A Short Introduction to Laser Physics

Abstract

To study the influence of light on the dynamics of an atom or a molecule experimentally, laser-light sources are used most frequently. This is due to the fact that laser light has well-defined and often tunable properties. The theory of the laser dates back to the 1950s and 1960s and by now is textbook material.

1.A Some Gaussian Integrals

Throughout this book, Gaussian integrals will be encountered. For complex valued parameters a and b with $\mathrm{Re}\,{a}\ge 0$, the following formulae hold:

$$\begin{aligned} \int _{-\infty }^{\infty }\mathrm{d}x\, \exp \{-a x^2\}= & {} \sqrt{\frac{\pi }{a}}, \end{aligned}$$

(1.29)

$$\begin{aligned} \int _{-\infty }^{\infty }\mathrm{d}x\, x \exp \{-a x^2\}= & {} 0, \end{aligned}$$

(1.30)

$$\begin{aligned} \int _{-\infty }^{\infty }\mathrm{d}x\, x^{2k}\exp \{-a x^2\}= & {} 1\cdot 3\cdots (2k-1)\left( \frac{1}{2a}\right) ^k\sqrt{\frac{\pi }{a}}, \end{aligned}$$

(1.31)

$$\begin{aligned} \int _{-\infty }^{\infty }\mathrm{d}x\, \exp \{-a x^2+b x\}= & {} \sqrt{\frac{\pi }{a}} \exp \left\{ \frac{b^2}{4a}\right\} , \end{aligned}$$

(1.32)

$$\begin{aligned} \int _{-\infty }^{\infty }\mathrm{d}x\, x \exp \{-a x^2+b x\}= & {} \left( \frac{b}{2a}\right) \sqrt{\frac{\pi }{a}} \exp \left\{ \frac{b^2}{4a}\right\} , \end{aligned}$$

(1.33)

$$\begin{aligned} \int _{-\infty }^{\infty }\mathrm{d}x\, x^2 \exp \{-a x^2+b x\}= & {} \left( \frac{1}{2a}\right) \left( 1+\frac{b^2}{2a}\right) \sqrt{\frac{\pi }{a}} \exp \left\{ \frac{b^2}{4a}\right\} . \end{aligned}$$

(1.34)

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

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

(1.35)

valid for positive definite symmetric matrices $\mathbf A$. Like 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.