18 August 2005 Linear canonical transforms, and speckle based metrology
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The Linear Canonical Transform (LCT) is a general transform which can be used to describe linear lossless quadratic phase systems (QPS). It can be shown that the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT) and the effect of a thin lens or Chirp Modulation Transform (CMT), are all special cases of the more general LCT. Using the Collins formula it is possible to represent these transforms as ABCD matrices. By cascading relevant matrices together, quite complicated bulk optical systems can be described in a compact manner. Digital Speckle Photography (DSP) 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 a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend this analysis to more general LCT systems. We demonstrate that a spherical illuminating wavefront can be conveniently described using matrix notation. We show that by changing the sphericity of wavefront we can change the domain of the LCT system. Hence by illuminating a target with a plane wavefront and then a spherical wavefront, we capture the motion in two separate LCT domains and we are thus in a position to fully determine the motion of a rigid body without a priori knowledge.
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Damien P. Kelly, Damien P. Kelly, Robert F. Patten, Robert F. Patten, Bryan M. Hennelly, Bryan M. Hennelly, Ying Liu, Ying Liu, Jennifer E. Ward, Jennifer E. Ward, John T. Sheridan, John T. Sheridan, "Linear canonical transforms, and speckle based metrology", Proc. SPIE 5879, Recent Developments in Traceable Dimensional Measurements III, 58790O (18 August 2005); doi: 10.1117/12.617465; https://doi.org/10.1117/12.617465

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