What could cause a diode laser to be emitting the half-harmonic of the fundamental frequency? I have a 405nm laser which is seemingly outputting a small portion of 810nm light.  I am wondering what mechanism this could be caused by.  Is this a down-conversion phenomenon or perhaps just another resonant frequency of the cavity that isn't filtered out well?
The laser is not a frequency-doubling laser.
 A: You haven't said precisely what instrument you are using, but what you are seeing is normal for grating monochromators.
Suppose we have light of one and only wavelength $\lambda$ incident normally on a grating.  Scattered light will be seen at the following angles, $\theta$:
$$ d \sin \theta = m \lambda \qquad \text{for } m=0, \pm 1, \pm 2, \pm 3, \ldots$$
where $d$ is the distance between rulings on the grating and $m$ is an integer which is called the order.  Any incident light will, with varying efficiency, be scattered into each of the orders.  Grating monochromators generally look for light scattered into the first order and infer the wavelength from the scattering angle.
But, note that a grating monochromator cannot distinguish between 810nm light in first order ($m$=1) and 405nm light in second order ($m$=2).  This is because $ 2\times 405 = 1\times 810$.  To avoid such issues, you must provide filters.  The numbers that you provide show that the filter is working well but it is not perfect.
In fixed monochromators, such as Ocean Optics sells, one normally does not have to think about the filters: they are built-in in front of the CCD array.  They, however, are never perfect filters.
Example
This photo shows light from a 632.8nm HeNe laser as it passes through a grating and onto a wall:

The bright spot at the center is $m=0$.  The spots immediately on either side are $m=1$ and $m=-1$.  The spots further out are $m=2$ and $m=-2$. If a monochromator is in the presence of a HeNe beam and is set to read 1265.6nm, it will inevitably see some light from $m=2$ of the 632.8nm beam.
More reading
ThorLabs, a vendor, provides a practical tutorial on gratings here.
