Laser Oscillators Described via the Dirac Notation

Here we derive the classical linewidth cavity equation

$\text{\Delta}\text{\lambda}\approx \text{\Delta}\text{\theta}{\left(\frac{\partial \text{\theta}}{\partial \text{\lambda}}\right)}^{-1}$ |
(9.1) |

using the Dirac notation approach. First, we notice that in this equation ∆θ is the beam divergence previously related to the uncertainty principle (see Chapter 3):

$\text{\Delta}p\text{\Delta}x\approx h$ |
(9.2) |

and (∂θ/∂λ)^{−1} is the overall cavity angular dispersion (Duarte, 2003).

We should also mention that Equation 9.1 is the single-pass version of the *multiple-pass* linewidth cavity equation (Duarte and Piper, 1984; Duarte, 1990, 2001):

$\text{\Delta}\text{\lambda}=\text{\Delta}{\text{\theta}}_{R}{(MR{\nabla}_{\text{\lambda}}{\text{\Theta}}_{G}+R{\nabla}_{\text{\lambda}}{\text{\Phi}}_{P})}^{-1}$ |
(9.3) |

where the multiple-return-pass beam divergence is given by (Duarte, 1989, 1990)

$\text{\Delta}{\text{\theta}}_{R}=\frac{\text{\lambda}}{\text{\pi}w}{\left(1+{\left(\frac{{L}_{\mathcal{R}}}{{B}_{R}}\right)}^{2}+{\left(\frac{{A}_{R}{L}_{\mathcal{R}}}{{B}_{R}}\right)}^{2}\right)}^{1/2}$ |
(9.4) |

*R* is the number of return-cavity ...

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