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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

An optical beam-splitter labeled with electric fields at the input and output ports. The directions of the arrows indicate the direction of propagation.

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

[1]
  1. ^ R. Loudon, The quantum theory of light, third edition, Oxford University Press, New York, NY, 2000.