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We consider general theoretical aspects of level crossings of multidimensional fluctuating functions. Examples of such functions are turbulent fields such as refractive-index functions or turbulence-aberrated fields such as laser intensity functions in free-space laser communications. From a practical point of view, it is important to consider level crossings because they correspond to the temporal instances or spatial occurrences of when or where a signal of interest reaches, exceeds, or falls below a particular threshold. For example, the statistics of level crossings for a laser communication signal at a threshold corresponding to the minimum detection signal are important in order to study the probability density of the extent of intervals of down-time for communication links. For 1-D signals, the concept of the level crossing scale is clear and well established as it is the extent of the interval between successive level crossings. However, for multidimensional fields, this concept cannot be utilized directly because it is not clear how to define or identify successive level crossings, and therefore level crossing scales, in multiple dimensions. We describe a theoretical formulation which enables a consistent definition of level crossing scales for multidimensional fields, i.e. consistent with the traditional 1-D definition. We use the recently-developed concept of the shortest-distance scale because the latter applies naturally to multiple dimensions. We define the probability density function of level crossing scales, in any number of dimensions, in terms of a derivative of the probability density function of shortest-distance scales. Analytically, we illustrate this approach using exact theoretical examples with 2-D objects and we also provide results for exponential, lognormal, and power-law level crossing statistics which are basic models for applications involving turbulence and free-space laser communications.
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