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CD-DA (Digital Audio) 3

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A Little Optics...

The Airy Disk and the Limits of Resolution

You might perhaps assume that light could be focussed into a single spot that is as small as we wish, only depending upon the optics. But there are limits. Instead of making a simple circular spot, "The image of a point formed by a perfect lens is a minute pattern of concentric and progressively fainter rings of light surrounding a central dot, the whole structure being called the Airy disk after George Biddell Airy, an English astronomer, who first explained the phenomenon in 1834." [Encyclopedia Brittanica online article on Optics]

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Airy Disk with Bessel function
From Prof. Alex Dzierba's Physical Optics Lab Manual, Indiana University, Bloomington

Because the Airy disk is the smallest "point of light" that an optical instrument is capable of revealing, two such points that are very close together will be indistinguishable. Thus the size of the Airy disk determines the instrument's resolving power. Abbe's Formula conveniently expresses the maxiumum resolving power as follows:

        x >= 0.61 lambda / (n sin theta)
        where x     is the minimum distance separating two objects
              lamda is the wavelength of the illumination
              n     is the index of refraction of the embedding medium
              theta is half the "angular aperture" of the lens. 

Many microscopes are designed to use a drop of oil or other high-index liquid between the lens and the subject in order to improve resolution, which is why the "n" term is important. "n" in air is 1.0.

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The term "n sin theta" is called the numeric aperture (NA), and is one of the standard optical parameters specified for microscope optics. We can think of it as a measure of light gathering ability. It is specifed for CD readers at 0.45. This is good -- it saves us the trouble of worrying about theta and using trig functions. Note: Several times I've seen Abbe's formula written as "d = lambda/(2 NA)", which is evidently an approximation.

The basic concept of Abbe's formula is that you can still resolve two points so long as the center of each respective Airy disk coincides with the first minima of the other Airy disk. In other words, they are separated by the radius of the central disk. This is commonly referred to as the "Rayleigh Criterion".

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The solid lines represent the diffraction patterns of the individual distributions. The dotted line represents the sum of the two light distributions.
The Airy Disk and You

Again, I'm no optics expert, but it seems to me that we should distinguish between (1) the Airy disk created by the laser "spot light" that is illuminating the Compact Disc surface and (2) our ability to resolve surface features in the reflected light. In the first instance, controlling the radius of the spot ensures that only certain features are illuminated at any one time. This reduces the task of reading the reflected to one of simple photo-detection. In the second instance, Abbe's Formula (based on the Airy disk) tells us the minimum separation distance in which we can distinguish two objects.

Let's go ahead and apply Abbe's formula to compute the minimum resolution. We know that NA = 0.45. But what value should we use for the wavelength? It is 780nm in air and 503nm (780/1.55) in polycarbonate. I'm not certain which one to use! So let's compute both numbers and see what happens.

          x = (0.61 * 503nm/0.45)  [Abbe resolution, wavelength in substrate]
            = 682nm resolution, 1.4 micron central disk diameter
          x = (0.61 * 780nm/0.45)  [Abbe resolution, wavelength in air]
            = 1057nm resolution, 2.1 micron central disk diameter.

There's a nice diagram in Kuhn (below) that shows the first nulls of the Airy disk falling on the adjacent tracks in order to minimize the amount of light reflected from those tracks. But if the first nulls fall on the adjacent tracks, then the central portion of the Airy disk is 3.2 microns in diameter (twice the track pitch of 1.6 microns). If the resolution of surface features uses the same optics (and it does), then I believe you'd have an Abbe resolution of 1.6 microns, which would be needlessly large. So I don't think this is correct. But what if it's the second nulls that fall on the adjacent tracks? Examination of Dzierba's Airy disk illustration above suggests that if the second nulls are 3.2 microns apart, then the diameter of the central disk is about 2 microns. This matches our calculations above using the wavelength in air.

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Diagram from Kuhn EE498 lecture notes
Illustrates FWHM and how the nulls of the Airy pattern align with the adjacent tracks. I think it must be the second nulls that are aligned, not the first nulls as shown.

Here's some more confusion: According to Michael C. McGoodwin (apparently citing Pohlmann), "the spot size d of a pickup (read) beam is defined as the half-intensity diameter (i.e., its full width at half maximum FWHM), or 0.61 x wavelength/NA". This, of course, is Abbe's formula, but I don't think it is the same as FWHM. Given the steepness of the edges, FWHM should be only slightly smaller than the whole central disk. Kuhn keeps citing a spot size of 1.7 microns (which I think he got from Pohlmann anyway). That's about 80% of the central disk diameter, and I'll bet that number is the FWHM.

So we have some evidence supporting our Abbe calculation using the wavelength in air. But I'm still not comfortable embracing a resolution of 1057nm. As will be seen shortly, the CD pickup is required to distinguish features that are 833nm apart. So it seems to me we need a theoretical resolution that is comfortably less than that. I have not yet found an adequate explanation anywhere, but it may be that the reflected light somehow affords a resolution closer to 682nm.

Regarding the width of all pits, I've heard different numbers from different reputable sources. Some (e.g. Kuhn, probably after Pohlmann) say pits are 500nm wide while others (e.g. Philips) say 600nm. As we have seen, either should provide adequate space between tracks. A larger pit width would provide greater contrast between pits and lands and therefore should be easier to read. It should also be easier to track because the the tracking beams will more quickly detect a differential when the tracking wanders. For curiosity's sake, this bears further investigation but doesn't otherwise make any difference to us.

Last Updated Monday October 15, 2001 17:58:09 PDT