Derive the evolution equations (10.48) for three-wave mixing. You could proceed as follows.

(a) Insert expressions (10.27) and (10.28) into the general wave equation (10.30) and extract the portions with frequency ω₃ = ω₁ + ω₂, thereby obtaining the generalization of Eq. (10.33):

(b) Explain why e^{i(k₃ · x−ω₃t)}f⁽³⁾ satisfies the homogeneous wave equation (10.50). Then, splitting each wave into its scalar field and polarization vector, A_i⁽ⁿ⁾ ≡ A(n)f_i⁽ⁿ⁾, and letting each A⁽ⁿ⁾ be a function of z (because of the boundary condition that the three-wave mixing begins at the crystal face z = 0), show that Eq. (10.53) reduces to

where α₃ is a constant of order unity that depends on the relative orientations of the unit vectors e_z, f⁽³⁾, and K̂₃. Note that, aside from α₃, this is the same evolution equation as for our idealized isotropic, dispersion-free medium, Eq. (10.34a). Show that, similarly, A⁽¹⁾(z) and A⁽²⁾(z) satisfy the same equations (10.34b) and (10.34c) as in the idealized case, aside from multiplicative constants α₁ and α₂.

(c) Adopting the renormalizations

with ζ_n so chosen that the photon-number flux for wave n is proportional to 

show that your evolution equations for A_n become Eqs. (10.48), except that the factor β' and thence the value of κ might be different for each equation.

(d) Since the evolution entails one photon with frequency ω₁ and one with frequency ω₂ annihilating to produce a photon with frequency ω₃, it must be that

(These are called Manley-Rowe relations.) By imposing this on your evolution equations in part (c), deduce that all three coupling constants κ must be the same, and thence also that all three β' must be the same; therefore the evolution equations take precisely the claimed form, Eqs. (10.48).

Equations

2 (A() ei k32-(31)) D - 2 xx/ (4() el (ksz-wist)). -2 dijk AD A() eikzz-wzt + A 2 A ei (k3z-w3t) (10.53)