Goerge Airy vs. Lord Rayleigh

AR principles

Pinhole photography appeals to many because of accessibility, unpredictability, and a relative low cost when starting out. The main principle of image formation is quite simple: a light passes through a small opening and upon reaching a screen in a dark room produces an image. This image, if projected onto a sensitized material, can then be recorded and further processed like any other. Simple enough.

As we scan examples of photographs produced through above approach, we easily see a ridiculously wild range of image resolution, contrast or both. While many believe that pinhole image is doomed to be blurred, the evidence counters that with examples of striking sharpness, especially considering there is no lens helping detail rendering.

Turns out, the subject of pinhole image quality is an engaging one. We can choose to skip the discussion altogether and just go with the flow, even if that flow appears to be asking for an adjustment. Or we can try and make some sens out of it. Explanation of theoretical principles governing light behavior, image formation, its resolution and perceived sharpness can be found in many a documents and articles. I have found however, that information is quite scattered though and I’m yet to come across one place that has it all. This unfortunately introduces some confusion to the matter, even the matter itself isn’t so overly complicated.

1-George Biddell Airy iraylei001p1

The two main personas always mentioned in the discussion on pinhole optimal size are George Airy and Lord Rayleigh. Both had sweated over relevant issues for years and each came up with his own interpretation, delivering formulas which differ by a factor of 1.225.

Which is right is the question, or does it matter?

The pinhole world appears to have agreed to disagree. While there is a consensus on having an optimal pinhole diameter for a given hole to film plane distance, at least two derivations of optimal size are in wide use (as per above).

It is interesting that while Rayleigh was well aware of what Airy came up with on light diffraction limitations with relation to resolvable detail, he chose a substantially larger hole diameter to achieve seemingly same resultant quality. Without a doubt there are two sides to this coin:

  • microscopic evaluation, perhaps one would call it an objective one
  • a real world evaluation based on what a human eye interprets as better, surely a more subjective approach

airy single disc airy double disc

Above is the Airy’s disc surrounded by the well agreed to Airy’s pattern of what light produces on a screen, shown of course at a microscopic level. We wish we’d only have the middle bright spot, but that is never the case, although the pattern gets more or less disturbed depending on several factors. Next to it is the double disc shown with good separation indicating good resolving ability.

Airy disc by definition is the smallest spot to which an optical system can focus a light beam. The size of the disk is resultant of the associated diffraction for any particular case. Diffraction is always present, but can be controlled to a degree, thus resolving ability of a system comes from how well the diffraction can be tamed (among other factors).

In pinhole image formation the Airy disc size is what limits resolution. Since it squarely depends on diffraction, study of pinhole size vs. diffraction effect (or its severity) was required.

Let’s go back a bit though and look at what would happen, if there were no diffraction at all and no wave lengths. As diagram below shows, we would be only discussing simple geometrics involving straight light rays, mostly rejected by a small opening, only few reaching the screen to form a point in an image.

simple pinhole geometrypdf_dwn

From simple straight light ray geometrics we now need to face the reality. Light being a form of electromagnetic waves with associated wave lengths, does not fully cooperate with our wants and needs. As the hole diameter gets smaller things get a little cranky in and past the hole. Wave length imposes its own limitations of the minimum aperture size, while diffraction wreaks further havoc on the projection screen.

Lord Rayleigh or John William Strutt (1842 – 1919), interestingly enough, was from Essex. I don’t know, if he had a secretary, but judging by the quality of his work, if he did, she would have defied all the jokes of today.

Rayleigh was a renown physicist with numerous credits to his name. What interests us in this discussion is the Rayleigh Criterion. 

Rayleigh Criterion

Resolution depends generally on aberration and diffraction characteristics of an image forming system. Pinhole imaging does not involve glass elements and aberration is not present. Diffraction is the key limiting element, but also wave length of light.

We can define resolution as ability to separate detail so it can be distinguished.

Once details begin to approach one another, resolution gradually degrades until details can no longer be distinguished.  Diagrams above show two points separated on the left, hardly distinguishable in the center, and blended into one on the right. The center disc & rings surrounding each point represent diffraction effect, and in the right image also affect of superimposing patterns. Rayleigh correlated diffraction and wave length into a single formula, which results in a minimum separation required to resolve a detail.

Rayleigh Criterion formula: β = 1.220 * λ/d

  • β – angle of separation in radians
  • λ – wave length of light
  • d – diameter of aperture

Examining formula we can see, that resolving limit of a particularly sized pinhole is a constant for the same wave length in angular terms. Its projection on film would increase with film plane moving away from the pinhole and decrease when moving in closer.

Taking above into account, we can check how a projection of two points of light would resolve through a pinhole of a specific size at a defined distance from projection screen.

pinhole at a distancepdf_dwn

It’s easy to see two relations:

  • change in PD will change the size of airy disk, smaller PD producing smaller disk
  • as film plane is moved away from original position (with PD remaining constant) the size of projected airy disk will correspondingly change, but how?

Let’s have a look at the same diagram with differing film plane position.

pinhole at a distance, bpdf_dwn

It is now becoming more apparent why optimal pinhole diameter depends on film to hole distance. As in the diagram above, moving film plane closer to pinhole while leaving pinhole size unchanged (with corresponding airy disk size also unchanged), would cause airy disks overlap degrading resolution.

How does airy disk size change with pinhole diameter?

  • Optimal Pinhole Diameter (OPD)
  • George Airy's diffraction-limited formula
    OPD = √(2.44*FHD*λ)
  • Rayleigh's formula
    OPD = √(3.66*FHD*λ)

  • Rayleigh Criterion Calculator
  • Airy Disk size formula
    β = 1.220 * λ/PD

Playing with above calculators we see something perhaps unexpected. The last diagram clearly showed airy disks beginning to overlap when film plane was moved closer to pinhole. This suggested a smaller pinhole with closer film plane, which in fact is the case coming from optimal pinhole diameter formulas. However, when substituting our NEW & smaller PD to airy disk calculator, we see actual INCREASE in airy disk size, which seems to be contradicting our earlier thoughts. Why is it?

The reason is relatively simple, yet easy to overlook.

We were wrong in earlier assumptions using simple geometric diagram to represent the behavior of light as it passes through a small aperture. Rayleigh’s Criterion is based on diffraction which gets more pronounced for smaller apertures. Light waves deflect more as they pass through a tighter opening causing increase in overall airy disk size.

The goal is to find the best compromise to effect best possible image quality.

… to be continued

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