In the book "Photonic Crystals, Molding the Flow of Light", the authors mention:

"Snell's laws are simply the combination of two conservation laws that follow from symmetry: conservation of frequency $w$ (from the linearity and time invariance of Maxwell's equations) and conservation of the component k$_{||}$ of k that is parallel to the interface.( from the continuous translational symmetry along the interface)"

I understand how the translational invariance leads to a conservation of k$_{||}$ in the medium on either side of the interface separately, but why should k$_{||1}$ be equal to k$_{||2}$ i.e k$_{||}$ remain conserved across the interface?

Snell's Law and momentum conservation

This thread provides answers with a classical intuition of billiard ball reflection, but I want to (1) understand how to use symmetries, and why, given that the two mediums are separately invariant under continuous translation, should the k$_{||}$ remain unchanged and (2) how do Maxwell's equations imply conservation of frequency


1 Answer 1


The main theorem connecting symmetry and conservation laws is Noether's theorem. Generally, it states that a quantity of $\left(\frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \dot{\mathbf{q}} - L \right) T_r - \frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \mathbf{Q}_r$ is conserved under perturbations in time and generalized coordinates $\mathbf{q}$ ($t \rightarrow t^{\prime} = t + \delta t, \mathbf{q} \rightarrow \mathbf{q}^{\prime} = \mathbf{q} + \delta \mathbf{q} ~)$. The Langrangian $L$ should be conserved (symmetric). $T_r, Q_r$ are some functions describing the perturbations: $$\delta t = \sum_r \varepsilon_r T_r $$ $$\delta \mathbf{q} = \sum_r \varepsilon_r \mathbf{Q}_r ~$$

As for me, this is not something you can understand intuitively since the perturbations could have a very complicated form. However, we can consider two easy cases:

1) $T=1, r=1, Q=0$, so simple linear translation in time. Then the Noether's quantity reads: $$\frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \dot{\mathbf{q}} - L$$. This is basically formula of hamiltonian, i.e. total energy of a system.

2) $T=0, r=1, Q=1$, so simple linear translation in a space coordinate. Then the Noether's quantity reads: $\frac{\partial L}{\partial \dot{q_k}}$, which is a momentum.

Now, talking about light, we should remember wave-particle duality and consider photons with momentum and energy. Momentum is wavevector $k$ and energy $\hbar \omega$. Linear translation in time and space for a photon just means a phase shift that couldn't affect a Lagrangian so it is conserved. Consequently, we have corresponding wavevector and frequency conservation laws.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.