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An iterative algorithm is presented to calibrate the shapes of the reference surfaces in three-flat test. Three measurements in three-flat test determine the even part of the flat surfaces. The odd part can be obtained by adding one or more rotation measurements and using analytical recovery algorithm. Here we present an iterative algorithm to obtain the odd part instead of special analytical recovery algorithm. The iterative algorithm is purely numerical, and only involved in data rotation operation. A simulation experiment proves the validity and high accuracy of the proposed algorithm method.
© 2014 Optical Society of America
1. Introduction
In order to improve measurement accuracy of plane interferometers, the absolute shape of the reference surface needs to be calibrated. Three-flat test is the most adopted method, in which the surfaces of three flats are compared interferometrically two by two. Three basic measurements can only give the profile of a single diameter [1], owing to the inversion that takes place when one flat is flipped front to back to face its counterpart. And to overcome this limitation, many methods introduce additional measurements, like rotations and translations, so that most of the surface can be retrieved by special analytical recovery algorithms [2–11].
Vannoni proposed an iterative algorithm that achieves reconstruction numerically [12–16], which overcomes the difficulties experienced with analytical recovery algorithms. And B. Gao presented a similar algorithm to Vannoni’s [17]. In their methods, three basic measurements and one rotation measurement are adopted. Starting from three trial surfaces and generating trial interferograms, the surfaces are recursively modified by repetitive rotation and flip manipulations until close coincidence between trail and given interferograms is achieved.
In fact, three basic measurements cannot only determine the profile of a single diameter, but also the even part of the whole surface with respect to the symmetry axis [8, 10]. In this paper, based on one added rotation measurement, an iterative algorithm aims at achieving the odd part of the flat surfaces. In section 2, the principle of the iterative algorithm method is described. In section 3, a simulation experiment is carried out to validate the proposed algorithm, and the reconstructed result is compared with Vannoni’s method. In section 4, application extendability of the algorithm is discussed.
2. Iterative algorithm method
In three-flat test, the three flats are named K, L, and M, whose shapes need to be calibrated. For each flat surface, a Cartesian coordinate system is considered, with the origin at the center and the z-axis coincident with the external normal. The flats are compared in pairs. Among basic three measurements, flat K is required to be flipped front to back about the y-axis. Three interferograms are produced according to
From Eq. (1), the even part of three flat surfaces can be calculated as the following [10]:
In the above, we indicate the operation of requiring the even part by a superscript “e”. One rotation measurement is added to obtain the odd part. In our case, the flat that is rotated is M. We indicate such a rotation by a subscript “R”. The interferogram after rotation is produced according to
From Eq. (1) and (3), the shear interferogram can be produced according to
The shear interferogram removes the information of rotationally symmetric components of surface M, and it contains the information of the rotationally nonsymmetric components [10]. If Ma is used to denote the rotationally nonsymmetric components of surface M, Eq. (4) can be written as:
The aim of the iterative algorithm is to obtain the rotationally nonsymmetric components of surface M, from which the odd part can be abstracted. And the steps of the iterative algorithm are as follows:
- 1. Initialize the shape of Ma at 0 corresponding to start with a perfectly plane surface.
- 2. Compute the synthetic shear interference patterns S according to Eq. (5).
In the above, the subscript “exp” is used to indicate the shear interferogram from experiments.
In the above, the subscript “-R” means a counter-rotation of the same amount as “R”, and “c” denotes the scaling factor. It is necessary that the scaling factor must be larger than one.
- 5. Substitute (Ma)new for Ma and back to step 2. The cycle is repeated until the difference computed in step 3 reaches a minimum.
Since the result Ma obtained from the above iterative algorithm denotes the rotationally nonsymmetric components of surface M, the odd part of surface M can be obtained as [8, 10]:
In Eq. (8), the superscript “o” denotes the operation of requiring the odd part. The whole shape of surface M can be achieved by combining the odd and even parts. And the shapes of K and L surfaces are further obtained by Eq. (1). Compared with Vannoni’s method, the iterative process of our method is only involved in data rotation operation, and flip operation is not needed, which is the advantage of our method.
3. Simulation experiment
A simulation experiment is performed, in which the absolute shape data from a 32-inch Fizeau interferometer are used as the shapes of surface K, L and M. Figure 1 shows the shape of surface M, of which the pixel resolution is 800 × 800. Three basic measurements and one rotation measurement are implemented. The rotation angle is 54 degree.
Above all, the rotation shear interferogram is produced from the measurement data, and the iterative algorithm is implemented by a computer with a 3100 MHz CPU. The scaling factor c used in the algorithm is 2. Processing is completed in approximately 26.2 seconds, and the number of iterations is 60. The rotationally nonsymmetric components of surface M is obtained, shown in Fig. 2. The odd part of surface M is subsequently required according to Eq. (8), shown in Fig. 3. Combined the even part of surface M obtained by three basic measurements, the whole shape is required, shown in Fig. 4. The reconstruction error is calculated, shown in Fig. 5, the PV and rms values of which are respectively 1.15 nm and 0.15 nm.
In order to observe the evolution of the iterative process, error rms values are calculated with the iteration number. In Fig. 6, the red curve denotes the variation of error rms values with the iteration number, from which it can be seen that the iteration process is convergent, and when the iteration number is beyond about 60, the variation is considerably small.
The proposed algorithm is compared to Vannoni’s algorithm. In Vannoni’s algorithm, the chosen scaling factor c is also 2. In Fig. 6, the blue curve denotes the variation of error rms values of Vannoni’s algorithm with the iteration number. If the iteration number is enough, such as 250, the reconstruction error of the proposed algorithm is a little greater than that of Vannoni’s algorithm, with error rms value 0.12 nm vs 0.07 nm of Vannoni’s algorithm. However, the convergence of the proposed algorithm is faster than that of Vannoni’s algorithm. In addition, due to be not involved in flip operation, the calculation time of one circle of the proposed method is less than that of Vannoni’s algorithm. If the iteration number is 60, the error rms value of the proposed method is 0.15 nm, and processing is completed in approximately 26.2 seconds. However for Vannoni’s algorithm, if the iteration number is the same, the error rms value and processing time are respectively 0.26 nm and 29.4 seconds. And if the error rms value decreases to less than 0.15 nm, the iteration number must be greater than 88, and the calculation time longer than about 42.9 seconds. Therefore if a small calculation period is needed, and reconstruction accuracy is also considered, the proposed algorithm has advantages compared to Vannoni’s algorithm.
4. Discussion
For the proposed iterative algorithm, the shear interferogram must contain sufficient information,otherwise some azimuthal frequencies cannot be recovered. For example, if 54 degree is chosen, the odd part of 20nθ frequencies can’t be recovered, where n denotes natural numbers. More rotation measurements benefit to provide more sufficient information to reconstruction. If two rotation measurements are performed, Eq. (7) can be written as
In Eq. (9), ΔS1 and ΔS2 respectively denote shear interferogram differences of two angles. Figure 7 shows the reconstructed error of surface M when two rotation angles of 37 degree and 110 degree are chosen. PV and rms values of the error are respectively 0.60 nm and 0.06 nm. The reconstructed error is reduced compared to that of one rotation measurement because the lost azimuthal frequencies are less.
In addition, since the proposed iterative algorithm can recover the rotationally nonsymmetric components of the rotated flat, it is applicable to some other situations on the reconstruction of the rotationally nonsymmetric components, such as two-flat test, sphere test and so on, where rotationally symmetric components is retrieved by other approaches.
5. Conclusion
Since the even part of the shapes can be required by three basic measurements in three-flat test, the added rotation measurements aim at achieving the odd part. In this paper, an iterative algorithm is presented to obtain the rotationally nonsymmetric components, from which the odd part can be abstracted. The simulation experiment shows that the proposed algorithm method has a considerable accuracy. Compared to Vannoni’s algorithm, the new algorithm has a faster convergence speed and less processing time. The proposed iterative algorithm is also applicable to the reconstruction of rotationally nonsymmetric components in two-flat test, sphere test and so on.
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