Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.

Kyle wrote:
Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.

Fermat's principle works fine for plane waves of any wavelength--you can
derive the law of reflection and Snell's law can be derived from it.
You need to be able to define what you mean by "the path that the light
takes", which is a geometric-optics concept, but for a plane wave in an
isotropic medium, the k vector works.  As long as the variations of n
happen on a scale >> lambda, Fermat's principle works fine for things
like schlieren as well, but otherwise you're right, it's basically valid
in the limit k->infinity.

There are other variational principles in optics that work for waves--my
thesis advisor was a big fan of them, back in the day.

Cheers,

Phil Hobbs

On Jul 10, 11:22 am, Phil Hobbs
pcdhSpamMeSensel... at (no spam) electrooptical.net> wrote:
Kyle wrote:
Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.
Fermat's principle works fine for plane waves of any wavelength--you can
derive the law of reflection and Snell's law can be derived from it.
You need to be able to define what you mean by "the path that the light
takes", which is a geometric-optics concept, but for a plane wave in an
isotropic medium, the k vector works. As long as the variations of n
happen on a scale >> lambda, Fermat's principle works fine for things
like schlieren as well, but otherwise you're right, it's basically valid
in the limit k->infinity.

There are other variational principles in optics that work for waves--my
thesis advisor was a big fan of them, back in the day.

Cheers,

Phil Hobbs

In the late 1940's Toraldo di Francia developed what he called
parageometrical optics. He proved that under some reasonable
assumptions the various diffracted orders follow both the principles
of Fermat and of Malus-Dupin. For reasons I do not understand this
theory never really got much support by other researcher. Can somebody
can explain why?

This is a quite accessible publication of his describing this theory
but he also wrote a book on EM:

Toraldo di Francia: Parageometrical Optics
JOSA , vol. 40, No.9, Sept 1950, pp600-602

Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.

rge11x wrote:
On Jul 10, 11:22 am, Phil Hobbs
pcdhSpamMeSensel... at (no spam) electrooptical.net> wrote:
Kyle wrote:
Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.
Fermat's principle works fine for plane waves of any wavelength--you can
derive the law of reflection and Snell's law can be derived from it.
You need to be able to define what you mean by "the path that the light
takes", which is a geometric-optics concept, but for a plane wave in an
isotropic medium, the k vector works.  As long as the variations of n
happen on a scale >> lambda, Fermat's principle works fine for things
like schlieren as well, but otherwise you're right, it's basically valid
in the limit k->infinity.

There are other variational principles in optics that work for waves--my
thesis advisor was a big fan of them, back in the day.

Cheers,

Phil Hobbs

In the late 1940's Toraldo di Francia developed what he called
parageometrical optics. He proved that under some reasonable
assumptions the various diffracted orders follow both the principles
of Fermat and of Malus-Dupin. For reasons I do not understand this
theory never really got much support by other researcher. Can somebody
can explain why?

This is a quite accessible publication of his describing this theory
but he also wrote a book on EM:

Toraldo di Francia: Parageometrical Optics
JOSA , vol. 40, No.9, Sept 1950, pp600-602

I'm not sure, having never read the paper, but from the sound of it, I
sort of suspect that it was superseded by the geometrical theory of
diffraction, which works for general objects and doesn't need to
separate the scattered light into orders.

Cheers,

Phil Hobbs

In article
6dd7a1db-3668-4daa-9fac-8fa134bdd38f at (no spam) f36g2000hsa.googlegroups.com>,
Kyle <kyle.m.douglass at (no spam) gmail.com> wrote:

Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.

The most accessible extension of Fermat's principle is in Feynman's "sum
over all possible paths" approach to the double slit experiment and
quantum mechanics. This method is not limited to a pair of slits.

Bill

On Jul 11, 6:31 pm, Phil Hobbs <pcdhSpamMeSensel... at (no spam) pergamos.net
wrote:
rge11x wrote:
On Jul 10, 11:22 am, Phil Hobbs
pcdhSpamMeSensel... at (no spam) electrooptical.net> wrote:
Kyle wrote:
Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity.
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.
Fermat's principle works fine for plane waves of any wavelength--you can
derive the law of reflection and Snell's law can be derived from it.
You need to be able to define what you mean by "the path that the light
takes", which is a geometric-optics concept, but for a plane wave in an
isotropic medium, the k vector works. As long as the variations of n
happen on a scale >> lambda, Fermat's principle works fine for things
like schlieren as well, but otherwise you're right, it's basically valid
in the limit k->infinity.
There are other variational principles in optics that work for waves--my
thesis advisor was a big fan of them, back in the day.
Cheers,
Phil Hobbs
In the late 1940's Toraldo di Francia developed what he called
parageometrical optics. He proved that under some reasonable
assumptions the various diffracted orders follow both the principles
of Fermat and of Malus-Dupin. For reasons I do not understand this
theory never really got much support by other researcher. Can somebody
can explain why?
This is a quite accessible publication of his describing this theory
but he also wrote a book on EM:
Toraldo di Francia: Parageometrical Optics
JOSA , vol. 40, No.9, Sept 1950, pp600-602
I'm not sure, having never read the paper, but from the sound of it, I
sort of suspect that it was superseded by the geometrical theory of
diffraction, which works for general objects and doesn't need to
separate the scattered light into orders.

Cheers,

Phil Hobbs

I think you are right, and that is what happened: GTD has taken over,
and parageometrical optics is by now completely forgotten judging it
being never mentioned in any textbook. I think that is unfortunate
because it seems to me very intuitive and simple. Of course, to
calculate the scattering off a complex body it would not be useful,
but di Francia used to it design microwave antennas and lenses with
scanning feeds. For example, using this technique Ronchi and di
Francia designed a wide angle microwave antenna with excellent
sidelobes.

Ronchi and di Francia:An Application of Parageometrical Optics to the
Design of a Microwave Mirror
IRE Trans. Ant. Prop. Jan 1958, pp129-133

Their technique was also used to design radar antenna:

Provencher: Experimental Study of a Diffraction Reflector
IRE Trans. Ant. Prop. May 1960, pp331-336

rge11x wrote:
On Jul 11, 6:31 pm, Phil Hobbs <pcdhSpamMeSensel... at (no spam) pergamos.net
wrote:
rge11x wrote:
On Jul 10, 11:22 am, Phil Hobbs
pcdhSpamMeSensel... at (no spam) electrooptical.net> wrote:
Kyle wrote:
Is Fermat's principle, the principle that states that the path that
light takes is an extremal, valid under the assumption that the
wavelength of light can't be ignored in a system, i.e. when
diffraction effects must be considered? My initial thought is no, but
that would seem like a very versatile and and useful theory would
cease to be of any importance outside of a limited area of validity..
Does it still retain any usefulness when a finite wavelength is
considered? Thanks for the input.
Fermat's principle works fine for plane waves of any wavelength--you can
derive the law of reflection and Snell's law can be derived from it.
You need to be able to define what you mean by "the path that the light
takes", which is a geometric-optics concept, but for a plane wave in an
isotropic medium, the k vector works.  As long as the variations of n
happen on a scale >> lambda, Fermat's principle works fine for things
like schlieren as well, but otherwise you're right, it's basically valid
in the limit k->infinity.
There are other variational principles in optics that work for waves--my
thesis advisor was a big fan of them, back in the day.
Cheers,
Phil Hobbs
In the late 1940's Toraldo di Francia developed what he called
parageometrical optics. He proved that under some reasonable
assumptions the various diffracted orders follow both the principles
of Fermat and of Malus-Dupin. For reasons I do not understand this
theory never really got much support by other researcher. Can somebody
can explain why?
This is a quite accessible publication of his describing this theory
but he also wrote a book on EM:
Toraldo di Francia: Parageometrical Optics
JOSA , vol. 40, No.9, Sept 1950, pp600-602
I'm not sure, having never read the paper, but from the sound of it, I
sort of suspect that it was superseded by the geometrical theory of
diffraction, which works for general objects and doesn't need to
separate the scattered light into orders.

Cheers,

Phil Hobbs

I think you are right, and that is what happened: GTD has taken over,
and parageometrical optics is by now completely forgotten judging it
being never mentioned in any textbook. I think that is unfortunate
because it seems to me very intuitive and simple. Of course, to
calculate the scattering off a complex body it would not be useful,
but di Francia used to it design microwave antennas and lenses with
scanning feeds. For example, using this technique Ronchi and di
Francia designed a wide angle microwave antenna with excellent
sidelobes.

Ronchi and di Francia:An Application of Parageometrical Optics to the
Design of a Microwave Mirror
IRE Trans. Ant. Prop. Jan 1958, pp129-133

Their technique was also used to design radar antenna:

Provencher: Experimental Study of a Diffraction Reflector
IRE Trans. Ant. Prop. May 1960, pp331-336

For antenna design, there's no serious competitor to the method of
moments, which turns a huge 3D problem into a very manageable 2-D
problem on the surfaces of the metal elements.  You wouldn't use GTD for
that, except possibly for things like diffraction at the edges of
supports and large reflectors.

Cheers,

Phil Hobbs

The most accessible extension of Fermat's principle is in Feynman's "sum
over all possible paths" approach to the double slit experiment and
quantum mechanics. This method is not limited to a pair of slits.

Bill

That's really Huyghens' principle, though...Feynman was a brilliant
showman as well as an excellent physicist, of course, and has had
everyone shaking their heads in wonder over that path stuff for years,
even though the basic idea was 300 years old.

Fermat's principle is purely geometrical.
'

As long as the variations of n happen on a scale >> lambda, Fermat's principle works fine for things like schlieren as well...

On Jul 10, 11:22 am, Phil Hobbs
pcdhSpamMeSensel... at (no spam) electrooptical.net> wrote:
As long as the variations of n happen on a scale >> lambda, Fermat's principle works fine for things like schlieren as well...

I didn't consider the variations in the index. This makes more sense
now. When I asked the question I had single slit diffraction in mind.
I hadn't considered the effects of an aperture, only the wavelength.

My thoughts concerning the "sum over histories" approach tend lie in
league with Phil. Here's why: assume we have a plane wave of
monochromatic light incident on a material whose index varies
periodically, acting essentially as a grating. Now, I understand that
the periodicity of the grating gives rise to what we observe as the
diffracted orders. These arise from plane waves whose k-vectors point
in the direction of each order. If we now take our optical path length
of a ray to be the optical distance from the start of the grating,
through the grating, and out of the grating to the location of each
order, then it would appear to me that each optical path length is in
fact different for the same incident wave. Thus, Fermat's Principle
appears to fail when explaining this effect whereas Feynman's has no
problem. This is what leads me to believe that the two ideas are in
fact different.

Any thoughts on this? Thanks as always.

Kyle

Fermat's principle is derivable from Huyghens' principle by the method
of stationary phase, but Huyghens' principle is not derivable from
Fermat's, because F. doesn't contain the idea of phase at all. That's
why it can't cover diffraction.

I agree with Bill that Huyghens' principle isn't quantitative, but it
does contain the basic physical idea used by Hamilton and later by
Feynman. Newton invented calculus of variations to solve Bernoulli's
brachistochrone problem after a hard day at the Mint--as he is reported
to have written, "I do not love to be teezed by foreigners about
Mathematicall things." I don't think it was applied to wave problems
before Hamilton, but I could easily be mistaken.