Laser cooling of atoms: more area or more power? I want to optimise a Magneto-Optical trap. Laser beams come from the the x, y and z directions (positive and negative) and slow the atoms down.
Would it be better to increase the beam spot-size (reducing the intensity) therefore slowing more atoms, or to have a smaller cooling region but with higher power and therefore more effective cooling?
Where should I start for a quantitative calculation?
 A: 
Optimizing the MOT beams 

Since the maximal force created by each beam on the atoms is limited by the saturation intensity $I_\text{sat}$ :
$$\tag{1}
F_\text{max}=\frac{\Gamma}{2}\hbar k\quad\text{when}\quad\frac{I}{I_\text{sat}}\geq 1
$$
you will always want to have enough power in the beams of your MOT so that you hit the saturation regime. Here, $k$ is the wavelength of your laser beam, $\Gamma$ is the linewidth of your atomic transition.
The second thing you wanna do if you want to maximize the number of atoms catched in the MOT is to have the largest MOT beams as possible. But since intensity decreases as $I(r)\sim P/r^2$ for a given optical power $P$, you should take care that you have enough power in order to stay in the saturation regime.
Let say that $\sigma$ is the waist of your MOT beams. What I mean is that the appropriate situation would be that :
$$
I\sim\frac{P}{\sigma^2}>I_\text{sat}
$$

Optimizing the magnetic gradient of the MOT

The other thing you can play with is the magnetic field gradient $b$ of your MOT. The resonance of an atom with a beam of your MOT at fixed detuning $\delta$ of the atomic transtion will occur for each couple of atomic position/velocity $(x,v)$ such that :
$$\tag{2}
\delta\pm kv=\mp\frac{\mu bx}{\hbar}
$$
$\mu$ being the magnetic moment of your atom.
Let's considere a situation where atoms are coming from $x=-\infty$ on the MOT. Since the distance on which you can catch atoms is finite and limited by the size $\sigma$ of your beams, an optimal situation seems to be when an atom with zero velocity at the position $x=+\sigma/2$ (the "left" edge of the MOT if you prefere) is resonant with the beam pushing it towards the center of the MOT $x=0$. This implies from (2) that :
$$\tag{3}
|\delta|=\frac{\mu b\sigma}{2\hbar}
$$
On the other side of the capture zone in $x=-\sigma/2$, that same beam is resonant with the atoms with velocity $v_i$ such that :
$$\tag{4}
|\delta|=kv_i-\frac{\mu b\sigma}{2\hbar}
$$
This implies from (3) :
$$\tag{5}
kv_i=2|\delta|=\frac{\mu b\sigma}{\hbar}
$$
Moreover the max incident velocity $v_i^\text{max}$ that you can catch with the MOT will be also limited by the saturation as the force applied on the atom is limited such that :
$$\tag{6}
F=m\frac{\mathrm{d}v}{\mathrm{d}t}=mv\frac{\mathrm{d}v}{\mathrm{d}x}=mv\frac{\mu b}{\hbar k}<F_\text{max}
$$
where $m$ is the mass of the atom.
This implies from (1) that :
$$
b<\frac{\Gamma(\hbar k)^2}{2\mu mv}
$$
with a maximum of capture when :
$$\tag{7}
b=\frac{\Gamma(\hbar k)^2}{2\mu m v_i^\text{max}}
$$
Replacing this expression of $b$ in the relation (5), you get :
$$
v_i^\text{max}=\sqrt{\frac{\hbar k \Gamma\sigma}{2m}}
$$
A: Check out this paper: 

Improved magneto-optic trapping in a vapor cell. K.E. Gibble, S. Kasapi and S. Chu. Opt. Lett. 17 no. 7, 526 (1992),
  PSU eprint.

You'll notice that the rate at which atoms below the capture velocity enter the volume of your cooling region increases with the beam diameter squared.  There is also a plot of the number of atoms trapped vs intensity.  Since it saturates at high intensity, you'll win in many cases with larger beams.  The beams that I used for my Rb MOT in grad school were more than 2 cm in diameter.  The saturation intensity for Rb is pretty low , so it was a win in this case.
