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

The coefficient of thermal expansion CTE is an important mechanical property for thin film materials. There are several problems that arise from the thermal expansion effect; for example, the mismatch of thermal expansion between the thin films and the underlying substrate may lead to residual stresses in the thin films. On the other hand, the thermal expansion effect can be exploited to drive microactuators. The CTEs of Al and Ti thin films were determined in the present study using the bilayer microcatilever technique. The contribution of this paper is to demonstrate the variation of the thin film CTE with the film thickness. The CTE of the Al thin film changes from 18.23= 10 y6 r8C to 29.97= 10 y6 r8C, when the film thickness increases from 0.3 to 1.7 mm. The CTE of the Ti thin film changes from 21.21= 10 y6 r8C to 9.04 = 10 y6 r8C, when the film thickness increases from 0.1 to 0.3 mm. Further, the concept that thin film CTE may be influenced by the defects in thin film materials is proposed. Thus, a desired thin film CTE can be obtained by tuning the film thickness as well as the deposition conditions. q 2000 Elsevier Science S.A. All rights reserved.

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On the thermal expansion coefficients of thin films

Partially coherent beams with a prescribed phase, state of polarization, and degree of coherence display many extraordinary propagation properties and are preferred in many applications, such as particle trapping, free-space optical communications, remote sensing, optical imaging, material thermal processing, image transformation, and optical encryption. In this review, we begin with describing conventional Gaussian Schell-model beams and partially coherent beams with prescribed phases, states of polarization, and degrees of coherence. Then we introduce theoretical models for various partially coherent beams and the methods for generating them. Finally, a brief summary is presented.

Generation of Partially Coherent Beams

Because of the quadratic relation between the optical field and intensity, an inherent non-linearity exists in almost all optical systems. A class of non-linear transformations which we call bilinear (quadratic with memory) is defined; its properties and relevance to optical imaging systems are discussed. For space-invariant systems, a generalized transfer function is defined which characterizes the bilinear system completely. We also examine the approximate linearization of bilinear systems for low-contrast images, and the propagation of noise through such systems. Special emphasis is given to the partially coherent system which is but a special case of this general bilinear system.

Optical Bilinear Transformations: General Properties

A theoretical model of the optical microscope based on the theory of partial coherence is used to predict the image profiles of lines on IC photomasks and assess factors contributing to measurement errors for different conditions of microscope operation. A comparison of experimental and theoretical image profiles is given, showing good agreement with theory for a 0.9 N.A. and linewidths as small as 0.5 microm. The primary sources of differences appear to be edge quality and accuracy of focus. The theory indicates that for well-corrected microscope optical systems, spectrally filtered to eliminate longitudinal chromatic aberration and chromatic difference of magnification, accurate determinations of linewidth may be made from the image profile using a threshold of 25% of maximum intensity (corrected for finite background transmittance in the opaque region). The correspondence between the edge location and the 25% threshold appears to be nearly invariant with small amounts of defocus and spherical aberration as well as variation in the numerical aperture of the condenser.

Linewidth measurement with an optical microscope: the effect of operating conditions on the image profile.

The theoretical irradiance is presented for the images of sharp-edged objects as a function of the coherence of the illumination of the object. These distributions are presented for optical systems possessing both square apertures and circular apertures. The irradiance for degraded edged objects is also determined for optical systems with square apertures. The influence of the degree of coherence upon contrast and edge gradient is considered in detail. Experimental results are given which show good agreement with the theoretical irradiance. In all cases the systems are assumed free from aberration and the quasimonochromatic assumption is employed.

Investigations of the Influence of the Degree of Coherence upon Images of Edge Objects

J L Harris has indicated that a general result from statistical decision theory can be used to derive a theoretical expression for predicting resolving power We show that by appropriate choice of tasks, it is possible to obtain, from this same result, other measures which have previously been employed as image-quality criteria Our results indicate that decision theory provides a common basis for deriving quality criteria for diverse tasks Prediction of image quality for object identification is also discussed

Theoretical Prediction of Image Quality

The effect of employing coherent illumination (such as a laser source in scanning optical systems) upon image content is investigated with particular reference to the microdensitometer system. The method of investigation involves the application of coherence theory to the imaging process in systems where the small-angle approximations can be made and the optics of the system are diffraction limited. The output of the system is determined for a sharp-edged object as the area illuminated in the object plane or the width of the illuminating slit is varied. A system configuration is found in which the edge image does not exhibit the usual ringing or diffraction fringes associated with coherent illumination.

Images of Coherently Illuminated Edged Objects Formed by Scanning Optical Systems

This paper presents the results of a design and experimental evaluation of two potential diffuse efflux collectors for microdensitometers designed to obtain linearity at high sample spatial frequencies. Use of a large numerical aperture condenser and an electroformed reflector were examined. The phase edge response, a generalized Collier's Q-factor, and susceptibility to stray light noise were measured. Either collector offers improved linearity, however, better rejection of stray light noise will be required for their practical use.

Experimental Evaluation of Efflux Optics for Linear Microdensitometry

Role of Object Contrast in Partially Coherent Image Formation

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