Proceedings of the
9th International Conference of Asian Society for Precision Engineering and Nanotechnology (ASPEN2022)
15 – 18 November 2022, Singapore
doi:10.3850/978-981-18-6021-8_OR-15-0066
Phase-Extraction Formula for Surface Profiling of Optical Flat using Wavelength-Tuning Interferometer
School of Mechanical Engineering, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, South Korea
ABSTRACT
Transparent parallel optical flats have been extensively used in production process of semiconductor industries. For improving the performance of semiconductor chips, the surface shape of the transparent parallel plate should be measured with nanoscale uncertainty. Atomic force microscopes (AFMs) have been widely used in the semiconductor industry to measure the surface profile of the optical devices precisely. However, the measurements using AFMs require a considerable amount of time to measure the entire surface profile distribution. To measure the surface shape of the transparent parallel plate, the phase-shifting interferometry using wavelength tuning has been widely used owing to its high resolution and noncontact measurement. In the phase-shifting technique, the phase difference between a target wave front and a reference wave front is changed linearly with wavelength tuning. The target phase can be calculated from the phase-extraction formula, the arctangent of the ratio between two combinations of the observed intensities. However, the measurement accuracy is degraded by not only phase-shift errors, but also by coupling errors between higher harmonic signals and phase-shift errors. The coupling errors can be the significant systematic errors when measuring the highly-reflective surface and optical thickness variation of the transparent plate. The phase-shift errors can occur because of the environmental uncertainties such as the temperature fluctuation and the floor vibration. These environmental uncertainties can be eliminated by using the special phase-extraction formula.
Systematic approaches for developing the phase-extraction formula have been proposed by several researchers based on the averaging method of successive samples, Fourier description of the sampling functions, and an analytical expansion using the linear equation group. In this study, the new 15-sample phase-extraction formula was developed that can suppress the coupling errors caused by harmonic signals up to the fourth-order and linear and nonlinear phase-shift errors. The 15-sample formula was derived using the linear equations consisting of the sampling amplitude of the phase-extraction formula. The characteristics of the proposed phase-extraction formula were visualized on the frequency domain and complex plane using the Fourier description and characteristic polynomial theory, respectively. In addition, the error-suppression capability of the proposed phase-extraction formula was confirmed by using the theory of the root mean square (RMS) error. The frequency response of the 15-sample phase-extraction formula was evaluated by using the concept of the signal-to-noise ratio (SNR) calculated using the frequency-transfer-function paradigm. Also, the sensitivity to the floor vibration of the newly developed 15-sample phase-extraction formula was evaluated quantitatively with comparing those of the other conventional phase-extraction formula. Finally, the surface profile of the transparent optical flat was measured by using the wavelength-tuning Fizeau interferometer and the newly developed 15-sample phase-extraction formula. The standard deviations and the systematic errors of the ripple distribution were discussed with comparing the other phase-extraction formula.
Keywords: Surface, Optical flat, Wavelength tuning, Fizeau interferometer
School of Mechanical Engineering, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, South Korea
ABSTRACT
Transparent parallel optical flats have been extensively used in production process of semiconductor industries. For improving the performance of semiconductor chips, the surface shape of the transparent parallel plate should be measured with nanoscale uncertainty. Atomic force microscopes (AFMs) have been widely used in the semiconductor industry to measure the surface profile of the optical devices precisely. However, the measurements using AFMs require a considerable amount of time to measure the entire surface profile distribution. To measure the surface shape of the transparent parallel plate, the phase-shifting interferometry using wavelength tuning has been widely used owing to its high resolution and noncontact measurement. In the phase-shifting technique, the phase difference between a target wave front and a reference wave front is changed linearly with wavelength tuning. The target phase can be calculated from the phase-extraction formula, the arctangent of the ratio between two combinations of the observed intensities. However, the measurement accuracy is degraded by not only phase-shift errors, but also by coupling errors between higher harmonic signals and phase-shift errors. The coupling errors can be the significant systematic errors when measuring the highly-reflective surface and optical thickness variation of the transparent plate. The phase-shift errors can occur because of the environmental uncertainties such as the temperature fluctuation and the floor vibration. These environmental uncertainties can be eliminated by using the special phase-extraction formula.
Systematic approaches for developing the phase-extraction formula have been proposed by several researchers based on the averaging method of successive samples, Fourier description of the sampling functions, and an analytical expansion using the linear equation group. In this study, the new 15-sample phase-extraction formula was developed that can suppress the coupling errors caused by harmonic signals up to the fourth-order and linear and nonlinear phase-shift errors. The 15-sample formula was derived using the linear equations consisting of the sampling amplitude of the phase-extraction formula. The characteristics of the proposed phase-extraction formula were visualized on the frequency domain and complex plane using the Fourier description and characteristic polynomial theory, respectively. In addition, the error-suppression capability of the proposed phase-extraction formula was confirmed by using the theory of the root mean square (RMS) error. The frequency response of the 15-sample phase-extraction formula was evaluated by using the concept of the signal-to-noise ratio (SNR) calculated using the frequency-transfer-function paradigm. Also, the sensitivity to the floor vibration of the newly developed 15-sample phase-extraction formula was evaluated quantitatively with comparing those of the other conventional phase-extraction formula. Finally, the surface profile of the transparent optical flat was measured by using the wavelength-tuning Fizeau interferometer and the newly developed 15-sample phase-extraction formula. The standard deviations and the systematic errors of the ripple distribution were discussed with comparing the other phase-extraction formula.
Keywords: Surface, Optical flat, Wavelength tuning, Fizeau interferometer
