# Maximum Operational Fluence Limits for Temporally Shaped Nanosecond Long Pulses
## 1 Introduction  
The control of the temporal shape of nanosecond pulses has found applications across various fields, ranging from industrial applications such as laser peening [1; 2; 3] to ultracold molecular formation [4; 5; 6]. In high energy laser facilities, temporal shaping has been used in the development of lasers themselves via optical parametric chirped-pulse amplification (OPCPA) techniques [7; 8; 9], and are used to create high density conditions [10] mostly for studies in warm dense matter [11; 12] and plasma physics [13], namely in inertial confinement fusion (ICF) [14].  
The maximum fluence at which a laser can operate is closely linked to the laser damage threshold, attributed to various causes [15; 16; 17; 18; 19; 20; 21; 22; 23; 24], including local thermal effects, thermally induced stress [18; 21; 22], electron avalanche [15], and factor mechanics [16] driving damage. In shorter time scales (ps to hundreds of ps), optical damage arises from multiphoton processes like multiphoton absorption [23], ionization, and electron avalanche [15; 24]. With even shorter pulses (in the fs regime), self-focusing and filamentation become crucial effects [24], potentially deforming the pulse's spatial and temporal profile and introducing hot spots, making damage more unpredictable.  
In the nanosecond time scale, damage is a probabilistic and cumulative process involving precursors [17; 21; 22; 25; 26] and imperfections in optics, such as nanoparticles, accumulating damage. Thermal heating acts as a determinant to the maximum energy achievable [20]. The accumulation is not linear and depends on diffusion [26], thermal factors [20], or other considerations. The process is wavelength-dependent [27] and even dependent on the morphology of defects [21].  
In larger laser systems, the complexity increases as different types of damage must be considered, such as bulk and surface damage [28; 29; 30]. Various optics will have differentmaximum operational fluence, with gratings being particularly sensitive to damage [31; 32; 33]. Moreover, damage can propagate between optics due to hot spots created by small damage sites earlier in the system or ghost reflections [31; 32].  
Irrespective of the origin of laser damage, it is fundamentally important to have prior knowledge of the maximum fluence at which the laser system can be operated safely, regardless of the pulse temporal profile. Testing all possible pulse shapes is impractical, necessitating the use of an indirect method. While a single point failure, where one optic is substantially weaker than the rest, simplifies this process, it is not always the case.  
In the Vulcan laser facility [34], the development of pulse shaping [35] prompted the evaluation of the energy deliverable in a nanosecond-shaped pulse. This becomes even more critical with the advent of high-repetition-rate ns lasers [36; 37]. Previous studies, considering damage energy proportional to a fractional power of pulse duration, do not align with the fluence limits observed in the Vulcan laser facility [20; 23; 26]. Due to the laser's nature, testing all facility elements is not feasible, necessitating the deduction of energy limits using the cumulative principle and the established maximum operational fluence.  
It is crucial to clarify that the objective is not to evaluate damage but rather to determine the maximum energy at which a laser can be operated. The focus is not on identifying the origin of damage or any other limitations to laser energy limits.