I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.

. The proof is on about page 50. I have not yet read it, and a few pages
preceding it are deliberately omitted. $145 for the book. I won't be buying

W. Watson writes:

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.

The first thing to appreciate, if one hasn't already, is that Fourier
analysis can be applied to the frequency, time, and/or spatial
characteristics of light. The underlying math is the same, but the
physical application is astonishingly different among these cases.

For example, one might digitally transform the spatial sampling of an image
to analyze for patterns, the frequency sampling of a beam through a lens to
measure aberration, and the time sampling through a fiber to measure pulse
delay and spreading over a distance.

In my distant past experience I dealt with Fourier series in very good
detail, and transforms as a mathematical subject, but never anything in
optics. However, I'm aware of their use in imaging with CCDs and astronomy.

Shortly after posting this, I came across a helpful source on the web
that hits right on my simple example of a lens.
http://sharp.bu.edu/~slehar/fourier/fourier.html>. What I found
interesting there is his statement that the lens actually performs the
transformation. The relationship is not obvious to me. In fact, I find
that amazing. I decided to ask good ole Google for proof. I was
surprised again to find a proof in a Google the book Elements of
Photonics at
http://books.google.com/books?id=CRfDHdahgz0C&pg=PA50&lpg=PA50&dq=lens+performs+a+fourier+tranform&source=web&ots=ktwy2709Cw&sig=eDje2xHKZvxw0NQOQmfhnsVDvkg&hl=en#PPA49,M1

. The proof is on about page 50. I have not yet read it, and a few
pages preceding it are deliberately omitted. $145 for the book. I won't
be buying it soon. Nevertheless, I plan to eek out whatever I can from
it. Another thing I find quite surprising is that none of the sources
I've looked at seemed to stress this lens ability to perform a
transform. Of course, I haven't been reading every word of material I've
examined.


Richard J Kinch wrote:
W. Watson writes:

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.

The first thing to appreciate, if one hasn't already, is that Fourier
analysis can be applied to the frequency, time, and/or spatial
characteristics of light. The underlying math is the same, but the
physical application is astonishingly different among these cases.

For example, one might digitally transform the spatial sampling of an
image to analyze for patterns, the frequency sampling of a beam
through a lens to measure aberration, and the time sampling through a
fiber to measure pulse delay and spreading over a distance.

W. Watson writes:

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.

The first thing to appreciate, if one hasn't already, is that Fourier
analysis can be applied to the frequency, time, and/or spatial
characteristics of light. The underlying math is the same, but the
physical application is astonishingly different among these cases.

For example, one might digitally transform the spatial sampling of an image
to analyze for patterns, the frequency sampling of a beam through a lens to
measure aberration, and the time sampling through a fiber to measure pulse
delay and spreading over a distance.

In article <Xns9A66B2E754804someconundrum@216.196.97.131>,
Richard J Kinch <kinch@truetex.com> wrote:

W. Watson writes:

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.
The first thing to appreciate, if one hasn't already, is that Fourier
analysis can be applied to the frequency, time, and/or spatial
characteristics of light. The underlying math is the same, but the
physical application is astonishingly different among these cases.

For example, one might digitally transform the spatial sampling of an image
to analyze for patterns, the frequency sampling of a beam through a lens to
measure aberration, and the time sampling through a fiber to measure pulse
delay and spreading over a distance.

The diffraction integral describing Fraunhofer diffraction is in the
form of a Fourier integral. For Fresnel diffraction, the same is truer
except that there is an quadratic exponential as a multiplier.

Thus, when someone was smart enough to notice that, the various
transform properties could be used to advantage.

Bill

W. Watson wrote:
In my distant past experience I dealt with Fourier series in very good
detail, and transforms as a mathematical subject, but never anything
in optics. However, I'm aware of their use in imaging with CCDs and
astronomy.

Shortly after posting this, I came across a helpful source on the web
that hits right on my simple example of a lens.
http://sharp.bu.edu/~slehar/fourier/fourier.html>. What I found
interesting there is his statement that the lens actually performs the
transformation. The relationship is not obvious to me. In fact, I find
that amazing. I decided to ask good ole Google for proof. I was
surprised again to find a proof in a Google the book Elements of
Photonics at
http://books.google.com/books?id=CRfDHdahgz0C&pg=PA50&lpg=PA50&dq=lens+performs+a+fourier+tranform&source=web&ots=ktwy2709Cw&sig=eDje2xHKZvxw0NQOQmfhnsVDvkg&hl=en#PPA49,M1

. The proof is on about page 50. I have not yet read it, and a few
pages preceding it are deliberately omitted. $145 for the book. I
won't be buying it soon. Nevertheless, I plan to eek out whatever I
can from it. Another thing I find quite surprising is that none of the
sources I've looked at seemed to stress this lens ability to perform a
transform. Of course, I haven't been reading every word of material
I've examined.


Richard J Kinch wrote:
W. Watson writes:

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.

The first thing to appreciate, if one hasn't already, is that Fourier
analysis can be applied to the frequency, time, and/or spatial
characteristics of light. The underlying math is the same, but the
physical application is astonishingly different among these cases.

For example, one might digitally transform the spatial sampling of an
image to analyze for patterns, the frequency sampling of a beam
through a lens to measure aberration, and the time sampling through a
fiber to measure pulse delay and spreading over a distance.


There are two similar but not identical senses in which a lens performs
a Fourier transform.

The first and easiest is that light coming in at an angle theta from the
axis of a lens of focal length f is all focused down to a point at
x=f*sin(theta). For monochromatic light of wavelength lambda, that
light has a radian spatial frequency (= transverse k vector) of
k_perp=2*pi*sin(theta)/lambda, so the lens is collecting all the light
with this spatial frequency and focusing it down to a single point, at a
position offset proportional to the spatial frequency. This is
precisely a Fourier transform operation, and apart from some spread in
the focused spot (due to finite aperture, finite wavelength, and
aberrations), and some nonlinearity of the frequency->position mapping
(due to geometrical distortion) the lens does it perfectly.

The second sense is that an object whose reflectance or transmittance is
sufficiently slowly varying, and whose surface relief is sufficiently
small, when illuminated with a plane wave, produces a far field pattern
(or angular spectrum) that is the Fourier transform of the object's
complex reflectance or transmittance function. This is an approximate
relation, and is the main approximation we make in Fourier optics.

Once the light has left the sample surface, its angular spectrum is what
it is, and the lens does a Fourier transform of that just as in the
first case.

Cheers,

Phil Hobbs

In article <Xns9A66B2E754804someconundrum@216.196.97.131>,
Richard J Kinch <kinch@truetex.com> wrote:

W. Watson writes:

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.
The first thing to appreciate, if one hasn't already, is that Fourier
analysis can be applied to the frequency, time, and/or spatial
characteristics of light. The underlying math is the same, but the
physical application is astonishingly different among these cases.

For example, one might digitally transform the spatial sampling of an image
to analyze for patterns, the frequency sampling of a beam through a lens to
measure aberration, and the time sampling through a fiber to measure pulse
delay and spreading over a distance.

The diffraction integral describing Fraunhofer diffraction is in the
form of a Fourier integral. For Fresnel diffraction, the same is truer
except that there is an quadratic exponential as a multiplier.

Thus, when someone was smart enough to notice that, the various
transform properties could be used to advantage.

Bill

The diffraction integral describing Fraunhofer diffraction is in the
form of a Fourier integral. For Fresnel diffraction, the same is true
except that there is an quadratic exponential as a multiplier.
Thus, when someone was smart enough to notice that, the various
transform properties could be used to advantage.

Who was the someone, historically? Probably not Fourier.

There's a seductive pedagogical neatness about that derivation, but I
think it's bad all through. Adding a paraxial lens (i.e. a parabolic
phase function), you can easily show that when d_i=f, the lens cancels the
quadratic phase terms in the Fresnel diffraction formula, leaving a pure
Fourier transform at the focus. Slick as can be--which is why it's been
used by generations of optics lecturers.

Unfortunately, that has left generations of students with the unshakable
conviction that Fourier optics is a strictly paraxial approach. This is
not true--Fourier optics works fine even at NA=1, if allowance is made for
polarization effects. Fourier optics is _so_ powerful and _so_ intuitive
that this loss of confidence is a great shame.

Who was the someone, historically? Probably not Fourier.

I've looked over some optics texts that use FTs to solve problems.
With my very basic understanding of the physics behind optics, it's
something of a puzzle how they do get used.
[snip]

"Phil Hobbs" wrote

There's a seductive pedagogical neatness about that derivation, but I
think it's bad all through. Adding a paraxial lens (i.e. a parabolic
phase function), you can easily show that when d_i=f, the lens cancels the
quadratic phase terms in the Fresnel diffraction formula, leaving a pure
Fourier transform at the focus. Slick as can be--which is why it's been
used by generations of optics lecturers.

Unfortunately, that has left generations of students with the unshakable
conviction that Fourier optics is a strictly paraxial approach. This is
not true--Fourier optics works fine even at NA=1, if allowance is made for
polarization effects. Fourier optics is _so_ powerful and _so_ intuitive
that this loss of confidence is a great shame.

Any reference to (a) Fourier Optics textbook(s) hich would go beyond the
paraxial domain (high NA) and include polarization effecs?

Thanks.


I don't know of one, unfortunately. I'm pretty sure there wasn't one

You guys might be interested in the fact that you can also do lossless
*discrete* Fourier transforms using *fiber optics* -- that is, you can
do ideal lossless DFTs using an array of single-mode optical fibers and
3 dB fiber couplers.

[Historical note: I felt quite proud of myself when I first guessed at
this idea; confirmed it analytically; then inquired of Joe Goodman, Jack
Gaskill, and a bunch of other Fourier and fiber optics gurus and
discovered that none of them had ever heard of it; and so published it a
couple of months before my 70th birthday:

[2] A. E. Siegman, "Fiber Fourier optics," Opt. Lett., vol. 26, pp.
1215--1217, 15 August 2001.

Abstract: The Fourier transform of a coherent optical image can be
evaluated physically by use of a single lens plus free-space
propagation, thereby providing the basis for the field of Fourier
optics. I point out that one can similarly evaluate the discrete Fourier
transform of a sampled or pixelated optical array physically by passing
the discrete array amplitudes through a network of single-mode fibers or
optical waveguides. A passive optical network that evaluates the fast
Fourier transform of a coherent array can be fabricated by use of
(N/2)log(2)[N] optical 3-dB couplers plus small added phase shifts.
Implementing such networks in fiber or integrated optical form could
provide the basis for a possible technology of fiber Fourier optics.

The problem was, I soon thereafter got a friendly note from Michel
Marhic pointing out that he had published essentially the same idea
several years earlier -- in the same journal -- and by the way he was
currently residing in an adjoining building at Stanford:

[1] M. E. Marhic, "Discrete Fourier transforms by single-mode star
networks," Opt. Lett., vol. 12, pp. 63--65, January 1987.

Notes: This clearly anticipates my 2001 Optics Letter on the same
subject.

Only compensation was that I think my presentation was maybe a bit
clearer and more general. So far as I know, this concept of "Fiber
Fourier Optics" has yet to go anywhere in practical applications.]