# Maximum Operational Fluence Limits for Temporally Shaped Nanosecond Long Pulses
## 3 Maximum Operational Fluence for Different Pulse Shapes  
In this study, we operate under the assumption that damage is cumulative. This aligns with methodologies applied elsewhere [26], where a phenomenological model is considered, involving the creation and diffusion of damage particles called _damageonium_ by [PERSON] et al. in [38]. This is mainly heat but can also be mechanical deformation or particles and as such propagates through the system. The cumulative effect eventually reaches a threshold where damage is produced. We justify this assumption based on the pulse duration's order of magnitude in this system, which has a minimum pulse duration of 200 ps and the smallest features on the order of 100 ps (limited by fast electronics). Please note that this is far from avalanche and multiphoton ionization phenomena [23], which mainly depend on peak intensity. Additionally, a significant reason to consider cumulative damage in large-scale lasers is that imperfections or damage in certain parts of the laser system can propagate forward or backward. This occurs either through simple diffraction or the creation of back reflections and ghosts [39]. With this condition, the accumulation of damage particles can be given by the following quantity D damaging the system [26; 38].  
\[D=\int g(t-\tau)I(t)dt \tag{2}\]  
Here, \(g(t)\) represents the impulse response function of the diffusion process of the damage particles [38]. Damage particles generated at each point of the pulse are proportional to the intensity and diffuse according to the function \(g(t)\) over time, relative to the pulse's initiation. The temporal intensity profile of the pulse is denoted by \(I(t)\) (Equation (2)). Damage initiation occurs once the cumulative damage, denoted as \(D\), reaches a certain probability threshold, \(D_{i}\). This threshold is reached when the fluence of the pulse attains the damage threshold fluence, \(\Phi_{D}\). The temporal intensity profile of such a pulse is expressed in Equation (3).  
\[I_{D}(t)=\frac{\Phi_{D}}{\int p(t)dt}p(t) \tag{3}\]  
where \(p(t)\) is the temporal shape of the pulse. This implies that the accumulated damaged particles at the point of damage initiation can be expressed as  
\[D_{i}=\frac{\Phi_{D}}{\int p(t)dt}\max_{\tau}\int g(t-\tau)p(t)dt \tag{4}\]  
At this juncture, we can utilize the fluence employed to define \(f(t)=g(t)/D_{i}\). Remarkably, the probability of damage initiation remains consistent for any pulse shape, irrespective of its form. By substituting this into Equation (4), the damage threshold can be expressed as  
\[\Phi_{D}=\frac{\int p(t)dt}{\max_{\tau}\int f(t-\tau)p(t)dt} \tag{5}\]  
The relationship between the maximum fluence and the probability of damage is delineated in Equations (1) and (2), assuming a square pulse--easily attainable through a shapeable laser [35]. Leveraging this understanding, we can compute the time-dependent \(f(t)\) and subsequently reintroduce it into Equation (5). This process is carried out to extend the generalization to any pulse shape and duration.  
Boundary conditions are applied at zero time. As a result, the maximum fluence is expressed as  
\[f(t)=\frac{d\Big{(}\frac{t}{\int\max(t)}\Big{)}}{dt}+\alpha\delta(t) \tag{6}\]where the constant value \(\alpha\) is  
\[\alpha=\lim_{t\to 0}\frac{t}{J_{max}(t)} \tag{7}\]  
One can then conclude that the damage threshold for any pulse shape has  
\[\Phi_{D}=\frac{\int p(t)dt}{\max_{\tau}\{\int p(t+\tau)\frac{d\left(\frac{t}{J_{ max}(t)}\right)}{dt}dt+\alpha\}} \tag{8}\]  
For each one of the functions in Equation (1), we have  
\[\begin{split} f_{1}(t)&=\frac{1}{c_{1}}\left[\frac{1 }{1-\exp-\frac{t}{c_{2}}}-\frac{t/c_{2}\exp-(t/c_{2})}{\left[1-\exp-(t/c_{2}) \right]^{2}}\right]\\ f_{2}(t)&=\frac{1}{c_{3}c_{4}^{2}}\frac{t}{\sqrt{1 +\left(\frac{t}{c_{4}}\right)^{2}}}\end{split} \tag{9}\]  
The \(\alpha\) constant is given by  
\[\begin{split}\alpha_{1}&=c_{2}/c_{1}\\ \alpha_{2}&=1/c_{3}\end{split} \tag{10}\]  
Regardless of the damage mechanism, Equations (6)-(8) remain valid.