Dielectric material in a Waveguide I'm in the middle of an experiment and before my final measurement I want to find out the electic properties of the material i'm working with. The question is:
If I have a metallic waveguide and I place a dielectric piece of material in the middle of it (the piece is thin) and I want to measure the transmission coefficient of this setup (the intensity of the propagating wave before and after the dielectric in the waveguide), how would the calculation go?
I know the modes in the waveguide but how do I take into account the dielectric slab?
Is there a way to solve this problem like a plane wave problem with transfer matrix?
Thanks.
I am adding a pictuire to show the waveguide with the dielectric. This is just a view of the inside of the waveguide, it will be closed from all sides obviously.

 A:  If the length of the slab  is more than a few wavelengths long and it is also homogeneous along the axis then you probably want to apply some mode matching technique to calculate the reflections from the "in" and "out" interfaces. There are explicit formulas for dielectric loaded waveguide modes (LSE and LSM) and using field continuity equations in the cross sectional area these modes can be matched with that of the TE and TM modes of the empty waveguide.
 If the dielectric obstacle is "short", less than a guide wavelength or so, then there are variational methods due to Schwinger to calculate the elements of the equivalent lumped element circuit between the "in" and "out" transmission lines. The variational calculus is also applicable to a dielectric slab with more complicated geometry and/or non-uniform permittivity.
 To measure the reflection and transmission coefficients of the dielectric loaded guide is quite easy: you need a network analyzer and make sure that the "in" and "out" guides can support only the single lowest order mode; do not forget to calibrate your system.
 Both mode matching and the Schwinger's variational method are discussed in detail in chapter 6 of Collin: Field Theory of Guided Waves.
