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Absolute testing of planarity and inhomogeneity with modified six-step method
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Abstract
We describe a modified six-step method to simultaneously measure the inhomogeneity of sample plate and the planarity of the four surfaces in an absolute manner, along with a high-efficiency iterative algorithm for data reduction. Combined with the iterative algorithm, the errors of inhomogeneity and flatness can be estimated with pixel-level spatial resolution in a fast and effective manner. The simulation and experiments prove the validity of the method and the measurement capability of reaching sub-nanometer accuracy. The method presented in this paper is cross-compared with traditional absolute testing method and the method of inhomogeneity. The difference between absolute plane measurements is less than 1.7 nm RMS, and the difference of inhomogeneity measurement accuracy is less than 2.3 nm RMS. The experimental results show that these two methods are highly consistent and have good repeatability, which verifies the accuracy of the methods proposed in this paper. Uncertainty analysis indicates that the proposed method improves the measurement uncertainty, compared with the classical transmission method.-
Keywords:
- optical testing /
- interferometry /
- absolute test /
- uncertainty evaluation
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References
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Overview
Overview: For many international projects, such as EUV lithography, Synchrotron Radiation Facility and Inertial Confinement Fusion, there are a variety of optical materials that must be produced with excellent optical flatness and refractive index homogeneity. Inhomogeneity is a measure of the variation in the refractive index within a material. Several methods can be used to measure the inhomogeneity based on standard phase measuring interferometry, including liquid immersion method and transmission method. Since the use of liquid is inconvenient for some applications, many researchers turn to develop techniques that can separate surface deviation caused by inhomogeneity from system errors (i.e. absolute test techniques). Absolute measurement of surface is a complex process that involves multiple measurements. As to absolute measurement of inhomogeneity, transmission method is the most popular. The transmission method with four measurements is able to remove the error contributions of the surfaces of the sample plate, as well as the errors of the return flat and interferometer. Another straightforward method called "window-flipping method", measuring the rear surface of the window by flipping the window, can avoid this problem. But the flipping method inherently has a big uncertainty introduced by systematic effects—the surface error of transmission flat cannot be removed completely. With the aid of absolute test techniques (such as three-flat test), it is possible to measure the surface in advance and store them for later use. But it is also a complex process. To simplify the whole measurement process, it is necessary to measure the absolute planarity and inhomogeneity of the sample plate at the same time. In this paper, a modified six-step method to simultaneously measure the inhomogeneity of the sample plate and the four surfaces in an absolute manner is presented. Unlike the classical transmission method, the modified flipping method can test a polished sample that has no wedge between the two surfaces (such as parallel transparent plates). With the help of a high-efficiency iterative algorithm for data reduction, the errors of inhomogeneity and flatness are estimated with pixel-level spatial resolution in a fast and effective manner without using the polynomial fitting. The example experiments prove the validity of the method and the measurement capability of achieving sub-nanometer RMS accuracy. Uncertainty analysis suggests that the modified six-step flipping method improves the measurement uncertainty, compared with the classical four-step transmission method. Furthermore, this method can be effectively applied to measure high quality polished samples and extended to measure large optical materials at oblique incidence (such as large optics in Synchrotron Radiation Facility, Inertial Confinement Fusion, and giant telescopes).
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Figure 1.
Six-step procedure for inhomogeneity evaluations, as well as absolute planarity measurements of the four surfaces K, L, M, N. (TF: Transmission flat, TF: Standard flat, SUT: Sample under test)
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Figure 2.
Example figure errors of original experimental surfaces K, L, M, N and figure error due to inhomogeneity of the sample plate
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Figure 3.
Example experimental results of simulation with six-step flipping test(the measurement noise of each measurement is 0.1 nm RMS)
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Figure 4.
(a) Retrieved figure errors of surfaces K, L, M, N and figure error due to inhomogeneity of the sample plate by the iterative algorithm; (b) The associated measurement error maps with respect to the original surfaces
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Figure 5.
Retrieved figure errors of (a) surface L (RMS= 18.61 nm), (b) surface M (RMS=4.87 nm), and (c) figure error due to inhomogeneity (RMS=19.47 nm) of the PPP by the proposed iterative algorithm
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Figure 6.
Retrieved figure errors of (a) surface L (RMS=19.71 nm), (b) surface M (RMS=4.62 nm), and (c) figure error due to inhomogeneity (RMS=19.67 nm) of the PPP by the proposed iterative algorithm
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Figure 8.
Coefficient of random error propagation versus refractive index
