the widely used approach to nonlinear optics is a **Taylor expansion** of the dielectric displacement field $\mathbf{D} = \epsilon_0\cdot\mathbf{E} + \mathbf{P}$ in a Fourier representation of the polarization $\mathbf{P}$ in terms of the dielectric susceptibility $\mathcal{X}$:

$\mathbf{P} = \epsilon_0\cdot(\mathcal{X}^{(1)}(\mathbf{E}) + \mathcal{X}^{(2)}(\mathbf{E},\mathbf{E}) + \dots)$ .

This expansion **does not work** anymore if the excitation field has components close to the resonance of the medium. Then, one has to take the whole quantum mechanical situation into account by e.g. describing light/matter interaction by a two-level Hamiltonian.

But this approach is certainly **not the most general one**.

## Intrinsically nonlinear formulations of electrodynamics

So, what kind of nonlinear formulations of electrodynamics given in a Lagrangian formulation are there?

One known ansatz is the **Born-Infeld model** as pointed out by Raskolnikov. There, the Lagrangian density is given by

$\mathcal{L} = b^2\cdot \left[ \sqrt{-\det (g_{\mu \nu})} - \sqrt{-\det(g_{\mu \nu} + F_{\mu \nu}/b)} \right]$

and the theory has some nice features as for example a maximum energy density and its relation to gauge fields in string theory. But as I see it, this model is an intrinsically nonlinear model for the free-space field itself and not usefull for describing nonlinear matter interaction.

The same holds for an ansatz of the form

$\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \lambda\cdot\left( F^{\mu\nu}F_{\mu\nu} \right)^2$

proposed by Mahzoon and Riazi. Of course, describing the system in **Quantum Electrodynamics** is intrinsically nonlinear and ... to my mind way to complicated for a macroscopical description for nonlinear optics. The **question** is: Can we still get a nice formulation of the theory, say, as a mean field theory via an effective Lagrangian?

I think a suitable ansatz could be

$\mathcal{L} = -\frac{1}{4}M^{\mu\nu}F_{\mu\nu}$

where $M$ now accounts for the matter reaction and depends in a nonlinear way on $\mathbf{E}$ and $\mathbf{B}$, say

$M^{\mu\nu} = T^{\mu\nu\alpha\beta}F_{\alpha\beta}$

where now $T$ is a nonlinear function of the field strength and might obey certain symmetries. The equation $T = T\left( F \right)$ remains unknown and depends on the material.

## Metric vs. $T$ approach

As pointed out by **space_cadet**, one might ask the question why the nonlinearity is not better suited in the metric itself. I think this is a matter of taste. My point is that explicitly changing the metric might imply a non-stationary spacetime in which a Fourier transformation might not be well defined. It might be totally sufficient to treat spacetime as Lorentzian manifold.

Also, we might need a simple spacetime structure later on to explain the material interaction since the polarization $\mathbf{P}$ depends on the matter response generally in terms of an integration over the past, say

$\mathbf{P}(t) = \int_{-\infty}^{t}R\left[\mathbf{E}\right](\tau )d\tau$

with $R$ beeing some nonlinear response function(al) related to $T^{\mu\nu\alpha\beta}$.

## Examples for $T$

To illustrate the idea of $T$, here are some examples.

For **free space**, $T$ it is given by $T^{\mu\nu\alpha\beta} = g^{\mu\alpha}g^{\nu\beta}$ resulting in the free-space Lagrangian $\mathcal{L} = -\frac{1}{4}T^{\mu\nu\alpha\beta}F_{\alpha\beta}F_{\mu\nu} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$
The Lagrangian of Mahzoon and Riazi can be reconstructed by

$T^{\mu\nu\alpha\beta} = \left( 1 + \lambda F^{\gamma\delta}F_{\gamma\delta} \right)\cdot g^{\mu\alpha}g^{\nu\beta}$.

One might be able to derive a Kerr nonlinearity using this Lagrangian.

So, is anyone familiar in a description of nonlinear optics/electrodynamics in terms of a gauge field theory or something similar to the thoughts outlined here?

Thank you in advance.

Sincerely,

Robert

### Comments on the first Bounty

I want to **thank** everyone actively participating in the discussion, especially **Greg Graviton**, **Marek**, **Raskolnikov**, **space_cadet** and **Willie Wong**. I am enjoying the discussion relating to this question and thankfull for all the nice leads you gave. I decided to give the **bounty** to Willie since he gave the thread a new direction introducing the **material manifold** to us.

For now, I have to reconsider all the ideas and I hope I can come up with a new revision of the question that should be formulated in a clearer way as it is at the moment.

So, thank you again for your contributions and feel welcome to share new insights.

This post imported from StackExchange Physics at 2014-04-01 16:36 (UCT), posted by SE-user Robert Filter