diffraction limiting?

[Ole]

One last try -- If the image was formed *only* by the "straight rays" with no bending to different destinations, the image would be nearly the size of the aperture. There is no question that it is not, at "pinhole aperture sizes, it will cover a considerably larger area.

Sorry Ed, but you'll have to continue. I've always thought that if it weren't for diffraction, the resolution on film would be equal to the size of the pinhole - all other things being ideal (perfect film, perfect hole, infinity distance an all that). So that the CoC=hole size.

And that diffraction (in a simple hole) would only serve to reduce this resolution, making the perfect pinhole size a compromise between the decreased hole size and the increased resolution.
Since there is no glass in a pinhole, spherical resolution doesn't enter into the equation at all. Chromatic aberration is obvious, as the effect of diffraction increases with decreasing wavelength. But not quite the same as in lens systems...

[Helen B]

Ed,

I'm not sure how far to get into this - up to now I admit that I have not been particularly careful and complete in my postings because I didn't expect that a refutation of the concept that there would be no image if there was no diffraction (my main reason for wading in) would cause this depth of reply, nor did I expect that anybody would be all that interested!

Briefly then, using the text you quote:

...If we make the diaphram smaller and smaller, then to our amazement, we see that the image spot becomes larger again. Careful study shows that there is a disc surrounded by weaker concentric rings, called Airy's disc, after its discoverer. This image is similar to the already familiar image with spherical abberations, but is more regular...."

"The image spot becomes larger again." Exactly! The circle formed by a point of light becomes larger because of diffraction. Airy's disc refers to the image (actually an interference pattern) made by a point source, so diffraction turns a point source into an enlarged point surrounded by rings. That's why it reduces resolution. In fact the Airy pattern is also known as the Point Spread Function (PSF).

The optimum pinhole size pretty much coincides with the diameter at which the geometric image of a point source is nearly the same diameter as the first dark ring of the Airy disc. This is not a surprise. It does mean that the image formed by an optimum pinhole can be adequately described in terms of Fraunhofer diffraction only. This is not the same as saying that the image would not exist if it weren't for diffraction. The 'diffraction only' explanation would not work if the pinhole was significantly larger. Experience shows that large pinholes do work - they just produce low resolution images. I guess that that is the main weakness I see in the 'image formed by diffraction only' argument.

[i]...One last try -- If the image was formed *only* by the "straight rays" with no bending to different destinations, the image would be nearly the size of the aperture. There is no question that it is not, at "pinhole aperture sizes, it will cover a considerably larger area.[i]

I don't agree that the image circle would be the size of the pinhole if it weren't for diffraction. Oblique rays can come though the pinhole, and anyone who has made a pinhole camera will know that the size of the image circle is highly dependent on the thickness of the material with the hole in it - which, along with the hole diameter, determines the cut-off angle. However, I would be very interested to learn of image circles that exceed the theoretical diameter calculated by geometric optics, and I admit to never having looked into that aspect of the pinhole cameras that I have made.

Anyway Ed, thanks for an interesting and most intelligent explanation of a viewpoint that I had not come across before.

Best,
Helen

[glbeas]

Helen is right about the pinhole imaging. It has nothing to do with diffraction. Visualise this...You are in a dark room with a very small window centered on the opposite wall. From the position you are at you can see a blob of light through it that is part of something. Move over some and you see another part. As you move around you can eventually see all of what is out there but in pieces. The classic description of how a pinhole camera works is similar. A ray of light bounces of the surface of an object and is partially absorbed. The remainder happens to enter the pinhole and its trajectory lands it on a certain spot on the film. Another ray hit adjacent to it and is altered and its trajectory lands it next to the previous rays landing site. And so on. Each ray has its own brightness and color and direction which when entering the pinhole detemines where on the film that spot of light hits and how bright and what color it is. They all merge like pixels on a dye transfer print into a recognizeable image in the camera and the film records it..

[Tom Hoskinson]

I have been lurking here, watching this thread. Here is some pertinent material I abstracted from a paper by Kazuo Sayanagi.

This information is from a paper titled: Pinhole Imagery, by Kazuo Sayanagi, published in September 1967 by the Journal Of The Optical Society Of America, Vol. 57, No. 9 1091-1099. In this paper, Sayanagi presents a modulation transfer function approach to calculating optimum pinhole diameters for various different conditions.

From Sayangi’s paper:

“The pinhole has been used as an imaging device for centuries. The oldest description of pinhole imagery I could find was written by the Arabian scholar Ibn Al-Haithan (A.D. 965-1038). He described the use of a pinhole to observe the projected image of a solar eclipse. He pointed out that the finest imaging with the pinhole can be obtained by using a very small hole and discussed the image quality when the size and shape of the hole are changed, as follows:”

“The image of the sun only shows this (crescent shape) property when the hole is very small. If the hole is larger the image changes, and the change is more marked with increasing size of the hole. If the hole is very large, the crescent shape of the image disappears altogether, and the light becomes round if the hole is round, quadrangular if it is quadrangular, and with any shaped opening you like, the image takes the same shape, always provided the hole is large and the receiving surface parallel to it.”

The first physical consideration of pinhole imagery, based on a mixed treatment of geometrical and physical optics, was written by Petzval (1857, 1859). He expressed the diameter D of the image point made by the pinhole as the sum of the geometrical diameter of the aperture d and the diameter of the diffraction pattern caused by the aperture d,

D = d + k l lambda / d

Where k is an optimal condition constant (Petzval chose k=2), l is the distance between the pinhole and the receiving plane and lambda is the wavelength of the incident light. The optimum diameter of the hole is defined so as to give the minimum diameter of the point image.

By solving the resulting partial differential equations, Petzval obtained:
d squared = k l lambda, with k = 2, where d is the optimum diameter of the pinhole.

Based on Fraunhofer diffraction in the presence of defocus, Lord Rayleigh’s analysis (1891) arrived at a k value of 3.6

Sayanagi (1967) used a modulation transfer function approach to calculate a general-purpose k value of 3.8

These approaches all assume an infinitely thin pinhole.

Equations don't work well in this text editor, Sean.

[Ed Sukach]

I was trying to say that the study of light - and diffraction was *far* more complex than the scope of APUG would allow.
It is *not* as simple as "The image is only formed by the straight rays..."

I'll suggest the book ... that I quoted verbatim et literatim:

A.C.S. van Heel
C.H.F. Velzel

What is Light?

Translated from the Dutch
by J.L.J. Rosenfeld

World University Library
McGraw-Hill Book Company
New York Toronto

(c) mrs. H.G. van Heel and C.H.F. Velzel 1968
Translation (c) George Weidenfeld and Nicolson Limited 1968
Library of Congress Catalog Card Number 67-24448
Phototyped by BAS Printers Limited, Wallop, Hampshire, England
Printed by Officine Grafiche Arnoldo Mondavori, Verona, Italy.

Another sample (struck with a sudden masochistic urge fro typing:

"Until now we have used as a model for the propagation of light the principles of Huygens, which we define as follows. One can imagine a wave front to originate out of the previous one by supporting each point in the latter to be a secondary source of spherical waves. The envelope of these spherical waves forms the new wave front. Notice that here, in contrast to our previous formulation of Huygens' principle, we are talking in the language of waves (italics mine -E.S.). But we have in the meantime discovered that waves show interference. Fresnel's addition to Huygens' principle consists, as we shall see in the following, in this, that he takes account of these interference phenomena. We should add a word about the range of validity of Fresnel's theory. The electromagnetic theory of light, which we shall describe later on, explains diffraction effects in an unforced manner and is, on deeper examination more precise. But Fresnel's theory of diffraction does, in most instances give the correct explanation even of details, and for this reason is considered by opticians to be a good working tool."

OK, class ... close your books. Time for a pop quiz...

Hmmm... another passage:
"This can be described as follows. It is unfortunatey a somewhat long and complicated tale, but it cannot be shortened and is most easily understood with the aid of a recipe, Think of a sphere about P which just touches V at Q and call the distance QP, l. Imagine further other spheres with P as centre whose radii are respectively ..."

No more ... Is Jim beam a good remedy for typer's (and brain) cramps?

[Tom Hoskinson]

I prefer Highland Park, Ed - but that's a different story.

[glbeas]

I was trying to say that the study of light - and diffraction was *far* more complex than the scope of APUG would allow.

Thats an understatement for sure.

It is *not* as simple as "The image is only formed by the straight rays..."

For practical purposes concerning pinhole cameras it is. Diffraction is the interference that limits how small you can make a pinhole for optimum sharpness. We aren't dealing with fancy diffraction lenses like is used on X-ray telescopes. The propogating sphere of light illuminating the surfaces is interfered with by the pinhole which allows a tiny spot through which is stopped at the film surface. Simple geometry suggests how the imaging takes place from this point.

I'll suggest the book ... that I quoted verbatim et literatim:

A.C.S. van Heel
C.H.F. Velzel

What is Light?

Translated from the Dutch
by J.L.J. Rosenfeld

World University Library
McGraw-Hill Book Company
New York Toronto

(c) mrs. H.G. van Heel and C.H.F. Velzel 1968
Translation (c) George Weidenfeld and Nicolson Limited 1968
Library of Congress Catalog Card Number 67-24448
Phototyped by BAS Printers Limited, Wallop, Hampshire, England
Printed by Officine Grafiche Arnoldo Mondavori, Verona, Italy.

Another sample (struck with a sudden masochistic urge fro typing:

"Until now we have used as a model for the propagation of light the principles of Huygens, which we define as follows. One can imagine a wave front to originate out of the previous one by supporting each point in the latter to be a secondary source of spherical waves. The envelope of these spherical waves forms the new wave front. Notice that here, in contrast to our previous formulation of Huygens' principle, we are talking in the language of waves (italics mine -E.S.). But we have in the meantime discovered that waves show interference. Fresnel's addition to Huygens' principle consists, as we shall see in the following, in this, that he takes account of these interference phenomena. We should add a word about the range of validity of Fresnel's theory. The electromagnetic theory of light, which we shall describe later on, explains diffraction effects in an unforced manner and is, on deeper examination more precise. But Fresnel's theory of diffraction does, in most instances give the correct explanation even of details, and for this reason is considered by opticians to be a good working tool."

OK, class ... close your books. Time for a pop quiz...

Hmmm... another passage:
"This can be described as follows. It is unfortunatey a somewhat long and complicated tale, but it cannot be shortened and is most easily understood with the aid of a recipe, Think of a sphere about P which just touches V at Q and call the distance QP, l. Imagine further other spheres with P as centre whose radii are respectively ..."

No more ... Is Jim beam a good remedy for typer's (and brain) cramps?

Looks like interesting reading. I wish I had time to delve into all the interesting sources I've encountered here. You've supplied some of the best.
I'm not saying you're wrong about anything Ed, just that you're overcomplicating the issue unneccesarily. Sometimes you need to drop to the simpler levels for the sake of clarity.
Jim Beam is great for those cramps purely on a medical basis. ;-)

[Helen B]

So, this all started so simply. I disagreed with the statement that a pinhole would not create an image if there was no diffraction. Do we all agree that the principal way in which a pinhole creates an image can be described by geometric optics, with the limit set by diffraction? Does that suit all parties?

And didn't Lord Rayleigh do well, getting a similar answer to Sayanagi (which is the same result I quoted, after assuming lambda, in case no-one had noticed). There are many ways to formulate the behaviour of light - the trick is to know which one is appropriate for the case under consideration.

Best,
Helen

[Tom Hoskinson]

Amen Helen! I think this particular horse is well and truly dead.

[Ole]

Amen Helen! I think this particular horse is well and truly dead.

...and full of holes as well