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[Phys-L] Re: Travel distance in a waveguide.



On 02/13/05 18:48, Michael Edmiston wrote:

My first question for this forum is... when a wave goes through a
waveguide, what is the effective length of the guide. Is it just the
physical length?

The answer to this narrow question is simple: The thing you
care about is the plain ordinary end-to-end length of the
guide. There is no deep physics here; mostly it's just
a matter of conventional definition. When somebody asks
the question about the speed of the wave in the guide,
you can assume they're asking about end-to-end black-box
properties.

My second question is... it is my understanding that an optical fiber
is constructed so the refractive index is continuously graded from a
higher refractive index at the center to a lower refractive index
near the outside.

A so-called step-index fiber has an index that goes
like this in cross section:

__________
| |
| |
_______| |_______
cladding core cladding


Meanwhile, a so-called graded index fiber goes more like
this:
______
/ \
/ \
______/ \______
cladding core cladding


Both kinds of fiber are available.

If this is true, what does it mean to refer to a
cable as having a particular refractive index?

Well, even in the graded-index fiber, there is a fairly
well-defined "core" index. (Knowing this number doesn't
tell you everything you need to know, of course.)

Putting my first two questions together, my third question becomes...
is this apparatus a fraud?

It depends on what you do with it. (Not everybody who
owns an axe is an axe-murderer.)

Has the company simply said the
refractive index of the fiber is 1.50 so the calculation ...
comes out correct?

I suspect they did. Yuck.

That is, might we say n = 1.50 is an effective
refractive index that we determine from knowing c, rather than
vice-versa?

I wouldn't call it a refractive index, "effective" or
otherwise. I might call it a fudge factor.

On 02/14/05 07:50, David Bowman wrote:

In a real sense *any* experiment that purports to actually measure c
is a 'fraud'. This is simply because c is a defined quantity--not a
measurable one.

I can't decide whether that's an important point or a trivial
one. It's true as stated ... but mostly it's a lesson about
terminology. I guess like many things, it seems deep until
you understand it, then it seems trivial in retrospect.

On the one hand, yes, there is a quantity denoted $c$ which is
a defined quantity, and therefore not measurable. This quantity
$c$ plays a very profound role in physics, including but certainly
not limited to the physics of light.

On the other hand, there is a quantity which we can call "the
speed at which this here light wave propagates from place to
place" which is not generally equal to $c$. We can denote it
by $s$ or by $c_1$ or whatever.

Here's the rub: It is traditional for $c$ to be called
"TheSpeedOfLight". This causes all sorts of confusion.
It overstates the connection between $c$ and light. A
generic light wave is neither necessary nor sufficient
for defining $c$. IMHO we really need a better way of
saying what $c$ is.s

I agree it is totally fraudulent to "measure" $c$. On
the other hand, there is nothing wrong with measuring
the speed at which EM waves propagate in fiber or in
microwave waveguide plumbing.

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

On 02/13/05 19:56, Bernard Cleyet wrote:

My understanding is that th fiber does work like a wave guide. I
suspect this is true if the relative dimensions are the same.

It does, but that's not the whole story.

There are two conceptually-independent bits of physics we
should be thinking about:
*) waveguide, and
*) index.
This gives us four possibilities:

1a: free propagation in vacuum 2a: free prop in refractive medium

1b: waveguide full of vacuum 2b: waveguide full of refr. medium

All but (1a) will propagate at speeds less than $c$.

The chase: the formula for the group speed down the guide is v sub g
= C sqrt[ 1- (frees space wavelength/2*waveguide dimension)^2]; the
wave guide dimension is for a rectangular guide with E perpendicular
(lowest mode).

That only applies to case (1b). A fiber is case (2b).

Let's be clear: A wave propagating along a fiber in the
Z direction
-- is slowed by the index of the material, *and*
-- is slowed by the fact that the wavefunction
is scrunched in the lateral directions (X and Y).

Both effects must be taken into account if we are to have
anything resembling an accurate description.

The formula BC quoted is super-easy to understand, and
indeed super-easy to derive from scratch, and also easy
to generalize to any mode, not just the lowest. The point
is that for a rectangular waveguide with perfectly shiny
walls, full of vacuum, in the interior we just have the
Maxwell equations. The boundary conditions do not change
the Maxwell equations! In the interior we can still
express the solutions as a superposition of sine waves,
and we still have omega^2 = kx^2 + ky^2 + kz^2. It is
trivial to determine kx and ky from the boundary conditions
and the mode number. So we are left with an expression
where kz and omega are the only variables.

When we try to get a "feel" for the behavior of guided
waves, the situation reminds me of the famous drawing
that can be seen either as a goblet or as two faces in
profile
http://www.cres.org/star/RubinGestalt.gif
... but you can't see it both ways at the same time.

A: One way of conceptualizing the guided wave is to
say that kx and ky don't count, so all we have is a
wave that moves in the Z direction with a nontrivial
dispersion relation kz(omega). This is 100% OK.
This correctly describes the black-box properties of
the guide. In this version, there is no zig-zagging.

B: The other way of conceptualizing it is to take away
the guide and set up a tricky superposition of waves
at funny angles, so that the superposition has nodes
at the places where the walls are going to be. Then
you can drop the walls into place without changing
anything in the interior of the guide. (In solid
state physics this would be called the "extended
zone scheme"). In this version, all modes (even
the lowest mode) have nontrivial zig-zagging.

==========

I can't resist pointing out the correspondence between
the waveguide equation and the massive scalar Klein-Gordon
equation. In the one case we have the waveguide cutoff
frequency, and in the other case we have the rest mass.

As they say: The same equations have the same solutions......