Quantum Optics for Engineers

9

Laser Oscillators Described via the Dirac Notation

9.1    Introduction

Here we derive the classical linewidth cavity equation

$\text{Δ}\text{λ}\approx \text{Δ}\text{θ}{\left(\frac{\partial \text{θ}}{\partial \text{λ}}\right)}^{-1}$

(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):

$\text{Δ}p\text{Δ}x\approx h$

(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):

$\text{Δ}\text{λ}=\text{Δ}{\text{θ}}_R{(MR{\nabla }_{\text{λ}}{\text{Θ}}_G+R{\nabla }_{\text{λ}}{\text{Φ}}_P)}^{-1}$

(9.3)

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

$\text{Δ}{\text{θ}}_R=\frac{\text{λ}}{\text{π}w}{\left(1+{\left(\frac{L_{\mathcal{R}}}{B_R}\right)}^2+{\left(\frac{A_R{L}_{\mathcal{R}}}{B_R}\right)}^2\right)}^{1/2}$

(9.4)

R is the number of return-cavity ...