W. Thomas Cathey

Extended depth of field through wave-front coding

We designed an optical-digital system that delivers near-diffraction-limited imaging performance with a large depth of field. This system is the standard incoherent optical system modified by a phase mask with digital processing of the resulting intermediate image. The phase mask alters or codes the received incoherent wave front in such a way that the point-spread function and the optical transfer function do not change appreciably as a function of misfocus. Focus-independent digital filtering of the intermediate image is used to produce a combined optical-digital system that has a nearly diffraction limited point-spread function. This high-resolution extended depth of field is obtained through the expense of an increased dynamic range of the incoherent system. We use both the ambiguity function and the stationary-phase method to design these phase masks.

New paradigm for imaging systems

We describe a new paradigm for designing hybrid imaging systems. These imaging systems use optics with a special aspheric surface to code the image so that the point-spread function or the modulation transfer function has specified characteristics. Signal processing then decodes the detected image. The coding can be done so that the depth of focus can be extended. This allows the manufacturing tolerance to be reduced, focus-related aberrations to be controlled, and imaging systems to be constructed with only one optical element plus some signal processing.

Extended depth of field and aberration control for inexpensive digital microscope systems.

We present a new application and current results for extending depth of field using wave front coding. A cubic phase plate is used to code wave fronts in microscopy resulting in extended depths of field and inexpensive chromatic aberration control. A review of the theory behind cubic phase plate extended depth of field systems is given along with the challenges that are face when applying the theory to microscopy. Current results from the new extended depth of field microscope systems are shown.

Phase plate to extend the depth of field of incoherent hybrid imaging systems

A hybrid imaging system combines a modified optical imaging system and a digital postprocessing step. We describe a spatial-domain method for designing a pupil phase plate to extend the depth of field of an incoherent hybrid imaging system with a rectangular aperture. We use this method to obtain a pupil phase plate to extend the depth of field, which we refer to as a logarithmic phase plate. Introducing a logarithmic phase plate at the exit pupil of a simulated diffraction-limited system and digitally processing the detectors output extend the depth of field by an order of magnitude more than the Hopkins defocus criterion. We also examine the effect of using a charge-coupled device optical detector, instead of an ideal optical detector, on the extension of the depth of field. Finally, we compare the performance of the logarithmic phase plate with that of a cubic phase plate in extending the depth of field of a hybrid imaging system with a rectangular aperture.

Passive ranging through wave-front coding: information and application

Passive-ranging systems based on wave-front coding are introduced. These single-aperture hybrid optical-digital systems are analyzed by use of linear models and the Fisher information matrix. Two schemes for passive ranging by use of a single aperture and a single image are investigated: (i) estimating the range to an object and (ii) detecting objects over a set of ranges. Theoretical limitations on estimator-error variances are given by use of the Cramer-Rao bounds. Evaluations show that range estimates with less than 0.1% error can be obtained from a single wave-front coded image. An experimental system was also built, and example results are given.

Aspheric optical elements for extended depth-of-field imaging

We provide experimental verification of the performance of an optical-digital imaging system that delivers near diffraction limited imaging over a wide depth of field. A custom aspheric optical element is used to modify an incoherent optical system so that the generated optical image is nearly independent of misfocus-induced blur. The resulting image, called an intermediate image, is not spatially diffraction limited. Digital processing of the intermediate image produces a final image that forms a close approximation to the diffraction limited image. The combined effect of the optical-digital system is to image objects independently of focus or range, that is, the system has an extended depth of field.

Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems

We describe a logarithmic phase filter to extend the depth of field of incoherent optical hybrid imaging systems with a rectangular aperture. By introducing this filter at the systems exit pupil and digitally processing the detectors output, we were able to extend the depth of field by an order of magnitude more than the Hopkins defocus criterion.

Aberration invariant optical/digital incoherent systems

We have developed a fundamental technique for control of important known and unknown lens aberrations. Control of lens aberrations through traditional means is very difficult in high-performance optical systems. Minimizing aberrations caused by deterministic design errors as well as statistical fabrication errors has often led to costly systems and fabrication techniques. By employing a special-purpose optical phase mask and digital signal processing we can form imaging systems that are invariant, or substantially insensitive, to a number of important lens aberrations.

Effect of detector noise in incoherent hybrid imaging systems

Hybrid imaging systems involve the joint design of an optical image-gathering module and digital processing algorithms to obtain a required final image. They have the potential to achieve imaging performance hitherto unobtainable by conventional imaging techniques. A reduction in the signal-to-noise ratio of the final image is one of their main disadvantages when one is considering linear signal processing. We analyze the effect of additive white noise at the detector on the performance of hybrid imaging systems under quasi-monochromatic incoherent illumination. We also show numerical results and computer-simulated images for an extended depth-of-field hybrid system.

Matrix description of near-field diffraction and the fractional Fourier transform

We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input. We show that for an input with N×N pixels, in an area d×d, it is necessary to evaluate the Fresnel diffraction pattern at distances z⩾d2/λN. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special phase matrix with fractional parameter ϵ. If the phase matrix is evaluated at ϵ=1, we find the discrete Fourier transform matrix.