What "adiabatic" means in context of photonic waveguides? I only know the thermodynamic definition that "adiabatic process is a process that doesn't exchange the heat or mass with the surroundings".
Does it mean, that for example "adiabatic waveguide/taper" is just a lossless waveguide/taper? Why won't authors of papers use just the word "lossless"?
Is adiabatic photonic device somehow better than a lossless one?
 A: Adiabatic usually refers to slow change in some parameter. This may be not obvious within thermodynamic context, where the terms has a very specific meaning, but the term is widely used outside of this field.
E.g., quasi-classical approximation in quantum mechanics (or Wentzel-Kramers-Brillouin approximation) is sometimes referred to as adiabatic approximation - since it originates when we assume that potentials are changing slowly on the scale of the de Broglie wave length.
Another well-known example is the envelope function of the Bloch states in solid state physics, resulting the effective mass approximation.
In electrodynamics (where I include studying waveguides) adiabatic may refer to different things, depending on the particular problem. One likely possibility is that the direction and width of the waveguide are changing slowly on the scale of the wave length, which often allows introducing position-dependent wave vector and other such features. In terms of the wave equation it often allows to reduce replace one of the second derivatives by the first-order derivative.
A: In an adiabatic transition the modes and their coupling are essentially unchanging. For the simplest example take a metal pipe (waveguide) and taper its end to a large radiating horn whose aperture is larger than that of the pipe's cross section.
If the taper is smooth and slow, i.e., adiabatic, then if the pipe had only a single mode propagating in it then there will be only a single mode  propagating in the taper while forming the radiation field in the horn's aperture. If the size of the aperture is such that in a pipe of similar dimension the ratio of the E and H fields would be close to $377\Omega$ then so it will be the radiation impedance of the horn and it will radiate into the space with very little reflection.
In this sense "adiabatic" means an unchanging or very slowly changing mode structure; it has nothing to do with loss or thermodynamics.
