Objective speckle displacement: an extended theory for the small deformation of shaped objects

This paper describes an extended and improved theory of the displacement of the objective speckle pattern resulting from displacement and/or deformation of a coherently illuminated diffuse object. Using the theory developed by Yamaguchi [Opt. Acta 28, 1359 (1981)], extended expressions are derived that include the influence of surface shape/gradients via the first order approximation of the shape as linear surface gradients. Both the original Yamaguchi expressions and the extended form derived here are shown experimentally to break down as the detector position moves away from the z-axis. As such, improved forms of the expressions are then presented, which remove some of the approximations used by Yamaguchi and can be used to predict the objective speckle displacement over a wide range of detector positions and surface slopes. Finally, these expressions are then verified experimentally for the speckle shifts resulting from object translations.