Jay Taylor's notes
back to listing indexSCIENCE HOBBYIST: Drawing holograms by hand
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W. Beaty, "Drawing Holograms by Hand", Proc. SPIE-IS&T Electronic
Imaging, Tung H. Jeong, ed., SPIE Vol. 5005, 156-167, 2003.
Copyright 2003 SPIE and IS&T |
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.
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.
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.
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.
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.
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.
When lit by an extended source, the overall result looks like the plastic
was abraded by sandpaper. A point source illuminator is required. In
this photo the sun is used as the light source, but a 200-watt spotlight
in a ceiling fixture will also
work. The entire collection of hologram-generation equipment is visible
in the photograph: a simple drafting compass with the graphite pencil
point replaced with a small iron nail.
Figure 7 shows examples of various effects that can be achieved. At the
top of the photo is a hologram of a deep plane of random stars with an
opaque square floating in the foreground. At the bottom is a tetrahedron
with one vertex penetrating the film plane and extending out into the air
as a real image (the scratch radii decrease to zero, then increase again
but with the scratches drawn upside down in order to produce a glowing
real-image line.) To the left is a cylindrical distribution of random
stars plotted by hand using trig tables. The cylinder also has a 3D curvy
line extending above which was sketched entirely freehand with little
effort. To the right is a 3D random starfield with a tiny galaxy in the
center, complete with axial plasma streams. In the center is a square
grid tilted in space (this grid was my first attempt at generating a fully
3D image rather than only creating flat image planes with constant depth.)
3.1 Speeding the production of arbitrary straight-line images
3.2 Pseudoscopic images
3.3 Controlling the horizontal limits of allowed viewing angles
3.3.1 Reconstructing images of opaque objects
For two adjacent scratches at the same vertical location (both scratches
part of the same horizontal band), if their radii are unequal, then if a
single point on each scratch gives the same viewing angle and also has the
same horizontal location, then that point on the scratch of larger radius
is the place where the occluded scratch should start (or should end).
Find the point that satisfies these conditions, and that is the proper
location for the end of the scratch. This recipe for producing opacity by
"early termination" of scratches is actually much easier than it sounds.
Some hands-on work with "scratch holography" will rapidly reveal how the
process works.
3.4 Image Distortions
A question arises regarding the reason for numerous similarities between
abrasion patterns and Rainbow Holograms. The answer is fairly simple.
Any diffraction grating made up of line-like fringes also behaves as a
parallel array of line-scatterers. Each scatterer in the array produces
astigmatic reflection, and it scatters a cone of rays. The cones from all
the scatterers overlap, and because of wave interference, most of the
scattered rays cancel. For reflection gratings, a zero-order ray still
reflects from the flat surface with equal angles, and the higher order
rays appear at angles where the many superposed cones of scattered light
all have relative phase of zero. Yet all the diffracted rays must also
lie upon the surface of the "scattering cone," since the diffracted light
is really just a superposed set of these "scattering cones."
Figure 9 shows the similarity between a line scatterer and a reflection
grating. A ray from a pointsource PS is reflected from these two optical
devices. In the grating diagram, the first order rays r1 and r1' lie on a
conical surface. Suppose the diffraction grating in Figure 9 was
illuminated by broadband white light. Each wavelength would produce
zero-order rays at a different angle, and the large number of rays would
tend to fill in the entire scattering cone. When a Rainbow hologram is
illuminated with white light, just such an effect occurs. When a
white-illuminated Rainbow hologram reconstructs an image, the hologram
fringe structure produces a set of these scattering cones, just as if the
hologram was actually composed of independent line-scatterers. In other
words, Rainbow holograms do not employ optical interference in order to
reconstruct an image. Instead they rely upon the optics of line-scatterer
arrays. Obviously some interference effects are present in a Rainbow
hologram, but they only impact the viewing angle, as well as producing the
familiar rainbow-colored artifact. The structure of the image is produced
only by the angle of the fringes, not by interference which critically
relies on fringe spacing.
In figure 10 we have a typical Rainbow hologram recording setup [5]. A parallel reference beam REF illuminates the hologram
plane H from below. Light from the same laser illuminates object O.
Object O sends an object beam through a narrow horizontal slit in aperture
plate AP, and this object beam is focused at the location of real image I.
The object beam then continues on to the hologram plane H, where it
illuminates a long narrow horizontal region. Optical interference
produces a fringe pattern in this region. This fringe pattern is
identical to that of a zoneplate lens, but it's a zoneplate lens which is
masked down into the shape determined by aperture plate AP. The
"horizontal band" shape of the interference pattern recorded on hologram H
is critical to the operation of Rainbow holograms.
Essentially Figure 10 depicts the recording of a single-point rainbow
hologram. As Siemens-Wapniarski and Givens [6] point
out, such a process can be repeated sequentially, object-point by
object-point, in order to "stamp" numerous zoneplate patterns onto
hologram H. The result is nearly identical to conventional holography;
only an undesirable phenomenon involving object-beam self-interference
will be lacking. Alternatively the entire object O can be illuminated
simultaneously to produce the usual rainbow hologram.
In figure 10 an interference pattern is recorded, and an enlarged view of
this pattern is shown in figure 11a. (To produce an erect image, the
hologram pattern from figure 10 has been rotated 180deg.) This pattern
has an obvious similarity to the curved scratches of "abrasion holograms."
Because Rainbow holograms reconstruct images via line-scatterer optics,
the optical interference produced between neighboring fringes can be
neglected. If we delete some fringes to increase the fringe-spacing in
figure 11a to resemble figure 11b, the hologram will still reconstruct the
same image point. If some distortion is tolerable, the fringe pattern can
even be changed to resemble figure 11c. This concept explains why
"abrasion holograms" have so many of the features of Rainbow holograms.
Rather than seeing these abrasion holograms as being crude stepchildren of
rainbow holography, we could reverse our perspective... Rainbow holography
is a unique method for harnessing coherent optics to conveniently produce
all the line-scatterers required by any arbitrary "abrasion hologram."
If a Rainbow hologram is wavelength-independent, then it follows that the
spacing of the fringes in its zone plate are size-independent. If we
could blow up a rainbow hologram to an enormous size, so its fringe
spacing was on the order of millimeters (or centimeters. or even meters!)
the hologram would continue to function. We could even replace the
hologram with an array of hand-drawn scratches forming the conic-section
shape of the fringes on the hologram. As long as the angles of those
scratches on the film plane were the same as the angles of the fringes in
the original hologram, the rainbow hologram would still be a rainbow
hologram. It would still reconstruct the same image. The rainbow-colored
artifact would vanish, and the allowed vertical viewing angle would
increase, but the shape of the virtual image would remain the same.
Doesn't this mean that we can readily produce Rainbow holograms the size
of roadside billboards using crude mechanical methods? I don't know. I
haven't tried it. (I don't see why it wouldn't work!) Also note that the
line scatterers only require a surface that produces specular reflection
in the dimension along the scratch; they don't have the far stricter
surface (or emulsion) requirements needed to produce optical interference
fringes. They can be scribed onto crude surfaces, and in theory they can
be made extremely large. Or we could even abandon scratches entirely, and
create Rainbow Holograms where the individual "fringes" take the form of
curved, polished metal wires or rods.
Taking yet another cue from Rainbow holography, we can alter a one-step
hologram setup
[5] and create something which incoherently records
white-light
holograms without employing coherent optics. In figure 12 the object O is
illuminated by a source of structured light SL. SL might be a
conventional slide projector with a multi-aperture plate. SL must project
tiny
spots of illumination onto object O, therefore the depth of field and
diffraction limits of the SL device are important. Light from one of
these spots on object O is focused by lens L to proudce real image I
positioned in front of the hologram recording emulsion H. Aperture plate
AP contains a narrow, curved slit with radius 2f; with radius of twice the
lens L focal length. As a result, a narrow curved line of light is
projected onto the hologram emulsion. The radius of the curve is
proportional to the distance between image I and hologram H. Each bright
point projected by SL onto object O produces a separate "curved scratch"
pattern on H. Note that the illumination might be incoherent white light,
and there is no reference beam involved. The resulting "abrasion
hologram" pattern could be etched into a metal plate in order to convert
the recorded lines into surface scratches (we might paint the etched
scratches with gloss black paint, so the paint's meniscus smoothes the
square-bottom etched cavities.) Rather than using a single curved slit in
plate AP, a series of vertically nested slits could be used. Or instead
the slit could be replaced by an aperture pattern resembling the
"fresnel-lens-like" scratch-segment pattern mentioned in section 3.4 and
depicted in figure 11b. The slit or slits are intended to project a
curved-line pattern onto H, so they must not be so narrow that they
produce significant interference effects. The aperture plate behaves as a
"single-point hologram", and the optical system "rubber stamps" numerous
copies of this pattern onto the film plane. The radius of curvature of
the recorded patterns is different for different depths of real image
point. This last aperture-plate idea is partly inspired by
Siemens-Wapniarski and Givens' paper[6], where they justify the practice of
producing synthetic holograms by recording single-point holograms on an
emulsion; recording one zoneplate at a time. Note that all of the above
description of a camera-device is only theory-informed speculation, and
this technique hasn't yet been explored empirically.
At present the main use for "abrasion holograms" is educational.
Students can easily produce them at almost no cost. Contrast this with
the complexity and expense of even the simplest holographic recording
method. In addition, the macroscopic size of the "fringes" in these
holograms gives students direct clues as to how Rainbow holograms
function. The concepts are no longer buried in the mathematics of 3D wave
interference. Also, scratch-holograms communicate some new insights as to
the nature of holography, they supply an "alternate mental model" which in
some situations might be useful in research. Finally, they provide an
interesting hobby where one can sit outdoors on a sunny day with a stack
of plastic plates and a set of dividers, while doing simple physics and
exploring a strange little niche in optical science.
Drawing Holograms by Hand
William J. Beaty, Box 351700, University of Washington,
Seattle WA 98195-1700
ABSTRACT
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.
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.1.1 A Chance Observation
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.
1.2 Independent Re-invention
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
[STEREO PAIR, CROSSEYED VIEWING]
3. PRODUCING 3D FIGURES USING ABRASION OPTICS
When generating virtual images composed entirely of straight lines
(such as the edges of the above cube), a simple method exists which
greatly speeds the process. First calculate the position and radii of the
scratches that produce the endpoints of the glowing line. Scribe these
scratches on the plastic. Next place the fulcrum point of your compass
exactly midway along a line drawn through the two fulcrum points
previously used to create the endpoint-scratches. Then set your compass
radius so it generates a scratch that is positioned exactly midway between
the two scratches forming the endpoints. Now scribe this "midpoint"
scratch. Next, repeat the process of finding midpoints between the three
existing scratches. You simply lay down the image points which divide the
line into 1/2, 1/4, 1/8, 1/16, etc. The scratches resemble the markings
on an English ruler, where new scratches are repeatedly placed between
existing scratches. When the scratches are dense enough (say 1mm or 0.5mm
apart) you're done. Observe the hologram under bright point source
illumination and you'll see a glowing line connecting the two endpoints
you initially calculated. Many additional hints and tricks can be found
on my website [8].
In figures 3 through 5 the illumination comes from a source positioned
vertically above the line-scatterer. If we instead illuminate the same
scatterer from below, the scattered rays focus together rather than
converging, and we obtain real images in front of the "film plane" rather
than virtual images located behind it. If the "cube hologram" in figure 6
is illuminated from below, the cube is brightly visible and still
maintains its general shape, but it turns inside-out and moves into the
air in front of the plastic plate. It resembles the upper corner of a
room rather than the projecting corner of a solid cube. In other words,
using a conjugate illumination beam will change virtual images into real
images, producing an effect that greatly resembles the pseudoscopic images
of conventional holography. This probably explains why some of the images
seen in car hoods appear to float in space above the abraded surface.
Since the virtual image depth is the same as the radius of the scratch,
then the angle of the circular arc is the same as the limits to the
viewing angle. See figure 8 and note the angles ? and ?'. Suppose we
want to produce an animation effect where a particular image "turns on"
and only becomes visible at certain viewing angles. To achieve this,
simply chose the appropriate angle ??when creating your curved scratches.
Images which are composed of short line-segment scratches of different
angles will "flash on" at different times as observers walk past. This
effect produces animation, but if it was used to encode a film of a
rotating object as "scratch holograms", it could also be used to change a
piece of cinema film into a "scratch multiplex hologram."
The hologram technique described in section 3 is based on the creation of
3D objects as collections of glowing spots. How can we reconstruct an
image of an opaque object which occludes deeper objects? Rainbow
holograms do this, so Scratch holograms should have that ability too.
Rainbow holograms produce opacity naturally because a shallower object
blocks the light from a deeper object, and the hologram of that portion of
the deeper object is never formed at the film plane. Therefore the key to
this effect is in section 3.3 above. When a shallow image of an opaque
object occludes a deeper object, we want the deeper image to "turn off."
To achieve this, we simply need to limit the angular extent of the
scratches which constitute the hologram of the deeper object. If a deep
image is created by a set of sandpaper marks on plastic, a shallower
opaque image needs to "erase" portions of those sandpaper marks. Note
that opacity issues are only relevant for scratches located in the same
horizontal band. The scratches are "holograms" of single points and they
act like horizontal stripes. A scratch on the left side might reconstruct
part of an opaque object which occludes a scratch on the right side. But
a scratch located near the middle can never create opacity for scratches
located higher or lower.
As Abramson discovers
[3], illumination which is significantly off-vertical
will cause the reconstructed image to become severely distorted. I suspect that Rainbow holograms would display just such distortions if they could produce visible images when the illumination angle is changed enormously. To solve this problem just do as the genuine rainbow holograms do automatically: pre-distort the scratch positions, so the resulting hologram is "programmed" to produce an accurate image only for a particular angle of illumination. In addition, distortion arises because the scratches ideally should take the form of extremely narrow horizontal bands, yet instead they curve downwards at the ends, and this produces unwanted vertical shifts at large horizontal viewing angles. Again take a hint from rainbow holography: rather than drawing continuous arc-scratches, instead break the scratches up into short line segments, then move them vertically until they resemble a cross section taken through a Fresnel lens. (Each scratch begins to resemble a horizontal array of pixels.) A third form of distortion arises because the scratches don't superpose linearly as light waves do, instead the overlapping scratches obliterate each other. We could eliminate this problem as before: by breaking each curved scratch up into "pixels", into numerous short line segments distributed in a straight horizontal band... but always moving the short scratches as necessary whenever two threaten to collide. (We might now have visions of mechanical non-diffractive "hologram printers" where the print head embosses a great number of narrow short lines with specular surfaces into plastic or metal. Or have visions of MEMS-based scratch-hologram plates covered with tiny reflective fibers which can be deflected to various angles by electrodes adjacent to each fiber.)
4. CONNECTIONS WITH RAINBOW HOLOGRAMS
5. A POSSIBLE "SINGLE FRINGE" HOLOGRAPHIC CAMERA
6. CONCLUSION
Created and maintained by Bill Beaty. Mail me at: .