An Improved Fitting Method for Predicting the Zernike Coefficient–Wavelength Curves

Broadband transmitted wavefront measurement is a detection method that uses transmitted wavefronts at a few specific wavelengths though a function of Zernike coefficients and wavelength to predict other transmitted wavefronts at any wavelength within a certain range. This method is used when the designed wavelength of the interferometer and the detection wavelength are different. The function of the Zernike coefficients and wavelength can be expressed by the Conrady formula which can be used to describe a monotonic function curve or a function curve with an inflection point. Two methods, i.e., solving and fitting, can be used to determine the coefficients of the Conrady formula. Owing to the inevitable errors in the actual measurements, the fitting method is preferred to determine the coefficients of the Conrady formula. However, by analyzing the Conrady formula, it was found that the curve obtained by fitting the Conrady formula is a monotonic curve. This method cannot be used to obtain a curve with an inflection point or a slow-changing curve in the long waveband. In this paper, we propose an improved data fitting approach that transforms the Conrady formula into a polynomial formula to fit a given set of data for determining the Conrady formula coefficients. The correctness of the method was verified via simulations. Finally, the feasibility of the proposed method was verified by changing the light source and using the Fizeau interferometer to measure the transmitted wavefront of the doublet lens. To ascertain the feasibility of using visible light to predict the invisible light band, a near-infrared 1064 nm light source was added to the experimental set-up to broaden the range of the transmitted wavefront prediction waveband. Experimental results showed that the conversion polynomial fitting method can be used to obtain monotonic curves and the curve with an inflection point and a slow-changing curve in the long waveband, thereby mitigating the limitation of directly fitting the Conrady formula using a set of data points. Furthermore, using this method, we verified that the visible light can be used to predict the transmitted wavefront at any wavelength in the invisible light band. This observation is highly significant for the practical applications of the broadband transmitted wavefront measurement technique.