2.1. Introduction
As described in our previous work, we have developed a one-dimensional numerical code for modeling energy, gain and spectral performance of an ytterbium-based regenerative amplifier, to which we have added routine for the spectral response of a generic Nd:glass amplifier following the first stage. The model for ytterbium-based amplification can be applied to different host materials and consists of two routines: the first one simulates the pump process and calculates the pump absorption and population inversion, and the second one simulates the overall amplification process calculating the pulse energy and spectrum after each pass through the gain medium. Both routines are ruled by the following discrete rate equations:
where
Nex is the excited state population,
Ngrd is the ground state population, (
z,
t) are the longitudinal position and time inside the gain medium,
I is the wavelength-dependent pulse intensity,
λ is the wavelength,
σa and
σe are respectively the absorption and emission cross-sections,
τf is the fluorescence lifetime,
h is Planck’s constant and
c is the speed of light in vacuum.
Ytterbium-doped laser media are described by a quasi-three-level model where the relevant laser transitions take place between the sub-levels of two manifolds, the ground-state 2F7/2 and the excited state 2F5/2. At room temperature, the sub-levels of the ground-state manifold are thermally populated due to the proximity between them, and in consequence reabsorption losses can be generated at the laser wavelength. This property has the drawback of making ytterbium lasers inefficient for very low pump intensities, but it can also be used beneficially in order to explore different sub-level transitions. This effectively means that one is able to tune the laser amplifier for different wavelengths by changing the pump power.
Two good candidates for efficient laser amplification near 1053 nm are Yb:CaF
2 and Yb:YAG, in the first case because of the relatively broad tunability [
10], and in the second because of the presence of a peak in the emission cross-section near this wavelength. We have used spectroscopic data for both materials (
Figure 1) [
13] and evaluated their amplification performance under different pump powers.
Table 1 lists the parameters that were used for the two materials. Where realistic input parameters for the pump source and the laser media were required, we have assumed typical values without loss of generality. The chosen pump pulse length matches the fluorescence lifetime of each gain media,
i.e., 2.4 ms for Yb:CaF
2 and 1 ms for Yb:YAG. It is assumed that the gain medium is longitudinally pumped in a two-pass configuration, in order to reach a high population inversion in Yb:CaF
2. The focused pump spot FWHM diameter is 700 μm. For Yb:CaF
2 we have considered 5 at% doping level and a length of 3.8 mm, while for Yb:YAG the doping level is 3 at% and the length is 8 mm. Furthermore no considerations about the damage threshold of the items in the cavity were taken into account.
Figure 1.
Absorption (solid lines) and emission (dashed lines) for Yb:CaF
2 (green) and Yb:YAG (blue). Adapted from Ref. [
13].
Figure 1.
Absorption (solid lines) and emission (dashed lines) for Yb:CaF
2 (green) and Yb:YAG (blue). Adapted from Ref. [
13].
Table 1.
Parameters used in the numerical simulations.
Table 1.
Parameters used in the numerical simulations.
Parameter | Yb:CaF2 | Yb:YAG |
---|
Pump pulse | | |
Power (W) | 46–66 | 20–30 |
Duration (ms) | 2.4 | 1 |
Beam diameter (mm) | 0.7 | 0.7 |
Wavelength (nm) | 938 | 938 |
Seed pulse | | |
Energy (pJ) | 100 | 100 |
Duration (ns) | 1 | 1 |
Beam diameter (mm) | 0.7 | 0.7 |
FWHM Bandwidth (nm) | 12 | 12 |
Gain medium | | |
Length (mm) | 3.8 | 8 |
Yb3+ concentration (at %) | 5 | 3 |
2.2. Regenerative Amplifier: Free-Running Cavity
In this section we will make a preliminary evaluation of the wavelength tunability for both media considering a free-running cavity, i.e., where amplified stimulated emission grows from noise rather than from a seed pulse. The amplified pulse is then extracted at the peak energy. The initial pulse traveling in the cavity has a rectangular spectral profile between 1020 and 1060 nm and an energy of 10−6 nJ, corresponding to ~1 photon for each wavelength slice used in the calculation. The pulse duration is 10 ns, which is similar to that obtained by cavity-dumping. We assume that there is no spectral filtering in the cavity (apart from the gain-induced modulation), so losses are equal for all wavelengths.
Figure 2 (Yb:CaF
2) and
Figure 3 (Yb:YAG) show the results in terms of output spectrum (left) and energy (right) performance
vs. pump power. For two discrete ranges of pump powers, 46–66 W for Yb:CaF
2 and 20–30 W for Yb:YAG, we have determined the maximum energy extractable and the number of roundtrips needed to achieve it.
Figure 2 (Yb:CaF
2) and
Figure 3 (Yb:YAG) show the results in terms of output spectrum (left) and energy (right) performance
vs. pump power. For two discrete ranges of pump powers, 46–66 W for Yb:CaF
2 and 20–30 W for Yb:YAG, we have determined the maximum energy extractable and the number of roundtrips needed to achieve it.
Figure 2.
Simulation of free-running cavity based in Yb:CaF2. Left: Output intensity vs. wavelength for a range of pump power; right: output energy and required number of passes vs. pump power.
Figure 2.
Simulation of free-running cavity based in Yb:CaF2. Left: Output intensity vs. wavelength for a range of pump power; right: output energy and required number of passes vs. pump power.
A common trend to both media is immediately apparent: for lower pump powers, there is a significant fraction of the amplified spectrum centered around 1048–1050 nm, corresponding to the case of a lower population inversion. As the pump power increases, gain shifting brings the spectral peak down to ~1030–1035 nm, resulting from the higher population inversion. The spectral distribution is however different for each case. For Yb:CaF
2 at the lowest pump power we observe a broad peak centered at 1048 nm. As the pump power is raised, the output energy is transferred to a second peak which forms around 1034 nm. At higher powers (54–58 W) it is possible to obtain a balance in the energy distribution between the two peaks. By increasing the pump power even further the longer wavelength peak will eventually vanish and the second one increases and slightly shifts towards lower wavelengths. This behavior confirms experimental results reported for different pump conditions by S. Ricaud
et al. [
14].
For Yb:YAG, a similar evolution is numerically witnessed, although the transition is more abrupt and the parameters are much more restrictive. The pump power required to overcome the lasing threshold (~20 W) is very close to that when the output spectrum is shifted to 1032 nm (~26 W), leaving a narrow range of pump powers and cavity loss combinations that allow amplification at 1049 nm. As a consequence of this, more than a thousand passes become necessary to reach high energies at this wavelength, as can be seen in
Figure 3 (right). In contrast, an Yb:CaF
2 amplifier operating at a similar wavelength would only require less than half the passes. This should be taken in consideration for the practical development of mJ-level amplifiers at 1050 nm based in Yb:YAG.
The behavior described above can be understood recurring to a model for the population inversion at the two different wavelength regions. Taking the case of Yb:YAG,
Figure 4 shows its energy level scheme, that can be considered a quasi-4 level. Both the ground level and the upper level are located within manifolds, separated by approximately 10,000 cm
−1. The absorption line at 941 nm is represented, as well as two transitions between (i) sublevels
f4 (10327 cm
−1) and
f3 (785 cm
−1), corresponding to the lasing wavelength 1048 nm, and (ii) sublevels
f4 and
f2 (612 cm
−1), corresponding to the wavelength 1029 nm. The figure also indicates the relative thermal occupancies for each manifold at a temperature of 300 K, assuming a Boltzmann distribution of the energy levels.
Figure 3.
Simulation of free-running cavity based in Yb:YAG. Left: Output intensity vs. wavelength for a range of pump power; Right: output energy and required number of passes vs. pump power.
Figure 3.
Simulation of free-running cavity based in Yb:YAG. Left: Output intensity vs. wavelength for a range of pump power; Right: output energy and required number of passes vs. pump power.
Figure 4.
Energy level scheme and laser transitions in Yb:YAG.
Figure 4.
Energy level scheme and laser transitions in Yb:YAG.
From the rate equations for the pump process, one can arrive at the following expression for the population difference, relating the population densities of the upper manifold
nU and lower manifold
nL, and the thermal occupancies of the relevant levels [
15]:
The full expressions for the population densities account for the absorption and stimulated emission cross-sections
σa and
σe, the fluorescence lifetime
τf and the pump rate
W, and may be found in the cited reference. The pump rate is defined as
where
Ppump is the pump power,
η is the excitation efficiency,
λp is the wavelength,
r is the radius of the excited region and
l is the length of the Yb:YAG crystal. By replacing the values used in our simulation and making Δ
n4m = 0, we find the pump power thresholds at which the population difference becomes positive, and amplification is possible (not taking cavity roundtrip losses into account). We arrive at the following values:
Pthr(1048 nm) = 11.2 W,
Pthr(1029 nm) = 25.2 W, closely matching the results of the simulation. This shows that there is a range of powers where amplification at the main peak wavelength is effectively suppressed due to the higher absorption, and operation around 1048 nm is possible. As soon as the upper pump threshold is reached, the larger gain cross-section at 1029 nm becomes prevalent.
2.3. Regenerative Amplifier: Seeded Cavity
We now consider the evolution of an input pulse inside a regenerative amplifier
i.e., a seeded cavity, for different pump powers. The input is chosen to be a Gaussian pulse centered at 1053 nm with a FWHM bandwidth of 12 nm, well inside the gain bandwidth of both media. Although the initial spectral response of the system is identical, one can expect that the evolution will be now be determined by these significantly different initial conditions. All other parameters are the same as in the previous section (
Table 1). Again, the output parameters are taken at the maximum pulse energy.
It is important to notice that since we are not considering any spectral filtering or hard spectral clip on the input pulse, its spectrum spreads infinitely, leaving a residual finite energy at lower wavelengths. For instance, for the parameters above the intensity at 1034 nm and 1032 nm will be respectively ~10−3 and ~2 × 10−4 times that at 1053 nm. Such values can be considered as noise inside the cavity, and from the results of the previous section we can expect their role in the amplification process to become more significant as the pump power increases.
Figure 5 (left) shows the results for Yb:CaF
2 for pump powers between 46–66 W. Contrary to the previous section we now observe that the output spectrum remains centered around 1049 mm for the entire pump power range. There is also a slight broadening (2.1 to 2.7 nm) towards the lower wavelengths due to the reduced reabsorption for higher population inversions. Besides, seeding the cavity allows reaching a higher output energy in a lower number of roundtrips for the same pump conditions due to the lower percentage of global losses.
Figure 5.
Simulation of regenerative amplification based in Yb:CaF2 in injection mode. Left: Output intensity vs. wavelength for a range of pump power; Right: output energy and required number of passes vs. pump power.
Figure 5.
Simulation of regenerative amplification based in Yb:CaF2 in injection mode. Left: Output intensity vs. wavelength for a range of pump power; Right: output energy and required number of passes vs. pump power.
The results for Yb:YAG are presented in
Figure 6. The major difference is that there is still an abrupt transition around 26–28 W where the output central wavelength changes from 1049 to 1032 nm, in spite of the injected signal. The reason for this difference between the two materials lies in their emission and absorption cross-sections. Yb:CaF
2 exhibits smoothly-varying curves between 1020 and 1080 nm (
Figure 1), such that small changes in the population inversion do not cause an abrupt transition between the lasing wavelengths. On the other hand, the very strong emission peak of Yb:YAG around 1030 nm (~8 × greater than that at 1050 nm) becomes dominant once the population inversion is favorable. Although it is possible to achieve mJ-level output energies at 1049 nm, more than 800 passes are required. One option could be the use of additional spectral filtering in the cavity by means of e.g., polarizers, etalons or dichroic mirrors, thereby allowing stronger pumping.
Figure 6.
Simulation of regenerative amplification based in Yb:YAG in injection mode. Left: Output intensity vs. wavelength for a range of pump power; Right: output energy and required number of passes vs. pump power.
Figure 6.
Simulation of regenerative amplification based in Yb:YAG in injection mode. Left: Output intensity vs. wavelength for a range of pump power; Right: output energy and required number of passes vs. pump power.
2.4. Ytterbium-Neodymium Hybrid Amplification Chain
The last results show that both materials can be considered promising candidates for milijoule-level regenerative amplification at ~1049 nm. Since the gap between this wavelength and the emission peak wavelength of Nd:glass (phosphate) at 1053 nm is very small, one can envisage their use in the pre-amplifier stages in a hybrid laser chain.
For evaluating this possibility, we used a simple routine based on the spectral response of an Nd:glass amplifier that is fed with the output data from the previous ytterbium-based amplification routine, such as energy and spectrum. The amplifier is modeled as a single-pass in an Nd:glass slab working far from saturation, described by the following unsaturated gain equation:
Here Iin and Iout are the spectral intensities of the pulse before and after the amplification, σe is emission cross-section of Nd:glass, N is the inverted population density, and L is the length of the gain medium.
For this hybrid configuration we chose the same realistic input parameters than for the previous simulations. The pump pulse, seed pulse and crystal parameters are those used in the previous section with the exception of the pump powers that were chosen to be 66 W for Yb:CaF2 and and 24 W for Yb:YAG, due to the favorable percentage of inverted population for amplification near 1053 nm and to prevent the generation of undesired high energy amplified spontaneous emission (ASE) that competed with the main pulse. We assume a 23.5 cm long Nd:glass with an inverted population density of 7.5 × 1018 cm−3.
Figure 7 shows the output spectra calculated for the two hybrid chain combinations. The pre-amplified and final pulse energies for Yb:CaF
2 were 13.4 mJ (after 219 passes) and 1.43 J while for Yb:YAG we obtained 1.4 mJ (822 passes) and 230 mJ respectively. For both cases there is strong gain narrowing during the pre-amplification stage, leading to a final bandwidth of 2 nm.
Figure 7.
Output spectra for a hybrid amplification chain with pre-amplifier based with Yb:CaF2 (left) and Yb:YAG (right). The solid black line represents the initial spectrum, the black dashed line the output spectrum from the pre-amplifier and the grey line the output spectrum after the power amplifier.
Figure 7.
Output spectra for a hybrid amplification chain with pre-amplifier based with Yb:CaF2 (left) and Yb:YAG (right). The solid black line represents the initial spectrum, the black dashed line the output spectrum from the pre-amplifier and the grey line the output spectrum after the power amplifier.
These results confirm that both Yb-doped media studied here are viable alternative pre-amplifier stages for a hybrid amplification chain operating at 1053 nm. However, the number of passes for a millijoule-level Yb:YAG amplifier is typically much larger.