Question

In this exercise we will derive the nonlinear interactions for the second harmonic using the scalar wave equation for the electric field (Eqn. 21.1-3) ot where ΕΞΕ(1,7) is the electric field. P-P(t, r)is the polarization n=n(a), cand Ho are the refractive index, vacuum speed of light and magnetic permeability. Mathematically, we can regard the wave equation as an inhomogeneous partial differential equation where the electric field wave is driven by the polarization of the media. The approximation we use to solve this difficult problem is called the First Born approximation. We model the incoming laser field as a plane wave of constant strength: -ikz-ar) COS where k, s the wavevector and ki = ni 2π/ λ-n, a / c with λί being the vacuum wavelength and n,-n(a); and E is a real constant. We assume that the polarization is instantaneously induced by the laser: ) The second order polarization - Now calculate the acceleration of the polarization (its second time derivative) which is relevant for the wave equation, use a complex exponentials formula. Hint: The acceleration of the polarization should be complex exponentials oscillating at +2w. These are the source for the second harmonic field!

Answer 1

Given The second order polarization p bar 6(2) = epsilon_0 x^(2) E^(2)_1 (e^-i(2kz-2wt) + e^i(2kz-2wt + 2) which can be written as p bar^?(2) = epsilon_0 x^(2) E^(2)_1 [2 cos^2 (2 k z - 2 w t)] [because e^ix + e^-ix/e + 2 = (2 cos x)^2] p bar^(2) = 2 epsilon_0 x^(2) E6(2)_1 [cos^2(2 k x - 2 w t)] Now first order derivative at p bar^(2) with respect to 't' dp bar^(2) = 2 epsilon_0 x^(2) E^(2)_1 [2 x - 2 w cos (2 k z - 2 w t)] second order derivative d^2p bar^(2)/dt^2 = -2 epsilon_0 x^(2) E^(2)_1 [2 times 4 w^2 sin (2 k z - 2 w t)] d^2 p bar^(2)/dt^2 = -16 w^2 epsilon_0 x^(2) E^(2)_1 [sin (2 k z - 2 w t)] So acceleration at polarization p bar^(2) d^2 p bar^(2)/dt^2 = -16 w^2 epsilon_0 x^(2) E^(2)_1[e^1(2kz-2wt) - e^-i(2kz-2wt)/2 i] d^2 p bar^(2)/dt^2 = -8 w^2/i epsilon_0 x^(2) E^(2)_1 [e^1(2kz-2wt) - e^-i(2kz-2wt)/2 i]