Question

The beam emerging from a ruby laser passes through a circular aperture $5.0 \mathrm{mm}$ in diameter. (a) If the spread of the beam is limited only by diffraction, what is the angular spread of the beam? (b) If the beam is aimed at the Moon, how large a spot would be illuminated on the Moon's surface?

Short answer

Answer: The angular spread of the laser beam is approximately 9.70°, and the size of the illuminated spot on the Moon's surface is about 6.52 * 10^7 meters.

Step-by-step solution

Identify the given information

The given information are: aperture diameter (D) = 5.0 mm and wavelength of the ruby laser (λ) = 694.3 nm.

Calculate the angular spread using the angular resolution formula

The angular spread (θ) of the laser beam can be calculated using the angular resolution formula: θ = 1.22 * (λ / D) where θ = angular spread λ = wavelength of the ruby laser beam (694.3 nm) D = diameter of the aperture (5.0 mm) Before calculating, convert the aperture diameter to the same unit as wavelength. D = 5.0 mm = 5000 nm Now, substitute the values in the formula: θ = 1.22 * (694.3 nm / 5000 nm)

Calculate the angular spread in radians and convert to degrees

θ = 1.22 * (694.3 nm / 5000 nm) = 1.22 * 0.13886 ≈ 0.16931 radians To convert radians to degrees, we can use the formula: θ (in degrees) ≈ θ (in radians) * (180 / π) θ (in degrees) ≈ 0.16931 * (180 / π) ≈ 9.70°

Calculate the size of the illuminated spot on the Moon's surface

To find the size of the illuminated spot on the Moon's surface, we need the distance between the Earth and the Moon. The average distance is approximately 384,400 km. We can calculate the size of the illuminated spot (S) using the formula: S = Distance × tan(angular spread) where S = size of the illuminated spot Distance = Distance from the Earth to the Moon (384,400 km) Angular spread = 9.70° Before calculating, convert the distance to the same unit as the aperture diameter. Distance = 384,400 km = 3.844 * 10^8 m Now, substitute the values in the formula: S = 3.844 * 10^8 m × tan(9.70°)

Calculate the size of the illuminated spot on the Moon's surface in meters

S = 3.844 * 10^8 m × tan(9.70°) ≈ 6.52 * 10^7 m So, the size of the illuminated spot on the Moon's surface is about 6.52 * 10^7 meters.