Distortion measurement of a lithography projection lens based on multichannel grating lateral shearing interferometry.

The optical distortion of the lithographic projection lens can reduce imaging quality and cause overlay errors in lithography, thus preventing the miniaturization of printed patterns. In this paper, we propose a technique to measure the optical distortion of a lithographic projection lens by sensing the wavefront aberrations of the lens. A multichannel dual-grating lateral shearing interferometer is used to measure the wavefront aberrations at several field points in the pupil plane simultaneously. Then, the distortion at these field points is derived according to the proportional relationship between the Z 2 and Z 3 Zernike terms (the tilt terms) and the image position shifts. Without the need for additional devices, our approach can simultaneously retrieve both the wavefront aberrations and the image distortion information. Consequently, it improves not only measurement speed and accuracy but also enables accounting for displacement stage positioning error. Experiments were conducted on a lithographic projection lens with a numerical aperture of 0.57 to verify the feasibility of the proposed method.

Digital filtering of ghost signal in phase measuring deflectometry.

We introduce a method of geometric screen modification to remove ghost reflections commonly observed in deflectometry optical testing. The proposed method modifies the optical layout and illumination source area to bypass the generation of reflected rays from the undesired surface. The layout flexibility of deflectometry allows us to design specific system layouts that avoid the generation of interrupting secondary rays. The proposed method is supported by optical raytrace simulations, and experimental results are demonstrated with convex and concave lens case studies. Finally, the limitations of the digital masking method are discussed.

Moire profilometry through simultaneous dual fringe projection for accurate phase demodulation: a comparative study.

In optical metrology, fringe projection and moire techniques have been widely used to measure the topography of objects. We can combine the advantages of the two techniques by applying a configuration of simultaneous dual projection in the fringe projection technique, which generates a superimposed fringe pattern containing a moire pattern that is phase modulated according to the topography. In this work, we present an analytic and comparative study of three methods to demodulate the phase of the moire pattern: the spatial, spatial-temporal, and temporal methods. Those methods consist of two steps: first, the moire pattern is extracted from the superimposed fringe pattern; next, the phase of the moire pattern is demodulated. The analytical results show that the resulting phase map has double phase sensitivity compared to that of the classical fringe projection technique. Experimental and numeric results prove the feasibility of this technique.

Least-squares gamma estimation in fringe projection profilometry.

This paper introduces a novel, to the best of our knowledge, method to estimate and compensate the nonlinear gamma factor introduced by the optical system in fringe projection profilometry. We propose to determine this factor indirectly by adjusting the least-squares plane to the estimated phase coming from the reference plane. We only require a minimal set of three fringe sinusoidal images to estimate the gamma factor. This value can be used to rectify computational legacy data and also to generate and project the new set of fringe patterns for which we perform the inverse gamma compensation. Experimental results demonstrate the feasibility of the proposed method to estimate and correct the gamma distortion.

Tukey's robust M-estimator for phase demodulation of interferograms with nonuniform shifts.

In this paper, we introduce an iterative scheme for phase demodulation of interferograms with nonuniformly spaced phase shifts. Our proposal consists of two stages: first, the phase map is obtained through a least squares fitting; second, the phase steps are retrieved using a statistical robust estimator. In particular, we use Tukey's biweighted M-estimator because it can cope with both noisy data and outliers in comparison with the ordinary least squares estimator. Furthermore, we provide the frequency description of the algorithm and the phase demodulation allowing us to analyze the procedure and estimation according to the frequency transfer function (FTF) formalism for phase-shifting algorithms. Results show that our method can accurately retrieve the phase map and phase shifts, and it converges by the 10th iteration.

Suppressing ripple distortions and spurious pistons in phase-shifting interferometry.

In this paper, we present new phase-shifting algorithms (PSAs) that suppress the ripple distortions and spurious pistons in phase-shifting interferometry. These phase errors arise when non-uniform phase-shifting interferograms are processed with PSAs that assume uniform phase shifts. By modeling the non-uniform phase shifts as a polynomial of the unperturbed phase-shift value $\omega_0$ω0, we show that the conditions for eliminating the ripple distortion and the spurious piston are associated with the $m$mth derivative of the PSA's frequency transfer function (FTF). Thus, we propose an approach to design robust algorithms based on the FTF formalism and we present four ready-to-apply PSAs formulas. Finally, our conclusions are supported by computer simulations.

Phase measurement of nonuniform phase-shifted interferograms using the frequency transfer function.

In this paper, we propose a phase measurement method for interferograms with nonuniform phase shifts. First, we measure the phase shifts between consecutive interferograms. Second, we use these values to modify the spectrum of the interferogram data. Then, by analyzing this spectrum, we design a suitable phase-shifting algorithm (PSA) using the frequency transfer function formalism. Finally, we test our PSA with experimental data to estimate the surface of an aluminum thin film. Our result is better than those obtained using the Fourier transform method, the principal component analysis method, and the least-squares PSA.

Spectral analysis for the generalized least squares phase-shifting algorithms with harmonic robustness.

We introduce the frequency transfer function (FTF) formalism for generalized least squares phase-shifting algorithms (GLS-PSAs), whose phase shifts are nonuniformly spaced. The GLS-PSA's impulsive response is found by computing the Moore-Penrose pseudoinverse. FTF theory allows analyzing these GLS-PSAs spectrally, as well as easily finding figures of merit such as signal-to-noise ratio (SNR) and harmonic rejection capabilities. We show simulations depicting that the SNR slightly decreases as the harmonic rejection robustness improves.

Phase-stepping algorithms for synchronous demodulation of nonlinear phase-shifted fringes.

In optical metrology synchronous phase-stepping algorithms (PSAs) estimate the measured phase of temporal linear-carrier fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using closed-loop, feedback, optical phase-stepped devices. On the other hand, open-loop phase-stepping devices usually give fringe patterns with nonlinear phase steps. The Fourier spectrum of linear-carrier fringes is composed of Dirac deltas only. In contrast, nonlinear phase-shifted fringes are wideband, spread-spectrum signals. It is well known that using linear-phase reference PSA to demodulate nonlinear phase stepped fringes, one obtains a spurious-piston. The problem with this spurious-piston is that it may wrongly be interpreted as a real thickness in any absolute phase measurement. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with nonlinear phase-stepping interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to synchronously demodulate nonlinear phase-stepped wideband fringes.

Dynamic 3D shape measurement by iterative phase shifting algorithms and colored fringe patterns.

In this work, we propose a novel technique to retrieve 3D shape of dynamic objects by the simultaneous projection of a fringe pattern and a homogeneous white light pattern, both coded in an RGB image. The first one is used to retrieve the phase map by an iterative least-squares method. The second one is used to match object pixels in consecutive images, acquired at various object positions. The proposed method successfully accomplishes the requirement of projecting simultaneously two different patterns. One extracts the object's information while the other retrieves the phase map. Experimental results demonstrate the feasibility of the proposed scheme.