Aperture synthesis allows a number of small apertures to operate cooperatively in the synthesis of a large full aperture
telescope. For earth-based systems, the effects of atmospheric turbulence, which introduces time-varying aberrations,
must somehow be corrected if good imagery is to be obtained. One correction scheme relies on a comparison, in a range
of overlap, of correctly-phased spatial frequency components with new components that are in error by unknown piston
(constant) and tip-tilt (linear) phase terms. Normally this method requires that the subapertures employed in the synthesis
be sufficiently small that phase aberrations beyond piston and tip-tilt be ignorable. Through the exploitation of lucky
imaging conditions, however, larger apertures can be used, with a subsequent increase in resolution and light-gathering
power for the optical system.
The lateral correlation properties of speckle fields have been shown to be useful in aligning multiple optical channels relative to one another. Relative rotational alignment can also be achieved using a sub-sectioning extension of this technique. In this paper, we examine the three dimensional correlation properties of speckle, and by so doing, create a technique that allows for absolute positioning of a single channel free space optical system on the optical axis without the need for markers or gratings.
Complex paraxial optical systems, consisting of multiple lenses and sections of free space propagation, can be described
using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as
the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp
Modulation Transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is
possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Apertures
are necessary in speckle based metrology systems to control the size of the speckle. We examine their effect on LCT
systems and show using the "generalized Yamaguchi correlation factor" that a useful interpretation of the system's behavior, using the LCT, may still be obtained. Furthermore, we experimentally demonstrate our ability to determine
simultaneous tilt and translation motion by capturing two, sequential, mixed doma in images with a single camera. We
also show how localized deformations in an object may be measured using this system.
It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp Modulation Transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. 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 the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin's matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating fields of varying curvature.
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.
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.
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