Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy

The phase-shifting point-diffraction interferometer (PS/PDI) was recently developed and implemented at Lawrence Berkeley National Laboratory to characterize extreme-ultraviolet (EUV) projection optical systems for lithography. Here we quantitatively characterize the accuracy and precision of the PS/PDI. Experimental measurements are compared with theoretical results. Two major classes of errors affect the accuracy of the interferometer: systematic effects arising from measurement geometry and systematic and random errors due to an imperfect reference wave. To characterize these effects, and hence to calibrate the interferometer, a null test is used. This null test also serves as a measure of the accuracy of the interferometer. We show the EUV PS/PDI, as currently implemented, to have a systematic error-limited reference-wave accuracy of 0.0028 waves (lambda/357 or 0.038 nm at lambda = 13.5 nm) within a numerical aperture of 0.082.     

Experimental comparison of a liquid-crystal point-diffraction interferometer (LCPDI) and a commercial phase-shifting interferometer and methods to improve LCPDI accuracy

We compare the phase measurements of a fused-silica witness sample made with a liquid-crystal point-diffraction interferometer (LCPDI) with measurements made with a Zygo Mark IV xp phase-shifting interferometer and find close agreement. Two phase-shift-error sources in the LCPDI that contribute to measurement discrepancies are frame-to-frame intensity changes caused by the dichroism of the dye and alignment distortions of the host liquid crystal. An empirical model of the phase-shift error caused by the host alignment distortions is presented and used to investigate the performance of two different phase-detection algorithms. It is suggested that by proper choice of LCPDI fabrication parameters and phase-acquisition methods, the device's accuracy can be significantly improved.     

Vibration-compensated interferometer for surface metrology

An advanced interferometer was built for surface metrology in environments with severe vibration. This instrument uses active control to compensate for effects of vibration to allow surface measurement with high-resolution phase-shifting interferometry. A digital signal processor and high-speed phase control from an electro-optic modulator allows phase measurements at 4000 Hz. These measurements are fed back into a real-time servo in the digital signal processor that provides a vibration-corrected phase ramp for the surface measurements taken at video rates. Unlike fringe locking, which compensates vibration to keep the phase constant, we show a true phase servo that allows the phase to be stabilized while it is ramped, enabling surface measurements using phase-shifting interferometry that requires multiple images with controlled phase shifts.     

Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer

A 300-mm-aperture digital phase-shifting Fizeau interferometer has been developed in-house for precision metrology of optical components fabricated by the optical workshop at Telecommunications and Industrial Physics, Commonwealth Scientific and Industrial Research Organization. We describe the procedures used in the calibration of the instrument. A reference data file representing the deviations from flatness of the reference surface is generated, measurement uncertainty estimated, and aberrations in the instrument assessed. Measurements on 250-mm-diameter uncoated optical surfaces have consistently shown short-term repeatability of 0.3-nm rms from measurement to measurement and allowed for absolute characterization of these surfaces to within a few nanometers.     

Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer

Recent technological innovations have enabled the development of a new class of dynamic (vibration-insensitive) interferometer based on a CCD pixel-level phase-shifting approach. We present theoretical and experimental results for an interferometer based on this pixelated phase-shifting technique. Analyses of component errors and instrument functionality are presented. We show that the majority of error sources cause relatively small magnitude peak-to-valley errors in measurement of the order of 0.002-0.005lambda. These errors are largely mitigated by high-rate data acquisition and consequent data averaging.     

Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor

The performance of ground-based optical imaging systems is severely degraded from the diffraction limit by the random effects of the atmosphere. Adaptive-optics techniques have been used to compensate for atmospheric-turbulence effects. A critical component in the adaptive-optics system is the wave-front sensor. At present, two types of sensors are common: the Hartmann-Shack wave-front sensor and the shearing interferometer wave-front sensor. In this paper we make a direct performance comparison of these two sensors. The performance calculations are restricted to common configurations of these two sensors and the fundamental limits imposed by shot noise and atmospheric effects. These two effects encompass the effects of extended reference beacons and sensor subaperture spacings larger than the Fried parameter r(0). Our results indicate comparable performance for good seeing conditions and small beacons. However, for poor seeing conditions and extended beacons, the Hartmann sensor has lower error levels than the shearing interferometer.     

Simple phase-shifting method in a wedge-plate lateral-shearing interferometer

A simple phase-shifting method in a wedge-plate lateral shearing interferometer is described. Simply moving the wedge plate in an in-plane parallel direction gives the amount of phase shift required for phase-shifting interferometry because the thickness of a wedge plate is not constant and varies along the wedge direction. This method requires only one additional linear translator to move the wedge plate. The required moving distance for a phase shift of the wave front with this method is of the order of a millimeter, whereas the typical moving distance for another method that uses a piezoelectric transducer is of the order of a wavelength. This method yields better precision in controlling the moving distance than do the other methods.     

Calibration of a Shack-Hartmann sensor for absolute measurements of wavefronts

We demonstrate a method with which to calibrate a Shack-Hartmann sensor for absolute wavefront measurement of collimated laser beams. Nearly perfect spherical wavefronts originating from a single-mode fiber were used as references. After the calibration, the uncertainty of the wavefront was less than lambda/100 peak to valley across a diameter of 6 mm. For example, this method allowed us to balance aberrations and prepare collimated beams with wavefronts that are plane to lambda/500 across 1 mm.     

Red-green-blue interferometer for the metrology of discontinuous structures

Discontinuous surface profiles, e.g., diffractive optical elements (DOEs), are commonly measured by white-light interferometry. White-light interferometry needs significantly more memory capacity and computer time than does phase-shifting interferometry; there are approximately ten times more frames to be taken to gather the required information about the object under test. But usually the grooves of the DOEs are too deep for single-wavelength phase-shifting interferometry. Here we show how phase-shifting techniques can be applied to DOEs. For this purpose three interference patterns are recorded simultaneously by a three-chip color CCD camera at three wavelengths (Red-green-blue). It is possible to calculate separately the optical path difference at each pixel from the three phase patterns modulo 2pi. The algorithms used and experimental results are presented.     

Shack-Hartmann sensor for fast infrared wave-front testing

Abstract

A Shack-Hartmann sensor has been designed for testing the wave front of CO₂ lasers. Fabrication of a lens array and a detector array with tight tolerances on position accuracy are essential steps. Parallel electronics allow for high-speed wave-front measurements with 1 kHz sampling frequency. The device has been used to investigate the behaviour of a high-power CO₂ laser. Besides the expected thermal drifts of beam direction at the beginning of laser action, periodic changes of beam direction, have been detected. The Shack-Hartmann sensor seems the appropriate device for controlling adaptive optics in high-power laser applications.     

Wavelength-multiplexed phase-locked laser diode interferometer using a phase-shifting technique

We propose a new signal-processing method for eliminating measurement errors that occur in the wavelength-multiplexed phase-locked laser diode interferometer. The basic idea proposed here is a very simple but effective way to improve measurement accuracy. With our scheme, the phase in the interference signal is strictly shifted by 2pi, which enables us to eliminate measurement errors. The equivalent wavelength ? is 80 mm, and the measurement accuracy reaches ~?/600. A step-height measurement was also carried out in the experiment.     

Misalignment effects of the Shack-Hartmann sensor

The Shack-Hartmann sensor uses a microlens array and a CCD camera for wave-front measurements. To obtain wave-front measurements with high accuracy, an accurate relative alignment of both is essential. The different states of misalignment of the Shack-Hartmann sensor are divided into groups and are treated theoretically and experimentally. Their effect on the accuracy of wave-front measurements is evaluated. In addition, a practical method for proper alignment of the Shack-Hartmann sensor is proposed.     

Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function

The estimation accuracy of a point-diffraction interferometer is examined with two phase-shifting schemes: spatial and temporal. Under the assumption of plane- or spherical-wave propagation through isotropic turbulence that can be accurately represented as a series of thin phase screens, results that are valid for any scintillation regime are obtained by use of the invariance with a propagation of the mutual coherence function. It is established that the estimation accuracy of the spatial phase-shifting point-diffraction interferometer is invariant with scintillation. Upper and lower bounds on the perfor mance of the temporal phase-shifting point-diffraction interferometer are developed. Wave optical simulation results are presented that validate the analytic predictions for the two phase-shifting schemes. The results and techniques presented can be used to assess the appropriate phase-shifting scheme given finite resources, such as a limited number of pixels in a detector array or a restricted detector frame rate.     

Detection of phase singularities with a Shack-Hartmann wavefront sensor

While adaptive optical systems are able to remove moderate wavefront distortions in scintillated optical beams, phase singularities that appear in strongly scintillated beams can severely degrade the performance of such an adaptive optical system. Therefore the detection of these phase singularities is an important aspect of strong-scintillation adaptive optics. We investigate the detection of phase singularities with the aid of a Shack-Hartmann wavefront sensor and show that, in spite of some systematic deficiencies inherent to the Shack-Hartmann wavefront sensor, it can be used for the reliable detection of phase singularities, irrespective of their morphologies. We provide full analytical results, together with numerical simulations of the detection process.     

Camera influence on the phase-measurement accuracy of a phase-shifting speckle interferometer

In a real phase-shifting interferometer the camera (i.e., a photodetector plus an analog-to-digital converter) cuts off intensities above some saturation level and provides a limited number of digitization steps. Owing to the intensity statistics of speckle fields, this might severely influence the accuracy of the calculated speckle phase. The optimum beam ratio and the modulation of the camera are computed. To calculate the standard deviation of the phase difference, first, we derive a relation that shows that the variances of the two measured phase frames are equal and that they must be added with the decorrelation-dependent variance. To obtain the minimum phase-measurement error of 25.1 mrad, it is found that the mean speckle intensity ought to be adjusted to be 0.058 times the saturation intensity of the camera and that the beam ratio is to be 5.7. The results are confirmed by computer simulation of a two-wavelength speckle interferometer.     

基于LMS算法的移相干涉仪环境振动控制器

环境振动严重影响移相干涉测量的过程。在运用光学外差法测得移相干涉仪环境振动信号的前提下,设计了采用高速DSP芯片的振动控制器,它对电压形式的振动信号进行采样,用自适应LMS算法分析振动信号,同时输出反馈控制信号。反馈控制信号驱动PZT光学移相器对振动引起的光程差(或波前相位)变化进行实时补偿。通过这种闭环控制,系统能够对幅频积小于100waves*Hz的环境振动进行有效补偿,实验中获得了稳定的移相干涉条纹,保证了光学测试的正确性。     

Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer

The white-light interferogram in a spectrally resolved white-light interferometer is decomposed in its constituent spectral components by a spectrometer and displayed along its chromaticity axis. A piezoelectric transducer phase shifter in such an interferometer can give a desired phase shift of pi/2 only at one wavelength. The phase shift varies continuously at all other wavelengths along the chromaticity axis. This situation is ideal for an experimental study of the phase error due to the phase-shift error in the phase-shifting technique, as it will be shown in this paper.     

Error propagation: a comparison of Shack-Hartmann and curvature sensors

Phase estimates in adaptive-optics systems are computed by use of wavefront sensors, such as Shack-Hartmann or curvature sensors. In either case, the standard error of the phase estimates is proportional to the standard error of the measurements; but the error-propagation factors are different. We calculate the ratio of these factors for curvature and Shack-Hartmann sensors in dependence on the number of sensors, n, on a circular aperture. If the sensor spacing is kept constant and the pupil is enlarged, the ratio increases as n(0.4). When more sensing elements are accommodated on the same aperture, it increases even faster, namely, proportional to n(0.8). With large numbers of sensing elements, this increase can limit the applicability of curvature sensors.     

移相干涉仪环境微扰的外差检测及信号处理

对环境微振动干扰进行补偿可减小移相干涉测量的误差,其中振动量的检测是实现振动补偿的前提。以声光调制器作为光学移频器,在移相式平面干涉仪中组合成外差干涉测振系统,可以实现光程差微小变化(范围为0到1/2波长)的实时检测。在外差信号处理中采用单片RF/IF相位测量芯片直接对两路40MHz模拟信号进行比相,简化了通常使用的数字测相方法,其精确测相的典型非线性值小于1度。用该系统实际测量了周期性振动和地面冲击振动对干涉仪的影响,获得了干涉仪所受微振动的幅度和相位。     

Reciprocal reflection interferometer for a fiber-optic Faraday current sensor

A reciprocal fiber-optic reflection interferometer for remote measurement of electrical current through the Faraday effect is described. The effects of polarization cross coupling because of nonideal elements are eliminated with a low-coherence source. Nonreciprocal birefringence phase modulation is employed for detection of the Faraday phase shift. The theoretical predictions are confirmed by measurements with a piece of straight fiber as the sensing element in a 100-turn solenoid. Currents from 0 to 40 A have been measured with a linear response and a noise limit of ~0.015 A/√Hz.