## Abstract

A new laser differential confocal ultra-large radius measurement (LDCRM) method is proposed for high-precision measurement of ultra-large radii. Based on the property that the zero point of a differential confocal axial intensity curve precisely corresponds to the focus points of focusing beam, LDCRM measures the vertex positions of the test lens and the last optical surface of objective lens to obtain position difference *L*_{1}, and then measures the vertex positions of the reflector and the last optical surface of objective lens to obtain the position difference *L*_{2}, finally uses the measured *L*_{1} and *L*_{2} to calculate the radius of test lens. This method does not require the identification of confocal position. Preliminary experimental results and theoretical analyses indicate that the relative uncertainty is 0.03% for a convex spherical lens with a radius of approximately 20 m. LDCRM provides a novel approach for high-precision ultra-large radius measurement.

© 2016 Optical Society of America

## 1. Introduction

Spherical lenses with ultra-large radii are widely used in large optical systems like laser fusion systems, astronomical telescopes and high-resolution observation systems. The radii of some spherical lenses in laser fusion programs measure up to 50 m. The imaging quality and performance of these optical systems depend on the accuracy of radius measurement. The existing radius measurement methods mainly include contact and noncontact methods.

Contact measurement methods use instruments like spherometers, profilometers, and coordinate measuring machines, which can scratch or deform the test surface. The measurement accuracy is only 0.2% for a test lens with radius 20 m [1,2]. Noncontact measurement methods mainly use shearing interferometry [3], Newton interferometry [4], Talbot interferometry [5], etc. Yun Woo Lee et al. used a half-aperture bidirectional shearing interferometer and directly compared the fringe widths in the two fields to obtain the radius of a test lens (10 m) with a relative measurement error of 0.2% [6]. K. V. Sriram et al. calculated the radius of a test lens (5 m) with an uncertainty of 0.5% using a technique based on Talbot interferometry and angle variation in moiré fringes [7].

A conventional method of noncontact radius measurements is the ‘cat’s eye and confocal’ method. The measurement principle is shown in Fig. 1. The radius *R* of a test lens is obtained by identifying the cat’s eye position A on the surface of the test lens and the confocal position B on the center of the test sphere [8]. Lars A. Selberg et al. used phase shifting interferometry to determine the cat’s eye position and confocal position of a test lens (*R* less than 100 mm) with a relative error of 0.001% [9]. In order to extend the range of radius measurement, Quandou Wang et al. devised a technique that combined Fizeau interferometry and a holographic zone plate to shorten the light path, and measured the radius of a test lens (10 m) with an error of 0.1% [10]. However, in practice, the interferometry fringes are affected by environmental factors such as temperature, vibrations, and airflow. Therefore, these above-mentioned methods have measurement uncertainties associated with environmental fluctuations [11,12]. In order to improve the measurement precision and tolerance to environmental factors, we previously proposed a laser differential confocal radius measurement method that uses the differential confocal focusing technique to identify the cat’s eye position and confocal position, with a relative measurement precision of 5 ppm [13].

However, the above measurement technique still has the following problems:

- 1) As shown in Fig. 1, when measuring the radius of a convex lens, focal length
*f ′*of the objective lens L_{O}needs to be larger than radius of the test lens. This increases the working distance of L_{O}and the moving distance of test lens, and lowers the tolerance to environmental disturbances. Therefore, this method is not suitable for measuring the radii of large convex lenses. - 2) As shown in Fig. 1, when the test lens has a small vibration, the deviation in focus point detected by positioning system at the confocal position B is much larger than that of cat’s eye position A [14,15]. Thus, this method is stringent in signal processing, adjustment precision and the fields of detectors.

The key issues that need to be addressed in ultra-large radius measurement techniques are, reducing the moving distance of test lens, and avoiding large deviations at confocal position.

Therefore, a new laser differential confocal ultra-large radius measurement (LDCRM) method is proposed in this paper which does not require the identification of the confocal position. This method uses differential confocal focusing techniques to obtain the radius of a test lens by precisely identifying the cat’s eye position and the vertex position on the last optical surface of the objective lens. Compared to the existing methods, LDCRM technique shortens the light path and the moving distance of a test lens. This method also extends the measurement range of radii and improves the tolerance to environmental disturbance.

## 2. LDCRM principle

As shown in Figs. 2 and 3, using the property that the zero point *Q* on the differential confocal axial intensity curve *I*(*u*, *u _{M}*) corresponds precisely to the focus point of the focusing beam, LDCRM measures the position differences

*L*

_{1}of a test lens (TL) by precisely identifying the vertex positions of TL and the last optical surface of L

_{O}, and then measures the position differences

*L*

_{2}of a reflector (RL) by precisely identifying the vertex positions of RL and the last optical surface of L

_{O}, finally the radius

*R*of the TL is calculated from the measured

*L*

_{1}and

*L*

_{2}, and thus, high precision measurement of the ultra-large radius is achieved.

#### 2.1 Position difference L_{1} measurement

As shown in Fig. 2, a laser beam is transmitted through a fiber to produce a divergent beam, which is then passed through a beam splitter (BS) and collimated into a parallel beam using a collimating lens (L_{C}). The parallel beam is then focused onto the cat’s eye position A by the objective lens (L_{O}). The reflected light from the TL passes through L_{O} and L_{C} again and reflected by the BS onto another beam splitter (L_{B}). The light is split by L_{B} into two components, which are received by detector 1 before the focus and detector 2 after the focus. Finally, LDCRM yields the differential confocal intensity curve *I*_{A}(*u*, *u _{M}*) through differential subtraction of two signals from detector 1 and detector 2. When the TL moves to position B, the converging beam is reflected by the TL and focused on the vertex position P on the last optical surface of L

_{O}. The reflected light from position P is also received by the detectors, and the differential confocal intensity curve

*I*

_{B}(

*u*,

*u*) is generated.

_{M}When the TL moves towards the cat’s eye position A, the differential confocal intensity curves *I*_{A}(*u, u _{M}*) generated by the differential confocal system is:

Here *λ* is the wavelength of the laser beam, *u* is the axial normalized optical coordinate, *D* is the effective aperture of collimating lens L_{C} and objective lens L_{O}, which is equal to the smaller aperture between L_{C} and L_{O}. *f _{O}* is the focal length of L

_{O},

*z*is the axial coordinate,

*M*is the axial offset,

*u*is the axial normalized optical coordinate of the offset and

_{M}*f*

_{C}is the focal length of L

_{C}.

When TL moves towards position B, the differential confocal intensity curves *I*_{B}(*u, u _{M}*) generated by the differential confocal system is:

The position difference *L*_{1} between A and B can be obtained by using the distance measuring instrument (DMI). It gives a precise measure of the distance between *Q*_{A} and *Q*_{B}, which correspond to zero points on the differential confocal intensity curves *I*_{A}(*u, u _{M}*) and

*I*

_{B}(

*u, u*), respectively.

_{M}#### 2.2 Position difference L_{2} measurement

As shown in Fig. 3, when RL moves to the cat’s eye position A, the focusing beam is reflected by RL. The reflected beam from RL is received by the differential confocal system, and the differential confocal intensity signal *I*_{A}(*u*, *u _{M}*) is generated. When RL moves to position C, the focusing beam is reflected by RL onto the vertex position P on the last optical surface of L

_{O}, and the differential confocal intensity curve

*I*

_{C}(

*u*,

*u*) is generated.

_{M}When RL moves towards cat’s eye position A, the differential confocal intensity curves *I*_{A}(*u, u _{M}*) generated by the differential confocal system is:

When RL moves towards position C, the light reflected by RL is focused onto P. When the distance between reflector RL and position C is *z*, the distance between the converging point of the focusing beam and the last surface of the L_{O} is 2*z*, then the differential confocal intensity curves *I*_{C}(*u, u _{M}*) received by the differential confocal system is:

The position difference *L*_{2} between A and C can be obtained using DMI to precisely measure the distance between *Q*_{A} and *Q*_{C}, corresponding to the zero points of differential confocal intensity curves *I*_{A}(*u, u _{M}*) and

*I*

_{C}(

*u, u*), respectively.

_{M}#### 2.3 Radius calculation

As shown in Fig. 4, the angle between the light axis *m* and light beam is *θ*, the distance between the cat’s eye position A and the vertex position P is *L*, the normalized aperture of light beam is *ρ*, the radius of TL in normalized aperture *ρ* is *R _{ρ}*, and the angle between light axis and

*R*is

_{ρ}*θ*

_{1}.

According to the geometrical optics theory, it follows that *L*, *ρ*, *L*_{1}, *L*_{2}, *θ, θ*_{1}, *D* and *R _{ρ}* satisfy Eq. (7) as follows:

The value of *R _{ρ}* can be calculated from Eq. (7) and radius

*R*of the TL is obtained by integrating

*R*with respect to variable

_{ρ}*ρ*from 0 to 1, such that,

From the measured position differences *L*_{1} and *L*_{2}, and using Eqs. (7) and (8), a high-precision measurement of *R* can be achieved.

## 3. System and experiments

#### 3.1 System design

From Eqs. (7) and (8), we obtain the relations between ∂*R*/∂*L*_{1} and *L*_{2}, and ∂*R*/∂*L*_{2} and *L*_{2} as shown in Fig. 5 when radii of TL are 5 m, 10 m, 15 m and 20 m, respectively.

It can be seen from Fig. 5 that the error transfer coefficients ∂*R*/∂*L*_{1} and ∂*R*/∂*L*_{2} decrease rapidly when the parameter *L*_{2} of objective lens *L*_{O} increases, and thereby it is beneficial to improve the measurement precision. However, increase in *L*_{2} extends the light path significantly and reduces the tolerance to environmental disturbance, thus making the setup more difficult to engineer. Considering the factors of measurement precision and engineering difficulty, an objective lens (Zygo Corporation) with parameter *L*_{2} = 522.5 ± 0.5 mm was chosen to be used as *L*_{O}.

The focusing sensitivity at zero-crossing point for different *u _{M}* and normalized radius

*v*

_{PH}of pinhole are shown in Figs. 6(a) and 6(b), respectively.

It can be seen from Fig. 6 that the focusing sensitivity is best when *u _{M}* = 5.21 and

*v*

_{PH}≤ 2.

Then, the main structure of LDCRM system, which includes differential confocal system, distance measuring system and control system is designed as shown in Fig. 7.

The actual LDCRM system is shown in Fig. 8.

In the LDCRM system, a He-Ne laser (*λ* = 632.8 nm) is used as the light source. An achromatic lens (Linos Corporation) with a focal length of 1250 mm and diameter of 150 mm is used as the collimating lens L_{C}, and the maximum measuring aperture of TL is *D*_{max} = 0.5 × 150 mm = 75 mm. An XL-80 laser interferometer (Renishaw Corporation) is used as the DMI and its relative measurement precision is 1 ppm. A high-accuracy air bearing slider made in our laboratory with a straightness of 0.3 μm and an effective moving range of 2000 mm is used as the motion rail. A convex lens with a radius of 20 m ( ± 1%) is used as TL in the following experiments.

#### 3.2 Experiments

The following experiments were done under the following conditions: the pressure is 102,540 ± 60 Pa, the temperature is 21.0 ± 0.5°C, and the relative humidity is 44 ± 5%.

### 3.2.1 Position difference *L*_{2} measurement

_{2}

As shown in Fig. 3, when the RL is moved to the cat’s eye position A, the measured differential confocal intensity curve is *I*_{A}(*z*) (shown in Fig. 9), and the zero point *Q*_{A} of *I*_{A}(*z*) obtained by linear fitting is at *z*_{A} = - 0.0281 mm. When the RL is moved to the position C, the measured differential confocal intensity curve is *I*_{C}(*z*) (shown in Fig. 9), and the zero point *Q*_{C} of *I*_{C}(*z*) obtained by linear fitting is at *z*_{C} = 522.5086 mm. The position difference is *L*_{2} = *z*_{C} − *z*_{A} = 522.5367 mm.

From ten measurements of the position difference *L*_{2} (shown in Fig. 10), we obtained an average value of *L*_{2} represented by *L _{2avg}* = 522.5353 mm and the repeatability

*σ*

_{L}_{2}

*=*0.0036 mm.

### 3.2.2 Position difference *L*_{1} measurement

As shown in Fig. 2, when the TL is moved to the cat’s eye position A, the measured differential confocal intensity curve is *I*_{A}(*z*) (shown in Fig. 11), and the zero point *Q*_{A} of *I*_{A}(*z*) obtained by linear fitting is at *z*_{A} = - 0.0154 mm. When TL is moved to position B, the measured differential confocal intensity curve is *I*_{B}(*z*) (shown in Fig. 11), and the zero point *Q*_{B} of *I*_{B}(*z*) obtained by linear fitting is at *z*_{B} = 508.7811 mm. The position difference is *L*_{1} = *z*_{B} − *z*_{A} = 508.7965 mm.

From ten measurements of the position difference *L*_{1} (shown in Fig. 12), we obtained the repeatability *σ _{L}*

_{1}

*=*0.0032 mm.

From Eqs. (7) and (8), we can obtain the radius of convex lens is *R =* 19889.3 mm, the average of *R* is *R _{avg}* = 19885.1 mm and its repeatability is

*σ*4.6 mm.

_{R}=## 4 Uncertainty analyses

#### 4.1 Uncertainty components u(L_{1})

### 4.1.1 Uncertainty *u*(*L*_{1})_{axial} caused by axial misalignment error

_{axial}

As shown in Fig. 13, in the LDCRM system, axis *m* of L_{O}, axis *t* of TL, and axis *l* of DMI should be aligned, but deviations in angles always exist in practice. The deviation in angle between *m* and *t* can be adjusted to be smaller than 10″, so that its effects on the measurements are negligible. If the angle between *l* and *m* is *β*, then the axial misalignment error is:

Angle *β* can be adjusted within 2′ when the slide movement range is 2000 mm and the position deviation of laser beam between the two ends of air bearing slider can be controlled within 1 mm by the careful adjustment. Angle *β* ∈ [0, *γ*] (*γ* = 2′ = 0.00058) obeys a uniform distribution, thus *σ _{axial}* obeys a projection distribution, and the probability density of the projection distribution is:

Its stand deviation is *D* = (3/20)*γ*^{2}, so the stand uncertainty u(*L*_{1})* _{axial}* introduced by

*σ*is:

_{axial}### 4.1.2 Uncertainty *u*(*L*_{1})_{offset} caused by two detectors with different offsets

_{offset}

As shown in Fig. 14, when the offsets of two detectors are different, the zero points *Q*_{A} and *Q*_{B} of differential confocal curves *I*_{A} and *I*_{B} will deviate from positions A and B, respectively. Thus, the measurement of *L*_{1} will change.

Let the offsets of detectors 1 and 2 be *M* and –*M* + *δ*, the axial normalized offsets be *u _{M}* and -

*u*, the zero points

_{M}+ u_{δ}*Q*

_{A}and

*Q*

_{B}from positions A and B in the measurements be Δ

*L*

_{A}and Δ

*L*

_{B}, and the normalized axial offset of Δ

*L*

_{A}and Δ

*L*

_{B}be Δ

*u*

_{A}and Δ

*u*

_{B}, respectively.

Then, the differential confocal intensity curves *I _{A}′*(

*u*,

*u*) and

_{M}*I*(

_{B}′*u*,

*u*) can be expressed based on Eq. (1) and (3), respectively, as:

_{M}According to Eq. (12) and (13), we obtain Δ*u*_{A} and Δ*u*_{B} as follows:

The actual offsets Δ*L*_{A} and Δ*L*_{B} can be obtained from Δ*u*_{A} and Δ*u*_{B} according to Eq. (2), and the error due to two detectors with different offsets is:

Where *f*_{o}′ is the focal length of L_{O}, and *f*_{o}′ = 1080 mm.

Assuming that the error obeys an uniform distribution, the uncertainty *u*(*L*_{1})* _{offset}* caused by two detectors with different offsets is:

In the LDCRM system, the offsets of detectors *δ* can be easily controlled within 0.02 mm by using the self-collimation technique when the positioning accuracy of adjustment motors used in the adjustment is better than 0.01 mm.

### 4.1.3 Uncertainty *u*(*L*_{1})_{DMI} caused by position measurement errors

_{DMI}

In a LDCRM system, the position difference *L*_{1} is measured using the XL-80 interferometer and its measurement error is:

The uncertainty *u*(*L*_{1})* _{DMI}* due to errors in distance measurement is:

### 4.1.4 Uncertainty *u*(*L*_{1})_{σ1} observed by repeated measurements

_{σ1}

Despite the aforementioned errors, some random factors, such as environmental disturbance and system noise, can also result in measurement errors. These errors can be expressed as the standard deviation *σ _{L}*

_{1}over repeated measurements. Thus, uncertainty

*u*(

*L*

_{1})

*observed by repeated measurements is:*

_{σ1}Considering the aforementioned uncertainty components, the combined uncertainty *u*(*L*_{1}) can be obtained by the following formula.

Hence, when *R* = 19885.18 mm, by substituting *β =* 2*′*, *δ* = 0.02 mm, *L*_{1} = 508.7965 mm, *L*_{2} = 522.5353 mm and *σ _{L}*

_{1}= 0.0032 mm into Eq. (11), (16), (18) and (20), we obtain

*u*(

*L*

_{1})

*= 2.58 × 10*

_{axial}^{−5}mm,

*u*(

*L*

_{1})

*= 1.14 × 10*

_{offset}^{−3}mm,

*u*(

*L*

_{1})

*= 5.08 × 10*

_{DMI}^{−4}mm, and

*u*(

*L*

_{1})

_{σ1}= 0.0032 mm, and

#### 4.2 Uncertainty component u(L_{2})

The uncertainty *u*(*L*_{2}) due to position difference *L*_{2} of RL can be written as:

Where σ* _{offset}* is the errors caused by two detectors with different offsets, σ

*is the alignment error, and σ*

_{axial}*is the distance measurement error. These errors have been discussed in detail in [16]. In addition, σ*

_{L}

_{L}_{2}is the stand deviation of Δ

*L*

_{2}from ten measurements. The existing errors in LDCRM are σ

*= 2.2 × 10*

_{axial}^{−4}mm, σ

*= 2.7 × 10*

_{offset}^{−3}mm, σ

*= 1.04 × 10*

_{DMI}^{−3}mm, and

*σ*

_{L}_{2}= 3.6 × 10

^{−3}mm after the careful adjustments and experiments. The stand uncertainty

*u*(

*L*

_{2}) calculated using Eq. (22) is:

#### 4.3 Combined uncertainty u_{c}(R)

Assuming that the aforementioned uncertainty components of radius *R* for TL measurements are independent of each other, the combined measurement uncertainty in LDCRM can be obtained by using Eq. (24) as follows:

*c*

_{1}and

*c*

_{2}are the uncertainty transfer coefficients of

*L*

_{1}and

*L*

_{2}, respectively.

The relative combined uncertainty is:

From Eqs. (7) and (8), we obtain:

Substituting Eqs. (21), (23) and (26) into Eq. (24), the combined uncertainty is:

Therefore, the relative combined uncertainty is:

Considering some negligible uncertainty components, the relative uncertainty of LDCRM is expected to be less than 0.03% for a test lens of approximately 20 m radius. Although the LDCRM has a stronger tolerance to environmental disturbance compared with the interferometry method, we still need to control the environmental disturbance to further improve the measurement precision.

## 5. Conclusions

In this paper, we proposed a method for high-accuracy measurement of the ultra-large radius for convex spherical surface. By using the zero points of differential confocal axial intensity curves, LDCRM measures the vertex positions of the test lens and the last optical surface of objective lens to obtain position difference *L*_{1}, and then measures the vertex positions of the reflector and the last optical surface of objective lens to obtain the position difference *L*_{2}, finally uses the measured *L*_{1} and *L*_{2} to calculate the radius of test lens. Compared to existing methods, LDCRM has the following advantages.

- 1) It uses reflection light path to avoid identifying the confocal position, which can greatly shorten the light path and the moving distance of test lens, and extends the measurement range.
- 2) It uses non-interference focusing techniques to determine the cat’s eye and vertex position precisely based on differential confocal axial intensity, which improves the tolerance to environmental interference.

Theoretical analysis and preliminary experimental results indicate that the relative uncertainty of LDCRM is less than 0.03% for a convex lens with an approximate radius of 20 m, thus providing a novel approach for high-precision measurement of the ultra-large radius.

## Funding

National Natural Science Foundation of China (No. 61327010 and 51405020); National Instrumentation Program (NIP, No.2011YQ04013602).

## References and Links

**1. **J. W. Gates, K. J. Habell, and S. P. Middleton, “A precision spherometer,” J. Sci. Instrum. **31**(2), 60–64 (1954). [CrossRef]

**2. **H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE **5638**, 387–394 (2005). [CrossRef]

**3. **Y. P. Kumar and S. Chatterjee, “Application of Newton’s method to determine the focal length of lenses using a lateral shearing interferometer and cyclic path optical configuration setup,” Opt. Eng. **49**(5), 053604 (2010). [CrossRef]

**4. **D. G. Abdelsalam, M. S. Shaalan, M. M. Eloker, and D. Kim, “Radius of curvature measurement of spherical smooth surfaces by multiple-beam interferometry in reflection,” Opt. Lasers Eng. **48**(6), 643–649 (2010). [CrossRef]

**5. **Y. Nakano and K. Murata, “Measurements of phase objects using the Talbot effect and Moire techniques,” Appl. Opt. **23**(14), 2296–2299 (1984). [CrossRef] [PubMed]

**6. **Y. W. Lee, H. M. Cho, and I. W. Lee, “Measurement of a long radius of curvature using a half-aperture bidirectional shearing interferometer,” Opt. Eng. **35**(2), 480–483 (1996). [CrossRef]

**7. **K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. **31**(1), 75–79 (1992). [CrossRef] [PubMed]

**8. **U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of spheres with interferometry,” CIRP Annals - Manufacturing Technol **53**(1), 451–454 (2004). [CrossRef]

**9. **L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. **31**(9), 1961–1966 (1992). [CrossRef]

**10. **Q. Wang, U. Griesmann, and J. A. Soons, “Holographic radius test plates for spherical surfaces with large radius of curvature,” Appl. Opt. **53**(20), 4532–4538 (2014). [CrossRef] [PubMed]

**11. **T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE **4451**, 432–447 (2001). [CrossRef]

**12. **C. Lin, S. Yan, Z. Du, G. Wang, and C. Wei, “Symmetrical short-period and high signal-to-noise ratio heterodyne grating interferometer,” Chin. Opt. Lett. **13**(10), 100501 (2015). [CrossRef]

**13. **W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal radius measurement,” Opt. Express **18**(3), 2345–2360 (2010). [CrossRef] [PubMed]

**14. **R. Sun, L. Qiu, J. Yang, and W. Zhao, “Laser differential confocal radius measurement system,” Appl. Opt. **51**(26), 6275–6281 (2012). [CrossRef] [PubMed]

**15. **T. L. Schmitz, C. J. Evans, A. Davies, and W. T. Estler, “Displacement Uncertainty in Interferometric Radius Measurements,” CIRP Annals - Manufacturing Technol **51**(1), 451–454 (2002). [CrossRef]

**16. **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]