There were some miscalculations and wrong units in the answers (b) and (d) like AI do. I would like you to reconsider the question and solve again!! I also want to know why you use the equation N=2L/lambda to find the number of modes, because I found N=LΔf/c as well. I would appreciate it if you could tell me why it's wrong.
Consider a gas of molecules inside a cylindrical cavity, which is 20 cm long and has a cross-sectional area of 0.1 cm^2. Assume that the molecules have an optical resonance at λ = 500 nm with a spectral width of 0.1 nm.
(a) How many cavity modes exist in the frequency band of this molecular resonance?
(b) How many longitudinal TEM_(q,0,0) cavity modes fall under the same bandwidth? (Include both polarizations.)
(c) Using the results from (a) and (b), estimate the probability of a spontaneous photon appearing in one of the polarized longitudinal modes.
(d) If the spontaneous emission rate for this molecular transition is γ_(sp) = 10^7 s^(-1), what is the absorption cross-section?