Jennifer E. Ward

Three-dimensional speckle size in generalized optical systems with limiting apertures

Correlation properties of speckle fields at the output of quadratic phase systems with hard square and circular apertures are examined. Using the linear canonical transform and ABCD ray matrix techniques to describe these general optical systems, we first derive analytical formulas for determining axial and lateral speckle sizes. Then using a numerical technique, we extend the analysis so that the correlation properties of nonaxial speckles can also be considered. Using some simple optical systems as examples, we demonstrate how this approach may be conveniently applied. The results of this analysis apply broadly both to the design of metrology systems and to speckle control schemes.

Controlling speckle using lenses and free space

The correlation properties of speckle fields are studied for general paraxial systems. The previous studies on lateral and longitudinal speckle size for the case of free-space propagation (Fresnel transform) are generalized to the case of the linear canonical transform. These results have implications for the control of speckle size, through appropriate design of optical systems, with particular relevance for speckle interferometry.

Metrology and the linear canonical transform

It is shown experimentally that both surface tilt and in-plane translation motion can be independently estimated using the speckle photographic correlation technique by capturing consecutive images in two linear canonical transform domains (using two different quadratic phase systems). A geometric interpretation, based on use of the Wigner distribution function is presented to describe the method and a simple matrix approach, based on the ABCD matrix, is used to quantify it. It is shown that the sensitivity and dynamic range of measurement of both tilt and translation are both variable and depend on the parameters of the ABCD matrix.

Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture

In speckle-based metrology systems, a finite range of possible motion or deformation can be measured. When coherent imaging systems with a single limiting aperture are used in speckle metrology, the observed decorrelation effects that ultimately define this range are described by the well-known Yamaguchi correlation factor. We extend this result to all coherent quadratic phase paraxial optical systems with a single aperture and provide experimental results to support our theoretical conclusions.

Lucky imaging and aperture synthesis with low-redundancy apertures

Lucky imaging, used with some success in astronomical and even horizontal-path imaging, relies on fleeting conditions of the atmosphere that allow momentary improvements in image quality, at least in portions of an image. Aperture synthesis allows a larger aperture and, thus, a higher-resolution imaging system to be synthesized through the superposition of image spatial-frequency components gathered by cooperative combinations of smaller subapertures. A combination of lucky imaging and aperture synthesis strengthens both methods for obtaining improved images through the turbulent atmosphere. We realize the lucky imaging condition appropriate for aperture synthesis imaging for a pair of rectangular subapertures and demonstrate that this condition occurs when the signal energy associated with bandpass spatial-frequency components achieves its maximum value.

An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems

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.

Linear canonical transforms, and speckle based metrology

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 LCTs 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 FRTs (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 LCTs, 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.

Speckle based metrology systems and linear canonical transform photography in mixed domains

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 FRTs (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 Collins 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.

Aperture synthesis imaging through turbulence with the aid of lucky imaging

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.