Modeling and optimization of single-pass laser amplifiers for high-repetition-rate laser pulses

We propose a model for a continuously pumped single-pass amplifier for continuous and pulsed laser beams. The model takes into account Gaussian shape and focusing geometry of pump and seed beam. As the full-wave simulation is complex we have developed a largely simplified numerical method that can be applied to rotationally symmetric geometries. With the tapered-shell model we treat (focused) propagation and amplification of an initially Gaussian beam in a gain crystal. The implementation can be done with a few lines of code that are given in this paper. With this code, a numerical parameter optimization is straightforward and example results are shown. We compare the results of our simple model with those of a full-wave simulation and show that they agree well. A comparison of model and experimental data also shows good agreement. We investigate in detail different regimes of amplification, namely the unsaturated, the fully saturated, and the intermediate regime. Because the amplification process is affected by spatially varying saturation and exhibits a nonlinear response against pump and seed power, no analytical expression for the expected output is available. For modeling of the amplification we employ a four-level system and show that if the fluorescence lifetime of the gain medium is larger than the inverse repetition rate of the seed beam, continuous-wave amplification can be employed to describe the amplification process of ultrashort pulse trains. We limit ourselves to this regime, which implies that if titanium:sapphire is chosen as gain medium the laser repetition rate has to be larger than a few megahertz. We show detailed simulation results for titanium:sapphire for a large parameter set.

Figure 1

Excitation geometry for the cylindrical model.

Figure 2

Four-level laser system.

Figure 3

Pulsed model cycle; details in the text.

Figure 4

Power transfer $P_{\mathrm{out}}-P_{\mathrm{in}}$ as function of the repetition rate for different seed power levels ($P_{\mathrm{in}}$), calculated with the pulsed amplifier model (solid lines) and the cw model (dashed lines). The inverse excited state lifetime of Ti:sapphire is $(3.2\hspace{0.5em}\mu$s$){}^{-1}=313\hspace{0.5em}$ kHz. For $f_{\mathrm{rep}}>1\hspace{0.5em}$MHz $~3{\tau }^{-1}$ the amplification becomes independent of $f_{\mathrm{rep}}.$

Figure 5

Cut-open side view of the geometry employed for the tapered-shell model. The beam propagates in the $z$ direction and initially has a Gaussian radial intensity distribution. The intensity in each shell is calculated according to the cw amplification model, whereas the propagation of the shape of individual shells follows the scaling based on Eqs. (25) and (26) [i.e., radius and thickness of the shells follow the radial size of a focused laser beam in the paraxial approximation $\propto w(z)$]. Due to the individual calculation of amplification in each shell, the beam profile can evolve away from Gaussian shape during propagation and amplification. Note that energy cannot be exchanged between shells in this model and that diffraction over multiple shells is ignored. This will be discussed in the text.

Figure 6

Experimental setup to measure the dependence of the amplification gain on the focal radius.

Figure 7

Gain as a function of focal radii. Pump and seed power are $0.5\hspace{0.5em}$ and $0.38\hspace{0.5em}$W. Pump and seed focal spot size are matched for each point. Points, experimental data; solid line, prediction of the tapered-shell model, with the only free parameter $\eta$ adjusted to be 0.34. The remaining parameters $\alpha =3.6\hspace{0.5em}{\mathrm{cm}}^{-1}$ and $L=2.4\hspace{0.5em}$mm were measured independently.

Figure 8

(Left column) Intensity inside the crystal as a function of radial and forward coordinate for a cut through the optical axis. The right column shows the instantaneous power (integrated radially) inside the crystal as a function of the propagation length. Focal positions of pump and seed beam are set to $z_{\mathrm{p}}=z_{\mathrm{s}}=1.5\hspace{0.5em}$mm. Three different seed power levels are simulated: 10 mW (a), 500 mW (b), and 10 W (c). Pump power is 50 W, $\eta =0.34$, and $\alpha =6.6\hspace{0.5em}$cm${}^{-1}$. The initial Gaussian beam parameters are such that without gain the seed and pump focal radii are $10\hspace{0.5em}\mu$m for all plots. The black solid line is the result of the tapered-shell model; the red dotted line is the result of a full-wave simulation (see Sec. 3d). (a) The on-axis intensity stays below saturation intensity until about propagation through half of the crystal. In (b), initially the on-axis intensity is below saturation intensity but surpasses it right after entering the crystal. (c) The intensity surpasses the saturation intensity for a large portion of the beam cross section right from the front edge of the crystal. The yellow and white lines show the $1/e^2$ intensity radius for given $z$ with and without gain, respectively. See text for further discussion.

Figure 9

Optimum $1/e^2$ focal sizes $w_{0\mathrm{s}}$ and $w_{0\mathrm{p}}$ of the pump (left) and seed beam (right) in $\mu$m. These contour plots are obtained by maximizing the output power varying both focus sizes along with their positions $z_{\mathrm{s}}$ and $z_{\mathrm{p}}$ for each combination of seed $P(z=0)$ and pump power $P_{\mathrm{p}}$. The crystal with parameters given in the main text is assumed to be much longer than the pump absorption length ${\alpha }^{-1}$.

Figure 10

Optimum focus position $z_{\mathrm{s}}$ and $z_{\mathrm{p}}$ measured from the crystals entrance facet at $z=0$ of the pump (left) and seed beam (right). These contour plots are obtained by maximizing the output power varying both focus sizes $w_{0\mathrm{s}}$ and $w_{0\mathrm{p}}$ along with their positions for each combination of seed $P(z=0)$ and pump power $P_{\mathrm{p}}$. The crystal with parameters given in the main text, is assumed to be much longer than the pump absorption length ${\alpha }^{-1}$.

Figure 11

Crystal length that yields an extracted power of $90\%$ of what is attainable with an infinitely long crystal. This contour plot is obtained by maximizing the output power by varying both focus sizes $w_{0s}$ and $w_{0p}$ along with their positions for each combination of seed $P(z=0)$ and pump power $P_{\mathrm{p}}$. The plot might be used to estimate the longest useful crystal.

Figure 12

Optimized output power (left) and gain (right). These contour plots are obtained by maximizing the output power varying both focus sizes $w_{0\mathrm{s}}$ and $w_{0\mathrm{p}}$ along with their positions $z_{\mathrm{s}}$ and $z_{\mathrm{p}}$ for each combination of seed $P(z=0)$ and pump power $P_{\mathrm{p}}$. The crystal with parameters given in the main text is assumed to be much longer than the pump absorption length ${\alpha }^{-1}$. Note the logarithmic spacing of the contour lines in the right plot.

Figure 13

Output power as a function of focal radius of the seed beam with pump focus size $w_{0\mathrm{p}}$ and focus positions $z_{\mathrm{p}}$ and $z_{\mathrm{s}}$ optimized for each point. Different pump power levels were employed for the calculation: $P_{\mathrm{p}}$ is 50 W (a), 20 W (b), 10 W (c), and 5 W (d). Note the kinks in (a) and (b). They originate from the crossing of two smooth curves for two local optima. At the intersection the global optimum jumps from one local optimum of the seed position before the crystal ($z_{\mathrm{s}}<0$) to another inside the crystal ($z_{\mathrm{s}}>0$) while the optimum pump focus position remains inside the crystal.