Driving a Laser Oscillation

Abstract

We investigate the role of electric polarization of a laser medium in order to obtain further insight into dynamical processes occurring in a laser. The reader, who does not wish to interrupt the description of a laser and its operation, may skip over this chapter.

Problems

9.1

Susceptibilities and polarization conductivities. Instead of our ansatz of an electromagnetic wave, $\tilde{E} =A\mathrm{e}^{\mathrm{i}(\omega t - k z)}$, we could use the ansatz $\tilde{E} =A \mathrm{e}^{\mathrm{i}(k z - \omega t)}$. Show that the imaginary part of the susceptibility changes sign but that the real field and the real polarization are the same in both cases. Discuss the corresponding polarization conductivity.

9.2

Linear dispersion.

• (a) Determine the linear dispersion $\mathrm{d}n/\mathrm{d}\omega$ of titanium–sapphire at a population difference $N_2 - N_1 =10^{24}$ m$^{-3}$.

• (b) Determine the shift of the resonance frequencies of a Fabry–Perot resonator (length 0.5 m) due to optical pumping of a crystal of 1 cm length, i.e., at a change of the population difference $N_2 - N_1 =0$ to $N_2 - N_1 =10^{24}$ m$^{-3}$.

9.3

Nonlinear dispersion of optically pumped titanium–sapphire.

• (a) Determine the nonlinear dispersion $\mathrm{d}^2n/\mathrm{d}v^2$ around the center frequency $\omega _0$.

• (b) Determine the nonlinear dispersion in the case that $N_2 - N_1 =10^{24}$ m$^{-3}$.

• (c) How large is the change of the refractive index in the frequency range $\nu _0 - \varDelta \nu _0/2,\nu _0$ and in the range $\nu _0,\nu _0 + \varDelta \nu _0/2$?

• (d) Determine the shift of the resonance frequencies (due to nonlinear dispersion of a crystal of 1 cm length of a Fabry-Perot resonator (length 0.5 m) due to optical pumping, i.e., at a change of the population difference $N_2 - N_1 =0$ to $N_2 - N_1 =10^{24}$ m$^{-3}$.

9.4

Drude theory. We obtain the Drude theory of the electric transport if we treat the electrons in a solid as free-electrons, i.e., if we set $\omega _0 =0$ in (9.42) and introduce the electron velocity $\mathrm{v} =\mathrm{d}x/\mathrm{d}t$. Then $\beta ^{-1} =\tau$ is the relaxation time of an electron: an electron (accelerated at time $t =0$) by an electric field loses its energy after the time $\tau$.

• (a) Derive the high frequency conductivities $\sigma _1(\omega )$ and $\sigma _2(\omega )$.

• (b) Determine the real part and imaginary of the high frequency mobility $\tilde{\mu } (\omega )$; $\tilde{\mathrm{v}}(\omega ) =\tilde{\mu }(\omega )\tilde{E}(\omega )$.

• (c) Determine the corresponding frequency-dependent susceptibilities and dielectric constants ($=$dielectric functions).

9.5

Perfect conductor of high frequency currents. We define a perfect conductor of high frequency currents as a conductor with free-electrons that have an infinitely long relaxation time. [A superconductor at temperatures that are small compared to its superconducting transition temperature $T_c$ can be a perfect conductor of high frequency currents at frequencies where $h\nu$ < $2\varDelta$; $2\varDelta$ is the superconducting energy gap; $T_c =7$ K for lead and 90 K for the high temperature superconductor YBa$_2$Cu$_3$O$_7$.]

• (a) Derive the high frequency conductivity of a perfect conductor.

• (b) Determine the dielectric function.

• (c) Calculate the values of $\sigma _2$ of an ideal conductor that contains free-electrons of a density $N =10^{28}$ m$^3$; $N =10^{25}$ m$^3$; $N =10^{22}$ m$^3$.

9.6

Show that the slowly varying amplitude approximation is valid if the change of the amplitude within a quarter of the period of a high frequency field is small compared to the amplitude of a high frequency field.

9.7

Rabi oscillation. An ensemble of two-level atomic systems that interact with a strong electric field can show an oscillation of the population inversion and, synchronously, an oscillation of the polarization. We assume that the frequency of the field is equal to the atomic resonance frequency and that transverse relaxation is absent. We furthermore assume that the only relaxation process is spontaneous emission of radiation but that the spontaneous lifetime $T_1$ is much larger than the period of the field. We describe the dynamics of the polarization by,

$$\begin{aligned} \frac{\mathrm{d}^2 P}{\mathrm{d}t^2}+\varDelta \omega _0 \frac{\mathrm{d}P}{\mathrm{d}t}+\omega _0^2P=\frac{2}{\pi }\varepsilon _0\hbar \omega _0 B_{21}^{\omega }\varDelta N E. \end{aligned}$$

(9.207)

An electric field $E =A\cos {\omega _0t}$ causes a polarization $P =B \sin {\omega _0t}$ that is 90$^\circ$ phase shifted relative to the field. We find, in slowly varying amplitude approximation, the equation

$$\begin{aligned} \frac{\mathrm{d}B}{\mathrm{d}t}+ \frac{\varDelta \omega _0 }{2}B(t) = \frac{1}{\pi }\varepsilon _0\hbar \omega _0 B_{21}^{\omega }A\varDelta N(t). \end{aligned}$$

(9.208)

The time dependence of the population difference is determined by the differential equation:

$$\begin{aligned} \frac{\mathrm{d}\varDelta N(t)}{\mathrm{d}t}+ \frac{\varDelta N(t) - \varDelta N_0}{T_1} = \frac{1}{2\hbar \omega _0} A B(t); \end{aligned}$$

(9.209)

the change of the population difference averaged over a period of the field, multiplied by the energy of a photon, is equal to AB/2.

• (a) Show that, under certain conditions, these two equations are equivalent to two second-order differential equations,

  $$\begin{aligned} \mathrm{d}^2 B/\mathrm{d}t^2+ \omega _\mathrm{R}^2 B =0, \end{aligned}$$

  (9.210)
  $$\begin{aligned} \mathrm{d}^2 \varDelta N/\mathrm{d}t^2+ \omega _\mathrm{R}^2 \varDelta N =0, \end{aligned}$$

  (9.211)

  where

  $$\begin{aligned} \omega _\mathrm{R}^2 = \varepsilon _0 b^0_{21}/(4\hbar \omega _0) A^2 \end{aligned}$$

  (9.212)

  and $\omega _\mathrm{R}$ is the Rabi frequency and, furtheremore, $\omega _{\text {R}} \ll \omega _0 ;$ this condition allows for application of the slowly varying envelope approximation. The differential equations are approximately valid if $\omega _\mathrm{R}\gg \omega _0$. The solutions are

  $$\begin{aligned} B =B_0 \sin \omega _\mathrm{R} t, \end{aligned}$$

  (9.213)
  $$\begin{aligned} \varDelta N = \varDelta N_0 \cos \omega _\mathrm{R} t. \end{aligned}$$

  (9.214)

  The amplitude of the polarization and the population difference oscillate with the Rabi frequency. The Rabi frequency is proportional to the amplitude of the electric field.

• (b) Make a draft of the time dependences of the population difference $\varDelta N$ and the amplitude of the polarization.

• (c) Calculate the Rabi frequencies for a medium with a naturally broadened line, with $T_1 =10^{-2}$ s. What is the minimum field amplitude necessary for the occurrence of a Rabi oscillation? (For more information about Rabi oscillations, see, for instance, [1, 5, 40]).

9.8

Start of laser oscillation.

Show, by use of the van der Pol (vdP) equation of a laser, that laser oscillation cannot start without an initial field [Hint: the vdP equation has two different solutions, depending on the initial conditions].

9.9

Write the van der Pol equation of a laser in dimensionless units, as well as the solution for the electric field.

9.10

Derive the van der Pol equation from the Lorenz-Haken equations.

9.11

The van der Pol equation.

Derive the van der Pol equation of a laser from the (general) van der Pol equation

$$ \frac{\hbox {d}^{\hbox {2}}y}{\hbox {d}\tau ^{2}}\hbox { }+\hbox { }\varepsilon {\hbox {(}}-{1} +\hbox { }y^{2}\hbox {) }\frac{\hbox {d}y}{\hbox {d}\tau }\hbox { }+\hbox { }y\hbox { }=\hbox { 0,} $$

where y is the dimensionless field, ${\varepsilon }$ ${\hbox {(}}$>0) is a parameter, and $\tau$ the dimensionless time [Hint: Make use of SVEA; see Problem 31.4].

9.12

Determine the amplitude of the polarization of a helium–neon laser medium that carries a polarization-current of 0.68 A m$^{-2}$ (see Example to Fig. 9.6).

9.13

Phase portrait .

• (a) Characterize onset of laser oscillation by a phase portrait.

  Hint: Make use of the solution A(t) of the van der Pol equation of a laser. The phase portrait is obtained for a plot of ${\dot{A}}$ (on the y axis) versus A (on the x axis), with the time t as a parameter that varies from t = 0 to $t\rightarrow \infty .$

• (b) Draw the phase portrait of a laser oscillation for the case that the gain is suddenly turned off.