Chiayu Ai

Effect of piezoelectric transducer nonlinearity on phase shift interferometry.

If the nonlinearity of the motion of a piezoelectric transducer (PZT) can be described as a quadratic function, the integrated intensity of one frame in phase shift interferometry can be calculated using the Fresnel integral. For a PZT with smaller nonlinearity, the rms phase error is almost linearly proportional to the quadratic coefficient The effects of PZT nonlinearity on the three- and the four-bucket algorithms are compared.

Absolute testing of flats by using even and odd functions

We describe a modified three-flat method. In a Cartesian coordinate system, a flat can be expressed as the sum of even-odd, odd-even, even-even, and odd-odd functions. The even-odd and the odd-even functions of each flat are obtained first, and then the even-even function is calculated. All three functions are exact. The odd-odd function is difficult to obtain. In theory, this function can be solved by rotating the flat 90°, 45°, 22.5°, etc. The components of the Fourier series of this odd-odd function are derived and extracted from each rotation of the flat. A flat is approximated by the sum of the first three functions and the known components of the odd-odd function. In the experiments, the flats are oriented in six configurations by rotating the flats 180°, 90°, and 45° with respect to one another, and six measurements are performed. The exact profiles along every 45° diameter are obtained, and the profile in the area between two adjacent diameters of these diameters is also obtained with some approximation. The theoretical derivation, experiment results, and error analysis are presented.

Effect of spurious reflection on phase shift interferometry

The phase errors caused by spurious reflection in Twyman-Green and Fizeau interferometers are studied. A practical algorithm effectively eliminating the error is presented. Two other algorithms are reviewed, and the results obtained using the three algorithms are compared.

Measurement of the inhomogeneity of a window

We describe three methods to measure the inhomogeneity of a window material. The first method immerses the window in a liquid between two planes. However, this method is inconvenient for some applications. The second method measures the optical figure of the front surface and then measures the return wavefront that transmits through the window and reflects from the rear surface of the window. The advantage of this method is that it can remove the contributions of both the surface figures and the return flat plus the system error of the interferometer. The disadvantage is that a small wedge must be fabricated between the two surfaces to eliminate spurious interference. The third method derives the inhomogeneity of the window material by measuring the optical figure of the front surface of the window and then flipping the mirror to measure the back surface. The advantage of this method is that it is not necessary to have a wedge between the two surfaces. The disadvantage of the window-flipping method is that the contribution of system error can increase.

Absolute testing of flats decomposed to even and odd functions

This paper describes a method for measuring the absolute flatness of flats. A function in a Cartesian coordinate system can be expressed as the sum of even-odd, odd-even, even-even, and odd-odd functions. Three flats are measured at eight orientations; one flat is rotated 180 degree(s), 90 degree(s), and 45 degree(s) with respect to another flat. From the measured results the even-odd and the odd-even functions of each flat are obtained first, then the even-even function is calculated. All three functions are exact. The odd-odd function is difficult to obtain. For the points on a circle centered at the origin, the odd-odd function has a period of 180 degree(s) and can be expressed as a Fourier sine series. The sum of one half of the Fourier sine series is obtained from the 90 degree(s) rotation group. The other half is further divided into two halves, and one of them is obtained from the 45 degree(s) rotation group. Thus, after each rotation, one half of the unknown components of the Fourier sine series of the odd-odd function is obtained. The flat is approximated by the sum of the first three functions and the known components of the odd-odd function. In the simulation, three flats (each is an OPD map obtained from a Fizeau interferometer) are reconstructed. The theoretical derivation and the simulating results are presented.

Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements

Large, high power laser systems such as that being constructed by Lawrence Livermore National Laboratories for the National Ignition Facility require accurate measurements of spatial frequencies of up to 2.5 lines/mm over a 100mm field of view.In order to ensure accurate measurements of the parts, the test apparatus must be well characterized. The systems transfer function (STF) of the interferometer under development to perform these measurements was calculated by comparing the power spectra of measurements of known phase objects to their theoretical power spectra. Several potential problem areas were identified and studied. Of primary concern was the effect on the STF of the rotating diffuser and incoherent relay system employed in most commercial laser Fizeau interferometers. It was determined that such an arrangement degraded the transfer function beyond acceptability. The other major concern was possible inability to measure certain frequencies due to propagation between the test piece and alignment of the system optics.Use of strictly coherent imaging and small propagation distance between the test piece and return flat, the system transfer function could be kept at acceptable levels within the range of interest.

Effect of retroreflection on a Fizeau phase-shifting interferometer

Phase errors in a Fizeau phase-shifting interferometer caused by multiple-reflected beams from a retroreflective optics, such as a corner cube and a right-angle prism, are studied. Single- and double-pass configurations are presented, and their measurement results are compared. An attenuator is not needed in a double-pass configuration because light is reflected by the retroreflective optics twice and the reference surface once and hence the intensities match. It is more accurate to test a corner cube or a right-angle prism in a double-pass configuration than in a single-pass configuration. Simulations and experimental results are presented.

Testing an optical window of a small wedge angle : effect of multiple reflections

Multiple reflections between two surfaces of a window introduce a fixed pattern error in optical measurements. One way to remove these spurious reflections is to use a reasonably large wedge so that the interference fringes formed by the two surfaces are too dense for the detector to resolve. However, this method does not work if the wedge angle is small, e.g., several arcseconds. By tilting both the window and the return mirror properly, it is possible to remove the effect of multiple reflections of a window. Theory and experimental results are presented.

Optical resolution of phase measurements of laser Fizeau interferometer

Accurate interferometric measurement of large laser slabs requires spatial frequencies of 1 mm/cycle to 33 mm/cycle over a 100 mm field of view to be passed by the system with no more than 25% loss in modulation. To eliminate noise and artifacts due to strictly coherent imaging, many commercial interferometers employ a rotating diffuser on an intermediate image plane and relay this image incoherently onto a detector. Unfortunately, this process may adversely affect the resolution of the instrument. Through measurement of a sinusoidal phase grating and fused silica step, the transfer function a laser Fizeau interferometer was measured for both a system with and without the incoherent relay system. Results are compared to those predicted by diffraction theory. Studies of the effects of defocus and propagation on the measurement were also made. Using strictly coherent imaging dramatically increases the systems ability to measure features of high spatial frequency and allows the measurement requirements for laser slabs to be met.

Testing stress birefringence of an optical window

This paper describes a method to measure the birefringence of an optical window. The transmitting wavefront includes the contributions from the two surfaces, the material inhomogeneity, and the birefringence. Because of the birefringence, the transmitting wavefront has different profiles for different orientations of polarization of linearly polarized beams. From this difference, the amount of phase difference for the fast and slow axes is obtained. Thus, the birefringence is calculated. With this method, the contributions from the two surfaces and the material inhomogeneity are removed. A laser rod was measured with different methods. The theoretical derivation, comparison of different methods, and experimental results are presented.