Beam propagation method: Difference between revisions

Revision as of 23:35, 28 June 2010

Revision as of 05:49, 19 May 2011

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to 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 [[waveguide]]s. 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'sthe 1970s. 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.<ref>{{citation|title=Integrated Photonics|author=Clifford R. Pollock, Michal. Lipson|year= 2003|publisher=Springer|id=ISBN 1402076355 |url=http://books.google.com/?id=DNJEoypcI6oC&pg=RA1-PA210&lpg=RA1-PA210&dq=%22Beam+propagation+method%22|isbn=9781402076350}}</ref>

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.

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==Principles==

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

<ref>Okamoto K. 2000 Fundamentals of Optical Waveguides (San Diego, CA: Academic) </ref>

<ref>EE290F: BPM course slides, Devang Parekh, University of Berkeley, CA</ref>

:<math>

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:<math>E(x,y,z,t)=\psi(x,y,z)exp(-j\omega t)</math>.

Now the spatial dependence of this field is written according to any one [[Transverse_modeTransverse mode|TE]] or [[Transverse_modeTransverse mode|TM]] polarizations

:<math>\psi(x,y) = A(x,y)exp(+jk_o\nu y)

</math>,

with the envelope

:<math>A(x,y)

</math> following a slowly varying approximation,

:<math>

\frac{\partial^2( A(x,y) )}{\partial y^2} = 0

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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.

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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 [http://www.elis.ugent.be/ELISgroups/lcd/research/bpm.php 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 [http://www.rsoftdesign.com/products.php?sub=Component+Design&itm=BeamPROP BeamPROP]: vector; commercial, free trial possible

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*[[Finite element method]]

*[[Maxwell's equations]]

*[[MoL| Method of Lines]]

*[[Light]]

*[[Photon]]

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[[Category:Electrodynamics]]

[[Category:Physical optics]]

{{electromagnetism-stub}}