Lasers and Beam Intensity

LASER stands for Light Amplification by Stimulated Emission of Radiation. Unlike ordinary light, laser light is:

• Monochromatic: single wavelength
• Coherent: all photons in phase
• Collimated: very low divergence

Stimulated Emission

An atom in an excited state can be triggered by an incoming photon to emit an identical photon — same wavelength, direction, and phase. This is stimulated emission, the mechanism behind laser amplification.

In a laser:

1. A gain medium (gas, crystal, semiconductor) provides atoms to excite.
2. A pump (electrical discharge, flash lamp, another laser) creates a population inversion — more atoms in the excited state than the ground state.
3. An optical resonator (two mirrors) traps the light, building up stimulated emission.
4. One mirror is partially transparent — the output coupler — letting the beam exit.

Laser Intensity

The intensity of a laser beam is the power per unit area:

$I = \frac{P}{A}$

For a circular Gaussian beam with diameter $D$ (radius $r = D/2$):

$A = \pi r^2 = \pi \left(\frac{D}{2}\right)^2$

$I = \frac{P}{\pi (D/2)^2}$

Example: A 5 mW laser pointer with beam diameter 2 mm:

$A = \pi (1 \times 10^{-3})^2 = \pi \times 10^{-6}\,\text{m}^2 \approx 3.14 \times 10^{-6}\,\text{m}^2$

$I = \frac{5 \times 10^{-3}}{3.14 \times 10^{-6}} \approx 1592\,\text{W/m}^2$

Even a low-power laser can reach intensities dangerous to the eye because the beam is so narrow.

Beam Area

The cross-sectional area of a circular beam of diameter $D$ (in mm) in mm²:

$A_{\text{mm}^2} = \pi \left(\frac{D}{2}\right)^2$

Your Task

Implement the intensity and area functions. Use $\pi$ from the math module.