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

Mathematical programming

Fast acquisition and high axial resolution are two primary requirements for three-dimensional microscopy. However, they are sometimes conflicting: imaging modalities such as confocal imaging can deliver superior resolution at the expense of sequential acquisition at different axial planes, which is a time-consuming process. Optical scanning holography (OSH) promises to deliver a good trade-off between these two goals. With just a single scan, we can capture the entire three-dimensional volume in a digital hologram; the data can then be processed to obtain the individual sections. An accurate modeling of the imaging system is key to devising an appropriate image reconstruction algorithm, especially for real data where random noise and other imaging imperfections must be taken into account. In this paper we demonstrate sectional image reconstruction by applying an inverse imaging sectioning technique to experimental OSH data of biological specimens and visualizing the sections using the OSA Interactive Science Publishing software.

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Three-dimensional microscopy and sectional image reconstruction using optical scanning holography

Inverse lithography technology (ILT) synthesizes photomasks by solving an inverse imaging problem through optimization of an appropriate functional. Much effort on ILT is dedicated to deriving superior masks at a nominal process condition. However, the lower k1 factor causes the mask to be more sensitive to process variations. Robustness to major process variations, such as focus and dose variations, is desired. In this paper, we consider the focus variation as a stochastic variable, and treat the mask design as a machine learning problem. The stochastic gradient descent approach, which is a useful tool in machine learning, is adopted to train the mask design. Compared with previous work, simulation shows that the proposed algorithm is effective in producing robust masks.

https://iopscience.iop.org/article/10.1088/2040-8978/12/4/045601/pdf

Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis

Optical lithography has enabled the printing of progressively smaller circuit patterns over the years. However, as the feature size shrinks, the lithographic process variation becomes more pronounced. Source-mask optimization (SMO) is a current technology allowing a co-design of the source and the mask for higher resolution imaging. In this paper, we develop a pixelated SMO using inverse imaging, and incorporate the statistical variations explicitly in an optimization framework. Simulation results demonstrate its efficacy in process robustness enhancement.

/pdf/pixelated-source-mask-optimization-for-process-robustness-in-498fazra2f.pdf

Pixelated source mask optimization for process robustness in optical lithography

Inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem. We show how the inverse lithography problem can be addressed as an obstacle reconstruction problem or an extended nonlinear image restoration problem, and then solved by a level set time-dependent model with finite difference schemes. We present explicit detailed formulation of the problem together with the first-order temporal and second-order spatial accurate discretization scheme. Experimental results show the superiority of the proposed level set-based ILT over the mainstream gradient methods.

Level-set-based inverse lithography for photomask synthesis

Level-set based inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem, interpreted with a time-dependent model, and then solved as a partial differential equation with finite difference schemes. This paper focuses on developing level-set based ILT for partially coherent systems, and upon that an expectation-orient optimization framework weighting the cost function by random process condition variables. These include defocus and aberration to enhance robustness of layout patterns against process variations. Results demonstrating the benefits of defocus-aberration-aware level-set based ILT are presented.

Robust level-set-based inverse lithography.

Source mask optimization (SMO) is a useful technique for printing the integrated circuit (IC) on a wafer with increasingly smaller feature size. However, complex SMO algorithms generally lead to undesirably long runtime resulting from an optimization of largely identical regions over the whole mask pattern. In this work, a weighted SMO scheme incorporating both an awareness of the hotspots and robustness against process variations is proposed. We show how optimal solutions are reached with fewer iterations by applying various degrees of correction in the corresponding regions. The proposed method includes identifying the hotspots and combining a weight matrix to the cost function for adjustment and control. Simulation results are compared with the mask optimization (under a fixed source) and conventional SMO to illustrate the performance improvement in terms of pattern fidelity, convergence rate and process window size.

Hotspot-aware fast source and mask optimization

Inverse mask synthesis is achieved by minimizing a cost function on the difference between the output and desired patterns. Such a minimization problem can be solved by a level-set method where the boundary of the pattern is iteratively evolved. However, this evolution is time-consuming in practice and usually converges to a local minimum. The velocity of the boundary evolution and the size of the evolution step, also known as the descent direction and the step size in optimization theory, have a dramatic influence on the convergence properties. This paper focuses on developing a more efficient algorithm with faster convergence and improved performance such as smaller pattern error, lower mean edge placement error, wider defocus band, and higher normalized image log slope. These improvements are accomplished by employing the conjugate gradient of the cost function as the evolution velocity, and by introducing an optimal time step for each iteration of the boundary evolution. The latter is obtained from an extended Euler time range by using a line search method. The authors present simulations demonstrating the efficacy of these two improvements.

/pdf/level-set-based-inverse-lithography-for-mask-synthesis-using-3j8g66d8jf.pdf

Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step

Optical proximity correction (OPC) is one of the most widely used Resolution Enhancement Techniques (RET) in mask designs Conventional OPC is often designed for a set of nominal imaging parameters without giving sufficient attention to the process variations caused by aspherical wavefront leaving the exit pupil of the lithography system As a result, the mask designed may deliver poor performance with process variations In this paper, we first describe how a general point spread function (PSF) with wave aberration can degrade the output pattern quality, and then show how the wave aberration function can be incorporated into an inverse imaging framework for robust input mask pattern design against aberrations A level-set-based time-dependent model can then be applied to solve it with appropriate finite difference schemes The optimal mask gives more robust performance against either one specific type of aberration or a combination of different types of aberrations

/pdf/aberration-aware-robust-mask-design-with-level-set-based-1715swwjhf.pdf

Yijiang Shen

Papers

Level-set-based inverse lithography for photomask synthesis

Robust level-set-based inverse lithography.

Hotspot-aware fast source and mask optimization

Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step

Aberration-aware robust mask design with level-set-based inverse lithography