Lithographic Pupil Plane Sensitivity Map for Aberration Quantification
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Solution Overview
Problem
Current methods for quantifying aberrations in lithographic apparatuses suffer from truncation errors due to the finite number of Zernike orders considered, particularly for highly localized illumination modes, leading to inaccuracies in determining lithographic effects such as offset and focus errors.
Innovation Solution
A method is introduced to determine a discrete, two-dimensional sensitivity map in the pupil plane, allowing for an all-orders analytical calculation that factors aberrations into two parts: the sensitivity map and the wavefront aberration map, eliminating truncation errors by representing the lithographic effect as an inner product of these maps.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If a finite number of Zernike orders are used to quantify aberrations, then the calculation complexity is reduced, but truncation errors occur leading to inaccurate lithographic effect measurements
Solution Approach 1:
The patent transforms the traditional one-dimensional Zernike polynomial expansion approach into a two-dimensional sensitivity map representation in the pupil plane. This dimensional transformation allows all Zernike orders to be simultaneously represented without truncation, as the sensitivity map captures the complete spatial frequency information of the optical system's response to aberrations.
Solution Approach 2:
The sensitivity map is pre-calculated and stored in the pupil plane before actual aberration measurement. This preliminary action enables the decoupling of system characteristics (sensitivity map) from actual aberration conditions, allowing rapid and accurate computation of lithographic effects without repeatedly performing complex integrations during measurement.
2Manufacturing precision
If highly localized illumination modes are used, then lithographic resolution is improved, but sensitivity to higher-order Zernike aberrations increases causing significant truncation errors
Solution Approach 1:
By representing both the illumination mode and sensitivity in the two-dimensional pupil plane, the patent captures the complete spatial distribution of sensitivity to all Zernike orders. This allows highly localized illumination modes to be accurately characterized without truncation, as the sensitivity map inherently accounts for all spatial frequencies excited by such illumination.
Solution Approach 2:
The patent changes the representation parameters from truncated Zernike coefficients to a continuous sensitivity map in the pupil plane. This parameter transformation allows the system to handle highly localized illumination modes that excite high spatial frequencies, as the sensitivity map preserves information about sensitivity to all orders of aberrations regardless of illumination localization.
3Measurement precision
If an all-orders analytical calculation is performed, then truncation errors are eliminated, but computational requirements increase significantly
Solution Approach 1:
The sensitivity map is pre-calculated and stored in the pupil plane before actual aberration measurement. This preliminary computation separates the system-specific calculations (done once) from the measurement-specific calculations (done rapidly for each aberration condition). The pre-computed sensitivity map enables fast evaluation of lithographic effects for any aberration state without repeating the full analytical integration.
Solution Approach 2:
The patent creates a computational model (sensitivity map) that copies the essential characteristics of the optical system's response to aberrations. This model allows rapid prediction of lithographic effects without performing the full analytical calculation each time, as the sensitivity map encapsulates the system's behavior for all possible aberration conditions.
Data Source
AI summary
A method for quantifying the effect of pupil function variations on a lithographic effect within a lithographic apparatus is disclosed. The method comprises: determining a discrete, two-dimensional sensitivity map in a pupil plane of the lithographic apparatus, wherein the lithographic effect is given by the inner product of said sensitivity map with a discrete, two-dimensional pupil function variation map of a radiation beam in the pupil plane. The pupil plane of a lithographic apparatus generally refers to the exit pupil of a projection system of the lithographic apparatus. Pupil function variations may comprise: relative phase variations within the pupil plane and/or relative intensity variations within the pupil plane.