Are some wavelengths 'longer' than others spatially, and as such explain things such as longitudinal modes?

Question

I'm currently learning lasers. In my research of mode-locking, I've stumbled across something called a longitudinal mode.

The allowed modes of the cavity are those where the mirror separation distance L is equal to an exact multiple of half the wavelength, λ:

L = (n λ) / 2

where n is an integer known as the mode order.

Does this mean the wavelength, the literal 450 nm length, say, has to be of that criterion? I realize increasing the wavelength will create a completely different sinusoid, and if that doesn't fit in the cavity where its 0 points are at either end, it won't lase? Because what I envision, in the first place, are photons bouncing from the gain medium, off the resonator, back through the gain medium, etc. If this is then an 'information' wave, for lack of a better word -- if its energy fits the criteria, if we took its sinusoid associated with its wavelength and extended it, if it could lase we would have 0 points at either resonator?

I guess what I'm trying to ask is, what is this wavelength in this case, and if it is what I'm thinking, then its wavelength does then have spacial significance? Is that what the wavenumber is, essentially?

What we definitely have in the laser system are photons bouncing off the gain medium to resonators and back. How on earth did this become turned into a standing wave? And how is it supposed to leak out if our abstraction we've made doesn't have nodes at either end of the mirror?

I know this is a mouthful, but I'd really appreciate any help.

Also, how does destructive interference occur if there are no 0 amplitude nodes at either end of the resonator in the first place? Having trouble visualizing that.ie, if our condition with L is not met?

Answer 1

I guess what I'm trying to ask is, what is this wavelength in this case, and if it is what I'm thinking, then its wavelength does then have spacial significance?

Wavelength is the actual distance in space between successive maxima of the field (E or M) in an EM wave.

Is that what the wavenumber is, essentially?

Wavenumber is $\frac{2\pi}{\lambda}$, so it tells you the number of cycles (measured in radians) of that wave that "fit" in a given distance.

For example, a HeNe laser typically outputs a wavelength of 632.8 nm. Let's say we have one with its output polarization in the +x direction.

That means there are 632.8 nm between locations with maximum positive E-field in the +x direction. Or between locations with maximum H-field in the -y direction.

And the corresponding wavenumber is $9.929\times10^6{\rm m}^{-1}$, meaning the phase of the wave evolves by 9,929,000 radians (~1,580,000 full cycles) as it propagates through 1 m of space.

What we definitely have in the laser system are photons bouncing off the gain medium to resonators and back. How on earth did this become turned into a standing wave?

Generally you should not think about photons when you are trying to understand how light propagates. Although photons are "particles" they don't move in straight lines like bullets. They propagate according to the wave equation, resulting in the same propagation behavior we expect from classical electromagnetics based on Maxwell's equations.

And how is it supposed to leak out if our abstraction we've made doesn't have nodes at either end of the mirror?

The assumption that the mirrors are perfect, therefore producing perfect nodes in the field pattern, is only an approximation. In fact, for there to be an output beam, the reflectivity of one mirror must be less than 1.0 exactly. But values above 0.90 are quite common.

This small change in reflectivity typically does not impact the field pattern enough to invalidate the analysis of the modes of the laser; or at least it causes no more error than other effects like thermal changes in the cavity dimension, thermal changes in the index of refraction of the medium in the cavity, etc.