Usuario:Felipebm/Método de propagación de haz

El Método de Propagación de Haz (BPM, del inglés Beam Propagation Method) se refiere a una técnica computacional usada en en Electromagnetismo para resolver la ecuación de Helmholtz para una onda estacionaria. El BPM funciona bajo la aproximación de la envolvente suave (slowly varying envelope approximation), para ecuaciones lineales y no-lineales.

El método de propagación de haz es una técnica de aproximación para simular la propagación de la luz en guia de onda que varían suavemente su perfil dieléctrico. Cuando una onda se propaga a lo largo de una guía de onda por una distancia muy larga (comparado con la longitud de onda), la simulación numerica rigurosa se hace muy difícil. El BPM se apoya en ecuaciones diferenciales ordinarias que también son lllamadas modelos de un solo sentido (one-way models). Estos modelos de un solo sentido implican solo una derivada de primer orden en la variable z (eje de la guia de ondas) y puede ser resuelto como un problema de valores iniciales. Este problema de valores iniciales no involucra el tiempo como una variable, por lo contrario es la variable espacial z la que evoluciona.​

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.

Principios

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

$(\nabla ^{2}+k_{0}^{2}n^{2})\psi =0$

with the field written as,

$E(x,y,z,t)=\psi (x,y,z)exp(-j\omega t)$.

Now the spatial dependence of this field is written according to any one TE or TM polarizations

$\psi (x,y)=A(x,y)exp(+jk_{o}\nu y)$,

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,

$\left[{\frac {\partial ^{2}}{\partial x^{2}}}+k_{0}^{2}(n^{2}-\nu ^{2})\right]A(x,y)=\pm 2jk_{0}\nu {\frac {\partial A_{k}(x,z)}{\partial z}}$

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 solution is 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 propagation direction.

Aplicaciones

BPM is a quick and easy method of solving for fields in integrated optical devices. It is typically used only in solving for intensity and modes within shaped (bent, tapered, terminated) waveguide structures, as opposed to scattering problems. These structures typically consist of isotropic optical materials, but the BPM has also been extended to be applicable to simulate the propagation of light in general anisotropic materials such as liquid crystals. This allows to analyze e.g. the polarization rotation of light in anisotropic materials, the tunability of a directional coupler based on liquid crystals or the light diffraction in LCD pixels.

BPM software

• RSoft's BeamPROP: vector; commercial, free trial possible
• Optiwave's OptiBPM: commercial, free trial possible
• FEAB (Finite Element Anisotropic Beam propagation method): academic, free version available

Ver también

• Método de diferencias finitas
• Método de los elementos finitos
• Ecuaciones de Maxwell
• Luz
• Fotón