Assignment 9, solutions:

Problem 1:

The SHG conversion efficiency is h given by
h = P_2w/P_w = 2(m₀/e₀)^3/2 ((wdl)²/n_o³) (sin²(Dkl/2)/(Dkl/2)²) P_w/A, 
where

For proper phase matching we have Dk = 0, sin²(Dkl/2)/(Dkl/2)² = 1.
For the Nd:YAG laser we have l = 1.06 10^-6 m, w = 1.78 10¹⁵ /s.
For KD^*P we have n_o = 1.49, n_e = 1.46 at 1.06 mm, d = 0.42 *(1/9) * 10^-22C³/(J V³).
With P_w/A = 2MW/(p(2.5 10^-3)²) and h = 0.2 we can solve for l.
l = 3 cm.

Problem 2:

The SHG conversion efficiency is h given by
h = P_2w/P_w = 2(m₀/e₀)^3/2 ((wdl)²/n_o³) (sin²(Dkl/2)/(Dkl/2)²) P_w/A.
For proper phase matching we have Dk = 0, sin²(Dkl/2)/(Dkl/2)² = 1.

Here
P_w/A = [(200 10^-3 J)/ )2 10^-8 s)]/(p(2 10^-3)²) = 10⁷ W /(1.26 10^-5 m²) = 7.96 10¹¹ W/m²,
l = 0.02 m, d = 0.45 *(1/9) * 10^-22C³/(J V³), w = 1.78 10¹⁵ /s, n_o = 1.49.
Solve for h.

h= 0.81
P_2w = hP_w = 8.1 MW. I_2w = P_2w /A = 6.43 10¹¹ W/m².

Problem 3:

(a)  The incident ray is a meridional ray.  Assume the photon is reflected.  
For the incident photon we have
p_xi = -psin(q₀ + q_i),  p_zi = -pcos(q₀ + q_i).
For the reflected photon we have, using the law of reflection, q_r = q_I,
p_xr = psin(q₀ - q_i),  p_zr = pcos(q₀ - q_i).

The momentum transferred to the sphere is Dp = p_i – p_f.
Dp_x = -p[sin(q₀ + q_i) + sin(q₀ - q_i)] = -2p sinq₀ cosq_i.
Dp_z = -p[cos(q₀ + q_i) + cos(q₀ - q_i)] = -2p cosq₀ cosq_i.
For a given q₀, when averaging over all angles f, Dp_x averages to zero.