A beam splitter is an optical device that splits a beam of light in two. It
is the crucial part of most interferometers.
Beam splitter designs 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 half of the light incident through one
"port" 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. 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 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 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 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. Transmitted waves
have no phase shift. Reflected waves entering from the reflective side are
phase-shifted by π, whereas reflected waves entering from the glass side have
no phase shift. This is due to the Fresnel equations, according to which
reflection causes a phase shift only when light passing through a material of
low refractive index is reflected at a material of high 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 coatings, where other phase shifts occur in all paths.
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
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
Use in experiments Several experiments in quantum theory
and relativity theory and other fields of physics use beam splitters; these
include: The Fizeau experiment of 1851 to measure
the speeds of light in water The Michelson-Morley experiment of 1887
to measure the effect of the luminiferous aether on the speed of
light The Hammar experiment of 1935 to refute
Dayton Miller's claim of a positive result from repetitions of the
Michelson-Morley experiment The Kennedy-Thorndike experiment of 1932
to test the independence of the speed of light and the velocity of the measuring
apparatus Bell test experiments to demonstrate
consequences of quantum entanglement and exclude local hidden variable theories
Wheeler's delayed choice experiment of 1978, 1984 etc., a series of thought
experiments in quantum mechanics to test what makes a photon behave as a wave or
a particle and when it happens The FELIX experiment to test the Penrose
interpretation that quantum superposition depends on space-time
curvature The many uses of the Mach–Zehnder
interferometer in areas such as quantum mechanics and quantum computation
Quantum mechanical description We consider two single mode input
fields, denoted by the annihilation operators which incident on the two
input ports of the beam splitter. The two output fields, denoted by are
linearly related to the input field by In order to obtain the entries of the
transformation matrix one must take into account that the commutation relations
of the fields hold. As is well-known from second quantization, one must make
sure that and
This, together with energy conservation yields the following set of constraints:
For a dielectric 50:50 beam splitter the reflected and transmitted beams differ
in phase by . Assuming the reflected beam suffers a phase shift, the input
and output fields are related by: The unitary transformation associated
with this transformation is Using this unitary, one can also write
the transforms as = Application for quantum computing=
In 2000 Knill, Laflamme and Milburn proved that it is possible to create
universal quantum computer solely with beam splitters, phase shifters,
photodetectors and single photon sources. The states that form a qubit in
this protocol are the one photon states of two modes, i.e. the states |01> and
|10> in the occupation number representation of two modes. Using these
resources it's possible to implement any single qubit gate and 2-qubit
probabilistic gates. The beam splitter is an essential component in this scheme
since it's the only one that creates entanglement between the Fock states.
References