The Active Medium: Energy Levels and Lineshape Functions

Abstract

We present a characterization of active media with respect to energy levels and line broadening.

Problems

4.1

Lineshape functions. At which frequency distance from the central line ($\nu _0 =4 \times 10^{14}$ Hz; $\varDelta \nu _{0}$ =1 GHz) does the lineshape function decrease by a factor of 100 (a) if the line has Lorentzian shape and (b) if the line has Gaussian shape?

4.2

Absolute number of two-level atomic systems. Determine the absolute number $N_\mathrm{tot}$ of two-level atomic systems for systems of different dimensionality .

• (a) Three-dimensional medium with a density $N =10^{24}$ m$^{-3}$ of two-level systems and a volume of 1 mm $\times$ 1 mm $\times$ 1 mm.

• (b) Two-dimensional medium with an area density $N^\mathrm{2D} =10^{16}$ m$^{-2}$ and an area of 1 mm $\times$ 1 mm.

• (c) One-dimensional medium with a line density $N^\mathrm{1D} =10^7$ m$^{-1}$ and a length of 1 mm.

4.3

Relate the lineshape function on the frequency scale and the lineshape function on the wave number scale.

4.4

Relate the dimensionless variables of the Lorentz resonance function expressed on the frequency scale and those expressed on the angular frequency scale.

4.5

Area under a Gaussian or Lorentzian curve.

• (a) Show that the width of a rectangular curve, which has the same height as a Gaussian curve and encloses the same area, is equal to

  $$\begin{aligned} \varDelta \nu _{0} /(2\sqrt{\ln 2} ) \approx 1.06 \times \varDelta \nu _{0} \approx \varDelta \nu _{0} . \end{aligned}$$

• (b) Show that the width of a rectangular curve, which has the same height as a Lorentzian curve and encloses the same area, is approximately equal to

$$\begin{aligned} (\pi /2)\varDelta \nu _{0} \approx 1.57 \times \varDelta \nu _{0} . \end{aligned}$$

4.6

Show that the integral over a narrow Lorentzian curve is approximately unity.

4.7

Derive the maximum value, Eq. (4.10), for the Lorentz resonance function.

4.8

Show that the width of a rectangular shape that has the same height and the same area as a Gaussian $\varDelta \nu _{0} /(2\sqrt{\ln 2} ) \approx 1.06 \times \varDelta \nu _{0}$, and, correspondingly, for a Lorentzian shape by $(\pi /2)\varDelta \nu _{0} \approx 1.57 \times \varDelta \nu _{0}$.