Objectives: The transmitted wavefront contains a wealth of information which is a very important evaluation index of optical systems as well as studying the optical field. The wavefront of the transmission optical system needs to be tested at specific wavelengths. From now on, wavefronts at only a few wavelengths can be accurately tested because of the limitation of instrument. With the development of technology, wavefront measurements of many of the much-needed optical systems are still not perfect, especially for some special-purpose bands, such as near-infrared. Therefore, a new idea for measuring the wavefront aberration of an optical transmission system in a broad wavelength range is proposed, using existing technologies. The wavefront can be tested at any wavelength by analyzing the functional relationship between transmitted wavefront Zernike coefficients and wavelength, which can indirectly reflect the variation of the wavefront and the wavelength.

Methods: The method of studying the relationship between transmitted wavefront Zernike coefficients and wavelengths by modeling optical systems and analyzing the curves. The first thing is to use Zemax design a monochromatic system, and introduce different errors to the system. Secondly, the Zernike-wavelength curves at 400nm to 1000nm were plotted from collecting the modeled systems with different simulation scenarios. Thirdly, the function of the Zernike-wavelength curves are fitted by Matlab curve fitting tool. And the solved or fitted curves using few data to compare the simulated curves. If the predicted curves are consistent with the simulated curves, it means the formula is a good fit for the functional relationship between Zernike coefficients and wavelengths. Finally, to verity the universality by analyzing more transmission systems with this formula.

Results: The function of Zernike-wavelength curves is obtained from the simulation and analysis. 1) The shape of the Zernike-wavelength curves in different simulation scenarios are similar, so the wavelength-dependent function for the Zernike coefficients are not affected by the errors. 2) The Zernike-wavelength curves of the simulated monochromatic system are monotonic, which can be expressed by many common formulas. However, only the power function and rational function provided in MATLAB fitting tool can fit Zernike-wavelength curves using few data. 3) The change of the transmitted wavefront Zernike coefficients with the wavelength is essentially caused by the change of the refractive index. Among many dispersion formulas, Conrady formula is suitable for fitting a curve to a sparse dataset because of it has the fewest coefficients. Then the first nine Zernike-wavelength curves of simulated monochromatic system are verified by solving Conrady formula. The maximum error of the calculated Zernike coefficients is within 1%. 4) Two main types of traditional transmission systems exist in addition to monochromatic systems, achromatic systems and apochromatic systems. The results show that Conrady formula can also describe Zernike-wavelength curves of partial achromatic systems, but it is not appropriate for apochromatic systems. Compared with other functions, Conrady formula is more suitable to express the Zernike- wavelength curves.

Conclusions：Based on the actual requirements of multi-wavelength wavefront testing, a method for studying the functional relationship between the transmitted wavefront Zernike coefficients and the wavelengths was established. The Conrady-Zernike formula was obtained through simulation and analysis. According to the concluded rules, measuring the transmitted wavefront at arbitrarily wavelength in a certain band can be initially solved in theory, which expands the application of the laser interferometer. Moreover, Zernike polynomials have a wide range of applications in optical design, aberration fitting, and so on. It can provide new research ideas and directions for these fields by the relationship between Zernike coefficients and the wavelength.