Electrical – Waveguides question

Tapering to a reasonable degree is preferable to avoid sharp nodes for a VSWR mismatch. Now, if you taper for too long, then you just have a large amount of line that is a sub-optimal impedance. It really is a matter of what you're shooting for in terms of insertion loss/VSWR/space tradeoffs. I always consult the microwaves101 website for nice rules of thumb but I usually go 5-10% of a wavelength if I have the room. It looks like you have a good two step taper there but you haven't provided any sizing so I can't be sure.

Sometimes, the circuit might want a capacitive stub on either side of an inductive length such as this.

Note: My experience tends to be in the 26 GHz and below range so please take my advice with a grain of salt.

The modes of a waveguide refer to the distribution of the electrical and magnetic fields across the cross-section of the waveguide.

The electric field going to (roughly) zero at high-conductivity surfaces (like the inner and outer conductor of a coaxial line) places boundary conditions on the differential equations describing the waveguide behavior. This leads to patterns in the way the fields distribute themselves across the waveguide area. If a certain E/M field pattern can propagate along the waveguide without changing, we call that a mode of the waveguide.

Generally there will be only one propagating mode at low frequencies, and additional modes will be able to propagate as the signal frequency increases.

The waveguide modes are important because different modes tend to propagate at different rates along the z-axis of the waveguide. Generally you want to operate a waveguide at a frequency where only a single mode is supported. At higher frequencies, a single pulse input into the guide might exit the other end highly distorted due to the different propagation velocities of the different modes.

Different mode structures can also create reflections at the connection between two transmission lines, even if both transmission lines have the same characteristic impedance.

I have also heard about single and multimode fibers in optical communication. Are they the same?

Yes, it's essentially the same. Of course, the frequencies at play are very different. And the boundary conditions are created by changes in dielectric constant of the material rather than conductive surfaces.