Beam Radius and Beam Quality of Laser Pulses
In this article I treat additional aspects concerning the beam radius and beam quality of laser light which come into play when the laser radiation is delivered in the form of pulses. Indeed, this deserves some thoughts, and some details are not trivial. Curiously, some articles on beam quality measurements on pulsed lasers do not even mention some of the central aspects discussed here.
Time-dependent Variations of Beam Shape
It were all quite easy if we could assume that the transverse profile of the laser beam stays constant during the full duration of the light pulse. The overall beam radius and beam quality would just match the values which one obtains at any time. But is that the case? Well, it depends:
- Mode-locked lasers usually have a nice diffraction-limited beam profile, which stays pretty much constant during a pulse. An exception are some lasers based on Kerr lens mode locking, where the very intense pulse peak generates a substantial amount of Kerr lensing. That is associated with significant, although usually not extreme temporal variations of beam shape concerning amplitude and phase.
- Q-switched lasers, particularly those producing pulses with very high pulse energies, are different in various aspects. Some of those lasers produce a diffraction-limited output, with power mostly in one or several fundamental resonator modes. In such cases, the beam shape will often not change that much during a pulse; it is possible, though, that the thermal lens in the laser crystal changes during the duration of a pulse, and that can have some effect. Besides, there are also transverse multimode lasers – particularly often among those with high energy output –, and there the temporal evolution of power in different transverse modes can be substantially different because the laser gain can be substantially mode-dependent. A typical situation would be that the fundamental mode comes up first, leading to a relatively small beam radius and high beam quality at the beginning of the pulse, but somewhat later on the higher-order modes come up as well, having both substantially larger beam diameters and larger divergence.
So there are really cases where the beam radius and beam quality change quite dramatically during a laser pulse.
How to Define Beam Radius and Beam Quality for Pulses?
Many textbooks, articles etc. discuss beam radius in beam quality only for continuous-wave radiation; it takes some thoughts to generalize this for pulses.
A first approach can be to consider beam radius and beam quality as time-dependent quantities, just applying the usual equations to the optical intensity at one point in time. However, we may also want such quantities characterizing the pulse as a whole.
For that purpose, it is quite natural to transform the used equations such that one just replaces optical intensities with fluence values. (Fluence is optical intensity integrated over time, with units e.g. of J/cm2.) For example, one can use the second moment of the fluence distribution (instead of the intensity profile) for calculating the beam radius according to ISO Standard 11146. In the same manner, one can obtain beam divergence and M2 factor as the common measure for beam quality.
The results obtained with such methods just give you what is relevant in cases where you do not resolve the pulse temporally. For example, they tell you how much the radiation is focused to a work piece in laser material processing without bothering about the temporal details.
What Do We Measure?
In most cases, beam radius and beam quality measurements are not time-resolved, at least not on the timescale of pulses, which are usually pretty short. However, such measurements should usually lead to results which are compatible with the definition explained above.
In many cases, the spatial beam parameters are actually measured using a large number of pulses in a pulse train with high pulse repetition rate. That is particularly the case for mode-locked lasers. By the way, measurements are often more difficult for Q-switched lasers, for which various commercial beam profilers are not suitable.
There are various possibilities of making time-resolved measurements even on very short time scales below the duration of a mode-locked pulse, but this is quite tricky stuff, and it will normally not be possible to obtain the required measurement systems commercially, at least not as a whole.
In numerical computations of laser pulse generation, for example, one naturally obtains time-dependent quantities and has the opportunity to study anything of interest. That could begin with qualitative checks – for example, to which extent the phenomenon of time-dependent mode powers materializes, and to what extent that could be mitigated e.g. by providing a gain profile which is spatially as flat as possible.
From calculated time-dependent intensity distributions, one can also calculate any qualities of interest like the beam radius and beam quality of the pulse as a whole. That can be quite tricky, however, in the very general case where even the position of the beam focus may change during a pulse. In many cases, a simplified approach will be sufficient, where one calculates the time-integrated beam quality factor by integrating the time-dependent quantity with the total optical power as a weight factor. Strictly speaking, however, one would need to calculate the fluence distributions in the near field and far field in order to compute beam radius, divergence and finally the beam quality from those.
This article is a posting of the Photonics Spotlight, authored by Dr. Rüdiger Paschotta. You may link to this page and cite it, because its location is permanent. See also the RP Photonics Encyclopedia.
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