Consider a waveguide (cylindrical or rectangular) which is terminated with a CPML layer in both ends, perpendicular on z-axis. The CPML layer is characterized by sz conductivity value, which changes with the distance inside the layer. The value of the maximum conductivity is given by
smax=-(m+1)lnR(0)/2nδ where m is the polynomial factor ,δ is the pml thickness and n the wave impedance in the medium.R(0) is the desired reflectivity of the layer.
In a waveguide where certain mode propagates, the wave impedance is different than the host material. But in the literature, smax is only defined using the host material wave impedance. Should it be replaced by the mode wave impedance or this is a characteristic of a medium, ignoring the nature of the waves propagate inside?
Hi,
PLMs are used for the absorption of plane or cylindrical waves and are much more robust than transparent boundary conditions (TBC) for incident angle variation. They are well adapted for propagative waves calculated with FDTD. The imaginary part of the PLMs is related to a streching of coordinates in order to modeling a semi-infinite space. The theory is exact only if the pml thickness is infinite. In practice, if the PMLs are to close to the waveguide, losses are artificially increase. The best is to enlarge the window in a such way the mode profil is closed to 0. PMLs are usefull only for leaky or radiation modes. If now, you replace in smax, the refractive index of the host material by the mode effective index, reflectivity at the boundary between the structure and the PML layer is not equal to R(0) and you loose advantage of impedance matching of the PMLs.
Regards
Imagine a 2D waveguide. You can write each waveguide mode as the sum of two plane waves bouncing between the two walls (the same picture holds for 3D rectangular and cylindrical waveguides). As the mode number increases these plane waves are increasingly more normally incident on the waveguide walls and consequently their angle of incidence on the air/PML interface gets more closer to pi/2 (the higher the mode number the more tangentially it is incident on the air/PML interface).
Using the R(\theta) of the PML for plane wave incidence (\theta is zero in the direction normal to the layer) you could set up an optimization problem to find the PML parameters so that R(\theta) is less than some desired value over the range of angles that correspond to the modes that are propagating in the waveguide (including \theta=0) and that you want your PML to absorb with R less than or equal to the desired value.
Thank you Dr Petropoulos. This is essentially what the alpha quantity declares in CPML formulation- the angle of incidence (more or less). I can understand the optimization problem . This seems an interesting problem but any solution will not be well-applicable in a overmoded waveguide. In other words...we cannot make the "Perfect" PML
The question is not on the angle of incidence..."Common" CPML uses medium wave impedance η. In the waveguide should be this impedance be replaced with modal impedance?
Dimitri, the answer to your last question is no (also see the other answer above mine).
And, yes, there is no perfect PML. The perfect absorbing boundary condition for a waveguide would be a time-domain Dirichlet-to Neumann map.
There is no way to specify angle in the PML. Any PML layer (terminated with a boundary condition) has a plane-wave reflection coefficient that is a function of angle of incidence and goes to 1 (or -1) for tangential incidence to the layer. That's why I suggested the optimization problem.
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