User:Thintommy/sandbox

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 E_c and E_d are linearly related to the inputs through

${\begin{bmatrix}E_{c}\\E_{d}\end{bmatrix}}={\begin{bmatrix}r_{ac}&t_{bc}\\t_{ad}&r_{bd}\end{bmatrix}}{\begin{bmatrix}E_{a}\\E_{b}\end{bmatrix}},$

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

$|E_{c}|^{2}+|E_{d}|^{2}=|E_{a}|^{2}+|E_{b}|^{2}.$

Requiring this energy conservation brings about the relationships between reflectance and transmittance

$|r_{ac}|^{2}+|t_{ad}|^{2}=|r_{bd}|^{2}+|t_{bc}|^{2}=1$

and

$r_{ac}t_{bc}^{\ast }+t_{ad}r_{bd}^{\ast }=0,$

where "$^{\ast }$" indicates the complex conjugate. Expanding, we can write each r and t as a complex number having an amplitude and phase factor; for instance, $r_{ac}=|r_{ac}|e^{i\phi _{ac}}$. The phase factor accounts for possible shifts in phase of a beam as it reflects or transmits at that surface. We then obtain

$|r_{ac}||t_{bc}|e^{i(\phi _{ac}-\phi _{bc})}+|t_{ad}||r_{bd}|e^{i(\phi _{ad}-\phi _{bd})}=0.$

Further simplifying we obtain the relationship

${\frac {|r_{ac}|}{|t_{ad}|}}=-{\frac {|r_{bd}|}{|t_{bc}|}}e^{i(\phi _{ad}-\phi _{bd}+\phi _{bc}-\phi _{ac})}$

which is true when $\phi _{ad}-\phi _{bd}+\phi _{bc}-\phi _{ac}=\pi$ and the exponential term reduces to -1. Applying this new condition and squaring both sides, we obtain

${\frac {1-|t_{ad}|^{2}}{|t_{ad}|^{2}}}={\frac {1-|t_{bc}|^{2}}{|t_{bc}|^{2}}},$

where substitutions of the form $|r_{ac}|^{2}=1-|t_{ad}|^{2}$ were made. This leads us to the result

$|t_{ad}|=|t_{bc}|\equiv T,$

and similarly,

$|r_{ac}|=|r_{bd}|\equiv R.$

It follows that $R^{2}+T^{2}=1$.

Now that the constraints describing a lossless beam-splitter have been determined, we can rewrite our initial expression as

${\begin{bmatrix}E_{c}\\E_{d}\end{bmatrix}}={\begin{bmatrix}Re^{i\phi _{ac}}&Te^{i\phi _{bc}}\\Te^{i\phi _{ad}}&Re^{i\phi _{bd}}\end{bmatrix}}{\begin{bmatrix}E_{a}\\E_{b}\end{bmatrix}}.$