# Microscopic origin of phase matching in second harmonic generation

In second harmonic generation (SHG), the second-order nonlinearity of a medium is used to double the frequency of a driving field. In Boyd's book "Nonlinear Optics", he derives the solution to the (macroscopic) Maxwell equations in case of a nonlinear polarization density $$P = \chi E + 2dE^2$$. The total field is described as the superposition of two planewave-like fields with frequency $$\omega$$ and $$2\omega$$: $$E(z,t) = E_1(z,t)+E_2(z,t) = A_1(z)e^{ik_1z-i\omega t}+A_2(z)e^{ik_2z-i2\omega t},$$ where $$k_{i} = n_{\omega_i}\frac{\omega_i}{c}$$ and $$A_i(z)$$ the slowly varying part of the amplitude. The wavevector mismatch is defined as $$\Delta k=2k_1-k_2.$$ The resulting set of coupled equations can be brought into the form $$\frac{du_1}{dz}=u_1u_2\sin\theta,\quad\frac{du_2}{dz}=-u_1^2\cos\theta,\quad\frac{d\theta}{dz}=\dots$$ where $$A_i(z) = u_i(z)e^{i\phi_i(z)}$$ and $$\theta(z)=2\phi_1-\phi_2+\Delta kz$$.

From these equations it can be seen that the "direction" of energy flow (i.e. wheter the intensity of the first or second harmonic field increases with z) depends on the sign of $$\cos\theta$$ and $$\sin\theta$$. From the perspective of SHG, $$\cos\theta = 1$$ is what one wants (this can be realized by having $$\Delta k=0$$ and no second harmonic present at the input of the crystal. If there is second harmonic light at the input, it has to have the "right" relative phase). So just the presence of the second harmonic wave with the "wrong" phase can reverse the process (from SHG to parametric down conversion).

I understand the derivation, but I don't have a physical intuition for this process reversal. It is often stated that the phase match condition can be understood as all dipoles along the path of the fundamental radiating with the right phase, such that they all constructively interfere. Another intuitive explanation is $$\Delta k=0$$ is to ensure momentum conservation. I find both of these arguments unsatisfactory. They both do not explain why to process is suddenly reversed, and there can be SHG even with $$\Delta k\neq 0$$ (just very inefficient).

The Hamiltonian describing this process is $$H = d\sum_{k_1,k_2}(a_2^\dagger a_1a_1 + a_1^\dagger a_1^\dagger a_2)C(\Delta k),$$ where $$C(\Delta k) = \int_V d^3r e^{i\Delta kr}$$ and $$V$$ is the volume of the nonlinear medium. In most books I have seen the argument that, for $$V=L^3$$ and $$L\gg \lambda$$, one writes $$C(k)\rightarrow \delta(k)$$, i.e. "momentum conservation is not violated". But this is not exact. There are still some contributions if $$\Delta k$$ is very small compared to $$L$$, and I think here is where this is all coming from. I would like to see, starting from this microscopic process, how phase matching emerges from the relative phase of the fundamental and second harmonic. I don't really know where to start. How does one even describe the relative phase of light in a quantum picture? I've tried calculating the time evolution of the elctric field operator, but I didn't really find a closed form.

Sorry for the lengthy question. Any help or resources are much appreciated!

I simply interpret this as an interference effect. Suppose that there is some small amount of second harmonic input together with the pump, and this field is out of phase with the second harmonic generated by the nonlinear response of the medium, then the two fields will clearly interfere destructively. This is not a "reversal" of the process, it is simply the the input field cancelling second harmonic that is generated. When the total amplitude of the second harmonic reaches zero it will start to grow, and assuming $$\Delta k = 0$$ it will grow until the pump has been depleted.

One way to understand the phase matching for $$\Delta k \neq 0$$ is as many successive SHG processes with some phase mismatch. In this case, the field generated in one SHG process will not interfere perfectly constructively with the next SHG process, and if the phase mismatch between two processes (not necessarily subsequent ones) is too large, then they will interfere destructively.

The destructive interference doesn't explain where the pump photons "are coming from" when the second harmonic is depleted, however due to energy conservation you can infer that they have to be generated. In the quantum picture, if you assume the SHG Hamiltonian then the unitarity of the process guarantees that the energy goes back to the pump.

That $$\Delta k$$ is not zero is also the case in the classical picture, since the amplitudes are dependent on $$z$$. Taking $$\delta(k)$$ is equivalent to considering the amplitudes slow.

What is more relevant here is that the classical description is dealing with coherent states, while in quantum case on usually limits the discussion to a number states (one photon disappears and two photons appear or vice versa.) If extended to coherent photon states (which have phase), the description would become very similar to the classical one (but I admit it requires some work.)

Perhaps related (regarding transition between classical and quantum description of electromagnetic processes):