Speckle

A laser spot on a wall, a painted surface or a sheet of paper looks grainy, and the grain seems to shimmer as the viewer moves. The surface is rough on the scale of a wavelength, so every point of the illuminated area scatters light with a random phase, and at any point of observation many of these waves add. Where they happen to add in phase the result is bright, where they cancel it is dark. The result is speckle, an interference pattern with random structure, and it appears whenever light of long coherence length meets a random medium.

Statistics and size

When many independent scatterers contribute, the field is a circular Gaussian random variable and the intensity follows a negative exponential distribution: the most probable intensity is zero, and the standard deviation equals the mean, so the speckle contrast $C = \sigma_I/\langle I\rangle$ is 1. Adding $N$ independent speckle patterns in intensity, from different wavelengths, angles, polarizations or times, reduces the contrast to $1/\sqrt{N}$: 0.5 for four patterns and 0.1 for a hundred.

The grain size is set by the illumination and viewing geometry. In free space, a spot of diameter $D$ observed at distance $z$ produces objective speckle of characteristic size about $1.22\lambda z/D$: 386 µm at 1 m from a 2 mm helium-neon spot. When the surface is imaged, the speckle in the image plane is subjective speckle, about the size of the imaging system's point spread function, so stopping a camera down makes the grains larger.

Where it matters

Speckle limits laser-illuminated imaging and displays, where it appears as noise on every image; laser projectors reduce it with moving diffusers, several wavelengths and vibrating screens. In multimode fibers, the output is a speckle pattern of the guided modes, which moves when the fiber is bent or heated; together with mode-dependent loss this creates modal noise, and mode scramblers are used to average it. Speckle adds noise to beam-profile measurements of coherent beams scattered by imperfect optics.

It is also a tool. Speckle interferometry and digital image correlation measure surface displacement and strain; laser speckle contrast imaging maps blood flow, because moving scatterers blur the pattern during an exposure; and speckle-based wavemeters and spectrometers read the wavelength from the pattern produced by a multimode fiber or diffuser.

Measurement

Speckle contrast is measured by recording the pattern on a camera whose pixels are smaller than the grains and computing the standard deviation divided by the mean over a uniformly illuminated region, after subtracting the dark background. The grain size is obtained from the width of the pattern's spatial autocorrelation.

References: J. W. Goodman, Speckle Phenomena in Optics, 2nd ed. (SPIE Press, 2020); J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).