The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

On the diffraction theory of optical images

Publisher Summary This chapter describes the self-imaging phenomenon and its applications. The self-imaging phenomenon requires a highly spatially coherent illumination. It disappears when the lateral dimensions of the light source are increased. When the source is made spatially periodic and is placed at the proper distance in front of the periodic structure, a fringe pattern is formed in the space behind the structure. The chapter discusses the theoretical and applicational aspects of the self-imaging phenomenon—that is, the property of the Fresnel diffraction field of some objects illuminated by a spatially coherent light beam. The applications of self-imaging are summarized in four main groups—namely, (1) image processing and synthesis, (2) technology of optical elements, (3) optical testing, and (4) optical metrology. The chapter describes the double diffraction systems using spatially incoherent illumination. The first periodic structure plays the role of a periodic source composed of a multiple of mutually incoherent slits. Depending on whether the periods of two periodic structures are equal, the Lau or the generalized Lau effect is discussed. Various applications of incoherent double-grating systems are described in the fields of optical testing, image processing, and optical metrology. After examining some cases of coherent and incoherent illumination, the general issue of spatial periodicities of optical fields and its relevance to the replication of partially coherent fields in space is discussed.

I The Self-Imaging Phenomenon and its Applications

Fractional Talbot images of optical gratings acting as periodic phase objects have been obtained by use of x rays of 0.069-nm wavelength from a third-generation synchrotron radiation source. Quantitative evaluation of the data obtained as a function of defocusing distance provides information on the lateral coherence of the beam as well as on the phase modulation in the object.

Fractional Talbot imaging of phase gratings with hard x rays.

An x-ray hologram was made by means of an x-ray laser and a laser-quality near normal incidence x-ray mirror. The high brightness and large coherence lengths of x-ray lasers now offer the potential for in vitro three-dimensional high-resolution holographic images of dynamically varying biological microstructures.

https://science.sciencemag.org/content/sci/238/4826/517.full.pdf

Demonstration of X-ray Holography with an X-ray Laser.

The passive and hybrid Q-switching and mode-locking of solid-state lasers, dye lasers, semiconductor lasers and gas lasers is reviewed. The dynamics of saturable absorbers and reverse saturable absorbers is illustrated. The nanosecond pulse generation by passive and hybrid Q-switching of low-gain active media is described. The picosecond and femtosecond pulse generation by passive and hybrid mode-locking in low-gain and high-gain active media is analysed. The performance data of passively and hybridly mode-locked cw femtosecond dye lasers are collected. The pulse shortening of ultra-fast pulses with saturable absorbers in intra-cavity and extra-cavity configurations is discussed.

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Passive Q-switching and mode-locking for the generation of nanosecond to femtosecond pulses

The formation and the contrast of Fourier images of a periodic object, especially a sinusoidally transparent object, are investigated under partially coherent illumination. It becomes apparent that the spatial coherence of light affects seriously the contrast of Fourier images and the positions of the Fourier image planes.

Effects of Spatial Coherence on Fourier Imaging of a Periodic Object

The van Cittert-Zernike theorem states that the spatial coherence over a space illuminated by an incoherent extended source is described by the Fourier transform of the intensity distribution over the source. The theorem is usually used in a restricted case of the spatial coherence in a plane parallel to the source plane and illuminated by an incoherent extended source of uniform intensity distribution. In this paper we re-examine the van Cittert-Zernike theorem by reviewing it in an original formulation and extend the theorem to the spatial coherence at any two points of a light field illuminated by an incoherent extended source having a non-uniform intensity distribution.

On the van Cittert-Zernike theorem

The following two problems have been investigated theoretically: systematic treatment of image formation in two-beam Fresnel holography from the viewpoint of spatial coherence, and influence of temporal coherence in two-beam Fraunhofer holography.

Some Effects of Spatial and Temporal Coherence in Holography

Image formation by in-line Fraunhofer holography is theoretically and experimentally treated from a viewpoint of coherence theory. It becomes apparent that the effective size of the hologram is reduced by partially coherent light and the reconstructed image is broadened by a degree of spatial and temporal coherence of the illuminating light. Throughout the paper a method of stationary phase is used for evaluating the propagation of the coherence function.

Hirofumi Fujiwara

Papers

Effects of Spatial Coherence on Fourier Imaging of a Periodic Object

On the van Cittert-Zernike theorem

Some Effects of Spatial and Temporal Coherence in Holography

Some Effects of Spatial and Temporal Coherence on In-line Fraunhofer Holography

A new holographic measurement of frequency sweeping of the output from a Q-switched ruby laser