Optical Aerial Image Calculation Using Binary Fraction Scaling
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Solution Overview
Problem
Conventional Fourier transform methods are time-consuming for optical aerial image calculations, and fast Fourier transform (FFT) cannot be directly applied due to the non-integer wavelength constraints in semiconductor technology.
Innovation Solution
The method involves scaling the pattern distribution by a specific constant, allowing the use of fast Fourier transform (FFT) for accelerated optical aerial image calculations, while maintaining the integrity of the original pattern distribution.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional Fourier transform is used for optical aerial image calculation, then calculation accuracy is maintained, but calculation time is excessively long
Solution Approach 1:
The patent changes the parameter of wavelength representation by expressing it as a binary fraction (ratio of two binary numbers) rather than a decimal value. This parameter transformation enables the use of FFT algorithms while maintaining calculation accuracy, thereby reducing calculation time from days to minutes without sacrificing precision.
2Productivity
If fast Fourier transform is used to accelerate calculation, then calculation speed is improved, but it cannot be directly applied due to wavelength constraints
Solution Approach 1:
The patent transforms the wavelength parameter into a binary fraction format (ratio of two binary numbers), which is compatible with the mathematical requirements of FFT algorithms. This parameter transformation allows FFT to be applied to semiconductor wavelengths (248nm, 193nm, etc.) that were previously incompatible, thereby achieving both high calculation speed and broad adaptability.
Solution Approach 2:
The patent introduces binary fraction representation as an intermediary mathematical form that bridges the gap between conventional wavelength values and FFT algorithm requirements. This intermediary representation allows the system to maintain physical accuracy while enabling efficient computational processing through FFT.
3Productivity
If FFT is applied with non-integer wavelength, then calculation efficiency is achieved, but wavelength must be expressed as exponential multiple of 2
Solution Approach 1:
The patent changes the wavelength parameter from decimal form to binary fraction form (ratio of two binary numbers). This parameter transformation increases calculation efficiency by enabling FFT usage while the binary representation naturally aligns with digital computer architecture, reducing implementation complexity despite the mathematical transformation required.
Data Source
AI summary
A method for calculating optical aerial images. The method includes following steps. A first pattern distribution in a spatial domain is multiplied by a scaling constant to scale the first pattern distribution to generate a second pattern distribution. A fast Fourier transform is performed on the second pattern distribution to generate a first spatial frequency spectrum distribution in a spatial frequency domain. The first spatial frequency spectrum distribution is multiplied by a pupil function to generate a second spatial frequency spectrum distribution. An inverse fast Fourier transform is performed on the second spatial frequency spectrum distribution to generate a first diffraction image distribution in the spatial domain. The first diffraction image distribution is divided by a scaling constant to scale the first diffraction image distribution to generate a second diffraction image distribution.