Making scale bars with no calculations (OT-->diffraction)

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rjlittlefield
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Post by rjlittlefield »

"Interact?"

Um, yeah, "interact". Superposition is a special kind of interaction from which the original waves eventually emerge unchanged. At least that's the way articles in the SIAM Journal talk about it. (SIAM = Society of Industrial and Applied Mathematics.) That model gives a nice transition to what happens with nonlinear media, in which the original waves may not emerge unchanged. More of those pesky "background and terminology" issues.

Anyway, you wrote:
It really is difficult to explain these sorts of issues when peoples background is limited, the way in which things tend to be explained is also overly complicated and impenetrable with maths thrown in for good measure. On the other hand it is very easy to enter into empty explanations, the kind of stuff that the TV and school is very good at churning out, they use terminology to replace real explanation. You can find yourself without enough information to truly understand, having instead to just take it for granted. I even find myself unable to understand explanations for things I already understand.
Amen to that!

So let's see if we can do better...

This has turned into a great discussion.

I think we're all on the same page about diffraction. That model using superposition of many circular waves is equivalent to what my grid cell simulation does. Information spreads in all directions all the time, in accordance with a 2nd order differential equation, and it's a bizarre feature of the way those zillions of contributions add up that causes waves to propagate. I am always fascinated that such interesting behavior comes from a calculation that just says (in its entirety):
Image
But we're not really here to discuss diffraction, we're here to discuss "why" images get fuzzy when you stop down too far. I put the "why" in scare quotes to emphasize that we're talking about models here. Clearly nothing more than the differential equation is required to predict the result, but, um, that doesn't mean it's the best way to think about it. :roll:

The more I think about the "spatial filtering" model, the more I like the feel of it. I'm still not very satisfied with those particular words, since both "spatial" and "filtering" mean something different to most photographers than they do in this conversation. But I think one can do a lot with a little using the general concept that the ability of waves to represent detail depends on their angle of incidence.

Something like this, perhaps...

"Light is a wave. If you look close enough, a beam of light has peaks and valleys. Visual detail in the subject is carried to the camera's sensor in the pattern of those peaks and valleys. The peaks and valleys in the light beam are all the same distance apart, but when the light beam hits the sensor, the distance between the peaks and valleys on the sensor depends on the angle of the beam. When the beam strikes the sensor at a fairly sharp angle, the peaks and valleys are close together, and this allows fine detail to be captured. But as the beam becomes more perpendicular to the sensor, the peaks and valleys move farther apart and some of their ability to represent fine detail is lost. If the beam is too close to perpendicular, then the peaks and valleys move too far apart to represent even the detail you want to see, and the image begins to look fuzzy. Stopping down the aperture too far blocks the light at steep angles that can form sharp images, leaving only the light that produces fuzzy and fuzzier images as it becomes more and more perpendicular."

ImageImage

I like the flavor of this explanation. It relies just enough on waves to get the idea across, without getting bogged down in superposition and tiny circular angels dancing on the head of a wavefront. (Sorry, a bit of passing whimsy there... :roll: )

No doubt this can be improved upon. Have at it!

BTW, I chose to say "beam" for a definite reason. I think that most photographers understand "ray" to be something with no width. But the light has to have width for these pictures to make sense. Just trying to think about it from the consumer's standpoint... :D

--Rik

Graham Stabler
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Post by Graham Stabler »

With water I suppose interact would not be so bad, as you might think of one wave lifting another but unless you believe in the ether that works less well for light. As I said, pedantry and certainly not as bad as saying the light interacts with the aperture.

I like your animation but at the same time I would prefer it if it included two beams interfering as one will not create anything for the sensor to see unless it can respond at the frequency of light. But it does give a flavor of the concept.

What are you programming in?

Cheers,

Graham

rjlittlefield
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Post by rjlittlefield »

With water I suppose interact would not be so bad, as you might think of one wave lifting another but unless you believe in the ether that works less well for light. As I said, pedantry and certainly not as bad as saying the light interacts with the aperture.
Hmm! I don't have any trouble saying/hearing that light interacts with the aperture! If it doesn't, then why does sticking in the aperture cause funny stuff to happen? (Google, "define: interact", To act together or affect each other. The shape of the wavefront is changed by size of the aperture, and the aperture -- or at least the stop that forms the aperture -- experiences a small force and even gets warm if the light is bright enough. Sure sounds like an interaction to me.)
Graham Stabler wrote:What are you programming in?
The grid-cell simulation is in Java. Java has a reputation for slow execution, but in fact it runs almost as fast as C/C++ when programmed correctly. This simulation runs at over 150 megaflops on my 4 year old Pentium 4 home machine (2.8 GHz). I've clocked an eigensolver on my office laptop at over 250 megaflops. This is all on one cpu.

The animated image in my previous post is just a sketch, hacked together in Photoshop.

"Flavor" is exactly what I was going for in that sketch.

It's hard to know what to draw to get the message across.

I played around this morning with interfering waves. Most of the pictures ended up more confusing than helpful. (Remember Moiré patterns?)

Here's the best I could come up with. One wave perpendicular to the sensor, interfering with a second wave at varying angle.

Image

I don't know what I think about these.

It's easy for me to imagine a newbie understanding that the distance between peaks and valleys in a single wave intersecting a plane will depend on angle. Maybe it's also easy for the newbie to understand that the distance between peaks and valleys in the interference pattern will depend on angle too, but it's not so easy to understand exactly how. Notice, for example, that the lines of nulls in the interference pattern do not have the same angle as either of the two interfering beams.

Of course (I say after considerable thought) there's a simple explanation using symmetry for why that has to be -- the axis of the interference pattern has to be midway between the axes of the beams. But then why is the distance between peaks and valleys of the interference pattern just the same as the distance between peaks and valleys of the slanted wave? (Yeah, I know, there's a simple explanation for that too. But these questions just keep coming...)

BTW, these latter illustrations are coming pretty much straight out of Java -- this time a code that just computes a bunch of phases and cosines and adds 'em up. The bottom part ("what the sensor sees") does come from a bit of Photoshop work so that I could manually choose the best y-location to pick up the interference pattern from. I could have done that computationally, but to start with it was easier to just do it visually.

--Rik

Graham Stabler
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Post by Graham Stabler »

rjlittlefield wrote: (Google, "define: interact", To act together or affect each other. The shape of the wavefront is changed by size of the aperture, and the aperture -- or at least the stop that forms the aperture -- experiences a small force and even gets warm if the light is bright enough. Sure sounds like an interaction to me.)
I did. The light that passes through the aperture doesn't interact with it and that would seem the most important light to consider. The reason it diffracts is because it is made finite (or at least more finite) in extent. Although in practice any beam of light is diffracting, even from the best HeNe, unless you have an infinitely wide one :)

I think your new diagrams would be more easy to understand with a greater wavelength (to better represent the gray scales). You can print these things on acetate and turn them by hand to see the effect. It is best if they are at the same incident angle but from differing directions, that way the interference will be in the line of the plane.

Graham

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Post by rjlittlefield »

Graham,

In the spirit of educational animations, you might be interested in this SIAM article titled "Prize-winning Video Brings Möbius Transformations to Life".

Other articles about this work can be found by Googling for appropriate terms. One good one from the authors' institution is here.

It notes that the video, "Moebius transformations revealed" can be viewed online at YouTube.

As of this moment, the view count is 1,302,394 and climbing.

Be sure to turn up the sound. There's no narration, but the background music adds greatly to the pleasure of watching this beautiful piece of work.

--Rik

Graham Stabler
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Post by Graham Stabler »

That's very neat indeed!

Though I often create animations of sorts with matlab it's quite a different thing to make it look so beautifull.

Graham

rjlittlefield
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Post by rjlittlefield »

It is beautiful. And it's effective. And yet, I just realized that it fails to really explain a key part of the theorem that it illustrates.

The theorem in question states that "All Möbius transformations can be obtained through inverse stereographic projection, followed by a rigid motion of the sphere of projection, followed by forward stereographic projection back to the plane."

But nowhere in the video do the words "stereographic" or "inverse" appear!

Well, the concept of "stereographic" simply corresponds to having that "light at the top" of the sphere, as the video puts it. And "inverse" simply means that the pattern initially painted on the sphere is whatever it needs to be for the stereographic projection to create a square.

Perhaps these aspects are completely obvious to the animation's originally intended audience. But I'm pretty sure they would be lost on most people picked at random -- at least if my own experience in teaching about projections in college geometry is any guide.

So while the video is both beautiful and illustrative, it's still far from being a standalone explanation. There is yet room for improvement... :D

--Rik

Graham Stabler
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Post by Graham Stabler »

You can see by the initial simple projection that the pattern is what is needs to be, it's self evident. Well it was to me and this is the first time I have ever heard of a Möbius transform. The point is that the flat plane can be created as a projection of a point source through a suitably patterned sphere an it shows it beautifully.

Is this thread now VOT :)

Graham

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