Laser physics question about Milonni book

[contempquant]

Hi,

I have the book by Peter Milonni, Laser Physics. Does anyone who has this know what the $$p(z)$$ and $$q(z)$$ represent in the equation on gaussian beam solution to a 'beamlike' wave on page 282, under the chapter 7.5 "Gaussian beams"?

$$\epsilon(\bar{r})=Ae^{ik(x^{2}+y^{2})/{2q(z)}}e^{ip(z)}$$

I can't work it out, and can't find any previous info on them in the book, are they just arbitrary constants?

 

[Redbelly98]

The book Lasers by Milonni & Eberly has the same equation in section 14.5, "Gaussian Beams", along with a derivation of q(z) and p(z).

They are assuming a solution to the paraxial equation of the form you wrote, where q and p are functions of z (not constants) that are to be worked out using the paraxial wave equation appearing on the previous page.

The Milloni & Eberly book proceeds with a derivation of what q and p are -- does your book not do this? At any rate, it turns out that q is defined in terms of the wavefront radius of curvature R and spot size w by

1/q(z) = 1/R(z) + iλ/(πw²(z))​

The factor e^ip(z) is a complex number, and gives the following information about the beam:

• The z-dependence of the on-axis (x=y=0) electric field magnitude, and
• Variations in the on-axis phase; more specifically, departures from the approximate e^ikz phase dependence.

I'm not really sure why he/they wrote it as e^ip(z), it seems they could more simply have written it as say f(z), without the exponential.