On top of that spectrum, you have the optical cavity modes which act like a filter. The spacing of the longitudinal modes depends on the length of the cavity, and the spectrum of those modes looks like a comb. The peaks of that comb spectrum will generally be narrower than the laser emission peak, and the result is that where the peaks of the comb overlap with the laser emission peak, light at that wavelength will be allowed through.

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Describelaser modes

Depending on the wavelength of a particular laser and the length of the cavity, it's possible to design lasers with multiple cavity modes falling within the emission peak, resulting in light with several distinct frequencies, or you can have lasers with only a single cavity mode falling within the emission peak. These "single mode" lasers are useful in some circumstance because they emit light in a narrow band of wavelengths, and the exact wavelength can even be tuned by varying the propertied of the cavity slightly so as to shift the cavity mode within the emission peak. This can be done either by carefully designing and manufacturing a cavity with a particular length, or for some types of laser it can be done dynamically by heating the laser material to change the length and/or refractive index.

Mode-lockedlaser

If I would speak of different modes of the HeNe-laser, would that mean, that there occur other wavelenghts next to $632.8\ \mathrm{nm}$ ? How is this possible? I only understand what a mode is in an optical cavity: It is simply an allowed resonance frequency of the cavity, e.g. a standing wave.

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Longitudinal mode

When an object becomes polarized, its center of positive charge becomes separated from its center of negative charge. Overall, there are just as many positive ...

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Why is it, that a laser can produce different modes? I take the HeNe-Laser as an example. The main wavelength stabilized and amplified is at $632.8\ \mathrm{nm}$. This originates simply from the neon emission.

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Laser modeswikipedia

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TE mode

Laser modespdf

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Apr 10, 2017 — Width of Subject = Subject Distance × Field of View ÷ 57 × Subject Width (i.e. 44% = 0.44). If we take the widest focal length of 24mm and do ...

As far as I understood, different laser modes correspond to different transversal distributions of the electromagnetic field. Therefore, different modes must have different frequencies.

The key point that I think you may have missed is that the emission line of the transition used to generate photons in any laser has a finite width. In the case of a helium neon laser, 632.8nm refers to the wavelength where the emission is brightest, but it doesn't mean that all the light comes at exactly that wavelength. There's a spectrum with a peak at that wavelength, but the peak has some width meaning that some light is emitted in a band of wavelengths to either side of 632.8nm.

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As we have mentioned (and can be seen in the equations) the wavelength of light is an important factor in the resolution of a microscope. Shorter wavelengths ...