William T. Rhodes

Resolution limits in practical digital holographic systems

We examine some fundamental theoretical limits on the abil- ity of practical digital holography DH systems to resolve detail in an image. Unlike conventional diffraction-limited imaging systems, where a projected image of the limiting aperture is used to define the system performance, there are at least three major effects that determine the performance of a DH system: i The spacing between adjacent pixels on the CCD, ii an averaging effect introduced by the finite size of these pixels, and iii the finite extent of the camera face itself. Using a theo- retical model, we define a single expression that accounts for all these physical effects. With this model, we explore several different DH record- ing techniques: off-axis and inline, considering both the dc terms, as well as the real and twin images that are features of the holographic record- ing process. Our analysis shows that the imaging operation is shift vari- ant and we demonstrate this using a simple example. We examine how our theoretical model can be used to optimize CCD design for lensless DH capture. We present a series of experimental results to confirm the validity of our theoretical model, demonstrating recovery of super- Nyquist frequencies for the first time.

Analytical and numerical analysis of linear optical systems

The numerical calculation of the Fresnel transform (FST) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel. The development of an efficient algorithm is further complicated by the fact that the output extent of the FST is dependent on the propagation distance. In this paper, we implement a recently proposed technique for efficiently calculating the FST in which we apply the Wigner distribution function and the space bandwidth product to identify suitable sampling rates. This method is shown to be suitable for all propagation distances. Our method can also be applied to describe the effect of a thin lens modeled as a chirp modulation transform (CMT). Combining our results for the FST and the CMT, we numerically calculate the light distribution at the output of both Cai-Wang and Lohmann Type-I optical fractional Fourier transform (OFRT) systems. Analytic solutions for the OFRT of rectangular window and circular apertures are presented. The analytical solutions are compared to experimental data and to numerical results for equivalent cases. Finally the numerical method is applied to examine the effect that apertured lenses, in the OFRT system, have on the output distribution.

Fundamental diffraction limitations in a paraxial 4-f imaging system with coherent and incoherent illumination

In the usual model of an imaging system, only the effects of the aperture stop are considered in determining diffraction-limited system performance. In fact, diffraction at other stops--those associated with different lens elements, for example--can also affect system performance and cause the imaging to be space variant, even in the absence of vignetting in the conventional ray optics sense. For the 4-f imaging system investigated in this paper, the severity of the space variance depends on the relative sizes of the two lens stops and the aperture stops. If the diameters of the lenses are equal, the aperture of the first lens has a greater effect on system performance than does that of the second.

Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis

One of the most important optical signal processing operations is the optical Fourier transform (OFT). Of the arrangements for implementation of the OFT, perhaps the most flexible is that for the scaled optical Fourier transform (SOFT), as it allows control over the scale of the output Fourier transform distribution. By means of an analysis in cylindrical coordinates, we examine some of the practical limits introduced by the use of a thin lens of finite aperture in the implementation of the SOFT. Using simple rules of thumb that are based on an examination of the phase and magnitude deviations from the ideal (infinite-lens) diameter case, we define a volume inside the geometric shadow, which we refer to as a sub-geometric shadow. We then show that inside this sub-geometric shadow errors introduced by diffraction can be quantified. 2006 Elsevier B.V. All rights reserved. OCIS: 070.0070; 070.2590; 050.1940

Time-average Fourier telescopy: a scheme for high-resolution imaging through horizontal-path turbulence

The problem of high-resolution imaging through long horizontal-path ground-level turbulence has gone unsolved since it was first addressed many decades ago. In this paper I describe a method that shows promise for diffraction-limited imaging through ground-level turbulence with large (meters) apertures and at large (kilometers) distances. The key lies in collecting image data in the spatial frequency domain via the method of Fourier telescopy and taking suitable time averages of the magnitude and phase of the Fourier telescopy signal. The method requires active illumination of the target with laser light, and the time averages required will likely be over many tens of seconds if not tens of minutes or more. The scheme will thus not be suitable for time-varying scenes. The basic scheme is described, and principle challenges briefly discussed.

Cross terms of the Wigner distribution function and aliasing in numerical simulations of paraxial optical systems

Sampling a function periodically replicates its spectrum. As a bilinear function of the signal, the associated Wigner distribution function contains cross terms between the replicas. Often neglected, these cross terms affect numerical simulations of paraxial optical systems. We develop expressions for these cross terms and show their effect on an example calculation.

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.

Closure phase and lucky imaging

Since its introduction by Jennison in 1958, the closure-phase method for removing the effects of electrical path-length errors in radio astronomy and of atmospheric turbulence in optical astronomy has been based on the non-redundant-spacing triple interferometer. It is shown that through application of lucky imaging concepts it is possible to relax this condition, making closure-phase methods possible with redundantly spaced interferometer configurations and thereby widening their range of application. In particular, a quadruple-interferometer can, under lucky imaging conditions, be treated as though it were a triple interferometer. The slit-annulus aperture is investigated as a special case.

High-security communication by coherence modulation at the photon-counting level

We show that key-specified interferometer path-length difference modulation (often referred to as coherence modulation), operating in the photon-counting regime with a broadband source, can provide a quantifiably high level of physics-guaranteed security for binary signal transmission. Each signal bit is associated with many photocounts, perhaps numbering in the thousands. Of great importance, the presence of an eavesdropper can be quickly detected. We first review the operation of key-specified coherence modulation at high light levels, illustrating by means of an example its lack of security against attack. We then show, using the same example, that, through the reduction of light intensities to photon-counting levels, a high level of security can be attained. A particular attack on the system is analyzed to demonstrate the quantifiability of the schemes security, and various remaining research issues are discussed. A potential weakness of the scheme lies in a possible vulnerability to light amplification by an attacker.

Projection-slice theorem: a compact notation

The notation normally associated with the projection-slice theorem often presents difficulties for students of Fourier optics and digital image processing. Simple single-line forms of the theorem that are relatively easily interpreted can be obtained for n-dimensional functions by exploiting the convolution theorem and the rotation theorem of Fourier transform theory. The projection-slice theorem is presented in this form for two- and three-dimensional functions; generalization to higher dimensionality is briefly discussed.