## Abstract

A new accurate method for long focal-length measurement based on Talbot interferometry is proposed. A divergent beam and two Ronchi gratings of different periods are employed, as the alternative of the collimated beam and two identical gratings, to achieve higher measurement accuracy. Moreover, with divergent beam, lenses of large aperture can be easily measured without scanning, which is required when it comes to traditional collimated beam. Numerical analysis and experiments were carried out. The results demonstrate the proposed method features remarkably high accuracy and repeatability.

© 2014 Optical Society of America

## 1. Introduction

Ultra-long focal-length lens (UFL) is now being widely used in large optical systems, such as space optical system, high energy laser system and laser fusion program. It is of great significance to develop a fast, real-time measurement system to satisfy the stringent requirement for ultra-precision long focal-length measurement. However, the accurate measurement is still a great challenge because it is difficult to find the exact position of the focus and the long measurement light-path can be easily affected by the disturbance of environment [1, 2]. Several approaches have been proposed in previous literatures. For example, in 1999, Meshcheryakov et al. [3] acquired a measured focal-length of 25 m with an error 0.1% by changing the position of a luminous slit with the help of an optical wedge. Moreover, DeBoo and Sasian [4, 5] applied a Fresnel-zone hologram to focal-length measurement with precision better than 0.01% when measured a 9 m focal-length lens. This technique is particularly useful for testing large and slow lenses. However, for lenses of large curvature, it is difficult to fabricate suitable hologram and the measurement precision is limited by optical lithography. Besides, a laser differential confocal technique for long focal-length measurement was presented by W. Zhao et al. [1, 6]. They measured the variation in position of the differential confocal focusing system (DCFS) focus with and without a UFL. The focal-length of UFL is obtained from the distance between the two focuses. The precision, depending on the reference lens, achieves about 0.01%. Although those methods have already reached high precision, the stringent requirement for environmental temperature, air disturbance and vibration is hardly achieved, and the machining precision of some optics components limits the further improvement of the measurement precision as well.

The most commonly used method to examine the focal-length of UFL is the moiré interferometry method, which is based on Talbot effect and the moiré technique. According to the Talbot effect theory [7–9], the period of the Talbot image of the Ronchi grating is related to the radius of the spherical wave when the Ronchi grating is illuminated by spherical wave. Moire´ pattern is formed when another Ronchi grating is put in the place of the Talbot image. Correspondingly the angle of the moire´ pattern is related to the radius of the spherical wave. This method is simple and relatively insensitive to environmental effects. Several articles [10–18] have discussed this method in details. In order to test UFL lens with large aperture, H. Changlun et al. [19] developed a scanning method to measure the focal length with the collimated lights projected on two equal-period gratings, and then X. Jin et al. [20] presented a calibration method to improve the measurement accuracy. The sensitivity of moiré interferometry is easily tuned by varying the crossed angle between two gratings, thus high measurement accuracy is achieved. However, this method requires ultra-high collimation of the incident light, which is difficult to satisfy. For example, less than 0.006 milli-radian of divergence angle is demanded to get 0.01% focal-length accuracy. Additionally, the inevitable cumulative errors induced by the scanning system make an enormous inaccuracy. Consequently, it is of very importance to find another way to fulfill the full-aperture measurement of the UFL lens.

In this paper, we propose and demonstrate a feasible and accurate method for long focal-length measurement to solve these problems. A divergent beam is employed instead of the traditional collimated beam to realize full-aperture measurement. To equip with the divergent beam, two Ronchi gratings of different periods are used as the alternative to the traditional identical gratings. In this way, the collimation process and scanning system are avoided, thus the stability and accuracy of the measurement are improved and the testing time is tremendously reduced accordingly. Furthermore, the measurement result is obtained through a simple calculation of the tilt angle of the moiré pattern, depending on neither precise adjusting mechanism nor troublesome manual adjustment.

This paper is constructed as follows. Description of the long focal-length testing system and the principle of this method will be presented in Section 2; Section 3 provides a numerical analysis of the proposed method; the proof test and results are demonstrated in Section 4 and Section 5 shows the discussions. This paper will be summarized in Section 6.

## 2. Measurement principle

A long-focal-length testing system (LFTS) that mainly consists of two gratings and a divergent beam generator is illustrated in Fig. 1. The planes of two gratings G1 and G2 are parallel to each other, and both set perpendicular to optic axis. A diffuse plate is placed close to the back of the grating G2 and a CCD camera focuses on the plate. A test lens is added in front of the grating G1. A divergent beam emitted from the divergent beam generator passes through the test lens and grating G1, then illuminates the grating G2, creates a moiré pattern on the diffuse plate. The moiré pattern is imaged by the CCD camera.

Figure 2 shows a schematic representation of the optical system, where (a) and (b) are the cases of without and with test lens, respectively. In Fig. 2(a), the focal-length of the divergent beam is denoted by${f}_{wave}$. The monochromatic divergent wave illuminates grating G1 and produces a Talbot image G1' at distance $Z=k{p}_{1}^{2}/\lambda $, where ${p}_{1}$ is the period of grating G1, $\lambda $ is the wavelength of laser source, and $k=1,2,\mathrm{3...}$is the Talbot plane number. However, as the divergent wave is illuminated, the Talbot image G1’ is magnified by a factor$\eta $. By superimposing the magnified Talbot image G1' on the second grating G2, which is placed at the Talbot distance$Z$, a moiré fringe pattern is formed after grating G2.

The angle $\alpha $between the moiré pattern’s fringe and the x axis is given by

Where ${p}_{2}$ and ${p}_{1}\text{'}$are the periods of grating G2 and the magnified Talbot image G1’, and $\theta $is the angle between gratings G1 and G2 in x-y plane.

According to the geometrical optics, in Fig. 2(a), the magnification factor $\eta $ can be obtained by

where${f}_{wave}$takes negative value, regarded as the focal-length of the beam illuminated on the grating G1, and Z is Talbot distance.According to the Eqs. (1) and (2), the focal-length ${f}_{wave}$can be expressed by

Then, as shown in Fig. 2(b), a test lens is inserted into the light path in front of the first grating G1. The new focal-length of the beam illuminating on the grating G1 is referred to as combined focal-length${f}_{com}$, which is combined by the focal-lengths of the testing lens ${f}_{lens}$and the divergent wave${f}_{wave}$. In this case, grating G2 has to be moved along the z-axis to find the corresponding Talbot distance $Z\text{'}$to match the combined new wave. Therefore, similar to Fig. 2(a), the combined focal-length is determined by the following equation:

In Eq. (5), $Z\text{'}$is the new Talbot distance between two gratings. The diagram of the test lens is shown in Fig. 2(b). The principle distances ${l}_{H}$and$l{\text{'}}_{H}$, and the distance from the back principle plane to grating G1 ${d}_{2}$can be expressed as:

Where, ${r}_{1}$ and ${r}_{2}$ are the radiuses of two surfaces of the lens, $d$is the thickness of the lens, and $n$is the refractive index. The distance${d}_{1}$, from front surface of lens to grating G1can be measured precisely by a grating ruler. When the lens parameters are confirmed by the preliminary measurement, the focal-length of the measured lens${f}_{lens}$is obtained by:

## 3. Numerical analysis

According to the previous formulas (1) to (7), the measurement result of the focal-length${f}_{lens}$ is mainly depends on the eight parameters${f}_{wave}$,${d}_{1}$,$Z\text{'}$,$\theta $,$\alpha $,${l}_{H}$,$l{\text{'}}_{H}$and$\beta $. Following, we present an analysis of the uncertainty of measured focal-length.

#### 3.1 Uncertainties analysis

The uncertainty of measured focal-length ($\delta $) can be expressed as follows,

In Eq. (8), $\partial {f}_{lens}/\partial p$ is operation of partial differential on parameter *p*, which is regarded as uncertainty transfer factor of parameter *p* and can be easily obtained from Eq. (7). $\Delta p$is the uncertainty of parameter *p*.

The uncertainties of each parameter are obtained as follow. The${f}_{wave}$,${d}_{1}$are measured by the grating ruler and the$Z\text{'}$is obtained by a precision displacement platform, the measurement uncertainties $\Delta {f}_{wave},\Delta {d}_{1},\Delta Z\text{'}$are better than 0.01mm. By using a calibration method [20], the uncertainties $\Delta \theta ,\Delta \alpha $ achieve 0.005°and 0.003°, respectively. The principal distances${l}_{H},$$l{\text{'}}_{H}$are determined by the ${r}_{1},$${r}_{2}$ and $d,$ the uncertainties of ${r}_{1},$${r}_{2}$ and$d$are ensured within 2.573 mm, 0.285 mm and 0.1 mm according to the specifications provided by manufacturer. Hence the uncertainties $\Delta {l}_{H}$and $\Delta l{\text{'}}_{H}$are estimated better than 0.1 mm and 0.03 mm.

The uncertainty of measured focal-length is very sensitivity to the parameter$\beta $. The uncertainty $\Delta \beta $ is worse than 0.001 under the consideration of the ± 0.1 um tolerance on the grating period, which is unacceptable for the requirement. Therefore we used an experimental method to determine the value $\beta $ and its uncertainty$\Delta \beta $. According to Eq. (3), the$\beta $can be expressed as:

Through Eq. (10) the value of$\beta $can be determined. In this way, the uncertainty$\Delta \beta $ is given by:

By substituting the uncertainties of parameters${f}_{wave},Z,\theta ,\alpha $, the uncertainty $\Delta \beta $ obtained is better than 0.0000089.

#### 3.2 Numerical simulation

Figure 3 shows the uncertainty of measured focal-length caused by parameter$\beta $simulated at several nominal focal-lengths using the experimental parameters of ${f}_{wave}\text{=}4000$mm,$Z\text{=15}00$ mm, ${d}_{1}\text{=}286$ mm, and $\theta ={0.3}^{o}$, and the nominal principle distances ${l}_{H}=44.22$ mm and $l{\text{'}}_{H}=14.74$ mm. This simulation also used the uncertainties of the parameters mentioned previously. As shown in Fig. 3(a), a large margin of errors are seen obviously when$\beta $is equal to 1, i.e. two identical gratings are not good choose to our system. There is minimum located at the point${\beta}_{0}=\text{0}\text{.72727}$, and away from this minimal point, the uncertainty increased rapidly. Figure 3(b) shows magnification plots around this minimal point to show clearly. At this minimal point, based on Eq. (3), we get$\alpha \approx \text{0}\text{.15}\xb0$, which is very close to the half angle of two gratings$\theta /2$. This feature indicates our system is very sensitive to the parameter$\beta $.

Hence, in order to realize the high accuracy, two gratings of different periods are necessary and the experimental parameters should be set to match well with the grating periods.

After confirm the values of the parameters, the uncertainty transfer factors of all the parameters and the uncertainties of measured focal-length caused by each parameter can be calculated. Here we give a sample when measuring a 15000 mm-focal-length lens. For ${f}_{wave}$,${d}_{1}$,$Z\text{'}$,$\theta $,$\alpha $,${l}_{H}$,$l{\text{'}}_{H}$and$\beta $, the uncertainty transfer factors are 13.75, −19.56, 7.33, 525.15 mm$/\xb0$, 1050.30 mm$/\xb0$, 6.42, −7.34 and −21325.75 mm respectively, and the uncertainties of measured focal-length caused by each parameter are 0.14, −0.20, 0.07, −0.03, 0.09, 0.64, −0.25 and −1.06 in unit of mm, respectively. The similar trend can be obtained in the cases of other focal-lengths. We can see that the uncertainty of measured focal-length in our method is sensitive to the parameters$\theta $,$\alpha $and$\beta $.

Simulations are also carried out to assess the uncertainties of different measured focal-lengths. Figure 4 shows the simulation results. For comparison, Fig. 4(a) is the result with traditional method using collimated light beam configuration [19], and Fig. 4(b) is that with the presented method.

According to Fig. 4(a), the relative uncertainties of the traditional method are 0.67%, 0.71% and 0.73% for nominal focal-lengths of 10000 mm, 20000 mm and 30000 mm, respectively. As shown in Fig. 4(b) the presented method achieves uncertainty of 0.005% when it comes to the 10000 mm focal-length lens. The uncertainties achieve 0.012% and 0.019% when 20000 mm focal-length lens and the 30000 mm focal-length lens are tested, respectively. Compared with the traditional method with collimated beam, the uncertainty of the presented method improves at least 50 times.

## 4. Experiments and results

The experimental setup for long-focal-length testing system consists of a divergent beam generator, a moiré pattern formation part, and a detection part. The divergent beam generator includes an infrared laser (wavelength$\lambda \text{=}1053$mm), a 0.65 NA micro objective (40 × ) and a pinhole. The moiré pattern formation part includes a diffuse plate and two gratings with periods of 200 um and 275 um. The grating G2 is connected with a 0.005°precision rotating platform (SA16A-RM) and then fitted on a 50 um precision displacement platform (PAKER-412XR). The detection part consists of two reflect mirrors and a 1628 × 1236 CCD camera. The focal-length of the imaging lens on camera is 35 mm and the distortion is less than 0.1%. The test lens is inserted in the front of grating G1 when tested. An autocollimation is utilized to ensure the parallelity of test lens and two gratings. Figure 5 is a real product photo of the LFTS.

The flow chart of the experimental steps is shown in Fig. 6: first we measure the distance between pinhole and the first grating G1 i.e. the focal-length of divergent beam ${f}_{wave}$with a grating ruler. The Talbot distance *Z* is obtained by the precision displacement platform. According to Eq. (10), the parameter $\beta $is calculated from the angle$\alpha $of the moiré fringe without test lens. Then we fix the test lens on the optic axis and measure the distance between the lens and grating G1${d}_{1}$. The crossed angle between the two gratings is set at$\theta ={0.383}^{o}$. The corresponding Talbot distance$Z\text{'}$is obtained by moving grating G2 until the tilt angle $\alpha $of dynamic fringe achieves 0.1915°. Finally, the focal-length of lens${f}_{lens}$could be computed according to Eq. (7).

Figure 7 shows the moiré pattern obtained in the experiments. Figure 7(a) is the image of fringes when the grating G2 is at Talbot distance$Z$without test lens. Then the test lens is inserted and Fig. 7(b) is obtained. As the grating G2 is not placed at the Talbot distance, the fringes are thick at this point. Figure 7(c) shows the image of fringes captured when grating G2 is moved to the corresponding Talbot distance$Z\text{'}$, which is the right place for calculation.

Twelve pieces of lenses are tested, and the results are shown in Table 1. In Table 1., columns from the left to right are the No. of test lens, nominal focal-length, measured focal-length, simulated uncertainty, difference between the measured focal-length and the nominal focal-length, and error, respectively. The results show the measured focal-length of the lenses agrees well with the nominal values, and the error is better than 0.16%.

In order to verify the new method works with high accuracy, we perform further comparative experiments. These lenses are also tested by the interferometry approach. To reduce the effect of air turbulence and electronic noise, 10 continuous measurement values are averaged for each measurement point. The results are shown in Fig. 8, in which the blue and red curves represent the errors of the interferometry approach and our LFTS, respectively. Good agreement between two systems is demonstrated. The difference of the errors between two systems is represented by the green curve, which is better than 0.06%, indicating the high relative precision of our system. Comparison between our method and the interferometry technique reveals that we could measure the long focal-length extremely accurately by using the proposed system.

The repeatability and the long-term stability of LFTS are also demonstrated. A 13500 mm focal-length test lens is measured repeatedly several times, and each time the test lens is re-inserted and then one group of 10 continuous measurements is done. Each spot in Fig. 9 indicates the measured focal-length, which is average value of the 10 continuous measurements. Figure 9(a) is the results of repeatability. The root-mean-square (RMS) error of repeatability is 0.1470 mm, better than 0.0011%. It is clearly that the installation error of our system is infinitesimal and only has little effect on the measuring results. Figure 9(b) shows the results of long-term stability, which consist of 40-groups tests with time interval of 2 hours. The RMS error of those values is 0.2468 mm, better than 0.0018%. The small RMS error indicates this method is of high accuracy and insensitive to the testing environment. The high-quality validity, repeatability and stability of the proposed new method are shown obviously.

Moreover, in general method, such as interferometry approach, the light path is as long as focal-length, making it difficult to focus. LFTS uses shorter light path and avoids the focusing process, tremendously reduces the testing time. It only takes 15 minutes for each lens measurement, while the interferometry approach takes more than 1 hour.

## 5. Discussion

The experimental results above are just performed in a general laboratory circumstance. However, the experiment environments, like temperature and vibration, affect the results. For example, as the distance changes with the fluctuation of the environmental temperature, we measure the distance between laser source and the first grating G1 by the grating ruler, and the tolerance for the maximum and minimum is about 0.2 mm during one day. Thus, if the constant temperature room and the air bearing table are applied, the accuracy and the stability can achieve a higher level.

The autocollimation utilized to ensure the parallelity of two gratings also makes a great contribution to reduce the installation error in the lens fixing process. The two crosshairs reflected from the front and back surfaces of the lens are superimposed only when the lens is perpendicular to the optical axis and the center of the lens is right on the axis. The accuracy of the autocollimation achieves 0.75″, which is fully satisfied the measurement requirement.

There are two ways to obtain the focal-length of the divergent wave ${f}_{wave}$in the experiment, the calculation from the moiré pattern and the grating ruler measurement. These two approaches agree well with each other and a calibration is implemented to improve the accuracy.

The actual principal point is hard to measurement in the experiment. Hence in our system the nominal${l}_{H}$,$l{\text{'}}_{H}$are employed, which may have some uncertainty. However, as shown in the numerical analysis, the uncertainty of measured focal-length caused by the${l}_{H}$and$l{\text{'}}_{H}$is less than 1 mm.

Note that, the aberration of lens may affect accuracy of angle observation. In Fig. 7(c) we can see the moiré pattern bends in the marginal areas. The measuring results are variational when using different aperture for calculation, such as 13498.9 mm for$512\times 512$pixels aperture and 13499.6 mm for full aperture, respectively (13500 mm focal length). This part will be discussed with other error consideration in the future publications.

## 6. Conclusion

A new method of long focal-length measurement based on Talbot interferometry is proposed. In this method, a divergent laser beam is employed as an alternative to the traditional collimated beam, and the scanning system in traditional method is not needed. This avoids collimation and scanning errors and reduces testing time remarkably. Meanwhile, two gratings of different periods are employed to match the divergent beam for higher measurement accuracy. From both numerical simulation and experiment, our method exhibits good performance in testing accuracy. In addition, the high efficiency of this method would be presented in testing large aperture lens. Note that, the aberration of lens may affect accuracy of angle observation, which will be discussed with other error consideration in the future publications.

## Acknowledgments

We gratefully thank L. Chai and Q. Li from the Research Center of Laser Fusion for their contributions. Portions of this work were performed under the auspices of the Research Center of Laser Fusion.

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