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.
Best Answer
One starts with Maxwell's equations and some boundary conditions involving derivatives and vector components. In general, this is hard to deal with but we begin making some approximations and assumptions.
We are interested only in fields oscillating at some frequency f. Therefore, all fields will be varying with time as \$exp(j\omega t)\$ where \$\omega=2\pi f\$ Then all time derivatives become multiplications by phase factors: calculus becomes algebra, at least along the time dimension. Oh, I just used complex numbers. That's another trick - taking the physical fields to be the real part of some mythical complex-valued field lets us use simpler algebra.
The waveguide has the same shape along some length. Just as we assume simple sinusoidal (or complex exponential) oscillation along time, we can do so also along the 'z' axis. (We take x,y as the plane perpendicular to the waveguide, where we define it cross-sectional shape, and z along the waveguide, as a common convention.) So Curl, Div and all that get simpler, with derivatives with respect to z becoming simple coefficients involving wavelength (or wavenumber, its reciprocal) and complex phase factors.
We're dealing with radiation only, not static electric or magnetic fields, not motor or generator dynamics, not high energy particle beams. We can say that the electric and magnetic fields are perpendicular for a wave of any given wave direction. When multiple waves are superimposed, maybe we can't say so any more, but not a problem because we solve for field patterns for pure waves first, then add together what we like. Anyway, the point is we can eliminate all magnetic field terms from the equations, replacing them with terms involving the electric field only.
Finally, we luck out in that the most easily manufactured waveguide shapes are rectangles or circles. We have a differential equation in two dimensions (x and y) for an electric field (a vector) and some boundary conditions applying to the parts of the field parallel or perpendicular to the wall of the waveguide. For a rectangle, we can treat the x part and the y part separately. For a circle, the radial and angular parts can be separated. The technique is called "separation of variables" and is very popular in electromagnetics, quantum mechanics, acoustics and basically almost everything else in physics. Result: a pair of simple ordinary differential equations, each easily solved.
For more complex shapes, or waveguides with ridges, holes, dents along the way, then it gets messy and we lose one or two of these nice simplifications. Time to turn it over to a computer, then.
Once we have the field pattern, the usefulness is in knowing where the field is strongest - you want to minimize obstructions, and use the best materials. Where the field is zero or fairly weak, you can put screws, welding seams, holes, or whatever without affecting the waves. Examination of the field patterns also helps with determining how well polarization is preserved within the structure, which is important to radio astronomers and for doubling information capacity in telecommunications. Finally, field patterns make for great illustrations to impress the public, congresscritters and investors .