Re: Diffraction, Fourier and Focal Planes

AiryDiscus wrote:

Jack Hogan wrote:

AiryDiscus wrote:

Jack Hogan wrote:

AiryDiscus wrote:

Jack Hogan wrote:

AiryDiscus wrote:

Jack Hogan wrote:

Enough wasteful diversions, back to substance. Where the rubber meets the road with regards to this sub-thread is the skill-testing question that was answered incorrectly below:

In the meantime I have a skill-testing question for you or anyone interested:

Let's say you have such an exit pupil and related function, a lens of focal length f and f-number N, focused at infinity. You use Fourier Optics to propagate the field at the exit pupil to the focal plane. How far precisely is that from the exit pupil? From the principal plane?

In the case that you wish to invoke the Fourier transforming property of a lens and use what I would call a focusing propagation, simply taking an FT, then the distance is one focal length.

Goodman would beg to differ. Vectors are once again being placed wantonly and without thought, just like some of the others I lamented about earlier , QED. Where is focal length measured from?

Here's some quotes from Goodman, on the section "Fourier transforming properties of lenses:"

Case of field 'at' the lens:

To find the distribution Uf(u, v ) in the back focal plane of the lens, the Fresnel diffraction formula, Eq. (4-17), is applied. Thus, putting z = f ,[...]

Case of field 'in front of' the lens:

Thus the amplitude and phase of the light at coordinates (u,v)are again related to the amplitude and phase of the input spectrum at frequencies (u/lambdaf ,v/lambdaf). Note that a quadratic phase factor again precedes the transform integral, but that it vanishes for the very special case d = f . Evidently when the input is placed in the front focal plane of the lens, the phase curvature disappears, leaving an exact Fourier transform relation!

So where is focal length measured from?

I suggest you to look in the other parts of the appendix of that book you have. If you want a different book, Welford is excellent.

'measure' implies a certain amount of rendering to reality. The focal length is the distance from the rear principle plane to the image plane. The language of "plane" is a bit casual, but these are all paraxial quantities anyway.

Here's a toy example that might help you in your crusade, which I'm sure likely exists in the good book.

A stop, one focal length from a thin lens of focal length and an object at infinity. The exit pupil is at -inf, and the object at -f.

There are several truthy priors, like the size of the airy disk, or where the image is in relation to the lens.

There is, again, no work to show (this is the fourier transforming property of a lens), unless you would like me to recreate proofs that have been known for what is now a very long time.

I encourage you to read the whole book, and not skim it around formulas you want to lift without context. Or take a course in the topic.

Prompt: using the variables at hand and any others that may be needed,

zi = f + ...

No definition of variables, vague open ended prompt. No thanks.

You really do not understand what I am asking, do you? What's the difference between z1 and zi?

Jack

You haven't defined anything What is even z1 an zi? Where is z0?

You have the same book I have. But let me help you: In Goodman z1 is measured from the principal plane, zi from the exit pupil.

So where is focal length measured from?

Actually, I have the fourth edition signed by him.

I gave you the definition of focal length, which you ignored.

So we are in agreement, focal length is measured from the second principal point, while Fourier Optic propagation is measured from the exit pupil, which, perusing Bill's tool, are in practice never exactly coincident. In fact there can be quite a difference, with consequent implications for key photographic figures like f-number and the distance of the imaging plane in Fourier Optics.

Now that you understand what I was saying, check how people in the know draw such diagrams or try to square them with f/D and you will understand my original comment about confusing terminology and wishing that folks be more precise and consistent in applying it.

Jack