1. Introduction

The energy and temporal shape of a laser pulse determines its basic parameters such as pulse width, peak power, and contrast ratio [1]. These parameters determine how it interacts with matter [2–4]. The control of these parameters can be achieved by using instant feedback and pulse modulation techniques in a master oscillator power amplifier (MOPA) structure [5–7]. To design or control such a system, an accurate mathematical model of the amplifier is essential. The modeling accuracy is especially important in high-power laser systems because an inaccurate model could miss or underestimate the high peak power pulses that could cause destructive optical damages on the system.

The depletion of stored energy is the fundamental cause of the change in pulse shape in energy-storage-type laser media. The first pulse amplification model that includes the gain depletion effect was developed by L. M. Frantz and J. S. Nodvik [8]. They derived the general photon density equations that describe arbitrary input pulses and arbitrary pumping conditions. Although this closed-form explicit equation is versatile, there is a restriction. The equation is only applicable with an amplifier that has no temporal pulse overlap in the medium. However, in many amplifier geometries, the temporal pulse overlap occurs when there are multiple passes in the amplifier.

Regarding the overlapping situation, modified Frantz–Nodvik equations have been reported for some specific geometries such as zig-zag slabs [9] and thin disks [10]. Conversely, for double-pass amplifiers that have multiple gain media, two methods utilizing the numerical approximation have been suggested [11,12]. One method used an iterative algorithm to solve the photon transfer equation [11]. The iteration starts with the assumption that no pulse overlaps because the initial boundary conditions cannot be found under the overlapping condition. Each result of the iteration is used as the initial value of the next iteration until the error becomes sufficiently small. The other method treats the gain medium as a combination of very thin media [12]. If the thickness of the thin segments is sufficiently small, the depletion effect can be ignored in each segment. With this assumption, the small signal gain equation can be used for the calculations.

In this paper, we considered pulse amplifications in a double-pass amplifier, which has multiple gain media. Under the pulse overlapping condition, we introduced a numerical calculation that directly uses the modified Frantz–Nodvik energy gain equation. The calculated energy and pulse shape agreed well with experimental and previously reported results. Furthermore, the error was maintained at a low value even in rough calculations, which makes it the fastest calculation method ever reported. In the following sections, we first explain the calculation method for the non-overlapping condition. Then, the calculation is extended for the pulse overlapping case. The computational cost and accuracy of the method are also assessed. Finally, comparisons between the experimental data and calculations are presented.

2. Numerical implementation

2.1. Non-overlapping case

The existing general photon density equation with an arbitrary input pulse is relatively complicated and contains infinite integrals [8]. In addition, the number density of photons and excited atoms are not practical for the experiment. In contrast, the uniform pumping and square input pulse assumptions allows the complex Frantz–Nodvik equation to be straightforward and useful, as shown in Eq. (1) [8,13].

where G_E is overall energy gain, g_o is the small signal gain, L is the length of the gain medium, J_in is the fluence of the input pulse, and J_sat is the saturation fluence. Although Eq. (1) is relatively simple and physically intuitive, it merely shows the averaged energy boost of a square pulse. Therefore, to use Eq. (1) in the calculation dealing with the temporal shape, we adopted two schemes.

Initially, we sliced an arbitrary input pulse, J_in (t), into N small segments of δt in the time-domain. Thus, $J_{in}^{(n)}\equiv J_{in}((n-1/2)\times \delta t)$, where n is an integer, and n ≥ 1. Then, the pulse length τ_pulse is defined as N × δt. In this scheme, as shown in Fig. 1(a), each segment is treated as a square pulse.

 

Fig. 1 Examples of a segmented (a) input and (b) output pulses of an amplifier. The outputs are calculated using (red bar) Eqs. (2) and (3) and (blue line) the general photon density equation n(x, t). (Example condition: Φ1.22 cm × L8.5 cm Nd:YAG single-pass amplifier, stored energy of 2 J, input pulse energy of 0.5 J×, super-Gaussian pulse shape, and pulse length of 5.5 ns)

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Secondly, we paired the stored energy equation with the segmented Frantz–Nodvik equation:

where J_sto is the equivalent fluence of the upper-level stored energy, J_ASE and J_pump are the fluence of the amplified spontaneous emission (ASE) and the pumping, γ = 1+g₁/g₂, and g₁ and g₂ are the degeneracy of the lower and the upper energy levels, respectively [13]. Equation (3) shows the paired equation and is simply the law of the conservation of energy in a gain medium. The shape distortion of the individual square segment is ignored. Nevertheless, as shown in Fig. 1(b), the envelope of $J_{out}^{(n)}$ clearly shows the evolution of the pulse shape. Furthermore, the envelope matches well with the calculated result of the original photon density equation: the line graph in Fig. 1(b).

It is also worth noting that the ASE and pumping during the amplification are added to Eq. (3). These two factors were ignored in the original Frantz–Nodvik equation. In addition, the dimension of the variables can be extended. For example, the fluences could be a function of space, time, and wavelength. With this extension, the spectral and spatial profile of the beam and pumping profile of the gain medium can be calculated. Such methods have worked well in previous research [14, 15].

2.2. Overlapping case: double-pass

Here, we consider the amplifier geometry shown in Fig. 2. When the round-trip time τ is shorter than the pulse length, the front and rear parts of the pulse will overlap in the medium. For the calculations, we assumed that the polarizations of the forward and backward beams were orthogonal. This is reasonable because, in most double-pass amplifiers, polarization rotators are used between the media and/or between the medium and the return mirror. These rotators are used for the output distinction and thermal effect compensation [16]. Another assumption we made is that the gain medium is modeled with infinitesimal thickness. In this case, the round-trip time is measured from the center of the medium. The validity of the second assumption is evaluated in section 3.

 

Fig. 2 A schematic of the double-pass amplifier

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For the computation, the round-trip time is discretized as τ_n ≡ Round[τ/δt]. When τ_n < N, the pulse overlap occurs. In addition, the time delay between J_in2 and J_out1 can be expressed by the following rules: $J_{in2}^{(n+{\tau }_n)}={\alpha }^{(n)}{J}_{out1}^{(n)}$, α⁽ⁿ⁾ ≈ 1 for the mirror, $J_{in1}^{(n)}=0$ for n > N, and $J_{in2}^{(n)}=0$ for n ≤ τ_n. All four pulses depicted in Fig. 2 can be calculated with the following modified Frantz–Nodvik equation:

In Eqs. (4)–(7), J_in2 is not necessarily connected with J_out1. If there is a second gain medium, J_out1 and J_in2 are the input and output of the other gain medium, respectively. In this case, α⁽ⁿ⁾ will be the energy gain of the other medium. In this manner, Eqs. (4)–(7) can be applied to the amplifiers that have more than one active medium.

Note that the propagation directions of the input beams are not distinguishable in Eq. (4). This makes sense for two reasons. Firstly, as mentioned earlier, we modeled the gain medium as infinitesimally thin. Secondly, the stimulated emission process is described only with the number of photons and excited ions that interact within the medium, as can be seen in Eq. (8) [13].

where N₂ is the number density of active ions in the excited states, B₂₁ is Einstein’s proportional constant of stimulated emission, ϱ is the radiation energy density function, and ν₂₁ is the wave frequency. In the next section, we apply Eqs. (2)–(7) to a double-pass dual-rod Nd:YAG amplifier. The calculation scheme of the algorithm, the error assessment, and the comparison with measurements are shown.

3. Result and discussion

3.1. Algorithm assessment

For the algorithm assessment, the dual-rod double-pass Nd:YAG amplifier shown in Fig. 3(a) is considered. The two Nd:YAG rods are identical and have a diameter of 12.2 mm, a length of 85 mm, and a doping concentration of 0.8%. The stored energy of the rod is controlled by a flashing voltage. A quartz rotator and quarter-wave-plate are used for the thermal effect compensation and the output distinction, respectively [16].

 

Fig. 3 (a) Schematic of dual-rod double-pass Nd:YAG laser amplifier (QR: quartz rotator, λ/4: quarter-wave-plate). (b) and (c) show the input pulses and change in stored energy of each medium. (d) and (e) show the final outputs of the amplifier that are calculated with different calculation schemes. (“Eq. (2)” is the calculation that do not include the pulse overlapping effect, “Eq. (4)” is the calculation that use the thin medium model, and “Eq. (4) w/EOPL” is the calculation that segments both the laser pulse and gain media.

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The assessments are conducted in two steps. In the first step, a relatively mild depletion condition (54% depletion of the stored energy of 0.5 J/rod) is chosen to show each pulse shape more clearly. As can be observed in Figs. 3(b) and 3(c), there are three temporal periods during the amplification. In the first period, the stored energy decreases slowly with only the weak forward pulse until the stronger backward beams arrive. In the second period, the strong backward pulse accelerates the depletion of stored energy. Lastly, when the backward beam is left alone, the depletion slows down. These distinct periods produce slope change regions on the amplified pulse. Furthermore, as shown in Fig. 3(d), the calculation that includes the pulse overlapping effect shows a more severe pulse shape distortion than the calculation that does not.

As mentioned in the previous section, we modeled a gain medium with infinitesimal thickness. To check the validity of this model, a calculation that meshed the gain medium was also conducted. The mesh size was decided as follows: δL = δt × c/n_YAG, where δt = 0.01 ns, c is the speed of light in free space, and n_YAG is the refractive index of the medium at a wavelength of 1064 nm. This meshing technique is identical to “the effective optical path length (EOPL) method” used in the small signal gain approximation report [12]. However, the small signal equation was not used, instead, Eqs. (4)–(7) were used to observe only the influence of the infinitesimal medium model. The results are shown in Figs. 3(d) and 3(e). In the macroscopic graph Fig. 3(d), the two results are not distinguishable. Even in the magnified view Fig. 3(e), there are only slight differences in the smoothness around the slope change region. Therefore, more systematic quantitative comparisons between different calculation schemes were conducted as a second step of the assessment.

In the second step, three calculation schemes are compared in terms of the pulse energy error and root mean square (RMS) pulse shape error. One scheme uses Eqs. (4)–(7) with the thin medium model while another scheme uses Eqs. (4)–(7) with the gain medium meshing technique (EOPL), and the other scheme uses the small-signal gain approximation with the EOPL technique [12]. The last scheme is added to compare the algorithm performance because it is known as the fastest method that can be used with multiple-medium double-pass amplifiers. As shown in Fig. 4, both errors are estimated as a function of δt. To define the errors, we used the result of Eqs. (4)–(7) that was calculated with the EOPL method and the very small δt (1 ps) as the reference. This reference setting is valid for two reasons. The first reason is that the two calculation schemes that use the EOPL method eventually converge with each other as δt decreases. Secondly, Eqs. (4)–(7) converge faster than the small-signal approximation. The depletion condition is also adjusted to reflect the more practical power amplifier condition: about an 80% depletion from the stored energy of 1.52 J/rod and the square shape input pulse, which has an energy of 30 mJ and pulse length of 10 ns.

 

Fig. 4 (a) Pulse energy errors and (b) RMS pulse shape errors of three different calculation schemes as a function of segment size. The inset of (a) shows the magnified view. The upper x-axes of the graphs show the corresponding segment number of the input pulse.

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As shown in the inset of Fig. 4(a), the thin medium model calculation has a small limit of error reduction. This limit originates from the small areal difference shown in Fig. 3(e), and it is the intrinsic error of the thin medium model. In this calculation, the intrinsic error for the pulse energy is 0.34%. In addition, the intrinsic RMS pulse shape error is 0.18% of peak and 4% of mean power. Therefore, we concluded that the thin medium model does not deflect the result by a large amount when it is used together with Eqs. (4)–(7). Furthermore, the thin medium model reduces the computation cost. As δt decreases, the pulse segment number N increases by the rational relation N = τ_pulse/δt. In the case of the thin medium model, the function call number is simply proportional to N. In contrast, when the EOPL method is used, the function call number is proportional to N × M, where M ≡ L/δL. Moreover, as noted in section 2.1, the dimension of the calculation would increase to produce the spectral and spatial profiles. Thus, the total calculation number difference increases.

It is also important to note that the major causes of error are different for the Frantz–Nodvik calculations and the small-signal gain approximation. When the segment size δt increases, the depletion effect cannot be ignored anymore. Hence, the errors of the small signal approximation rapidly increase with δt. Similarly, the errors also increase when an intense segment (a strong peak) appears. In contrast, the Franz–Nodvik calculations trace the pulse envelope well for all segment sizes. The errors mostly originate from the sharp peaks or rapid changes that cannot be delicately expressed with thick bar-shaped segments. Additionally, all the graphs of Fig. 4 show a somewhat discrete feature because the ’round’ function is used to segment the pulse length, the gain medium, and the optical distances. For the same reason, the pulse energy error becomes very small at some specific segment sizes, for example, the divisors of the gain medium length: 0.52 ns, 0.26 ns, and so on. In addition, the pulse energy error of the thin medium model (red dashed line) is lower than that of the thickness-considered model (blue line) when δt is large. This is because the round function is used less in the thin medium model calculation.

3.2. Comparison with the measured data

Lastly, we compared the calculations with the measurement data. To check the influence of the pulse overlap, the calculated results that do not consider the overlapping are shown. Furthermore, all the calculations were performed with the thin medium assumption and δt = 0.2 ns. For the experiment, an amplifier, depicted in Fig. 3(a), was used. Another Nd:YAG laser (Quanta-Ray-Lab150, Spectra-Physics Inc.) provided the seed beam. The input pulse energy was controlled by using a polarization beam splitter and half-wave-plate. To produce a top-hat-like input beam, the 10 mm (FWHM) Gaussian-like beam of the seed laser expanded 1.5 times and the central part of the expanded beam was used. Each gain medium was pumped by two flash-lamps operating at 1300 V. The stored energy was estimated to be 1.52 J/rod with an optical to optical chamber efficiency of 6%. In addition, in separate experiments, the effective clear aperture of the amplifier was measured to be 91% and 79% of the rod cross-sectional area for the single- and double-pass cases, respectively. The results are summarized in Fig. 5.

 

Fig. 5 (a) The output energy of the single- and double-pass amplifier as a function of input pulse energy. The point graphs show the measurement data, and the line graphs are the calculated results. “Eq. (4)” and “Eq. (2)” represent the calculations that do and do not consider the pulse overlapping, respectively. (b) The total extracted energy from the gain media as a function of input energy. The x-axis uses a log scale. (c) The calculated output pulse shapes when the amplifier is fully saturated with 1 J of input pulse energy. (d)–(f) The measured input and output, and calculated outputs when the input energy is 10.9 mJ. The measured output corresponds to the empty circle point of graph (a).

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Firstly, as shown in Fig. 5, the calculation of the extended Frantz–Nodvik equation with the thin medium model matches well with the experimental data. The energy curve, shown in Fig. 5(a), the peak position shift, and the narrowed pulse width are well represented, as shown in Figs. 5(d)–5(f). In addition, for the influence of the pulse overlap, the extraction efficiency decreased, as can be seen in Figs. 5(a) and 5(b). Although the extraction efficiencies of the overlap and non-overlap cases become identical when the amplifier is fully saturated, the expected shape is quite different, as shown in Fig. 5(c). It shows that the pulse shape distortion by the depletion effect is more severe when the overlapping occurs.

4. Conclusion

We developed a pseudo-analytical method to accurately and efficiently simulate the amplification of laser pulses in the case of the overlap of two polarization-orthogonal pulses by modifying the Frantz–Nodvik equation. The developed method could be applied to a double-pass amplifier that has multiple gain media and an arbitrary input pulse shape. Despite modeling the gain medium with an infinitesimal thickness, the intrinsic errors of the method was only 0.34% for the pulse energy and 0.18% of peak power (4% of mean power) for the temporal shape. In addition, the errors were maintained at a low value even in very rough calculations. Consequently, the lowest computation cost ever reported was achieved among the methods. Finally, the experimental results agreed well with the calculated results. When it was compared with the non-overlapping amplification, a more severe pulse shape distortion and lower energy extraction efficiency were observed.