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Understanding diffraction


timarmes

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The original inquiry was "understanding diffraction ... in lens systems" not "understanding the RATIO OF diffraction TO IMAGE FORMING LIGHT ... in lens systems" but since the latter is all we care about, why not concentrate on that? AND, why not specify, completely and unambiguously that the latter is what we are talking about. So let us talk about all three elements in reference to each other:

 

- diffraction,

 

- apertures

 

- the image plane.

 

Let's not forget the definition of diffraction -- the bending of light as it passes over an edge. The more edge, as there is in a large aperture, the more diffraction. The less edge, as there is in smaller apertures, the less diffraction.

 

Now, let's also remind ourselves of another definition, aperture -- the hole that image forming light passes through on it's way to the image plane. The larger the aperture, the more image forming light will hit the image plane. The smaller the aperture, the less image forming light will hit the image plane.

 

Combine these -- the definition of diffraction and the definition of aperture -- and you get, AT the image plane:

 

- larger apertures = more diffraction but also w-a-y more image forming light AT the image plane, and so the RATIO of diffraction to image forming light is actually quite small AT the image plane. We call this "less diffraction" when we mean a "smaller diffraction/signal ratio".

 

- smaller apertures = less diffraction but also w-a-y less image forming light AT the image plane, and so the RATIO of diffraction to image forming light is actually quite large AT the image plane. We call this "more diffraction" when we mean a "larger diffraction/signal ratio".

 

So, earlier on this thread: "...Diffraction is greater for smaller apertures, less for large apertures..." Oops! Probably meant: "Diffraction/SIGNAL RATIO is greater for smaller apertures, less for large apertures." Or perhaps, "Diffraction EFFECTS RECORDED AT THE IMAGE PLANE are greater for smaller apertures, less for largeer apertures."

 

And, that's all we care about -- the actual ratio of diffraction to image forming signal AT the image plane.

 

The fact that there is less total diffraction at smaller apertures than at larger apertures is somewhat irrelevant, as, at smaller apertures, there is even w-a-y less image forming light, also!

 

Let me also edit an earlier post: "...The reason the circumference of an aperture is not mentioned in any of the formulas that define diffraction, only the DIAMETER of the aperture, is that diameter can be used to calculate the effect of the greater circumference and the resulting greater diffraction, but also the w-a-y greater image forming light."

 

As already mentioned, the diffraction goes down arithmetically with the circumference from larger to smaller apertures, but the image forming light goes down geometrically with the area from larger to smaller apertures.

 

Formulae mean nothing if they do not accurately represent what real-world facts we are trying to measure or predict. Formulae support, not prove!

 

Also earlier on this thread, let me identify and give credit where credit is due for some links and WHY there are relevant or not:

 

http://www.gonda.ucla.edu/bri_core/na.htm = Numerical Aperture and Resolution from Francon, M. 1961. Book "Progress in Microscopy". Hmm ... microscopy ... relevancy?

 

http://www.cambridgeincolour.com/tutorials/diffraction-photography.htm = Sean McHugh offers support for lesser diffraction to signal ratio with longer lenses than with shorter lenses, presuming the aperture is farther from the image plane in a longer lens than the aperture to image plane distance is in a shorter lens.

 

http://photography.about.com/od/basics/a/bpaperture.htm = Peter Marshal about.com on Photography, generalized and simplified definitions of photo gear - aperture, auto manual, iris, boke[h], relative apertures, depth of field, diffraction limits, optimum apertures, including this gem which supports my point all along: "...when the lens is stopped down, a greater fraction of the light passing through it will pass close to the edge of the iris..." Nice paraphrase, Peter Marshal.

 

http://www.kenrockwell.com/tech/focus.htm = Ken Rockwell ... well. I'm not sure why Ken is EVER relevant to any discussion of photography -- he's a photographer, not a word-smith, as he freely admits! Here he chats about large format stuff unrelated to photographic subject content.

 

= = = = = = = = = =

 

Oh, and Roger Hicks, below is from your reference and for our archives of this thread -- Google searches for [define:numerical aperture] and [define:na] (non photo- optical- definitions removed):

 

Definitions of numerical aperture on the Web:

 

* The number that expresses the light gathering ability of a fiber. Related to acceptance angle. http://www.cetpak.com/Technical/glossary.htm

 

* The numerical aperture of a microscope objective is a measure of its ability to gather light and resolve fine specimen detail at a fixed object distance. All modern microscope objectives have the numerical aperture value inscribed on the lens barrel, which allows determination of the smallest specimen detail resolvable by the objective and an approximate indication of the depth of field. http://www.microscopyu.com/articles/formulas/formulasindex.html

 

* The sine of the acceptance angle of a fibre multiplied by the refractive index of the medium from which the light is entering (air = 1). http://www.fibreoptech.co.uk/glossary.asp

 

* A unitless measure of the ability of a lens to gather and focus light. NA = n sin q, where q is the angle of the light as it narrows to the focal point. A numerical aperture of 1 implies no change in parallel light beams. The higher the number, the greater the focusing power and the smaller the spot. http://www.dvdmadeeasy.com/glossary/n.html

 

* A unitless number referring to the light gathering ability of a fiber and is defined as the Sine of half acceptance angle. http://www.fontcanada.com/gloss.html

 

* In optical fiber, the sine of the maximum acceptance half-angle, q max , times the refractive index of the core (assuming an air-to-core interface). The larger the NA, the greater the amount of light that is accepted into the fiber for propagation to the distal end. http://www.polymicro.com/techsupport/techsupport_glossary.htm

 

* (NA): The light-gathering ability of a fiber; the maximum angle to the fiber axis at which light will be accepted and propagated through the fiber. The measure of the light-acceptance angle of an optical fiber. NA = sin a, where a is the acceptance angle. NA is also used to describe the angular spread of light from a central axis, as in exiting a fiber, emitting from a source, or entering a detector. http://web1.mtnl.net.in/~powertel/glossery.htm

 

* The product of the angle formed by the cone of on-axis rays and the index of refraction of the medium in which the cone resides. With higher numerical aperture, more light will be collected. In a diffraction limited system, the numerical aperture is directly proportional to the resolution of the optical system. http://www.navitar.com/zoom/zoom_glossary.htm

 

* The characteristic of a fiber optic strand which defines its acceptance of impinging light. The degree of openness, light gathering ability, and angular acceptance are other terms describing this characteristic. http://www.nuhorizons.com/Glossary/Optoelectronics.html

 

* A ratio that describes the cone of light emitted by the condenser or accepted by the objective lens. Objectives with a larger NA have greater resolving power. (a 100x oil lens with a 1.25 NA can resolve smaller objects than a 100x lens with a 0.95 NA) http://www.bi-optic.com/vocab.html

 

* a measure of the acceptance angle of a lens. Higher numerical aperture means the lens gather more diffraction orders yielding higher resolution but at the expense of depth of focus. http://www.icknowledge.com/glossary/n.html

 

* (fiber optic) The light gathering ability of a fiber, defined as the sine of half the angle that contains 90% of the optical power that is captured by the fiber. http://connectors.tycoelectronics.com/glossary/glossary-n.stm

 

* This is a measurement of the light-gathering capacity of a fiber ? specifically, it measures how light spreads out after leaving the fiber. Numerical aperture also can be used to estimate the acceptance cone or angle when coupling to a fiber. http://www.corning.com/photonicmaterials/products__services/specialty_fiber/photosensitive/glossary.asp

 

* A figure that, in effect, indicates the capacity of an optical fibre to receive light. http://www.networkingsolutions.co.uk/acatalog/gl.html

 

* The numerical aperture (NA) of an optical fibre defines the characteristic of the fibre in terms of its acceptance of impinging light. It can be calculated and expressed as an index (a number). The higher the number the more light than can be accepted. http://www.aefos.com/html/glossary/n.htm

 

* The angle at which a fibre will gather light and propagate it down the core http://www.opticalfibresuk.com/glossery_of_terms.htm

 

* In microscopy, the numerical aperture, AN, of an objective is: where I is the index of refraction of the medium in which the lens is working (1.0 for air, up to 1.56 for oils), and a is the angular aperture of the lens. It is basically a measure of the diameter of the aperture compared to the focal length. In photography, the f-number expresses the same relationship. http://en.wikipedia.org/wiki/Numerical_aperture_(microscopy)

 

Definitions of na on the Web:

 

* The number that expresses the light gathering ability of a fiber. Related to acceptance angle. http://www.cetpak.com/Technical/glossary.htm

 

* Abbreviation of numerical aperture. The higher the value is the higher the resolution is. http://www.mo-forum.gr.jp/english/glossary/

 

* A unitless measure of the ability of a lens to gather and focus light. NA = n sin q, where q is the angle of the light as it narrows to the focal point. A numerical aperture of 1 implies no change in parallel light beams. The higher the number, the greater the focusing power and the smaller the spot. http://www.dvdmadeeasy.com/glossary/n.html

 

* Simply, a non-dimensional number that indicates the ability of a fibre or other device to receive light input. Specifically, the sine of the half angle of the acceptance or radiance cone of an optical fibre, multiplied by the refractive index of the material in contact with the fibre face. http://www.interconnect.co.za/links.html

 

* In optical fiber, the sine of the maximum acceptance half-angle, q max , times the refractive index of the core (assuming an air-to-core interface). The larger the NA, the greater the amount of light that is accepted into the fiber for propagation to the distal end. http://www.polymicro.com/techsupport/techsupport_glossary.htm

 

* The product of the angle formed by the cone of on-axis rays and the index of refraction of the medium in which the cone resides. With higher numerical aperture, more light will be collected. In a diffraction limited system, the numerical aperture is directly proportional to the resolution of the optical system. www.navitar.com/zoom/zoom_glossary.htm

 

* The characteristic of a fiber optic strand which defines its acceptance of impinging light. The degree of openness, light gathering ability, and angular acceptance are other terms describing this characteristic. http://www.nuhorizons.com/Glossary/Optoelectronics.html

 

* The sine of the vertex angle of the largest cone of meridional rays that can enter or leave an objective, multiplied by the refractive index of the medium in which the vertex is located. In air the NA must be less than 1. http://www.pmel.org/Surface-Glossary.htm

 

* A ratio that describes the cone of light emitted by the condenser or accepted by the objective lens. Objectives with a larger NA have greater resolving power. (a 100x oil lens with a 1.25 NA can resolve smaller objects than a 100x lens with a 0.95 NA) http://www.bi-optic.com/vocab.html

 

* A mathematical formula devised by Ernst Abbe for the direct comparison of dry and all types of immersion objectives for resolving power. Numerical Aperture is the sine of half the angular aperture of the objective multiplied by the refractive index of the medium between the front lens and the cover glass. NA ranges from 0.1 to 0.95 for dry objectives and up to 1.4 for oil immersion lenses. http://www.visioneng.com/technology/glossary.htm

 

Hmmm .. another source of heat more than light for me -- drat! I think I'll go out and shoot something! =8^o

 

Click!

 

Love and hugs,

 

Peter Blaise peterblaise@yahoo.com http://www.peterblaisephotography.com/

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<i>The more edge, as there is in a large aperture, the more diffraction. The less edge, as there is in smaller apertures, the less diffraction.</i>

<p>

Eh? Where does this come from? Alan Davenport is right: for circular apertures (or any hole) the diameter is what's important.

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.

 

Earlier in this thread:"...The more edge, as there is in a large aperture, the more diffraction. The less edge, as there is in smaller apertures, the less diffraction..."

 

Someone responded: "...Eh? Where does this come from?..."

 

Peter Blaise responds: ... from the DEFINITION of diffraction - see above in the miles and miles of postings to this thread. Diffraction is where light bends when going over an edge.

 

Someone responded further: "... Alan Davenport is right: for circular apertures (or any hole) the diameter is what's important..."

 

Peter Blaise responds: Important for what? Please be specific.

 

Diameter, as mentioned, is easily a part of any equation where the circumference of a hole (where the diffraction is) must be calculated AND the area of the hole (where the signal, or image forming light is) must also be calculated. The ratio between the circumference and the area is surprisingly identical to the ratio of diffraction to signal (image forming light) hitting the image plane! Go figure! =8^o

 

The dependence on "diameter" in a formula does not change the definition of diffraction.

 

As we use smaller apertures, the circumference (where the diffraction is) goes down arithmetically, and the area (where the "signal" or image forming light is) goes down geometrically. At some point, the ratio between them at the image plane favors the diffraction, hence the phrase "diffraction limited". I think the goal is to discover if there is actually ONE aperture for any particular lens, and what is that aperture where a lens's resolving power becomes diffraction limited. I think no one has yet offered specific examples of lenses and subjects and image enlargement where this can be illustrated.

 

Anyway, does ANYONE have ANY image where the ONLY thing wrong is "diffraction"? Otherwise, the image would be perfect? I certainly don't. I'd love to see some samples. Otherwise, this is an academic discussion. The only reason I'm hanging in here so long is to keep up the good fight against basing ANY of our photography decisions on fallacies, errors or misunderstanding.

 

Peter Blaise peterblaise@yahoo.com http://www.peterblaisephotography.com/

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Peter Blaise, who writes at excessive length, remarked "I think the goal is to discover if there is actually ONE aperture for any particular lens, and what is that aperture where a lens's resolving power becomes diffraction limited. I think no one has yet offered specific examples of lenses and subjects and image enlargement where this can be illustrated."

 

40/4.5 Luminar. This lens is diffraction limited wide open. Image quality deteriorates markedly on stopping down at all. At the magnifications the lens is intended to be used at, stopping down reduces overall sharpness because the gain of DoF on stopping down is swamped by diffraction. Come to think of it, this is true of all of the Luminars.

 

55/2.8 AIS MicroNikkor. This lens is diffraction limited at f/4. Same thing. At normal distances, stopping down below f/4 hurts little, but when the lens is reversed and used above around 5:1 stopping down below f/4 is pure loss. Re hurts little, in an old test MP found that at normal distances the lens resolved best at f/5.6, went quite mushy at f/32.

 

You have to understand the tradeoff between sharpness in the plane of best focus and sharpness in depth. Most of us give up one to get the other. Copying flat subjects is a prominent exception.

 

Do a search in Usenet for Brian Caldwell's comments on diffraction limted optics. He posts as BC in several of the rec.photo.equipment.* newsgroups, also in sci.optics.*

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<i>Diffraction is where light bends when going over an edge.</i>

<p>

Your reasoning is right for some geometries (such as a linear obstruction) but not all (like an aperture). To see why geometry matters, it helps to think about why "light bends when going over an edge." There are a couple ways to do this, probably most people would use <a href=http://www.mathpages.com/home/kmath242/kmath242.htm> Huygen's wavelets </a>, where destructive interference through a small aperture is less complete than through a larger one (and is only truly complete through an aperture of infinite size), generating greater spreading and stronger fringes -- what we would call more diffraction. <i>Why</i> the Huygen-Fresnel priciple works is an open question ("where do little wavelets come from, mommy?") but it does work, and work quantitatively.

<p>

Another way to look at it is to step back and ask what you're trying to do. A perfect imaging system would render a point in the object plane as a point in the image plane. To get that point (a spatial delta function) in the image plane you need a transverse field with a white wavenumber spectrum. Any obstruction (such as finite size) means that you cannot reconstruct a point in the image plane, and the smaller the aperture the worse the reconstruction -- we'd call that more diffraction. The actual pattern of the diffraction can be varied by changing the wavenumber spectrum (beyond the crude clipping of a window, that is, the <a href=http://mathworld.wolfram.com/ApodizationFunction.html> "apodization" </a> mentioned by Alex Lofquist) but you cannot eliminate the diffraction itself.

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(a) Let's NOT comment on each other's posting styles.

 

(b) Thanks to a previous poster here for providing personal experience that your own 2 lenses (they are YOUR OWN lenses, right?) appear to have objectionable diffraction after certain apertures. I presume you decided for yourself after much experimentation and trying different apertures and subject distance, lighting and contrast ratios and film and subject matter and enlargement sizes? How about image samples? Can you assist us in how we might make a similar assessment of our own lenses regarding the threshold (if there be any) of being "diffraction limited"? Thanks!

 

© Maybe it's just me, but the reason I quote extensively is that I find it unreasonable to expect each participant now and in the future to independently go out and serendipitously find and read the exact same resource to which I am referring. I searched Google for [bC diffraction inurl:rec.photo.equipment] and [bC diffraction inurl:sci.optics] and found nothing. Please quote whatever you think is relevant for us to read rather than hope we will ever find the same thing that impressed you!

 

(d) No pictures?

 

Click!

 

Love and hugs,

 

Peter Blaise peterblaise@yahoo.com http://www.peterblaisephotography.com/

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.

 

Earlier in this thread: "... Your reasoning is right for some geometries (such as a linear obstruction) but not all (like an aperture)..."

 

Peter Blaise responds: Please tell me how a ray of light knows which of the different and supposedly competing or conflicting laws of geometry you want it to obey or disobey at any one time? I've asked this before but no one has yet to offer a theory of a super-light-speed inter-communications between waves/rays/photos of light.

 

I put it to you that each individual ray of light does not know nor care about anything except that it alone as hit an edge and diffracts accordingly.

 

Our observations are of the collective effect through a lens and (a roughly circular) aperture, and I am postulating that our observation criteria do not in the least concern any individual wave/ray/photon of light on it's trajectory, whose individual path is ignorant of the paths of other waves/rays/photons.

 

Or, does anyone have a lens with an aperture that is so small at to approach the size of a single wave/ray/photon such that the one ray can actually tell that it is tryign to s-q-u-e-e-z-e through a hole because that one wave/ray/photon is making contact with opposite sides of the hole at the same time? What is that, f/18,446,744,073,709,551,616 or something close? ;-)

 

Why are we spending so much time on the academic trivia on this? Does no one have personal experience and sample photos to share?

 

= = = = = = = =

 

Well, I DID go out and shoot something -- and I brought 'em back alive. That's the beauty of photography for me! I was out shooting wild flowers and butterflies and insects and stuff at the 40-acre Winkler Botanical Preserve here in Alexandria, Northern Virginia, US -- see satellite map below (lots of parks around where I call home) -- while my system at home was batch-scanning 20-year old shots of Arches, Capital Reef, Dead Horse Point, Moab, Utah, US and other travels over the years -- see http://data2.itc.nps.gov/parksearch/state.cfm?st=UT

 

Sorry, but neither experience turned up useful examples of diffraction to share.

 

Click!

 

Love and hugs,

 

Peter Blaise peterblaise@yahoo.com http://www.peterblaisephotography.com

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Ooops - here the elusive map to the Winkler Botanical Preserve:

 

http://www.maps.google.com/

 

and enter

 

[5400 Roanoke Avenue, Alexandria, Virginia, 22311 USA]

 

and I like the hybrid view with roads and sattelite.

 

Enjoy -- maybe I'll see you there?

 

Click!

 

Love and hugs,

 

Peter Blaise peterblaise@yahoo.com http://www.peterblaisephotography.com/

 

PS - photo.net seems adverse to letting me upload an image ... oh well. So much for full participation. Maybe I need more 1,000 words to stand in for the pictures that are missing? ;-)

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<i>Please tell me how a ray of light knows which of the different and supposedly competing or conflicting laws of geometry you want it to obey or disobey at any one time? I've asked this before but no one has yet to offer a theory of a super-light-speed inter-communications between waves/rays/photos of light.</i>

<p>

You're making this more complicated than it really is. Some terminology first -- not to get into semantics, but I don't know what you mean by "ray". Photons and waves are both well defined, but you have to be careful about usage because there is a crucial difference, namely that waves are at field (as in "light is an oscillatory electromagnetic field") level and photons at energy (field^2) level, so that in the photon description phase information is lost - you can kludge it back in, but for interference effects such as diffraction working at field level is much more natural. You seem to be describing light as a stream of photons, which is fine if you keep the above in mind. Otherwise you'll have to work mighty hard to understand interference. Think about light also as a wave and not only does interference naturally follow, but so will the dependence on geometry (aka "boundary conditions")

<p>

<i>I put it to you that each individual ray of light does not know nor care about anything except that it alone as hit an edge and diffracts accordingly.

<p>

Our observations are of the collective effect through a lens and (a roughly circular) aperture, and I am postulating that our observation criteria do not in the least concern any individual wave/ray/photon of light on it's trajectory, whose individual path is ignorant of the paths of other waves/rays/photons. </i>

<p>

No offense but I don't think you believe that. Not unless you don't believe in interference. Not unless you don't believe in refraction. And you still haven't explained <i>why</i> it is that light diffracts at an edge.

<p>

Re: actual results from real lenses. I submit that for the vast majority of photographic work (high resolution 2-D imaging such as copy work and astrophotography excepted) such results are just as irrelevant as theory. How many times do we have to stop down for DOF, how many of us use heavy camera stands (or even beefy tripods), how many of us put a loupe on the ground glass for critical focusing, how many of us use the biggest format we can carry (since if we're talking ultimate sharpness we clearly don't care about DOF)? And ultimately, photography is about vision, not replication, and diffraction limited performance is important only to the extent that that helps express a vision. After all, some of the strongest visual images were done using a paintbrush and palette knife.

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Earlier in this thread: "... you still haven't explained why it is that light diffracts at an edge..."

 

Peter Blaise responds: ... and I don't really care! All we know is that is does, and so that is what we are trying to incorporate into our photographic decisions. IF light diffracts at an edge, THEN as the edge becomes a significant portion of our lens setting, as it does at smaller apertures, we should be prepared to expect the resulting diffraction to become a greater part of our captured image.

 

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Earlier n this thread: "...Think about light...as a wave...and..."boundary conditions"...naturally follow..." [edited]

 

Peter Blaise responds: Ahh, so we do agree after all! I imagine that most of this thread is just describing the same general understanding differently and THINKING that we disagree when in actuality, we agree, after all, essentially!

 

Earlier in this thread: "...in the photon description [of light]...information is lost..."

 

Peter Blaise responds: By whom?

 

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Peter Blaise wrote earlier in this thread [clarified and distilled]: "...I put it to you that...light does not know nor care about anything except that it...[has] hit an edge and diffracts accordingly...[and light is not in the least concerned with] our observation criteria..."

 

A response: "...I don't think you believe that..."

 

Peter Blaise responds: Light doesn't care what I believe.

 

Earlier in this thread: "...you don't believe in interference...[?]"

 

Peter Blaise responds: Those are OUR observations. The light itself could care less that we see it as if it were "putting on a show" for us, so to speak. You're not suggesting atom smashing, are you? I believe that atom smashing in particle accelerators is all about small particles actually "interfering" with each other -- having a good head-on crack up! That's not what most of us are trying to do with our photography. If you think the light inside interference effects "knows" that those interference effects are observable by you and I, then I ask again, how does light "know" these things? Have we wandered into a discussion of the meaning of consciousness?

 

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Peter Blaise asked earlier: "Does no one have personal experience and sample photos to share [illustrating diffraction effects]?

 

Earlier in this thread, perhaps as a response: "...some of the strongest visual images were done using a paintbrush and palette knife..."

 

Peter Blaise responds: So I guess that's a "no" to illustrative photographic examples of diffraction effects, eh?

 

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Peter Blaise further responds: Reading through my library I'm struck with the following:

 

K. G. Birch wrote in "The Focal Encyclopedia of Photography", 1969, page 445: "...Diffraction: bending of light rays around opaque objects, specially apparent if the opaque object has a sharp edge... Diffraction: when a wave front meets a knife edge, it generates a new wave front at the edge itself. This spreads round the obstacle, leading to some light being apparently bent round the edge..."; and on page 845: "...it will be seen from the optical transfer function of a perfect lens that all spatial frequencies above zero lines per millimetre the lens performance is impaired, a necessary condition imposed by diffraction at the edge of the lens opening..."

 

Peter Blaise responds: Don't ya just love the references to "wave" and "ray" and "edge" when discussing and defining diffraction?!? ;-) I do not offer this reference as proof, but merely as one more reference, encouraging us all to keep researching and expanding our resources.

 

Also from "The Focal Encyclopedia of Photography", 1969, page 34: "...Airy disc. Central area of the image of a point source of light focused by a lens free from aberrations. The Airy disc is surrounded by concentric rings of light of feebler intensity which can in practice be ignores. The effect is an optical phenomenon connected with the wave nature of light and is called after its discoverer, G. B. Airy, Astronomer Royal in 1830..."

 

Peter Blaise responds: Optical phenomenon? In other words, we see it, but the light itself does not "know" it's putting on a show for us, eh?

 

-----------

 

I wish I could share a diffraction-limited image. Anyone? Anyone?

 

Click!

 

Love and hugs,

 

Pete Blaise peterblaise@yahoo.com http://www.peterblaisephotography.com/

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In the event that you're interested in and are willing to learn about the physical aspects of light (as opposed to photographic, which encompasses primarily artistic and documentary aspects) let me suggest someday sitting down with a good optics book. Hecht and Zajac's <i>Optics</i> is an excellent introduction and recommended for any photographer interested in the physics of light. If you're interested in something more advanced than H&Z just ask, I've got a whole stack.

<p>

Re: diffraction limited images: all it takes is a little <a href=http://www.pinhole.org/gallery/index.cfm> looking </a> <a href=http://hubblesite.org> around</a>.

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