# Laser beam divergence

How can I calculate the divergence of a laser beam by diffraction? I want to find the following relation,

$$\Delta\Omega = \frac{\lambda^2}{A},$$

where $\lambda$ is the wavelength and $A$ is the area.

this is the laser in a cavity.

• What exactly is the question? You want a derivation of the diffraction equation? Or do you want to see how this equation is applied in a special case? – Johannes Sep 20 '14 at 13:51
• really i wanna find angular laser beam divergence by diffraction relations. ΔΩ=λ^2/A = (∆θ)^2 – Natali Sep 20 '14 at 14:30
• Presumably $\Omega$ is the solid angle, in which case your expression is essentially the square of the expression for the width of an Airy disk. – John Rennie Sep 20 '14 at 15:27

I will assume that you are asking about laser beams in the fundamental, diffraction limited Gaussian mode. The standard expression for the divergence angle of a Gaussian beam in the far field is (see the Wikipedia page on Gaussian beams) $$\theta=\frac{\lambda}{\pi\omega_0}$$ where $\omega_0$ is the so-called waist size of the Gaussian beam. From here you can calculate the solid angle subtended by the beam which is given, in the small $\theta$ limit, as $$\Theta\simeq\pi\theta^2=\frac{\lambda^2}{\pi\omega_0^2}=\frac{\lambda^2}{A},$$ where $A$ is the area of the beam's waist.