Peter Dirksen

Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.

Novel aberration monitor for optical lithography

The aberration monitor allows independent determination of spherical, coma, astigmatism and three point in a single experiment using existing equipment. The monitor consists of a circular phase object, with a diameter of approximately (lambda) /NA and a phase depth of (lambda) /2. Due to the relative large diameter, the image prints as a narrow ring into the resist. Without aberrations its contours are concentric circles. Aberrations deform the ring in a characteristic way. A detailed analysis of the ring shape through focus identifies the aberrations of the projection lens. A linear aberration model is compared with simulations. Experimental results of various aberrations are shown and ar correlated to line width measurements and interferometric lens data.

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

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.

Assessment of optical systems by means of point-spread functions

Publisher Summary This chapter presents the computation of the point-spread function of optical imaging systems and the characterization of these systems by means of the measured three-dimensional structure of the point-spread function. The point-spread function, accessible in the optical domain only in terms of the energy density or the energy flow, is a nonlinear function of the basic electromagnetic field components in the focal region. That is why the reconstruction of the amplitude and phase of the optical far-field distribution that produced a particular intensity point-spread function is a nonlinear procedure that does not necessarily have a unique solution. Since the 1970s, the quality of optical imaging systems (telescopes, microscope objectives, high-quality projection lenses for optical lithography, space observation cameras) has been pushed to the extreme limits. At this level of perfection, a detailed analysis of the optical point-spread function is necessary to understand the image formation by these instruments, especially when they operate at high numerical aperture. In terms of imaging defects, it allowed to suppose that the wavefront aberration of such instruments is not substantially larger than the wavelength λ of the light. In most cases, the aberration even has to be reduced to a minute fraction of the wavelength of the light to satisfy the extreme specifications of these imaging systems. The past work on point-spread function analysis and its application to the assessment of imaging systems is presented in the chapter. This includes discussions on: the theory of point-spread function formation, energy density and power flow in the focal region, quality assessment by inverse problem solution, and quality assessment using the extended Nijboer–Zernike diffraction theory.

Impact of high order aberrations on the performance of the aberration monitor

The aberration ring test is used to determine the low and high order lens aberrations. The method is based on two key elements: the linear response of ART to aberrations and the use of multiple imaging conditions. Once the model parameters are determined by means of simulations, the Zernike coefficients are solved from a set of linear equations. The Zernike coefficients thus obtained are correlated to interferometric lens data and to line width measurements.

Aberration retrieval using the extended Nijboer-Zernike approach

We give the proof of principle of a new experimental method to determine the aberrations of an optical system in the field. The measurement is based on the observation of the intensity point-spread function of the lens. To analyze and interpret the measurement, use is made of an analytical method, the so-called extended Nijboer-Zernike approach. The new method is applicable to lithographic projection lenses, but also to EUV mirror systems or microscopes such as the objective lens of an optical mask inspection tool. Phase retrieval is demonstrated both analytically and experimentally. The extension of the method to the case of a medium-to-large hole sized test object is presented. Theory and experimental results are given. In addition we present the extension to the case of aberrations comprising both phase and amplitude errors.

Concise formula for the Zernike coefficients of scaled pupils

Modern steppers and scanners have a projection lens whose numerical aperture (NA) can be varied so as to optimize the image performance for certain lithographic features. Thus a variable fraction of the aberrations is actually involved in the imaging process. In this letter, we present a concise formula for the NA scaling of the Zernike coefficients. In addition, we apply our results to the Strehl ratio.

Zernike representation and Strehl ratio of optical systems with variable numerical aperture

We consider optical systems with variable numerical aperture (NA) on the level of the Zernike coefficients of the correspondingly scalable pupil function. We thus present formulas for the Zernike coefficients and their first two derivatives as a function of the scaling factor ε ≤ 1, and we apply this to the Strehl ratio and its derivatives of NA-reduced optical systems. The formulas for the Zernike coefficients of NA-reduced optical systems are also useful for the forward calculation of point-spread functions and aberration retrieval within the Extended Nijboer–Zernike (ENZ) formalism for optical systems with reduced NA or systems that have a central obstruction. Thus, we retrieve a Gaussian, comatic pupil function on an annular set from the intensity point-spread function in the focal region under high-NA conditions.

Latent image metrology for production wafer steppers

A new latent image metrology technique is discussed that determines best focus with a precision of (sigma) equals 20 nm. This technique uses the existing alignment system of an ASM-L wafer stepper and requires no hardware or software modifications. The user just needs a standard chrome reticle. It can operate for machine setup at the factory, but also in-process for fully automatic self calibration of focus and tilt. A typical measurement takes a few minutes.

Improved wafer stepper alignment performance using an enhanced phase grating alignment system

Processes such as chemical mechanical polishing and spin coating can result in the asymmetric deformation of alignment marks. In this paper, the effects of such asymmetric mark deformations on the accuracy of the stepper alignment system are investigate. An advanced phase grating alignment system is presented which is more robust against the above mentioned process-induced alignment deviations. The potential of the new alignment system will be illustrated with result of both numerical simulations and experimental measurements. Various process modules that are known to cause mark deformations have been investigated.