A Laser Theory

Abstract

In this chapter, we present simple laser equations describing the dynamics of laser oscillation. The equations are coupled rate equations relating the populations of the laser levels and the photon density.

Problems

8.1

Threshold condition. Evaluate the threshold condition of a titanium–sapphire laser operated as cw laser. The data of the laser: Fabry–Perot resonator $L = 10$ cm, filled with the active titanium-sapphire crystal (gain cross section $\sigma _{21} = 3 \times 10^{-23}$ m$^2$; frequency $\nu = 360$ THz); reflectivity of the output coupling mirror $R = 0.98$; cross-sectional area of the laser beam $a_1 a_2 = 0.5$ mm$^2$.

8.2

Photon density, output power and efficiency. Determine the density of photons in the laser resonator and the laser output power of the laser described in Problem 8.1, for a pump power that is 10 times larger than the threshold pump power. Evaluate the efficiency of conversion of a pump photon into a laser photon.

8.3

Oscillation onset time.

• (a) Show that the oscillation onset time is always large compared to the period $2 \pi /\omega$ of the laser field. [Hint: make use of the data of Table 7.1.]

• (b) Estimate the oscillation onset time of the titanium-sapphire laser (described in Problem 8.1).

8.4

Formulate the threshold condition in the case that the length $L^{\prime }$ of the active medium is smaller than the length of the laser resonator. Is the condition $GV = 1$ still valid?

8.5

Estimate the laser linewidth of a semiconductor laser of a wavelength of 0.8 $\upmu$m and an output power of 1 mW; loss factor $V_1 = 0.3$ and volume of the active $\text{ medium } = 10^{-13}$ m$^3$.

8.6

Coherence length. Monochromatic laser radiation consists of radiation of a line (halfwidth $\varDelta \lambda$) at the laser wavelength $\lambda$.

• (a) Determine the coherence length $l_\mathrm{coh}$. [Hint: use as criterion that the number of wavelengths of radiation at $\lambda - \varDelta \lambda /2$ and $\lambda + \varDelta \lambda /2$ differs by 1.]

• (b) Determine the coherence length of radiation generated by a semiconductor laser.

• (c) Determine $l_\mathrm{coh}$ of radiation generated by a highly stabilized helium–neon laser.

• (d) Determine the coherence length of the radiation of a hypothetical continuous wave laser at a frequency of $4 \times 10^{14}$ Hz that is stabilized with a relative accuracy of 10$^{-16}$.