Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system

Abstract

Taking the classical Ignatowsky/Richards and Wolf formulas as our starting point, we present expressions for the electric field components in the focal region in the case of a high-numerical-aperture optical system. The transmission function, the aberrations, and the spatially varying state of polarization of the wave exiting the optical system are represented in terms of a Zernike polynomial expansion over the exit pupil of the system; a set of generally complex coefficients is needed for a full description of the field in the exit pupil. The field components in the focal region are obtained by means of the evaluation of a set of basic integrals that all allow an analytic treatment; the expressions for the field components show an explicit dependence on the complex coefficients that characterize the optical system. The electric energy density and the power flow in the aberrated three-dimensional distribution in the focal region are obtained with the expressions for the electric and magnetic field components. Some examples of aberrated focal distributions are presented, and some basic characteristics are discussed.

On the computation of the nijboer-zernike aberration integrals at arbitrary defocus

“…a result that was announced in [6], formula (10), and proved in [8], Appendix A. Accordingly, there holds…”

“…In [8], the Nijboer-Zernike approach has been further extended so as to include aberrated optical systems with high numerical aperture using a vectorial diffraction formalismà la Ignatowskyi/Richards-and-Wolf. The basic integrals to be considered can then be expressed in terms of…”

“…As a rule of thumb one should include in the infinite series over l some 3f terms to get sufficiently accurate results. The approach has been extended further in [8] so as to cover the cases of high numerical aperture which requires computation of field components as well as inclusion of the radiometric effect and the state of polarization. Application of the method in a lithographic context is considered in [9]- [10], where the inverse problem of estimating the coefficients β nm from measurements of the (intensity) point-spread function in the focal region is solved.…”

“…Thus we insert Bauer's formula (8) into the integral at the right-hand side of (2) and interchange integral and summation. Then one is faced with the problem of writing products R 0 2k R m n as a linear combination of Zernike polynomials with upper index m so as to be able to apply (11).…”

Through-Focus Point-Spread Function Evaluation for Lens Metrology using the Extended Nijboer-Zernike Theory

“…Recent research has focused on the incorporation of high-NA vector diffraction into the ENZ-theory [8], on the effects of image blurring due to latent resist image diffusion during the post-exposure bake and on the influence of lateral smear by mechanical vibrations [9]. All these effects tend to obscure the real lens contribution to image degradation and they have to be taken into account in the retrieval process.…”