Another point about the phase: it's very difficult to measure. Of the four quantities we'd like to measure about light, until our group's work, humankind could only measure the spectrum of ultrafast light. Everything else varies too fast to be measured.
We'd be done if we could measure the spectral phase. Perhaps we could use the spectrum and some additional information, like the fact that a pulse has zero intensity outside a small interval in time (that is, that it's a pulse).
But this simply doesn't work.
The Fundamental Theorem of Algebra says that we can factor polynomials. Interestingly, it's also what prevents us from solving the 1D Phase-Retrieval Problem! The ability to factor polynomials allows ambiguities: multiple spectral phases that correspond to the same data. If this sounds weird, it is. But hang in there; we'll have more to say about this on a future page.