Pinhole optics and format size

I'm trying to get my head around what effect format size itself has on image quality when it comes to pinhole performance. Please understand that I'm only trying to consider format size itself, and I already know that format size effects other aspects of camera design. We have flexibility when designing pinhole cameras, so to directly attempt to compare the merits of different format sizes, I will attempt to isolate format size and hold other things equal, at something close to optimal while I'm holding them equal. So please don't say something simple like "well larger formats let you use larger pinholes". I think we need to take that into account from the get-go. So we attempt to isolate the effects of format size so that we can compare them to other format sizes directly.

Of course we all know there is a Rayleigh formula for determining the theoretically optimum pinhole size. I will assume this formula is 'good'.



PINHOLE SIZE
The only variable terms in the equation besides the mysterious 1.9 constant and the wavelength term (which can be considered a constant if we don't try to get fancy with integrating over the visual spectrum) are the "focal length" and the pinhole size, d (note that we are not considering angle-of-view or format size yet). So basically we can say that for any given focal length, there is one unique optimum pinhole size. But since,in the formula, f is proportional to the square of d, we can't say that there is an optimal "f-stop" for pinhole cameras. If we double the focal length of a camera, we don't double the diameter of the pinhole, we only increase it by a factor of 1.414.

So there is one optimal F-stop for any given focal length, but not an optimum f-stop for all pinhole cameras. And we can see that for long focal lengths, the optimal F/stop given by the Rayleigh formula is going to be a larger number. As we continue making the focal length larger, we continue making the pinhole bigger, but at a slower rate, so the f/d ratio will be larger for larger focal lengths. Using only Rayleigh-optimal pinholes, we would expect the camera to get slower as we increase the focal length.

IMAGE CIRCLE
A second dimension in camera design involves the size of the image circle. For a given format size, there is a minimum focal length (completely independent of the pinhole size) that will result in even illumination of the film. This is due only to the combined effects of the smaller effective area of the pinhole and the longer distance to the film as you move away from the center of the image circle. This minimum focal length is strictly dependent on the size of the format and the degree of light falloff that is acceptable. I have done calculations to approximate the largest image circle diameter for which light hitting a flat film plane falls by 1 stop at the edges compared to the center of the film. This falloff occurs for all focal lengths and all pinhole sizes at an angle of 37* from the axis or a total angle-of-view of 74 degrees. I come up with the following relationship between focal length and image size for the maximum image circle for 1 stop of light falloff, assuming a round pinhole in infinitely thin material and flat film:

max image circle diameter for 1 stop falloff=2*focal length*tan(37 degrees)
or equivalently
minimum focal length for 1 stop light falloff= (image circle diameter)/(2*tan(37*))

Remember this is completely independent of the pinhole size, and I did this analysis myself so it could be wrong. But even if I am wrong with the numbers, it's still true that approximately, there is a fixed minimum focal length for every format size, if you accept a certain amount of light dropoff at the edges.

So we can see that IF we take the minimum focal length that gives no more than 1 stop of light falloff as optimal, AND use only Rayleigh-optimal pinholes, that any image format size will have only ONE optimum pinhole size and ONE optimum focal length associated with it. Now, we can finally compare image format sizes to each other using these conditions.

So how do they compare? We've already seen that larger format sizes at these optimal conditions will have larger f-numbers and be slower than shorter focal lengths/format sizes. But this doesn't tell me much about image quality. The parameters that we now can see that are completely attributable to format size have to do with

1. Larger formats have larger f-numbers--they are slower--at optimum conditions.
2. Larger formats require less enlargement for a given output size.

Now, considering only these two factors, you tell me if larger formats are better or worse.