Conventional Phase-shifting Algorithms Research Articles

Conjugate complex function coupling strategy of interferometric field: Synthetic-wavelength interferometric fringe extraction in simultaneous dual-wavelength interferometry

Simultaneous dual-wavelength interferometry (SDWI) supplies a better solution to extend the measurement range of traditional single-wavelength interferometry with higher efficiency. To decompose the needed longer beat-frequency synthetic-wavelength (λS) information from the generated moiré fringe pattern in SDWI, a conjugate complex function coupling strategy (CCFC) of interferometric field is proposed in this study. By the Fourier transform and the half-band filter in one side of the spectrum, the positive half spectrum is obtained. The inverse Fourier transform and conjugate operation are performed on the half spectrums to obtain the two conjugate complex functions of the interferometric field. Coupled by the multiplication for the conjugate complex functions, the lower beat-frequency synthetic-wavelength interferometric fringe pattern is derived directly. And if the SDWI is implemented with the π/2 phase shift at λS, the obtained phase-shift interferogram for λS could be demodulated by the conventional phase-shift algorithm. Compared with other spatial-domain Fourier transform demodulation theory, the half-band filter used in CCFC method is appropriate for the different spectral width and carrier spatial frequency. The necessary introduced linear carrier is merely about 0.0947 times of the carrier in the former numerically. And it is only used to pursue the separation between the ±1 order phase lobes instead of the separation between the different wavelength phase lobes. The feasibility and applicability of the CCFC method are verified using simulation and experimental results.

Precise Measurement of the Surface Shape of Silicon Wafer by Using a New Phase-Shifting Algorithm and Wavelength-Tuning Interferometer

In wavelength-tuning interferometry, the surface profile of the optical component is a key evaluation index. However, the systematic errors caused by the coupling error between the higher harmonics and phase shift error are considerable. In this research, a new 10N − 9 phase-shifting algorithm comprising a new polynomial window function and a DFT is developed. A new polynomial window function is developed based on characteristic polynomial theory. The characteristic of the new 10N − 9 algorithm is represented in the frequency domain by Fourier description. The phase error of the new algorithm is also discussed and compared with other phase-shifting algorithms. The surface profile of a silicon wafer was measured by using the 10N − 9 algorithm and a wavelength-tuning interferometer. The repeatability measurement error across 20 experiments was 2.045 nm, which indicates that the new 10N − 9 algorithm outperforms the conventional phase-shifting algorithm.

Surface profilometry of silicon wafers using wavelength-tuned phase-shifting interferometry

Surface profile is one of the key evaluation indexes in the optical industry. With the development of optical technology, wavelength-tuned phase-shifting interferometry has been widely used for measuring the surface profiles of optical components based on the high accuracy of this technique. However, coupling errors caused by higher harmonics and phase-shift errors may lead to systematic errors in the calculated phase. In this research, to suppress coupling errors effectively, we propose an 8N - 7 phase-shifting algorithm consisting of a novel polynomial window function and discrete Fourier transform. By locating eight multiple roots on the characteristic diagram, the polynomial window function of the proposed algorithm is derived based on the theory of characteristic polynomials. The characteristic polynomial of the 8N - 7 algorithm is estimated based on a Fourier representation. Phase errors are discussed and compared to the errors in other algorithms. The results indicate that the 8N - 7 algorithm has the smallest phase error among the compared algorithms and can compensate for up to sixth-order nonlinearity in phase measurements. Finally, the proposed algorithm was applied to profile the surface of a silicon wafer. The results revealed that the standard deviation across 20 experiments was 3.231 nm, which is much smaller than the standard deviations of conventional phase-shifting algorithms.

Phase retrieval from the phase-shift moiré fringe patterns in simultaneous dual-wavelength interferometry

Simultaneous dual-wavelength interferometry (SDWI) could extend the measured range of each single-wavelength interferometry. The moiré fringe generated in SDWI indirectly represents the information of the measured long synthetic-wavelength phase, thus the phase demodulation is rather arduous. To address this issue, we present a method to convert the moiré fringe pattern into a synthetic-wavelength interferogram (moiré to synthetic-wavelength, MTS). After the square of the moiré fringe pattern in the MTS method, the additive moiré pattern is turned into a multiplicative one. And the synthetic-wavelength interferogram could be obtained by a low-pass filtering in spectrum of the multiplicative moiré fringe pattern. Therefore, when the dual-wavelength interferometer is implemented with the π/2 phase shift at a sequence of synthetic-wavelength phase-shift interferograms with π/2 phase shift could be obtained after the MTS method processing on the captured moiré fringe patterns. And then the synthetic-wavelength phase could be retrieved by the conventional phase-shift algorithm. Compared with other methods in SDWI, the proposed MTS approach could reduce the restriction of the phase shift and frame numbers for the adoption of the conventional phase-shift algorithm. Following, numerical simulations are executed to evaluate the performance of the MTS method in processing time, frames of interferograms and the phase shift error compensation. And the necessary linear carrier for MTS method is less than 0.11 times of the traditional dual-wavelength spatial-domain Fourier transform method. Finally, the deviations for MTS method in experiment are 0.97% for a step with the height of 7.8 μm and 1.11% for a Fresnel lens with the step height of 6.2328 μm.

Color-fringe pattern profilometry using an efficient iterative algorithm

Colored fringe pattern approach has been proposed to overcome the limitations of the sequential phase-shifting fringe pattern profilometry for dynamic measurements. However, the use of colored fringe patterns leads to a mayor problem, the color cross-talk, which has been considered to be equivalent to reconstructing the phase map from the fringe patterns with unknown phase shifts. Therefore, phase error is introduced when conventional phase-shifting algorithms with fixed phase shift values are used to retrieve the phase. To overcome the cross-talk issue, we propose the use of a fringe pattern normalization algorithm to achieve the intensity modulation balance among the different color channels and then the use of an advanced iterative algorithm to retrieve the phase. Simulations and experimental results show that the proposed procedure can significantly reduce the influence of the color cross-talk.

Phase-measuring algorithm to suppress inner reflection of transparent parallel plate in wavelength tuning Fizeau interferometer

Optical thickness variation is a fundamental characteristic of transparent optical devices. When measuring this variation by using a wavelength-tuning Fizeau interferometer, the measurement accuracy depends on the phase-shifting algorithm being used. Therefore, the phase-shifting algorithm should compensate for any errors incurred during the measurement, including the phase-shift error, coupling errors, and bias modulation of intensity. Among these errors, however, the coupling errors between the higher harmonics resulting from the inner reflections of the transparent plate and phase-shift error have not previously been considered. This paper presents a derivation of a 19-sample phase-shifting algorithm that can compensate for the miscalibration and 1st-order nonlinearity of the phase shift, coupling errors, and bias modulation of intensity during wavelength tuning. The characteristics of the 19-sample algorithm were estimated with respect to the Fourier representation in the frequency domain. The phase error of measurements performed using the 19-sample algorithm was discussed and compared with that of measurements obtained using other conventional phase-shifting algorithms. Finally, the optical thickness variation of a fused silica parallel plate was obtained using a wavelength-tuning Fizeau interferometer and the 19-sample algorithm. The measurement accuracy was discussed by comparing the ripples in the crosstalk noise with those calculated using other phase-shifting algorithms. The experimental results indicated that the optical thickness variation measurement accuracy for the fused silica plate was approximately 2nm.

Error-compensating phase-shifting algorithm for surface shape measurement of transparent plate using wavelength-tuning Fizeau interferometer

When measuring the surface shape of a transparent sample using wavelength-tuning Fizeau interferometry, the calculated phase is critically determined by not only phase-shift errors, but also by coupling errors between higher harmonics and phase-shift errors. This paper presents the derivation of a 13-sample phase-shifting algorithm that can compensate for miscalibration and first-order nonlinearity of phase shift, coupling errors, and bias modulation of the intensity, and has strong suppression of the second reflective harmonic effect. The characteristics of the 13-sample algorithm are estimated with respect to Fourier representation in the frequency domain. The phase error of measurement performed using the 13-sample algorithm is discussed and compared with those of measurements obtained using other conventional phase-shifting algorithms. Finally, the surface shape of a fused silica wedge plate obtained using a wavelength tuning Fizeau interferometer and the 13-sample algorithm are presented. The experimental results indicate that the surface shape measurement accuracy for a transparent fused silica plate is 3nm. The accuracy of the measurement is discussed by comparing the amplitudes of the crosstalk noise calculated using other conventional algorithms.

Color-fringe pattern profilometry using a generalized phase-shifting algorithm.

In order to overcome the limitations of the sequential phase-shifting fringe pattern profilometry for dynamic measurements, a color-channel-based approach is presented. The proposed technique consists of projecting and acquiring a colored image formed by three sinusoidal phase-shifted patterns. Therefore, by using the conventional three-step phase-shifting algorithm, only one color image is required for phase retrieval each time. However, the use of colored fringe patterns leads to a major problem, the color crosstalk, which introduces phase errors when conventional phase-shifting algorithms with fixed phase-shift values are utilized to retrieve the phase. To overcome the crosstalk issue, we propose the use of a generalized phase-shifting algorithm with arbitrary phase-shift values. The simulations and experimental results show that the proposed algorithm can significantly reduce the influence of the color crosstalk.

Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry

In phase-shifting Fizeau interferometers, the phase-shift error and multiple-beam interference are the most common sources of systematic error affecting high-precision phase measurements. The nonsinusoidal waveforms can be minimized by applying synchronous detection with more than 4-sample. However, when the phase-shift calibration is inaccurate, these algorithms cannot eliminate the effects of nonsinusoidal characteristics. Moreover, when measuring the surface profile of highly reflective samples, the calculated phase is critically determined not only by the decrease in the fringe contrast but also by the coupling error between the harmonics and phase-shift error. In this paper, the phase errors calculated by conventional phase-shifting algorithms were estimated by considering the coupling error. We show that the 4N−3 algorithm, comprising the polynomial window function and the DFT term, has the smallest phase error among the conventional phase-shifting algorithms. The surface profile of the highly reflective silicon wafer was measured using a wavelength-tuning Fizeau interferometer and the 4N−3 algorithm.

An INSPECT Measurement System for Moving Objects

Noncontact optical imaging is frequently used in the inspection and metrology of stationary objects, including in particular the reconstruction of the 3-D surface profile. A technique, known as phase-measuring profilometry, involves projecting a sinusoidal pattern and then inferring the height of various points on the object by measuring the resulting phase changes at the respective locations. However, this method cannot be directly applied to systems involving moving objects, as the translation and the perspective geometry effect manifest as errors in the height calculations. In this paper, we report on an imaging and numerical surface profilometry with error compensation technology (INSPECT) measurement system that is tailored for moving objects. We model the imaging system that considers the nonlinear perspective geometry effect, and simplify to a first-order equation using Taylor series expansion. With this, we generalize the conventional phase shift algorithm, and develop the optimization procedures that can compute the height information effectively. We apply this technology to the INSPECT measurement system in semiconductor manufacturing and show significant improvement in accuracy and robustness.

Blind phase error suppression for color-encoded digital fringe projection profilometry

Color-encoded digital fringe projection profilometry (CDFPP) has the advantage of fast speed, non-contact and full-field testing. It is one of the most important dynamic three-dimensional (3D) profile measurement techniques. However, due to factors such as color cross-talk and gamma distortion of electro-optical devices, phase errors arise when conventional phase-shifting algorithms with fixed phase shift values are utilized to retrieve phases. In this paper, a simple and effective blind phase error suppression approach based on isotropic n-dimensional fringe pattern normalization (INFPN) and carrier squeezing interferometry (CSI) is proposed. It does not require pre-calibration for the gamma and color-coupling coefficients or the phase shift values. Simulation and experimental works show that our proposed approach is able to effectively suppress phase errors and achieve accurate measurement results in CDFPP.

Fourier analysis of two-stage phase-shifting algorithms

Differential phase-shifting algorithms (DPSAs) and sum phase-shifting algorithms (SPSAs) recover directly the phase difference and the phase sum, respectively, encoded in two patterns. These algorithms can be obtained, for instance, by an appropriate combination of phase-shifting algorithms (PSAs), which makes unnecessary the previous calculation and subtraction or addition of each individual optical phase by means of conventional PSAs. A filtering process in the frequency domain is presented that allows us to obtain in a simple and elegant manner a qualitative characterization with a Fourier description of the two-stage phase-shifting evaluation that reveals possible phase shifter miscalibration errors and unexpected harmonics in the signal.

Flatness measurement by a grazing Ronchi test

The Ronchi test with a LCD amplitude sinusoidal grating is used for testing nominally flat surfaces. We prove that it is possible to measure flat surfaces without using a reference element by modifying the common optical Ronchi set up. The Ronchi rulings are computer generated and displayed on the LCD. By displaying various phase-shifted rulings and capturing the corresponding images, the phase is obtained with the conventional phase-shifting algorithms. Theoretical and experimental results are shown.

Sinusoidal liquid crystal display grating in the Ronchi test

The Ronchi test with a LCD amplitude sinusoidal grating is used for testing convergent optical systems. The rulings are computer- generated and displayed on the LCD. By displaying various phase- shifted rulings and capturing the corresponding images, the phase is obtained with the conventional phase-shifting algorithms. For reference, experimental results are compared with those on a commercial Fizeau interferometer. © 2003 Society of Photo-Optical Instrumentation Engineers.

Design and assessment of symmetrical phase-shifting algorithms

Conventional phase-shifting algorithms based on a least-squares estimate use N samples over an incomplete period of the sampled waveform. We introduce a class of phase-shifting algorithms having N + 1 samples symmetrically disposed over one full period of the sampled waveform. Fourier analysis techniques are used to derive these algorithms and modify them to improve their performance in the presence of phase-shift errors. The algorithms can be used in phase measurement systems having periodic, but not necessarily sinusoidal, waveforms.