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I am currently studying Laser Systems Engineering by Keith Kasunic. Chapter 1.2.1 Temporal Coherence says the following:

Axial (or longitudinal) modes are determined by the geometrical fit (or resonance) of a given wavelength in the laser cavity. That is, if the two mirrors that define a laser cavity are nearly planar and perfectly reflecting – an ideal assumption, given that one mirror will be designed not to be so that light can escape the cavity as output power – then Fig. 1.12 shows that an integer number $m = 1, 2, 3$, etc., of half-wavelengths of the electric field fit in the cavity length $L$: $$m \dfrac{\lambda}{2n} = L \ \ \ \ \ \text{[m]} \tag{1.4}$$ where the refractive index $n$ of the gain medium is included to account for the reduction in wavelength in comparison with its free-space ($n = 1$) value. With the exception of a specific type of semiconductor laser known as a vertical-cavity surface-emitting laser (VCSEL) with a cavity length $L \approx \lambda$, the number of half-wavelengths is large in practice. For example, for a HeNe laser emitting at $\lambda = 633$ nm with $L = 100$ mm and $n \approx 1$, $m = 2L/\lambda = 2 \times 0.1 \ \text{m} / 633 \ \text{nm} = 315,955$ half-wavelengths. enter image description here

I understand the cavity length $m \dfrac{\lambda}{2n} = L$, as a standalone, but what is meant by "half-wavelengths" of the electric field, and how is this represented mathematically, in terms of electromagnetism and electromagnetic waves? And, relatedly, how does the $m$ values, as shown in figure 1.12, influence the electromagnetic waves themselves, beyond the simple equation of the cavity length $m \dfrac{\lambda}{2n} = L$ (that is, where in the mathematics of electromagnetic waves does the $m \dfrac{\lambda}{2n} = L$ come into play)?

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  • $\begingroup$ It's very simple, a half-wavelength is a distance equal to half the wavelength. If the wavelength is 632 nm, then a half-wavelength is 316 nm. $\endgroup$
    – The Photon
    Dec 14 '20 at 6:07
  • $\begingroup$ @ThePhoton But that doesn't answer my questions with regards to the context of electromagnetic waves. I'm looking for more detail within that context. $\endgroup$ Dec 14 '20 at 6:39
  • $\begingroup$ There's nothing special about EM waves in explaining this term. If 1 meter is 100 cm, then a half meter is 50 cm. If one wavelength is 632 nm, then a half wavelength is 316 nm. It's represented mathematically by $\lambda/2$, just like you used it in your post. $\endgroup$
    – The Photon
    Dec 14 '20 at 6:41
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    $\begingroup$ They're being over-specific. The wavelength would be the same if you measured the magnetic field. The diagram shows a case where 6 wavelengths (i.e. 12 half-wavelengths) fit in the cavity (labeled m=12) and a case where 5 wavelengths (10 half-wavelengths) fit labeled m=10, and a case where 4 wavelengths (8 half-wavelengths) fit, labeled m=8. In each case the number m is equal to the number of half-wavelengths in the cavity. $\endgroup$
    – The Photon
    Dec 14 '20 at 6:47
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    $\begingroup$ Presumably n is the index of refraction of the material in the cavity and $\lambda$ is the vacuum wavelength, so that the actual wavelength in the material is $\lambda/n$. $\endgroup$
    – The Photon
    Dec 14 '20 at 6:48
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the cavity length must be a multiple of the half wavelength of the laser (and the laser is a wave in the electric field). That's the 2 in the divisor of the formula. The "m" tells you that in a given cavity length several frequencies can resonate, so for example, if 100Ghz can resonate, then so do 200,300,400GHz etc...

BTW that drawing is misleading, I believe a standing wave must have a potential of 0 at the wall of the cavity, that's related to the fact that only an integer number of half wavelength of a radiation can resonate.

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