Quasi-three-level Laser Gain Media
Early theoretical models for laser gain media distinguished four-level and three-level laser gain media, which exhibit rather different characteristics e.g. in terms of threshold pump power when used in a laser. Later on, it was found that there can be a kind of intermediate situation, where one has four relevant energy levels, but with the second-lowest level being only slightly above the ground state (see Figure 1). Despite having four levels, one does not obtain typical four-level behavior because that second-lowest energy level has a substantial population (e.g. a couple of percent of the ground state population) by thermal excitation according to the Boltzmann distribution (usually considering room temperature).
As a consequence of that, the unpumped gain medium causes some reabsorption loss at the laser wavelength, and transparency (zero net gain) is reached only for some finite pump intensity. For higher pump intensities, there is positive net gain, as required for laser operation or signal amplification.
There are many examples of such behavior in the context of solid-state lasers and particularly for fiber lasers and fiber amplifiers. Here, one has to deal with Stark level manifolds, where the optical transitions between those may or may not be substantially overlapping in frequency space:
- A simple example are Yb3+:YAG laser crystals, where the laser-active ions (Yb3+) exhibit only two Stark level manifolds (2F7/2 and 2F5/2), as shown Figure 2, and the sub-levels can be identified. All laser transitions end on the higher sub-levels of the ground state manifold, and exhibit only slightly longer wavelengths than the pump transitions. That small wavelength difference implies a low quantum defect, which is beneficial in terms of power conversion efficiency and consequently also leads to a relatively weak heat generation, so that detrimental effects such as thermal lensing are relatively weak. On the other hand, the significant population in those end levels implies that positive net gain is only achieved for rather high pump intensities. Therefore, such laser gain media are hardly suitable for lamp pumping, and also only to a limited extent for side pumping. End-pumped Yb:YAG lasers, however, can reach quite high power conversion efficiencies, for example as thin-disk lasers with multi-kW output powers and efficiencies above 60%.
- Ytterbium ions can also be used in glasses, e.g. as Yb-doped fibers. Here, the different sub-levels cannot be identified because the optical transitions strongly overlap. A beneficial aspect of that is the large gain bandwidth. However, the effective transition cross-sections are accordingly reduced, which also results in lower gain. Depending on the signal or laser wavelength, reabsorption effects can be strong or weak. They are rather weak for wavelengths of 1060 nm or longer, and in that case were nearly has the characteristics of a four-level gain medium, while quasi-three-level behavior becomes increasingly substantial for shorter wavelengths. There are even lasers and amplifiers operating at 975 nm, where the system behaves like a three-level system.
- A similar behavior is found in erbium-doped fiber amplifiers, only with signal or laser wavelengths in the 1.5-μm region, and usually with the use of a higher-lying pump level and a correspondingly larger quantum defect. In-band pumping around 1.45 μm is also possible, however.
Excitation-dependent Gain Spectra
An important fact is that the spectral shape of the optical gain in a quasi-three-level laser medium depends on the excitation level because this affects the balance between emission and reabsorption. This is relevant for lasers and amplifiers:
- For relatively weak pumping, an amplifier may exhibit gain mostly at longer wavelengths, while the net gain becomes negative for shorter wavelengths, where reabsorption still dominates. For stronger pumping, the gain is shorter wavelength becomes positive and often rises more rapidly, so that the gain maximum is finally obtained there. As an example, Figure 3 shows gain spectra for erbium ions for different excitation levels.
- The emission wavelength of a quasi-three-level laser often depends on the resonator losses: high losses require a higher gain, and thus a higher excitation level, and consequently a shorter wavelength of maximum gain. (Similarly, the wavelength of maximum gain can be reduced by reducing the doping concentration because this also implies a higher excitation density.) For low resonator losses, however, the laser would operate at a longer wavelength because a positive net gain is sooner reached in that region.
Problems for the Laser Efficiency
In principle, quasi-three-level gain media can have a very low quantum defect and thus be highly efficient. However, reabsorption causes problems which often lead to a reduced power efficiency:
- Considering an end-pumped laser with unidirectional pumping, it is clear that the pump intensity at the output end (concerning the pump direction) must still be high enough to achieve positive net gain there. That implies a certain residual pump power, which cannot be absorbed. Therefore, even if the slope efficiency is very high in terms of absorbed pump power, it will be substantially lower with respect to incident pump power. Bidirectional pumping is no solution to that, except if it is realized by back-reflecting the pump power at the output end – which, however, is often not practical. Note that the problem with incomplete pump absorption is particularly severe when operation with a high excitation level – for example, for exploiting gain at relatively short wavelengths – is required.
- Apart from incomplete pump absorption, one also has increased losses by fluorescence because part of the excited ions are needed just to overcome the reabsorption.
On the other hand, reabsorption does not directly imply a loss of energy because it contributes to the upper-laser-level population. Its detrimental effects for the power efficiency are only the indirect ones as explained above.
In high-power fiber lasers and amplifiers, which are mostly based on Yb-doped fibers, the quasi-three-level nature of ytterbium is largely avoided by operation at relatively long laser or signal wavelengths – typically longer than 1060 nm or even around 1100 nm. Although the quantum defect is then somewhat higher, the efficiency can still exceed 80%. Losses by fluorescence are relatively weak due to operation with rather high signal intensities, and pump absorption can be quite complete because of the weaker we absorption or signal or laser light at long wavelengths.
In the literature, a profound misunderstanding of various aspects related to three-level gain media is sometimes encountered:
- It is sometimes not understood that the reabsorption does not constitute a real loss of energy because it excites ions into the upper level, so that stimulated emission can again occur at a later time. Nevertheless, three-level characteristics tend to decrease the laser efficiency for other reasons, as explained above.
- Another frequent mistake is to believe that (according to Einstein's original simple model) there are equal transition cross-sections for stimulated emission and reabsorption, although this does not hold for effective transition cross-sections for transitions between level manifolds with non-degenerate Stark levels, as often occur in solid-state laser gain media.
- The saturation fluence is often not calculated correctly: it is the photon energy divided by the sum of emission and absorption cross-sections (not the emission cross-section alone).
- It is not true that reabsorption in a three-level gain medium (whether in an unpumped region or not) acts like a saturable absorber, leading to Q-switching effects and the like. For such effects, the cross-sections of the absorber would have to be larger than those of the gain medium.
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