aplanatic cardoide refraction

[math...@gmail.com]

http://www.encyclopediaofmath.org/index.php/Descartes_oval

Where to learn about refraction across cardoide surfaces? that is, case c = 0 in above Encyclopedia link? It is stated that Cartesian ovals have been studied well for now a couple of centuries, but no images depict the source, target and the path, or how Snell's Law is obeyed. The Newtonian polar cases are perhaps better known [ r*(1 + eps cos(th))]= constant optical path length,but what about the Limaḉons and Cardoides[ constant r/(1 + eps cos(th))]?

Regards,
Narasimham

[Salmon Egg]

In article <860a7b7d-628f-48e4...@googlegroups.com>,

math...@gmail.com wrote:

> http://www.encyclopediaofmath.org/index.php/Descartes_oval
>
> Where to learn about refraction across cardoide surfaces? that is, case c = 0
> in above Encyclopedia link? It is stated that Cartesian ovals have been
> studied well for now a couple of centuries, but no images depict the source,
> target and the path, or how Snell's Law is obeyed. The Newtonian polar cases
> are perhaps better known [ r*(1 + eps cos(th))]= constant optical path

> length,but what about the Lima?ons and Cardoides[ constant r/(1 + eps
> cos(th))]?

To my mind, the Wikipedia article is a bit easier to understand just
what a Cartesian oval i. It is a generalization of an ellipse where the
distances from the foci are weighted differently. Mathematicians tend to
be more interested in esoteric properties than applied mathematicians.

http://en.wikipedia.org/wiki/Cartesian_oval

Although they are equivalent, my guess is the use of Fermat's
variational principle, will be a better way of approaching understanding
than will Snell's law will be.

--

Sam

Conservatives are against Darwinism but for natural selection.
Liberals are for Darwinism but totally against any selection.

[Narasimham Gudipaty]

Were any aplanat Limaḉon lenses & mirrors analysed, fabricated and tested so far? I mean is the assumption of Descartes found wrong or unexplored for centuries ?

Regards
Narasimham

[Narasimham Gudipaty]

On Wednesday, May 1, 2013 11:08:44 PM UTC+5:30, Salmon Egg wrote:

> > http://www.encyclopediaofmath.org/index.php/Descartes_oval
>
> >
>
> > Where to learn about refraction across cardoide surfaces? that is, case c = 0
>
> > in above Encyclopedia link? It is stated that Cartesian ovals have been
>
> > studied well for now a couple of centuries, but no images depict the source,
>
> > target and the path, or how Snell's Law is obeyed. The Newtonian polar cases
>
> > are perhaps better known [ r*(1 + eps cos(th))]= constant optical path
>
> > length,but what about the Lima?ons and Cardoides[ constant r/(1 + eps
>
> > cos(th))]?
>
>
> To my mind, the Wikipedia article is a bit easier to understand just
>
> what a Cartesian oval i. It is a generalization of an ellipse where the
>
> distances from the foci are weighted differently. Mathematicians tend to
>
> be more interested in esoteric properties than applied mathematicians.
>
>
> http://en.wikipedia.org/wiki/Cartesian_oval
>
>
>
> Although they are equivalent, my guess is the use of Fermat's
>
> variational principle, will be a better way of approaching understanding
>
> than will Snell's law will be.

> Sam

Thanks Sam. Fermat time minimization could help set up the Cartesian oval's differential equation. In this connection,is it possible to get or see Newton's work involving bi-normals mentioned in Wikipadia?

Narasimham

[Helpful person]

On May 3, 5:00 am, Narasimham Gudipaty <mathm...@gmail.com>

>
> Thanks Sam. Fermat time minimization could help set up the Cartesian oval's  differential equation. In this connection,is it possible to get or see Newton's work involving bi-normals mentioned in Wikipadia?
>
> Narasimham

I don't have it with me but I believe there is a good description in
Born and Wolfe.

http:..www.richardfisher.com

[Narasimham Gudipaty]

Thanks. In Born & Wolfe,some pages are omitted., no clue from the
available index.May be in Newton's Opticks?

Narasimham

http://books.google.co.in/books?id=oV80AAAAIAAJ&printsec=frontcover&dq=born+wolfe&hl=en&sa=X&ei=aN2DUei1IIiErQeW04H4CQ&redir_esc=y#v=onepage&q=born%20wolfe&f=false

[Salmon Egg]

In article <b3f68174-1326-429b...@googlegroups.com>,

Narasimham Gudipaty <math...@gmail.com> wrote:

>
> Thanks Sam. Fermat time minimization could help set up the Cartesian oval's
> differential equation. In this connection,is it possible to get or see
> Newton's work involving bi-normals mentioned in Wikipadia?
>

I have not read about bi-normals. My guess is that part of Newton's work
is not useful. His particle approach to optics was fundamentally wrong,
but was the best available at the time. The modern quantum theory of
light fits in more closely to what became the quantized version of the
wave theory of light.

Please do not take this statement as a disparagement of Newton's work.
He could not possibly know everything including stuff that seems pretty
simple to us now.

[Narasimham Gudipaty]

On May 3, 9:54 pm, Salmon Egg <Salmon...@sbcglobal.net> wrote:
> In article <b3f68174-1326-429b...@googlegroups.com>,

>  Narasimham Gudipaty <mathm...@gmail.com> wrote:

> > Thanks Sam. Fermat time minimization could help set up the Cartesian oval's
> > differential equation. In this connection,is it possible to get or see
> > Newton's work involving bi-normals mentioned in Wikipadia?
>
> I have not read about bi-normals. My guess is that part of Newton's work
> is not useful. His particle approach to optics was fundamentally wrong,
> but was the best available at the time. The modern quantum theory of
> light fits in more closely to what became the quantized version of the
> wave theory of light.
>
> Please do not take this statement as a disparagement of Newton's work.
> He could not possibly know everything including stuff that seems pretty
> simple to us now.

In geometric optics Newton with his more powerful differential form
approach easily looked
through Descartes' error when he stuck to his own cartesian form to
speculate....

There are no Limacon aplanats in existence todate or ever, I think.

Narasimham

[Phil Hobbs]

Well, you could consider actually buying the book like the rest of us.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

[Helpful person]

On May 3, 2:25 pm, Phil Hobbs

>
> Well, you could consider actually buying the  book like the rest of us.
>
> Cheers
>
> Phil Hobbs
>

Born and Wolfe makes a hefty paperweight. I don't use it anymore as i
now do more engineering than physics. However, it was well read when
i was at college. One of the best optics books ever written, up there
with Conrady and Twyman.

http://www.richardfisher.com

[Salmon Egg]

In article <SalmonEgg-5B47D...@news80.forteinc.com>,

Salmon Egg <Salm...@sbcglobal.net> wrote:

> Please do not take this statement as a disparagement of Newton's work.
> He could not possibly know everything including stuff that seems pretty
> simple to us now.

Newton knew enough geometrical optics to come up with the reflecting
telescope! I do not know, but he may have realized a paraboloid was the
best shape to use. If so, I also do not know if he was able to figure
its surface.

[Phil Hobbs]

Great book. I have a sixth edition in the Pergamon reinforced paperback
(a la Landau and Lifshitz). Last time I used it really seriously I was
trying to do some fancy crystal optics stuff. L&L "Media" is really
great on that too.

[Narasimham Gudipaty]

On Wednesday, May 1, 2013 7:30:33 PM UTC+5:30, Narasimham Gudipaty wrote:
> http://www.encyclopediaofmath.org/index.php/Descartes_oval
>
>
>
> Where to learn about refraction across cardoide surfaces? that is, case c = 0 in above Encyclopedia link? It is stated that Cartesian ovals have been studied well for now a couple of centuries, but no images depict the source, target and the path, or how Snell's Law is obeyed. The Newtonian polar cases are perhaps better known [ r*(1 + eps cos(th))]= constant optical path length,but what about the Limaḉons and Cardoides[ constant r/(1 + eps cos(th))]
>

> Narasimham

Hallo optics enthusiasts and professionals,

I am still curious about it, want to make a calculation when source of beam in dense medium and focussed in light medium.

Although shorter length modern aplanats by lens combination may be there now, purely from a historical viewpoint is any aplanat as obtainable by Fermat-Descartes approach leading to Cartesian ovals ever been described, made and tested ? that's is my query.

Fermat-Descartes approach for aplanat ovals ( former minimum weighted time, latter weighted distance constancy ) are both proved wrong with one Newton's differential treatment master strike, or so it appears to me ..Is it correct?

What is now the accepted standard to be without sph. aberration in refraction ?

Regards
Narasimham

[boxman]

The calculations of cartesian ovals is detailed in Introduction to
Nonimaging Optics by Julio Chaves. See chapter 17 sections 11-13.
Cartesian ovals are used often in non-imaging optical designs. Most LED
collimator lenses use a cartesian oval cross section to capture the
light that is not collected by the TIR surfaces to enable collimation of
the LED in that region.

[Narasimham Gudipaty]

Thanks. So, in fact is it a point to point imaging effected through a
Cartesian aplanatic in the refracting portion? (Am unable to get to
see the book).

Regards
Narasimham

[Narasimham Gudipaty]

On Thursday, May 9, 2013 6:41:03 PM UTC+5:30, Narasimham Gudipaty wrote:
> On May 7, 6:40 pm, boxman <box...@voyager.net> wrote:

....

> Thanks. So, in fact is it a point to point imaging effected through a
> Cartesian aplanatic in the refracting portion? (Am unable to get to
> see the book).

Or,if used here in non-imaging application then wave-front does not arrive at target in the same phase as at source,right? I am trying to find at least one example anywhere.. as per Descartes' original expectation.

Regards
Narasimham

[boxman]

I'm not sure I understand your question completely, but I will explain
what I can to see if it helps.

If you have a point source F in a medium of refractive index n1, you can
concentrate the light emitted from that source onto a point G in a
medium of refractive index n2 using a refractive surface that is a
cartesian oval. For this case, the optical path length from F to G is a
constant and is given by n1*t + n2*s = K where t is the length of the
ray in medium 1, s is the length of the ray in medium 2 and K is a
constant. Using that relationship, you can derive parametric equations
for a given set of points that are a function of the angle phi which is
the angle of the ray to the optical axis passing through the points.

The Limacon of Pascal falls out of these equations when you have the
special case where K^2-f^2*n2^2=0 where f = distance along optical axis
from cartesian oval to point G (i.e. the focal length) and n2 is less
than n1.

Don't know if that helps, but feel free to explain further what you are
looking for.

[Narasimham Gudipaty]

On Saturday, May 11, 2013 1:43:15 AM UTC+5:30, Boxman wrote:

> On 5/9/2013 5:37 PM, Narasimham Gudipaty wrote:
>
> > On Thursday, May 9, 2013 6:41:03 PM UTC+5:30, Narasimham Gudipaty
> > wrote:
> >> On May 7, 6:40 pm, boxman <box...@voyager.net> wrote:
> >> Thanks. So, in fact is it a point to point imaging effected through

-----

> > Regards Narasimham
>
> I'm not sure I understand your question completely, but I will explain
>
> what I can to see if it helps.

Thanks. I have begun ovals derivation and getting some clarity now.

My main query or doubt was and continues to be: Is sin i / sin r a constant at

all points during refraction across interface of dense to rare media?

> If you have a point source F in a medium of refractive index n1, you can
>
> concentrate the light emitted from that source onto a point G in a
>
> medium of refractive index n2 using a refractive surface that is a
>
> cartesian oval. For this case, the optical path length from F to G is a
>
> constant and is given by n1*t + n2*s = K where t is the length of the
>
> ray in medium 1, s is the length of the ray in medium 2 and K is a
>
> constant. Using that relationship, you can derive parametric equations
>
> for a given set of points that are a function of the angle phi which is
>
> the angle of the ray to the optical axis passing through the points

I adopted the same Fermat optical length approach to derive and get the form

cos(phi) = C + r/A + B/r in polar co-ordinates (r,phi). Am using your notation

in what follows. n1 > n2.

There are two oval types as the above is a quadratic.

1) egg type n1*t + n2*s > 0 and

2) dimpled ovals n1*t - n2*s > 0, for constant optical path DIFFERENCE. They are

from Limacon family. r = a + b cos(phi) There is inversion possible between

them.

But are the latter in use? I mean to bring light from dense medium to focus

(aplanatically) to a point into the rarer medium are there any applications at

all?

> The Limacon of Pascal falls out of these equations when you have the
>
> special case where K^2-f^2*n2^2=0

i.e., K = f * n2

Depending on K > = < f * n2 we have G forming inside the egg oval, on the oval

or outside the oval. Is this correct? There are 3 cases and in one case rays are

fully inside dense medium like in an ellipse.

> where f = distance along optical axis

f = FG ? That is distance between pole F and 'target' destination G,distance f

apart?

> from cartesian oval

or the pole F?

> to point G (i.e. the focal length) and n2 is less than n1.
>
> Don't know if that helps, but feel free to explain further what you are
>
> looking for.

It is helping very much and thanks again for your patient response.

Regards
Narasimham

[Narasimham Gudipaty]

On Saturday, May 11, 2013 1:43:15 AM UTC+5:30, Boxman wrote:

Dear Boxman and all others,

Requesting for your specific comments to clear up it fully. The following seems to me right.I worked on it.Shall much appreciate your responses.

Which of the following four statements are true?

(Spherical aberration free aplanat lens of single material of refractive index > 1 , say 1.5. Assume dense medium lens at left and rarer medium (air) at right on the x-axis. Light travels from left to right in situations mentioned below, i.e., 1 & 3 emerging out of dense medium, 2 & 4 entering into dense medium)

1. Planar wave-front from inside the lens focuses to a point in air F with shrinking /converging spherical waves. For this to happen, lens shape is a hyperbola.

2. Planar wave-front from outside the lens focuses to a point F inside lens with shrinking /converging spherical waves. For this to happen, lens shape is an ellipse.

In 1 and 2 above expanding wave-fronts in reverse order give rise to straight planar beams.

3. Spherical expanding wave-front issuing from focus F1 inside the lens focuses to a point F2 in air with shrinking /converging spherical waves. For this to happen, lens shape is a Cartesian Oval of dimpled ball type.

4. Spherical expanding wave-front front issuing from focus F1 from outside the lens focuses to a point F2 in the lens with shrinking /converging spherical waves. For this to happen, lens shape is a Cartesian Oval egg type.

In 3 and 4 above F1 and F2 can be swapped.

Regards
Narasimham G.L.

[Narasimham Gudipaty]

To make what I mean is clearer, slightly change it.

> Requesting for your specific comments to clear up it fully. The following seems to me right.I worked on it.

>Shall much appreciate your responses.

>  Which of the following four statements are true? If untrue what should be correct?

>
> (Spherical aberration free aplanat lens of single material of refractive index > 1 , say 1.5.

 Assume dense medium lens at left / rarer medium (air) at right on
the x-axis for cases 1 & 3,

and dense medium lens at right/ rarer medium (air) at left on the x-
axis for 2 & 4.

Light travels from left to right in the four situations.

( this makes 1 & 3 emerging out of dense medium, 2 & 4 entering into
dense medium).

> 1.      Planar wave-front from inside the lens focuses to a point in air F with shrinking /converging spherical waves. For this to happen, lens shape is a hyperbola.
>
> 2.      Planar wave-front from outside the lens focuses to a point F inside lens with shrinking /converging spherical waves. For this to happen, lens shape is an ellipse.
>

> In 1 and 2 above expanding wave-fronts.If taken direction of light is reversed it gives rise to straight planar beams.
>
> 3.      Spherical expanding wave-front issuing from focus F1 inside the lens focuses to a point F2 in air with shrinking /converging spherical waves. For this to happen, lens shape is (near to axis portions of) a Cartesian Oval of dimpled ball type.
>
> 4.      Spherical expanding wave-front front issuing from focus F1 from outside the lens focuses to a point F2 in the lens with shrinking /converging spherical waves. For this to happen, lens shape is ( near to axis portion of) a Cartesian Oval egg type.
>
> In 3 and 4 above F1 and F2 can be swapped if direction of light is reversed.

Regards
Narasimham

[J-C]

<math...@gmail.com> wrote in message
news:860a7b7d-628f-48e4...@googlegroups.com...

http://www.encyclopediaofmath.org/index.php/Descartes_oval

Where to learn about refraction across cardoide surfaces? that is, case c =
0 in above Encyclopedia link? It is stated that Cartesian ovals have been
studied well for now a couple of centuries,

actually about 350 yrs

but no images depict the source, target and the path, or how Snell's Law is
obeyed.

Make an effort and look harder including in the writings od Descartes and
Huygens which are easily available on the net.

==========================================================

You might want to have a look at this article

Aplanatic (or Cartesian) Optical Surfaces by J P Southall

http://books.google.com/books?id=NTvWAAAAMAAJ&pg=PA614&dq=%22cartesian+ovals%22+southall&hl=en&sa=X&ei=CYWaUcOhJce70AH-moCoDA&ved=0CC0Q6AEwAA#v=onepage&q=%22cartesian%20ovals%22%20southall&f=false

and at J P Southall's book

The Principles and Methods of Geometrical Optics

although a century old it has a lot of interesting (and in fact
still quite current) stuff. Fermat, Snell, Huygens, Fresnel etc
are still valid in normal conditions. Attempt at humor, insert
smiley here.

http://books.google.com/books?id=JH4tAAAAYAAJ&pg=PR8-IA1&dq=southall+optics&hl=en&sa=X&ei=oIqaUYKODrfG4AOQsYCACw&ved=0CE8Q6AEwBA

Also by the same author

http://books.google.com/books?id=JRoJAAAAIAAJ&printsec=frontcover&dq=southall+optics&hl=en&sa=X&ei=oIqaUYKODrfG4AOQsYCACw&ved=0CDgQ6AEwAA

Cardioids have no application in optics, except as reflecting surfaces in
certain special condensers for microscopes, although you see them often
(in your coffee cups say).

Not aware of any applications of limacons.

As to Descartes ovals they are pretty useless in practice for a number of
reasons, while they maybe stigmatic they do not fulfill the sine condition
for example, therefore they can't be used for imaging extended objects

But they are perpetually resurrected by well meaning people.

[Narasimham Gudipaty]

On May 21, 2:17 am, "J-C" <malaking@buwaya> wrote:
> <mathm...@gmail.com> wrote in message
----

> As to Descartes ovals they are pretty useless in practice for a number of

> reasons, while they may be stigmatic they do not fulfill the sine condition
> for example, therefore they can't be used for imaging extended objects.

Thanks.(Working in India I have only a limited or ready access to such
good old book sources. Apart from internet, trying to derive it
afresh).

But under what situations is Snell's law not valid ? Why cartesian
ovals, with optical length invariance considered,do not obey the law?
What situation makes it to re-conform ? ( I mean, when did the law
become illegal :) ?).

Narasimham

[J-C]

"Narasimham Gudipaty" <math...@gmail.com> wrote in message
news:5a1082a5-b794-4fbb...@a15g2000pbu.googlegroups.com...

===============================================

The references I gave you are entirely free and I am sure that you can
download them in India as in most elsewhere.

There are many more such "Google Books" texts in optics which are
available for free at various levels and even if old they provide a good
start at least for elementary (and in fact not so elementary) start in
optics, geometrical and physical.

You don't have to re-invent, re-discover or re-derive anything.

And Google Books is only one of many such internet resources,
many quite recent and up to date.

Please choose two or three books and start to study them earnestly.

We are very fortunate to have the internet at our disposal, Srinivasa
Ramanujan had very few books but he did study whatever he had very
hard.

I am sorry but since I do not understand your question I am unable
to give you an answer, now or later.

[Narasimham Gudipaty]

On Wednesday, May 22, 2013 1:42:42 AM UTC+5:30, J-C wrote:
> "Narasimham Gudipaty" <mathma..@gmail.com> wrote in message
>
> news:.... @a15g2000pbu.googlegroups.com...

Thanks a lot for guidance and encouragement. Sorry for insufficent expression. My aim was to see how to design a transmitter/receicer optical combo, source inside denser medium between two foci, modifying Snell's law to suit differetial equation formulation.

http://i44.tinypic.com/2wgtymx.jpg
http://i40.tinypic.com/cuqzq.png

I have come this far, Snell's law not being obeyed was in fact a matter of concern, but now ok.Optical path length tallies. Pl comment.

Regards
Narasimham

[Narasimham]

The ray trace of an egg shaped Cartesian lens (a shallow version is a
convex lens) is available, far out rays converge at other focus after
entering dense medium.

Can someone point me to a ray trace of a Cartesian lens (whose shallow
version is a concave lens) where rays appear to diverge from (issue
out from) same side focus after entering dense medium?

Some indication is here concerning interior and exterior ovals, but I
like to look fora single ray trace.

http://www.sciencedirect.com/science/article/pii/S0030401804003104

TIA
Narasimham