Intrinsic permutation in nonlinear susceptibilities and pulses

The nonlinear susceptibilities have an intrinsic permutation symmetry. This symmetry treats two frequency components that are equal differently than it treats two frequency components that are slightly different.

Typically, the susceptibility is found by assuming the electric field is a Fourier sum of plane waves:

$ E(t) = sumlimits_{omega = -infty}^{infty} tilde{E}(omega_n) e^{iomega_n t} $

Implicitly, this makes an assumption:

$ E(omega_n) = frac{1}{T} intlimits_{-T/2}^{T/2}dt E(t) e^{-i omega_n t}$

Where $omega_n = 2pi n/T$. This converges to the Fourier integral in the limit that $T rightarrow infty$.

Unfortunately, the third order polarization is written as (from Boyd, pg. 13):

$ P^{(3)}(omega) = 3 chi^{(3)}(omega; omega, -omega, omega) |E(omega)|^2E(omega) + sumlimits_{omega_i,omega_jneqomega}left(6chi^{(3)}(omega; omega, -omega_i, omega_i) |E_i(omega_i)|^2E(omega) + 6chi^{(3)}(omega; omega, -omega_j, omega_j) |E_j(omega_j)|^2E(omega) right)$

My issue lies with the “$6$” and “$3$” here, and how they interact with the sum, as $Trightarrowinfty$.

If I have a pulse, my $E$ field is some distribution over frequencies, and I would find the total third order polarization by summing over all the frequencies present in my pulse. We assume the pulse is Gaussian for simplicity (though any pulse that is sufficiently smooth should do). We also note that the susceptibilities $chi^{(3)}(omega_i, omega_j, omega_k)$ are continuous in the $omega$s (so to first order, the susceptibility doesn’t change much when the frequencies change very little).

For small $T$, the first term ($chi^{(3)}(omega,omega,-omega)$) will dominate. The spacing between $omega_n$ and $omega_{n+1}$ is large enough that the electric field component $E_j(omega_j)$ for $omega_j neq omega$ is small enough to make the second two terms small.

However, as $T$ gets large, the frequency difference between $omega_j$ and $omega$ will get smaller, and for a sufficiently short pulse (compared to $T$), $E_j(omega_j)$ and $E(omega)$ will be roughly the same magnitude. In this case, the first term will be 1/2 as large as the second two terms.

The problem is that the polarization then becomes dependent on $T$, getting LARGER at $T$ gets larger, due to the fact that, in the infinite $T$ limit, the “3” has no effect on the sum/integral, but the “$6$” terms do, while in the small $T$ limit, the “3” term dominates.

How do I address this discrepancy? Both Boyd and Butcher and Cotter describe this intrinsic permutation symmetry, and assign different “permutation numbers” to $chi^{(3)}(omega, – omega, omega)$ and $chi^{(3)}(omega, -omega_i, omega_i)$.