Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Beam splitter
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Classical lossless beam splitter== For beam splitters with two incoming beams, using a classical, lossless beam splitter with [[electromagnetic wave equation|electric fields]] ''E<sub>a</sub>'' and ''E<sub>b</sub>'' each incident at one of the inputs, the two output fields ''E<sub>c</sub>'' and ''E<sub>d</sub>'' are linearly related to the inputs through :<math> \mathbf{E}_\text{out} = \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} = \tau\mathbf{E}_\text{in}, </math> where the 2Γ2 element <math>\tau</math> is the beam-splitter transfer matrix and ''r'' and ''t'' are the [[reflectance]] and [[transmittance]] along a particular path through the beam splitter, that path being indicated by the subscripts. (The values depend on the polarization of the light.) If the beam splitter removes no energy from the light beams, the total output energy can be equated with the total input energy, reading :<math> |E_c|^2+|E_d|^2=|E_a|^2+|E_b|^2. </math> Inserting the results from the transfer equation above with <math>E_b=0</math> produces :<math> |r_{ac}|^2+|t_{ad}|^2=1, </math> and similarly for then <math>E_a=0</math> :<math> |r_{bd}|^2+|t_{bc}|^2=1. </math> When both <math>E_a</math> and <math>E_b</math> are non-zero, and using these two results we obtain :<math> r_{ac}t^{\ast}_{bc}+t_{ad}r^{\ast}_{bd}=0, </math> where "<math>^\ast</math>" indicates the complex conjugate. It is now easy to show that <math>\tau^\dagger\tau=\mathbf{I}</math> where <math>\mathbf{I}</math> is the identity, i.e. the beam-splitter transfer matrix is a [[unitary matrix]]. Each ''r'' and ''t'' can be written as a [[complex number]] having an amplitude and phase factor; for instance, <math>r_{ac}=|r_{ac}|e^{i\phi_{ac}}</math>. The phase factor accounts for possible shifts in phase of a beam as it reflects or transmits at that surface. Then we obtain :<math> |r_{ac}||t_{bc}|e^{i(\phi_{ac}-\phi_{bc})}+|t_{ad}||r_{bd}|e^{i(\phi_{ad}-\phi_{bd})}=0. </math> Further simplifying, the relationship becomes :<math> \frac{|r_{ac}|}{|t_{ad}|}=-\frac{|r_{bd}|}{|t_{bc}|}e^{i(\phi_{ad}-\phi_{bd}+\phi_{bc}-\phi_{ac})} </math> which is true when <math>\phi_{ad}-\phi_{bd}+\phi_{bc}-\phi_{ac}=\pi</math> and the exponential term reduces to -1. Applying this new condition and squaring both sides, it becomes :<math> \frac{1-|t_{ad}|^2}{|t_{ad}|^2}=\frac{1-|t_{bc}|^2}{|t_{bc}|^2}, </math> where substitutions of the form <math>|r_{ac}|^2=1-|t_{ad}|^2</math> were made. This leads to the result :<math> |t_{ad}|=|t_{bc}|\equiv T, </math> and similarly, :<math> |r_{ac}|=|r_{bd}|\equiv R. </math> It follows that <math>R^2+T^2=1</math>. Having determined the constraints describing a lossless beam splitter, the initial expression can be rewritten as :<math> \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}. </math><ref name="Loudon">R. Loudon, The quantum theory of light, third edition, Oxford University Press, New York, NY, 2000.</ref> Applying different values for the amplitudes and phases can account for many different forms of the beam splitter that can be seen widely used. The transfer matrix appears to have 6 amplitude and phase parameters, but it also has 2 constraints: <math>R^2+T^2=1</math> and <math>\phi_{ad}-\phi_{bd}+\phi_{bc}-\phi_{ac}=\pi</math>. To include the constraints and simplify to 4 independent parameters, we may write<ref name="teich">{{cite journal |last1=Campos |first1=Richard |last2=Bahaa |first2=Saleh |last3=Malvin |first3=Teich |title=Quantum mechanical lossless beam splitter: SU(2) symmetry and photon statistics |journal=Physical Review A |date=Aug 1, 1989 |volume=40 |issue=3 |pages=1371β1384 |doi=10.1103/PhysRevA.40.1371|pmid=9902272 |bibcode=1989PhRvA..40.1371C }}</ref> <math>\phi_{ad}=\phi_0+\phi_T, \phi_{bc}=\phi_0-\phi_T, \phi_{ac}=\phi_0+\phi_R</math> (and from the constraint <math>\phi_{bd}=\phi_0-\phi_R-\pi</math>), so that :<math> \begin{align} \phi_T & = \tfrac{1}{2}\left(\phi_{ad} - \phi_{bc} \right)\\ \phi_R & = \tfrac{1}{2}\left(\phi_{ac} - \phi_{bd} +\pi \right)\\ \phi_0 & = \tfrac{1}{2}\left(\phi_{ad} + \phi_{bc} \right) \end{align} </math> where <math>2\phi_T</math> is the phase difference between the transmitted beams and similarly for <math>2\phi_R</math>, and <math>\phi_0</math> is a global phase. Lastly using the other constraint that <math>R^2+T^2=1</math> we define <math>\theta = \arctan(R/T) </math> so that <math>T=\cos\theta,R=\sin\theta</math>, hence :<math> \tau=e^{i\phi_0}\begin{bmatrix} \sin\theta e^{i\phi_R} & \cos\theta e^{-i\phi_T} \\ \cos\theta e^{i\phi_T} & -\sin\theta e^{-i\phi_R} \end{bmatrix}. </math> A 50:50 beam splitter is produced when <math>\theta=\pi/4</math>. The [[#Phase shift|dielectric beam splitter]] above, for example, has :<math> \tau=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, </math> i.e. <math>\phi_T = \phi_R =\phi_0=0</math>, while the "symmetric" beam splitter of Loudon <ref name="Loudon" /> has :<math> \tau=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}, </math> i.e. <math>\phi_T = 0, \phi_R =-\pi/2, \phi_0=\pi/2</math>.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)