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 Ec and Ed 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}, $$ 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
 * E_c|^2+|E_d|^2=|E_a|^2+|E_b|^2.

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

r_{ac}t^{\ast}_{bc}+t_{ad}r^{\ast}_{bd}=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}. $$