Analysis and correction of spherical aberrations in long focal length measurements
Introduction
Ultra-long focal length lenses (UFL) are widely used in large optical systems, such as space optical systems, high-energy laser devices, and laser fusion programs. However, long focal length lenses are difficult to be determined accurately, due to their small numerical apertures (N.A.s), large focal depths, and long light path measurements, which are easily affected by environmental disturbances [1], [2]. Several high-precision methods are currently used to measure long focal length lenses. For example, Meshcheryakov et al. [3] inserted an optical wedge to the light path with a luminous slit and then removed it. They acquired a measured focal length of 25 m with an error 0.1%. DeBoo and Sasian [4], [5] used a Fresnel-zone hologram to measure a 9 m lens with precision better than 0.01%. This technique is particularly suitable for lenses with slow curvature. However, it is difficult to fabricate proper holograms and the measurement precision is limited by optical lithography when measuring lenses with large curvature. Besides, Zhaoet al. [1], [6] proposed a laser differential confocal technique for long focal length measurement. The focal length was obtained through measuring the variation in position of differential confocal focusing system (DCFS) focus. The precision of this method is about 0.01%, depending on the reference lens in DCFS. However, these methods require stringent environmental temperature, air disturbance, and vibration conditions. To overcome these difficulties, moiré interferometry method based on the Talbot effect and moiré technique are used due to their high measurement accuracy [7], [8], [9], [10], [11].
The Gaussian equation is typically applied to calculate the focal length of lenses. However, it does not consider errors caused by spherical aberrations, so this measurement system requires improvement. In this paper, a highly accurate error correction method for spherical aberrations in long focal length measurements is presented, with a detailed analysis of long focal length measurements based on divergent light and Talbot interferometry. Moiré fringes are transformed into frequency domains, and a new method based on weighted averages is proposed to improve the precision of moiré fringe angles. The modified values are compared with experimental results of interferometry measurements to prove the proposed method’s validity and reliability.
Section snippets
Measurement principle
The working principle of the long focal measurement system with divergent beam is shown in Fig. 1. axis is the direction of the optical axis. The testing lens is placed 4000 mm away from the focal plane of microscope objective. A uniform divergent beam with pupil of 610 mm and focal length of −4000 mm illuminates the testing lens. Then, a square Ronchi grating (G1) with size of 430 and period of 200 um is placed 100 mm behind the testing lens. At the position of G1’s Talbot distance,
Generation of spherical aberrations
Due to spherical aberrations in testing lenses, disagreements between real image points and ideal Gaussian image points are inevitable. As shown in Fig. 2, is the distance between the object point and the front surface vertex of the testing lens . is the distance between the Gaussian image point and the back surface vertex of the testing lens . is the distance between the real image point and the back surface vertex of the testing lens (). is the distance between the
Correction results
The prototype of the system is shown in Fig. 6. First, we corrected the data measured with light located at different distances from the testing lens, as shown in Table 1 and Fig. 7. Decreasing the distance between the light source and lens increased the errors caused by spherical aberrations, that is, the errors between the nominal focal length and the measurement accuracy. After the data was corrected using our system, the focal lengths measured with light located at different distances from
Conclusions
A highly-precise error correction method for spherical aberrations in long focal length measurements based on divergent light and Talbot interferometry was presented in this paper. Simulations and experiments validated the method and its advantages. Moreover, the experimental results indicated that this method’s relative error was approximately 0.007% when measuring a lens with a nominal focal length of 13,500 mm. We also compared the focal lengths of 10 different lenses corrected by our method
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The funding source of this research is ‘National Natural Science Foundation of China ’, with funding code ’6100700’.
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