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# Beam propagation method

Beam Propagation Method (BPM) refers to a computational technique in Electromagnetics, usedto solve the Helmholtz 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 of light in slowly varying optical waveguides. It is essentially the same as the so-called Parabolic Equation (PE) method in underwater acoustics. 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 order derivative 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] :$\left( abla^2 + k_0^2n^2\right)psi = 0$with the field written as, :$E\left(x,y,z,t\right)=psi\left(x,y,x\right)exp\left(-jomega t\right)$.

Now the spatial dependence of this field is written according to any one TE or
TM polarizations:$psi\left(x,y\right) = A\left(x,y\right)exp\left(+jk_o u y\right)$,with the envelope :$A\left(x,y\right)$ following a slowly varying approximation, :$frac\left\{partial^2\left( A\left(x,y\right) \right)\right\}\left\{partial y^2\right\} = 0$

Now the solution when replaced into the Helmholtz equation follows,:$left \left[frac\left\{partial^2 \right\}\left\{partial x^2\right\} + k_0^2\left(n^2 - u^2\right) ight\right] A\left(x,y\right) = pm 2 jk_0 u frac\left\{partial A_k\left(x,z\right)\right\}\left\{partial z\right\}$

With the aim to calculate the field at all points of space for all times, we only need to compute the function$A\left(x,y\right)$ for all space, and then we are able to reconstruct $psi\left(x,y,z\right)$. 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

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