Laser differential confocal radius measurement

A new laser differential confocal radius measurement (DCRM) is proposed for high precision measurement of radius. Based on the property of an axial intensity curve that the absolute zero precisely corresponds to the focus of the objective in a differential confocal system (DCS), DCRM uses the zero point of the DCS axial intensity curve to precisely identify the cat'seye and confocal positions of the test lens, and measures the accurate distance between the two positions to achieve the high-precision measurement of radius of curvature (ROC). In comparison with the existing measurement methods, DCRM proposed has a high measurement precision, a strong environmental anti-interference capability and a low cost. The theoretical analyses and preliminary experimental results indicate that DCRM has a relative measurement error of better than 5ppm. ©2010 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (180.1790) Confocal microscopy; (220.4840) Testing. References and links 1. P. Becker, H. Friedrich, and K. Fujii, “WGiardini, G Mana, A Picard, H-J Pohl, H Riemann and S Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009). 2. D. Malacara, “Optical Shop Testing, 2nd edition,” (Wiley-Interscience, 1992), Chap17. 3. L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992). 4. Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001). 5. Y. Pi, and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. 32(9), 1063–1065 (2007). 6. U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004). 7. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009). 8. T. L. Schmitz, C. J. Evans, A. D. Davies and W. T. Estler, “Displacement Uncertainty in Interferometric Radius Measurements,” CIRP Annals Manufacturing Technology, 51, 451–454 (2002). 9. T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-ofcurvature,” Proc. SPIE 4451, 432–447 (2001). 10. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004). 11. W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005). 12. L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001). 13. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1999), Chap. 9. 14. A. Davies, and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44(28), 5884–5893 (2005). 15. T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).


Introduction
Sphere component is widely used in precise optical systems as the most essential optical component, such as violet lithography, astronomical telescope and laser fusion program. The radius of curvature (ROC) is one of the most basic and important parameters in the sphere surface measurement, and the high precision ROC measurement is still a difficulty [1]. At present, the main methods used for ROC measurement include spherometer, Foucault test, traveling microscope, focus retrocollimated interferometry, autocollimation, simple fiberoptic autocollimation method and figure measuring interferometer(FMI), and so on [2][3][4][5][6]. In the aforementioned methods, FMI is the most well-known method used for the ROC accurate measurement, which uses the interference fringe to identify the cat's eye and confocal positions, and measures the accurate distance between the two positions to obtain the ROC. The common FMI uses PZT for phase-shifting, and has disadvantages as below. 1) It is difficult for the measurement beam to converge to a point due to the diffraction and the focaldepth restricts the accuracy of identification at the cat's-eye and confocal positions. 2) The environmental vibration and airflow have an adverse effect on the stability of interference fringe at the cat's-eye and confocal positions [7,9]. To improve the anti-interference capability, the instantaneous phase shifting interferometry was developed, but it is expensive and complex.
And therefore, a low-cost, compact laser differential confocal radius measurement (DCRM) is proposed, which has a high accuracy of the identifications at cat's-eye and confocal points and a stronger anti-interference capability by using the differential confocal intensity detection [10,11]. In addition, DCRM is easy to be combined with a pupil filter and it can shorten the focal-depth to further improve the accuracy of the identifications at the cat'seye and confocal positions [12].

DCRM principle
Based on the property that the zero point of the axial intensity curve corresponds to the focus of the objective in a differential confocal system (DCS) established by ourselves [10,11], DCRM as shown in Fig. 1 uses the null points O A and O B of DCS axial intensity curves I A and I B to make precise identifications at the cat's-eye position A and confocal position B of the test sphere, and measures the distance between the two null positions to achieve the ROC accurate measurement. As shown in Fig. 1, L C is collimating lens, PBS is polarized beam splitter, P is λ/4, L 1 is standard focusing lens, L 2 is collecting lens, BS is beam splitter, S is point light source, D 1 and D 2 are detectors, P 1 and P 2 are pinholes, R is test lens, M is the offset of pinhole from the focus of L 2 .
When the test lens R is moved near the cat's-eye position A along the optical axis, the measurement beam is reflected along the backtracking by the lens R and then reflected on the collecting lens L 2 and BS by PBS, and the two measurement beam split by BS are received by detectors D 1 and D 2 , respectively. Near the cat's-eye position A, the overlap area of the measurement beam and the test surface is so small that the effect of the figure error on the DCS intensity curve is negligible. Then the signal I A (v,φ,u,u M ) obtained through the differential subtraction of I A1 (v,φ,u, + u M ) and I A2 (v,φ,u,-u M ) received by the detectors D 1  ρ θ ρ θ ρ π θ ρ π θ ρ ρ θ π ρ θ ρ θ ρ π θ ρ π π where ρ is the radial normalized radius of a pupil, θ is the angle of variable ρ in the polar coordinate, u is the axial normalized optical coordinate, v is the lateral normalized optical coordinate, φ is the angle of variable v in the polar coordinate, z is the axial displacement of the object, r 0 is the radial coordinate of the objective, D/f´ is the relative aperture of DCS objective L 1 , p 1 (ρ,θ) is the pupil function of L 1 , p 2 (ρ,θ) is the pupil function of L 2 and p C (ρ,θ) is the pupil function of L C .
When L 1 and L 2 are the same in apertures, p C (ρ,θ) = 1, p 1 (ρ,θ) = 1, p 2 (ρ,θ) = 1, And as shown in Fig. 1, the zero point O A of intensity curve I A (0,0,u,u M ) precisely corresponds to the cat's-eye position A of the test lens. Near the confocal position B, the overlap area of the measurement beam and the test surface is so great that the effect of the figure error Φ(ρ,θ) on the DCS intensity curve cannot be ignored, which introduced the wave-front error of 2Φ(ρ,θ) in the measurement. When the test lens R is moved near the confocal position B along the optical axis, the signal I B (v,φ,u,u M ,Φ) obtained through the differential subtraction of I B1 (v,φ,u, + u M ,Φ) and I B2 (v,φ,u,-u M ,Φ) received by detectors D 1 and D 2 is [10,11] 2 B1 B2 And as shown in Fig. 1, the zero point O B of intensity curve I B (0,0,u,u M ) precisely corresponds to the confocal position B of the test lens.
2.2 Effect of Φ(ρ,θ) on the accuracy of identification at confocal position B As shown in Fig. 2, figure error Φ(ρ,θ) can make the average radius r avg (best-fit sphere) of the test lens R deviate from r and the DCRM intensity response curve change in shape under the high precision ROC measurement. ( , ) + cos cos + cos where A 040 is the primary spherical aberration coefficient, A 022 is the astigmatism coefficient, A 031 is the coma coefficient, A 120 is the curvature of field coefficient and A 111 is the distortion coefficient.

Effect of Φ(ρ,θ) on identification sensitivity
Sensitivity S B (0,0,0,C,Φ) at zero point O B can be obtained by differentiating Eq. (4) on u and is Sensitivity curve S B (0,0,0,C,Φ) obtained from Eq. (9) is shown in Fig. 4. It can be seen from Fig. 4 that DCRM sensitivity S B (0,0,0,C,Φ) decreases as the aberration coefficient A n increases and |S B (0,0,0,C,Φ) |>0.27 which is half of that without aberrations when A 040 <0.7λ and A 022 <0.3λ. And so, the accuracy of identification by DCRM entirely satisfies measurement requirements of the accuracy for the ROC measurement.
It is therefore that the accuracy of the measurement by DCRM is high and the effect of aberrations on DCRM is small as the surface figure precision of sphere measured is high when it is used for the high-precision radius measurement.

DCRM system using virtual pinhole for tracking
As shown in Fig. 1, DCRM requires that the pinhole center is coincident with the optical axis and the center of the Airy disk coming from the collecting lens L 2 exactly passes through the pinhole. In the measurement procedure, it is difficult to make the optical axes in coincidence between the test lens R and DCRM, and the axes offset between them results in the lateral offset of the focus of collecting lens L 2 . The lateral offset of the pinhole from the focus has an effect on DRCM intensity curve I(z) result in an effect on the sensitivity and precision of the identification. And the alignment error arising from the measurement line and datum line incur Abbe errors in DCRM.
In order to reduce the assembly difficulty of the test lens and measurement error caused by the assembly of the lens, a virtual pinhole (VPH) using CCD detection as shown in Fig. 5 is substituted for a conventional physical pinhole (PH). The Airy disk on the L 2 focal plane is magnified and imaged on the CCD detector by microscope objective MO, and the centre coordinates P(x,y) of the Airy disk is obtained using image g(m,n) from CCD based on Eq. (10).
where m and n the pixel coordinate of the image. And then, the VPH with a given radius r VPH is centered at point P(x, y), the grey summation of pixels within the circle of VPH is calculated, and the circle is equivalent to the physical pinhole of a confocal microscopy system and its size and shape can be adjusted by the software.
DCRM system (DCRMS) using CCD tracking detection by VPH is established as shown in Fig. 6 based on the measurement principle shown in Fig. 1 and VPH technique. In the measurement procedure, DCRM processes image g(m,n) using Eq. (10) to obtain the intensity centre of the Airy disk in real time and automatically track the center position P(x,y) of the Airy disk. It makes the PH centers coincident with the centre of Airy disk to eliminate the effect of PH lateral offset on position precision at confocal point and simplify the adjustment of the test lens R.

VPH size
As pinhole size decreases, the system sensitive increases while the energy received by the detector decreases. Hence, in a high system sensitive area, a larger pinhole should be selected to ensure enough energy passes through the pinhole. Analysis and calculation indicate that the pinhole size must satisfy Eq. (11), and an optimum pinhole should be in the range of 3-10µm in diameter [10,11].
where d is the pinhole diameter, λ, the optical wavelength, sinα, the numerical aperture of L 2 .
If the MO magnification is N, the pixels corresponding to the VPH diameter satisfies where p is the pixel size of CCD.

VPH offset
As shown in Fig. 7 From Eq. (13) and Eq. (14), It can be seen from Eqs. (2) and (15)  (16) It is obvious that in the DCRMS light-path adjustment, ∆z A and ∆z B caused by the offset of the detectors are the same in the size and direction so that they can counteract each other. And so, the adjustment difficulty of the detectors reduces greatly in DCRMS because the measurement error caused by δ is small.

Illumination collimation system
As shown in Fig. 8, it is difficult in the DCRMS light-path adjustment for the illumination collimation system to ensure the pinhole and the focus of collimation lens L C in coincidence. This changes the defocused pupil function p C (ρ,θ) of collimating lens L C so that the curves I A (0,0,u,u M ) and I B (0,0,u,u M ,Φ) are translated along the axial direction.
Assuming the PH offset is ∆z C while its corresponding normalized offset is u C , and the offset of zeros O A and O B from focuses A and B are respectively ∆z A and ∆z B in the measurement while their normalized value are u A and u B . Then defocused pupil function p C (ρ,θ) of the collimating lens L C satisfies 2 ( / 2) ( , ) From Eqs. (18) and (19), It can be seen from Eq. (2) and Eq. (20) that when the offset ∆z A and ∆z B of points O A and O B are equal, they can counteract each other. The radius measurement error caused by ∆z C is (21) Hence, it greatly reduces the adjustment difficulty of DCRMS detectors because the measurement error caused by ∆z C is very small.

Nonlinear calibration of CCD detectors
The zero point is obtained through the differential subtraction of the normalized VPH data, so the difference of two CCDs in intensity response has little effect on the position of the zero point. However, the nonlinearity of the intensity response has an effect on the determination of the zero point. Due to the discreteness of data in the measurement, the zero point can hardly be captured exactly, so the CCD nonlinearity must be calibrated and a linear curve can be fitted by the data near the zero point. And the calibration setup on CCD nonlinearity is shown in Fig. 9. The test chart in accordance with ISO14524:99 are placed at the outlet of the integrating sphere, and a set of grey values are shown in Table 1. A set of grey data shown in Table 1 is obtained from the test chart using MTV-2821 CCD being calibrated. And the nonlinear calibration function R −1 (y) shown in Fig. 10 are obtained through third-order polynomial fitting to the grey data, where x is the normalized luminance of the test chart and y is the normalized grey value of CCD output. The nonlinear calibration on the image g(m,n) coming from CCD is made using R −1 (y) in DCRMS and the response curves before and after calibration are shown in Fig. 11. It can be seen from Fig. 11 that the VPH response curve I(z) is more coincident with the theory in shape after the calibration, its side lobe is suppressed obviously, its linearity is improved and its half width at half of a maximum is diminished.

Adjustment and correction of axial alignment errors
As shown in Fig. 12, the measurement axis d of a distance measurement interferometer (DMI), the motion axis of translation stages m and radius axis of lens measured t should be coincident, whereas the angles between them always exist in practice. The angle between axes t and m is β and the angle between axes m and d is γ, the ROC measurement error is [3] (1 cos cos ) (22) It can be seen from Eq. (22) that β and γ should be diminished to reduce the ROC measurement error σ axial .
The angle γ is diminished through the careful alignment of DMI measurement axis d and motion axis of translation stage m and it is adjusted only at the time of establishing the system. And the correction of angle β is essential because it is produced by the placement of the test lens and should be adjusted for each test lens.
The adjustment of angle β in DCRMS is shown in Fig. 12, the center coordinate of the spot p A (x,y) is obtained using VPH when the test lens is translated to the cat's-eye position A, then the center coordinate of the spot p B (x,y) is obtained using VPH when the test lens is translated to the confocal position B, and thereby obtaining the lateral displacement ε of the spot centre. Angle β is obtained using the following equations, and DMI tan 2r where n is the pixel number corresponding to the lateral displacement of spot center from position A to position B, f L1 is the focal length of standard focusing lens L 1 and f L2 is the focal length of collecting lens L 2 .
In the practical measurement, ROC measurement error caused by angle β can be corrected using Eq. (25).
where r DMI is the radius measured by DMI. The ROC measurement error σ axial and the adjustment difficulty of the test lens reduce greatly because it caused by angle β can be corrected.

Measurement errors
The main errors having effects on ROC measurement are the distance measurement error of laser interferometer, position errors at cat's-eye position A and confocal position B, wavefront aberrations of sphere measured and axial alignment errors, and so on [3,14,15].

Distance measurement error σ L
The distance AB between cat's-eye position A and confocal position B can be obtained by DMI, and its measurement precision is highly dependent on the deadpath error [8,9]. For the distance measurement, the variation of atmosphere in the laboratory is compensated by monitoring the temperature, pressure, and humidity real-timely. The measurement errors consist of: 1) σ Pa for the barometer (pa), 2) σ K for the thermometer (°C), and 3) σ H for the hygrometer (% relative humidity).The distance measurement error satisfies: where σ nair is the refraction index error of air, L deadpath is the difference between the DMI measurement and reference path lengths.

Synthesis error
Considering the effect of the aforementioned several errors on the measurement results, the total measurement error σ r is When the parameters of the test lens are r = 100mm, D = 20mm and PV = 0.1λ, the magnification of objective used for VPH is N = 10, the CCD pixel size is p = 8µm and signal noise ratio is SNR = 150:1, the errors obtained using Eqs. (26)~(31) are σ L ≈0.1µm, σ z ≈0.1µm, σ axial ≈0.01µm and σ figure ≈0.01µm, respectively.
The system relative measurement error can be better than 5ppm with the consideration of environment and some negligible errors.

Experiment setup
The experimental setup shown in Fig. 13 is established based on Fig. 6 to verify the validity of DCRM proposed. The environmental conditions in the measurement laboratory are: pressure = (102540 ± 60) Pa, temperature = (20 ± 0.2) °C and relative humidity = (40 ± 4) %. In the experiment, the objective lens of M-20X, N.A. = 0.40 and f L1 = 9.0mm produced by NEWPORT is used as standard focusing L 1 , the focal length of collecting lens L 2 is f L2 = 160mm, the CCD used is MTV-2821, the magnification of VPH microscope objective is N = 10 × , DMI used is X80 laser interferometer produced by RENISHAW, and motion rail used is high-precision air bearing slider of 300mm in range and 0.1µm in straightness. The test sphere used is concave, its diameter is 42mm and the ROC of its front surface is r = −36.697mm obtained by the traveling microscope with the uncertainty of 0.002mm [2].

Experiment
As shown in Fig. 6, when the test lens R is moved along the optical axis under the measurement, DCRMS uses the null O A of DCS intensity curve I A to precisely determine the cat's-eye position A. The null O A of DCS intensity curve I A as shown in Fig. 14 corresponds to the cat's-eye position A, and the position coordinate corresponding to O A is 36.69542mm. Then, the pixel coordinate of the spot centre obtained by VPH p A (x,y) is (376.6, 309.1). The air sensor of X80 is used to monitor the temperature, pressure and humidity realtimely for compensating the laboratory environment. The errors are: 1) σ Pa = 50pa for the barometer, 2) σ K = 0.1°C for the thermometer, and 3) σ H = 3% for the hygrometer. The DCRM deadpath is L deadpath = 0.6m. The distance measurement error obtained by Eq. (26) (39) The relative aperture of the objective lens M-20X is 0.87, the CCD pixel size is p = 8µm and signal noise ratio is SNR = 150:1, the null error obtained using Eq. (28) is