Assignment 7, solutions:

Problem 1:

(a)  For a Gaussian beam of a particular wavelength, the product d₀q is constant.  For a TEM₀₀ mode d₀ depends on the beam divergence angle as d₀ = 4l/pq, where l is the wavelength of the radiation.

Here d₀ = 10^-3 m, l = 532 nm, so q = 6.77 10^-4 rad.

The variation of the beam diameter in the vicinity of the beam waist is given by d² = d₀² + q²z², where d is the diameter at a distance ±z from the waist along the beam axis.  A large distance from the waist we have d = qz. 

d(z = 100 km) =10⁵ q = 67.7 m.

(b)  I = (20 W)/(pd²/4) = 5.55 mW/m² = 0.55 mW/cm².

Problem 2:

For Laser amplification the condition on the mirror separation is that one round trip contains an integral number of wavelengths 2L = ml, where m is an integer.  The corresponding resonant frequencies are f_m = c/λ = mc/(2L.  The separation between resonance frequencies is Df = c/2L.  

(a)  Dl = 2L/m - 2L/(m+1) = 2L/(m(m+1)) » 2L/m².

(b)  Df = c/(2L) = (3 10⁸/0.6) Hz = 500 MHz,  

Dl = 2L/m²,  m = 2L/l = 1.167 10⁶, Dl = 4.4 10^-13 m.

(c)  Df = c/(2L) = (3 10⁸/6 10^-4) Hz = 500 GHz,

Dl = 2L/m²,  m = 2L/l = 896, Dl = 7.48 10^-10 m. 

Problem 3:

Assume a source emits waves with wavelength l ± Dl.  Waves with wavelength l and l + Dl, emitted in phase, will destructively interfere after some optical path length l_c = l²/(2pDl); l_c is called the coherence length.

(a) Df = c/(2L) = (3 10⁸/3) Hz = 100 MHz.

(b)  There are roughly (1500 MHz)/(100 MHz)   = 15 modes oscillating in the cavity, so the laser's output approximates that of an incoherent source with the same bandwidth, i.e. the same range of wavelengths l + Dl.