A Comparison of Laser Oscillators and Quasiclassical Solid State Oscillators

Abstract

We present three types of quasiclassical oscillators that are able to generate microwave radiation of high frequency: Gunn oscillator (used as source of radiation up to $\sim$200 GHz); superlattice oscillator (in development, up to 200 GHz); resonant-tunnel diode oscillator (demonstrated up to 700 GHz). These oscillators are solid state oscillators, driven by active media.

Problems

31.1

Equivalent circuit.

• (a) Replace the equivalent circuit of Fig. 31.5 by a parallel resonant circuit; the active device has the negative admittance $G_\text {d}$ and the loss resistor the admittance G.

• (b) Derive the differential equation for the high frequency voltage.

• (c) Discuss the dependence of the negative admittance of the device on the voltage across the device.

• (d) Show that the output power of the oscillator is $P_\mathrm{out} = (1/2)G \hat{U}^2$, where $G + G_\mathrm{neg} =0$ is the condition of steady state oscillation.

31.2

Electric polarization.

• (a) Determine the electric polarization of a dipole domain consisting of a positive area charge $\rho l e$ at $x = 0$ and a negative charge of density $\rho$ in the range 0, l.

• (b) Determine the polarization of a dipole domain consisting of a negative charge of area density $\rho l e$ at $x = x_0$ and a positive charge of density $\rho$ in the range $x_0, x_0+l$.

31.3

Van der Pol oscillator.

• (a) Evaluate for a van der Pol oscillator (with the data: $a= 10^{-2}$ $\Omega ^{-1}$; $b = 10^{-2}$ $\Omega ^{-1}$ V$^{-2}$; $\omega _0 = 2 \pi$ $\times$ 10$^{10}$ Hz; $C= 1$ pF; and $G= 1 \Omega ^{-1}$) the small-signal net growth coefficient and show that it is small compared to $\omega _0$.

• (b) Determine the voltage amplitude for the steady state oscillation.

• (c) Determine the current amplitude for the steady state oscillation. [Hint: make use the relation $\cos ^3 \alpha = \frac{3}{4} \cos \alpha + \frac{1}{3} \cos 3 \alpha$ and neglect the term with $3 \alpha$.]

31.4

Van der Pol equation.

• (a) Show that the van der Pol equation can be written in dimensionless units,

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

  where y is the voltage in dimensionless units, $\tau = \omega _0 t$ the dimensionless time and $\epsilon$ the small-signal net gain coefficient in dimensionless units.

• (b) Solve the van der Pol equation for $\epsilon \ll 1$ at steady state oscillation. [Hint: make use of the relation $\cos ^3 \tau = \frac{3}{4} \cos \tau + \frac{1}{3} \cos 3 \tau$ and neglect the term with $\cos 3 \tau$.]

31.5

Which of the following differential equations describe a self-sustained oscillation?

• (a) $\dfrac{\mathrm{{d}} ^2 y}{\mathrm{{d}} t^2} + \dfrac{\mathrm{{d}} y}{\mathrm{{d}} t} + y = 0.$

• (b) $\dfrac{\mathrm{{d}} ^2 y}{\mathrm{{d}} t^2} - \dfrac{\mathrm{{d}} y}{\mathrm{{d}} t} + y = 0.$

• (c) $\dfrac{\mathrm{{d}} ^2 y}{\mathrm{{d}} \tau ^2} + \varepsilon (- 1 + y^2) \dfrac{\mathrm{{d}} y}{\mathrm{{d}} \tau } + y = 0.$

31.6

Compare a classical oscillator and a laser oscillator. (A classical oscillator has a stable I–V characteristic, Fig. 31.12, while a laser oscillator has a current-density-field characteristic that varies during onset of oscillation, Fig. 9.7.)

31.7

Show that the impedance $Z(\omega )$ of a resonance electrical circuit has Lorentzian lineshape.