Introduction:

The self interference of many random coherent waves scattered from a rough object surface or propagated through a medium of random refractive index fluctuations results in a granular structure known as speckle pattern. This physical phenomenon has very close parallels in other branches of physics, for example, coherent imaging, radar astronomy, synthetic aperture radar, acoustical imagery etc. The grainy noise due to the high coherence of the laser beam was found to set a fundamental limitation in high resolution hologram microscopy and coherent imagery. Therefore, techniques were devised to reducing the graininess in the image either by introducing small vibrations while recording or by making the source partially coherent. The turning point occured when it was realised that speckles could be used as a random pattern that carried information about the object surface. Therefore, speckle has the rare distinction of being both a nuisance and a useful measuring tool.

In the last decade with the emergence of better measurement capabilities, newer recording mediums(as an example photorefractive medium) and the development of fast computers and interfaces have initiated numerous developments and innovations in the field of speckles. The surge in the work on speckle metrology has resulted in books and several review articles [1-10]. Some of the books that have appeared in the recent past cover the latest developments in speckle methods and applications: Holography and Speckle Interferometry, Speckle Metrology edited by Sirohi, Optical shop testing, edited by Malacara, and Optical Metrology by Gasvik.

The attractiveness of speckle methods lies in the ability to measure large deformations, reduce sensitivity both in speckle photography and speckle interferometry, variable sensitivity to in-plane and out-of-plane displacements, displacement derivatives, and shape measurement. The change introduced can be physical change in the shape of the object or the change in the refractive index.

Some statistical properties:

As a general case we discuss here the statistics of laser speckles to give us a better understanding of some properties of speckle pattern. The first order statistics deals with the probability density function of irradiance at a single point for an ensemble average of scatterers. The amplitude of the light wave at a given point of observation is considered to have contributions from different scattering regions of the surface. This pattern is identical to the classical pattern of random walk in a complex plane. It is assumed that the scattering regions are statistically independent of each other and the phases are uniformly distributed between and -. It has been shown that the irradiance at a point for linearly polarized Gaussian speckle follows the negative exponential statistics. The most probable brightness of the speckle is zero. The contrast of the speckle pattern is a ratio of the variance to the average irradiance. Using this definition, it can be easily shown that the contrast of a polarized speckle is always unity. Speckle patterns can be added both on an amplitude and intensity basis. In amplitude addition, there is no alteration in the statistics except for a scaling factor. Addition on the intensity basis changes the statistics and the most probable brightness of the speckle is not zero. The single point statistics as dealt is insufficient to describe other properties such as coarseness of the spatial structure of the speckle pattern. Therefore, one studies the second order statistics.

The speckle pattern formed due to self interference among scattered waves propagating in space is termed objective speckle. Speckle pattern observed in the image of a diffuse object illuminated by a coherent beam is termed subjective speckle pattern.. Speckle work in partial coherent light is important in stellar speckle interferometry and in reduction of speckle noise [1]. Changes in the speckle pattern can be related to the changes in the object surface either using objective (free-space) or subjective (image-plane) speckles. The average size of a speckle, if circular area of illumination or circular aperture is used in the imaging system.

In an imaging system, it is assumed that the size of the speckle incident on the lens pupil is extremely small when compared to the diameter of the lens. So the lens surface is treated as a diffuse surface. The autocorrelation and power spectral density are calculated as such by applying the lens parameters. The autocorrelation is independent of any aberration present in the imaging system. Aberrations only effect the phase of the intensity incident at the scattering spot.

References

1. Dainty, J. C, "Introduction", in Laser Speckle and Related Phenomenon, Ed. Dainty, J. C., Second edition, Springer Verlag, 1984.

2. Stetson, K. A., "A review of speckle photography and speckle interferometry", Opt. Eng., Vol. 14, 1975, 482-489.

3.Ennos, A. E., "Speckle interferometry", Prog. Optics, Vol. 16,1978, 233-288.

4. Francon, M., Laser Speckle and Application to Optics, Academic Press, New York, 1979.11. Francon, M., Laser Speckle and Application to Optics, Academic Press, New York, 1979.

5. R.K. Erf. Speckle Metrology, Academic Press, New York, 1978.

6. Vest, C. M., Holographic Interferometry, John Wiley and Sons, New York, 1979.

7. Jones, R., and Wykes, C., Holographic and Speckle Interferometry, Cambridge University Press, London 1983.

8. Sirohi, R. S., Selected Papers on Speckle Metrology, SPIE, Milestone Ser. MS 35, SPIE Optical Engineering Press, Washington, D.C., 1991.

9. Sirohi, R. S., Speckle Metrology, Marcel Dekker, New York, 1993.

10. Joenathan, C., "Speckle Photography, Shearography, and ESPI", in Optical methods for testing , ed. P. Rastogi, R. Tech, London, 1997.