Eigenmode expansion
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Eigenmode expansion (EME) is a computational electrodynamics modelling technique. It is also referred to as the mode matching technique or the bidirectional eigenmode propagation method (BEP method). Eigenmode expansion is a linear frequency-domain method. It offers very strong benefits compared with
FDTD Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to ...
, FEM and the
beam propagation method 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 ...
for the modelling of
optical waveguides An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber waveguides, transparent dielectric waveguides made of plastic and glass, liquid light g ...
, and it is a popular tool for the modelling linear effects in fiber optics and silicon photonics devices.


Principles of the EME method

Eigenmode expansion is a rigorous technique to simulate electromagnetic propagation which relies on the decomposition of the electromagnetic fields into a basis set of local
eigenmodes A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
that exists in the cross section of the device. The eigenmodes are found by solving
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
in each local cross-section. The method can be fully vectorial provided that the mode solvers themselves are fully vectorial. In a typical waveguide, there are a few guided modes (which propagate without coupling along the waveguide) and an infinite number of radiation modes (which carry optical power away from the waveguide). The guided and radiation modes together form a complete basis set. Many problems can be resolved by considering only a modest number of modes, making EME a very powerful method. As can be seen from the mathematical formulation, the algorithm is inherently bi-directional. It uses the scattering matrix (
S-matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
) technique to join different sections of the waveguide or to model nonuniform structures. For structures that vary continuously along the z-direction, a form of z-discretisation is required. Advanced algorithms have been developed for the modelling of optical tapers.


Mathematical formulation

In a structure where the optical
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
does not vary in the z direction, the solutions of Maxwell's equations take the form of a plane wave: : E(x,y,z) = E(x,y)e^ We assume here a single wavelength and time dependence of the form \exp(i \omega t) . Mathematically E(x,y) e^ and \beta are the eigenfunction and eigenvalues of Maxwell's equations for conditions with simple harmonic z-dependence. We can express any solution of Maxwell's equations in terms of a superposition of the forward and backward propagating modes: E(x,y,z)= \sum_^M H(x,y,z)= \sum_^M These equations provide a rigorous solution of Maxwell's equations in a linear medium, the only limitation being the finite number of modes. When there is a change in the structure along the z-direction, the coupling between the different input and output modes can be obtained in the form of a scattering matrix. The scattering matrix of a discrete step can be obtained rigorously by applying the boundary conditions of Maxwell's equations at the interface; this requires calculation of the modes on both sides of the interface and their overlaps. For continuously varying structures (e.g. tapers), the scattering matrix can be obtained by discretising the structure along the z-axis.


Strengths of the EME method

* The EME method is ideal for the modelling of guided optical components, for fibre and integrated geometries. The mode calculation can take advantage of symmetries of the structure; for instance cylindrically symmetric structures can be modelled very efficiently. * The method is fully vectorial (provided that it relies on a fully vectorial mode solver) and fully bidirectional. * As it relies on a scattering matrix approach, all reflections are taken into account. * Unlike the beam propagation method, which is only valid under the
slowly varying envelope approximation In physics, slowly varying envelope approximation (SVEA, sometimes also called slowly varying asymmetric approximation or SVAA) is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a per ...
, eigenmode expansion provides a rigorous solution to Maxwell's equations. * It is generally much more efficient than
FDTD Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to ...
or FEM as it does not require fine discretisation (i.e. on the scale of the wavelength) along the direction of propagation. * The scattering matrix approach provides a flexible calculation framework, potentially allowing users to only re-calculate modified parts of the structure when performing parameter scan studies. * It is an excellent technique to model long devices or devices composed of metals. * Fully analytical solutions can be obtained for the modelling of 1D+Z structures.


Limitations of the EME method

* EME is limited to linear problems; nonlinear problems may be modelled using iterative techniques. * EME may be inefficient for modeling structures requiring a very large number of modes, which limits the size of the cross-section for 3D problems.


See also

*
Computational electromagnetics Computational electromagnetics (CEM), 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 computer ...


References

{{reflist, refs= {{cite journal , author= G.V. Eleftheriades , title=Some important properties of waveguide junction generalized scattering matrices in the context of the mode matching technique , journal=IEEE Transactions on Microwave Theory and Techniques , volume=42 , issue=10 , pages=1896–1903 , year=1994 , doi=10.1109/22.320771 , bibcode=1994ITMTT..42.1896E {{cite book , author= J. Petracek , title=2011 13th International Conference on Transparent Optical Networks , chapter=Bidirectional eigenmode propagation algorithm for 3D waveguide structures , pages=1–4 , year=2011 , doi=10.1109/ICTON.2011.5971039 , isbn=978-1-4577-0881-7 {{cite journal , author= D. Gallagher , url=http://www.photond.com/files/docs/leos_newsletter_feb08_article.pdf , title=Photonics CAD Matures , journal=LEOS Newsletter , year=2008


External links


Improved Formulation of Scattering Matrices for Semi-Analytical Methods That is Consistent with Convention
Electrodynamics Computational electromagnetics Numerical linear algebra