Carrier Frequency Calculation of Interference Fringes at Low Spatial Frequencies

Objective The carrier frequency of the fringes corresponds to is the tilt and constant terms of the phase distribution. Meanwhile, in interferometry, the carrier frequency solution of fringes is significant. It can be adopted for the calibration of phase-shift devices in interferometers and for phase extraction. Second, the carrier frequency parameter is required to correct the retrace errors in interferometers. Additionally, even in the absolute measurement of the two flats, the carrier frequency parameters of the fringes are also employed. Thus, the carrier frequency parameters of the fringes can be utilized in all aspects of interferometry. At present, the carrier frequency parameter solution of fringes can be divided into two categories, with one being the absolute parameter solution method, such as the image processing and Fourier transform methods. This kind of method only employs a single-frame interferogram to compute the absolute value of the carrier frequency, but it has many limitations, including the low computational accuracy of the image processing method and proneness to the singular solution. Meanwhile, the Fourier transform method is only applicable in the case of the high carrier frequency, and cannot be applied to the low spatial frequency interferometric fringes. The Fourier transform method is only applicable to the case of high carrier frequency, and for low spatial frequency interference fringes, its spectrum is coupled with the zero frequency, which is difficult to separate with the large solving error. In response to the limitations of the single-frame method, we carry out the research on the phase-shift method, which is a class of relative parametric solution methods. Its essence is a random tilt phase-shift algorithm, which is mainly adopted for phase solution, but incidentally, the phase-shift between the interferograms or the relative value of the carrier frequency parameter can also be obtained. Methods First, the carrier frequency parameters (f(x), f(y) and f(z)) are estimated. Then, the interference model is approximated by omitting the higher-order terms of the phase. In such conditions, we can construct a linear fit to solve fz. After obtaining fz, we can obtain parameter f(x) by selecting a row of elements and constructing a new fit. Similarly, we can select a column of elements and obtain the parameter f(y). Finally, considering that the estimated carrier frequency parameters have errors, the above process is iterated repeatedly to find the accurate parameter values. Results and Discussions Simulations show that the method is applicable to many cases, and despite the even background distribution of the interferogram, the carrier frequency can be accurately calculated. The maximum error can be better than 0.01 lambda, and the minimum can be up to 0.002 lambda, lambda is 632.8 nm. The obtained carrier frequency is employed in phase-shift interferometry, and the root mean square error of the phase can be up to 0.0002 lambda. In the experiments, the proposed method has a carrier frequency error of less than 0.007 lambda, compared with the phase-shift interferometry. When the calculated carrier frequency is adopted for phase-shift interferometry, there is no significant ripple in the phase. Finally, this method has an error in principle because of the phase omission. In principle, the phase should be satisfied to be much smaller than the carrier frequency. With the ratio of the peak to valley (PV) of the phase to the carrier frequency parameter defined as c, the discussion shows that when c<0. 5, the prerequisite that the phase is "much smaller than" the carrier frequency can be approximately satisfied, and the error of this method is better than 0.025 lambda. When c<0. 25, the accuracy can be further improved to 0.01 lambda. Conclusions A new carrier frequency calculation method of fringes is proposed, and the simulation and experiments show that the method is widely applicable, with the error of the calculated carrier frequency better than 0.01. in almost any case. It is worth noting that in adopting the proposed method, it is necessary to satisfy the prerequisite that the higher-order phases in the interferometric fringes are much smaller than the carrier frequency. Additionally, the discussion shows that for the general case, only 2-3 fringes in the interferogram are needed to realize the accurate carrier frequency solution, and even for some high-precision planar phases, only one fringe is necessary. The proposed method also has a wide range of applications, and the carrier frequency accuracy of the method can fully satisfy the phase solution in the phase-shift interferometry without any obvious ripple in the phase. Compared with the existing methods, our method has the following advantages such as simple iteration, high running efficiency, and applicability to the case of uneven background of the interferogram. Meanwhile, it is a method of calculating the absolute parameters of the carrier frequency of a single-frame interferogram, and the carrier frequency can be accurately calculated for the interferometric fringes of almost any spatial frequency.