Speckle pattern

A speckle pattern is a random intensity pattern produced by the mutual interference of a set of wavefronts. This phenomenon has been investigated by scientists since the time of Newton, but speckles have come into prominence since the invention of the laser and have now found a variety of applications.

Where speckle occurs
A familiar example is the random pattern created when a laser beam is scattered off a rough surface - see picture. A less familiar example of speckle is the highly magnified image of a star through imperfect optics or through the atmosphere (see speckle imaging). A speckle pattern can also be seen when you look at sunlight scattered by your fingernail.

The speckle effect is observed when radio waves are scattered from rough surfaces such as ground or sea, and can also be found in ultrasonic imaging. In the output of a multimode optical fiber, a speckle pattern results from a superposition of mode field patterns. If the relative modal group velocities change with time, the speckle pattern will also change with time. If differential mode attenuation occurs, modal noise results.

How speckle occurs
The speckle effect is a result of the interference of many waves, having different phases, which add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly. If each wave is modelled by a vector, then it can be seen that if a number of vectors with random angles are added together, the length of the resulting vector can be anything from zero to the sum of the individual vector lengths—a 2-dimensional random walk, sometimes known as a drunkard's walk.

When a surface is illuminated by a light wave, according to diffraction theory, each point on an illuminated surface acts as a source of secondary spherical waves. The light at any point in the scattered light field is made up of waves which have been scattered from each point on the illuminated surface. If the surface is rough enough to create pathlength differences exceeding one wavelength, giving rise to phase changes greater than 2π, the amplitude, and hence the intensity, of the resultant light varies randomly.

If light of low coherence (i.e. made up of many wavelengths) is used, a speckle pattern will not normally be observed, because the speckle patterns produced by individual wavelengths have different dimensions and will normally average one another out. However, speckle patterns can be observed in polychromatic light in some conditions. .

Subjective speckles
The speckle pattern formed by imaging an object with a rough surface, illuminated by a coherent beam is called a “subjective speckle pattern”. The image above is a subjective speckle pattern since it was formed by imaging the area illuminated by a laser beam. Any movement of the imaging system changes the detailed structure of the subjective speckle pattern.

Each point in the image can be considered to be illuminated by a finite area in the object. The size of this area is determined by the diffraction-limited resolution of the lens which is given by the Airy disk whose diameter is 2.4λu/D where u is distance between the object and the lens, and D is the diameter of the lens aperture. (This is a simplified model of diffraction-limited imaging).

The light at neighbouring points in the image has been scattered from areas which have many points in common and the intensity of two such points will not differ much. However, two points in the image which are illuminated by areas in the object which separated by the diameter of the Airy disk, have light intensities which are unrelated. This corresponds to a distance in the image of 2.4λv/D where v is the distance between the lens and the image. Thus, the ‘size’ of the speckles in the image is of this order. This can be demonstrated by changing the lens aperture when viewing a speckle pattern or by looking at the laser spot on a surface through a very small hole, when the speckles will be very big.

Objective speckles
The speckle pattern formed by the propagating scattered waves is called an “objective speckle pattern”. This can be observed by shining a laser beam onto a metal surface, and using a piece of paper to catch the scattered light.

The relative phases of the light contributing to the overall intensity changes across the speckle field. When angle of scattering changes such that the relative path difference between light scattered from the centre of the illuminated area compared with light scattered from the edge of the illuminated changes by λ, the intensity becomes uncorrelated. Dainty derives an expression for the mean speckle size as λz/L where L is the width of the illuminated area and z is the distance between the object and the location of the speckle pattern.

Applications of Speckle
When lasers were first invented, the speckle effect was considered to be a severe drawback in using lasers to illuminate objects, particularly in holographic imaging because of the grainy image produced. It was later realized that speckle patterns could carry information about the object's surface deformations, and this effect is exploited in holographic interferometry and electronic speckle pattern interferometry. The speckle effect is also used in stellar speckle astronomy, speckle imaging and in eye testing using speckle.

Reference
Speckle Interferometría de moteado Speckle Speckle Спекл