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SECTION TITLE: Classical Lossless Beam Splitter
NEED TO UPLOAD THE RIGHT FIGURE (BeamSplitter.png, but renamed?)
THE FIGURE displays a simple diagram of a beam-splitter with electric fields incident at both its inputs. The two output fields Ec and Ed are linearly related to the inputs through
where the 2 × 2 element is the beam-splitter matrix. r and t are the reflectance and transmittance along a particular path through the beam-splitter, that path being indicated by the subscripts.
Assuming the beam-splitter removes no energy from the light beams, the total output energy can be equated with the total input energy, reading
Requiring this energy conservation brings about the relationships between reflectance and transmittance
and
where "" indicates the complex conjugate.
Expanding, we can write each r and t as a complex number having an amplitude and phase factor; for instance, . The phase factor accounts for possible shifts in phase of a beam as it reflects or transmits at that surface. We then obtain
Further simplifying we obtain the relationship
which is true when and the exponential term reduces to -1. Applying this new condition and squaring both sides, we obtain
where substitutions of the form were made. This leads us to the result
and similarly,
It follows that .
Now that the constraints describing a lossless beam-splitter have been determined, we can rewrite our initial expression as