In optical interferometry, phase-shifting algorithms can calculate the phase maps of the geometrical properties from measured samples with high accuracy. The phase-shifting algorithms require several interferograms captured from dynamic adjustable interferometers, with shifting occurring at uniform intervals. It is crucial to maintain the uniform shift intervals in phase-shifting algorithms because nonuniformities in intervals cause nonlinearities in the system and degrade the accuracy of measurement. However, experimental conditions inevitably introduce errors into dynamic adjustments due to environmental uncertainties. To address these drawbacks, we propose a novel method, shifted-interferogram generator, that eliminates the need for dynamic adjustments by utilizing a static optical interferometer with a single frame of interferogram. The proposed interferogram shifter involves capturing a single frame of an interferogram and employing deep learning to generate interferograms with ideal shifting intervals. For assessment of the proposed method, a single interferogram indicating surface shape of an optical flat was captured by Fizeau interferometer and given to the shifted-interferogram generator. The generated shifted interferograms were compared with the ideally shifted interferograms which contain exact shifting intervals and the results shows high similarities. Subsequently, the generated interferograms were used to extract phase, resulting in significantly improved accuracy comparable to more complex multi-step phase-shifting algorithms. This approach not only enhances measurement accuracy but also simplifies experimental setups, reducing susceptibility to environmental uncertainties.
Wavelength-tuning Fizeau interferometer has been widely applied to measurements of optical surfaces. However, when measuring highly reflective surfaces, such as silicon wafer surfaces, a coupling error between higher harmonics and phase-shift miscalibration, can cause a significant systematic error in the calculated phase. In this study, a new 11-frame phase-extraction algorithm is derived for the suppression of the coupling error. The 11-frame algorithm was visualized in the frequency domain and complex plane, and its compensation capability was confirmed by numerical error analysis. Finally, the surface shape of the silicon wafer was measured by applying the 11-frame algorithm and Fizeau interferometer.
The optical thickness is an important property of transparent plates when fabricating critical components. Spatially nonuniform errors, which are major factors in the measurement of the absolute optical thickness, can be brought about by phase errors related to the nonlinear error term. In this study, equations and a new sampling window for suppressing spatially uniform errors were constructed. Using these equations and new sampling amplitudes, we designed a new phase-shifting algorithm with 17 samples. Herein, the characteristics and advantages to the measurement of highly reflective surfaces when applying this new 17-sample algorithm are presented based on a Fourier representation. In addition, the superiority of the new algorithm in terms of its error control is demonstrated by comparing the errors occurring from a varying object phase, as well as root mean square errors, with those from different conventional algorithms.
Wavelength-tuning interferometry has been widely used for measuring the thickness variation of optical devices used in the semiconductor industry. However, in wavelength-tuning interferometry, the nonlinearity of phase shift causes a spatially uniform error in the calculated phase distribution. In this study, the spatially uniform error is formulated using Taylor series. A new 9-sample phase-shifting algorithm is proposed, with which the uniform spatial phase error can be eliminated. Finally, optical-thickness variation of transparent plate is measured using the proposed algorithm and wavelength-tuning Fizeau interferometer.
Indium tin oxide (ITO) thin films have been widely used in displays such as liquid crystal displays and touch panels because of their favorable electrical conductivity and optical transparency. The surface shape and thickness of ITO thin films must be precisely measured to improve their reliability and performance. Conventional measurement techniques take single point measurements and require expensive systems. In this paper, we measure the surface shape of an ITO thin film on top of a transparent plate using wavelength-tuning Fizeau interferometry. The surface shape was determined by compensating for the phase error introduced by optical interference from the thin film, which was calculated using the phase and amplitude distributions measured by wavelength-tuning. The proposed measurement method achieved noncontact, large-aperture, and precise measurements of transparent thin films. The surface shape of the sample was experimentally measured to an accuracy of 40 nm, mainly limited by the accuracy of the reference surface of 30 nm.
In this study, a 6N – 5 phase shifting algorithm comprising a polynomial window function and discrete Fourier transform is developed for the simultaneous measurement of the surface shape and optical thickness of a transparent plate. The characteristics of the 6N – 5 algorithm were estimated by connection with the Fourier representation. The phase error of the measurements performed using the 6N – 5 algorithm is discussed and compared with those of measurements obtained using other algorithms. Finally, the surface shape and optical thickness of a transparent plate were measured simultaneously using the 6N – 5 algorithm and a wavelength tuning interferometer.
In phase-shifting Fizeau interferometers, phase-shift errors and multiple-beam interference are the most common sources of systematic error affecting high-precision phase measurements. Nonsinusoidal waveforms can be minimized by applying synchronous detection with more than four samples. However, when 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 errors. In this study, we calculate phase errors using phase-shifting algorithms that take into account the coupling error. We show that the 4N – 3 algorithm, which consists of a polynomial window function and a discrete Fourier transform term, results in the smallest phase error. As a demonstration, the surface profile of a highly reflective silicon wafer is measured using a wavelength-tuning Fizeau interferometer and the 4N – 3 algorithm.
For the surface shape measurement of a semiconductor with a highly reflective index, it is important to effectively
suppress the harmonic signals from multiple reflections. In application, the phase extraction algorithm should have a
maximum value when there is no phase-shift miscalibration. In this presentation, a new 4N - 3 phase extraction algorithm,
which has the ability to suppress harmonic signals and exhibits a fringe contrast maximum value when there is no phaseshift
error, was derived. This new 4N - 3 algorithm consists of a new polynomial window function and a discrete Fourier
transform term and has the ability to compensate for 2nd-order nonlinearity in the phase shift. The suppression ability of the new polynomial window function is compared with other conventional window functions. The sampling functions of
the new 4N - 3 algorithm have much smaller amplitudes in the vicinity of the detection frequency than does synchronous
detection or other phase extraction algorithms with conventional window functions.
Interferometric surface measurement of parallel plates presents considerable technical difficulties owing to multiple beam interference. To apply the phase-shifting technique, it is necessary to use an optical-path-difference-dependent technique such as wavelength tuning that can separate interference signals in the frequency domain. In this research, the surface shape and optical thickness variation of a lithium niobate wafer for a solid Fabry-Perot etalon during the polishing process were measured simultaneously using a wavelength-tuning Fizeau interferometer with a novel phase shifting algorithm. The novel algorithm suppresses the multiple beam interference noise and has sidelobes with amplitudes of only 1% of that of the main peak. The wafer, which was in contact with a supporting glass parallel plate, generated six different interference fringes that overlapped on the detector. Wavelength-tuning interferometry was employed to separate the specific interference signals associated with the target different optical paths in the frequency domain. Experimental results indicated that the optical thickness variation of a circular crystal wafer 74 mm in diameter and 5-mm thick was measured with an uncertainty of 10 nm PV.
Nonlinearity and non-uniformity of phase-shifts significantly contribute to the error of the evaluated phase in phase-shifting interferometry. However, state of the art error-compensating algorithms can counteract the linear mis-calibration and first-order nonlinearity associated with the phase-shift. We describe an error expansion method that is utilized to construct a phase-shifting algorithm that can compensate the second-order nonlinearity and non-uniformity of phase-shifts. The conditions for eliminating the effect of second-order nonlinearity and non-uniformity of phase-shifts are given as a set of linear equations for the sampling amplitudes. We developed a novel 11-sample phase-shifting algorithm that can compensate for the second-order nonlinearity and non-uniformity of phase-shifts and is robust up to a 4th harmonic. Experimental results show that the surface shape of a transparent plate could be measured with a precision of 1 nm, over the 120-mm-diameter aperture.
The interference fringe order of a transparent glass plate was determined using a three-surface wavelength-tuning Fizeau
interferometer and an excess fraction method. We employed multiple-surface interferometry considering the potential for
simultaneous measurement of the surface shape and geometric thickness. The optical thickness signal was separated
from the three interference signals in the frequency space. A frequency selective phase-shifting algorithm and a discrete
Fourier analysis detected the phase of the modulated interference fringes. The optical thickness obtained by wavelengthtuning
Fizeau interferometry is related to the group refractive index. In contrast, the optical thickness deviation obtained
by the phase-shifting technique is related to the ordinary refractive index. These two kinds of optical thicknesses were
synthesized with the help of the dispersion relation of a fused-silica glass. Finally, the interference fringe order was
determined using an excess fraction method that could eliminate the initial uncertainty of the refractive index.
The surface flatness and the uniformity in thickness and refractive index of a mask-blank glass have been
requested in semiconductor industry. The absolute optical thickness of a mask-blank glass of seven-inch square
and 3mm thickness was measured by three-surface interferometry in a wavelength tuning Fizeau interferometer.
Wavelength-tuning interferometry can separate in frequency space the three interference signals of the surface
shape and the optical thickness. The wavelength of a tunable laser diode source was scanned linearly from 632
nm to 642 nm and a CCD detector recorded two thousand interference images. The number of phase variation of
the interference fringes during the wavelength scanning was counted by a temporal discrete Fourier transform.
The initial and final phases of the interferograms before and after the scanning were measured by a phase
shifting technique with fine tunings of the wavelengths at 632 nm and 642 nm. The optical thickness defined by
the group refractive index at the central wavelength of 337 nm can be measured by this technique. Experimental
results show that the cross talk in multiple-surface interferometry caused a systematic error of 2.0 microns in the
measured optical thickness.
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