From: mcn@cfa203.HARVARD.EDU (Charley Noecker, Prec Astron)
Newsgroups: sci.optics
Subject: Re: Interferometry question
Date: 11 Oct 92 03:16:44 GMT
Article-I.D.: cfa203.1992Oct11.031644.19634
Organization: Smithsonian Astrophysical Observatory, Cambridge, MA,  USA

Apologies to the net if you've already seen this; I've been having
trouble posting to Usenet lately.

From article <1at8h1INNkaj@agate.berkeley.edu>, 
by aephraim@physics.Berkeley.EDU (Aephraim M. Steinberg):
> In article <1992Oct2.140430.24476@cfa203.harvard.edu> 
> mcn@cfa203.HARVARD.EDU (Charley Noecker, Prec Astron) writes:

>> {my original letter deleted}
 
> I don't believe the relation holds for beamsplitters with loss.
> I did once come up with the following limit for the phase difference (I
> treated only the case where the beam splitter is symmetric, so there is
> a unique phi_t and a unique phi_r, which must differ by pi/2 if the
> beam-splitter is lossless):
> 
> cos (phi_r - phi_t) <= alpha / [2 |r| |t|] 
> 
> where r and t are the amplitude reflection and transmission coefficients
> and alpha is the power loss for a single incident beam.
> 
> Furthermore, if you assume that the total power loss is the same for any
> combination of incident beams (i.e., doesn't involve any interference
> effects), you CAN show that the phase shift is still always pi/2.  I am
> told that this is not generally true in complex structures, but I would
> imagine if the loss is due to scattering off of randomly distributed
> impurities or something like that, this should hold.
> -- 
> Aephraim M. Steinberg                    | "WHY must I treat the measuring
> UCB Physics                              | device classically??  What will
> aephraim@physics.berkeley.edu            | happen to me if I don't??"
>                                          |       -- Eugene Wigner

I did manage to prove the relation

	phi_r1 + phi_r2 - phi_t1 - phi_t2 = pi 

for a lossless beamsplitter and for a particular kind of lossy
beamsplitter: a thin partially-reflecting interface (like an index
step) with a layer of absorber on each side, possibly with different
total absorption; |r_1| is not always equal to |r_2| but by symmetry
|t_1| = |t_2| and phi_t1 = phi_t2.  

I'd like a proof/disproof of this phase rule using a beamsplitter
description which is general enough to include real multilayer or 
metal beamsplitter coatings.

The energy conservation condition I applied was that the sum of the
output beam intensities referred to the reflecting interface must be
equal to the sum of the input beam intensities also referred to the
reflecting interface:

    I_1 * abs_1  +  I_2 *abs_2   =   I_3 / abs_1  +  I_4 / abs_2

That is, the input intensity I_1 is reduced by the factor abs_1 before
it reaches the reflecting interface, and the outgoing beam on the same
side is reduced by the same factor abs_1 again before reaching the
observer as I_3; likewise for the other side of the beamsplitter.
This is essentially requiring all lost energy to be proportional to a
sensible linear combination of the beam __intensities__ (e.g. not
E-fields).  The phase rule above resulted from this energy
conservation requirement.

The sum of I_3 and I_4 is no longer constant under these conditions;
however, the sinusoidal variations of I_3 and I_4 are still exactly
opposite in phase.  The missing energy in the sum is deposited in the
absorbers on either side of the beamsplitter.  When I_3 reaches a
maximum, more energy is dissipated in the absorber on that side; when
I_4 reaches a maximum, the dissipation moves to that side.  If abs_1 >
abs_2, then the total lost light power reaches its max/min exactly
when I_3 reaches its max/min.  

Now the beamsplitter described above still isn't much like a real one.
I'd like to be sure that nothing unexpected happens when it's made of
-- alternating layers of dielectrics, each with a small amount of
	absorption, or
-- a thin layer of metal, perhaps with some paramagnetic or
	ferromagnetic contaminants.
The energy conservation condition would then have to be some condition
on the integral of local absorptivity times local light intensity.

The goal is to know whether there is ___EVER___ an exception to
the rule that the two intensities I_3 and I_4 vary exactly oppositely
in phase (and how bad is the violation).  We are in the precision
interferometry business, trying to develop a space-borne astrometric
optical interferometer with 5 micro-arcsec resolution; this requires
knowing all of the funny physics at the few-picometer level.  We too
will certainly want symmetric beamsplitters for suppression of drift
and systematics, so it is OK to limit the analysis to this situation.
(NB: the phase shift (phi_r - phi_t) is not necessarily pi/2 if the
beamsplitter isn't symmetric).

Thanks to Aephraim Steinberg for his comments; does anyone have
anything to add?

Charley Noecker
noecker@cfa.harvard.edu