Photon arithmetic for laser problems

1. Energy of a photon: E = hc/λ (J) = 1240/λ(nm) (eV).
2. Number of photons per second = power / energy of one photon.
3. Number of photons in a pulse = pulse energy / energy of one photon.
4. Peak power of a pulse = energy / pulse duration.
5. Intensity of a beam = power / cross-sectional area.
6. Ranging: distance = c × (round-trip time)/2.

Example 6.1. A 3 mW laser pointer emits green light of wavelength 532 nm. Photon energy = 1240/532 = 2.33 eV = 3.73 × 10⁻¹⁹ J. Photon rate = 3 × 10⁻³/3.73 × 10⁻¹⁹ = 8.0 × 10¹⁵ photons per second.

Example 6.2. A pulsed ruby laser emits 0.5 J in 0.5 ms. Peak power = 0.5/5 × 10⁻⁴ = 1000 W. If this pulse is concentrated on an area of 1 mm², the intensity is 1000/10⁻⁶ = 10⁹ W/m², which is enough to evaporate metal.

Example 6.3. A laser beam leaves a source with diameter 2 mm and after 100 m its diameter is 2.4 mm. The intensity falls by (2/2.4)² = 0.69, i.e. to 69 % of its initial value, whereas a point source would have fallen by a factor 10⁴ at the same distance of 100 m compared with 1 m.

Example 6.4. An atom has an excited level at 2.30 eV above the ground level. It is stimulated to emit. Wavelength = 1240/2.30 = 539 nm (green), and the output photon is in phase with the stimulating one.