A new method for three-dimensional image formation is proposed in which the distribution of complex amplitude at a plane is measured by phase-shifting interferometry and then Fresnel transformed by a digital computer. The method can reconstruct an arbitrary cross section of a three-dimensional object with higher image quality and a wider viewing angle than from conventional digital holography using an off-axis configuration. Basic principles and experimental verification are described.

Phase-shifting digital holography

We first provide an overview of 3-D shape measurement us- ing various optical methods. Then we focus on structured light tech- niques where various optical configurations, image acquisition tech- niques, data postprocessing and analysis methods and advantages and limitations are presented. Several industrial application examples are presented. Important areas requiring further R&D are discussed. Finally, a comprehensive bibliography on 3-D shape measurement is included, although it is not intended to be exhaustive. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)00101-X)

Overview of three-dimensional shape measurement using optical methods

Digital holography is an emerging field of new paradigm in general imaging applications. We present a review of a subset of the research and development activities in digital holography, with emphasis on microscopy techniques and applications. First, the basic results from the general theory of holography, based on the scalar diffraction theory, are summarized, and a general description of the digital holographic microscopy process is given, including quantitative phase microscopy. Several numerical diffraction methods are described and compared, and a number of representative configurations used in digital holography are described, including off-axis Fresnel, Fourier, image plane, in-line, Gabor, and phase-shifting digital holographies. Then we survey numerical techniques that give rise to unique capabilities of digital holography, including suppression of dc and twin image terms, pixel resolution control, optical phase unwrapping, aberration compensation, and others. A survey is also given of representative application areas, including biomedical microscopy, particle field holography, micrometrology, and holographic tomography, as well as some of the special techniques, such as holography of total internal reflection, optical scanning holography, digital interference holography, and heterodyne holography. The review is intended for students and new researchers interested in developing new techniques and exploring new applications of digital holography.

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Principles and techniques of digital holographic microscopy

We applied phase-shifting digital holography to microscopy by deriving the complex amplitude of light scattered from microscopic three-dimensional objects through a microscope objective by video camera recording, phase-shifting analysis, and computer reconstruction. This method requires no mechanical movement and provides a flexible display and quantitative evaluation of the reconstructed images. A theory of image formation and experimental verification with specimens are described.

Three-dimensional microscopy with phase-shifting digital holography

We develop and test a nonlinear optimization algorithm for solving the problem of phase retrieval with transverse translation diversity, where the diverse far-field intensity measurements are taken after translating the object relative to a known illumination pattern. Analytical expressions for the gradient of a squared-error metric with respect to the object, illumination and translations allow joint optimization of the object and system parameters. This approach achieves superior reconstructions, with respect to a previously reported technique [H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004)], when the system parameters are inaccurately known or in the presence of noise. Applicability of this method for samples that are smaller than the illumination pattern is explored.

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Phase retrieval with transverse translation diversity: a nonlinear optimization approach

We describe several results characterizing the Hubble Space Telescope from measured point spread functions by using phase-retrieval algorithms. The Cramer-Rao lower bounds show that point spread functions taken well out of focus result in smaller errors when aberrations are estimated and that, for those images, photon noise is not a limiting factor. Reconstruction experiments with both simulated and real data show that the calculation of wave-front propagation by the retrieval algorithms must be performed with a multiple-plane propagation rather than a simple fast Fourier transform to ensure the high accuracy required. Pupil reconstruction was performed and indicates a misalignment of the optical axis of a camera relay telescope relative to the main telescope. After we accounted for measured spherical aberration in the relay telescope, our estimate of the conic constant of the primary mirror of the HST was - 1.0144.

Hubble Space Telescope characterized by using phase-retrieval algorithms

A laser radar system for three-dimensional (3-D) lensless imaging is analyzed in theory and experiment. 3-D imaging is accomplished by making use of the relationship between the angular and wavelength dependence of the scattered light and an object's 3-D Fourier transform. The concept is demonstrated by obtaining a 3-D image of an extended object by using a charge-coupled device detector array and an argon-ion laser with a tunable intracavity etalon.

Three-dimensional lensless imaging using laser frequency diversity.

A dual focal length ophthalmic lens is formed from a birefringent material with its fast and slow axes perpendicular to the user's visual axis. The dual focal property arises due to the differing indices of refraction of the birefringent material for light polarized parallel to the fast and slow axes. Light emanating from far objects having one polarization and light emanating from near objects having the opposite polarization are both focused onto the user's retina. Depending upon which object is being viewed, an in-focus and a blurred image appear simultaneously on the user's retina. The ability of the user's eye/brain system to distinguish between the two images provides bifocal action from a single lens. The invention may take the form of a spectacle, contact, or intraocular implant lens.

Bifocal ophthalmic lens constructed from birefringent material

A system and method for three-dimensional imaging utilize a lens to perform a two-dimensional Fourier transform of an interference pattern while focusing the pattern on a two-dimensional detector array which is positioned in the image plane of the lens. This allows immediate previewing of the imaged object for proper positioning. Coherent energy beams are utilized to create a series of interference patterns, or image-plane holograms, each at a different frequency of the source energy beams. Furthermore, at each frequency, the relative phase between an object and a reference energy beam is varied to capture the complex values associated with the interference patterns. After capturing and storing the various interference patterns, a computer performs a one-dimensional Fourier transform, or other simplified processing to generate the three-dimensional image of the object. The image resolution extends to the micron range making this system and method easily adaptable to a variety of three-dimensional inspection applications.

System and method for holographic imaging with discernible image of an object

One algorithm proposed for recovering an object’s Fourier phase from the phase of its bispectrum is the least-squares solution of the overdetermined system of equations that relate the Fourier and bispectrum phases. Although the least-squares technique is robust with respect to noise, it fails when the bispectrum phase is known only modulo 2π. A technique is presented for unwrapping the modulo 2π bispectrum; one can then obtain an accurate least-squares phase solution. Simulations that demonstrate the technique and test sensitivity to noise are given.

Joseph C. Marron

Papers

Hubble Space Telescope characterized by using phase-retrieval algorithms

Three-dimensional lensless imaging using laser frequency diversity.

Bifocal ophthalmic lens constructed from birefringent material

System and method for holographic imaging with discernible image of an object

Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase