Inertia alignment of phase-shifting algorithms for high-numerical-aperture spherical testing in Fizeau interferometry.

In Fizeau interferometry for high-numerical-aperture spherical surface tests, the mechanical phase shift becomes spatially nonuniform within the observation aperture. We divided the aperture into annular regions and calculated the object phase using several algorithms designed for different phase shifts. The division substantially decreased the nonuniformity; however, it caused bias errors at the regional boundaries in the measured phase. The error is due to the different error coefficients of the algorithms for the phase-shift nonlinearity. A convolution technique that modifies a sampling window to align the error coefficients of a set of algorithms is proposed. The technique is experimentally shown to minimize phase measurement errors.

Hard x-ray phase-contrast microscopy using a Gabor hologram without a zero-order term.

In order to achieve a nanometer-scale resolution in an x-ray microscopy system, a Gabor-type hologram was produced by eliminating the zero-order term of the object diffraction pattern. In this system, a Fresnel zone plate was used for strong illumination of an object, and the zero-order diffraction was physically eliminated by a center stop. An accurate phase plate of π/2 in the Zernike method was numerically created, and the phase-contrast image was realized. The theoretical resolution was 19.8 nm. We proved that a gold nanocube with a size of 50 nm can be reconstructed with the predicted resolution.

Simultaneous measurement of the surface shape and thickness for an optical flat with a wavelength-tuning Fizeau interferometer with suppression of drift error.

Two types of phase-shifting algorithms were developed for simultaneous measurement of the surface and thickness variation of an optical flat. During wavelength tuning, phase-shift nonlinearity can cause a spatially nonuniform error and spatially uniform DC drift error. A 19-sample algorithm was developed that eliminates the effect of the spatially uniform error by expanding the 17-sample algorithm with characteristic polynomial theory. The 19-sample algorithm was then altered to measure the surface shape of the optical flat by rotation of the characteristic diagram. The surface shape and thickness variation were measured with these two algorithms and a wavelength-tuning Fizeau interferometer.

Interferometric profile measurement of optical-thickness by wavelength tuning with suppression of spatially uniform error.

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. The characteristics of 9-sample algorithm is discussed using Fourier representation and RMS error analysis. Finally, optical-thickness variation of transparent plate is measured using the proposed algorithm and wavelength-tuning Fizeau interferometer and the error is compared with 7-sample algorithm.

Canceling the momentum in a phase-shifting algorithm to eliminate spatially uniform errors.

In phase-shifting interferometry, phase modulation nonlinearity causes both spatially uniform and nonuniform errors in the measured phase. Conventional linear-detuning error-compensating algorithms only eliminate the spatially variable error component. The uniform error is proportional to the inertial momentum of the data-sampling weight of a phase-shifting algorithm. This paper proposes a design approach to cancel the momentum by using characteristic polynomials in the Z-transform space and shows that an arbitrary M-frame algorithm can be modified to a new (M+2)-frame algorithm that acquires new symmetry to eliminate the uniform error.

Interferometric measurement of surface shape by wavelength tuning suppressing random intensity error.

In this research, the susceptibility of the phase-shifting algorithms to the random intensity error is formulated and estimated. The susceptibility of the random intensity error of conventional windowed phase-shifting algorithms is discussed, and the 7N-6 phase-shifting algorithm is developed to minimize the random intensity error using the characteristic polynomial theory. Finally, the surface shape of the transparent wedge plate is measured using a wavelength-tuning Fizeau interferometer and the 7N-6 algorithm. The experimental results indicate that the surface shape measurement accuracy for the transparent plate is 2.5 nm.

Compensation for correlated error in multilayer interferometer.

Wavelength tuning interferometry is used to measure and estimate the surface shape of a sample. However, in multilayer interferometry (e.g., a lithium niobate [LNB] crystal wafer attached to a supporting plate), the correlated error between the higher harmonics and the phase-shift error causes considerable error in the calculated phase. In this study, the correlated errors calculated by various types of windowed phase-shifting algorithms are analyzed in connection with the characteristic polynomial theory and Fourier representation of the algorithms. The surface shape and optical thickness variation of the LNB wafer are measured simultaneously using the windowed phase-shifting algorithms. The results are compared in terms of the observed ripples and measurement repeatability. The experimental results show that the 4N-3 algorithm is optimal and possesses the smallest repeatability error.

Phase-shifting algorithm inside an optical cavity for absolute length measurement.

Phase-shifting interferometry inside an optical cavity is useful for measuring the length of a cubic object and the diameter of a metal sphere. A new error-compensating phase-shifting algorithm is proposed, with which a spatially uniform phase error caused by nonlinear phase modulation can be eliminated. It is shown that the conventional algorithm design for harmonic analysis cannot compensate for this dc error unless the sampling weight definition is extended to a complex number. Specifically, a 3% quadratic nonlinearity in the phase modulation is found to yield a systematic error of 1.5 nm in the object length in many conventional algorithms. A new 13-frame algorithm is described that applies a modified discrete Fourier window and can reduce the dc error to less than 0.5 nm.

Immersion liquid techniques in solid particle characterization: A review.

Chemical, physical and optical properties of small solid particles are widely utilized in our everyday merchandises. For example, tailored particles embedded in paper or cosmetics improve the visual appearance of the products substantially. As a consequence of the small size of particles, one particle characterization tool is a microscope. It may provide e.g. the particle size, shape and the refractive index. The determination of the refractive index, using the microscope, typically exploited the so-called immersion liquid method. In this review, we provide an overview of non-imaging immersion matching techniques including immersion liquid set, the temperature, the wavelength, the double variation and the liquid evaporation methods. The basic features, benefits and limitations of each technique have been described followed by examples of potential applications in a quality monitoring of particle suspensions and colloids in industry.

Simultaneous measurement of surface shape and optical thickness using wavelength tuning and a polynomial window function.

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 with suppression of the coupling errors between the higher harmonics and phase shift error. The characteristics of the 6N - 5 algorithm were estimated by connection with the Fourier representation in the frequency domain. 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.

Measurement of optical thickness variation of BK7 plate by wavelength tuning interferometry.

This paper presents the derivation of a 17-sample phase-shifting algorithm that can compensate the miscalibration and first-order nonlinearity of phase shift error, coupling error, and bias modulation of the intensity and satisfy the fringe contrast maximum condition. The phase error of measurements performed using the 17-sample algorithm is discussed and compared with those of measurements obtained using other algorithms. Finally, the optical thickness variation of a BK7 optically transparent plate obtained using a wavelength tuning Fizeau interferometer and the 17-sample algorithm are presented. The experimental results indicate that the optical thickness variation measurement accuracy for the BK7 plate was 3 nm.

Surface measurement of indium tin oxide thin film by wavelength-tuning Fizeau interferometry.

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 5.13 nm.

Absolute optical thickness measurement of transparent plate using excess fraction method and wavelength-tuning Fizeau interferometer.

The absolute optical thickness of a transparent plate 6-mm thick and 10 mm in diameter was measured by the excess fraction method and a wavelength-tuning Fizeau interferometer. The optical thickness, defined by the group refractive index at the central wavelength, was measured by wavelength scanning. The optical thickness deviation, defined by the ordinary refractive index, was measured using the phase-shifting technique. Two kinds of optical thicknesses, measured by discrete Fourier analysis and the phase-shifting technique, were synthesized to obtain the optical thickness with respect to the ordinary refractive index using Sellmeier equation and least-square fitting.

Multiple-surface interferometry of highly reflective wafer by wavelength tuning.

The surface shape and optical thickness variation of a lithium niobate (LNB) wafer were measured simultaneously using a wavelength-tuning interferometer with a new phase-shifting algorithm. It is necessary to suppress the harmonic signals for testing a highly reflective sample such as a crystal wafer. The LNB wafer subjected to polishing, which is in optical contact with a fused-silica (FS) supporting plate, generates six different overlapping interference fringes. The reflectivity of the wafer is typically 15%, yielding significant harmonic signals. The new algorithm can flexibly select the phase-shift interval and effectively suppress the harmonic signals and crosstalk. Experimental results indicated that the optical thickness variation of the LNB wafer was measured with an accuracy of 2 nm.

Design of phase shifting algorithms: fringe contrast maximum.

In phase shifting interferometry, the fringe contrast is preferred to be at a maximum when there is no phase shift error. In the measurement of highly-reflective surfaces, the signal contrast is relatively low and the measurement would be aborted when the contrast falls below a threshold value. The fringe contrast depends on the design of the phase shifting algorithm. The condition for achieving the fringe contrast maximum is derived as a set of linear equations of the sampling amplitudes. The minimum number of samples necessary for constructing an error-compensating algorithm that is insensitive to the jth harmonic component and to the phase shift error is discussed. As examples, two new algorithms (15-sample and (3N - 2)-sample) were derived that are useful for the measurement for highly-reflective surfaces.

Discontinuous surface measurement by wavelength-tuning interferometry with the excess fraction method correcting scanning nonlinearity.

Wavelength-tuning interferometry can measure surface shapes with discontinuous steps using a unit of synthetic wavelength that is usually larger than the step height. However, measurement resolution decreases for large step heights since the synthetic wavelength becomes much larger than the source wavelength. The excess fraction method with a piezoelectric transducer phase shifting is applied to two-dimensional surface shape measurements. Systematic errors caused by nonlinearity in source frequency scanning are fully corrected by a correlation analysis between the observed and calculated interference fringes. Experiment results demonstrate that the determination of absolute interference order gives the profile of a surface with a step height of 1 mm with an accuracy of 12 nm.

Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer.

Wavelength-scanning interferometry permits the simultaneous measurement of variations in surface shape and optical thickness of a nearly parallel plate. Interference signals from both surfaces of the test plate can be separated in frequency space; however, these frequencies are shifted from the expected frequency by the refractive-index dispersion of the test plate and any nonlinearity that is due to wavelength scanning. Conventional Fourier analysis is sensitive to this detuning of the signal frequency and suffers from multiple-beam interference noise. We propose new wavelength-scanning algorithms that permit a large tolerance for dispersion of the test plate and nonlinearity of scanning. Two 19-sample algorithms that suppress multiple-interference noise up to the second order of the reflectance of the test plate are presented. Experimental results show that the variation in surface shape and optical thickness of a glass parallel plate of 250-mm diameter was measured with a resolution of 1-2 nm rms.

Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry.

The interferometric surface measurement of single or stacked parallel plates presents considerable technical difficulties due to multiple-beam interference. To apply phase-shifting methods, it is necessary to use a pathlength-dependent technique such as wavelength scanning, which separates interference signals from various surfaces in frequency space. The detection window for frequency analysis has to be optimized for maximum tolerance against frequency detuning due to material dispersion and scanning nonlinearities, as well as for suppression of noise from other frequencies. We introduce a new class of phase-shifting algorithms that fulfill these requirements and allow continuous tuning of phase detection to any frequency of interest. We show results for a four-surface stack of nearparallel plates, measured in a Fizeau interferometer.

The long-term effect of replacement therapy in a short girl with autoimmune atrophic thyroiditis of prepubertal onset.

A 9 yr 11 mo old girl was admitted to our hospital because of short stature. Her growth rate gradually decreased and her height was 120 cm (-2.5 SD) on admission. The mother's and father's heights were 157 cm (-0.2 SD) and 163 cm (-1.3 SD), respectively. Her bone age was retarded (6 yr 10 mo). An MRI indicated pituitary enlargement, which mimicked adenoma. Evaluation of the pituitary-thyroid axis and thyroid function proved she had primary hypothyroidism (T3 0.5 ng/ml, T4 1.0 µg/dl, TSH 1,030 µU/ml). These findings, thyroid autoantibody (anti-microsome antibody 400 xs) and histopathology (moderate fibrosis and mild lymphocytic infiltration) suggested acquired hypothyroidism due to autoimmune atrophic thyroiditis of prepubertal onset. Since the evaluation, she has been treated with levothyroxine. The pituitary enlargement disappeared within 3 mo after levothyroxine replacement. The growth rate increased and her height reached 153.2 cm (-1.0 SD) during 10 yr replacement (at 19 yr 11 mo of age). An improvement in her final height was obtained by long-term thyroid hormone replacement therapy. Enough endocrinological study and repeated MRI evaluation are necessary in cases of pituitary enlargement which mimics adenoma before considering surgery.

Measurement of air turbulence for on-machine interferometry.

There is increasing demand for in situ shape measurements performed on ultraprecision processing machines. One major source of error during interferometric measurements performed on machines is fringe displacement due to external disturbances. We have developed an interferometer equipped with an electro-optic phase modulator that measures the phase of interference fringes before they are displaced by air turbulence. The frequency characteristics of air turbulence induced by a heat source are derived from successive measurements of a test surface. Experimental results show that the phase of the interference fringes can be accurately measured in the presence of air turbulence when the intensity of the fringes is sampled at a speed of several hundred hertz.