How to not be seen

 kw: book reviews, nonfiction, science, optics, visibility, invisibility

In a video you may have seen (watch it here before continuing; spoiler below),

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…titled "Selective Attention Test", you are asked to keep careful watch on certain people throwing basketballs.

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…Several seconds in, someone wearing a gorilla suit walks into the middle of the action, turns to the camera, beats its chest, then walks back out of the scene. When this is shown to people who've never heard of it, about half report seeing the "gorilla", and half didn't see it. 

This is called Inattentional Blindness. It is used by stage magicians, whose actions and talk in the early part of a performance direct the audience's attention away from what is happening right in front of them. A magician can't be content with misdirecting half of the audience; the goal is 100%. This is often achieved!

But what if someone wants to vanish from plain sight, without benefit of a flash of fire or smoke (the usual prop for a vanishing act)? Optical science researcher Gregory J. Gbur might have something to say about that in his book Invisibility: The History and Science of How Not to be Seen.

Much of the history Dr. Gbur draws upon is found in science fiction. It seems that every scientific discovery about optics and related fields was fodder for science fiction writers to imagine how someone could be made invisible. This cover image from a February 1921 issue of Science and Invention (edited and mostly written by Hugo Gernsback, later to write lots of science fiction and edit Amazing Stories) shows the rays from something similar to an X-ray machine making part of this woman invisible.

I looked for this cover image online and found an archive of S&I issues. However, the issues were apparently produced with various covers for different regions, and the version in the archive had a cover touting a different application of X-rays. However, the article on page 1074, referred to in the cover shown above, does discuss whether X-rays or something like them can be used to provide invisibility, and also shows another way that structures inside the body may be seen.

Here the "transparascope" makes certain tissues transparent, allowing the viewing of others. IRL, the development of CT scanning and MRI scanning, fifty-odd years later, were required to achieve such views. The invisibility beam of the cover image has so far proved elusive.

Invisibility sits in the broader realm of "how not to be seen." The book shows in detail that the technologies that have been developed to hide or cloak objects can only work perfectly over very narrow ranges of light wavelength (and by analogy, waves in water and other media), and usually a narrow range of viewing angle. Is perfection needed? That depends…

In the late 1960's I worked for a defense contracting company, mainly as an optical technician. I was loaned to a related project as an experimental subject. The team was gathering data on the limits of human vision, detecting the contrast between a lighted object in the sky (an aircraft) and the sky. This was the Vietnam War era. 

The experimental setup was a room with one wall covered with a screen on which versions of "sky blue" were projected. At the center was a hole and various targets were set in this hole. They simulated the look of a dark or darkish object in the sky, and each target had several lighted spots, little lamps. The lamps' color and brightness could be adjusted. I was instructed to tell what I could see. The first day I was there, the background target was black, and the lamps were small and bright. The targets had differing numbers of lamps and their brightness would be adjusted to reduce the visibility of the overall target. This tested acuteness of vision; how many lamps on a certain size target would "fuzz together" and seem to illuminate its entire area? 

For most people, the "fuzz" angle is 1/60th of a degree. When you look up at a Boeing 737 at 30,000 ft elevation, its length of about 130 feet means it subtends and angle of about 1/4 degree. It would take two rows of 25 lamps along the fuselage, and at least 10 lamps, or 10 pairs of lamps, along each wing, to counter-illuminate it and reduce its visibility. That's a lot. A B-52 bomber is 20 feet longer and its engines are huge, like misplaced chucks of fuselage.

On another day, the target's background color was a blue color somewhat darker than the "sky". The target had the optimum size and spacing of lamps to seem of more-or-less uniform brightness, and the brightness and color of the lamps were varied. This tested our color acuity; how far could the colorimetry of the target-lamp combination vary to remain invisible or minimally visible?

This image simulates the second kind of target-lamp combination If you look at this image from a sufficient distance, the simulated target will nearly disappear, or for you it may vanish completely. This works best if you either take off your glasses or look through reading lenses, to defocus the image.

The average color and brightness of the simulated target are a close match to the surrounding sky-blue color. Thus, if an aircraft's belly is painted a medium blue, and a sufficient number of lamps are mounted on it and controlled by an upward-looking system, it can seem to vanish against the sky as long as it is high enough that the angular distances between the lamps is smaller than the circle of confusion (1/60th degree) of the eyes of an observer below.

This set of letter-targets is similar to a different test. Each letter has a little different color and brightness than the "sky". The 5 letters here make up the word "ROAST", but are not in order. For this test the sky color would be adjusted to see which letters were least and most visible. In both panels you will probably see three or four letters, but one or two that are not seen in one panel will be seen in the other.

In the end, it was all for nought. The sky is too variable, and human vision is also variable. There are three kinds of color blindness, and six kinds of "anomalous color vision"; any of these renders visible a target that "normal" eyes cannot see. It's kind of the opposite of those color-blindness tests with pastel "bubbles" that show the letter K to "normies" but the letter G to most color blind people. Also, wearing polarized glasses changes the perceived color of the sky, and tilting your head makes a dramatic difference in the color. Anyone with shades on would see the aircraft easily.

A further drawback of these tests was that no Asians' eyes were tested. In my regular job at the time, we were developing an infrared light source that Asians could not see. The near-infrared lamps used for night vision goggles and SniperScopes were invisible to Anglos, but quite visible to the Vietnamese. Several American snipers lost their lives when they turned on their SniperScope and a bullet came back instantly. What eventually worked was not a different light source but hypersensitive image amplification, the "starlight scope".

My wife is Asian. Certain items that look green to me she tells me are blue. Away from the green-blue boundary, she and I agree on the colors of objects.

The later chapters of Invisibility describe experiments and simulations that could lead to effective cloaking. There is even an appendix that shows a home tinkerer how to make a couple of kinds of visual cloaks that work in at least one direction. Full-surround cloaking is still out of reach, but who knows?

This book earns my "fun book of the year" award. Well written and very informative.

It takes big eyes to see fine detail, part 4

kw: musings, physics, light, visibility

This continues a discussion I began here. I think the point is established that if you want to see fine detail, the solution is not a smaller "eye" but a larger one. But I have an error to correct.

In the prior post, I stated the Rayleigh criterion for the resolution of a diffraction-limited optical system, which is based on the Airy formula

x = 1.22 λl/d

where l/d is the focal ratio, such as 4:1 for an f/4 system.

I neglected to mention that this formula is only valid if the light path between the object and the viewing lens is vacuum (or air, to a close approximation). In the case of high-powered optical microscopes, the space between the object and the objective lens is filled with an oil that has the same index of refraction as the glass of the slide and of the lens, typically about 1.5. The symbol for refractive index is n, so in an optical microscope with an oil-immersion objective, the formula is

x = 1.22 λl/nd

This replacement of λ with λ/n resolves the difference I'd stated between the Rayleigh criterion and the Abbe limit: 0.866/1.5 = 0.58, which means that an oil-immersed lens with N.A. = 1.4 (f/0.71) will resolve details as small as 0.58λ, which is 0.35µ for incandescent light (effective λ = 0.6µ) and 0.28µ for daylight (effective λ = 0.48µ).

My calculations for ultraviolet viewing remain unchanged because immersion oils absorb UV light.

Now, before we consider even smaller wavelengths, there is a technology that can resolve very small details for certain objects, using visible light. That is Near-field Scanning Optical Microscopy, or NSOM.

A very (very!) smooth sample can be scanned with a narrow glass fiber tip, just a few nanometers across. As long as the fiber tip is within a few nm of the surface being scanned, the spot of light will effectively be very much smaller than a wavelength. In practice, resolutions of around 10nm have been achieved. This is fifty times as good as imaging with lens optics. This technology is not for the faint of heart or poverty of pocketbook. Just drawing a glass fiber so it necks down to 3-5 nm diameter is a challenging task. Handling it to mount it and use it without breaking it…tricky. If you don't need the optical response, electron microscopes are cheaper, both SEM and TEM.

The concept to grasp for what follows is that of disturbance. We don't think of it this way, but even using visible light to view an object disturbs the object. The way light is reflected requires disturbing the outer electrons of a substance, which either absorb or re-emit the photons that bang into them. Visible photon energies are low, however, and don't typically cause atoms to be shifted out of position. Even x-rays and electrons a thousand times as energetic as visible photons seldom force atoms to jump aside.

But the electron bombardment, in particular, cause a disturbance in electron orbits. When a high-powered electron microscope is used to image atoms, we are seeing them slightly distorted, because what we are seeing is an image of the threshold position of the electron cloud around each atom's nucleus, at some equilibrium position as blown by the "wind" of electrons we are viewing with.

Is viewing the atoms, or their outer electron clouds, the ultimate achievable? Not at all. Hitting a small enough sample with a sufficiently dense electron beam, we can strip off the outer electrons, ionizing the atoms in the sample, but the ions will quickly neutralize by grabbing electrons from a cloud that builds up around the sample. Ion imaging may have its place, but I haven't heard of it being used. Nonetheless, there may be a way to use such a technique to probe various electron orbitals. X-rays are more typically used to do that, however, and the technology is well understood. We won't go further with that here.

Another reason for disturbing atoms enough to ionize them is related to probing the mechanics of the structure of a single atom's electron cloud. In this small realm, the order and spacing of each electron's orbital cloud are governed by quantum mechanics. For such studies to be meaningful, the atoms need to be isolated, so charged gases and plasmas are used. Atoms so disturbed emit and absorb electromagnetic radiation of all kinds, so this is the realm of spectroscopy, which covers a vast range of energies, from quite low (medium infrared starting at about 3µ) through visible and ultraviolet to x-rays with wavelengths of less than 1nm and energies of 100 KeV or more.

The next barrier is the atomic nucleus. What does it take to disturb it enough to learn about its innards? Ernest Rutherford began to show the way by bombarding gold foil with alpha particles, which are totally ionized helium nuclei. He was throwing one kind of nucleus at another, though he didn't know it. The α particles had an energy near 4 MeV, and when one came close to a gold nucleus, it bounced right back, or to the side. So four million volts isn't enough to do more than get a rough fix on the location of a nucleus.

As it turns out, once the proton was discovered (it is a bare hydrogen nucleus), it was found to weigh about 1,800 times as much as an electron. It takes many MeV to do more than make a proton bounce around like a smacked ping pong ball. Today's particle physics, developed since the 1950s, is in the business of finding out how the components of protons work together. It takes millions of MeV to have much chance of cracking open a proton (AKA p⁺), and its energetic components are very shy, splitting into showers of other particles within 10^-24 second.

The diameter of a nucleus is so small that, it has been often written, its size compared to the electron cloud is similar to a grain of sand hovering at the center of a cathedral or airplane hanger. An iron atom is about 10^-10m in diameter, and its nucleus is a bit smaller than 10^-13m, a thousand times smaller.

How big a machine is required to "see" details this small, and smaller? As it turns out, the finer the details you want to see, the bigger the machine it takes. Particle Accelerators are the tool we need to produce a probe that can see inside a proton. Accelerators are of two types: linear and circular. An ordinary TV set, and a small electron microscope, are both accelerators of the linear type, that accelerate electrons to energies of 30-50 KeV. Early accelerators that fit in a lab with a high ceiling, known as Cockcroft-Walton accelerators, could produce electrons with energies of a few MeV, and they are 4-5m tall. Both these kinds of accelerator are one-push types: make lots of volts and let electrons "fall" from one end to the other.

Higher energies are produced by letting the accelerated electrons fly through a hole and giving them another push with radio waves. The biggest of these, the Stanford Linear Accelerator (SLAC) imparts energies up to 50 GeV to electrons. But linear is a one-shot kind of technology. Once the electrons fly out the end, you're done with them. Much higher energies are had by running particles in circles, so you can push them over and over again, then divert them to hit your target.

The modern tool for this is the synchrotron. The old Cosmotron was 23m in diameter and imparted a bit more than 3 GeV to protons. The main synchrotron that accelerates electrons, the LEP, gives them the same energy as SLAC, but it can run beams in opposite directions so as to crash them head-on, for more energetic collisions. It can do the same with positrons, or with positrons one way and electrons the other. Lotsa possibilities! But protons, being 1,800 times as heavy, achieve much higher energies. The largest proton accelerator, the LHC (which operated briefly late last year, and is now being repaired!), should achieve 7 TeV (million MeV) per proton. But the LHC is designed to accelerate heavy nuclei containing many protons and neutrons, for purposes of learning more about how the quarks and gluons that make up a whole nucleus interact. The LHC is 8,500m in diameter, or 8km.

Putting the few figures I've given above with others, I produced this chart:

In the legend, p+ is for proton and e- is for electron. For electrons, getting to a certain energy takes about the same size machine, whether linear or circular. Circular machines have losses, however, because turning the electrons in a circle causes bursts of electromagnetic energy to leak away. This accounts for the somewhat different scale between the red and blue lines.

The green line shows proton energies in a few synchrotrons. The straightness of the line allows us to determine a formula, for gaining more energy if our investigations in the future require it:

E_max = 53.2 MeV x Diameter^1.3, for Diameter in meters.

Though we are producing particles with TeV scale energies, more is always desired. The 10^-24s time scale of quark-gluon interactions requires a wavelength of 3x10^-16m, or a particle energy of 41 GeV. In practice, it takes many times this energy, most of which is used to create "resonance" particles that spall away, so some is left over to push a couple of quarks a few trillionths of a meter apart so you can get a reaction you can measure.

How much energy is "enough"? Some Cosmic rays are more energetic than the output of the biggest accelerator we could build on earth. Let us use the formula above to calculate the energy from an accelerator 14,700km in diameter: we get 1.1x10¹⁷ eV, or 110,000 TeV. The most powerful cosmic ray so far detected had an energy of 3x10¹⁸ eV, and an accelerator to produce such particles would need to be 4 million km in diameter.

Until new technologies allow more efficient circular accelerators, then, we are limited to some thousands of TeV. The wavelength of ~100,000 TeV particles is as short as 10^-22m, and they can interact on time scales as brief as 10^-30 second. And that's the finest resolution achievable from machines that fit onto the surface of the earth.

It takes big eyes to see fine detail, part 3

kw: musings, physics, light, visibility

This is the third in a series of posts that began here. In that post, it was first determined that the resolution in the sensor plane depends only on the focal ratio (f-stop or f/number). A f/4 lens-sensor (film, retina, or digital camera chip) system will record ("see") details no finer than 3µ apart when the effective wavelength is 0.6µ, no matter how large or small the lens is. Thus a physically larger camera with f/4 optics will record more total detail than a smaller one. All this was based on "ordinary" photography and viewing, with the camera-to-subject distance being greater than the lens-to-sensor distance.

In the second post, this situation was reversed, and we examined photomicrography and the limits of optical magnification. The absolute minimum f/number for any lens is f/0.5, and the practical limit is f/0.6, while the standard for microscope optics (other than rare special products) is f/0.71. At that ratio, the lens diameter is 1.4 times the distance between the object being viewed and the optical center of the lens. The smallest details such a system can view are 0.52µ apart using incandescent light (effective wavelength 0.6µ) and 0.42µ apart when using a bluer, more daylight-like light (0.48µ).

Microscopists quote the Abbe Limit, which claims resolution close to half the wavelength for a f/0.71 system, but I have not been convinced by my own experience. The Rayleigh criterion I rely on predicts instead 0.866 times the effective wavelength for an f/0.71 system. Whichever criterion we prefer, though, how do we see stuff smaller than that? We use "light" with a shorter wavelength.

Two technologies are currently used to image microscopically using wavelengths shorter than 0.4µ or 400nm (From this point, wavelengths will be expressed in nanometers, since a micron (µ) is large on these scales).

• Ultraviolet light has been used, but is not practical below 200nm for two reasons. Firstly, air absorbs far-UV light shorter than 200nm, and secondly, fused quartz absorbs shorter than 160nm. UV microscopes are designed for use at a single wavelength, because there are few transparent materials that could be coupled with fused quartz to prepare "achromatic" lenses. Rather than a compound microscope, a simple design of objective is used to cast an image directly on a UV-sensitive sensor. The focal distance determines the magnification. Using 200nm UV light in air the practical resolution limit is 173nm.
• Electron microscopes of two main types have much, much better resolution than this, and can now image atoms directly. This is because electrons at modest voltages have much shorter wavelengths than visible and UV light, yet they are relatively easy to focus using electrostatic or electromagnetic lenses. The two types are transmission (called TEM) and scanning (SEM).

When I first read about electron microscopes in the early 1950s (TEM was invented in 1931), I found it very exciting. However, once I learned how to determine the wavelength of electrons, I was puzzled why even the million-volt TEMs could not "see" atoms. Then I learned about aberrations.

I was familiar with optical aberrations because of my interest in astronomy and telescopes; a friend of my father's was making a refracting telescope, and explained why it takes careful matching of the curvatures of four surfaces, on two pieces of glass, to get a lens that has acceptable correction of both chromatic and spherical aberration. A single spherical surface does not focus parallel light to a single point, but to a focal ray that is elongated along the optical axis. For a single wavelength of light, two spherical surfaces can correct most of this spherical aberration. Multi-wavelength (and thus multi-colored) light is also focused at different distances, and it takes a second lens with somewhat different optical parameters from the first, to make a doublet that corrects most of the color error.

Electrons of a single wavelength are easy to produce, so "color" correction is not needed. However, spherical aberration of simple electron lenses is extreme. So much so, if we had to use optical systems with visible light that had that level of aberration, we could not see anything smaller than about a millimeter!

A technical note: the wavelength of any particle, including an electron or photon, is related to its energy by a simple constant. Particle energies are expressed in electron-Volts, or eV. The constant is 1.24 eV-µ (for use in the visible and IR ranges) or 1240 ev-nm (for use with shorter wavelengths). Thus green light with a wavelength of 0.5µ has an energy of 2.48eV (1.24/0.5), and an electron accelerated in a 1,000-volt field has an energy of 1,000eV and a wavelength of 1.24nm (1240/1000).

It seems then a simple matter to use the 30,000-eV electrons produced by any old-fashioned TV set to see atoms. They have a wavelength of 1240/30000 = 0.04nm, smaller than most atoms—atom radii are in the range of 0.05-0.3nm. Until recently, it took a ten-million-volt TEM to clearly image atoms. Spherical aberration has been a severe constraint. In the past few years, however (see the TEAM 0.5 page), methods have been developed to correct the aberrations and permit direct imaging of even smaller atoms.

Though it is smaller than a 10,000,000V machine, which had to be twenty feet long just to avoid arc-over, the TEAM 0.5 microscope is still about three feet high, twice the height of optical microscopes.

In a follow-up post, we'll make the jump to some really small things scientists want to "look" at, for which technologies other than lenses must be used to produce an image.

It takes big eyes to see fine detail, part 2

kw: musings, physics, light, visibility

In the prior post I explained that larger eyes can see, and larger cameras can record, more detail from a particular object or scene. Briefly, the size of the smallest spot that can be produced by a diffaction-limited lens has a radius expressed by the Airy formula,

x = 1.22 λf/d

Given an effective wavelength (λ) for "average" visible light of 0.6µ, this formula tells us that the radius x of the smallest spot (on film or digital sensor) that the lens can make depends only on its focal ratio f/d. A small lens with a short focal length can cast an image on a only a small area, comparable to the lens-to-sensor distance. A larger lens at a greater distance from the sensor can cast a useful image on a sensor of larger area. Since the spot size, which limits the useful pixel resolution, is the same for both lenses, the larger lens will capture more detail from the object being viewed or imaged.

Continuing with visible light, let us consider how to see finer and finer detail in extreme close-up mode. When the lens gets closer to the object than it is to the sensor, things get a bit more complex, but only a bit. In the diagram below, the small "F" on the left is being magnified by a lens to produce a real image, shown as a larger "F".

The real image to the right can be seen if you put your eye a suitable distance further to the right, and if you put camera film or a digital sensor in the real image's location, the "prime focal plane" it will record the image. In the arrangement shown, the real image is four times the size of the object.

To analyze the detail we might be able to record, there are two regions of interest:

• The Object-to-Lens distance and Lens diameter establish the Objective focal ratio, which determines the smallest detail on the Object that can be imaged. In microscopy, this is usually expressed as its inverse, the Numerical Aperture, or N.A. For this diagram, the Objective focal ratio is 2, so the N.A. is 0.5. Using the formula above, the smallest detail seen will have a radius of 1.22x0.6µx2 = 1.46µ.
• The Lens-to-Image distance and Lens diameter establish the Image Focal Ratio, which determines the effective pixel resolution of the system. The magnification here is 4, meaning the lens-to-image distance is 8 times its diameter, so this is a f/8 system. The blur circle is thus 1.22x0.6µx8 = 5.86µ, or about 6µ in radius.

To take good advantage of this system, a digital camera sensor needs pixels 6µ x 6µ or smaller. Note that in magnifying the image, the lens has also magnified the blur circle by an equal amount. This is a 4x objective. A 40x objective will magnify the blur circle by 40x, to about 60µ if the objective focal ratio is f/2. Viewing with the eye and recording with a sensor will both need some mediation.

How much detail can the human eye see? People who are not nearsighted typically look at something no closer than 10" (250mm). Normal vision has an angular resolution (at the point of attention) of about a minute of arc, which means we can resolve items no finer than 1/3,500 of the distance from the eye. Vision called 20/20 is based on resolution closer to 1/1,000, or about 3.5 minutes of arc. Photographs printed at your local drug store have a resolution of 200 pixels per inch, or 8/mm. Held at a close distance, where they still "look pretty good", the finest details are about 1/2,000 the distance to the eye. Let us use this ratio to understand what we need to do to see everything the objective lens above is putting into the real image it presents to us.

If you place your eye ten inches from the real image, you can see it. The smallest details visible to you will be about 1/8mm or 1/200 inch apart; 1/8mm is 125µ. The original 4x system is producing a real image with details as small as about 6µ, so a lot of the detail in the image is wasted.

That is why the compound microscope has an eyepiece. A 10x eyepiece is placed one inch (25mm) from the real image and lets you see the image ten times larger. In this instance, you can now see details in the image as small as 12.5µ. To see "everything that is there", you need a 20x eyepiece, which lets you see 6.25µ details, which is getting close. At that point, the effective magnification is 80x. So how do microscopes with effective magnification of 1000x or so work?

First, we have to get a lens that has a lot smaller Objective focal ratio, smaller than f/1. It was discovered about a century ago that there is a point inside a glass sphere for which spherical aberration was nullified. Simply speaking, that means an easy-to-produce lens can magnify a lot. To get the object "inside" a spherical lens, one side of the sphere is ground flat, and an immersion medium such as oil is used to "extend" the lens so it surrounds the object:
Ratios as low as f/0.55 are possible, but f/0.71 is most common with glasses that need minimal color correction. At f/0.71, the Rayleigh criterion at 0.6µ means that detail on the Object can be seen as small as 1.22x0.6µx0.71 = 0.52µ. What magnification is then possible? Recall that the 1/2,000 criterion means that human eyes can see 1/8mm or 125µ. 125/0.52 = 240x. That is the maximum "useful" magnification using visible light.

Most high-power oil-immersion lenses are designed to provide a real image 100x larger than the area of the object being examined. So when you use a 10x eyepiece, you're seeing that area magnified 1,000x. However, if you examine the image critically (I have done so), you will realize that the edges between light and dark objects do not appear knife-sharp. Try a lower-power eyepiece: 5x looks a lot sharper, though of course everything appears smaller. You can't get 2.5x eyepieces, but if you could, the image you see would be well matched to the ability of your eye. The value of looking at the larger, blurrier image is that you can scan it more effectively for the subtle details, which might be missed at a lower magnification. As a reference, an E. coli cell has a diameter near 2µ and a length of about 5µ, so these critters look pretty small even at 1,000x.

A bit of extra resolution can be had by using a shorter wavelength. I've used, with little explanation, an effective wavelength of 0.6µ. That is the effective wavelength for incandescent light, which is rather orange, though we get used to feeling that it is "white enough". But if we throw away most (not all) of the red-orange light with a filter, we'll have a bluish light that is also "white enough" but has a shorter effective wavelength. My microscope came with a "daylight" filter that shifts the effective wavelength to about 0.48µ. With this filter, the ultimate resolution of the 100x objective is not 0.52µ but 0.42µ, allowing a magnification at maximum sharpness of 300x, and quite a bit more detail is then visible at 1,000x.

A slightly mathematical note to finish this portion: Microscopes have been standardized for decades at a 160mm "tube length", which fixes the distance between the real image and the objective lens at 160mm. The actual size of the microscope depends on the way the lenses are attached. A f/0.71 objective lens (N.A. 1.40) thus has a distance from the object (the microscope slide) of 1.6mm and a diameter of 2.25mm. The lens-to-image focal ratio is 160/2.25 or f/71 (100 times f/0.71). With my blue filter in place, the pixel size of the real image is 42µ (100 times 0.42µ). If I put a digital camera sensor in the focal plane, where the real image is, a lot of its resolution will be wasted. That is why microscopes set up for photography use a relay lens that shrinks the image, usually by a factor of 8, so that it is well-suited to recording by a sensor whose pixels are in the 5µ-10µ range.

These are the limits to seeing fine detail with visible light. Though I'd intended to explore shorter wavelength light (and other radiations), this post is long enough already.

It takes big eyes to see fine detail

kw: musings, physics, light, visibility

After reviewing The Lightness of Being recently (post here), I began to consider how it takes such large machines to study the tiniest things. Then I spoke with someone who was asking about diffraction, and I explained that it is a visible manifestation of Heisenberg's uncertainty principle.

[Airy Disk, image from the Wikipedia Commons. For a discussion, click here.] This shows the blur circle that results when parallel light of a single wavelength passes through a circular opening and is then focused to a "point" by a perfect lens. The radius of the first dark ring is given by the formula

The ratio f/d is just the focal ratio, or f-number of the lens. What this firstly shows is that the absolute size of the Airy Disk is independent of the size of the lens, for a given ration of focal length f to lens diameter d. Thus, when your camera reports it is using f/4 to record a scene, the smallest dot its lens can make on the film or digital sensor is 2.44 times 4 times the brightest wavelength. If we use 0.6μ (600 nanometers, nm) as the "effective" wavelength for whitish light, that smallest dot is about 5.86μ. Let's call it 6μ. If you work out the uncertainty in the position of a photon that results from a given diameter lens, the Heisenberg formula will give the same answer, about 6μ when the ratio of the lens diameter to its distance from the recording medium is 1/4.

Most point-and-shoot digital cameras these days have pixels of 3μ x 3μ or smaller, which is why these small cameras try to use larger lens openings in the range f/2.5-f/3. The sensors in my digital SLR are closer to 7μ x 7μ, because the sensor is larger. That allows me to use smaller lens openings for some depth of field and still have a lot of detail in the image. This camera is more than twice the size and ten times the weight of a more ordinary pocket camera.

OK, I am getting ahead of myself here. I wrote above that the dot size depends only on the f/number of the lens. That is just the beginning of the story. Let's consider an f/4 system, because we have worked out the relevant figures just above. The blur circle is about 6μ in diameter. The "Rayleigh criterion" for resolving detail states that the center of one spot must be no closer to the center of the next than the radius of the blur circle. In such a situation, one bright center is on top of the dark ring of the next center. Two such spots in an f/4 system are thus 3μ apart.

Now let us consider three possible "eyes" with a focal ratio of f/4. First, we'll discuss a camera with a lens-and-sensor arrangement similar to the human eye in size. This is the situation with many pocket digital cameras. The sensor might record 10 mpx (megapixels), or 3650x2740, and have a size of 6mm x 4.6mm (6,000μ x 4600μ). The individual pixels are only 1.6μ x 1.6μ. The lens focal length for "normal" focus is about the size of the image sensor's diagonal dimension, or 8mm (8,000μ). At f/4, the lens diameter is 2mm (2,000μ) (The lens is bigger than this, allowing it to record images whose blur circle is closer to the size of the image pixels, but it can "stop down" to f/6 or f/8 to exclude excess light or allow greater depth of field).

No matter what the size of the pixels is, the smallest details that can be resolved are in the range of 3μ apart on the sensor. Effectively, 6,000/3 = 2,000 and 4,600/3 = 1,533, such that the true image "detail size" is 3mpx. Only for focal ratios near f/2 can you capture all the detail that sensor can resolve.

Now let us consider a "spider eye", with a lens-to-retina distance of about a millimeter. I picked a spider's eye because it is not segmented like the eyes of most insects, but is a tiny lens-and-retina eye. If such an eye has a lens diameter of 1/4mm (250μ), it will also be able to resolve details that are 3μ apart on the retina. But the largest possible retina that can be kept a millimeter from the lens has a size of less than 3mm x 3mm. Let us suppose the "sweet spot" or middle region is 1.5mm x 1.5mm (1500μ x 1500μ). Let us further assume that the nerve cells in the retina are smaller than 3μ (they are probably close to 2μ in diameter). Then it will be, in effect, a 500x500 pixel array, or about a quarter megapixel. That is sufficiently detailed for many uses, but is actually a bit blurrier than analog TV. Actually, the eye of a real spider uses a spherical lens almost in contact with its retina, so its focal ratio is f/1 or even smaller. This allows for more detailed seeing than an f/4 eye at such a small size, and works in darker environments.

Now let us consider a large landscape camera, of the type often called Graphic; Graflex was a favorite older brand of large camera. These typically were a foot long or more, and recorded on film sizes of 4"x5" (100mm x 125mm) or larger. Such cameras are still made, and some photographers still use large sheet film in them. They make stunning images. One can also get scanning backs for them to record digital landscapes, but they are very costly. How will such a costly system perform at f/4?

Parameters: Focal length about 12" or 300mm, diameter 3" or 75mm. A scanning back or film cassette size of 125mm x 100mm (125,000μ x 100,000μ) is commonly used with a lens of this length. Scanning backs have pixel sizes ranging from 5μ to 10μ, because they have resolution aplenty. A 100mm scanner with a 125mm travel and 10μ pixels will be using each pixel to the fullest. Its final image size is 12,500x10,000, or 125Mpx. That is more than forty times the detail than what we found for the digital camera considered above, and hundreds of times the detail that can be captured by the spider eye.

Now suppose you are using these three systems to look at an object just one meter from the camera lens, and about half a meter (500mm) across. The spider eye will cover the item with 500 pixels (or fewer) each way, seeing details no smaller than a millimeter. The small digital camera will record an image that is effectively 2000 pixels the large way, seeing details as small as 1/4 millimeter. This is about how well your eye does from half a meter away also. The Graphic camera will record near-microscopic details, with 12,500 x 10,000 pixels, recording details smaller than 1/20 mm, or 50μ.

Just considering visible light, it becomes clear that larger cameras are needed to record finer details. To see smaller details than this, or to resolve smaller features of far-away objects, the same principle holds, as I'll get into in the next day or two.