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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?

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The wavelength can't change much because it comes from an atomic transition. According to Wikipedia the wavelength normally only varies by 0.001nm. How accurately do you need to know the wavelength? – John Rennie Dec 17 '13 at 11:52
@JohnRennie Many thanks for you comment. I have to measure the wavelength using the diffraction grating and say if I can be sure that the light is monochromatic. The only justification that I have thought is the if the light isn't monochromatic, I'll see more maximums of the same order. But I have to keep in mind the resolution power of the grating, so I'm asking if, using those values of D and N, I can be sure about monochromaticity.. Many thanks again! – sunrise Dec 17 '13 at 12:27
Your experiment can only measure a range of frequencies, and it can only measure frequencies with a limited resolution. So you can't say the light is monochromatic, only that extra frequencies must lie outside your experimental limits. This is a problem all experimental physicists face - welcome to the world of experimental science! :-) – John Rennie Dec 17 '13 at 12:44

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.

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Many thanks for your answer! But I haven't a high resolution spectrometer.. Please, read my reply to JohnRennie in the original post.. thanks again – sunrise Dec 17 '13 at 12:32
@sunrise If you don't have a HR spectrometer I don't see other trustful way of measuring the width of the emission. One thing is for certain and you don't need to make an experiment: "the emission band is not infinitesimally thin". – cinico Dec 17 '13 at 13:50

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.

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I have studied the Fabry-Perot interferometer, but I don't know what is a Fabry-Perot etalon. I have searched on the net, but I haven't understood where I have to put the lens and the reflecting surfaces. Do I need reflecting surfaces to clean up? Many thanks! – sunrise Dec 17 '13 at 13:24
@sunrise - not an easy thing to build from scratch. The FP etalon requires two optically flat surfaces mounted exactly parallel at a specified distance (generally N half-wavelengths) apart. There are wedged versions, e.g.… – Carl Witthoft Dec 17 '13 at 13:54
I haven't understood where I should put the laser in respect with the two surfaces... But if I don't use the reflecting surfaces and use two lens (collimating and focusing,…), can it have some sense? – sunrise Dec 17 '13 at 14:15
Yep, if your best grating doesn't meet your needs than the only thing you can do experimentally is get a higher dispersion device. My current school has a cute little (circa 20 cm optical path) Michelson interferometer with some absurdly high dispersion sitting on a shelf in the prep room. Small, easy to understand and it really works. – dmckee Dec 17 '13 at 17:24

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).

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