Re: [Phys-l] Transparencies and phase retardation of wave

[John Denker <jsd@av8n.com>]

On 04/07/2007 09:34 AM, alex brown wrote:
> [1] I am having problems with understanding how phase and intensity works and would appreciate
> your comments on my summary and questions below of what happens when light passes through a 
> transparency.

OK.


> With reference to Hecht p618 I understand that a transparency does not cause amplitude modulation
>  of a wave, just a phase modulation.  We cannot see the effect of this by viewing the phase 
> retarded wave on a screen.  This phase retarded wave can be thought of as consisting of the 
> incident wave and a diffracted wave from the transparency(called phase object by Hecht). Why does
>  he just say one diffracted wave and not a whole load of plane waves travelling in different 
> directions representing modes?  Like you would see from a line of souces oscillating in phase.
>
> [2] Hecht states that there is a phase difference between the two of pi/2....
>
> Why is it this and not another value?
>
> and so one wave lags the other in terms of the phasor diagram.  

Note that I consider [1] to be mostly a physics question, in
contrast to [2] which is mostly a question about the underlying
mathematics.

If I understand the question aright, the central issue is the
definition of "phasor component".

To describe any sinusoidal function of anything:
 a) it suffices to specify the phase and amplitude.  Call this
  polar coordinates.
 b) it suffices to specify the two phasor components.  Call this
  Cartesian coordinates.

Here's the picture:
  http://www.av8n.com/physics/img48/argand-phasor.png
where
  (A, theta) = (amplitude, phase)
  (p, q)     = phasor components

Note that (a) and (b) are just two representations of the same
thing.  Switching from one to the other is just a change of
coordinates.  We are talking about the coordinates of a point
in the /phase space/ of the system ... and I am using the term
phase space in a very precise technical sense.  Liouville's
theorem and all that.

One phasor component is perpendicular to the other by construction,
by definition.  FWIW you could span the phase space using two
/non-perpendicular/ basis vectors ... but there would be no
advantage in doing so.

I repeat that this applies to any sinusoidal function of anything.
 -- This includes but is not limited to sinusoidal waves.
 -- This includes but is not limited to modes of the EM field.

> I now have two waves of the same 
> frequency whose E fields are perpendicular to each other.  

No.  Phasor components are perpendicular in phase-space, not
in position-space or polarization-vector-space.

> The sum of these, which we see at the 
> screen, produces the equation of the ellipse.  

No.

An ellipse in the xy plane can arise if x is varying like
cos(phase) and y is varying like sin(phase).  But here
(so far anyway) we aren't talking about the xy plane.
We're talking about a *scalar* which happens to be a
sinusoidal function of something else.

Forget about the EM field, forget about polarization, and
forget about transparent media;  instead consider the
sinusoidal motion of a one-dimensional mass-on-a-spring
oscillator.  This sinusoidal motion has a phase and an
amplitude, and it has phasor components.  If you change
the phase of this oscillator, this can be described by
a change in the phasor components.  There can't be an
ellipse in 1D.

  The overall EM field can be broken down into 2N
  harmonic oscillators, which can be considered
  independent 1D scalars for present purposes.
  Each of these 2N oscillators has its own phase
  and amplitude ... or (equivalently) its own
  phasor components.  The number 2N is meant to
  cover the two polarizations and the N different
  spatial modes, i.e. N different k-vectors.

> Using the phase difference given the light is 
> circularly polarised at the screen.  

This needlessly confuses phase with polarization.

I repeat:  consider a single mode, which implies a single
polarization (and a single k-vector).  Each such mode has
its own phase and amplitude.  A transparent phase-shifter
will not change the polarization (or the k-vector).

> We can produce an amplitude modulation at the screen by introducing a further phase retardation 
> so the two E fields cause interference pattern... 

No, we can't.

There is no interference between one phasor component and
the other.  The two phasor basis vectors are orthogonal,
so the cross term (the interference term) necessarily
vanishes.

To say the same thing in symbols, A^2 = p^2 + q^2 with no
cross term, i.e. no pq interference term, as you can see
from the picture
   http://www.av8n.com/physics/img48/argand-phasor.png