Method for the calculation of partially coherent imagery

Kintner, Eric C.

doi:10.1364/ao.17.002747

Cited by 68 publications

(22 citation statements)

References 5 publications

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“…(6) and (7), and the change of variable suggested above, lead directly to the following exact finite sum for the intensity. Denoting   X as the greatest integer less than X:…”

Section: Periodic Objects: Hopkins' Integrationmentioning

confidence: 96%

“…Kintner showed that, for the 1D case where the centers of the pupils are constrained to lie on the horizontal (or vertical) axis, and where σ<1, the TCC geometry takes the form of one of 4 distinct configurations. 6 If we denote the intersection of the offset pupils by D and the source by S , then these configurations are as follows:…”

Section: Generalization Of Kintner's Algorithmmentioning

confidence: 99%

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<title>Exact computation of scalar 2D aerial imagery</title>

Gordon

2002

SPIE Proceedings

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An exact formulation of the problem of imaging a 2D object through a Köhler illumination system is presented; the accurate simulation of a real layout is then not time-limited, but memory-limited. The main idea behind the algorithm is that the boundary of the region that comprise a typical TCC is made up of circular arcs, and therefore the area -which determines the value of the TCC -should be exactly computable in terms of elementary analytical functions. A change to integration around the boundary leads to an expression with minimal dependence on expensive functions such as arctangents and square roots. Numerical comparisons are made for a simple 2D structure.

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“…(6) and (7), and the change of variable suggested above, lead directly to the following exact finite sum for the intensity. Denoting   X as the greatest integer less than X:…”

Section: Periodic Objects: Hopkins' Integrationmentioning

confidence: 96%

Section: Generalization Of Kintner's Algorithmmentioning

confidence: 99%

<title>Exact computation of scalar 2D aerial imagery</title>

Gordon

2002

SPIE Proceedings

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“…2 where the pupil (the element constraining the numerical aperture of the system) of the entire lens system is located in the fourier plane of the mask image Mack (2006); Levinson (2001). This allows us to make abstraction of the lens system and the source and approximate the entire projection process using Fourier optics Kintner (1978). This includes an extended source, or even off-axis illumination.…”

Section: Optical Projection Lithography Simulationmentioning

confidence: 99%

“…-First, a partially coherent modulation transfer function (MTF) of the optical system is calculated from the wavelength, numerical aperture and the shape of the extended source Kintner (1978). The MTF is only calculated for the relevant spatial frequencies and the result is cached for later use.…”

Section: Optical Projection Lithography Simulationmentioning

confidence: 99%

Closed-loop modeling of silicon nanophotonics from design to fabrication and back again

et al. 2008

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We present a method for component-centric modeling of silicon nanophotonics, where a closed optimization loop allows to take the effects of the fabrication process into account during the design of nanophotonic components. This enables black-box component descriptions with functional parameters. Underlying mask layouts of the components can then automatically be optimized for their actual performance, and not just for their geometric layout. To simulate the effect of fabrication, we developed a projection lithography simulator which was included inside the optimization loop. This method was applied to the design of a 1-dimensional distributed Bragg mirror

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“…Kintner 9 showed that, for the 1D case where the centers of the pupils are constrained to lie on the horizontal (or vertical) axis, and where σ 0 < 1, the TCC geometry takes the form of one of 4 distinct configurations. If we denote the intersection of the offset pupils by D and the source by S, then these configurations are as follows:…”

Section: Derivation Of the Geometrical Configurationsmentioning

confidence: 99%

Lithographic image simulation for the 21st century with 19th-century tools

Gordon¹,

Rosenbluth

2004

SPIE Proceedings

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Simulation of lithographic processes in semiconductor manufacturing has gone from a crude learning tool 20 years ago to a critical part of yield enhancement strategy today. Although many disparate models, championed by equally disparate communities, exist to describe various photoresist development phenomena, these communities would all agree that the one piece of the simulation picture that can, and must, be computed accurately is the image intensity in the photoresist.The imaging of a photomask onto a thin-film stack is the only phenomenon in the lithographic process that is described fully by well-known, definitive physical laws. Although many approximations are made in the derivation of the Fourier transform relations between the mask object, the pupil, and the image, these and their impacts are well-understood and need little further investigation.The imaging process in optical lithography is modeled as a partially-coherent, Köhler illumination system. As Hopkins has shown, we can separate the computation into 2 pieces: one that takes information about the illumination source, the projection lens pupil, the resist stack, and the mask size or pitch, and the other that only needs the details of the mask structure. As the latter piece of the calculation can be expressed as a fast Fourier transform, it is the first piece that dominates.This piece involves computation of a potentially large number of complex quantities called Transmission Cross Coefficients (TCCs), which are correlations of the pupil function weighted with the illumination intensity distribution. The advantage of performing the image calculations this way is that the computation of these TCCs represents an up-front cost, not to be repeated if one is only interested in changing the mask features, which is the case in Model-Based Optical Proximity Correction (MBOPC).1 The down side, however, is that the number of these expensive double integrals that must be performed increases as the square of the mask unit cell area; this number can cause even the fastest computers to balk if one needs to study medium-or long-range effects. One can reduce this computational burden by approximating with a smaller area, but accuracy is a concern, especially when building a model that will purportedly represent a manufacturing process. This work will review the current methodologies used to simulate the intensity distribution in air above the resist and address the above problems. More to the point, a methodology has been developed to eliminate the expensive numerical integrations in the TCC calculations, as the resulting integrals in many cases of interest can be either evaluated analytically, or replaced by analytical functions accurate to within machine precision. With the burden of computing these numbers lightened, more accurate representations of the image field can be realized, and better overall models are then possible.

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Method for the calculation of partially coherent imagery

Cited by 68 publications

References 5 publications

<title>Exact computation of scalar 2D aerial imagery</title>

<title>Exact computation of scalar 2D aerial imagery</title>

Closed-loop modeling of silicon nanophotonics from design to fabrication and back again

Lithographic image simulation for the 21st century with 19th-century tools

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