## Wednesday, June 03, 2009

### 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.