Question

Consider a He-Ne laser oscillating at the wavelength of λ = 632.8 nm with a Doppler-broadened gain linewidth of Δν_g = 1.7×10⁹ Hz. (The gain linewidth is the width of the Gaussian distribution of the gain of the laser medium. For gas lasers it is usually determined by Doppler broadening.) Assume that the laser operates with a resonator of length L = 50 cm. Calculate the number of modes in which the laser can emit laser light. Hint: calculate the number of standing waves for the given resonator that fall within the gain linewidth.

Answer 1

Operating wavelength = λo = 632.8nm =632.8 × 10^-9 m

Length of the tube = L = 50cm = 0.5m

Let n be the mode number corresponding to λo, then

n = L/(1/2 * λo) = 0.5m / (1/2 * 632.8 × 10^-9 m)

n = 1264222.2

However n must be an integer so 1264222 and 1264223 would be two possible modes that are within the spectrum

The n1 and n2 values corresponding to the shortest λ1 and longest λ2 wavelengths are

Delta λ = ∆λ = 1.7×10⁹ Hz =  1.72 × 10^-16 m

n1 = L / [1/2(λo - (1/2 *Delta λ))] = 0.5 / 1/2(632.8 × 10^-9 - (1/2 *1.76 × 10^-12 m))] =  1580280.3

n2 = L / [1/2(λo + (1/2 *Delta λ))] = 0.5 / 1/2(632.8 × 10^-9 + (1/2 *1.76 × 10^-12 m))] =  1580275.9

Thus n values are: 1580275, 1580276, 1580277, 11580278, 1580279, 1580280 There are about six modes.