A theory of partially coherent imaging is presented. In this theory, a singular matrix P is introduced in a spatial frequency domain. The matrix P can be obtained by stacking pupil functions that are shifted according to the illumination condition. Applying singular-value decomposition to the matrix P generates eigenvalues and eigenfunctions. Using eigenvalues and eigenfunctions, the aerial image can be computed without the transmission cross coefficient (TCC). A notable feature of the matrix P is that the relationship between the matrix P and the TCC matrix T is T=P†P, where † represents the Hermitian conjugate. This suggests that the matrix P can be regarded as a fundamental operator in partially coherent imaging.

Computation theory of partially coherent imaging by stacked pupil shift matrix

Standard simulations of optical projection systems for lithography with scalar or vector methods of Fourier optics make the assumption that the wafer stack consists of homogeneous layers. We introduce a general scheme for the rigorous electromagnetic field (EMF) simulation of lithographic exposures over non-planar wafers. Rigorous EMF simulations are performed with the finite-difference time-domain (FDTD) method. The described method is used to simulate several typical scenarios for lithographic exposures over non-planar wafers. This includes the exposure of resist lines over a poly-Si line on the wafer with orthogonal orientation, the simulation of “classical” notch problems, and the simulation of lithographic exposures over wafers with defects.

Rigorous simulation of exposure over nonplanar wafers

This paper physically compares two different matrix representations of partially coherent imaging with the introduction of matrices E and Z, containing the source, object, and pupil. The matrix E is obtained by extending the Hopkins transmission cross coefficient (TCC) approach such that the pupil function is shifted while the matrix Z is obtained by shifting the object spectrum. The aerial image I can be written as a convex quadratic form I=⟨ϕ|E|ϕ⟩=⟨ϕ|Z|ϕ⟩, where |ϕ⟩ is a column vector representing plane waves. It is shown that rank(Z)≤rank(E)=rank(T)=N, where T is the TCC matrix and N is the number of the point sources for a given unpolarized illumination. Therefore, the matrix Z requires fewer than N eigenfunctions for a complete aerial image formation, while the matrix E or T always requires N eigenfunctions. More importantly, rank(Z) varies depending on the degree of coherence determined by the von Neumann entropy, which is shown to relate to the mutual intensity. For an ideal pinhole as an object, emitting spatially coherent light, only one eigenfunction—i.e., the pupil function—is enough to describe the coherent imaging. In this case, we obtain rank(Z)=1 and the pupil function as the only eigenfunction regardless of the illumination. However, rank(E)=rank(T)=N even when the object is an ideal pinhole. In this sense, aerial image formation with the matrix Z is physically more meaningful than with the matrix E. The matrix Z is decomposed as B†B, where B is a singular matrix, suggesting that the matrix B as well as Z is a principal operator characterizing the degree of coherence of the partially coherent imaging.

Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients

This paper discusses the physical difference between two image formation approaches for partially coherent imaging. In one approach, the pupil function is shifted according to illumination condition as in the transmission cross coefficient (TCC) approach, whereas in the other approach, the object spectrum is shifted. Although the two approaches result in identical images, they are built on distinct physical models. Eigenfunction analysis reveals that the TCC approach is built on an artificial optical model only for image calculation. Therefore, the two approaches are not interchangeable except for image calculation. Such an example is found in calculating the entropy in an imaging system.

Two models for partially coherent imaging

In this thesis, we present applications in optics involving the diffraction theory of light for two advanced technologies. We have used a rigorous vectorial diffraction method to model: (i) the imaging of mask structures in extreme ultraviolet lithography, and (ii) ultrashort pulse propagation through small apertures. We have explained the chronological development of the method, the full derivation to implement the method, and we have also given several educational examples. Therefore, the method is not only a tool to calculate the effects but also to understand the physics behind the results. The first objective is to study the potential and challenges of the optical lithographic technique at the extreme ultraviolet wavelength region (at about Λ = 13.5 nm). We have investigated the application of the phase mask concept at this wavelength range, in order to increase the resolution (decrease the printed size) in the image plane of the projection system. Together with our optimised designs, the phase mask concept is necessary to be applied in the extreme ultraviolet lithography technique, when the desired image resolution becomes comparable to the dimension of the wavelength. In the discussion regarding our second objective, we have demonstrated that small slit apertures illuminated by the two fundamental polarisation modes (transverse electric and transverse magnetic) introduce polarisation-dependent dispersion effect that can be used for interesting applications. When using a double slit aperture, similar to the Young's double-slit experiment, the interference pattern resulting from the diffracted light from both slits can be properly explained. The interference pattern is influenced by several effects, i.e. the waveguiding effect of the slit that is polarisation-dependent, the diffraction by the slit aperture, and the generation of surface-bounded waves on the metal that appears for the transverse magnetic incident wave. The demonstrated high spatial and temporal information can be exploited for more advanced applications in optics, such as microscopy, spectroscopy, femtochemistry, etc.

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Vectorial diffraction of extreme ultraviolet light and ultrashort light pulses

A method based on optimal expansions is described for making an order of magnitude speed up in the analysis of partial coherence effects of scattering in optical lithography, inspection, and alignment. This method expands the incident mutual intensity from the illumination into coherent nonuniform plane waves whose effects can be added incoherently. For three‐dimensional structures, the CPU time is reduced by an order of magnitude over the uniform plane wave approach of Abbe. For two‐dimensional structures, the number of simulations with the decomposition technique has been found to be about three times smaller than with Abbe’s approach. The method has been incorporated into TEMPEST and shown to give accurate results and reduced CPU time in applications of imaging an attenuated phase shift mask, patterning of gates over an active area well in silicon, and inspection of a trench in silicon dioxide, where the CPU time savings are most significant due to large NA’s and high σ’s. Results for an embedded phase...

R. J. Socha

Papers

Propagation effects of partial coherence in optical lithography