# Beam splitter

Schematic illustration of a beam splitter cube.
1 - Incident light
2 - 50% Transmitted light
3 - 50% Reflected light
In practice, the reflective layer absorbs some light.
Beam splitters

A beam splitter is an optical device that splits a beam of light in two. It is the crucial part of most interferometers.

In its most common form, a cube, it is made from two triangular glass prisms which are glued together at their base using polyester, epoxy, or urethane-based adhesives. The thickness of the resin layer is adjusted such that (for a certain wavelength) half of the light incident through one "port" (i.e., face of the cube) is reflected and the other half is transmitted due to frustrated total internal reflection. Polarizing beam splitters, such as the Wollaston prism, use birefringent materials, splitting light into beams of differing polarization.

Aluminum coated beam splitter.

Another design is the use of a half-silvered mirror, a sheet of glass or plastic with a transparently thin coating of metal, now usually aluminium deposited from aluminium vapor. The thickness of the deposit is controlled so that part (typically half) of the light which is incident at a 45-degree angle and not absorbed by the coating is transmitted, and the remainder is reflected. A very thin half-silvered mirror used in photography is often called a pellicle mirror. To reduce loss of light due to absorption by the reflective coating, so-called "swiss cheese" beam splitter mirrors have been used. Originally, these were sheets of highly polished metal perforated with holes to obtain the desired ratio of reflection to transmission. Later, metal was sputtered onto glass so as to form a discontinuous coating, or small areas of a continuous coating were removed by chemical or mechanical action to produce a very literally "half-silvered" surface.

Instead of a metallic coating, a dichroic optical coating may be used. Depending on its characteristics, the ratio of reflection to transmission will vary as a function of the wavelength of the incident light. Dichroic mirrors are used in some ellipsoidal reflector spotlights to split off unwanted infrared (heat) radiation, and as output couplers in laser construction.

A third version of the beam splitter is a dichroic mirrored prism assembly which uses dichroic optical coatings to divide an incoming light beam into a number of spectrally distinct output beams. Such a device was used in three-pickup-tube color television cameras and the three-strip Technicolor movie camera. It is currently used in modern three-CCD cameras. An optically similar system is used in reverse as a beam-combiner in three-LCD projectors, in which light from three separate monochrome LCD displays is combined into a single full-color image for projection.

Beam splitters with single mode fiber for PON networks use the single mode behavior to split the beam. The splitter is done by physically splicing two fibers "together" as an X.

Arrangements of mirrors or prisms used as camera attachments to photograph stereoscopic image pairs with one lens and one exposure are sometimes called "beam splitters", but that is a misnomer, as they are effectively a pair of periscopes redirecting rays of light which are already non-coincident. In some very uncommon attachments for stereoscopic photography, mirrors or prism blocks similar to beam splitters perform the opposite function, superimposing views of the subject from two different perspectives through color filters to allow the direct production of an anaglyph 3D image, or through rapidly alternating shutters to record sequential field 3D video.

## Phase shift

Phase shift through a beam splitter with a dielectric coating.

A beam splitter that consists of a glass plate with a reflective dielectric coating on one side gives a phase shift of 0 or π, depending on the side from which it is incident (see figure).[1] Transmitted waves have no phase shift. Reflected waves entering from the reflective side (red) are phase-shifted by π, whereas reflected waves entering from the glass side (blue) have no phase shift. According to Fresnel equations there is only a phase shift when light incident from low refractive index to high refractive index (n = refractive index). This is the case in the transition of air to reflector, but not from glass to reflector. This does not apply to partial reflection by conductive (metallic) coatings, where other phase shifts occur in all paths (reflected and transmitted).

## Classical Lossless Beam Splitter

We consider a classical lossless 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

$|E_c|^2+|E_d|^2=|E_a|^2+|E_b|^2.$

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

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

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

$\begin{bmatrix} E_c \\ E_d \end{bmatrix} = \begin{bmatrix} Re^{i\phi_{ac}}& Te^{i\phi_{bc}} \\ Te^{i\phi_{ad}}& Re^{i\phi_{bd}} \end{bmatrix} \begin{bmatrix} E_a \\ E_b \end{bmatrix}.$[2]

## References

1. ^ Zetie, K P; S F Adams and R M Tocknell, How does a Mach–Zehnder interferometer work?, retrieved 13 February 2014
2. ^ R. Loudon, The quantum theory of light, third edition, Oxford University Press, New York, NY, 2000.