From: rwc124@rsphy1.anu.edu.au (Roderick Vance)
Newsgroups: sci.optics
Subject: Re: Interferometry question
Date: 12 Oct 1992 23:37:20 GMT
Organization: Optical Sciences Centre, Australian National University
Distribution: world
Reply-To: rwc124@rsphy1.anu.edu.au (Roderick Vance)
NNTP-Posting-Host: 150.203.15.51

In article <1992Oct11.031644.19634@cfa203.harvard.edu>,
mcn@cfa203.HARVARD.EDU (Charley Noecker, Prec Astron) writes:

|> 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.
|

...Some  lines deleted ....

|> 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
|> 

I took part in an experiment a few years back with an optical fibre Mach
Zehnder
interferometer, wherein the beamsplitter and recombiners were 2x2
symmetric
directional couplers. A clear violation of the much-discussed phase rule
was seen;
in our experiments:

	phi_r1 + phi_r2 - phi_t1 - phi_t2 = about 170 degrees

This effect was highly repeatable.

Our explanation for the above was a variable coupler radiation loss, which 
depended on the coupler's input conditions. A very, very simple analytical
model of this effect can be made as follows. The system of two coupler 
input fibres has two bound eigenfields, namely the modes of each fibre
in isolation,
and the coupler has three eigenfields, one of which cannot be supported
by 
the two fibres in isolation. Thus this third eigenfield models the
radiation field.
(I know the radiation field comprises an uncountable continuum of modes,
but
i did say that this was a SIMPLE model!). Then the space of coupler
eigenfields
is a three dimensional vector space and the space bound eigenfields of
the two
fibres in isolation is a two dimensional subspace. It is now simple
geometry to
show that the excitation of the radiation eigenfield of the coupler
depends on the
relative phase between the two coupler inputs. The power driving this
radiation
eigenfield is lost at the coupler output - hence the variable loss that
gives rise to
the 170 degree phase difference above. 

It is a simple matter to show that the above would happen in a Michelson
interferometer
as well - the effect didn't seem important at the time so we never tried
to see
this experimentally. 

I hope that someone finds the above interesting.

Thanks for your attention

Roderick Vance
Optical Sciences Centre
Australian National University
two of which are linear superpositions of the Now, if you 
Newsgroups: sci.optics