# Beam propagation method

**Beam Propagation Method**(BPM) refers to a computational technique inElectromagnetics , usedto solve theHelmholtz equation under conditions of a time-harmonic wave. BPM works under slowly varying envelope approximation, for linear and nonlinear equations.The

**beam propagation method**(BPM) is an approximation technique for simulating the propagation oflight in slowly varying opticalwaveguide s. It is essentially the same as the so-calledParabolic Equation (PE) method in underwateracoustics . Both BPM and the PE were first introduced in 1970's. When a wave propagates along a waveguide for a large distance (larger compared with the wavelength), rigorous numerical simulation is difficult. The BPM relies on approximate differential equations which are also called the one-way models. These one-way models involve only a first orderderivative in the variable z (for the waveguide axis) and they can be solved as "initial" value problem. The "initial" value problem does not involve time, rather it is for the spatial variable z. [*citation|title=Integrated Photonics|author=Clifford R. Pollock, Michal. Lipson|year= 2003|publisher=Springer|id=ISBN 1402076355 |url=http://books.google.com/books?vid=ISBN1402076355&id=DNJEoypcI6oC&pg=RA1-PA210&lpg=RA1-PA210&ots=o4ORvoHrCJ&dq=%22Beam+propagation+method%22&ie=ISO-8859-1&output=html&sig=kY2xdP7q7iv5MdX_3vzd63LmOpg*]The original BPM and PE were derived from the slowly varying envelope approximation and they are the so-called paraxial one-way models. Since then, a number of improved one-way models are introduced. They come from a one-way model involving a square root operator. They are obtained by applying rational approximations to the square root operator. After a one-way model is obtained, one still has to solve it by discretizing the variable z. However, it is possible to merge the two steps (rational approximation to the square root operator and discretization of z) into one step. Namely, one can find rational approximations to the so-called one-way propagator (the exponential of the square root operator) directly. The rational approximations are not trivial. Standard diagonal Padé approximants have trouble with the so-called evanescent modes. These evanescent modes should decay rapidly in z, but the diagonal Padé approximants will incorrectly propagate them as propagating modes along the waveguide. Modified rational approximants that can suppress the evanescent modes are now available. The accuracy of the BPM can be further improved, if you use the energy-conserving one-way model or the single-scatter one-way model.

**Principles**BPM is generally formulated as a solution to Helmholtz equation in a time-harmonic case, [

*Okamoto K. 2000 Fundamentals of Optical Waveguides (San Diego, CA: Academic)*] [*EE290F: BPM course slides, Devang Parekh, University of Berkeley, CA*] :$(\; abla^2\; +\; k\_0^2n^2)psi\; =\; 0$with the field written as, :$E(x,y,z,t)=psi(x,y,x)exp(-jomega\; t)$.Now the spatial dependence of this field is written according to any one TE or

TM polarizations:$psi(x,y)\; =\; A(x,y)exp(+jk\_o\; u\; y)$,with the envelope :$A(x,y)$ following a slowly varying approximation, :$frac\{partial^2(\; A(x,y)\; )\}\{partial\; y^2\}\; =\; 0$Now the solution when replaced into the Helmholtz equation follows,:$left\; [frac\{partial^2\; \}\{partial\; x^2\}\; +\; k\_0^2(n^2\; -\; u^2)\; ight]\; A(x,y)\; =\; pm\; 2\; jk\_0\; u\; frac\{partial\; A\_k(x,z)\}\{partial\; z\}$

With the aim to calculate the field at all points of space for all times, we only need to compute the function$A(x,y)$ for all space, and then we are able to reconstruct $psi(x,y,z)$. Since the solutionis for the time-harmonic Helmholtz equation, we only need to calculate it over one time period. We can visualize the fields along the propagation direction, or the cross section waveguide modes.

The master equation is discretized (using various centralized difference, crank nicholson scheme etc) and rearranged in a causal fashion. Through iteration the field evolution is computed, along the propagationdirection.

**Applications**BPM is a quick and easy method of solving for fields in integrated optical devices. It is typicallyused only in solving for intensity and modes within shaped (bent, tapered, terminated) waveguidestructures, as opposed to scattering problems. It is not suited as a generalized solution to Maxwellequations, like the FDTD or FEM methods.

**ee also***

Computational electromagnetics

*Finite-difference time-domain method

*Finite element method

*Maxwell's equations

* Method of Lines

*Light

*Photon **References**

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Laser beam profiler**— A laser beam profiler captures, displays, and records the spatial intensity profile of a laser beam at a particular plane transverse to the beam propagation path. Since there are many types of lasers ultraviolet, visible, infrared, continuous… … Wikipedia**Monte Carlo method for photon transport**— Modeling photon propagation with Monte Carlo methods is a flexible yet rigorous approach to simulate photon transport. In the method, local rules of photon transport are expressed as probability distributions which describe the step size of… … Wikipedia**Non-line-of-sight propagation**— Non line of sight (NLOS) or near line of sight is a term used to describe radio transmission across a path that is partially obstructed, usually by a physical object in the innermost Fresnel zone. Many types of radio transmissions depend, to… … Wikipedia**Convolution for optical broad-beam responses in scattering media**— Photon transport theories, such as the Monte Carlo method, are commonly used to model light propagation in tissue. The responses to a pencil beam incident on a scattering medium are referred to as Green’s functions or impulse responses. Photon… … Wikipedia**History of scientific method**— The history of scientific method is inseparable from the history of science itself. The development and elaboration of rules for scientific reasoning and investigation has not been straightforward; scientific method has been the subject of… … Wikipedia**Computational electromagnetics**— Computational electromagnetics, computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computationally… … Wikipedia**BPM**— steht im Gesundheitswesen für: Berufsverband der Fachärzte für Psychosomatische Medizin und Psychotherapie Deutschlands BPM steht in der Technik für: Beam Position Monitor, Gerät zur Erfassung der Strahllage von zumeist optischen Anlagen und… … Deutsch Wikipedia**BPM**— is a three letter acronym that may refer to: Business * Bank Pertanian Malaysia * Banca Popolare di Milano * Bataafse Petroleum Maatschappij, A Dutch oil company * BPM Energy, an Irish Energy drink * BPM (time service), People s Republic of China … Wikipedia**Wireless energy transfer**— or wireless power is the transmission of electrical energy from a power source to an electrical load without artificial interconnecting conductors. Wireless transmission is useful in cases where interconnecting wires are inconvenient, hazardous,… … Wikipedia**Space-based solar power**— Left: Part of the solar energy is lost on its way through the atmosphere by the effects of reflection and absorption. Right: Space based solar power systems convert sunlight to microwaves outside the atmosphere, avoiding these losses, and the… … Wikipedia