

FFTs and Optical Lenses - skierscott
http://scottsievert.github.io/blog/2014/05/27/fourier-transforms-and-optical-lenses/

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mgraczyk
Interesting topic, but one thing should be made clear for people without an
engineering/applied mathematics background.

The FFT is not the same as the fourier transform. Most of the time the
distinction is irrelevant, but in this case it truly does matter. The
transformed signal measured with the lens is the result of a sampled
continuous space fourier transform (CSFT). The FFT computed is the discrete
space fourier transform. The two are not always equivalent.

I bring this up because the author uses the terms "fourier transform" and
"FFT" interchangeably.

~~~
skierscott
You are correct; the FFT or DFT is not the same thing as the continuous-time
Fourier transform (CFT).

This technical report[1] examining the relation between the DFT and CFT shows
in eq. 20 that the DFT is just the CFT evaluated at w=k _2_ pi/(N*T). As N
becomes large (number of pixels is large), this approaches the CFT.

I left this detail out; I wanted to put this in terms the reader knew.

[1]:[http://bsp.pdx.edu/Reports/BSP-
TR0201.pdf](http://bsp.pdx.edu/Reports/BSP-TR0201.pdf)

~~~
panic
If you just left out one "F" from "FFT" it would be more precise and less
confusing. Is there any reason to bring the Fast Fourier Transform algorithm
up at all?

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frozenport
I work with optics, there is a lot wrong in this article. I would hold the
upvotes until these problems are fixed.

#1. This relationship only holds at focal points

#2. Nothing to do with Fast Fourier Transform

#3. No mention of complex part of FT

#4. No derivation of why this is the case

~~~
gsteinb88
Also, ignores that Fourier optics only holds for the paraxial approximation
(i.e. the small angle approximation). I'd put the rule of thumb somewhere
around a numerical aperture of 0.3 [0] -- beyond that, polarization effects
start to come into play, and beyond 0.5-0.6, it becomes an issue in, for
example, microscopy.

[0]
[http://en.wikipedia.org/wiki/Numerical_aperture](http://en.wikipedia.org/wiki/Numerical_aperture)
\-- Similar to f-number from photography, and describes how big the aperture
of the lens is in relation to its focal length. Specifically, it's the sine of
the angle from the optical axis to the ray extending from the focal point to
the edge of the lens, for a single lens system.

~~~
frozenport
>>around a numerical aperture of 0.3

From experience you are being overly conservative. Foremost polarization
effects depend on the rigidity of your surface and the bandwidth of your light
source. The correction is minimal. Most importantly oil immersion 1.4 NA is
common and nobody complains.

>>Also, ignores that Fourier optics

Although there is another good point. After making the born approximation you
get this kind of dish shape in the Fourier transform, but on the other hand
the dish is compensated on both sides of the lense

~~~
gsteinb88
Ah yeah, to be fair it's not a _huge_ correction, but in what I do (single-
mode confocal microscopy) it definitely becomes an issue around NA=0.3 when
trying to characterize spot sizes and collection efficiency functions. Namely,
the overlap integrals can change significantly if your source is also
polarized. Also, I'm probably more sensitive to these things than most since
I'm at the single-photon level usually.

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tanvach
Wow this brings back some memories! I spent a long time doing research in this
area. You can do very cool stuff by realizing that spatial light pattern
propagates as Fourier transform with a "spherical" term in the integral
(having a perfect convex lens at the focal point cancels this term, hence
Fourier transform. Also if laser propagates to sufficient large distance, the
term vanishes. A lensless, diffractive projector will always be in focused).

Note that you need a coherent, planar light source like an expanded laser
beam.

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quarterwave
One interesting point is why is the far-field amplitude of light diffracted by
the lens a spatial Fourier transform? The best explanation I know of is in
Chapter 21 of the Feynman Lectures on Physics (vol 2), accessible online
(thanks to CalTech) at
[http://www.feynmanlectures.caltech.edu/II_21.html#Ch21-S3](http://www.feynmanlectures.caltech.edu/II_21.html#Ch21-S3)

In short: Moving charges create electromagnetic radiation (light). Since light
travels at non-infinite speed, when a wiggle of light reaches us, we're
actually seeing the imprint of the motion of the source charges at an earlier
point in time. This earlier time is clearly related to how far the observer is
from the source. Hence, if the distance is r then at time t we would be seeing
the wiggle of the source charge at an earlier time = t - r/c. For a collection
of source charges we need to integrate the delayed source potential over the
volume of the source, so it's intuitively clear why that would be a Fourier
transform.

There's technicalities of far-field and 1/r fall-off, which I've glossed over,
you can find full details in Chapter 21. The final line is a gem: 'You will
not, then, be surprised to find that the laws of electricity and magnetism are
already correct for Einstein's relativity. We will not have to “fix them up,”
as we had to do for Newton's laws of mechanics.'

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chrisBob
My favorite example of using a lens for a FT is creating a dark field image.
You just put a small beam block in the center of a pupil plane and block all
of the low frequency components. You end up with bright lines only at the
edges of things which are the high frequency components of the image.

Also "google images" isn't an image source, and just because you found
something on the internet doesn't mean it is free to use.

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d136o
There are already several comments here from people who are knowledgeable on
the topic, so I might not be fully correct in the following:

I think x-ray crystallography applies similar techniques to try to work
backwards from an image of diffracted light to the crystallized molecular
structure that would have created the (transform) of light to yield the image.
For example, you have some compound, say DNA, and want to know its molecular
shape. I think one of the methods that Rosalind Franklin and company used was
to take crystallized DNA, shoot x-rays at it, and study the resulting
diffraction pattern(s) to determine that DNA had to be a helical structure
with atoms bonding at such and such angles. And if that's not clear enough I
hope it doesn't escape you to note that it immediately suggests a mechanism
for DNA replication hah.

That's hastily written and just tying random facts from undergrad so feel free
to correct/add/disprove at will I am sure there are some commenters who know
way more in much more detail. I do miss the full time learning days!

~~~
SixSigma
The took the photos but Francis Crick used LSD to help him imagine what the
shadow was they were looking at.

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aortega
Yes, FFT is an algorithm, a computer-optimized implementation of the Fourier
transform.

Speaking of optics, the famous slit-experiment, where you can see the
diffraction patterns of light passing through one or more slits, is the
fraunhoffer-pattern, the 2D fourier transform of the slit:
[http://physics.stackexchange.com/questions/94852/why-is-
the-...](http://physics.stackexchange.com/questions/94852/why-is-the-
fraunhoffer-pattern-the-fourier-transform-of-the-slit)

------
neltnerb
I think the most memorable thing I learned in quantum mechanics is that
quantum mechanical scattering is also exactly the fourier transform. Suddenly
braggs law became much more sensible.

~~~
frozenport
>>is also exactly the fourier transform

No, this is only under the 1st born approximation, in the far-field and
without the 'dish' shape intrinsic to propagation. A lot of times this model
doesn't work. Also now you are in the momentum space which doesn't map
immediately to the x,y,z coordinates you might have started with. Also the
evanescent fields that the Ewald sphere didn't include...

~~~
neltnerb
Hehe, well, thanks for the correction. Yes on all of these things, I'm sure.

I should have said "the exact solution for the most basic justifiable
approximation". Honestly, it's been 9 years since I last did quantum
seriously, so that was the extent that I remembered.

Deriving Bragg's law in front of my quals committee and refusing to use
"mirror planes" despite it being a Materials Science Ph.D. was just viscerally
satisfying. It may not have been absolutely correct math, but at least it was
way more justifiable than the typical accepted "solution" in my field =)

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jwise0
Ben Krasnow also did a good piece on this, showing a system that he built with
a 4F correlator. Of course, his piece is full of things that look like things
you could build at home if you had infinite resources on hand ... but it
really does drive home just how possible it is to build this system, and
provides a very intuitive view of how it works.

His piece is:
[https://www.youtube.com/watch?v=wcRB3TWIAXE](https://www.youtube.com/watch?v=wcRB3TWIAXE)

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ocfnash
A very interesting application of this point of view was a proposed projector
of Light Blue Optics. The projector would have the property that it was near
100% efficient: it would essentially steer the light to the desired locations
(rather than blocking it out where it was not desired).

The idea is outlined briefly here:
[http://redfrontdoor.org/blog/?p=409](http://redfrontdoor.org/blog/?p=409)

~~~
gjm11
(Mathematician at Light Blue Optics here.)

Unfortunately that near-100%-efficiency wasn't quite realised in practice, for
several reasons. I think I can describe some of them without giving away
anything commercially sensitive:

1\. The spatial light modulator we used didn't give anything like the 180
degrees of phase rotation outlined there, which means that a large fraction of
the light landing on it passed straight through.

2\. Having only two phase states leads, as mentioned in that page, to a
conjugate image with as much light in it as the one we actually want.

3\. All that random noise does indeed average out nicely when you have many
"subframes". But since there's no such thing as negative light, parts of the
image that are meant to be black will inevitably end up something other than
black. So there's a loss of contrast, which means that some of the light in
the image isn't really doing you much good.

4\. The optical design has a bunch of lenses and mirrors and things in it, and
every surface is an opportunity to lose a little bit of light.

The actual optical efficiency figure was, let's say, somewhat less than 100%.
(Also, for every frame we displayed we had to compute a lot of Fourier
transforms, and the compute hardware takes power too. Which wouldn't matter
for large mains-powered projectors, but is more of an issue when you're trying
to make a small low-power device for mobile use.)

We had some next-generation technology in the works that would (if brought to
completion) have fixed most of these issues and produced better efficiency
along with better image quality -- but then we made the (very sensible)
decision to get out of the picoprojection market completely.

~~~
ocfnash
Fascinating, I've wondered a few times over the past year or so why the ideas
were shelved.

Each of your points is very illuminating. It's such a nice idea that it's a
shame it turned out not to be worth pursuing from a business point of view.

Perhaps some day somebody will take up those next-generation ideas!

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Ono-Sendai
We use the FFT to simulate aperture diffraction in our computer-generated
images. Example:
[http://www.indigorenderer.com/sites/default/files/features_a...](http://www.indigorenderer.com/sites/default/files/features_aperture_diffraction.jpg)

