Speckles, or laser hot spots, are the intensity maxima of optically smoothed laser beams that seed instabilities, making a quantitative statistical description of speckles of interest. Earlier statistical theories estimated the number of speckles above an intensity level set for optically smoothed beams produced by random phase plates, using an ansatz that relates intensity maxima to the maxima of the real and imaginary components of the underlying complex Gaussian electric field. Here, we count speckles directly from the laser intensity field, treating it as a ${\chi }_2^2$ random field and imposing the local maximum conditions without a single-component ansatz. We evaluate the theory for square, circular, annular, and Gaussian aperture spectra, and include induced spatial incoherence as a temporal smoothing mechanism. Monte Carlo simulations confirm the theory and show improved accuracy relative to the earlier ansatz-based formulation. Applications include a simple stimulated Brillouin scattering reflectivity model using the resulting speckle statistics, and a comparison between speckle-driven and thermal noise density-fluctuation spectra.

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• Fluctuations & noise
• Laser-plasma interactions
• Stimulated Brillouin & Raman scattering in plasmas
• Extreme event statistics
• Stochastic analysis methods