4.1 Analytic solution to coupled equations

In high-pulse-energy laser systems, efficient energy extraction from the laser gain medium requires that the laser fluence should approach the medium’s saturation fluence. Such laser operation depletes the inversion leading to temporally dependent saturation of the gain, which, in a pulsed laser system, causes the output pulse to become temporally distorted. Analytic solutions to the coupled-rate and energy-transport equations for a homogeneously saturating thin slab are used in PSOPS to determine the time-dependent gain within each laser disk at discrete locations across the laser aperture. The four-level equations can be expressed as^{[Reference Martin and Milam37]}

(1)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D6FF}n_{3}}{\unicode[STIX]{x1D6FF}t}=W_{p}n_{0}+c\unicode[STIX]{x1D719}\unicode[STIX]{x1D70E}_{23}n_{2}-\frac{n_{3}}{\unicode[STIX]{x1D70F}_{32}},\end{eqnarray}$$

(2)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D6FF}n_{2}}{\unicode[STIX]{x1D6FF}t}=c\unicode[STIX]{x1D719}\unicode[STIX]{x1D70E}_{12}n_{1}-c\unicode[STIX]{x1D719}(\unicode[STIX]{x1D70E}_{21}+\unicode[STIX]{x1D70E}_{23})n_{2}-\frac{n_{2}}{\unicode[STIX]{x1D70F}_{21}}+\frac{n_{3}}{\unicode[STIX]{x1D70F}_{32}},\qquad\end{eqnarray}$$

(3)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D6FF}n_{1}}{\unicode[STIX]{x1D6FF}t}=c\unicode[STIX]{x1D719}\unicode[STIX]{x1D70E}_{21}n_{2}-c\unicode[STIX]{x1D719}\unicode[STIX]{x1D70E}_{12}n_{1}+\frac{n_{2}}{\unicode[STIX]{x1D70F}_{21}}-\frac{n_{1}}{\unicode[STIX]{x1D70F}_{10}},\end{eqnarray}$$

(4)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D6FF}n_{0}}{\unicode[STIX]{x1D6FF}t}=-W_{p}n_{0}+\frac{n_{1}}{\unicode[STIX]{x1D70F}_{10}},\end{eqnarray}$$

(5)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D6FF}t}+c\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D6FF}z}=c\unicode[STIX]{x1D719}(\unicode[STIX]{x1D70E}_{21}n_{2}-\unicode[STIX]{x1D70E}_{12}n_{1}-\unicode[STIX]{x1D70E}_{23}n_{2}),\end{eqnarray}$$

where $n_{i}(n=0,1,2,3)$ are the population densities of the levels, $c$ is the speed of light in the medium, $\unicode[STIX]{x1D719}$ is the photon number density, $W_{p}$ is the pumping rate, $\unicode[STIX]{x1D70E}_{23}$ is the effective excited-state absorption cross section out of the upper laser level, $\unicode[STIX]{x1D70E}_{12}$ and $\unicode[STIX]{x1D70E}_{21}$ are the effective stimulated absorption and emission cross sections, respectively, and $\unicode[STIX]{x1D70F}_{32}$, $\unicode[STIX]{x1D70F}_{21}$ and $\unicode[STIX]{x1D70F}_{10}$ are the decay times of the pump level, laser transition and terminal laser level, respectively. We assume that the transition from the pump band into the upper laser level occurs very rapidly and that the lifetime of the upper laser level is long compared to the laser pulse’s transit time^{[Reference Campbell and Suratwala38]}. We also assume that temporal variations in the inversion produced by the pump are small during the stimulated emission process that amplifies the laser pulse^{[Reference McMahon, Emmett, Holzrichter and Trenholme39]}. The main discharge current pulse to the amplifier flashlamps has a width of greater than $300~\unicode[STIX]{x03BC}\text{s}$, and its peak is timed to coincide with the arrival of the laser pulse in the amplifier, whereas the maximum laser pulse width on OMEGA EP is currently 10 ns and a round-trip time through the beamline amplifier disks is approximately 700 ns. We assume that there is negligible excited-state absorption or other up-conversion losses ($\unicode[STIX]{x1D70E}_{23}\sim 0$) since the available inversion and the gain profile in the OMEGA EP disks are dominated by amplified spontaneous emission (ASE) depumping^{[Reference Guch and Murray40]}, which is taken into account in the PSOPS model by using the measured SSG. As will be shown, the use of the measured beamline SSG in the model and the ability to optimize the model’s fitting parameters between SSG measurements effectively take into account inversion losses that limit the available gain. The terminal-level decay time $\unicode[STIX]{x1D70F}_{10}$ is only a significant contributor to the amplification process for pulse widths $\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D70F}_{10}$ and does not contribute significantly to most OMEGA EP pulse shapes, but it will be taken into account by using an empirically derived modification to the saturation fluence^{[Reference Bibeau, Trenholme and Payne41]}. We therefore assume for now that the terminal-level lifetime is much shorter than the pulse width of the laser such that $n_{1}\sim 0$. It will be shown that a further modification to the saturation fluence also takes into account glass host matrix contributions to inhomogeneous broadening^{[Reference Pennington, Milam and Eimerl42]}. With these simplifying assumptions, the relevant equations in the frame of the laser pulse are

(6)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}n_{2}(z,t)}{\unicode[STIX]{x1D6FF}t} & = & \displaystyle -\frac{I(z,t)}{\hslash \unicode[STIX]{x1D714}}\unicode[STIX]{x1D70E}_{21}n_{2}(z,t),\end{eqnarray}$$

(7)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}I(z,t)}{\unicode[STIX]{x1D6FF}z} & = & \displaystyle I(z,t)\unicode[STIX]{x1D70E}_{21}n_{2}(z,t),\end{eqnarray}$$

where $I(z,t)=c\unicode[STIX]{x1D719}(z,t)\hslash \unicode[STIX]{x1D714}$ is the laser pulse intensity at photon frequency $\unicode[STIX]{x1D714}$, $\hslash$ is Planck’s constant, and the dependence on propagation distance and time is explicitly shown. These coupled equations in ($z,t$) can be reduced to coupled equations in $t$ only and solved to obtain the time-varying partially saturated gain in the time frame of the laser pulse. This partially saturated gain can be expressed in terms of the input or output pulses of the laser amplifier and the initial gain prior to arrival of the pulse as^{[Reference Siegman43, Reference Siegman44]}

(8)$$\begin{eqnarray}G_{\text{in}}(t)=\frac{1}{1-(1-G_{0}^{-1})\exp [-F_{\text{in}}(t)/F_{\text{sat}}]}\end{eqnarray}$$

or

(9)$$\begin{eqnarray}G_{\text{out}}(t)=1+(G_{0}-1)\exp [-F_{\text{out}}(t)/F_{\text{sat}}],\end{eqnarray}$$

where $G_{0}$ is the initial gain, $F_{\text{sat}}$ is the saturation fluence,

(10)$$\begin{eqnarray}F_{\text{sat}}=\frac{\hslash \unicode[STIX]{x1D714}}{\unicode[STIX]{x1D70E}_{21}},\end{eqnarray}$$

and

(11)$$\begin{eqnarray}\displaystyle F_{\text{in}}(t)\equiv \displaystyle \int _{t_{0}}^{t}I_{\text{in}}(t^{\prime })\,\text{d}t^{\prime }, & & \displaystyle\end{eqnarray}$$

(12)$$\begin{eqnarray}\displaystyle F_{\text{out}}(t)\equiv \displaystyle \int _{t_{0}}^{t}I_{\text{out}}(t^{\prime })\,\text{d}t^{\prime }. & & \displaystyle\end{eqnarray}$$

The integrals in Equations (11) and (12) are taken from the starting time $t_{0}$ of the pulse up to the normalized time $t$ within the pulse. The output pulse can be determined from a given input pulse using Equation (8) as

(13)$$\begin{eqnarray}I_{\text{out}}(t)=G_{\text{in}}(t)I_{\text{in}}(t),\end{eqnarray}$$

whereas Equation (9) can be used to determine the input pulse required to produce a desired output pulse,

(14)$$\begin{eqnarray}I_{\text{in}}(t)=I_{\text{out}}(t)/G_{\text{out}}(t).\end{eqnarray}$$

Note that the inversion dependence in $z$ need not be considered and that for each time $t$ in the pulse, the gain is dependent only on the initial gain $G_{0}$ and the accumulated energy per unit area from time $t_{0}$ compared to the saturation fluence. The gain dependence in $z$ could, in principle, be calculated, which may be important for subsequent passes in cases where the terminal-level lifetime is long^{[Reference Lowdermilk and Murray45–Reference Zuegel and Seka47]}; however, as will be shown, the use of the initial gain $G_{0}$ for each of the multiple passes through a laser disk is a valid approximation for the OMEGA EP beamline. We note that a similar semi-analytic approach to determine input and output pulse shapes has been used in a quasi-three-level, all-fiber amplifier chain^{[Reference Schimpf, Ruchert, Nodop, Limpert, Tünnermann and Salin48]}.

4.2 Multi-pass beamline amplification

In the absence of losses or higher-order effects contributing to pulse-shape distortion, Equations (8)–(14) imply that a precise knowledge of the initial gain $G_{0}$ and saturation fluence $F_{\text{sat}}$ is sufficient to determine the input or output pulse shapes. In the PSOPS model, the time-dependent gain is determined at discrete locations across the laser aperture, where the initial gain $G_{0}(x,y)$ is taken as the measured SSG of each disk, and the saturation fluence is inferred from prior beamline input versus output energy measurements under gain-saturation conditions. For multi-pass amplification, each laser amplifier disk is treated as a thin slab and either Equation (13) or (14) is applied iteratively per disk for forward or backward prediction, respectively^{[Reference Lowdermilk and Murray45]}. Therefore, for forward prediction, the output intensity of disk $k$ is given by

(15)$$\begin{eqnarray}I_{k}(t,x,y)=\unicode[STIX]{x1D6FD}^{2}\cdot G_{k}(t,x,y)I_{k-1}(t,x,y),\end{eqnarray}$$

where

(16)$$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-18.0pt}G_{k}(t,x,y)\nonumber\\ \displaystyle & & \displaystyle \hspace{-12.0pt}\quad =\frac{1}{1-\{1-[G_{0}(x,y)]^{-1}\}\exp [-F_{k}(t,x,y)/F_{\text{sat}}]},\end{eqnarray}$$

(17)$$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-18.0pt}F_{k}(t,x,y)=\displaystyle \int _{t_{0}}^{t}\unicode[STIX]{x1D6FD}\cdot I_{k-1}(t^{\prime },x,y)\,\text{d}t^{\prime },\end{eqnarray}$$

and a per-disk-surface loss factor $\unicode[STIX]{x1D6FD}$ is included in the model to account for passive losses. The SSG and saturation fluence at any location ($x$, $y$) within the beam are both taken as constant for the propagation distance through a single disk. Following the repeated application of Equations (15)–(17) through the entire beamline, frequency conversion to the third harmonic uses look-up tables from MIXER calculations^{[Reference Craxton32, Reference Craxton33]}. For backward prediction starting with the UV-beam profile, pulse shape and energy, these tables provide the amplified IR intensity at the end of the beamline from which the beamline input intensity is recursively calculated using the following equations:

(18)$$\begin{eqnarray}\displaystyle I_{k}(t,x,y) & = & \displaystyle I_{k+1}(t,x,y)/\unicode[STIX]{x1D6FD}^{2}\cdot G_{k}(t,x,y),\end{eqnarray}$$

(19)$$\begin{eqnarray}\displaystyle G_{k}(t,x,y) & = & \displaystyle 1+[G_{0}(x,y)-1]\exp [-F_{k}(t,x,y)/F_{\text{sat}}],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(20)$$\begin{eqnarray}\displaystyle F_{k}(t,x,y) & = & \displaystyle \displaystyle \int _{t_{0}}^{t}[I_{k+1}(t^{\prime },x,y)/\unicode[STIX]{x1D6FD}]\text{d}t^{\prime },\end{eqnarray}$$

where $I_{k}$ ($t,x,y$) is the input intensity of the $k$th disk.

4.3 Saturation fluence considerations

The saturation fluence for Nd:glass laser media given by Equation (10) must be modified to account for an inhomogeneous broadening effect in the laser glass^{[Reference Pennington, Milam and Eimerl42]} and for bottlenecking of the terminal level of the lasing transition for pulse widths close to the terminal-level lifetime $\unicode[STIX]{x1D70F}_{10}$, where $\unicode[STIX]{x1D70F}_{10}\sim 0.25$ ns for Nd-doped phosphate laser glasses^{[Reference Bibeau, Trenholme and Payne41, Reference Bibeau and Payne49]}. The effective saturation fluence of the OMEGA EP beamline is dependent on the beamline’s output fluence and has been inferred from prior fits to gain-saturation data, as shown in Figure 3. The optimization fits from which the plot in Figure 3 was derived were done to the measured UV power using PSOPS forward predictions while adjusting the loss term $\unicode[STIX]{x1D6FD}$ and effective cross-section term $\unicode[STIX]{x1D70E}_{21}$ in the model, and the fit values have been scaled to account for Brewster angle incidence on the disks in order to compare with literature values. To reduce the effect of pulse width on the determination of saturation fluence, shot data included only pulse widths of 2 ns or greater; to reduce the contribution of measurement error and pulse-shape temporal modulation, only beam output energies of greater than ${\sim}2~\text{kJ}$ were used, where the pulse-shape distortion from gain saturation was large^{[Reference Yarema and Milam50]}. The measured effect is consistent with that reported for LHG-8^{[Reference Martin and Milam51]} and for LG-750^{[Reference Pennington, Milam and Eimerl42]}, where the composition is similar to that of the LHG-8 laser glass used in OMEGA EP^[52]. The inferred saturation fluence for any given output fluence in Figure 3 must be interpreted as an average over all amplifier disks, where significant saturation occurs through only the last pass of the cavity and booster amplifiers. Nonetheless, in Section 5, we show that the use of this saturation fluence as a fitting parameter for the model has provided robust agreement between model simulations and measurements.

Figure 3. Plot of the inferred saturation fluence versus beamline output fluence from optimization fits to OMEGA EP beamline 3 data for nine-main-amplifier and seven-booster-amplifier configuration. Only shots with pulse widths and beam output energies ${\geqslant}2~\text{ns}$ and ${\geqslant}2~\text{kJ}$, respectively, were used in the fit.

Assuming negligible pumping during the laser pulse width $\unicode[STIX]{x1D70F}$, an additional modification to the saturation fluence is used to account for bottlenecking of the terminal laser level^{[Reference Bibeau, Trenholme and Payne41]}:

(21)$$\begin{eqnarray}F_{\text{sat}}^{\prime }=\frac{\hslash \unicode[STIX]{x1D714}/\unicode[STIX]{x1D70E}_{\text{em}}}{\unicode[STIX]{x1D6FE}(R)},\end{eqnarray}$$

where

(22)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}(R)=1+K\cdot B(R)\end{eqnarray}$$

and $\unicode[STIX]{x1D70E}_{\text{em}}$ is the effective emission cross section determined from the optimization fits; $\unicode[STIX]{x1D6FE}(R)$ is an empirical scaling factor whose value is dependent on the ratio $R=\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{10}$; $K=0.35$ is an effective degeneracy ratio for the upper and lower Nd manifolds in Nd-doped phosphate glass; the empirical function $B(R)$ was determined in Ref. [Reference Bibeau, Trenholme and Payne41]. The dependence of $[\unicode[STIX]{x1D6FE}(R)]^{-1}$ on laser pulse width $\unicode[STIX]{x1D70F}$ is plotted in Figure 4. This correction to the saturation fluence generally has a small effect on the predicted beam output energy for most OMEGA EP pulses. For a 1-ns square pulse, the modified saturation fluence can provide an improvement in predicted beam output energy of the order of 1% but this is generally less than the error between simulated and measured energies on the first shot of the day (see Section 6.1). Greater improvement would be expected for shorter pulses, but an accurate assessment is currently limited by the impulse response of the regen pulse measurement (see Section 5).

Figure 4. Pulse-width dependence of empirical scaling factor $[\unicode[STIX]{x1D6FE}(R)]^{-1}=[1+K\cdot B(R)]^{-1}$ to account for finite lifetime of the terminal laser level in Nd-doped phosphate laser glass, as per Ref. [Reference Bibeau, Trenholme and Payne41].

For subsequent passes of the laser pulse through an amplifier disk, some amount of gain recovery might be expected, owing to drain of the terminal level. In this case, the terminal-level population would be reduced to $\exp (-T/\unicode[STIX]{x1D70F}_{10})$ of its value immediately after passage of the pulse, where T is the round-trip time between passes^{[Reference Lowdermilk and Murray45, Reference Trenholme and Goodwin53]}. In the OMEGA EP beamline, the shortest round-trip time, and therefore the least amount of gain recovery, would be expected for the main cavity disk closest to the beamline deformable mirror (DM, Figure 1), where the round-trip time between the disk and the DM is 22 ns. In this case, $\exp (-22~\text{ns}/0.25~\text{ns})\sim 10^{-38}$, thereby justifying the assumption of full gain recovery between passes and the use of the same initial gain distribution for each pass of a beamline disk. The beamline SSG measured with a 10-ns square pulse is used in the PSOPS model, as described in Section 4.4. Using the measured SSG effectively takes into account inversion losses such as from ASE depumping, flashlamp or reflector degradation and gain recovery for subsequent passes through the amplifier resulting from terminal-level drain.

4.4 Small-signal gain measurements

The spatially dependent, single-pass SSG for each amplifier disk used in the model is derived from full-system SSG measurements for the seven-disk booster-amplifier configuration and for each main-amplifier configuration to be used during shot operations. The SSG is measured by dividing calorimetrically calibrated beam-fluence measurements at the injection plane and beamline output. The resulting beam ratio for an amplified shot in the small-signal regime is divided by the same ratio with the amplifier disks unpumped, thereby eliminating the contribution from passive loss. The single-disk SSG $G_{0}(x,y)$ is taken as the geometric mean for each measurement,

(23)$$\begin{eqnarray}G_{0}(x,y)=\left\{\frac{[F_{\text{out}}(x,y)/F_{\text{in}}(x,y)]_{\text{pumped}}}{[F_{\text{out}}(x,y)/F_{\text{in}}(x,y)]_{\text{unpumped}}}\right\}^{1/N},\end{eqnarray}$$

where $F_{\text{in}}(x,y)$ and $F_{\text{out}}(x,y)$ are the temporally integrated, measured beamline input and output fluence profiles, respectively, and $N=d\cdot p$, where $d$ is the number of disks configured and $p$ is the number of passes through the disk. A separate SSG measurement is made for the main cavity- and booster-amplifier sections of the beamline. The single-disk SSG is taken as the geometric mean for each measurement. For example, Figure 5 shows a 15th-order Legendre fit to the measured nine-disk, four-pass cavity total SSG (Figure 5(a)) and the inferred single-disk SSG (Figure 5(b)). Figure 6 shows column-averaged, horizontal lineouts through the single-disk SSG distributions for different numbers of main-amplifier disks fired. The cavity disk at the end of the amplifier chain is pumped only in the odd-disk configuration and the asymmetric distribution of flashlamp light across the end disk results in the asymmetric gain profile seen in Figure 6(a)^{[Reference Alger, Erlandson, Fulkerson, Horvath and Jancaitis29]}.

Figure 5. (a) Fifteenth-order Legendre fit to the measured total small-signal gain for beamline 1, nine-main-amplifier configuration and (b) its $N\text{th}$ root (geometric mean) for the nine-disk, four-pass cavity where $N=36$. The single-disk small-signal gain shown in (b) is used in the PSOPS model when nine main amplifiers are configured for beamline 1.

Figure 6. Horizontal lineouts of column-averaged, beamline 1 single-disk, small-signal gain maps for different numbers of main-amplifier disks fired. (a) Odd number of cavity amplifiers; (b) even number of cavity amplifiers.

4.5 Model calibration and simulation methods

The optimization from which Figure 3 was derived was performed to the measured UV power while adjusting the loss and effective cross-section terms as fitting parameters. The frequency-conversion model has been previously calibrated to achieve excellent agreement between simulated and measured third-harmonic conversion efficiency. For the IR beamline section of the model, two different forms of the model have been used and have achieved similar accuracy.

The first method is an extension of the optimization done in Figure 3. When the model begins to show a loss of accuracy, a single optimization fit is performed if a recent high-energy shot is available. A check is done to ensure that the $\unicode[STIX]{x1D70E}_{\text{em}}$ fit value is within the previously determined range of inferred values. This calibration can be done within a few minutes between OMEGA EP shots. Once the model is calibrated, subsequent forward predictions use a quasi-Newton method to determine the correct saturation fluence where a low-resolution spatial grid is propagated in the forward direction using an initial value of saturation fluence. The average fluence of the output beam is then used to adjust the saturation fluence per the slope in Figure 3 iteratively until a convergent value of saturation fluence is determined. This final value of the saturation fluence is then scaled for pulse-width dependence and used for the final forward simulation, which may be of higher spatial resolution. The quasi-Newton method used for forward predictions is illustrated by the flowchart in Figure 7. For backward simulation starting with a UV beam and pulse, the backward-simulated IR beamline output fluence is used directly to calculate the required saturation fluence per the most recent $\hslash \unicode[STIX]{x1D714}/\unicode[STIX]{x1D70E}_{\text{em}}$ fit value and slope from Figure 3.

Figure 7. Flowchart illustrating the quasi-Newton method used for PSOPS forward predictions with a low-resolution grid. On the first iteration, the spatially-averaged output beam fluence $F_{\text{out}}$ is compared to $F_{\text{out}}$ from the calibration shot. The saturation fluence $F_{\text{sat}}$ is then scaled per the previously measured $F_{\text{sat}}$ vs. $F_{\text{out}}$ slope. Pulse-width correction to $F_{\text{sat}}$ is applied before the final (high resolution) simulation.

In the second method, the local saturation fluence for each grid point and amplifier disk is determined per the equation in Figure 3 for both forward and backward simulations. For the forward simulation, Equation (16) takes the form,

(24)$$\begin{eqnarray}\displaystyle & & \displaystyle G_{k}(t,x,y)\nonumber\\ \displaystyle & & \displaystyle =\frac{1}{1-\{1-[\unicode[STIX]{x1D6FC}\cdot G_{0}(x,y)]^{-1}\}\exp [-F_{k}(t,x,y)/F_{\text{sat},k}(x,y)]},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where

(25)$$\begin{eqnarray}F_{\text{sat},k}(x,y)=\frac{m\displaystyle \int _{-\infty }^{\infty }\unicode[STIX]{x1D6FD}\cdot I_{k-1}(t,x,y)\,\text{d}t+b}{\unicode[STIX]{x1D6FE}(R)},\end{eqnarray}$$

and $m=0.082$ and $b=3.741$ are the fit values shown in the plot of Figure 3; $\unicode[STIX]{x1D6FC}$ is a multiplier on the SSG; the correction for the finite lifetime of the terminal laser level is explicitly shown. Optimization to the measured UV power is performed in the forward direction using the per-disk-surface loss term $\unicode[STIX]{x1D6FD}$ and the factor $\unicode[STIX]{x1D6FC}$ as fit variables. For the backward simulation, Equation (19) becomes

(26)$$\begin{eqnarray}\displaystyle G_{k}(t,x,y) & = & \displaystyle 1+[\unicode[STIX]{x1D6FC}\cdot G_{0}(x,y)-1]\nonumber\\ \displaystyle & & \displaystyle \times \,\exp [-F_{k}(t,x,y)/F_{\text{sat},\text{k}}(x,y)],\end{eqnarray}$$

where

(27)$$\begin{eqnarray}F_{\text{sat},k}(x,y)=\frac{m\displaystyle \int _{-\infty }^{\infty }[I_{k+1}(t,x,y)/\unicode[STIX]{x1D6FD}]\text{d}t+b}{\unicode[STIX]{x1D6FE}(R)}.\end{eqnarray}$$

In practice, we have found similar accuracy with both methods, although the first method (quasi-Newton) has generally been more robust for a broad range of energies and pulse shapes. The results shown in the following sections have therefore used this method. The simulation and calibration sequence can thus be summarized as follows.

1. (1) Backward simulation using requested UV pulse shape, energy, and beam profile to calculate required regen pulse shape, stage energies, system throttles, and laser diagnostic filtrations.

2. (2) Forward simulation using measured inputs to confirm results of backward simulation and refine, if necessary.

3. (3) Take full system UV shot, and check model calibration by comparing measured and post-shot simulated UV energies and pulse shapes. Calibrate model, if necessary.

6.1 Improvements to UV energy and pulse-shape accuracy

Drifts in system performance can lead to noticeable deviations between simulated and achieved pulse shapes and energies, which can be minimized with an agile system model such as PSOPS. For example, Figure 11 shows how optimization of the injected pulse shape can be done for small changes in system performance. Although the nominal AWG pulse shape is determined prior to shot day, small changes in regen performance on shot day can significantly affect the UV pulse shape. For example, the left plot in Figure 11(a) shows a prediction of the expected UV pulse based on the pre-shot PSM measurement of the regen pulse that departs from the ideal pulse shape near the end of the pulse. Based on this prediction, the AWG waveform was modified to provide the compensated pre-shot prediction shown on the right in Figure 11(a). The post-shot UV simulation showed excellent agreement with the measurement (Figure 11(b)).

Figure 11. (a) Predicted and requested UV pulse shapes showing how day-to-day changes in regen performance are compensated using PSOPS predictions; (b) post-shot UV pulse simulation, measurement and corresponding energies (beamline 3 shot 22,254). On-target UV energy: 775 J requested, 751 J measured, 752 J simulated.

Although the AWG adjustment is not currently a closed loop, the regen performance is typically sufficiently stable so that only minor adjustments are required. Closed-loop AWG waveform optimization will be implemented in the near future. In addition to changes in regen performance, small changes in beamline gain and losses may result in approximately 5% discrepancy between the measured and simulated UV energy on the first shot of the day, in which case the model can be calibrated before the second shot based on the measured pulse power, as described in Section 4. The optimized simulated UV energy is typically within ${\sim}1\%$ of the measurement on the second shot of the day.

6.2 Improvements to experimental flexibility

PSOPS has also enhanced laser facility flexibility by enabling users to adjust requested UV pulse shapes and energies between laser shots within a predefined range that is determined uniquely for each experimental campaign. The allowed range of energy and pulse-shape modification is assessed with respect to the laser system’s fluence limits, the range of energy and pulse shapes planned for the day and the likelihood of maintaining each beamline’s 90-minute shot cycle. In the example shown in Figure 12(a), a significant increase in the slope of the UV pulse was desired following a laser shot while maintaining 220 J of UV on-target energy. In Figure 12(b), different energies were desired while maintaining the original normalized design pulse shape that produced 500 J of UV on-target energy. These requests were based on each user’s real-time analysis of experimental data and were accommodated in each case by adjusting the front-end pulse shape and throttles per the PSOPS pre-shot prediction.

Figure 12. Examples showing facility flexibility enabled by PSOPS. (a) Based on an analysis of data from the previous shot, a significant increase in the slope of the UV pulse was desired while maintaining 220-J UV on-target energy. This request was accommodated by front-end pulse-shape and throttle adjustments prior to the next shot per the PSOPS pre-shot prediction. On-target UV energy: 220 J requested, 222 J measured (beamline 4 shot 20,647). (b) Different energies were requested while maintaining the original normalized design pulse shape that produced 500-J UV on-target energy. The measured UV on-target energies are shown in the label (beamline 3).

Most of the discrepancies between the requested and measured pulse shapes in Figure 12 can be understood by noting that the corresponding IR beam intensities entering the frequency-conversion crystals were close to the small-signal regime for frequency tripling, where the third-harmonic intensity is proportional to the third power of the IR intensity^{[Reference Craxton33]}. Thus, small uncorrected deviations from the ideal regen pulse shapes produced significantly larger deviations in the frequency-converted pulse shapes. We expect that the automated adjustment of the AWG waveform will allow improved pulse-shape optimization to be performed within OMEGA EP’s 90-minute shot-cycle time.

6.3 Increased effective pulse-duration range

Currently, OMEGA EP’s regens can accommodate single beamline pulse widths of up to 10 ns. However, improved system modeling in conjunction with precision timing allows the technique of pulse stitching to achieve up to a $4\times$ increase in effective pulse duration. With pulse stitching, as illustrated in Figure 13, pulse shapes from different beamlines can be precisely combined on target to form a single composite pulse shape. The composite 27-ns ramped pulse shape shown in Figure 13 was formed by incoherent addition of the individual pulses, separated by the temporal delay between them. Prior to the shot, PSOPS is used to predict the composite pulse, given the specified beam-to-beam temporal delay (Figure 13(a)). The measured composite pulse shown in Figure 13(b) was formed using the individual beamline pulse-shape measurements and the measured beam-to-beam UV pulse timing.

Figure 13. (a) Pre-shot prediction and (b) measurement of approximately 27-ns composite UV pulse formed by incoherent addition of the individual beamline pulse shapes and beam-to-beam timing (shot 31,182).

We note that all four long-pulse beams are derived from the same single-frequency oscillator, and the focusing optic assemblies on the OMEGA EP target chamber are mounted such that the beams form a cone of approximately $23^{\circ }$ half-angle with the vertex of the cone at target chamber center. Therefore, the incoherent sum used to form the composite pulse does not take into account high-frequency modulation that may occur on target in the pulse overlap regions.

6.4 Improved system alignment

PSOPS has been used as a tool to optimize the alignment of beam-shaping apodizers in the Sources front end (see Figure 1). PSOPS predictions of the effect of small changes in beam centering and rotation on the amplified near-field beam uniformity can be used as a guide to optimize the apodizer alignment without requiring amplified shots. This has resulted in a better understanding of required tolerances for centering and rotating both the beam-shaping apodizer and the apodized injected beam with respect to the gain profile of the beamline. As an example, Figure 14 shows the measured effect that identifying and correcting a small error in apodizer alignment has on the amplified IR near-field beam. Using the measured injected near-field beam, PSOPS forward simulations were used to predict the amplified beamline output near-field profile and to correct the apodizer’s alignment with respect to the gain profile of the beamline within a 10-minute shot cycle. A 0.49-mm shift of the apodizer improved the output beam quality, both in terms of the fluence contrast (from 13.1% to 9.4%) and the peak-to-mean fluence ratio (from 1.46:1 to 1.43:1). Fluence contrast is defined as the standard deviation of the fluence divided by the mean fluence value.

Figure 14. Measured beamline 3 output IR near-field beam-fluence profile (a) before moving the beam-shaping apodizer (contrast $=13.1\,\%$, peak to mean $=$ 1.46:1) and (b) after moving the apodizer by 0.49 mm (contrast = 9.4%, peak to mean $=$ 1.43:1). Contrast is defined in the text. The apodizer adjustment was guided by PSOPS simulations.

PSOPS has also been used to perform iterative, multi-axis optimization of the apodizer alignment to reduce the peak fluence of the frequency-converted UV beam. Figure 15 shows that a rotation of the apodizer by $3.5^{\circ }$, followed by lateral shifts of 0.36 mm (horizontal) and 0.24 mm (vertical) reduced the peak-to-mean UV-beam fluence from 3.74:1 (Figure 15(a)) to 2.76:1 (Figure 15(d)) using the same beamline saturation conditions. The improved near-field profile shown in Figure 15(d) was achieved using iterative PSOPS predictions followed by a single amplified UV shot. By limiting near-field beam fluence, fluence-limited damage may be avoided, leading to enhanced energy performance. The quality of the regen beam that was incident on the apodizer contributed to the residual nonuniformity seen in Figure 15(d), suggesting that further reduction in UV-beam peak fluence may be possible by improving the regen-output beam quality, for which the model can also be used as a guide owing to its accurate and rapid prediction capability. For example, PSOPS has been used during front-end qualification as a post-shot analysis tool to assess whether residual nonuniformity of the front-end near-field beam is acceptable for the given shot campaign, or whether additional laser facility time should be used to improve the beam uniformity. These measurements, guided by PSOPS simulations, have provided a better understanding of laser-system contributions to the amplified UV-beam profile and, importantly, have demonstrated that small adjustments to apodizer alignment are often sufficient to correct near-field beam nonuniformity in lieu of designing and manufacturing new apodizers.

Figure 15. Beamline 3 UV near-field beam-fluence profiles showing PSOPS-guided, multi-axis optimization of beam-shaping apodizer alignment to reduce the peak fluence in the amplified UV beam. A rotation of the apodizer by $3.5^{\circ }$, followed by lateral shifts of 0.36 mm (horizontal) and 0.24 mm (vertical) reduced the peak-to-mean UV-beam fluence. (a) Measured UV near-field beam before moving apodizer with peak-to-mean fluence of 3.74:1. (b) PSOPS simulation of (a). (c) PSOPS prediction after moving apodizer. (d) Measured UV beam after moving apodizer with peak-to-mean fluence of 2.76:1. The improved near-field profile shown in (d) was achieved using iterative PSOPS predictions followed by a single amplified UV shot.