manVsPhD

As a photonics researcher entering his last year of the PhD, got any resources for looking for jobs in photonics?

These definitions really depend on the context but I'll give it a try. A wave is a solution to a wave equation. A wave equation is a second order partial differential equation of a certain form that ties the second time derivative of a scalar or vector to its second derivative in space. Any form of a solution to those equations is a wave, but usually there is a convenient basis to work with where the resulting waves are simple. In many cases that would be the Fourier basis where the solutions are complex exponents.

In electrostatics an electric field is the force per unit charge acting at a given point in space. It's basically a way to measure force. Same for the magnetic field in magnetostatics. Things get more complicated than that in electrodynamics (and even more when one considers quantum field theory... electrodynamics is a classical field theory) but imagine you have a periodic array of weights tied to springs. A field is any excitation of that chain, i.e. I displace one of the weights a little and let it go. Taking that setup to a continuum limit I end up with a continuous variable wave equation but the excitation is essentially the same - I locally push something using some sort of probe, or in other words, this probe creates a field by applying force. You can see how the definition of the field in electrodynamics is actually pretty much the same as the definition of a wave. The chain defines a partial differential equation, whose solutions are waves. The quantity we solve the partial differential equation for is the field (or potential, which defines the field, but that's for a different post).