How to be sure that a He-Ne laser light is monochromatic How can I be sure that the emission of a He-Ne laser contains only one single mode of laser cavity?
The only thing that I know is that if I use a diffraction grating and the light isn't monochromatic, I'll see maximums of the same order $m$ at different angles, but I also know that if wavelengths are very close I may not see them. I have to mind the resolutive power of the grating ($R=mN$).
If N1=1000 lines/mm and N2=500 lines/mm and the grating paces are D1=10^-6 m and D2=2*10^-6 m, will I see different maximums if the light isn'tmonochromatic?
Do you know other ways to know if the light of a He-Ne is monochromatic?
 A: You should keep in mind that true monochromatic light is not possible due to uncertainty principle. The emission will be always a band with a certain width which depends on temperature and other technological factors.
The best thing to do is to use a high resolution spectrometer and take a spectrum of your laser, taking the necessary precautions not to damage the detector, keeping the slits as closed as possible and optimizing you alignment.
Even with all the precautions, you will have a "slight" broadening of your line due to the experimental equipment.
But, as JohnRennie said, you are already expecting an atomic transition thus a very narrow emission band.
A: It depends on "how monochromatic" a source you need for your current use.  Further, you can have multiple modes of a single wavelength.  Using a Fabry-Perot etalon can clean up things a bit.
But if your question is not how to achieve, but rather how to evaluate, your source, then you will be limited by the resolution of your spectrometer, or the peak-spacing of your FP etalon, etc.  
A: Your laser cavity is a Fabry-Pérot interferometer. The free spectral range tells you how close two neighboring laser modes can get:
$\Delta \nu=\frac{c}{2nl}$ (for a linear resonator, length l, refractive index n).
The resolution of your spectrometer needs to be smaller than this free spectral range. You can increase the free spectral range by either building a shorter cavity or by introducing an etalon inside the laser cavity. Like the laser cavity, it has a free spectral range which is very large due to its thinness.
As already mentioned by Carl Witthoft, there can be several modes of the same wavelength (apertures or mode-selective pumping can suppress unwanted modes).
