A Short Introduction to Laser Physics

Abstract

In this chapter, we start by reviewing Einstein’s derivation of Planck’s radiation law, whereby the Einstein coefficients for stimulated and spontaneous radiative processes are introduced. Generalizing to the case of non-equilibrium, conditions for laser activity are derived. After a brief discussion of possible realizations of the laser principle, we discuss the generation of short laser pulses by the superposition of a fundamental and side-band frequencies. The carrier envelope phase is then discussed as well as the characterization of laser pulses by windowed Fourier transforms that concludes this chapter.

Appendix: Some Gaussian Integrals

Throughout this book, Gaussian integrals will be encountered. For complex valued parameters a and b with $\operatorname{Re} {a}\geq0$, the following formulae hold:

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm{d}x \exp\bigl\{-a x^2\bigr\} =&\sqrt{\frac {\pi}{a}} \end{aligned}$$

(1.28)

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm{d}x x \exp\bigl\{-a x^2\bigr\} =&0 \end{aligned}$$

(1.29)

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm{d}x x^2\exp\bigl \{-a x^2\bigr\} =& \biggl(\frac{1}{2a} \biggr)\sqrt{ \frac{\pi}{a}} \end{aligned}$$

(1.30)

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm{d}x \exp\bigl\{-a x^2+b x\bigr\} =&\sqrt{\frac {\pi}{a}} \exp \biggl\{ \frac{b^2}{4a} \biggr\} \end{aligned}$$

(1.31)

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm{d}x x \exp\bigl\{-a x^2+b x\bigr\} =& \biggl(\frac{b}{2a} \biggr)\sqrt{ \frac{\pi}{a}} \exp \biggl\{\frac{b^2}{4a} \biggr\} \end{aligned}$$

(1.32)

$$\begin{aligned} \int_{-\infty}^{\infty}\mathrm{d}x x^2 \exp\bigl \{-a x^2+b x\bigr\} =& \biggl(\frac{1}{2a} \biggr) \biggl(1+ \frac{b^2}{2a} \biggr)\sqrt {\frac{\pi}{a}} \exp \biggl\{ \frac{b^2}{4a} \biggr\} . \end{aligned}$$

(1.33)

A generalization of one of the formulae above to the case of a d-dimensional integral that is helpful is

$$\begin{aligned} \int\mathrm{d}^dx\exp\{-\boldsymbol {x}\cdot\mathbf{A}\boldsymbol {x}+\boldsymbol {b}\cdot \boldsymbol {x}\}= \sqrt{\frac{\pi^d}{\det\mathbf{A}}} \exp \biggl\{\frac{1}{4}\boldsymbol {b}\cdot \mathbf{A}^{-1}\boldsymbol {b} \biggr\} , \end{aligned}$$

(1.34)

valid for positive definite symmetric matrices A. As in the 1d-case, it can be proven by using a “completion of the square” argument. Furthermore, the convention that non-indication of the boundaries implies integration over the whole range of the independent variables has been used.