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Beam splitter
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{{Short description|Optical device which splits a beam of light in two}} {{Refimprove|date=May 2014}} [[File:Beam-splitter.svg|255px|thumb|Schematic illustration of a beam splitter cube.<br> '''1''' - Incident light<br>'''2''' - 50% transmitted light<br>'''3''' - 50% reflected light<br>In practice, the reflective layer absorbs some light.]] [[File:Beamsplitter-2.jpg|255px|thumb|Beam splitters]] A '''beam splitter''' or '''beamsplitter''' is an [[optical instrument|optical device]] that splits a beam of [[light]] into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as [[Interferometry|interferometers]], also finding widespread application in [[fibre optic]] [[telecommunications]]. ==Designs== In its most common form, a cube, a beam splitter is made from two triangular glass [[prism (optics)|prism]]s which are glued together at their base using polyester, [[epoxy]], or urethane-based adhesives. (Before these synthetic [[resin]]s, natural ones were used, e.g. [[Canada balsam]].) 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 [[reflection (physics)|reflected]] and the other half is transmitted due to [[Total internal reflection#Frustrated_TIR|FTIR (frustrated total internal reflection)]]. [[polarizer|Polarizing beam splitters]], such as the [[Wollaston prism]], use [[birefringent]] materials to split light into two beams of orthogonal [[polarization (waves)|polarization]] states. [[File:Flat metal-coated beamsplitter.png|thumb|left|Aluminium-coated beam splitter.]] Another design is the use of a half-silvered mirror. This is composed of an optical substrate, which is often a sheet of glass or plastic, with a partially transparent thin coating of metal. The thin coating can be [[aluminium]] deposited from aluminium [[vapor]] using a [[physical vapor deposition]] method. 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 or substrate material, 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 (North America)|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 [[sputtering|sputter]]ed 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 ([[thin-film interference]]), 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 spotlight]]s to split off unwanted [[infrared]] (heat) radiation, and as [[output coupler]]s in [[laser construction]]. A third version of the beam splitter is a [[dichroic prism|dichroic mirrored prism]] assembly which uses [[dichroism|dichroic]] [[optical coating]]s 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 camera]]s 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]] [[Image projector|projectors]], in which light from three separate monochrome LCD displays is combined into a single full-color image for projection. Beam splitters in [[Passive optical network|PON networks]] are often made with [[single-mode optical fiber]], by exploiting [[evanescent wave]] coupling between a pair of fibers to share the beam between them.<ref>{{cite web |last1=Paschotta |first1=Rüdiger |title=Beam Splitters |url=https://doi.org/10.61835/mjw |website=RP Photonics Encyclopedia |publisher=RP Photonics AG |access-date=15 March 2025 |doi=10.61835/mjw}}</ref> The splitter is constructed by fusing together the two parallel bare fibers at one point.<ref>{{cite web |last1=Paschotta |first1=Rüdiger |title=Tutorial: Passive Fiber Optics, Part 8: Fiber Couplers and Splitters |url=https://www.rp-photonics.com/tutorial_passive_fiber_optics8.html |website=RP Photonics Encyclopedia |publisher=RP Photonics AG |access-date=15 March 2025}}</ref> Arrangements of mirrors or prisms used as camera attachments to photograph [[stereoscopy|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 [[periscope]]s 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 [[Active shutter 3D system|sequential field 3D]] video. ==Phase shift== [[File:Wavesplitter1.GIF|thumb|Phase shift through a beam splitter with a dielectric coating.]] Beam splitters are sometimes used to recombine beams of light, as in a [[Mach–Zehnder interferometer]]. In this case there are two incoming beams, and potentially two outgoing beams. But the amplitudes of the two outgoing beams are the sums of the (complex) amplitudes calculated from each of the incoming beams, and it may result that one of the two outgoing beams has amplitude zero. In order for energy to be conserved (see next section), there must be a phase shift in at least one of the outgoing beams. For example (see red arrows in picture on the right), if a polarized light wave in air hits a [[dielectric ]] surface such as glass, and the electric field of the light wave is in the plane of the surface, then the reflected wave will have a phase shift of π, while the transmitted wave will not have a phase shift; the blue arrow does not pick up a phase-shift, because it is reflected from a medium with a lower refractive index. The behavior is dictated by the [[Fresnel equations]].<ref>{{citation |author1=Zetie, K P |author2=Adams, S F |author3=Tocknell, R M |title=How does a Mach–Zehnder interferometer work? |url=https://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_zehnder00.pdf |access-date=13 February 2014}}</ref> This does not apply to partial reflection by conductive (metallic) coatings, where other phase shifts occur in all paths (reflected and transmitted). In any case, the details of the phase shifts depend on the type and geometry of the beam splitter. ==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 2×2 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=1371–1384 |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>. == Use in experiments == Beam splitters have been used in both [[thought experiment]]s and [[Experimental physics|real-world experiments]] in the area of [[quantum mechanics|quantum theory]] and [[relativity theory]] and other fields of [[physics]]. 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 (hypothetical) [[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]] (from ca. 1972) to demonstrate consequences of [[quantum entanglement]] and exclude [[local hidden-variable theory|local hidden-variable theories]] * [[Wheeler's delayed choice experiment]] of 1978, 1984 etc., to test what makes a photon behave as a wave or a particle and when it happens * The [[Free-orbit experiment with laser interferometry X-rays|FELIX]] experiment (proposed in 2000) to test the [[Penrose interpretation]] that [[quantum superposition]] depends on [[spacetime curvature]] * The [[Mach–Zehnder interferometer]], used in various experiments, including the [[Elitzur–Vaidman bomb tester]] involving [[interaction-free measurement]]; and in others in the area of [[quantum computation]] == Quantum mechanical description == In quantum mechanics, the electric fields are operators as explained by [[second quantization]] and [[Fock state|Fock states]]. Each electrical field operator can further be expressed in terms of [[Mode (electromagnetism)|modes]] representing the wave behavior and amplitude operators, which are typically represented by the dimensionless [[creation and annihilation operators]]. In this theory, the four ports of the beam splitter are represented by a photon number state <math>|n\rangle</math> and the action of a creation operation is <math>\hat{a}^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle</math>. The following is a simplified version of Ref.<ref name="teich" /> The relation between the classical field amplitudes <math>{E}_{a},{E}_{b}, {E}_{c}</math>, and <math>{E}_{d}</math> produced by the beam splitter is translated into the same relation of the corresponding quantum creation (or annihilation) operators <math>\hat{a}_a^\dagger,\hat{a}_b^\dagger, \hat{a}_c^\dagger</math>, and <math>\hat{a}_d^\dagger</math>, so that :<math> \left(\begin{matrix} \hat{a}_c^\dagger\\ \hat{a}_d^\dagger \end{matrix}\right)= \tau \left(\begin{matrix} \hat{a}_a^\dagger\\ \hat{a}_b^\dagger \end{matrix}\right) </math> where the transfer matrix is given in [[#Classical lossless beam splitter|classical lossless beam splitter]] section above: :<math> \tau=\left(\begin{matrix} r_{ac} & t_{bc}\\ t_{ad} & r_{bd} \end{matrix}\right) =e^{i\phi_0}\left(\begin{matrix} \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{matrix}\right). </math> Since <math>\tau</math> is unitary, <math>\tau^{-1}=\tau^\dagger</math>, i.e. :<math> \left(\begin{matrix} \hat{a}_a^\dagger\\ \hat{a}_b^\dagger \end{matrix}\right)= \left(\begin{matrix} r_{ac}^\ast & t_{ad}^\ast\\ t_{bc}^\ast & r_{bd}^\ast \end{matrix}\right) \left(\begin{matrix} \hat{a}_c^\dagger\\ \hat{a}_d^\dagger \end{matrix}\right). </math> This is equivalent to saying that if we start from the vacuum state <math>|00\rangle_{ab}</math> and add a photon in port ''a'' to produce :<math>|\psi_\text{in}\rangle=\hat{a}_a^\dagger|00\rangle_{ab}=|10\rangle_{ab},</math> then the beam splitter creates a superposition on the outputs of :<math>|\psi_\text{out}\rangle=\left(r_{ac}^\ast\hat{a}_c^\dagger+t_{ad}^\ast\hat{a}_d^\dagger\right)|00\rangle_{cd}=r_{ac}^\ast|10\rangle_{cd}+t_{ad}^\ast|01\rangle_{cd}.</math> The probabilities for the photon to exit at ports ''c'' and ''d'' are therefore <math>|r_{ac}|^2</math> and <math>|t_{ad}|^2</math>, as might be expected. Likewise, for any input state <math>|nm\rangle_{ab}</math> :<math> |\psi_\text{in}\rangle=|nm\rangle_{ab} =\frac{1}{\sqrt{n!}}\left(\hat{a}_a^\dagger\right)^n\frac{1}{\sqrt{m!}}\left(\hat{a}_b^\dagger\right)^m|00\rangle_{ab} </math> and the output is :<math> |\psi_\text{out}\rangle =\frac{1}{\sqrt{n!}} \left(r_{ac}^\ast\hat{a}_c^\dagger+t_{ad}^\ast\hat{a}_d^\dagger\right)^n \frac{1}{\sqrt{m!}} \left(t_{bc}^\ast\hat{a}_c^\dagger+r_{bd}^\ast\hat{a}_d^\dagger\right)^m |00\rangle_{cd}. </math> Using the [[Binomial theorem#Multi-binomial theorem|multi-binomial theorem]], this can be written :<math> \begin{align} |\psi_\text{out}\rangle &=\frac{1}{\sqrt{n!m!}} \sum_{j=0}^n \sum_{k=0}^m \binom{n}{j} \left( r_{ac}^\ast \hat{a}_c^\dagger \right)^j \left( t_{ad}^\ast \hat{a}_d^\dagger \right) ^{(n-j)} \binom{m}{k} \left( t_{bc}^\ast \hat{a}_c^\dagger \right)^k \left( r_{bd}^\ast \hat{a}_d^\dagger \right) ^{(m-k)} |00\rangle_{cd} \\ &=\frac{1}{\sqrt{n!m!}} \sum_{N=0}^{n+m} \sum_{j=0}^N \binom{n}{j} r_{ac}^{\ast j} t_{ad}^{\ast (n-j)} \binom{m}{N-j} t_{bc}^{\ast (N-j)} r_{bd}^{\ast (m-N+j)} \left(\hat{a}_c^\dagger\right)^N \left( \hat{a}_d^\dagger\right)^{M}|00\rangle_{cd}, \\ &=\frac{1}{\sqrt{n!m!}} \sum_{N=0}^{n+m} \sum_{j=0}^N \binom{n}{j} \binom{m}{N-j} r_{ac}^{\ast j} t_{ad}^{\ast (n-j)} t_{bc}^{\ast (N-j)} r_{bd}^{\ast (m-N+j)} \sqrt{N!M!} \quad |N,M\rangle_{cd},\end{align} </math> where <math>M=n+m-N</math> and the <math>\tbinom{n}{j}</math> is a binomial coefficient and it is to be understood that the coefficient is zero if <math>j\notin\{ 0,n \}</math> etc. The transmission/reflection coefficient factor in the last equation may be written in terms of the reduced parameters that ensure unitarity: :<math> r_{ac}^{\ast j} t_{ad}^{\ast (n-j)} t_{bc}^{\ast (N-j)} r_{bd}^{\ast (m-N+j)} =(-1)^j\tan^{2j}\theta(-\tan\theta)^{m-N}\cos^{n+m}\theta\exp-i\left[(n+m)(\phi_0+\phi_T)-m(\phi_R+\phi_T)+N(\phi_R-\phi_T)\right]. </math> where it can be seen that if the beam splitter is 50:50 then <math>\tan\theta=1</math> and the only factor that depends on ''j'' is the <math>(-1)^j</math> term. This factor causes interesting interference cancellations. For example, if <math>n=m</math> and the beam splitter is 50:50, then :<math> \begin{align} \left(\hat{a}_a^\dagger\right)^n\left(\hat{a}_b^\dagger\right)^m &\to \left[\hat{a}_a^\dagger\hat{a}_b^\dagger\right]^n \\ &= \left[\left(r_{ac}^\ast\hat{a}_c^\dagger+t_{ad}^\ast\hat{a}_d^\dagger\right) \left(t_{bc}^\ast\hat{a}_c^\dagger+r_{bd}^\ast\hat{a}_d^\dagger\right) \right]^n \\ &= \left[\frac{e^{-i\phi_0}}{\sqrt{2}}\right]^{2n} \left[\left(e^{-i\phi_R}\hat{a}_c^\dagger+e^{-i\phi_T}\hat{a}_d^\dagger\right) \left(e^{i\phi_T}\hat{a}_c^\dagger-e^{i\phi_R}\hat{a}_d^\dagger\right) \right]^n \\ &= \frac{e^{-2in\phi_0}}{2^n}\left[e^{i(\phi_T-\phi_R)} \left(\hat{a}_c^\dagger\right)^2 +e^{-i(\phi_T-\phi_R)}\left(\hat{a}_d^\dagger\right)^2 \right]^n \end{align} </math> where the <math> \hat{a}_c^\dagger \hat{a}_d^\dagger </math> term has cancelled. Therefore the output states always have even numbers of photons in each arm. A famous example of this is the [[Hong–Ou–Mandel effect]], in which the input has <math>n=m=1</math>, the output is always <math>|20\rangle_{cd}</math> or <math>|02\rangle_{cd}</math>, i.e. the probability of output with a photon in each mode (a coincidence event) is zero. Note that this is true for all types of 50:50 beam splitter irrespective of the details of the phases, and the photons need only be indistinguishable. This contrasts with the classical result, in which equal output in both arms for equal inputs on a 50:50 beam splitter does appear for specific beam splitter phases (e.g. a symmetric beam splitter <math>\phi_0=\phi_T=0,\phi_R=\pi/2</math>), and for other phases where the output goes to one arm (e.g. the dielectric beam splitter <math>\phi_0=\phi_T=\phi_R=0</math>) the output is always in the same arm, not random in either arm as is the case here. From the [[correspondence principle]] we might expect the quantum results to tend to the classical one in the limits of large ''n'', but the appearance of large numbers of indistinguishable photons at the input is a non-classical state that does not correspond to a classical field pattern, which instead produces a statistical mixture of different <math>|n,m\rangle</math> known as [[Photon statistics#Poissonian light|Poissonian light]]. Rigorous derivation is given in the Fearn–Loudon 1987 paper<ref>{{cite journal|last1=Fearn|first1=H.|last2=Loudon|first2=R.|date=1987|title=Quantum theory of the lossless beam splitter|journal=Optics Communications|volume=64|pages=485–490|doi=10.1016/0030-4018(87)90275-6|number=6|bibcode=1987OptCo..64..485F }}</ref> and extended in Ref <ref name="teich" /> to include statistical mixtures with the [[density matrix]]. === Non-symmetric beam-splitter === In general, for a non-symmetric beam-splitter, namely a beam-splitter for which the transmission and reflection coefficients are not equal, one can define an angle <math>\theta</math> such that <math>\begin{cases} |R| = \sin(\theta)\\ |T| = \cos(\theta) \end{cases}</math> where <math>R</math> and <math>T</math> are the reflection and transmission coefficients. Then the unitary operation associated with the beam-splitter is then <math> \hat{U}=e^{i\theta\left(\hat{a}_{a}^{\dagger}\hat{a}_{b}+\hat{a}_{a}\hat{a}_{b}^{\dagger}\right)}. </math> === Application for quantum computing === In 2000 Knill, Laflamme and Milburn ([[KLM protocol]]) proved that it is possible to create a 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 ([[Fock state]]) of two modes. Using these resources it is possible to implement any single qubit gate and 2-qubit probabilistic gates. The beam splitter is an essential component in this scheme since it is the only one that creates [[Quantum entanglement|entanglement]] between the [[Fock states]]. Similar settings exist for [[continuous-variable quantum information|continuous-variable quantum information processing]]. In fact, it is possible to simulate arbitrary [[Bogoliubov transformation|Gaussian (Bogoliubov) transformations]] of a quantum state of light by means of beam splitters, phase shifters and photodetectors, given [[Squeezed coherent state|two-mode squeezed vacuum states]] are available as a prior resource only (this setting hence shares certain similarities with a Gaussian counterpart of the [[KLM protocol]]).<ref>{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title= Simulating arbitrary Gaussian circuits with linear optics|journal=Physical Review A|date=2018|volume=98|issue=6 |page=062314|doi=10.1103/PhysRevA.98.062314|arxiv=1803.11534|bibcode=2018PhRvA..98f2314C }}</ref> The building block of this simulation procedure is the fact that a beam splitter is equivalent to a [[Squeezed coherent state#Operator representation|squeezing transformation]] under ''partial'' [[T-symmetry|time reversal]]. ==Diffractive beam splitter== {{excerpt|Diffractive beam splitter}} ==Reflection beam splitters== [[File:InfraTec Beamsplitter Prinzip.jpg|thumb|Principle drawing of a reflection beam splitter in a pyroelectric sensor (four optical channels)]] Reflection beam splitters reflect parts of the incident radiation in different directions. These partial beams show exactly the same intensity. Typically, reflection beam splitters are made of metal and have a broadband spectral characteristic. Due to their compact design, beam splitters of this type are particularly easy to install in [[infrared detector]]s.<ref>[https://www.infratec.eu/sensor-division/service-support/glossary/beam-splitters ''Beam Splitters''] on www.infratec.eu</ref> At this application, the radiation enters through the [[aperture]] opening of the detector and is split into several beams of equal intensity but different directions by internal highly reflective microstructures. Each beam hits a sensor element with an upstream optical filter. Particularly in [[Nondispersive infrared sensor|NDIR gas analysis]], this design enables measurement with only one beam with a minimal beam cross-section, which significantly increases the interference immunity of the measurement. ==See also== *[[Power dividers and directional couplers]] ==References== {{commons category|Beam splitters}} {{reflist}} [[Category:Mirrors]] [[Category:Optical components]] [[Category:Microscopy]]
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