Speckle (interference)

Speckle, speckle pattern, or speckle noise designates the granular structure observed in coherent light, resulting from random interference. Speckle patterns are used in a wide range of metrology techniques, as they generally allow high sensitivity and simple setups. They can also be a limiting factor in imaging systems, such as radar, synthetic aperture radar (SAR), medical ultrasound and optical coherence tomography.^[1][2][3][4] Speckle is not external noise; rather, it is an inherent fluctuation in diffuse reflections, because the scatterers are not identical for each cell, and the coherent illumination wave is highly sensitive to small variations in phase changes.^[5]

Speckle patterns arise when coherent light is randomised. The simplest case of such randomisation is when light reflects off an optically rough surface. Optically rough means that the surface profile contains fluctuations larger than the wavelength. Most common surfaces are rough to visible light, such as paper, wood, or paint.

The vast majority of surfaces, synthetic or natural, are extremely rough on the scale of the wavelength. We see the origin of this phenomenon if we model our reflectivity function as an array of scatterers. Because of the finite resolution, at any time we are receiving from a distribution of scatterers within the resolution cell. These scattered signals add coherently; that is, they add constructively and destructively depending on the relative phases of each scattered waveform. Speckle results from these patterns of constructive and destructive interference shown as bright and dark dots in the image.^[6]

Speckle in conventional radar increases the mean grey level of a local area.^[7] Speckle in SAR is generally serious, causing difficulties for image interpretation.^[7][8] It is caused by coherent processing of backscattered signals from multiple distributed targets. In SAR oceanography, for example, speckle is caused by signals from elementary scatterers, the gravity-capillary ripples, and manifests as a pedestal image, beneath the image of the sea waves.^[9][10]

The speckle can also represent some useful information, particularly when it is linked to the laser speckle and to the dynamic speckle phenomenon, where the changes of the spatial speckle pattern over time can be used as a measurement of the surface's activity, such as which is useful for measuring displacement fields via digital image correlation.

Although scientists have investigated this phenomenon since the time of Newton,^[11] speckles have come into prominence since the invention of the laser.^[12]

The speckle effect is a result of the interference of many waves of the same frequency, having different phases and amplitudes, which add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly. If we model each wave by a vector, we can then see that if we add a number of vectors with random angles 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. In the limit of many interfering waves, and for polarised waves, the distribution of intensities (which go as the square of the vector's length) becomes exponential ${\textstyle P(I)={\frac {1}{\langle I\rangle }}\exp \left({\frac {-I}{\langle I\rangle }}\right)}$, where ${\displaystyle \langle I\rangle }$ is the mean intensity.^{[1][2][13][14]}

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 path-length 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, we can observe speckle patterns in polychromatic light in some conditions, such as with partially coherent light,^[15] or spatially coherent light.^[16]

Speckle patterns have been used in a variety of applications in microscopy,^[20][21] imaging,^[22][23] and optical manipulation.^[24][25][26]

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. Researchers later realized that speckle patterns could carry information about the object's surface deformations, and exploited this effect in holographic interferometry and electronic speckle pattern interferometry.^[27] Speckle imaging and eye testing using speckle also use the speckle effect.

Speckle is the chief limitation of coherent lidar and coherent imaging in optical heterodyne detection.

In the case of near field speckles, the statistical properties depend on the light scattering distribution of a given sample. This allows the use of near field speckle analysis to detect the scattering distribution; this is the so-called near-field scattering technique.^[28]

When the speckle pattern changes in time, due to changes in the illuminated surface, the phenomenon is known as dynamic speckle, and it can be used to measure activity, by means of, for example, an optical flow sensor (optical computer mouse). In biological materials, the phenomenon is known as biospeckle.

In a static environment, changes in speckle can also be used as a sensitive probe of the light source. This can be used in a wavemeter configuration, with a resolution around 1 attometre,^[29] (equivalent to 1 part in 10¹² of the wavelength, equivalent to measuring the length of a football field at the resolution of a single atom^[30]) and can also stabilise the wavelength of lasers^[31] or measure polarization.^[32]

The disordered pattern produced by speckle has been used in quantum simulations with cold atoms. The randomly-distributed regions of bright and dark light act as an analog of disorder in solid-state systems, and are used to investigate localization phenomena.^[33]

In fluorescence microscopy, a sub-diffraction-limited resolution can be obtained in 2D from saturable/photoconvertible pattern illumination techniques like stimulated emission depletion (STED) microscopy, ground state depletion (GSD) microscopy, and reversible saturable optical fluorescence transitions (RESOLFT). Adapting speckle patterns for use in these applications enables parallel 3D super-resolution imaging.^[34]

Speckle is considered to be a problem in laser based display systems like the Laser TV. Speckle is usually quantified by the speckle contrast. Speckle contrast reduction is essentially the creation of many independent speckle patterns, so that they average out on the retina/detector. This can be achieved by,^[35]

• Angle diversity: illumination from different angles
• Polarization diversity: use of different polarization states
• Wavelength diversity: use of laser sources which differ in wavelength by a small amount

Rotating diffusers—which destroys the spatial coherence of the laser light—can also be used to reduce the speckle. Moving/vibrating screens or fibers may also be solutions.^[36] The Mitsubishi Laser TV appears to use such a screen which requires special care according to their product manual. A more detailed discussion on laser speckle reduction can be found here.^[37]

Synthetic array heterodyne detection was developed to reduce speckle noise in coherent optical imaging and coherent differential absorption LIDAR.

Speckle patterns can also be observed over time instead of space. This is the case of phase sensitive optical time-domain reflectometry, where multiple reflections of a coherent pulse generated at different instants interfere to produce a pseudorandom time-domain signal.^[43]

Speckle interference pattern may be decomposed in the sum of plane waves. There exist a set of points where amplitude of electromagnetic field is exactly zero. Researchers had recognized these points as dislocations of wave trains.^[44] We know these phase dislocations of electromagnetic fields as optical vortices.

There is a circular energy flow around each vortex core. Thus each vortex in the speckle pattern carries optical angular momentum. The angular momentum density is given by:^[45]

${\displaystyle {\begin{aligned}{\vec {\mathbf {L} }}\left({\vec {\mathbf {r} }},t\right)&={\vec {\mathbf {r} }}\times {\vec {\mathbf {S} }}\left({\vec {\mathbf {r} }},t\right)\\{\vec {\mathbf {S} }}\left({\vec {\mathbf {r} }},t\right)&=\epsilon _{0}c^{2}{\vec {\mathbf {E} }}\left({\vec {\mathbf {r} }},t\right)\times {\vec {\mathbf {B} }}\left({\vec {\mathbf {r} }},t\right).\end{aligned}}}$

Typically vortices appear in speckle pattern in pairs. These vortex - antivortex pairs are placed randomly in space. One may show that electromagnetic angular momentum of each vortex pair is close to zero.^[46] In phase conjugating mirrors based on stimulated Brillouin scattering optical vortices excite acoustical vortices.^[47]

Apart from formal decomposition in Fourier series the speckle pattern may be composed for plane waves emitted by tilted regions of the phase plate. This approach significantly simplifies numerical modelling. 3D numerical emulation demonstrates the intertwining of vortices which leads to formation of ropes in optical speckle.^[48]

• Diffusing-wave spectroscopy
• Gaussian noise
• Laser speckle contrast imaging
• Salt and pepper noise