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

[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.)

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.



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]

(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

[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]

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]

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...

--Rik

[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