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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Tue Jun 20, 2006 5:54 pm Post subject:
The biggest mistake in classical optics



We consider a glass system with tree glass plates (all having weak
neglectable absorption). All of these three
glass plates shall have an different refraction index, that means non of
these three indexes shall be equal.
Now we put these three plates together and let light fall perpendicular onto
the surface.
We look into standard books like Max Born or other good optical books and
find the solution for this problem or at least the way the sokutions are
built. We get simple Formulas for the reflection and transmission
coefficient, they belong to a Formula System containig Fresnels formulas or
Drudes Formulas (if you have complex indices with absorption). But here we
restrict to the above mentioned system of three glas plates
with different real index.
Now you can make the following: If you let the thickness of the middle glass
plate go to zero. What happens:
If you do this, the formulas for the system of the two outer glas plates is
not the formula, you would get
if you are doing the same process for the two outer glas plates. The formula
where you let the thickness
of the middle plate go to zero still contains the index of the middle plate
when you let go thickness d to zero.
This is a paradox which shows, that something on the theory is wrong. It is
not a small mistake, it is a big error. The theory is wrong !
But what is wrong with this theory, i will answer it here. When i say you
that it took me more than 20 years to
find the simple error, then laugh !
Let us consider a third arrangement: We have our three glass tubes same as
above but we separate them
from each other. What we do: We separe them in vacuum ( Index 1 ). Now we
make the following:
We take for this also the Drude theory and then shift the plates ( in
between the vacuum ) together.
Now we have a second formula for three plate system which is different from
the first which stands in the Books. This formula has one big advantage
against that one in the books:
If you now let the thickness d of the middle glass plate go to zero, then
you get a solution where the index of the middle glass plate plays no roll
any more ! And this formula you get ( the two plate formula) is identical
with the solution you derive when you have a two plate system with a vaccum
layer in between land let the thickness of the vacuum layer go to zero !
The soltions of this theory are fully self consistent !
Thats the biggest mistake of the old optics of layers at all. The old optics
of layers is wrong, except you have
exact vacuum between all plates. In this case the old theory is right.
This finite vacuum layers have been forgotten in the old optics. 

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Herman Family science forum Guru Wannabe
Joined: 14 Jun 2005
Posts: 173

Posted: Wed Jun 21, 2006 4:42 am Post subject:
Re: The biggest mistake in classical optics



"Josef Matz" <josefmatz@arcor.de> wrote in message
news:4498351f$0$4498$9b4e6d93@newsread2.arcoronline.net...
Quote:  We consider a glass system with tree glass plates (all having weak
neglectable absorption). All of these three
glass plates shall have an different refraction index, that means non of
these three indexes shall be equal.
Now we put these three plates together and let light fall perpendicular
onto
the surface.
We look into standard books like Max Born or other good optical books and
find the solution for this problem or at least the way the sokutions are
built. We get simple Formulas for the reflection and transmission
coefficient, they belong to a Formula System containig Fresnels formulas
or
Drudes Formulas (if you have complex indices with absorption). But here we
restrict to the above mentioned system of three glas plates
with different real index.
Now you can make the following: If you let the thickness of the middle
glass
plate go to zero. What happens:
If you do this, the formulas for the system of the two outer glas plates
is
not the formula, you would get
if you are doing the same process for the two outer glas plates. The
formula
where you let the thickness
of the middle plate go to zero still contains the index of the middle
plate
when you let go thickness d to zero.
This is a paradox which shows, that something on the theory is wrong. It
is
not a small mistake, it is a big error. The theory is wrong !
But what is wrong with this theory, i will answer it here. When i say you
that it took me more than 20 years to
find the simple error, then laugh !
Let us consider a third arrangement: We have our three glass tubes same as
above but we separate them
from each other. What we do: We separe them in vacuum ( Index 1 ). Now we
make the following:
We take for this also the Drude theory and then shift the plates ( in
between the vacuum ) together.
Now we have a second formula for three plate system which is different
from
the first which stands in the Books. This formula has one big advantage
against that one in the books:
If you now let the thickness d of the middle glass plate go to zero, then
you get a solution where the index of the middle glass plate plays no roll
any more ! And this formula you get ( the two plate formula) is identical
with the solution you derive when you have a two plate system with a
vaccum
layer in between land let the thickness of the vacuum layer go to zero !
The soltions of this theory are fully self consistent !
Thats the biggest mistake of the old optics of layers at all. The old
optics
of layers is wrong, except you have
exact vacuum between all plates. In this case the old theory is right.
This finite vacuum layers have been forgotten in the old optics.

The world can now rest comfortably.
Michael 

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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Thu Jun 22, 2006 9:07 am Post subject:
Re: The biggest mistake in classical optics



"Herman Family" <ecalptsudwaseht.in.reverse@frontiernet.net> schrieb im
Newsbeitrag news:V64mg.4031$Oh1.2158@news01.roc.ny...
Quote: 
"Josef Matz" <josefmatz@arcor.de> wrote in message
news:4498351f$0$4498$9b4e6d93@newsread2.arcoronline.net...
We consider a glass system with tree glass plates (all having weak
neglectable absorption). All of these three
glass plates shall have an different refraction index, that means non of
these three indexes shall be equal.
Now we put these three plates together and let light fall perpendicular
onto
the surface.
We look into standard books like Max Born or other good optical books
and
find the solution for this problem or at least the way the sokutions are
built. We get simple Formulas for the reflection and transmission
coefficient, they belong to a Formula System containig Fresnels formulas
or
Drudes Formulas (if you have complex indices with absorption). But here
we
restrict to the above mentioned system of three glas plates
with different real index.
Now you can make the following: If you let the thickness of the middle
glass
plate go to zero. What happens:
If you do this, the formulas for the system of the two outer glas plates
is
not the formula, you would get
if you are doing the same process for the two outer glas plates. The
formula
where you let the thickness
of the middle plate go to zero still contains the index of the middle
plate
when you let go thickness d to zero.
This is a paradox which shows, that something on the theory is wrong. It
is
not a small mistake, it is a big error. The theory is wrong !
But what is wrong with this theory, i will answer it here. When i say
you
that it took me more than 20 years to
find the simple error, then laugh !
Let us consider a third arrangement: We have our three glass tubes same
as
above but we separate them
from each other. What we do: We separe them in vacuum ( Index 1 ). Now
we
make the following:
We take for this also the Drude theory and then shift the plates ( in
between the vacuum ) together.
Now we have a second formula for three plate system which is different
from
the first which stands in the Books. This formula has one big advantage
against that one in the books:
If you now let the thickness d of the middle glass plate go to zero,
then
you get a solution where the index of the middle glass plate plays no
roll
any more ! And this formula you get ( the two plate formula) is
identical
with the solution you derive when you have a two plate system with a
vaccum
layer in between land let the thickness of the vacuum layer go to zero !
The soltions of this theory are fully self consistent !
Thats the biggest mistake of the old optics of layers at all. The old
optics
of layers is wrong, except you have
exact vacuum between all plates. In this case the old theory is right.
This finite vacuum layers have been forgotten in the old optics.
The world can now rest comfortably.

Yes it is simple and everybody can understand it. But the meaning of it is
very very fundamental even
it is so "easy".
Josef


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Edward Green science forum addict
Joined: 21 May 2005
Posts: 95

Posted: Thu Jun 22, 2006 10:16 pm Post subject:
Re: The biggest mistake in classical optics



Josef Matz wrote:
<...>
That's a nice story.
My question to you is, have you been able to track down the implied bad
assumption in the old method? Given the finite wavelength of light, it
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it? There must be some implicit assumption about the
properties of interfaces which goes bad in the limit. Maybe the usual
interface boundary conditions imply a surface layer at least a few
wavelengths thick. When we let the sheet get thinner than this, we are
violating our assumptions, and so wind up with a ghost effect from a
sheet of zero thickness. Including intervening vacuum layers avoids
this pitfall, somehow. Just a guess. 

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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Sat Jun 24, 2006 3:06 am Post subject:
Re: The biggest mistake in classical optics



"Edward Green" <spamspamspam3@netzero.com> schrieb im Newsbeitrag
news:1151014587.202498.190470@u72g2000cwu.googlegroups.com...
Quote:  Josef Matz wrote:
...
That's a nice story.

Ok thanks for compliment
Quote:  My question to you is, have you been able to track down the implied bad
assumption in the old method? Given the finite wavelength of light, it

Yes. the old method is wrong.
Quote:  can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it? There must be some implicit assumption about the

Take for example a thin gold layer on glass. The old theory has different
reflexion
transmission and polarization properties, interference in the layers. This
is covered
in serious books by means of thickness dependent index of gold or whatever
material.
this is a measurement result (thickness dependence of index) based on wrong
theory.
Infact these measurement curves indicate that with the new theory such an
assumption
can be overcome. The index is not thickness dependent in prnciple for
thicknesses
less than about 1/10 of the wavelenght.
Quote:  properties of interfaces which goes bad in the limit. Maybe the usual
interface boundary conditions imply a surface layer at least a few
wavelengths thick. When we let the sheet get thinner than this, we are

No a few atomic layers is sufficient to reach homgene material. Smaller
coating thickness
than 1 atomic layer you loose macroscopic homogenity. Thus a more
complicated three
dimensional theory necessary for very thin layers. But if the wavelength is
big compared to
the atom radius (or the avarage atomic distance between atoms for smaller
than 1 atomic layer)
of the substances, then the new formulas hold even for this cases.
Quote:  violating our assumptions, and so wind up with a ghost effect from a
sheet of zero thickness. Including intervening vacuum layers avoids
this pitfall, somehow. Just a guess.

The vacuum layer is theoretical necessary to get energy fluxes for absortive
layers. Such a theory
exists now. Even the Nimtz double prism experiment can now be calculated.
Result:
light can passes the gap between the two prisms (when total reflection
condition is fulfilled)
as inhomogene waves with velocities very much higher than c (in vacuum gap).
So Nimtz is makroskopic electrodynamics ! Some old theories break down now.
The existence of the vacuum layers can be easily veryfied experimentally
with antike optical
experimenters stuff.
Josef Matz 

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p.kinsler@ic.ac.uk science forum beginner
Joined: 11 Sep 2005
Posts: 8

Posted: Sun Jun 25, 2006 9:43 pm Post subject:
Re: The biggest mistake in classical optics



Edward Green <spamspamspam3@netzero.com> wrote:
Quote:  Josef Matz wrote:
...
My question to you is, have you been able to track down the implied bad
assumption in the old method? Given the finite wavelength of light, it
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it? There must be some implicit assumption about the
properties of interfaces which goes bad in the limit. Maybe the usual
interface boundary conditions imply a surface layer at least a few
wavelengths thick. When we let the sheet get thinner than this, we are
violating our assumptions, and so wind up with a ghost effect from a
sheet of zero thickness. Including intervening vacuum layers avoids
this pitfall, somehow. Just a guess.

I'm not sure exactly what standard approach is being described
here (I'm too lazy to look it up right now), but it no doubt
relies on fixing fields and their gradients at the interface.
Thus you might not need an assumption of wavelengths, just one
of a layer, which couldn't be got rid of with the usual
limit.
Hmm, is this an Efield only standardapproach we're talking about
here?

+
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +442075947520 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/ 

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Timo Nieminen science forum Guru Wannabe
Joined: 12 May 2005
Posts: 244

Posted: Mon Jun 26, 2006 11:22 am Post subject:
Re: The biggest mistake in classical optics



On Fri, 22 Jun 2006, Edward Green wrote:
Quote:  Given the finite wavelength of light, it
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it?

No.
Take an interface between two media, calculate reflection and transmission
coefficients (either amplitude or power, your choice), and then take 3
media, and ditto. Take the limit as the thickness of the middle medium
approaches zero, and you get the 2 media case.
For example, if it's a highindex layer between 2 identical media, then
there's a 1/2 wave phase shift between the reflections from the front and
back of the interface, and these destructively interfere as the thickness
> 0. In the limit, no reflection; the middle medium becomes completely
transparent. This is even the case when you have total internal reflection
 tunnelling across a 0 thickness barrier gives 100% transmission.
Haven't read the original post in detail, but IME there is no problem, if
you do it properly. Do it wrongly, and it's easy to get the wrong result,
especially when dealing with lossy or gain media. Don't talk about angles,
talk about wavevectors instead.

Timo Nieminen  Home page: http://www.physics.uq.edu.au/people/nieminen/
Eprints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html 

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Timo Nieminen science forum Guru Wannabe
Joined: 12 May 2005
Posts: 244

Posted: Mon Jun 26, 2006 11:35 am Post subject:
Re: The biggest mistake in classical optics



On Sat, 24 Jun 2006, Josef Matz wrote:
Quote:  "Edward Green" <spamspamspam3@netzero.com> schrieb:
My question to you is, have you been able to track down the implied bad
assumption in the old method? Given the finite wavelength of light, it
Yes. the old method is wrong.
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it? There must be some implicit assumption about the
Take for example a thin gold layer on glass. The old theory has different
reflexion
transmission and polarization properties, interference in the layers. This
is covered
in serious books by means of thickness dependent index of gold or whatever
material.

I've not dealt with very thin layers myself, but for layers thick enough
so that you can assume locality and linearity, there's no problem. Are you
talking about errors due to assumptions of locality, or errors due to
incorrect treatment of nonlocality, or something else?
Boundary conditions for the components of E and H parallel to the
interface, parallel component of the wavevector being the same on both
sides of any plane interface, what is the problem?

Timo Nieminen  Home page: http://www.physics.uq.edu.au/people/nieminen/
Eprints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html 

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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Tue Jun 27, 2006 11:22 am Post subject:
Re: The biggest mistake in classical optics



"Timo A. Nieminen" <timo@physics.uq.edu.au> schrieb im Newsbeitrag
news:Pine.WNT.4.64.0606262127020.1388@serene.st...
Quote:  On Sat, 24 Jun 2006, Josef Matz wrote:
"Edward Green" <spamspamspam3@netzero.com> schrieb:
My question to you is, have you been able to track down the implied bad
assumption in the old method? Given the finite wavelength of light, it
Yes. the old method is wrong.
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it? There must be some implicit assumption about the
Take for example a thin gold layer on glass. The old theory has
different
reflexion
transmission and polarization properties, interference in the layers.
This
is covered
in serious books by means of thickness dependent index of gold or
whatever
material.
I've not dealt with very thin layers myself, but for layers thick enough
so that you can assume locality and linearity, there's no problem. Are you
talking about errors due to assumptions of locality, or errors due to
incorrect treatment of nonlocality, or something else?

Yes its something else. Its the way the reflection formulas are formed. In
the old theory
there are quotients of media indexes on both sides determining the
reflection and transmissin formulas.
In the new theory these formulas have not this quotiens, because one medium
always is vacuum (index 1).
Quote:  Boundary conditions for the components of E and H parallel to the
interface, parallel component of the wavevector being the same on both
sides of any plane interface, what is the problem?

This is right. nevertheless the complete reflection coefficient depends on
if you use vacuum layers or not.
(same for transmission coefficient). If you compare simple experiments you
will find the theory with
vacuum layers to be the right one. The one which brings right results for
thin layers and the one which
is the only possible to calculate energy fluxes in absorbing layers. Thats
the main benefit of the new
theory. And of shure the thicknes dependence of index for thin metallic
layers is identified not to be
a physical reality, but a misinterpretation due to the use of wrong
formulas.
The vacuum layers make all that possible and of course a layer system with
every layer followed by a
vacuum layer is a physical one. And in you compress the vacuum layers to
zero you have the Drude
layer system but another formula set for the reflexion and transmission
coefficients. Two different formulas
for the same physical system  clear at least one must be wrong. And as i
told already everything is
speaking for that the Drude formulas  not containing vacuum layers  are
the wrong formulas.
Now you can say: The new formulas are in principle also Drude formulas. This
is right. Its the principle
that at each interface a infinitesimal thin vacuum layer must be assumed in
order to get the right
reflection and transmission coefficients. This indeed is the fundamental
new. Drude in principle just
did not know this addinional principle. Thats all.
And together with the principle of calculating tunnel fluxes and tunnel spin
fluxes you get the full theory.
This at least is how i see things presently. Also Nimtz double prism
experiment comes out with this theory!
Thanks so far
Josef Matz


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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Tue Jun 27, 2006 11:23 am Post subject:
Re: The biggest mistake in classical optics



"Timo A. Nieminen" <timo@physics.uq.edu.au> schrieb im Newsbeitrag
news:Pine.WNT.4.64.0606262112530.1388@serene.st...
Quote:  On Fri, 22 Jun 2006, Edward Green wrote:
Given the finite wavelength of light, it
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it?
No.

Yes it can
Quote: 
Take an interface between two media, calculate reflection and transmission
coefficients (either amplitude or power, your choice), and then take 3
media, and ditto. Take the limit as the thickness of the middle medium
approaches zero, and you get the 2 media case.
For example, if it's a highindex layer between 2 identical media, then
there's a 1/2 wave phase shift between the reflections from the front and
back of the interface, and these destructively interfere as the thickness
> 0. In the limit, no reflection; the middle medium becomes completely
transparent. This is even the case when you have total internal reflection
 tunnelling across a 0 thickness barrier gives 100% transmission.
Haven't read the original post in detail, but IME there is no problem, if
you do it properly. Do it wrongly, and it's easy to get the wrong result,
especially when dealing with lossy or gain media. Don't talk about angles,
talk about wavevectors instead.

Timo Nieminen  Home page: http://www.physics.uq.edu.au/people/nieminen/
Eprints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html



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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Wed Jun 28, 2006 4:46 pm Post subject:
Re: The biggest mistake in classical optics



<p.kinsler@ic.ac.uk> schrieb im Newsbeitrag
news:plc3n3e3o.ln1@delillo.lsr.ph.ic.ac.uk...
Quote:  Edward Green <spamspamspam3@netzero.com> wrote:
Josef Matz wrote:
...
My question to you is, have you been able to track down the implied bad
assumption in the old method? Given the finite wavelength of light, it
can't really make any difference if we interpose a infinitesmal vacuum
layer or not, can it? There must be some implicit assumption about the
properties of interfaces which goes bad in the limit. Maybe the usual
interface boundary conditions imply a surface layer at least a few
wavelengths thick. When we let the sheet get thinner than this, we are
violating our assumptions, and so wind up with a ghost effect from a
sheet of zero thickness. Including intervening vacuum layers avoids
this pitfall, somehow. Just a guess.
I'm not sure exactly what standard approach is being described
here (I'm too lazy to look it up right now), but it no doubt
relies on fixing fields and their gradients at the interface.
Thus you might not need an assumption of wavelengths, just one
of a layer, which couldn't be got rid of with the usual
limit.
Hmm, is this an Efield only standardapproach we're talking about
here?

You can just use the E  field for doing all calculations, you can also
calculate and take the H field.
The only difference to Drudes theory really is, that you shift vacuum layers
in at each interface.
Then let the thickness of this layers go to zero. Thats all.
With this you can get a metall layer optic with energy formulas which does
not exist up to now.
You also can calculate energy densities, group velocities, tunnel fluxes in
layers if you know a
few things more. But the vacuum layer concept is necessary for. Yes i can
show now that
energy flux theorems are only possible with vacuum layer concept  without
never!
Josef Matz


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p.kinsler@ic.ac.uk science forum beginner
Joined: 11 Sep 2005
Posts: 8

Posted: Thu Jun 29, 2006 1:25 pm Post subject:
Re: The biggest mistake in classical optics



Josef Matz <josefmatz@arcor.de> wrote:
Quote:  Hmm, is this an Efield only standardapproach we're
talking about here?
You can just use the E  field for doing all calculations, you can also
calculate and take the H field.

Here is my guess for a resolution to the inconsitency you
report. This is all off the top of my head so may be imperfect,
I don't have time to check details at the moment.
If you calculate using only the E field (or alernatively only
the B field) you need to specify the boundary conditions beween
the layers as matching both the value and gradient of the field
appropriately. It is my suggestion that the gradient matching
criteria means that the limit of a vanishingly thin layer
doesn't work properly. In such a limit, the gradient in the vanished
layer has no meaning, but the condition is still trapped in the
model.
If you instead decide to match E and B values across the interface,
you don't need a gradient condition (as E gradient depends on B,
B gradient on E), and the limit will work.

+
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +442075947520 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/ 

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Josef Matz science forum Guru Wannabe
Joined: 08 May 2005
Posts: 255

Posted: Thu Jun 29, 2006 6:18 pm Post subject:
Re: The biggest mistake in classical optics



Ok. A thin layer i handle as bulk material too. This is fulfilled at least
for a 10 atomic layer very good.
As usual for electromagnetic waves crossing continua media, you have 4
steadyness conditions:
E(tangential), H(tangential), B(normal) and D(normal) are steady. But as you
see with vacumm layers
always one layer of the involved two is vacuum. Therefore you have more
layers than in old Drude theory.
This layers go into limit thickness zero to get the right reflection and
transmission formulas.
You can name these formulas new Drude formulas or Matz formulas or new
Fresnel formulas for that
specific case.
Gradients are normally not viewed upon. But interesting if you have a
gradiend in index
(for example graded index fibers). There you have a vacuum layer in
principle at every point of the
gradient. Or you can say: Vacuum is everywhere and matter(index) is
everywhere where is a gradient.
Vacuum is not surrounding matter as it was vied in the last century physics.
Now vacuum and matter
are at the same points at the same time. This explains very good that there
are two models of the
so called makroskopic maxwell equations. One taking the complex index and
one settling on vacuum
with polarization properties of medium.
Ok the principle that vacuum penetrates matter in form of vacuum layers
along the gradients is a new
fundamental principle in physics. If you want you can call it the
relativistic invariant aether. Probably
not only useful in optics and coatings.
There is probable much more behind this principle  beeing a general new
principle in electrodynamics
(and maybe also atom and particle physics).
God himself give us the option to use them (the vacuum layers) in our
thoughts or not. And he gives you also the answer if you you think about. If
you neglegt them or do not know them you make mistakes in some physical
fields. One of them is Optics of layers.
Josef Matz
<p.kinsler@ic.ac.uk> schrieb im Newsbeitrag
news:2v0dn3die.ln1@delillo.lsr.ph.ic.ac.uk...
Quote:  Josef Matz <josefmatz@arcor.de> wrote:
Hmm, is this an Efield only standardapproach we're
talking about here?
You can just use the E  field for doing all calculations, you can also
calculate and take the H field.
Here is my guess for a resolution to the inconsitency you
report. This is all off the top of my head so may be imperfect,
I don't have time to check details at the moment.
If you calculate using only the E field (or alernatively only
the B field) you need to specify the boundary conditions beween
the layers as matching both the value and gradient of the field
appropriately. It is my suggestion that the gradient matching
criteria means that the limit of a vanishingly thin layer
doesn't work properly. In such a limit, the gradient in the vanished
layer has no meaning, but the condition is still trapped in the
model.
If you instead decide to match E and B values across the interface,
you don't need a gradient condition (as E gradient depends on B,
B gradient on E), and the limit will work.

+
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +442075947520 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/



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