DRAFT
I was sitting in Christmas Mass, sweating in midsummer, and the reflection of the lights off the oscillating fans pointed out to me the immense unrealized potential for mechano-optical laser displays, like the kind we used for psychedelic light shows back in my childhood. In particular, we can use them for archival of moving pictures.
However, it looks like opacity holograms will continue to be more practical and have comparable information density.
Point a laser pointer obliquely at a mirror-surfaced cylinder rotating around its axis. The laser pointer bounces off the cylinder and makes a spot on the wall, at an angle that depends on its angle to the surface; because the surface is curved, it diverges a bit in the plane perpendicular to the cylinder's axis. It doesn't move as the cylinder rotates.
Now suppose we cut a flat mirror-finish facet into the cylinder, parallel to the axis, such that it will rotate into the beam at some point. When this happens, the spot on the wall will jump to a new position as the surface under the beam abruptly changes angle; then, as the cylinder continues rotating, the rotation of that surface will move the spot perpendicular to the cylinder's axis; and finally, when the beam runs off the other edge of the facet, the spot will jump back to where it was originally.
If we rotate the cylinder fast enough that the motion isn't visible, it will draw a short dash on the wall around the point where the beam normally ends up.
The angle subtended by the dash, as seen from the center of the cylinder, is exactly twice the angle subtended by the facet.
If the facet, instead of being flat, describes a logarithmic spiral centered at the center of the cylinder, then instead of a dash, we will just be drawing another point on the wall.
If, instead of making it a spiral, we angle the facet so that it's not flat and parallel to the axis of the cylinder, but conical instead, we can displace the point parallel to the axis as well.
By combining conicality and spirality, we can place the point anywhere on the wall.
From here on, I'm going to use "up" and "down" to mean the directions on the wall parallel to the cylinder's axis, "right" to mean the direction in which the point reflected from a flat axis-parallel facet would be moving, and "left" to mean its opposite.
If a facet, rather than being perfectly flat, is slightly concave, we can focus the initially-collimated beam to a smaller point on the wall, if we know the focal distance to the wall, so that the point can be smaller than the illuminated part of the facet. This actually isn't going to be useful for the rest of what I'm discussing here.
By forming a series of many such facets on the circle that the beam describes on the surface of the cylinder, we can draw many points on the wall in quick succession, forming an arbitrary image --- indeed, one that changes over time. If the facets are smaller than the spot on the cylinder illuminated by the beam, several of them will be illuminated simultaneously, and indeed we can run several parallel tracks of them. Smaller facets allow us to program a larger number of points.
There isn't a guarantee that the facets will form an approximately cylindrical shape if we put them edge to edge. Consider, for example, the case where the animation is just a single stationary point to the left of center; your shape is a logarithmic spiral. And, of course when one point is significantly above or below its predecessor, there is a gap between the facets, which must be bridged in some fashion. You can do this Fresnel-lens-style by introducing discontinuities in the surface that won't be illuminated, so that the aforementioned single-point-left-of-center has the shape of a circular-saw blade rather than a continuous spiral.
If you bias the image to be in one particular direction from the centerline, such that an unbroken surface encoding it would be roughly a logarithmic spiral expanding as it rotates, you can ensure that the discontinuities introduced by the Fresnel-reflector operation are never illuminated, so you don't lose any light to them.
Edges of facets will tend to diffract: a planar wavefront reflected from the center of a flat facet will be reflected as a planar wavefront, but the part that reflects from the edge of that flat facet will form cylindrical wavefronts spreading out from the edge. If the facet is very large compared to the wavelength of light, the diffraction from the edges will not be significant, but if it is small, a significant amount of incident light will end up reflected as stray light by this mechanism. Worse, if these edges occur at regular intervals (like the sawblade shape mentioned previously) the diffracted light will form planar wavefronts going in other directions, but divided up by color, like any other diffraction grating. It's possible that you could use this mechanism to get an arbitrary color picture from a white light source, but I'm not sure you have enough degrees of freedom; alternatively, randomizing the discontinuities should break up such patterns and result in the stray light being just a generalized wash.
Also, though, the divergence of the reflected beam is diffraction-limited: a small facet is like a small aperture, and so you end up with the projected point being an Airy disk. This is what imposes the information capacity limit on this medium: if you try to light up more pixels by making smaller facets, you have more stray light and you unavoidably light them up with bigger spots.
So how many pixels can you have?
This depends on the Airy limit. Suppose we use 600nm as our relevant wavelength, which is a slightly orangish yellow and very nearly as sensitive as our daytime spectral sensitivity peak of 555nm.
So how far off the center axis of a facet at right angles to the light source do you have to go before the far edge of the facet is a whole wavelength further from you than the near edge, so that you're in the first null? Consider if we try to make our pixel facets 20μm across. The sine of that angle is 600nm / 20 microns, or about 1/32 to 1/40, so it turns out it's about 1.4 to 1.8 degrees --- and the whole bright beam reflected from that facet, inside that first null, is therefore a cone of 2.8 to 3.6 degrees. (1.22 λ/d is the Airy formula typically given for a round aperture, which is a little smaller than the 2 λ/d I'm using here.)
To put that in more normal terms, a visible-wavelength light beam passing through a 20-micron aperture will be diffracted into a cone with about 3 or 4 degrees of divergence.
(To see if I was smoking crack here, I checked the Wikipedia article on "beam divergence". It says, "Gaussian laser beams are said to be diffraction-limited when their radial beam divergence [half the cone angle] is close to the minimal possible value, given by θ = λ/πw, where λ is the laser wavelength and w is the radius of the beam at its narrowest point, which is called the 'beam waist'." In this case, λ/w is about 1/16 to 1/20, so our divergence in radians should be a πth of that, which is only slightly narrower than the answer I got above, and the difference may depend on where you set your cutoff.)
This stands in sharp contrast to what we normally expect from, say, a megapixel display, which is pixels of something like 2 arcminutes, 0.03 degrees, two orders of magnitude better. You can improve things by putting the projector closer to the wall than your eye is, but you're still only going to get reasonable resolution over about one radian in each dimension, before things start to kind of sketch out around the edges due to the super oblique angles.
If you want to get megapixel-sharpness images out of this thing, you're going to have to use bigger facets so that you can get milliradian-level resolution. So you can't get 2500 pixels per square millimeter in a useful way.
You can use curved facets if your image is made out of lines; the curvature is equivalent to a bunch of facets that are very narrow in one dimension, and so scatter the light more in that direction, along the line of light, at right angles to the long skinny facet.
A mixed model may provide the best information density: use small facets toward the center of the beam to direct the overall brightness of a scene to the right places (with necessarily high divergence) and larger or curved facets, perhaps in the dimmer outer parts of the beam, to draw outlines with milliradian-resolution definition. Such a facet might be 10 microns by 1mm, diffracting the incident beam into a line about 100 milliradians by 1 milliradian, in some arbitrary orientation, placed in some arbitrary place on the screen. You have space for, say, 75 such lines in a square millimeter, with another 400 facets in the range of 24 microns across, each directing some brightness onto an area something like 30 milliradians in size. 1-4 square millimeters should be enough to encode a "frame" of video, and getting a spot of light down to 1-4 millimeters is easily feasible. It isn't necessary to get it up to 1-4 millimeters, because instead of using a diffuse spot, you can just scan the spot quickly over a long distance of the surface.
This means that a second of animation is something like 100 square millimeters, or a square centimeter, of reflective medium.
There's an additional degradation of resolution in the left-right direction caused by the beam striking the reflected medium being of nonzero size and the medium rotating through it, but I don't think that's important for two reasons. First, there's no fundamental obstacle stopping you from making that beam be however small you decide to make your individual facets; second, you can diminish that rotation arbitrarily by using arbitrarily large cylinders, or even using a flexible cylinder that you depress to be locally flat so that its movement under the beam is purely translational, or rotational around its focal point on the wall. If you don't take such measures, though, a 1mm-diameter beam illuminates about 1/50 radian on a 10cm-diameter cylinder, so you get a left-right smearing of about 1/25 radian from the rotation.
As a point of comparison, fancy professional laser shows often use galvanometer-driven mirrors capable of 25000 points per second, and what I've proposed above (475 "points" per square millimeter and 100 square millimeters per second) is about 47500 points per second; so this is clearly capable of producing visually arresting animations.
If we don't have a laser pointer, we can use a sunbeam through a hole the size of the desired beam. The divergence of the beam, and thus the resolution of the projected image, will be limited by the angular size of the sun in the sky, about half a degree. This resolution can be improved substantially, at the cost of brightness, by putting a pinhole between the hole and the sun, blotting out most of the sun's disk. If the pinhole is fairly near, that will introduce substantial nonplanarity in the light wavefront, which will change the effective focal length of concave facets.
A milliradian-sized pinhole to blot out most of the sun will blot out about 74/75 of it, reducing the illuminance available to light the image by about 20dB, down from, say, 100 kilolux down to, say, 1000 lux at the point where the beam hits the facets, so the illuminance on the screen will be something like 1 lux, if it's being spread out over a screen area around 1000 times the size of the illuminated area on the facets. This probably means that such an animation will require a dark room for viewing by sunlight at full resolution without concentrating the sunlight using some kind of nonimaging optics.
(Of course, if you just illuminate it with direct sunlight, you'll get the image anyway, just blurred by convolution with the sun's disk.)
So far I've been talking about cylinders, but all of this continues to work just as well for animation if we're talking about a disc, too, or even just a flat sheet. Any illuminated circle on the surface corresponds to some image; as the circle shrinks, it will include images from fewer and fewer physically adjacent frames of video until it starts including only certain drawing elements of a single frame; and any path the circle takes over the surface will produce an animation. Purely translational motion also eliminates the rotational left-right intraframe smearing mentioned in an earlier section, which can easily reach tens of milliradians.
Suppose that, instead of shooting for milliradian-resolution line drawings, you just want to put an animation of readable text onto your surface. Maybe you only need to be able to display a word or two at a time, or even a letter. Does this help? Remember, before, we were looking at something like only 100 vectors per square millimeter, because of milliradian-level sharpness requirements at visible wavelengths.
How many vectors do you need per letter?
I designed a 6-pixel-tall proportional pixel font a while back, with the objective of conserving text with laser-printed microscopic letters on paper, which encoded my 4.45-megabyte test Bible in 4866x19254, or 21 black-and-white pixels per letter, which means that each letter averages 3.5 pixels wide, including the space needed between the letters to make the text readable. It's black on white, and it's about 29% black, so that's about 6.1 black pixels per letter. Based on that, let's figure that we probably need about 4 vectors per letter and a resolution of about 0.05 radians.
Well, this lets us use the originally-hoped-for 20-micron-wide facets, which do in fact give us about 0.05-radian resolution. For lines, again, we can use narrow facets that diffract light into a streak in the desired direction, so that they actually take up less space. For example, a lowercase "i" might require a full 20-micron-square facet for the dot, but only a 20-micron-wide by 6-micron-tall facet for the vertical line beneath it.
If we figure that the average facet is then half of that 20-micron-square configuration, then we can get about 5000 vectors per square millimeter, or 1250 letters. That is similar to my laser-printed microprint approach, which at 1200dpi only gets 2232 pixels per square millimeter, which works out to about 1100 letters; but it doesn't need a microscope to read it. (A laser-printed 1200dpi pixel is about 20 microns in size, so this comparability is not totally surprising.)
Now, though, we run into a different problem. An average word might need only about 20 vectors to display it, but it's going to be projected over 0.3 by 0.9 radians, which means that you only have room for about three or four words to be displayed at a time. But that's 60-80 vectors, which fit into a 200-micron-square area. Maybe you can get a laser pointer down to 200 microns square, but getting a sunbeam that small is hard.
If, instead, we figure that we need something more like 0.015-radian resolution, and thus our facets need to be 60 microns across when they're full points, we can fit 30 or 40 words onto the screen at a time. But now we only have about 500 vectors per square millimeter, which is also about 40 words. 40 words per square millimeter is somewhat inferior to microfilm, but dramatically higher than traditional printing's 0.014 words per square millimeter.
It's probably best to organize the words such that they "scroll" around the screen like the words in an old ytalk session: rather than attempting to move old words, after filling up the last line, you wrap around to the top and put new words on the top line. If your spot of light is too small, you'll have less than a 30-40-word screenful of text visible, while if it's too large, you'll have new words overlapping old words, maybe several screensful. Ideally, as you scan your light beam over the text, new words fade in shortly after old words fade out. You can display some kind of position indicator in some fixed part of the screen; it can be quite a long streak, so it can use up very little space on the medium.
You can use this technique to make a super awesome solar sundial which not only projects the time, to the minute, onto the wall, but also tells you what day of the year it is, although typically you'll have two choices for that.
How would you go about fabricating such a demandingly-shaped reflector? If it's a one-off, I think you can probably use electrochemical machining, which is sort of the opposite of electroplating --- you make your workpiece the anode, pumping electrons out of it, which it then hungrily obtains from negative ions floating around in water, which then combine with metal atoms from the surface; you limit the current flow to a tiny part of its surface area by moving the cathode around very close to it; and you wash away the electrolyte rapidly enough to prevent the metal ions from the anode from being reduced onto the cathode. Due to anodic leveling, it produces a mirrorlike finish.
You probably want to use a sort of rake-shaped cathode to produce the desired surface complexity, dragging many wire points over the surface, each with its height controlled separately to within a fraction of a micron, say, 50 nanometers. They should probably be spaced something like 10 microns apart, 100 of them to a millimeter. Each one will tend to cut out a spherical shell into the metal around it, cutting a cylindrical trench through the metal as it moves. We want to keep the deviation from the desired shape of the metal down below, say, 100 nanometers, to avoid fucking up the light wavefront too much, and there's 5 microns from the center to the edge of the trench; acos(1 - (100nm/5 microns)) = 0.2 radians, so we want those wire points to be carving out 25-micron-radius spheres. That's going to make it a little tricky to control the angles of 10-micron-wide facets, but I think it's doable.
A 25-micron inter-electrode gap is smaller than is typical for electrochemical machining, but not outrageously so; 80 to 800 microns is typical. Some "pECM" processes use an inter-electrode gap of as little as 10 microns, along with vibrating electrodes and pulsed current.
We can do a little electropolishing afterwards to try to anodically smooth the surface, but we'd like to be removing very little material. Electropolishing is also going to round off the Fresnel-reflector knife edges and result in more stray light.
How much overall material do we need to remove?
Suppose our facets have angles varying over more or less a quarter of a radian in every direction from the overall surface, and that we can generally choose to alternate them to minimize discontinuities. In the above, the biggest facets were up to a millimeter in size. If we're cutting millimeter-sized facets at a quarter-radian from horizontal, we have to cut up to a quarter-millimeter deep! On average, we might be removing 100 microns of the surface of the material, 100 milliliters per square meter, or 0.01 milliliters per square centimeter.
If we're doing this in aluminum (which may not be the highest-quality choice, but millimeter-thick gold is expensive) that's 27 milligrams per square centimeter. Aluminum is conveniently 27 grams per mole, so that's a millimole per square centimeter. We have to pump off three electrons from each aluminum atom in order to turn it into aluminum chloride, so we need three millimoles of electrons per square centimeter. A coulomb is only 6.2e18 electrons, while a millimole is 6.0e20, so that's 97 coulombs per square centimeter, plus whatever gets wasted in electrolyzing the water; at 1 ampere, that's about a minute and a half per square centimeter, or 280 micrograms per second.
Some random paper about the material removal rate of electrochemical machining (Sudiarso, Latifah, Ramdhani, and Mahardika 2013) says that at 0.6 to 1.09 amps on aluminum 1100 with a 2-mm-diameter brass cathode, at 15 volts, they were able to remove 79 micrograms of aluminum per second. That's a factor of almost four less efficient than the calculation I pulled out of my ass above, which is far more accurate than that calculation had any damned right to be.
That also works out to be about (79 micrograms/sec) / (0.8 amps * 15 volts) = 6 or 7 micrograms of aluminum removed per joule. 27 milligrams per square centimeter is then 4 kJ/cm^2. This is a very small amount of energy.
How can you control the position of 100 metal points per millimeter to within 50 nanometers, with a total range of vertical motion of 250 microns (5000 times the resolution)? This seems very challenging indeed. The most promising approach would seem to be using a small number of points, such as 10 to 30, with a separate piezoelectric actuator for each one; a small hydraulic actuator might also work.
I conclude that it's probably feasible to do electrochemical machining of these surfaces, but it requires making some advances in electrochemical machining.
Electrical discharge machining (EDM) is reportedly somewhat more advanced than ECM and can typically produce even better surface finishes, but it doesn't seem like it is applicable to this process, because it consumes its electrodes. Wire EDM is a common process, using a consumable wire electrode, but it runs the wire through the workpiece and then discards it; so it can't be used to make concave shapes. Perhaps it might be possible to perform a kind of wire EDM that consumes the end of the wire, completely, rather than merely spark-eroding the side of the wire somewhat.
It might make more sense to press or stamp the facets into the surface using a hard tool, one press per facet, particularly if you're using a soft metal like aluminum or gold. You need five degrees of freedom (two to control the reflection angle of the facet) with 10-micron positioning precision, and, in the Z-axis, enough force to stamp plastically 125 microns deep into the surface.
For reproduction, there are existing models of mass-production of reflective metallized surfaces with micron-level detail, specifically for diffraction gratings. Master gratings are cut into glass on a ruling engine at the Grating Lab in upstate New York, which has three ruling engines, one made by the hands of Michelson himself; and then casts are taken in plastic from the glass, and then other casts from the casts, and so on. Each grating can only withstand a few casting operations, perhaps five or ten, before being damaged enough to degrade its quality, but you can make several generations of them, which is sufficient to supply the world demand for lab-quality diffraction gratings. The final gratings are metallized, typically by vacuum deposition of aluminum.
Exactly the same process would work for these animations, although the particular resin and hardening process used for the gratings seems to be a secret.
