Difference between revisions 111804236 and 111804241 on dewiki

A '''parametric oscillator''' is a simple harmonic oscillator whose parameters
(its resonant frequency <math>\omega</math> and damping <math>\beta</math>)
are variable

:<math>
\frac{d^{2}x}{dt^{2}} + \beta(t) \frac{dx}{dt} + \omega^{2}(t) x = 0
</math>

(contracted; show full)In general, the variations in damping and frequency are relatively small perturbations

:<math>
\beta(t) = \omega_{0} \left[b + g(t) \right]
</math>

:<math>
\omega^{2}(t) = \omega_{0}^{2} \left[
  1 + h(t) \right]
</math>

where <math>\omega_{0}</math> and <math>b\omega_{0}</math> are constants,
namely, the time-averaged oscillator frequency and damping, respectively.  
The transformed frequency can be written in a similar way

:<math>
\Omega^{2}(t) = \omega_{n}^{2} \left[  1 + f(t) \right]
</math>

where <math>\omega_{n}</math> is the [[natural frequency]] of the damped 
harmonic oscillator

:<math>
\omega_{n}^{2} \equiv \omega_{0}^{2} \left( 1 - \frac{b^{2}}{4} \right)
</math>

and 

:<math>
\omega_{n}^{2} f(t) \equiv \omega_{0}^{2} h(t) - 
\frac{1}{2\omega_{0}} \left( \frac{dg}{dt} \right)
- \frac{b}{2} g(t) - \frac{1}{4} g^{2}(t)
</math>

Thus, our transformed equation can be written

:<math>
\frac{d^{2}q}{dt^{2}} + \omega_{n}^{2} \left[  1 + f(t) \right] q = 0
</math>

Remarkably, the independent variations <math>g(t)</math> and 
<math>h(t)</math> in the oscillator damping and resonant frequency,
respectively, can be combined into a single pumping function 
<math>f(t)</math>.  The converse conclusion is that any form 
of parametric excitation can be accomplished by varying either
(contracted; show full)
:<math>
\frac{dr}{dt} = \left( \alpha_{\mathrm{max}} \cos 2\theta \right) r
</math>

:<math>
\frac{d\theta}{dt} = -\alpha_{\mathrm{max}} 
\left[
  \sin 2\theta - \sin 2\theta_{\mathrm{eq}} \right]
</math>

where we have defined for brevity

:<math>
\alpha_{\mathrm{max}} \equiv \frac{f_{0} \omega_{n}^{2}}{4\omega_{p}}
</math>

:<math>
\sin 2\theta_{\mathrm{eq}} \equiv \left( \frac{2}{f_{0}} \right) \epsilon
</math>

and the detuning

:<math>
\epsilon \equiv \frac{\omega_{p}^{2} - \omega_{n}^{2}}{\omega_{n}^{2}}
</math>

The <math>\theta</math> equation does not depend on <math>r</math>, and 
linearization near its equilibrium position <math>\theta_{\mathrm{eq}}</math> 
shows that <math>\theta</math> decays exponentially to its equilibrium

:<math>
\theta(t) = \theta_{\mathrm{eq}} + 
\left( \theta_{0} - \theta_{\mathrm{eq}} \right) e^{-2\alpha t}
</math>
where the decay constant 
<math>\alpha \equiv \alpha_{\mathrm{max}} \cos 2\theta_{\mathrm{eq}}</math>.
(contracted; show full)which represents a simple harmonic oscillator (or, alternatively, 
a bandpass filter) being driven by a signal 
<math>-\omega_{n}^{2} f(t) q</math> that is proportional to 
its response <math>q</math>.  


Assume that <math>q(t) = A \cos \omega_{p} t</math> already 
has a
n oscillation at frequency <math>\omega_{p}</math> and 
that the pumping <math>f(t) = f_{0} \sin 2\omega_{p}t</math> has
double the frequency and a small amplitude <math>f_{0} \ll 1</math>.
Their product <math>q(t)f(t)</math> produces two driving signals,
one at frequency <math>\omega_{p}</math> and the other at
frequency <math>3 \omega_{p}</math>

:<math>
(contracted; show full)external driving force is at the mean resonant frequency
<math>\omega_{0}</math>, i.e., 
<math>E(t) = E_{0} \sin \omega_{0} t</math>.
The equation then becomes

:<math>
\frac{d^{2}x}{dt^{2}} + b \omega_{0} \frac{dx}{dt} + 

\omega_{0}^{2} \left[  1 + h_{0} \sin 2\omega_{0} t \right] x = 
E_{0} \sin \omega_{0} t
</math>

whose solution is roughly

:<math>
x(t) = \frac{2E_{0}}{\omega_{0}^{2} \left( 2b - h_{0} \right)} \cos \omega_{0} t
(contracted; show full)
* [[Optical parametric oscillator]]
* [[Optical parametric amplifier]]

[[Category:Oscillators]]
[[Category:Amplifiers]]
[[Category:Dynamical systems]]
[[Category:Ordinary differential equations]]