Can you reverse transform a thin film diffraction pattern?

[John_H]

Thank you for your insights on this.

Suppose our kitchen table has a double glass top. Here and there it produces Newton's rings type interference patterns.

I understand that by reverse transforming an interference pattern you can recover an image.

If I were to somehow do this (optically, or computationally) to one of the ring interference patterns on the tabletop, would I recover an image of the kitchen ceiling? Maybe the floor?

Another version of the same problem. Say I look over the side of a boat and see my face reflected in the water. Then, in refueling the outboard motor, say -- I spill a thin film of gasoline onto the surface of the pond. If I were to somehow FT that colorful interference pattern on the water, would I recover an image of my face, peering down into it?

John

 

[Danger]

That's one of the coolest questions I've seen in a while.
Keep in mind that I know nothing about this subject, but my gut impression is that you can't recover an image from an oil slick. At least, not a clear one. I'm basing that solely upon the fact that it's constantly in motion.

 

[John_H]

Here is one -- pattern.

I guess the problem could be cleaned up and simplified if we dispense with the water and use wet pavement, as in this photo. In this pattern, the surface is solid, all the light is arriving from the "upper world" and you get rid of the problem of light reflected from below the pond's surface. (Or below the glass kitchen tabletops). To idealize the problem further I guess you could smooth out the pavement.

The basic question still comes out about the same. Is there an image that could be extracted from this thin film interference pattern by performing a Fourier Transform? Or Reverse transform, convolution, whatever?

More directly, does a thin film type of interference pattern contain the information it takes to form an image?

Maybe a way to get at it is to try to figure out how, or if, thin film interference patterns differ from, say, double or multiple aperture interference patterns.

Thank you for your help on this. John

 

[Danger]

Your thanks is premature. I already gave it all that I've got. I don't even know what a 'Fourier Transform' is.

 

[John_H]

I collected some links that seem to pertain.

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/oilfilm.html#c1

http://www.reindeergraphics.com/index.php?option=com_content&task=view&id=212&Itemid=158

http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Properti-2.html

At the practical level, a Fourier Transform is best understood as something that you can do. You can do it to light, with a lens, almost instantaneously.

You can also do it with a computer by running a program.

It has endless of applications in engineering, but one of the things you can do with it is transform an image into the "frequency domain" -- visually an interference pattern. And then, if you want, transform it back again, into a literal image.

Fourier declared in the early 19th century that he could describe any periodic function as a series, or summation, of sine and cosine terms, each term modified with difference constants. There are some nice applets on the net that illustrate this process. Running a Fourier Transform consists performing an integration derived from the Fourier series.

The transform is readily reversible. If you change a sign in the exponent and run it again, you can flip back and forth between an image and an interference pattern.

Anyway, that is my understanding of it so far -- no guarantees but very hard won lore. Books I have been able to find on the subject of Fourier optics are essentially opaque. It is a like learning a foreign language from scratch. There is no step one.

On the other hand, it is fascinating to see the outcome. I especially like that Reindeer graphics work.

John

 

[Integral]

I collected some links that seem to pertain.

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/oilfilm.html#c1

http://www.reindeergraphics.com/index.php?option=com_content&task=view&id=212&Itemid=158

http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Properti-2.html

At the practical level, a Fourier Transform is best understood as something that you can do. You can do it to light, with a lens, almost instantaneously.

Yes, a lens can be seen as a form of a analog computer, it transforms a interference pattern at the focal plane into a image. A hologram is simply a "picture" of an interfernce pattern. A laser, enables the lens in your eye to perform the transform.

You can also do it with a computer by running a program.

Only for very simple interference patterns.

It has endless of applications in engineering, but one of the things you can do with it is transform an image into the "frequency domain" -- visually an interference pattern. And then, if you want, transform it back again, into a literal image.

Fourier declared in the early 19th century that he could describe any periodic function as a series, or summation, of sine and cosine terms, each term modified with difference constants. There are some nice applets on the net that illustrate this process. Running a Fourier Transform consists performing an integration derived from the Fourier series.

The transform is readily reversible. If you change a sign in the exponent and run it again, you can flip back and forth between an image and an interference pattern.

I think you have missed something, or I am not understanding you. Generally there is a bit more to an inverse transform then a sign change.

Anyway, that is my understanding of it so far -- no guarantees but very hard won lore. Books I have been able to find on the subject of Fourier optics are essentially opaque. It is a like learning a foreign language from scratch. There is no step one.

On the other hand, it is fascinating to see the outcome. I especially like that Reindeer graphics work.

John

There are lots of good presentations of Fourier Transforms, I guess if you do not have the appropriate math background they would be pretty hard to follow. You need a good understanging of infinite series and Integral calculus.

I do not believe that you would be able to recover images from your table top with these methods. What you would recover from the interference pattern is information concerning the surfaces from which it was created. That is if you could capture the correct information and do an inverse transform, you would learn that the pattern was created by parallel surfaces with a separtion the thickness of your glass plate.

Generally interference patterns containing real image information are way to complex to be amenable to computer analysis.

 

[John_H]

forward and reverse transforms, general form

Thank you for your reply. To clarify the forward and reverse tranform procedures, here is an excerpt from Wikipedia's treatment.

View attachment wiki forward and reverse.doc