The output of a laser is 3.000 W when pumped at 7 times the threshold. The optical mode in the laser is a cylinder with a diameter of 0.8 mm, and the output mirror transmission is 0.25 %. Determine the saturation intensity of the laser transition

To determine the saturation intensity of the laser transition, we need to calculate the threshold power and the output power of the laser.

1. Calculate the threshold power:
Since the laser is pumped at 7 times the threshold, we can calculate the threshold power (P_threshold) by dividing the output power (P_output) by 7:
P_threshold = P_output / 7 = 3.000 W / 7 ≈ 0.4286 W

2. Calculate the area of the optical mode:
The optical mode in the laser is a cylinder with a given diameter. The formula to calculate the area (A) of a circle is:
A = π * (diameter/2)^2

Substituting the values:
A = π * (0.8 mm / 2)^2 ≈ 0.5027 mm^2

3. Calculate the output mirror transmission:
The output mirror transmission is given as a percentage. We need to convert it to a decimal value by dividing by 100:
T = 0.25% / 100 = 0.0025

4. Calculate the saturation intensity:
The saturation intensity (Isat) can be calculated using the following formula:
Isat = (4 * P_threshold) / (c * A * T)

Where c is the speed of light in vacuum (approximately 3 x 10^8 m/s).

Substituting the values:
Isat = (4 * 0.4286 W) / (3 x 10^8 m/s * 0.5027 mm^2 * 0.0025)

To avoid unit inconsistencies, we should convert millimeters (mm) to meters (m):
Isat = (4 * 0.4286 W) / (3 x 10^8 m/s * 0.5027 x 10^-6 m^2 * 0.0025)

Simplifying the calculations:
Isat ≈ 22789.88 W/m²

Therefore, the saturation intensity of the laser transition is approximately 22789.88 W/m².