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Four-level and Three-level Laser Gain Media

Definition: laser gain media without/with reabsorption from the lower laser level

More general term: laser gain media

German: Vierniveau- und Dreiniveau-Lasermedien

Categories: optical materials, laser devices and laser physics

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URL: https://www.rp-photonics.com/four_level_and_three_level_laser_gain_media.html

Optical amplification in the gain medium of a laser or laser amplifier arises from stimulated emission, where the input light induces transitions of laser-active ions from some excited state to a lower state and acquires the energy related to the reduction in excitation energy.

Different numbers of electronic states may be involved, with important implications to various characteristics of a laser or amplifier. In the simplest cases, one is dealing with well-defined energy levels e.g. of isolated atoms or ions. In solid-state laser gain media, the situation is somewhat more complicated, involving Stark level manifolds. The explanations in this article start with simple energy levels but also consider Stark level manifolds later on.

Three-Level Systems

It was realized early on that for laser amplification one requires at least three energy levels; two levels are not sufficient, since optical pumping can then never result in a population inversion which provides gain. Simultaneous pump absorption and signal amplification can not occur.

three-level system
Figure 1: Energy level diagrams of a three-level system. The horizontal lines indicate energy levels; the higher a line, they higher the corresponding energy.

One more energy level is required for pumping, see the left part of Figure 1. Pumping transfers atoms (or ions) from the ground state into a level above the upper laser level, from which the atoms can (usually by non-radiative transitions) get into the upper laser level. An important aspect is that the atoms cannot be pushed back to the ground state by stimulated emission with pump radiation, since their excitation energy is too low for that – they cannot “see” the pump light any more. Therefore, with sufficiently intense pumping it is possible to reach a population of the upper laser level which is well above 50%, and thus higher than that of the lower laser level (the ground state); a population inversion is reached. In that situation, stimulated emission at the laser wavelength dominates over re-absorption on that laser transition, so that a positive net gain results, which can be used for amplification or laser operation.

Because the population inversion requires that more than half of the atoms are in the upper laser level, pumping with fairly high optical intensity is required. In case of a laser, the threshold pump power will be correspondingly high, which is often problematic.

A popular example of a three-level laser medium is ruby (Cr3+:Al2O3), as used by Maiman for the first laser, although the situation is in reality somewhat more complicated due to the Stark level manifolds (see below).

Pure three-level laser gain media are seldom used, while quasi-three-level media (see below) are quite common, particularly in the context of fiber lasers and fiber amplifiers.

Four-Level Systems

four-level system
Figure 2: A typical four-level system. The laser transition ends on a level above the ground state, which is quickly depopulated.

A far lower threshold pump power can be achieved with a four-level laser medium, where the lower laser level is well above the ground state (see Figure 2) and is quickly depopulated e.g. by multi-phonon transitions (in case of a solid-state medium) or by collisions (in a gas). Ideally, no appreciable population density in the lower laser level can occur even during laser operation, since the lower laser level is very short-lived. In that way, reabsorption of the laser radiation is largely avoided (provided that there is no absorption on other transitions). This means that there is no absorption of the gain medium in the unpumped state, and a positive net gain is achieved already for a rather low population in the upper laser level. The gain usually rises linearly with the absorbed pump power.

The most popular four-level solid-state gain medium is Nd:YAG. All lasers based on neodymium-doped laser gain media, except those operated on the ground-state transition around 0.9–0.95 μm, are four-level lasers.

Neodymium ions can also be directly pumped into the upper laser level, e.g. with pump light around 880 nm for Nd:YAG. While this reduces the quantum defect and thus possibly increases the laser efficiency, it also opens the possibility of stimulated emission of pump radiation reducing the upper-state population. The latter is not necessarily a problem, since a quite low upper laser level population is sufficient. Even though effectively only three levels are involved, the term three-level system would not be used here.

Quasi-Three-Level Systems

quasi-three-level system
Figure 3: A quasi-three-level system.

A quasi-three-level laser medium is one with a kind of intermediate situation, where the lower laser level is so close to the ground state (see Figure 3) that an appreciable population in that level occurs in thermal equilibrium at the operating temperature. As a consequence, the unpumped gain medium causes some reabsorption loss at the laser wavelength, and transparency is reached only for some finite pump intensity. For higher pump intensities, there is gain, as required for laser operation.

For more details, see the article on quasi-three-level laser gain media.

Stark Level Manifolds

Atoms and ions as used in laser gain media exhibit Stark level manifolds. Depending on how strong the interactions of atoms with their neighborhood are, and how much the nature of that neighborhood varies between different atoms, one may or may not be able to spectroscopically resolve the different Stark levels.

Often, in laser models for solid-state laser gain medium one deals with whole Stark level manifolds, rather than with all their sub-levels. The strength of optical transitions is then characterized with effective transition cross sections, which are intrinsically temperature-dependent. One may still apply the basic reasoning concerning three-level or four-level transitions in such cases, but keeping in mind that some details get more complicated. For example, the effects of different level degeneracies may need to be taken into account. Also, it can happen e.g. that the ground state manifold has its degeneracy lifted so much that one obtains a quasi-three-level system instead of a three-level system: one exploits laser transitions ending on the highest sub-levels of the ground state manifold, which have only a limited thermal population.

Smooth Transition Between Four-level and Three-level Characteristics

There can actually be a smooth transition from three-level to four-level gain characteristics with increasing laser wavelength. For example, erbium-doped glass (see Figure 4) shows strong three-level behavior around 1535 nm but nearly four-level behavior for long wavelengths e.g. beyond 1600 nm. Similarly, ytterbium-doped glasses exhibit pronounced three-level characteristics for wavelengths below ≈ 1040 nm. For operation at such short wavelengths, a large inversion density is required for overcoming the reabsorption loss. For longer wavelengths, as sometimes used particularly in fiber lasers, there is hardly any reabsorption, and in a long fiber only a very low excitation density may be required to obtain sufficient gain.

erbium gain
Figure 4: Gain and absorption (negative gain) of erbium (Er3+) ions in germano-alumino-silicate glass for excitation levels from 0 to 100% in steps of 20%. Strong three-level behavior (with transparency reached only for > 50% excitation) occurs at 1530 nm. At longer wavelengths (e.g. 1580 nm), a lower excitation level is required for obtaining gain, but the maximum gain is smaller.

Further Remarks

Pronounced three-level behavior is inevitable for gain media with a very small quantum defect, because this enforces a small energy spacing between the lower laser level and the ground state, so that thermal population of the lower laser level is significant.

By reducing the temperature of the laser crystal, it is possible to obtain less pronounced three-level characteristics, i.e., a reduced degree of reabsorption on the laser wavelength. This is essentially because the population in higher-lying sublevels of the ground state manifold is reduced. As an example, Yb:YAG has pronounced three-level characteristics at 1030 nm when operated at room temperature, while essentially four-level characteristics are obtained for cryogenic operation at 77 K (the temperature of liquid nitrogen).

Note that the gain media of semiconductor lasers actually also behave like three-level lasers, exhibiting losses in the unpumped state and a shape of the gain spectrum which depends on the excitation density.

Questions and Comments from Users

2020-06-15

Suppose you have a quasi three-level system with the lower laser level being one of the Stark split levels of the ground state. Normally, this state sees thermal population due to the Boltzmann distribution. Now assume the system is inverted and starts lasing. The ions from the upper laser level transition into the lower laser level. This – in theory – limits the extracted energy to 1/2 of the initial population above transparency, but it depends on the ability of the lower level to thermally equalize faster or slower than the extraction. Suppose you have a Q-switched laser producing  10–30 ns pulses. Does the lower state equalize fast enough (i.e. in <1–10 ns) so the extracted energy is higher than the 1/2 of the initial population?

Answer from the author:

First of all, the energy limitation is usually weaker than by a factor of 2, since due to the Boltzmann distribution at moderate temperatures you need much less than 50% excitation for obtaining positive net gain.

The relaxation within a Stark level manifold is usually fast enough in a Q-switched laser – it occurs on a picosecond timescale.

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Bibliography

[1]P. P. Sorokin and M. J. Stevenson, “Stimulated infrared emission from trivalent uranium”, Phys. Rev. Lett. 5 (12), 557 (1960), doi:10.1103/PhysRevLett.5.557 (the first four-level laser)
[2]W. P. Risk, “Modeling of longitudinally pumped solid-state lasers exhibiting reabsorption losses”, J. Opt. Soc. Am. B 5 (7), 1412 (1988), doi:10.1364/JOSAB.5.001412
[3]J. O. White, “Parameters for quantitative comparison of two-, three-, and four-level laser media, operating wavelengths, and temperatures”, IEEE J. Quantum Electron. 45 (10), 1213 (2009), doi:10.1109/JQE.2009.2020607
[4]Blog article: What is different for quasi-three-level lasers?

(Suggest additional literature!)

See also: laser physics, quasi-three-level laser gain media, laser gain media, ytterbium-doped laser gain media, rare-earth-doped laser gain media, laser transitions, Stark level manifolds, solid-state lasers, reciprocity method, effective transition cross sections, The Photonics Spotlight 2006-08-12
and other articles in the categories optical materials, laser devices and laser physics


Dr. R. Paschotta

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics Consulting GmbH. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!

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