# What does the width of the 2D Fourier transform of a speckle pattern mean?

Speckle patterns can be studied by capturing an image of the speckle pattern produced on a rough surface when light from a laser passes through an aperture stop.

When analysing this speckle pattern, a 2D Fourier transform of the image can be plotted which produces a semi-triangular gaussian-like distribution. What does the width of this Fourier transform tell you?

The power spectral density of a rough speckle pattern (a rough pattern has a phase change exceeding $$2\pi$$) is proportional to the intensity auto-correlation of the illumination on the rough surface: $$S(\omega,\nu)\propto\int I(x,y)I(x-\lambda z \Omega,y- \lambda z \nu)\,dxdy$$), where $$z$$ is the path length from the rough surface to the observation plane. For rough surfaces the size of the speckles are determined by the illuminating laser and not the surface.