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Abstract
We developed a theoretical model for the relationship between the input and amplified laser beams of energy stability and spatial uniformity in the amplification process. 10 Hz, 8 ns, 1064 nm Nd:YAG Q-switched resonator with Nd:YAG main amplifier was employed for the experiment. The theoretical model simulation and Frantz-Nodvik simulation were performed by utilizing the obtained beam image, acquired energy from the experiment, and stored fluence from the gain medium. The result indicated that the fluctuation of the spatial distribution in a single beam influences the stability of temporally distributed energy during the amplification process of the laser beam, thereby improving energy stability.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser beam amplification occurs when the photon undergoes stimulated emission and population inversion of the gain medium. The quality of the amplified beam characteristics is represented by two categories: spatial and temporal characteristics. The spatial characteristics of laser beam are analyzed in a single shot, represented by spatial uniformity. Energy stability is commonly implemented in multiple shots to analyze the temporal characteristics of the laser beams. Both characteristics play key roles in material processing and science fields, such as pulsed laser annealing [1–3], pulsed laser deposition [4,5] and Ti:sapphire laser pumping [6,7]. Improving spatial uniformity and energy stability has been approached from an independent perspective. However, spatial uniformity and energy stability are statistical characteristics of single shot and multiple shots in the temporal domain. Thus, it is necessary to achieve an analytical solution to the way the statistical properties of the spatial distribution of a beam for a single shot are related to properties of temporally distributed energy for multiple shots.
A Laser with spatially uniform and temporally stable energy is the preferred quality. Spatial uniformity is the ratio of the deviation of the fluence distribution to the average fluence, defined by international standards [8]. The spatial uniformity of the laser beam is degraded by the interference of the optical components, birefringence, thermal lens of the components, and alignment errors [9]. Modifying the optical path of each local region using diffractive optics elements and beam shapers is one method to improve the spatial uniformity [10,11], while the other method is enhancing uniformity through the mixing of local beams using homogenizer [12–14]. In addition, improving the structure of the cavity is another method to improve uniformity [15–17]. However, the energy stability is derived from the shot-to-shot intensity fluctuation divided by the average energy of the laser, expressed as root-mean-square (RMS) stability [18] where the highest stability has been reported $\sigma =$ 0.27$\%$[19]. The factors that affect energy stability are the emission frequency stability, thermal effect of each optical component, alignment error, mechanical vibrations, and stability of the energy source which can increase shot-to-shot intensity fluctuations [20]. Frequency stabilization using an optical modulator is a method to stabilize the energy stability [21]. Gain medium surface processing or new types of gain medium is another method to compensate for thermal effects to minimize the fluctuation [22], or utilizing optical components that are insensitive to thermal effects is an alternative method [20,23].
In this study, the strongly coupled relationship between energy stability and spatial uniformity during amplification is validated. The energy distribution in the temporal domain contributes to calculating energy stability. Spatial uniformity, however, is RMS error of unit time energy distribution in the spatial domain for one shot. From a statistical point of view, a multi-shot is a set of single shots, which implies that energy stability is dependent on spatial uniformity. In addition, thermal effect and misalignment errors are factors that affect energy stability and spatial uniformity [9,20], which suggest the mutual relationship between energy stability and spatial uniformity. We experimentally identify the relationship between spatial uniformity and energy stability based on gain saturation effect. Moreover, it has been theoretically proven that numerical simulations performed and experimentally verified that spatial uniformity and energy stability are highly coupled during pulse amplification.
2. Relationship between energy stability and spatial uniformity in amplification process
In this study, we derived theoretical models of the relationship between the energy stability and spatial uniformity of laser beams through an amplification mechanism. The details of the derivation of the formula are included in Supplement 1. We defined the spatially distributed input beam intensity($j_s^i$) for a single pulse. We quantified the intensity distribution($j^i_{s,t}$) with periodic interval $\Delta T=1/f_{rep}$ in Eq. (1). The quantified intensity distribution can be defined using the average input beam energy distribution ($j_a^i$) and spatial fluctuation ($\delta j_s^i$). The shot-to-shot beam energy fluctuation is a non-dimensional unit($\delta _t$).
The energy stability of the input and amplified beams ($\sigma _T^i, \sigma _T^o$) is defined by Eq. (2), which is denoted as ISO 11554 [18]. Here, $E_a^i, E_a^o$ are the beam average energies of the input and amplified beams, respectively; $E_T^i, E_T^o$ are the energies of each shot for the input and amplified beams, respectively; $N_T$ is the number of shots.
The spatial uniformity of the input and amplified beams ($\sigma _s^i, \sigma _s^o$) is defined by Eq. (3), which is denoted as ISO 13694 [8]. $j_a^i, j_a^o$ are the beam average fluences of the input and amplified beams, respectively; $j_s^i, j_s^o$ are the fluences of a single pixel for the input and amplified beams; $N_s$ is the number of pixels, which is the product of horizontal pixels and vertical pixels of CMOS sensor.
From Eq. (4), we considered the stored fluence in gain medium $J_{sto}$ as uniformly pumped for the amplification. The Frantz-Nodvik equation is expanded into a Taylor series using a finite-difference approximation for the spatially distributed fluence of a spatially fluctuating input beam ($j_a^i+\delta j_s^i$) in Eq. (5). In addition, the Frantz-Nodvik equation was expanded into Eq. (6) for the temporally fluctuating input beams, and $\delta A$ represents the area of the surface element.
Through Taylor’s approximation, the spatial uniformity and energy stability of the amplified output laser can be expressed by Eq. (7) and (8).
Equation (8) emphasizes that RMS error of spatial distribution is coupled to the RMS error of energy temporal distribution. Eq. (8) clearly shows that the energy stability is affected by the input energy stability and energy itself, as well as by the spatial uniformity of the input beam. However, Eq. (7) suggests that the spatial uniformity of the amplifier laser beam is un-affected by the energy stability, but by the input beam uniformity and input energy. We validated the equation through an experiment using a theoretical model.
3. Experiment
3.1 Experimental setup
We utilized a 1064 nm, 10 Hz Nd:YAG Q-switched resonator with a main amplifier system to analyze the spatial uniformity and energy stability relationship between the input and amplified laser beams, as shown in Fig. 1. The structure of the Q-switched Nd:YAG resonator was a negative-branch unstable resonator pumped by two flash lamps in the vertical direction. The main amplifier utilized an Nd:YAG rod, and the pumping method is the same as that of the Q-switched resonator. The gain medium was pumped by the flashlamp driver (PS5050, EKSPLA; PFN spec.: capacitance 60 $\mu$F, inductance 100 $\mu$H, and pulse duration FWHM 163 $\mu$s). The pump radiation from the flashlamps was about 400 $\mu$s at a 10 Hz repetition rate. All of the Nd:YAG rods used in the experiment have the following characteristics; rod diameter $\phi$=12.2 mm, rod length L=85 mm, saturation fluence $J_{sat}$ =0.66 $J/cm^2$, doping concentration 0.8 $\%$. A quartz polarization rotator (QR) was installed between the resonator and main amplifier to compensate for thermally induced depolarization [9,27]. The experiment was conducted using the following procedure. Energy was applied inside the resonator for a total of 9 sections from 133 to 819 mJ. The stored energy in the Nd:YAG of the main amplifier was 1.90 J, which was numerically calculated using the same methods as in a previous study [26].
Fig. 1. Schematic of experimental setup. HR: high reflectivity mirror, PC: pockels cell, TFP: thin film polarizer, GRM: Gaussian reflectivity mirror, QR: quartz polarization rotator. (a) Two complementary metal oxide semiconductor(CMOS) image sensors are installed at the end of the resonator and main amplifier. (b) and (c) show the location for energy measurement utilizing a pyroelectric energy meter.
In this experiment, the beam profile and energy were obtained and evaluated based on ISO 11554 and 13694 standards. The beam profiles were recorded simultaneously before and after entering the main amplifier from a complementary metal oxide semiconductor (CMOS) sensor (Mako G-040, Allied Vision Technologies), as shown in Fig. 1(a). The beam energy was measured at each rear position of the resonator and main amplifier using a pyroelectric detector (QS25LP, Gentec-EO), as shown in Fig. 1(b) and (c). At least 100 beam profiles were recorded to evaluate the spatial characteristics. To evaluate energy stability, we obtained energy from at least 600 pulses. Figure 2 shows a typical example of the fluence histograms and fluence distributions for the input and amplified laser beams. The fluence distribution was obtained using the average of the measured energy and beam profiles.
Fig. 2. Typical example of input and amplified beam histogram and profile. (a) input beam fluence histogram ($j_s^i$, black line) and input beam fluence distribution(right image). (b) Amplified beam fluence histogram ($j_s^o$, red line) and amplified beam fluence distribution(right image).
3.2 Numerical implementation
The simulation was conducted using two methods; the Frantz-Nodvik equation (Eq. (4)) and a theoretical model (Eq. (7) and (8)). Each simulation was numerically calculated utilizing a two-dimensional (2D) fluence distribution and temporal energy distribution.
3.2.1 Frantz-Nodvik simulation
We perform Frantz-Nodvik simulation to compare the theoretical model (Eq. (7) and (8)). Figure 3(a) shows the procedure of Frantz-Nodvik simulation. Energy distribution in the temporal domain, which is the input energy distribution ($E_T^i$), was obtained from a pyroelectric energy meter. The amplified energy distribution in the temporal domain ($E_T^o$) is derived by Frantz-Nodvik equation (Eq. (4)), then utilized into Eq. (2) for acquiring energy stability ($\sigma _T^o$).
Fig. 3. Numerical implementation procedure of (a) Frantz-Nodvik simulation. (b) theoretical model simulation
For obtaining spatial uniformity, the 2D fluence distribution from the resonator ($j_s^i$) was obtained by combining the acquired normalized fluence distribution from the CMOS sensor with the measured energy from the pyroelectric energy meter. The Fresnel diffraction does not show a significant difference in the 2D fluence distribution when the distance between the image plane and the Nd:YAG surface is less than 200 mm. The amplified 2D fluence distribution ($j_s^o$) is derived from input 2D fluence distribution ($j_s^i$), by Frantz-Nodvik equation Eq. (4). The spatial uniformity of the amplified 2D fluence distribution ($\sigma _s^o$) is acquired by Eq. (3).
3.2.2 Theoretical model simulation
Figure 3(b) shows the schematic flow chart of the theoretical model simulation. First, we obtained energy distribution in temporal domain ($E_T^i$) and input 2D fluence distribution ($j_s^i$), by the same method in the Frantz-Nodvik simulation. Then, the spatial uniformity of input beam ($\sigma _s^i$) and energy stability of input beam ($\sigma _T^i$) is derived by using Eq. (2) and (3). Utilizing theoretical model (Eq. (7) and (8)), the amplified beam spatial uniformity ($\sigma _s^o$) and amplified energy stability ($\sigma _T^o$) is acquired.
4. Results and discussion
Figure 4(a) shows the relationship between spatial uniformity and energy stability according to the Frantz-Nodvik equation(Eq. (4)), the theoretical model (Eq. (7) and (8)) and the experimental results. The most stable energy stability values were 0.52% and 0.58% for the input and amplified beams, respectively. Compared with the experimental values from the simulation, the Frantz-Nodvik simulation and the theoretical model matched well. The error of the energy stability estimated with Frantz-Nodvik simulation was 0.47%, and the error of spatial uniformity is 1.5%. For (Eq. (7) and (8)), the errors in energy stability and spatial uniformity were 0.48% and 2.5%, respectively. The numerically calculated results using the Frantz-Nodvik simulation best fit the experimental results, and the errors in the theoretical model for spatial uniformity and energy stability result from the omission of more than the tertiary term of the Taylor expansion. In addition, stored fluence distribution in the gain medium causes non-uniform gain distribution due to vertical pumping. The Frantz Nodvik simulation results(Fig. 4, blue triangle dot line) would match well to the experiment result(Fig. 4, red square dot line), considering the pumping fluence non-uniformity.
Fig. 4. (a) Energy stability ($\sigma _T$) and spatial uniformity($\sigma _s$) (b) Energy stability ($\sigma _T$) and measured energy. (c). Spatial uniformity ($\sigma _s$) and measured energy. A series of conventions are accepted in (a)-(c): black square dot line: experimental results for laser beam from the resonator with spatial uniformity($\sigma _s^i$) and energy stability($\sigma _T^i$), red square dot line: experimental results for amplified beam from main amplifier with spatial uniformity($\sigma _s^o$) and energy stability($\sigma _T^o$), blue triangle dot line: Frantz-Nodvik equation numerically calculated utilizing Eq. (4), and grey circle dot line: theoretical model calculated by utilizing Eq. (7), (8).
From the experiment, the spatial uniformity and energy stability of the input and amplified beams improved when the energy increased, and tended to be proportional. In contrast, in the case of energy stability, both the input and amplification of the experimental results do not improve further from a specific value $\sigma _T = 0.5\%$, which might be related to environmental factors, such as ambient air flow, temperature and humidity. In addition, because high-power lasers cannot propagate directly into CMOS sensor due to laser induced damage threshold, polarization information is in-accurately feed into the sensor, resulting in errors in the spatial uniformity calculation through simulations. It is possible that the increase in the applied energy of the Nd:YAG Q-switched laser causes the growth of heat generation, which leads to occur nonlinear effects, such as thermal depolarization and the thermal lens effect [28,29] degrading the beam quality and energy stability.
$\sigma _T$ denotes the shot-to-shot energy stability of the laser beam resulting from multiple shots, and $\sigma _s$ is the spatial uniformity of the laser beam representing the spatial characteristics of a single beam. The amplification mechanism improves both energy stability and spatial uniformity, as shown in Fig. 4(b) and (c). The improvement in stability and uniformity during amplification is caused by the gain saturation effect. The gain value exhibited nonlinearity according to the input beam fluence. The fluctuation of the amplified beam is compressed once the deviation of the input beam is amplified with a saturated gain; thus, spatial uniformity and energy stability are improved. Fig. 5(b) and 5(c) present clear examples of the gain saturation effect.
Fig. 5. (a) Typical example for gain saturation in fluence distribution. Amplification for fluence distribution with different gain compresses the deviation, thereby improving spatial uniformity. (b) Typical example for the way amplification compresses energy stability. Gain curve nonlinearity compresses root-mean-square (RMS) energy stability. (c) The principle how different gain works to improve spatial uniformity.
The spatial uniformity of the amplified beam depends on the spatial uniformity of the input beam and input energy from Eq. (7). In contrast, the energy stability depends on the energy stability of the input beam, energy, and the spatial uniformity, as shown in Eq. (8). Specifically, the single-beam characteristics of the input beam and multi-shot characteristics significantly influence the multiple shots characteristics of the amplified beam, which is strong evidence of how spatio-temporal characteristics are coupled. Figure 6 extrapolates the manner in which the input beam spatial uniformity regulates the amplified beam energy stability. The second derivative of the Frantz-Nodvik equation is always negative in Eq. (8); thus, for the most stable energy, the input beam should be uniform and has the optimal energy. In addition, the results suggest that modifying the spatial shape of the input beam before it propagates into the main amplifier can improve the output beam quality.
Fig. 6. Ratio of input and amplified beam energy stability according to input beam spatial uniformity.
5. Conclusion
The relationship between spatial uniformity and shot-to-shot energy stability during amplification was presented. We demonstrated a mathematical proof by numerically calculating the Frantz-Nodvik equation and theoretical estimations. The experiment was conducted with a 1064 nm, 10 Hz, 8 ns, flash lamp-pumped Nd:YAG Q-switched laser with a main amplifier to validate our theory. The theoretical model and the Frantz-Nodvik simulation closely matched the experimental results. The spatial uniformity of the amplified beam is affected by the spatial uniformity of the input beam and input average energy. The energy stability of the amplified beam is influenced by spatial uniformity, energy stability, and average energy of the input beam. The results suggest that amplified beam energy stability is dependent on the spatial uniformity of the input beam, where single-beam characteristics are coupled with energy stability, which is a multi-beam characteristics.
Funding
Korea Institute for Advancement of Technology (P0008763); National Research Foundation of Korea (2021R1I1A3051341).
Disclosures
The authors declare that they have no conflicts of interest.
Data availability
The data underlying the results presented in this study are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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