Application and Optimisation of the Spatial Phase Shifting ...

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2 Statistical Properties of Speckle Patterns
When a rough object is illuminated coherently, e.g. by a laser, the light field scattered back from it
acquires a random, grainy structure. The object can be considered "rough" as soon as the surface height
variations are on the scale of the light's wavelength. The irregular light field extends into space, and at
each spatial point we find a coherent superposition of many scattered elementary waves that all have
random intensities and phases. This produces a speckle pattern whose spatial intensity and phase structure
is random as well. Speckle noise is what makes holographic interferometry and ESPI measurements
inherently more noisy than those of classical interferometry. But the speckle effect is not restricted to
electromagnetic radiation; it has also received some attention in ultrasound research [Bur78, Wag83,
Hon97].
To get an idea of the phenomenon, we will consider the properties of speckle patterns in this chapter.
These are of course treated with the tools of statistics, and a wealth of knowledge has been collected since
the first pioneering studies [All63, Gol65, Low70, McKe74]. We begin with the first-order statistics of
intensity and phase and their gradients, putting some emphasis on the 1-D gradients that play an important
role for SPS. The gradient statistics provide useful facts for changes of the speckle field over distances
well below the coherence length, or speckle size; to get a description of the field for two points that are
arbitrarily far apart, we need the explicit second-order statistics. These are particularly important for SPS.
The discussion is restricted to the so-called fully developed speckle patterns, since these are generated by
the great majority of objects that are not optically smooth; in fact, the scatterers to produce partially
developed speckle patterns have to be specially prepared [Cha79, Tak75, Kad85, Mol90a]; a good general
survey on this topic is [Tak86]. Moreover, we assume the light to be perfectly monochromatic and
polarised. The treatment will be valid for free-space propagation (objective speckles) as well as image
fields (subjective speckles), provided the object´s microstructure is not resolved (see 2.2.1).
2.1 Experimental set-up
Where appropriate, we illustrate the findings by experimental results from a large-objective-speckle
interferometer with spatial phase measurement that was built as shown in Fig. 2.1 [Kun97]. Large
subjective speckles would be rather dark due to the small aperture needed; and also, since most apertures
are polygons, one would obtain anisotropic speckles. Of course, it is possible to design subjective-speckle
interferometers, and experimental findings for image-plane speckles produced by weak scatterers have
been reported in [Kad85].
The basic set-up is of Mach-Zehnder type. In contrast to [Kol99], our geometry should compensate for the
spherical part of the scattered field, so that we measure its speckled part only. This is indispensable if we
are to find out something about phase gradients. The adjustment of the interferometer therefore requires
special care, since the curvatures of the wavefronts should match exactly when they are brought back