Impact of random spatial noise of a laser beam on the probability of fluence to exceed the threshold value

The analytical expressions of the probability for a two-dimensional random function (for example, fluence of a laser beam with noise) to exceed the threshold value over a given area have been derived. The probability is fully determined only by two parameters. The first one, the product of the beam area and the squared second normalized spectral moment, has the meaning of the equivalent number of noise modes and is determined only by the spectral properties of the noise. The second parameter, on the contrary, does not depend on the noise spectrum; it is determined only by the ratio of the standard deviation to the difference between the threshold and mean values of fluence. The numerical simulations are in a good agreement with the theory. The results are generalized to several cases of practical interest.