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

We show that the orbital angular momentum can be used to unveil lattice properties hidden in diffraction patterns of a simple triangular aperture. Depending on the orbital angular momentum of the incident beam, the far field diffraction pattern reveals a truncated optical lattice associated with the illuminated aperture. This effect can be used to measure the topological charge of light beams.

Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Light's Orbital Angular Momentum

The probability distribution p(n, T) of the number of counts n from a photoelectric detector illuminated by coherent light for a time T is studied, by associating photons stochastically with Gaussian random waves. The cumulants of the distribution are derived and it is shown to be of the expected form for a boson assembly in a limited volume of phase space. The distribution depends strongly on the degeneracy of the light beam. It approaches the Poisson form for classical particles at low degeneracies and the distribution characteristic of classical waves at high degeneracies. The analysis leads, incidentally, to an expression for the extent of the unit cell of phase space in the direction of the beam. It is argued that this should be adopted as the measure of coherence length.

https://iopscience.iop.org/article/10.1088/0370-1328/74/3/301/pdf

Fluctuations of Photon Beams: The Distribution of the Photo-Electrons

This paper describes the fundamentals of phase-only liquid crystal on silicon (LCOS) technology, which have not been previously discussed in detail. This technology is widely utilized in high efficiency applications for real-time holography and diffractive optics. The paper begins with a brief introduction on the developmental trajectory of phase-only LCOS technology, followed by the correct selection of liquid crystal (LC) materials and corresponding electro-optic effects in such devices. Attention is focused on the essential requirements of the physical aspects of the LC layer as well as the indispensable parameters for the response time of the device. Furthermore, the basic functionalities embedded in the complementary metal oxide semiconductor (CMOS) silicon backplane for phase-only LCOS devices are illustrated, including two typical addressing schemes. Finally, the application of phase-only LCOS devices in real-time holography will be introduced in association with the use of cutting-edge computer-generated holograms. The capabilities and applications of phase-only liquid crystal on silicon (LCOS) technology are reviewed by scientists in China and the United Kingdom. Zichen Zhang and co-workers from Tsinghua University in Beijing and the University of Cambridge describe how an electronically controlled liquid-crystal pixel array on a silicon backplane can be useful for manipulating the phase of coherent light. Such LCOS phase modulators have many potential applications, including real-time holography, head-up displays, wavelength-selective switches and reconfigurable optical add-drop multiplexers. Commercial devices currently offer array sizes of up to about 1-inch diagonal with pixel pitches in the range 6–20 μm. However, the technology is still immature; the liquid-crystal materials and silicon control backplane need to be optimized in order to realize the potential of such devices.

/pdf/fundamentals-of-phase-only-liquid-crystal-on-silicon-lcos-hysjyx9vfk.pdf

Fundamentals of phase-only liquid crystal on silicon (LCOS) devices

Smart pixels are discussed in terms of the following neurally inspired functions: memory, sensors, intrapixel circuitry, interpixel communication, and optical outputs. The types of liquid crystals and electronic circuits used in VLSI/liquid crystal arrays, and their electrooptic properties are reviewed. Examples of electrically addressed and optically addressed liquid crystal smart pixels, and results from a 128*128 VLSI/FLC SLM are presented. This device has a response time of 150 mu s, a contrast ratio of 8:1 measured on a row containing more than 50 pixels, and a frame rate of 4.7 kHz. Improvements of these devices are discussed in terms of what the silicon technology can yield in the future. A few examples in which the liquid crystal on silicon technology has been applied are presented. >

Smart spatial light modulators using liquid crystals on silicon

The construction of a 50- × 50-pixel spatial light modulator based on an active silicon backplane and using the hybrid field effect in nematic liquid crystals as the light modulating process is described. The design and electrical evaluation of the pixel array, which is fabricated in 1.5-μm nMOS and has an individual memory cell within each pixel, are discussed. The performances of a 16 × 16 prototype SLM and the new 50- × 50-pixel device are compared to provide an indication of progress toward high performance spatial light modulators with onboard pixel memory.

Development of a spatial light modulator: a randomly addressed liquid-crystal-over-nMOS array.

Using a series expansion in Zernike polynomials to express the pupil function of an optical system, a means for computing the Optical Transfer Function has been found which avoids explicit numerica...

A New 'Analytic' Method for Computing the Optical Transfer Function

The development of a ferroelectric liquid-crystal-over-single-crystal-silicon spatial light modulator is described. The reflective SLM has an array of 176 X 176 pixels over a clear aperture of 5.28 mm X 5.28 mm. Prototype devices driven from a specially designed high speed frame store have been operated at frame rates of approximately equals 1 kHz.

A high performance spatial light modulator

An nMOS-addressed liquid-crystal array has been designed, primarily for use as a spatial light modulator in the Fourier plane of a coherent optical processor. Specification of the performance of such filters in an ideal system is considered briefly, and a description is given of the operating parameters of our prototype device. Details are presented of the VLSI design for the integrated circuit, which consists of a square array of 16 × 16 pixels covering a total active area of 3.2 × 3.2 mm2 on a silicon chip. An outline is also given of a simple interface circuit, built from discrete components, which allows the liquid-crystal filter to be controlled through the 8-bit port of a microcomputer. When illuminated with noncoherent light, the device acts as a liquid crystal display with ‘on-board’ memory.

Evaluation of an nMOS VLSI array for an adaptive liquid-crystal spatial light modulator

A simple condition is shown to be necessary and sufficient to suppress edge-ringing in optical imaging. This criterion is valid in the presence of aberrations and apodization, and for all cases of (spatially stationary) partially coherent illumination. The edge-ringing tendencies of an optical system may be assessed through the use of two new performance functions which form a Fourier-transform pair. One of these performance functions is related to the ‘transmission cross-coefficient’ which appears in the theory of partially coherent imaging. In the coherent limit this performance function reduces to the amplitude transfer function (pupil function), and in the incoherent limit it reduces to the Optical Transfer Function (OTF). The use of this performance function for the general assessment of partially coherent imaging systems is suggested.

R.M. Sillitto

Papers

Development of a spatial light modulator: a randomly addressed liquid-crystal-over-nMOS array.

A New 'Analytic' Method for Computing the Optical Transfer Function

A high performance spatial light modulator

Evaluation of an nMOS VLSI array for an adaptive liquid-crystal spatial light modulator

Edge-ringing in Partially Coherent Imaging