# How many cavity round trips typically occur inside laser resonators?

Is it possible to estimate the number of cavity round trips occurring inside typical laser resonators, e.g. in a laser diode or a He-Ne laser?

Motivation: I've come across a short blog text, where the author draws an analogy between laser cavities and popular "infinity mirror" toys in order to explain the generation of spatial coherence, i.e. why many lasers represent point sources to a good approximation. The high spatial coherence is explained by an increase in the effective distance between the lasing volume of the active medium and the output coupler, resulting in a reduced apparent angular size of the source volume.

Assuming an etalon type cavity consisting of two parallel planar mirrors, I want to test this analogy by estimating the spatial coherence for different cavity lengths $$L_{cavity}$$ and comparing the results to specifications of commonly available laser types.

From $$L_{cavity}$$ and the number of round trips $$N_{roundtr}$$, one could deduce a "virtual" distance of the lasing volume to the output mirror

$$D_{appar} = N_{roundtr} \cdot L_{cavity}$$

and, with the diameter $$d_{active}$$ of the lasing volume, estimate its apparent angular size when seen from the output mirror as

$$\alpha_{appar} = 2 \ arctan \left( \dfrac{d_{active}}{2 D_{appar}} \right)$$

By this analogy, one could perhaps also make a rough estimation of the degree of collimation achieved after $$N_{roundtr}$$ cavity round trips, i.e. of the beam divergence stated in most datasheets.

PS.: I'm not a laser specialist, so any expert opinion or correction on this topic would be welcome.

• I'd like to answer this but there are a lot of topics covered here and a few things I find confusing. First off, when we talk about spatial coherence there is longitudinal coherence (i.e. points separated along the direction of propagation are in phase with eachother) or transverse coherence (points which are separated transverse to the direction of propagation are in phase with each other). In the question can you clarify which of these you're interested in at each point in the question? Commented Mar 14, 2023 at 7:10
• Next, I would not say lasers are like point sources. A point source emits a spherical wave while a laser emits a wave which typically has a very very small angular extent. In short, what I'll say, before giving a full answer, but maybe to help you refine the question, is that (1) the average number of round trips light makes is related to how non-reflective the mirrors are (this can be used to calculate the finesse) and (2) for laser cavities, the geometry of the output beam is related to the curvatures and spacings between the two cavity mirrors. Commented Mar 14, 2023 at 7:13
• Note that lasers never use two planar mirrors in their lasing cavity like some of the etalons in the pages you linked. At least one of the mirrors is always curved. The use cases for etalons are different than the use cases for laser cavities. Commented Mar 14, 2023 at 7:14
• And sorry, one final thing, the divergence of the light coming out of a laser is NOT related to the finesse, i.e., the number of round trips made in the cavity. Number of round trips is related to how far the mirrors are from perfect reflectivity while beam divergence is related to cavity mirror geometry (curvature and spacing). They are two orthogonal questions. I can explain more in an answer I guess if the question is refined just a little. Commented Mar 14, 2023 at 7:16
• Lasers by Siegman is the canonical reference for all of this: amazon.com/Lasers-Siegman/dp/0935702113 Commented Mar 14, 2023 at 7:16

I'll try to address the direct question asked in the title of this question:

"How many cavity round trips typically occur inside laser resonators?"

The cavity finesse roughly captures the answer to this question. No mirror is perfect. If power $$P_0$$ is incident on a mirror then some percentage $$R$$ of that power will be reflected, some percentage $$T$$ will be transmitted, and some percentage $$L$$ will be scattered (off of surface defects) or absorbed in the mirror material. By conservation of energy we have $$1 = R + T + L$$.

intuitively, if 1% of the light is lost from both mirrors then you'd expect the beam to make approximately 100 round trips before it is pretty much totally out of the cavity. This is basically what the Finesse says. Suppose we have two mirrors with $$R_1, L_1, T_1$$ and $$R_2, L_2, T_2$$. The formula for Finesse (which can be approximately though of as "the number of round trips"

$$\mathcal{F} = \frac{2\pi}{T_1 + T_2 + L_1 + L_2}$$

Finesse is limited, in practice, by how perfectly we can design mirrors with no loss or transmission. The highest finesse cavities in the world probably reach $$\mathcal{F} \approx 10^6$$, meaning transmission and losses are at the single part-per-million (ppm) level.

Depending on the specific commercial laser you will probably find finesses anywhere from 100 to 100,000 or more. The finesse will depend on various design criteria, and higher is not always better depending on what you want.

When it comes to lasers, the finesse is related to the longitudinal coherence of the laser. That is, how correlated is output light at one position and time with light at the same position but another time. Or equivalently, how correlated is output light at one position and time with output light at the same time but another position along the propagation direction of the beam?

The free spectral range of a cavity tells us how far apart in frequency space its repeated resonances are. It is given (in cyclic frequency units) by:

$$f_{FSR} = c/2 L_{cav}$$

Where $$L_{cav}$$ is the length of the cavity (not to be confused with mirror losses $$L$$). The cavity linewidth (in cyclic frequency units) is given by $$\nu = f_{FSR}/ \mathcal{F}$$

$$\nu$$ is related to the coherence time by $$\nu \propto \frac{1}{t_{coh}}$$ and the coherence length by $$\nu \propto c/l_{coh}$$. So we can see increasing the length of the cavity or the finesse of the cavity. So I think there is something to be said along the lines that: the cohereence of light coming out of a cavity is related to the duration of time or length of travel that it spends within the cavity resonator.

This might answer some parts of the question but I'm not sure if it answers more than the Wikipedia page on Fabry-Perot Interferometers.

The question asks a lot of stuff about what I would call "transverse" spatial coherence. And unfortunately it is hard for me to make sense of these parts of the question. The transverse spatial profile of a cavity is related to the curvature and distance between the mirrors. That is, the geometry of the output beam depends on the geometry of the cavity mirrors. Changing the finesse of the cavity (i.e. mirror reflectivity) and thus number of round trips has no effect on the shape of the output beam of light. This is at least true for optical cavities and lasers. In these cases at least one (usually both or all) cavity mirrors are concave to sort of refocus light back into the cavity when it bounces of the mirrors. etalons with two truly planar surfaces are not used as cavity resonators but rather serve other kinds of purposes.

See "Lasers" by Siegman for a much more thorough treatment of all of this.

Response to first comment. Above I've been a little glib about "if $$\mathcal{F} = 100$$ then the light makes roughly 100 round trips before it's 'gone'". The more precise mathematical statements (more rigorous derivations can be found in the reference) is like this. The field in the cavity decays like $$e^{-\pi \nu t}$$ and the energy decays like $$e^{-2\pi \nu t}$$. Focusing on the energy, the $$1/e$$ time for this decay is $$t_{1/e} = \frac{1}{2\pi \nu} = \frac{\mathcal{F}}{2\pi f_{FSR}}$$ The cavity round trip time is $$\tau = 1/ f_{FSR}$$ so the $$1/e$$ number of round trips is $$N_{1/e} = t_{1/e}/\tau = \frac{\mathcal{F}}{2\pi} = \frac{1}{T_1 + T_2 + L_1 + L_2}$$ Which is actually pretty satisfying. The total round trip energy loss probability is $$L_{TOT} = T_1 + T_2 + L_1 + L_2$$ so the energy in the cavity as a function of round trips is $$E = (1-L_{TOT})^N = \left(1-\frac{L_{TOT}N}{N}\right)^N \approx e^{-L_{TOT} N}$$ I guess if you read this last section in reverse it's a little bit of a hand-wavey proof that the energy decays like $$e^{-2\pi \nu t}$$.

Transverse Profile:

I'll assume the cavity mirrors have spherical surfaces. The modes inside of an optical cavity can be expanded as Hermite Gaussian modes. The modes have a waist which is located at some position (within the cavity if both cavity mirrors are concave, or outside the cavity if one of the mirrors is convex) in space and which has some waist size. These two parameters fully describe the mode. For optical cavities, the modes which survive are exactly those modes whose wavefronts at the locations of the cavity mirrors have radii of curvature that match the radii of curvature of the cavity mirrors. It turns out that if you constrain a Gaussian beam (for a certain wavelength) to have fixed radii of curvature at two positions in space this constrains the size and position of that beam's focus. The output beam is just whatever that mode looks like propagating out past the output mirror. One note is that for certain mirror curvatures and positions there is no possible Gaussian mode supported by that cavity geometry. For more details see the Lasers textbook. But note that the geometry of the mode in the cavity and the output mode is fully determined ONLY by the length of the cavity and the radii of curvature of the two cavity mirrors. I emphatically repeat that that changing the mirror reflectivities (and therefore the cavity finesse and number of round trips) has NO IMPACT WHATSOEVER on the transvere profile of the output beam.

• Hi, thanks a lot for your answer and the reference, I need to read and think a bit about it and will reply shortly. Just one comment/question upfront: Wouldn't a loss of 1% per round trip and a total of 100 round trips correspond to $0.99^{100} \approx 0.37$, i.e. about 63% loss? So, are you interpreting the ratio of initial to final energy as a probability and drawing a "probabilistic cut-off" at 37% energy remaining inside the cavity? Commented Mar 15, 2023 at 12:03
• @srhslvmn See my edit at the bottom. The answer is essentially yes. we have $37\% \approx 1/e$ and, yes, we're using $1/e$ as the "probabilistic cut-off" we're using to put a number to "number of round trips before the light decays out". Commented Mar 15, 2023 at 13:12
• The relationship to coherence is that, since the temporal response is $e^{-2\pi \nu t}$, in frequency we find a resonance peak with a width related to $\nu$. This means that the "coherence time" for light coming out of the cavity is $\approx 1/\nu$ (or maybe just exact equality, I'm not exactly sure, I usually just look at line shapes and don't think about coherence times). Commented Mar 15, 2023 at 13:20
• Thx a lot. I'll have to digest the relation between finesse and coherence time...but concerning your point that "The transverse spatial profile of a cavity is related to the curvature and distance between the mirrors. That is, the geometry of the output beam depends on the geometry of the cavity mirrors. Changing the finesse of the cavity (i.e. mirror reflectivity) and thus number of round trips has no effect on the shape of the output beam of light.", I'm thinking that this point might be exactly what I'm asking! Because I'm suspecting a connection precisely between said quantities: Commented Mar 15, 2023 at 14:30
• 1.) The cavity geometry, e.g. the cavity length, is one determinant of the beam geometry. 2.) The cavity finesse is a determinant of the number of possible round trips. => But if the number of round trips translates into an effective cavity length (the "infinity mirror"), then the finesse is actually also a determinant of the beam geometry, since a high finesse results in a high round trip number, which equates to a long effective cavity length...a geometric parameter that impacts spatial coherence! And if we now say that finesse it also linked to temporal coherence, ... Commented Mar 15, 2023 at 14:36