual image locus is
positioned at the vertices of the cones: it is distributed along the core
of the line-scatterer itself. A second virtual image is shown in fig. 4.
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Figure 4
The intersection of lines extended from sets of cones of rays produces an
astigmatic
virtual image of the light source: an ellipse-shaped locus of points in the YZ plane
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When the scatterer in fig. 4 is lit from above, and for small values of
angle a, the intersection of the rays cast backwards from the cones
forms a virtual image in space. Rays a1, a2, etc. from each cone are
projected backwards as a1', a2', etc. They intersect with corresponding
rays from other cones, and the intersection takes the form of a U-shaped
locus in the YZ plane. The rays form a tilted cone intersecting a
plane, so the locus is a conic section, and for finite illuminator
distances the locus should be an ellipse. The major axis of this
ellipse-shaped virtual image is aligned with the rays from the distant
source PS, with the nearer focus coincident with the center of the
line-scatterer, while the distant focus coincides with the location of
the illuminator.
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Figure 5
Light from a distant source creates a virtual image point P
when observed by eyes positioned at O1 and O2
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Human stereopsis lets observers perceive either of the two virtual images.
In figure 5 an observer's eyes are positioned at O1 and O2, and they
observe the ellipse virtual image described in figure 4. However, the
observer is able to see only a tiny segment of this ellipse by looking
through the curved scatterer, as if the scatterer behaves as a
slit-aperture. This slit masks the vertical extent of the ellipse
virtual image locus, so the observer's eyes at O1 and O2 perceive only a
single bright point P positioned somewhere behind the curved scatterer.
A second point lying on the ellipse-image in the positive half of the Z
plane is not seen. This second point can be observed only if the eyes are
moved to the opposite side of the XY plane. Or in other words, the
line-scatterer produces both a reflection-mode image sent to one side of
the XY plane, and a transmission-mode image sent to the other. This
resembles conventional hologram optics.
The source of illumination in fig. 5 is required to have very limited
extension, otherwise the observed image point becomes a horizontal line
segment and the image suffers an astigmatic blur effect. This blur is
similar to that observed in Rainbow holograms [4]. The
blur is reduced when radius r is made shorter (less blur when the virtual
image P is very close to the film plane.) Thus we see another similarity
to conventional holography.
Note that the curved scratch scatters light in the vertical Y
dimension as well, so if human eyes are positioned at points O3 and
O4 in figure 5, stereopsis would force them to perceive the light
as coming not from point P, but instead from the vertices of the
cones of rays located within the curved scatterer itself. For this
reason the image produced by the curved scatterer can have
horizontal parallax only. If viewed with eyes turned 90 degrees,
the image lacks depth: it appears at the location of the scratches
on the surface of the plate. Obviously this is a major similarity
to Rainbow holography.
What is the depth (focal length f) of point P in figure 5? Knowing that P
lies on the ellipse-shaped locus in figure 4, the value for f must vary
with the position of the observer and with the angle of the rays from the
distant point source. In figure 5 we have an observer positioned
broadside to the curved scatterer, with eyes at O1 and O2. Inspecting the
diagram from a position above it, we find that certain rays from each cone
described in fig 4 can be extended through the cone vertex to converge on
the virtual image point P in figure 5. Angle b is equal to angle
a since the lines forming them are both part of the same cone of
rays. Since the side of the triangle opposite to angle b is also
shared by the side of a second triangle opposite to angle a, we
have identical triangles, therefore focal length f simply equals radius of
curvature r of the scatterer. This is only true for
vertically-illuminated "broadside" views of the scatterer under vertical
illumination and for small angles a (i.e. for observer distance >>
interocular spacing.) . For example, if the light from the scratch was
viewed from above (i.e. looking down from the location of point source
PS), the curved scatterer would act instead like a spherical mirror, and
the focal length measured from the scatterer to the virtual image point
would then lie between the scatterer and point C. Focal length f would
then have the usual value of 0.5r. On the other hand, if the observer's
eyes O1, O2 are moved downward in the negative Y direction, the image
point P migrates upwards along the ellipse, and the image depth increases
(it eventually becomes nearly the same as the distance to point source
PS.) But for viewing angles close to those shown in figure 5, the virtual
depth remains close to the value for r.
Virtual depth of a point approximates the scratch radius r. This is a
useful result. Suppose we were to employ a double-pointed compass (a
dividers) to scribe a curved scratch onto a plastic plate. Make the
scratch resemble a circular arc as shown in figure 5. We could hang this
plate on a wall, illuminate it with a distant point source placed
vertically above the plate, then observe the plate with two eyes oriented
horizontally. Wed see a glowing spark of light shining from within the
curved scratch. The virtual depth of the glowing spot would be the same
as the radius of the scratch. Now suppose we lay down several hundred
similar scratches, each with a different XY position, plus a Z position as
set by the spacing of the dividers. Could we not draw arbitrary objects
or scenes in 3D as sets of glowing points? This actually works very
well.
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