# 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?

## 1 Answer

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.

For proof see the original paper by Goldfischer ("Autocorrelation Function and Power Spectral Density of Laser-Produced Speckle Patterns" -- I think it should be open source since it is an OSA paper). For a more in-depth discussion see the book "Laser speckle and related phenomena" by Goldfischer et. al.