Linear canonical transforms and speckle based metrology

Abstract

A Linear Canonical Transform (LCT) is a general integral transform which can be used to describe a whole host of complex paraxial optical systems. It can be shown that Fourier Transform (FT), Fractional Fourier Transform (FRT), Chirp Multiplication Function (CMT), (which is used as a model for a thin lens), and the Fresnel Transform (FST) are all specific cases of LCT's and are particularly important in optics. Using the Collins ABCD matrix formula it is possible to represent the above integral transforms in matrix notation. Furthermore since most bulk optical systems can be built using lenses of different curvatures (CMT) and free space propagation (FST) it becomes straight forward, to describe optical systems using matrix notation, (which is interchangeable with LCT integral notation). Speckle Photography (SP) can be used in the analysis of surface motion in combination with an optical LCT. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of the metrology system. Using a novel correlation technique it is possible to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously. In this paper these ideas are extended to more general LCT's, which allow the consideration of more complicated bulk optical systems. Combined with correlation techniques we experimentally demonstrate our ability to determine both, the magnitude and direction, of tilting (rotation) and translation motion of a surface over a greater range and sensitivity than previous OFRT techniques allowed.