## Abstract

A new laser differential confocal ultra-long focal length measurement (LDCFM) method is proposed with the capability to self-calibrate the reference lens (RL) focal length and the axial space between the test lens and RL. Using the property that the focus of laser differential confocal ultra-long focal length measurement system (LDCFS) precisely corresponds to the null point of the differential confocal axial intensity curve, the proposed LDCFM measures the RL focal length *f*
_{R}′ by precisely identifying the positions of the focus and last surface of RL, measures the axial space *d*
_{0} between RL and test ultra-long focal length lens (UFL) by identifying the last surface of RL and the vertex of UFL last surface, and measures the variation *l* in focus position of LDCFS with and without test UFL, and then calculates the UFL focal length *f*
_{T}′ by the above measured *f*
_{R}′, *d*
_{0} and *l*. In addition, a LDCFS based on the proposed method is developed for a large aperture lens. The experimental results indicate that the relative uncertainty is less than 0.01% for the test UFL, which has an aperture of 610 mm and focal length of 31,000 mm. LDCFM provides a novel approach for the high-precision focal-length measurement of large-aperture UFL.

© 2015 Optical Society of America

## 1. Introduction

Ultra-long focal length lens (UFL) with large-aperture is widely used in laser fusion program, such as the National Ignition Facility in Unite States, the Gekko XII in Japan, the Laser Magajoule in French, and the Shenguang III in China [1, 2]. In the development and adjustment of laser fusion system, the requirement for the relative accuracy of UFL focal length measurement is less than 0.02%. However, the accurate focal-length measurement of large aperture UFL is still a great challenge for the following reasons:

- The UFL has a small numerical aperture (N.A.) to make the focal range long due to the diffraction effect, so it is difficult to precisely identify the focus.
- The UFL has a long focal length, which makes the precise distance measurement difficult.
- The UFL has a long measurement light-path and is thus easily affected by environmental disturbance.

Now, the exsiting methods used for the focal length measurement of UFL can be classified into two categories.

One is based on moiré technique and Talbot effect. For example, the moiré deflectometry methods [3, 4] used the relations between the focal length of test UFL and the angle of moiré fringes formed by two gratings behind the test UFL to obtain the UFL focal length, and the relative theoretical measurement error was determined to be less than 0.1%. The Talbot interferometry [5, 6] used Tabot interferometry and Fourier analysis technique to improve the discriminant accuracy of moiré pattern angle through removing the noise caused by grating lines, and the experiment showed that a focal length of 240 mm can be evaluated with a measurement error of less than 0.3%. Based on Tabot interferometry, in [7], C. Hou et al. developed a scanning method to measure the long focal-length for large aperture UFL and have obtained a relative measurement error of better than 0.13% for the UFL with an aperture of 150 mm and focal length of 18,000 mm. Then, in [8], J. Luo et al. used a divergent beam and two gratings of different periods instead of collimated beam and two identical gratings, to achieve long focal-length measurement, and its relative measurement error is determined to be less than 0.0018% for the UFL with focal length of 13,500mm. However, in the practical engineering application, especially for the measurement of large aperture test lens, the angles of moiré fringes are seriously affected by environmental factors such as temperature, vibration and airflow. Therefore, the measurement method is stringent on the environment.

The other is the combination lens long focal-length measurement. The measurement principle is shown in Fig. 1
, which obtains the UFL focal length *f*
_{T}′ by measuring the focus variation *l* in position of measurement system with and without UFL, the focal length *f*
_{R}′ of reference lens RL and the axial space *d*
_{0} between UFL and RL.

Based on combination method, in [1], T.G. Parham et al. used a Fizeau phase-shifting interferometer to measure the focus variation *l* to obtain the UFL focal length, and the experiment showed that a focal length of 30,000mm can be measured with a relative precision of 0.02%. In [9], Brian DeBoo and Sasian applied Fresnel holographic lens to the system instead of RL and have obtained the relative error better than 0.02% for the UFL with focal length of 9,000mm.

As shown in Fig. 1, the measurement precision of combination method is affected not only by the measurement precision of *l*, but also by the precision of *f*
_{R}′ and *d*
_{0}. However, *f*
_{R}′ and *d*
_{0} must be measured by using a non-contact method because the UFL in laser fusion system is coated with anti-reflection film.

In the combination measurement shown in Fig. 1, the key to achieving high-precision measurements of *l*, *f*
_{R}′ and *d*
_{0} is to exactly determine the positions of A, B, C and D by using a non-contact method.

The methods in [1, 9] used a phase-shifting interferometer to identify the positions of focuses A and B, so the measurements were easily affected by environmental disturbances. Furthermore, they cannot directly measure *f*
_{R}′ or *d*
_{0}. So, the measurement environment requirements are stringent.

Therefore, a new laser differential confocal ultra-long focal length measurement meathod (LDCFM) that utilizes self-calibration of *f*
_{R}′ and *d*
_{0} is proposed in this paper. Furthermore, a laser differential confocal ultra-long focal length measurement system (LDCFS) for UFL with large aperture is developed by using LDCFM, which could achieve precise measurements of *l*, *f*
_{R}′, and *d*
_{0} in the same system.

Compared with the existing UFL focal-length measurement methods, the proposed method has higher measurement precision, better stability and stronger tolerance of environmental interference.

## 2. LDCFM principle

As shown in Fig. 2
, the LDCFM uses the property that the focus of LDCFS precisely corresponds to the null points *Q*
_{A} and *Q*
_{B} of the differential confocal axial intensity curves *I*
_{A} and *I*
_{B}, to precisely measure the RL focal length *f*
_{R}′ by identifying the exact positions of RL focus and last surface, to measure the lens space *d*
_{0} between RL and UFL by determining the last surface of RL and the vertex of UFL last surface, and to measure the focus variation *l* in position of LDCFS with and without the measured UFL, and thereby calculating the UFL focal length *f*
_{T}′ by the above measurements of *f*
_{R}′, *d*
_{0} and *l*.

#### 2.1 Combination LDCFM

As shown in Fig. 2, the proposed LDCFM involves the following processes: the light coming from a laser passes through microscope objective (MO) and is focused on the pinhole (PH) to produce a point light source. The divergent beam from light source is transmitted to a beam splitter (BS) and collimated into parallel beam by collimating lens (L_{C}). The parallel beam is then focused onto the surface A of reflector (L_{R}) by reference lens (RL), and the reflected light from L_{R} passes through RL and L_{C} again, and is reflected by L_{B} onto BS. The light is divided by BS into two components, and they are received by detector 1 before the focus and detector 2 after the focus, respectively. Finally, LDCFM yields the differential confocal intensity curve *I*(*u*,*u _{M}*) through differential subtraction of two signals from detector 1 and detector 2.

When UFL is not inserted between L_{C} and RL, the differential confocal intensity response *I*
_{A}(*u*,*u _{M}*) received by the differential confocal system is:

*u*is the axial normalized optical coordinates,

*z*is the axial coordinates,

*u*is the axial normalized optical coordinates of the offset,

_{M}*M*is the axial offset,

*λ*is the beam wavelength,

*f*

_{C}is the focal length of collimating lens. When UFL is not inserted,

*f*is the RL focal length. Otherwise,

*f*is the focal length of combination lens.

When UFL is inserted between L_{C} and RL, the focus position changes from point A to point B, and the differential confocal intensity response *I*
_{B}(*u*,*u _{M}*) received by the differential confocal system is:

The focus variation *l* between focuses A and B can be obtained by using the DMI to precisely measure the distance between *Q*
_{A} and *Q*
_{B}, which correspond to the null points of differential response curves *I*
_{A}(*u*,*u _{M}*) and

*I*

_{B}(

*u*,

*u*), respectively.

_{M}When the RL is a flat convex lens, the UFL back-focal-distance (BFD) *f*
_{TBFD}′ can be obtained from the following combination focal length formula:

For the thickness *b*
_{1}, refractive index *n*
_{1} and radii *r*
_{11} and *r*
_{12} given by preliminary measurements, the effective focal length *f*
_{T}' of UFL can be calculated as follows:

*n*

_{1}is the UFL refractive index,

*b*

_{1}is the UFL thickness, and

*r*

_{11}and

*r*

_{12}are the radii of curvature of the front and back surfaces of the UFL, respectively.

According to Eq. (5), in order to obtain *f*
_{T}
*′*, *f*
_{R}
*′* and *d*
_{0} must also be precisely measured, in addition to focus variation *l*.

#### 2.2 Focal length measurement of RL

As shown in Fig. 3
, LDCFS uses the null points *Q*
_{A} and *Q*
_{E} of differential confocal intensity cuves *I*
_{A}(*u*,*u _{M}*) and

*I*

_{E}(

*u*,

*u*) to find the exact positions of focus A and last surface S, respectively, and to obtain

_{M}*f*

_{R}

*′*by measuring the distance Δ

*L*

_{1}between

*Q*

_{A}and

*Q*

_{E}.

When L_{R} moves towards position A, the differential intensity curve *I*
_{A}(*u*,*u _{M}*) received by the LDCFS is:

When L_{R} moves to position E, the light reflected by L_{R} is focused onto S. When the distance between reflector L_{R} and position E is *z*, the distance between the converging point of the measurement beam and the last surface of the RL is 2*z*, so the differential intensity curve *I*
_{E}(*u*,*u _{M}*) received by the LDCFS is:

The distance Δ*L*
_{1} between positions A and E can be obtained by using the DMI to precisely measuring the corresponding distance between the null points *Q*
_{A} and *Q*
_{E} of differential response curves *I*
_{A}(*u*,*u _{M}*) and

*I*

_{E}(

*u*,

*u*) .

_{M}According to the focal length formula for a flat convex lens, the focal length *f*
_{R}
*′* of RL can be calculated as follows:

*f*

_{RBFD}

*′*,

*n*

_{0}, and

*b*

_{0}are the back focal length, refractive index, and thickness of the RL, respectively, and the errors caused by

*n*

_{0}and

*b*

_{0}can be considered to be negligible.

#### 2.3 Axial space measurement of combination lens

As shown in Fig. 4
, LDCFS uses the null points *Q*
_{F} and *Q*
_{G} of differential confocal curves *I*
_{F}(*u*,*u _{M}*) and

*I*

_{G}(

*u*,

*u*) to find the exact positions of the RL last surface S and the vertex of the UFL last surface P, respectively, the DMI is used to obtain the distance between these two points, and then, LDCFS determines the lens space

_{M}*d*

_{0}by ray-tracing method.

According to ray tracing theory, *d*
_{0} can be calculated as follows:

*r*

_{01}is the radius of RL convex surface,

*b*

_{0}is the RL thickness,

*n*

_{0}is the RL refractive index,

*U*

_{0}is the half of the combination lens aperture angle, and the errors caused by

*r*

_{01},

*U*

_{0},

*b*

_{0}, and

*n*

_{0}can be considered to be negligible.

## 3. Instrument design

Based on the proposed LDCFM above, the LDCFS main structure is designed as shown in Fig. 5 , which includes differential confocal system, collimating lens, 4D-adjuster stage, reference lens, distance measuring system and control system.

The LDCFS is designed as shown in Fig. 6 based on the main structure shown in Fig. 5 and is developed as shown in Fig. 7 .

As shown in Fig. 5, the differential confocal system is mainly composed of laser resource, two intensity detectors and beam splitter, and the wavelength of laser resource is 1.053μm to match the work wavelength of UFL. The detector is a pinhole detector with the diameter of only 15 μm, so it can inhibit the disturbance of the stray light effectively. Besides, all of the optical components of the differential confocal system are integrated inside the LDCFS shell, so the effects on measurements caused by environmental variations including stray light and fluctuation of the light path are further suppressed.

The effective apertures of reference lens and collimating lens are all designed to be Φ610 mm in order to satisfy the measurement requirements of the 430 mm × 430 mm UFL aperture, and an aspherical structure is used in the RL and in the collimating lens to decrease system aberration caused by these lens.

Distance measuring system mainly included distance measurement interferometer DMI, precise air-bearing slider and driving motor. An XL-80 Renishaw dual-frequency interferometer is used as the DMI, which has a relative distance measurement error of less than 1 × 10^{−6}. The air-bearing slider has a straightness of better than 0.2μm and movement range of 4,000mm.

Control system is composed of motor driving system, computer and control software. The reflector and the DMI pyramid move along the air-bearing slider when motor driver controls the movement of air-bearing bushing, and the DMI precisely measures the position of reflector to achieve system scaning and positioning. The computer simultaneously obtains the differential confocal intensity curve by processing the Airy disk received by detector1 and detector2, and the null point of this differential confocal intensity curve corresponds exactly to LDCFS focus.

In order to improve the LCFS stability, the confocal system, collimating lens, 4D-adjuster stage, reference lens, and distance measuring system are installed collinearly in turn on a 8000 mm × 600 mm × 400 mm marble base.

## 4. Uncertainty analysis

#### 4.1 Uncertainty transfer coefficients

The uncertainty transfer coefficients of *d*
_{0}, *f*
_{R}′, and *l* are obtained by differentiating Eq. (4) on *d*
_{0}, *f*
_{R}′ and *l*, respectively.

When *f*
_{R}′ = 2800 mm, *d*
_{0} = 420 mm, and 10000 mm ≤ *f*
_{T}′ ≤ 50000 mm, the calculated uncertainty transfer coefficients of *d*
_{0}, *f*
_{R}′, and *l* are as shown in Fig. 8
.

It can be seen from Eq. (11) and Fig. 8 that the measurement errors of *l* and *f*
_{R}
*′* have large effects on the *f*
_{T}′ measurement, while the measurement error of *d*
_{0} has small effect.

#### 4.2 Uncertainty evaluation

### 4.2.1 Uncertainty caused by *d*_{0}

It is known from [10] that the relative measurement error of aixal space is less than 0.02%, so the stand uncertainty of *d*
_{0} can be described according to the uniform distribution as follows when *d _{0}*≈420mm.

### 4.2.2 Uncertainty caused by *f*_{R}′

According to the uniform distribution, the combined stand uncertainty of *f*
_{R}′ can be written as:

*is the detector offset error, σ*

_{offset}*is the alignment error, and σ*

_{axial}*is the distance measurement error. These errors have been discussed in detail in [11]. In addition, σ*

_{L}_{Δ}

_{L}_{1}is the stand uncertainty of Δ

*L*

_{1}from 20 measurements. The existing errors in LDCFS are determined to be σ

*= 0.3 μm, σ*

_{L}*= 12.5 μm, σ*

_{offset}*= 0.03 μm, and σ*

_{axial}_{Δ}

_{L}_{1}= 7.31 μm after the careful adjustment and experiments. The stand uncertainty of reference lens

*f*

_{R}′is obtained using Eq. (14).

### 4.2.3 Uncertainty caused by *l*

The causes of uncertainty in *l* are maily the errors caused by two detectors with different offsets, the distance measurement error, and axes alignment errors.

(1) Errors caused by two detectors with different offsets

As shown in Fig. 9
, when the offsets of two detectors are different, the null points *Q*
_{A} and *Q*
_{B} of differential confocal curves *I*
_{A} and *I*
_{B} deviate from positions A and B, respectively.Thus, the measurement of *l* changes.

Let the offsets of two detectors from the L_{C} focus be *M* and -*M* + *σ*, and the offsets of *Q*
_{A} and *Q*
_{B} from positions A and B in the measurement process be Δ*l*
_{A} and Δ*l*
_{B}, respectively. Then, the error σ* _{M}* caused by σ can be obtained based on Eqs. (1) and (2) as:

When the reflector L_{R} moves to the positions A and B, the axial normalized optical coordinate *u*
_{A} and *u*
_{B} can be described as, respectively:

The actual offsets Δ*l*
_{1} and Δ*l*
_{1} of positions A and B are, respectively:

Therefore, the error *σ _{M}* caused by the two detectors with different offsets is:

For the LDCFS with L_{C} focal length of 2,800mm, the deviation *σ* can be easily controlled within 50μm by the repeated adjustment.

(2) Distance measurement error

As shown in Fig. 2, the distance between positions A and B is measured by the XL-80dual-frequency Renishaw interferometer, and its measurement error is:

(3) Axes misalignment errors

As shown in Fig. 10
, the axis of RL is considered as the reference axis, the angle between axes of UFL and RL is *α,* the angle between the axes of DMI and RL is *β*. Then, the measurement errors σ* _{α,β}* caused by misalignments between axes can be calculated as:

Angle *α* can be controlled within 1′ by using the self-collimation when three distance sensors and a reflector is used in adjustment, angle *β* can also be controlled within 1′ by the careful adjustment when a slide with movement range of 4,000mm is used.

Besides the aforementioned errors, random errors caused by environment and system noise can be expressed as the standard deviation σ* _{l}* through the several repeated measurements. Thus, according to the uniform distribution, the combined stand uncertainty

*u*(

*l*) is:

#### 4.3 Combined stand uncertainty of f_{T}′

Considering the effects of three uncertainty *u*(*f*
_{R}′), *u*(*d*
_{0}), and *u*(*l*), the total combined stand uncertainty *u*(*f*
_{T}′) is:

*u*(

_{rel}*f*

_{T}′) is

It is verified by repeated experiments that *σ* = 50 μm, *α* = 1′, *β* = 1′ and σ* _{l}* = 12 μm in LDCFS. By using Eqs. (11) and (12)-(25),

*u*

_{c}(

*f*

_{T}′) and

*u*(

_{rel}*f*

_{T}′) can be calculated as shown in Table 1 , when

*f*

_{T}′ is 12,000mm, 20,000mm, 31,000mm, 40,000mm or 50,000mm.

It can be seen from Table 1 that the relative uncertainty of 0.0084% can be achieve with LDCFS when the focal length of UFL is between 12,000mm and 50,000mm.

## 5. Experiments

As shown in Fig. 7, a flat convex lens is used as the RL, which has an aperture of Φ610mm, focal length of 2800 ± 3 mm, *b*
_{0} = 119.79 mm, *n*
_{0} = 1.5067, and *r*
_{01} = −1418.985 mm. A square lens with a aperture of 430 mm × 430 mm and focal length of approximately 31.2 m is used as the UFL with *b*
_{1} = 46.5 mm, *n*
_{1} = 1.5067, *r*
_{11} = 9377 mm, and *r*
_{12} = 28133 mm.

The experimental process is shown in Fig. 11 .

As shown in Fig. 3, when UFL is not inserted between L_{C} and RL, the L_{R} is moved to the position A of RL focus, the measured differential confocal intensity curve is *I*
_{A}(*z*) in Fig. 12
, and the precise position A of RL focus corresponding to null point *Q*
_{A} is *z*
_{A} = −0.5131mm. When the L_{R} is moved to the position E, the measured differential confocal intensity curve is *I*
_{E}(*z*) in Fig. 12, and the precise position S of RL last surface corresponding to the null point *Q*
_{E} is *z*
_{E} = 1358.4987mm. Thus, the RL back-focal-length is *f*
_{RBFD}′ = 2|*z*
_{A}-*z*
_{E}| = 2718.0236mm.

The average focal length of RL is *f*
_{R}′ = 2797.522 mm and the repeatability is σ_{R} = 14.6 μm from twenty measurements.

As shown in Fig. 2, when UFL is inserted between L_{C} and RL, the L_{R} is moved to the position B of combination lens focus, the measured differential confocal intensity curve is *I*
_{B}(*z*) in Fig. 13
, the precise position B of combination lens focus corresponding to the null point *Q*
_{B} is *z*
_{B} = 233.2757 mm. Then, UFL is removed from LDCFS and the L_{R} is moved back to the position A of RL focus, the measured differential confocal intensity curve is depicted by the *I*
_{A}(*z*) in Fig. 13 and the precise position A of RL focus corresponding to the null point *Q*
_{A} is *z*
_{A} = −0.0141 mm. Thus, the variation in focus is *l* = *z*
_{B} - *z*
_{A} = 233.2898 mm.

Similarly, the coordinates of positions F and G are *z*
_{F} = −286.4262 mm and *z*
_{G} = −0.0002 mm, respectively. The distance between positions F and G is Δ*L*
_{2} = 286.4264 mm, the axial space between UFL and RL is *d*
_{0} = 418.29 mm, as calculated by using Eq. (9).

From 20 measurements as shown in Fig. 14(*a*)
, the repeatability of *l* is σ* _{l}* = 10.89 μm, the average focal length of UFL is

*f*

_{T}′ = 31,218.34 mm, and the repeatability of

*f*

_{T}′ is σ

_{T}= 1.56 mm.

In order to further verify the stability and reproducibility of LDCFS, ten comparative measurements with UFL repeatly inserted and adjusted at different time are obtained which includes twenty measurements in one point, and the averages *f*
_{Tavg}′ from ten measurements are shown in Fig. 14(*b*).

Substituting *u*(*d _{0}*) = 4.8 μm,

*u*(

*f*

_{R}) = 7.9 μm and σ

*= 10.89 μm into Eqs. (23), (24) and (25), the combined stand uncertainty*

_{l}*u*

_{c}(

*f*

_{T}) is:

*f*

_{T}′ is:

Considering some negligible errors, the relative stand uncertainty of LDCFS can be less than 0.01%.

## 6. Conclusions

In this paper, a new LDCFM is proposed with the self-calibration capability of the RL focal length and axial space between UFL and RL. Based on the prosposed method, a LDCFS with large aperture is developed to verify its validity. Compared with the existing method, the LDCFM has the following advantages.

- It can achieve the high-precision self-calibration of RL focal length by using differential reflection-confocal focusing technology, thus eliminating the effect of RL machining error and further improving the measurement accuracy.
- It can achieve the high-precision self-calibration of axial space between UFL and RL by using differential reflection-confocal chromatography focusing technology, thus proventing damage to the UFL coating film.
- It greatly shortens the length of light-path by using combination lens focal length measurement and obviously reduces the volume of LDCFS, so the environmental anti-interference capability is improved effectively.

The experiments indicate that LDCFM can achieve a relative measurement precision of less than 0.01% with the focal length of 31,000mm and thus provide a novel approach for the high-precision focal-length measurement of large-aperture UFL.

## Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China (No. 61327010) and National Instrumentation Program (NIP, No.2011YQ04013602).

## References and links

**1. **T. G. Parham, T. J. McCarville, and M. A. Johnson, “Focal length measurements for the National Ignition Facility large lenses,” Optical Fabrication and Testing (OFT 2002) paper: OWD8 http://www.opticsinfobase.org/abstract.cfm?URI=OFT-2002-OWD8*.* [CrossRef]

**2. **C. Jin, S. Liu, Y. Zhou, X. Xu, C. Wei, and J. Shao, “Study on measurement of medium and low spatial wavefront errors of long focal length lens,” Chin. Opt. Lett. **12**, S21203 (2014).

**3. **J. C. Bhattacharya and A. K. Aggarwal, “Measurement of the focal length of a collimating lens using the Talbot effect and the moiré technique,” Appl. Opt. **30**(31), 4479–4480 (1991). [CrossRef] [PubMed]

**4. **I. Glatt and O. Kafri, “Determination of the focal length of nonparaxial lenses by moire deflectometry,” Appl. Opt. **26**(13), 2507–2508 (1987). [CrossRef] [PubMed]

**5. **Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. **24**(19), 3162–3166 (1985). [CrossRef] [PubMed]

**6. **P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, “Measurement of focal length with phase-shifting Talbot interferometry,” Appl. Opt. **44**(9), 1572–1576 (2005). [CrossRef] [PubMed]

**7. **H. Changlun, B. Jian, H. Xiyun, S. Chen, and Y. Guoguang, “Novel method for testing the long focal length lens of large aperture,” Opt. Lasers Eng. **43**(10), 1107–1117 (2005). [CrossRef]

**8. **J. Luo, J. Bai, J. Zhang, C. Hou, K. Wang, and X. Hou, “Long focal-length measurement using divergent beam and two gratings of different periods,” Opt. Express **22**(23), 27921–27931 (2014). [CrossRef] [PubMed]

**9. **B. DeBoo and J. Sasian, “Precise focal-length measurement technique with a reflective Fresnel-zone hologram,” Appl. Opt. **42**(19), 3903–3909 (2003). [CrossRef] [PubMed]

**10. **W. Zhao, R. Sun, L. Qiu, L. Shi, and D. Sha, “Lenses axial space ray tracing measurement,” Opt. Express **18**(4), 3608–3617 (2010). [CrossRef] [PubMed]

**11. **J. Yang, L. Qiu, W. Zhao, and H. Wu, “Laser differential reflection-confocal focal-length measurement,” Opt. Express **20**(23), 26027–26036 (2012). [CrossRef] [PubMed]