Error analysis for a laser differential confocal radius measurement system Xu Wang,1 Lirong Qiu,2 Weiqian Zhao,2,* Yang Xiao,2 and Zhongyu Wang1 1
School of Instrumentation Science & Opto-electronics Engineering, Beihang University, Beijing 100191, China 2
Beijing Key Lab for Precision Optoelectronic Measurement Instrument and Technology, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China *Corresponding author: [email protected] Received 8 October 2014; revised 20 November 2014; accepted 28 December 2014; posted 6 January 2015 (Doc. ID 224604); published 5 February 2015
In order to further improve the measurement accuracy of the laser differential confocal radius measurement system (DCRMS) developed previously, a DCRMS error compensation model is established for the error sources, including laser source offset, test sphere position adjustment offset, test sphere figure, and motion error, based on analyzing the influences of these errors on the measurement accuracy of radius of curvature. Theoretical analyses and experiments indicate that the expanded uncertainty of the DCRMS is reduced to U 0.13 μm 0.9 ppm · R (k 2) through the error compensation model. The error analysis and compensation model established in this study can provide the theoretical foundation for improving the measurement accuracy of the DCRMS. © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (180.1790) Confocal microscopy; (220.4840) Testing. http://dx.doi.org/10.1364/AO.54.001078
1. Introduction
Sphere components are widely used in precision optical systems, such as violet lithography, astronomical telescope, and laser fusion program. These components are also used as standard components in calibrating precise instruments that include a coordinate measuring machine, a profilometer, and laser scanner. Nevertheless, difficulty is encountered when radius of curvature (ROC) is measured in a highly precise manner [1]. At present, common ROC measurement methods may be classified into two categories, namely, contact and noncontact measurements. Contact measurement methods, including spherometer [2] and coordinate measuring machine [3], limit the further improvement of measurement accuracy and expansion of application because the test sphere is easily scratched and deformed because of contact. Interferometric ROC measurement [4], which is a noncontact 1559-128X/15/051078-07$15.00/0 © 2015 Optical Society of America 1078
APPLIED OPTICS / Vol. 54, No. 5 / 10 February 2015
measurement method, can obviously improve measurement accuracy; however, this method is easily influenced by environmental disturbance, including vibration, temperature, and airflows [5,6]. To improve the measurement accuracy and antiinterference capability of the existing ROC measurement methods, we proposed a laser differential confocal radius noncontact measurement method [7], and developed a corresponding measurement system with the measurement repeatability between 4 and 12 ppm [8]. However, many factors affect the measurement accuracy of the laser differential confocal radius measurement system (DCRMS). These factors include laser source offset, distance measurement error of distance measurement interferometer (DMI), misalignment between axes of differential confocal system and DMI, adjustment offset of test sphere position, radial translation, and rotation error of slider motion, and figure of test sphere [5,9–12]. To further improve the measurement accuracy of the DCRMS, the effects of these aforementioned factors on the ROC measurement results are analyzed and their mathematic
models are established; moreover, the measurement uncertainty of the DCRMS is estimated.
Z Z 1 2π 1 I B u; uM p ρ; θ · po ρ;θ · po ρ;θ π 0 0 c jρ2 2uuM juρ2 2 4 11D∕f o 2 ρ2 8
2 ρdρdθ
2. DCRMS Principle
· pc ρ;θ · e
The principle of DCRMS is shown in Fig. 1, which uses the property that the null points OA and OB of the axial intensity curves I A and I B correspond to the focus P of the objective lens Lo to accurately identify the cat’s eye A and confocal B positions of the test sphere T [13], and uses DMI to measure the distance between the two null points, thus, a high precision ROC measurement is accomplished [7]. When a test sphere T is moved near the cat’s eye position A or confocal position B, the measurement beam is reflected along the backtracking by the test sphere T, and then reflected on BS2 by BS1. Two measurement beams split by BS2 are received by detector 1 placed after the focus of collimating lens Lc with offset M and detector 2 placed before the focus of collimating lens Lc with offset M, respectively. The differential confocal intensity response curves I A at the cat’s eye position A and I B at the confocal position B, shown in Fig. 1, are obtained by the subtractions of the two normalized intensity signals from the two detectors, as shown in Eqs. (1) and (2). The laser source S is assumed as a point source, and geometrical approximation of the laser beam transformation is adopted during analyses [14].
Z Z 1 2π 1 − p ρ; θ · po ρ; θ · po ρ;θ π 0 0 c 2 jρ2 2u−uM juρ2 2 411D∕f o 2 ρ2 8 · pc ρ;θ · e ρdρdθ ; (2)
Z Z 1 2π 1 p ρ; θ · po ρ; θ · po ρ; π θ I A u; uM π 0 0 c 2 2 2 jρ2 2uuM jλf 2 − o2 u Rρ πD ρdρdθ · pc ρ; π θ · e 2 Z Z 1 2π 1 − p ρ; θ · po ρ; θ · po ρ; π θ π 0 0 c 2 2 2 jρ2 2u−uM jλf 2 − o2 u Rρ 2 πD ρdρdθ ; (1) ·pc ρ; π θ · e
Laser differential confocal system
Lc
BS1
and
u
π D 2 z; 2λ f o
(3)
where ρ is the radial normalized radius of the pupil, θ is the angle of the variable ρ in the polar coordinate, z is the axial displacement between the test sphere T and null point position, u is the axial normalized optical coordinate of the variable z, uM is the normalized offset of the detectors from the focus of the collimating lens Lc, D is the diameter of the objective lens Lo, f o is the focal length of the objective lens Lo, pc ρ; θ is the pupil function of collimating lens Lc, and po ρ; θ is the pupil function of the objective lens Lo. In the scanning process, the position coordinates zA and zB of the null points OA and OB of the differential confocal intensity response curves corresponding to the cat’s eye position A and confocal position B are recorded by DMI. The ROC of test sphere T is calculated using Eq. (4). R zA − zB :
(4)
DCRMS is mainly composed of a laser differential confocal system, a distance measurement system and a control system, which all have effects on the ROC measurement results. Some effects are independently caused by one part, and some are caused
Lo
Distance measurement system Cat’s eye A Confocal B T
S
DMI
P Airy disk Detector 1
Slider
BS2 p1
zA
+M Detector 2 p2
Airy disk -M
Guide
zB
R
Control system I
I2(u,-uM) I1(u,+uM)
OA
OB
IA
IB
I(u,uM)=I1(u,+uM)-I2(u,-uM)
u u
Fig. 1. DCRMS principle. S, laser source; BS, beam splitter; Lc, collimating lens; Lo, objective lens; P, focus of Lo; T, test sphere; M, offset of detectors from focus of Lc. 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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3. Error Analyses
In DCRMS, the synthetic effects of the laser source offset Δs , the distance measurement error Δl, the angle γ of optical axes between the differential confocal system and the DMI, the position adjustment offset of the test sphere Δa , the slider motion error Δm, the test sphere figure Φρ; θ on the ROC measurement are expressed as R f R RDMI ; Δs ; Δl ; γ; Δa ; Δm ; Φρ; θ;
(5)
where, RDMI is the ROC measurement result obtained by the DMI. The effects of these factors on the ROC measurement accuracy are described in detail. A.
Laser Source Offset Δs
As shown in Fig. 2, when the axial offset of the laser source S from the focus Oc of collimating lens Lc is Δs , the distances from the convergence points p1 and p2 to focal plane 1 and 2 of the collimating lens Lc are Δzp1 and Δzp2 , respectively. The conjugate points of p1 and p2 on the objective lens Lo are point p0 , and the conjugate point of the laser source S on the objective lens Lo is S0 . Consequently, point p0 is the image of point S0 by the test sphere when the DCRMS moves to the null points near the cat’s eye position A or confocal position B. Δzp1 and Δzp2 are expressed as Δzp1 Δzp2 −Δs f c ∕f c 2Δs ;
(6)
where, f c is the focal length of the collimating lens Lc. The distance L between S0 and p0 is calculated using Eq. (7). L−
2Δs f 2o : f 2c
(7)
At the cat’s eye position A, the distance Lce between points S0 and p0 is obtained by ray tracing using Eq. (8). Z Lce
arctanD∕2f o 4−R−z 0
ce sinφ−αce cos αce tan φ dφ; sin2αce −φD∕2f o 2
(8)
and αce arcsin
R zce sin φ ; R
(9)
where φ is the divergence angle of the source point S0 , and zce is the distance between S0 and the test sphere T at the cat’s eye position A. At the confocal position B, the distance Lcf between S0 and p0 is calculated as follows: Z Lcf
arctanD∕2f o 4z 0
cf
Rsinφ αcf cos αcf tan φ dφ; sin2αcf φD∕2f o 2 (10)
and αcf arcsin
−zcf R · sin φ ; R
(11)
where zcf is the distance between S0 and the test sphere T at the confocal position B. Let Lce L;
(12)
Lcf L:
(13)
Then, zce is obtained by using Eqs. (7), (8) and (12), and zcf is obtained by using Eqs. (7), (10) and (13). The measurement deviation of ROC caused by laser source offset is expressed as ΔRs zce − zcf − R:
(14)
Using Eq. (14), the measurement deviation of ROC caused by the laser source offset Δs is shown in Fig. 3. It can be seen from Fig. 3 that the measurement deviation of ROC increases as the laser source offset Δs increases, and the measurement deviation of ROC increases as the test ROC decreases with the same Δs . In practice, the laser source offset is less than 0.7 mm. The maximum measurement deviations of ROC for the objective lens Lo with the different focal length are shown in Fig. 4. The measurement Measurement deviation of ROC (μm)
by an interaction effect of several parts. Therefore, the discovery and compensation of the effects on the ROC measurement are crucial in improving the measurement accuracy of the DCRMS.
R=-100mm R=-50mm R=-10mm R=10mm R=50mm R=100mm
fc=1000mm fo=150mm Laser source offset (mm)
Fig. 2. Effect of laser source offset Δs . 1080
APPLIED OPTICS / Vol. 54, No. 5 / 10 February 2015
Fig. 3. Effect of laser source offset on ROC measurements.
angle γ is less than 40 in the DCRMS, and then the ROC measurement uncertainty caused by angle γ is expressed as
Maximum deviation of ROC (µm)
0.5 0.4 0.3 0.2
σ 3 0.35 × 10−6 · R:
0.1 0
Adjustment Offset Δa of Test Sphere Position
D.
-0.1 fo=65mm fo=150mm fo=330mm fc=1000mm
-0.2 -0.3 -0.4 -0.5
-200 -150 -100 -50 0 50 100 150 200 ROC (mm)
Fig. 4. Maximum measurement deviation of ROC caused by laser source offset.
deviation of ROC decreases as the focal length of objective lens Lo decreases, and the maximum measurement deviation of ROC can be controlled within 0.1 μm for the appropriate objective lens. Then, the ROC measurement uncertainty caused by the laser source offset Δs is obtained as σ 1 0.05 μm: B.
(18)
As shown in Fig. 6, the optical axes of the test sphere T and the differential confocal system should be coincident. But in practice, the adjustment offset Δa of test sphere from the differential confocal system still exists, and the effect of the offset Δa on the ROC measurement is analyzed as follows. At the cat’s eye position A, the effect of a little offset Δa on Airy disk position may be ignored [8]. At the confocal position B, the offset dx; dy of the axes between the test sphere and the differential confocal system is obtained by recording the Airy disk center pixel position nx ; ny and calculating dx and dy using Eqs. (19) and (20), respectively.
An XL-80 laser interferometer produced by RENISHAW Company is used as DMI in the DCRMS, and the measurement accuracy of the machine is 0.5 ppm (k 2), then, the DMI measurement uncertainty is calculated using Eq. (16). 0.5 ppm · RDMI 0.25 × 10−6 · RDMI : 2
(19)
dy
ny · p · f o ; 2N · f c
(20)
where p is the pixel size of the CCD, and N is the magnification of the microscopy objective of the detector. Therefore, the adjustment offset Δa of test sphere position is calculated using Eq. (21).
(16) Δa
C. Angle γ of Optical Axes between Differential Confocal System and DMI
As shown in Fig. 5, the optical axes of the differential confocal system and DMI should be aligned in theory, whereas the angle γ between them always exists in practice. The measurement deviation of ROC caused by angle γ is obtained using Eq. (17) [5]. ΔRγ R1 − cos γ:
nx · p · f o ; 2N · f c
(15)
Distance Measurement Error Δl
σ2
dx
(17)
Practically, the angle γ can be adjusted through aligning the convergence points of DMI beam in the laser differential confocal system with the laser source S, when the objective lens Lo is not mounted, and the aligning accuracy is up to 1 mm. Hence, the
q dx2 dy2 :
(21)
The measurement deviation of ROC caused by Δa is obtained as Δa ΔRa R 1 − cos arctan : RDMI
(22)
The effect of Δa on the ROC measurement may be compensated by tracking the center position of the Airy disk. The compensated adjustment accuracy of the test sphere position is U dx U dy 8 × 10−6 · f o. Cat’s eye A R S RDMI Focal plane 1
Center
Confocal B T Δa T axis System axis nx n y
Focal plane 2 Image on focal plane
Fig. 5. Effect of angle of optical axes.
Fig. 6. Effect of adjustment offset of test sphere position. 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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ΔR
RDMI
xDMI ; yDMI , and the distance vector between the differential confocal system optical axis and the slider motion axis is xm ; ym , when the test sphere T mounted on the slider moves from the confocal position B to the cat’s eye position A, the measurement deviations of ROC caused by εx and εy are obtained, using Eqs. (26) and (27), respectively:
x, y
T DMI axis System axis
R
DMI
y
z
Motion axis Cat’s eye A
Confocal B
x
O
Fig. 7. Effects of translational errors δx and δy .
ΔRεx −yDMI · tan εx −
Hence, the ROC measurement uncertainty caused by Δa is: σ 4 3.2 × 10−11 · f 2o · R−1 DMI : E.
(23)
Motion Error Δm
As shown in Figs. 7–9, the slider motion error Δm consists of radial translation errors δx and δy in the x and y directions and rotation errors εx, εy and εz on x, y, and z axes during z motion, respectively. Their effects on the ROC measurement are described as follows: 1. Radial Translation Errors δx , δy As shown in Fig. 7, the test sphere T has radial translation δx and δy caused by the straightness of the slider when the test sphere T mounted on the slider moves from the confocal position B to the cat’s eye position A along the z axis. The measurement deviation of ROC caused by δx and δy is: ΔRδxy
q 8 < −R − R2 − δ2x δ2y ; q : −R R2 − δ2 δ2 ; x y
R0
:
(24)
The highly precise air-bearing slider used in DCRMS exhibits a traveling range of 1.3 m and the straightness of 1 μm. Then the ROC measurement uncertainty caused by the radial translation δx and δy is: σ 5 ≈ 10−6 · R−2 mmjR∶mm:
q R2 y2m 1−cos εx 2 −jRj ;
(26) q ΔRεy xDMI · tan εy R2 x2m 1 − cos εy 2 − jRj : (27) Figure 9 shows the effect of the rotation error εz on the ROC measurement. When the test sphere T mounted on the slider moves from the confocal B position to the cat’s eye A position, the radial deviation of the curvature center of the test sphere T caused by εz is: q (28) Δεz x2m y2m εz : Therefore, the measurement deviation of ROC caused by εz is: ΔRεz
p 2 − Δ2εz; −R − pR 2 −R R − Δ2εz ;
(29)
In the distance measurement system of DCRMS, xm 230 mm, and the alignment accuracy of ym, xDMI and yDMI is 2 mm. The motion rotation error of the highly precise air-bearing slider is below 500. Hence, the ROC measurement uncertainties caused by the rotation errors εx, εy and εz are obtained, respectively. σ 6 σ 7 ≈ 0.03 μm;
(30)
σ 8 ≈ 1.5 × 10−8 · R−2 mmjR∶mm:
(31)
(25)
2. Rotation Errors εx , εy and εz Figure 8 shows the effect of rotation errors εx and εy on the ROC measurement. Supposing that the distance vector between the DMI measurement axis and the differential confocal system optical axis is
F.
Test Sphere Figure Φρ; θ
Only several primary aberrations that exhibit distinct effects on the ROC measurement are considered, and the test sphere figure Φρ; θ is [15]: ΔR z
RDMI
DMI axis
T
Δz
APPLIED OPTICS / Vol. 54, No. 5 / 10 February 2015
y xm,ym
Motion axis Cat’s eye A
Fig. 8. Effects of rotation errors εx and εy .
DMI
R
System axis
1082
R0
z
z
O Confocal B
Fig. 9. Effects of rotation error εz.
x
Φρ; θ a40 ρ4 a31 ρ3 cos θ a22 ρ2 cos2 θ 2
a20 ρ a11 ρ cos θ;
R RDMI − ΔRs − ΔRγ − ΔRa − ΔRδxy
where a40 is the primary spherical aberration coefficient, a31 is the coma coefficient, a22 is the astigmatism coefficient, a20 is the curvature of field coefficient, and a11 is the distortion coefficient. The test sphere figure Φρ; θ can make the average ROC Ravg of the best-fit sphere deviate from the theoretical value r and also make the intensity response curves of DCRM change shape. The offset Δr of Ravg from r caused by Φρ; θ is [7]: Δr ≈
2π 2a40 a22 : λ
− ΔRεx − ΔRεy − ΔRεz − ΔRΦ :
(32)
(33)
(37)
Assuming the effects of the aforementioned errors on ROC measurements are independent, the total measurement uncertainty of the DCRMS is obtained by using Eq. (38). v u 9 uX (38) σ2: σ t R
i
i1
Substituting Eqs. (15), (16), (18), (23), (25), (30), (31) and (36) to Eq. (38), respectively, the total measurement uncertainty of DCRMS is superior to
The intensity response curve I B at the confocal position B with Φρ; θ is expressed as
σ R 0.067 μm 0.45 ppm · R:
Z Z jρ2 2uu 2 juρ2 1 2π 1 M 2 411D∕f o 2 ρ2 −j2kΦρ;θ 8 I B u; uM ; Φ e ρdρdθ ·e π 0 0 Z Z jρ2 2u−u 2 juρ2 1 2π 1 M 2 411D∕f o 2 ρ2 8 · e−j2kΦρ;θ ρdρdθ : − e π 0 0
Consequently, the null point offset of the intensity response curve is approximately equal to Δr [7], that is to say, the null point of the intensity response curve corresponds to the curvature center of bestfit sphere. Therefore, the effect of Φρ; θ on identifying confocal position B is negligible. At the cat’s eye position A, the overlap area of the measurement beam and the test sphere surface is extremely small, so that the overlap area can be considered as an ideal mirror with no aberration. Therefore, the measurement deviation of ROC is the distance between the overlap area and the best-fit sphere, thus, ΔRΦ −δfigure :
(35)
This deviation can be corrected precisely through the test sphere figure obtained by the phase shift interferometer. The figure measurement accuracy of the ZYGO GPI interferometer is up to λ∕20, and the ROC measurement uncertainty caused by the Φρ; θ after compensation is: σ9 G.
λ 0.016 μm: k · 20
The ROC measurement result after compensation is obtained by substituting Eqs. (14), (17), (22), (24), (26), (27), (29), and (35) to Eq. (5).
(34)
For k 2, the expanded uncertainty of the DCRMS is obtained by using Eq. (40). U kσ R 0.13 μm 0.9 ppm · R:
(40)
4. Experiments
In order to verify the measurement accuracy of DCRMS, the test sphere used is a high-precision standard ball with a diameter of 50 mm, which could be measured by a length measuring machine (LMM). And its nominal diameter measured by LMM is Φ 50.0527 mm, and the expanded uncertainty of the used LMM is U 0.08 0.4L μmk 2 L∶m. The ROC of the standard ball is repeatedly measured for 10 times using DCRMS. Table 1 shows the measurement results obtained. The pixels position corresponding to the Airy disk center at the confocal position B is nx ; ny 20; 15, then the measurement deviation of ROC is obtained using Eqs. (19)–(22). Table 1.
(36)
Errors Synthesis
(39)
Number 1 2 3 4 5
ROC Measurement Results
Result (mm)
Number
Result (mm)
25.02638 25.02641 25.02638 25.02635 25.02642
6 7 8 9 10
25.02654 25.02648 25.02658 25.02650 25.02642
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5. Conclusion
The error analysis and compensation model of DCRMS are established in this study, and the theoretical analyses and experiments indicate that: (1) The major error sources of DCRMS include laser source offset, distance measurement error, angle of optical axes between the differential confocal system and DMI, and rotational errors of the slider on axes x and y; (2) The expanded uncertainty of DCRMS is U 0.13 μm 0.9 ppm · Rk 2. Fig. 10. Standard ball configuration.
The established model for error analysis and compensation provides a crucial theoretical foundation to perform high-precision ROC measurement and optimize the DCRMS. The work was supported by the National Instrumentation Program of China (NIP, No. 2011YQ04013606) and the National Natural Science Foundation of China (No. 61327012). References
Fig. 11. ROC compensation.
measurement
results
obtained
ΔRa 4.5 × 10−2 nm:
through
(41)
The figure of the standard ball obtained by a ZYGO GPI interferometer is shown in Fig. 10, where PV 0.09λ and δfigure −0.019λ. And the figure measurement uses the same NA and fixed form with ROC measurement to make the test region the same approximately. Figure 11 shows the compensated ROC of the standard ball, where the mean value of the ROC is calculated as R 25.02644 mm:
(42)
The offset from the nominal ROC is ΔR R −
Φ 0.09 μm: 2
(43)
The expanded uncertainty of the DCRMS for this standard ball is obtained by Eq. (40): U 0.13 μm 0.9 ppm · R 0.15 μm:
(44)
Figure 11 shows that the nominal ROC is between the lower and upper limits of the measurement result obtained using DCRMS. Consequently, experimental result agreed well with the theoretical result. 1084
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1. P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H. J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 092002 (2009). 2. J. W. Gates, K. J. Habell, and S. P. Middleton, “A precision spherometer,” J. Sci. Instrum. 31, 60–64 (1953). 3. K. Yoshizumi, T. Murao, J. Masui, R. Imanaka, and Y. Okino, “Ultrahigh accurate 3-D profilometer,” Appl. Opt. 26, 1647–1653 (1987). 4. L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31, 1961–1966 (1992). 5. T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432–447 (2001). 6. T. L. Schmitz, C. J. Evans, A. Davies, and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” CIRP Ann 51, 451–454 (2002). 7. W. Q. Zhao, R. D. Sun, L. R. Qiu, and D. G. Sha, “Laser differential confocal radius measurement,” Opt. Express 18, 2345–2360 (2010). 8. R. D. Sun, L. R. Qiu, J. M. Yang, and W. Q. Zhao, “Laser differential confocal radius measurement system,” Appl. Opt. 51, 6275–6281 (2012). 9. A. Davies and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44, 5884– 5893 (2005). 10. 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, 6692–6700 (2008). 11. X. Ding, B. Hong, X. Sun, and W. Liu, “Approach to accuracy improvement and uncertainty determination of radius of curvature measurement on standard spheres,” Measurement 46, 3220–3227 (2013). 12. U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” CIRP Ann 53, 451–454 (2004). 13. W. Q. Zhao, J. B. Tan, and L. R. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12, 5013–5021 (2004). 14. J. M. Yang, L. R. Qiu, W. Q. Zhao, R. J. Shao, and Z. G. Li, “Measuring the lens focal length by laser reflection-confocal technology,” Appl. Opt. 52, 3812–3817 (2013). 15. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley Interscience, 2007), Chap. 13, pp. 501–504.
School of Instrumentation Science & Opto-electronics Engineering, Beihang University, Beijing 100191, China 2
Beijing Key Lab for Precision Optoelectronic Measurement Instrument and Technology, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China *Corresponding author: [email protected] Received 8 October 2014; revised 20 November 2014; accepted 28 December 2014; posted 6 January 2015 (Doc. ID 224604); published 5 February 2015
In order to further improve the measurement accuracy of the laser differential confocal radius measurement system (DCRMS) developed previously, a DCRMS error compensation model is established for the error sources, including laser source offset, test sphere position adjustment offset, test sphere figure, and motion error, based on analyzing the influences of these errors on the measurement accuracy of radius of curvature. Theoretical analyses and experiments indicate that the expanded uncertainty of the DCRMS is reduced to U 0.13 μm 0.9 ppm · R (k 2) through the error compensation model. The error analysis and compensation model established in this study can provide the theoretical foundation for improving the measurement accuracy of the DCRMS. © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (180.1790) Confocal microscopy; (220.4840) Testing. http://dx.doi.org/10.1364/AO.54.001078
1. Introduction
Sphere components are widely used in precision optical systems, such as violet lithography, astronomical telescope, and laser fusion program. These components are also used as standard components in calibrating precise instruments that include a coordinate measuring machine, a profilometer, and laser scanner. Nevertheless, difficulty is encountered when radius of curvature (ROC) is measured in a highly precise manner [1]. At present, common ROC measurement methods may be classified into two categories, namely, contact and noncontact measurements. Contact measurement methods, including spherometer [2] and coordinate measuring machine [3], limit the further improvement of measurement accuracy and expansion of application because the test sphere is easily scratched and deformed because of contact. Interferometric ROC measurement [4], which is a noncontact 1559-128X/15/051078-07$15.00/0 © 2015 Optical Society of America 1078
APPLIED OPTICS / Vol. 54, No. 5 / 10 February 2015
measurement method, can obviously improve measurement accuracy; however, this method is easily influenced by environmental disturbance, including vibration, temperature, and airflows [5,6]. To improve the measurement accuracy and antiinterference capability of the existing ROC measurement methods, we proposed a laser differential confocal radius noncontact measurement method [7], and developed a corresponding measurement system with the measurement repeatability between 4 and 12 ppm [8]. However, many factors affect the measurement accuracy of the laser differential confocal radius measurement system (DCRMS). These factors include laser source offset, distance measurement error of distance measurement interferometer (DMI), misalignment between axes of differential confocal system and DMI, adjustment offset of test sphere position, radial translation, and rotation error of slider motion, and figure of test sphere [5,9–12]. To further improve the measurement accuracy of the DCRMS, the effects of these aforementioned factors on the ROC measurement results are analyzed and their mathematic
models are established; moreover, the measurement uncertainty of the DCRMS is estimated.
Z Z 1 2π 1 I B u; uM p ρ; θ · po ρ;θ · po ρ;θ π 0 0 c jρ2 2uuM juρ2 2 4 11D∕f o 2 ρ2 8
2 ρdρdθ
2. DCRMS Principle
· pc ρ;θ · e
The principle of DCRMS is shown in Fig. 1, which uses the property that the null points OA and OB of the axial intensity curves I A and I B correspond to the focus P of the objective lens Lo to accurately identify the cat’s eye A and confocal B positions of the test sphere T [13], and uses DMI to measure the distance between the two null points, thus, a high precision ROC measurement is accomplished [7]. When a test sphere T is moved near the cat’s eye position A or confocal position B, the measurement beam is reflected along the backtracking by the test sphere T, and then reflected on BS2 by BS1. Two measurement beams split by BS2 are received by detector 1 placed after the focus of collimating lens Lc with offset M and detector 2 placed before the focus of collimating lens Lc with offset M, respectively. The differential confocal intensity response curves I A at the cat’s eye position A and I B at the confocal position B, shown in Fig. 1, are obtained by the subtractions of the two normalized intensity signals from the two detectors, as shown in Eqs. (1) and (2). The laser source S is assumed as a point source, and geometrical approximation of the laser beam transformation is adopted during analyses [14].
Z Z 1 2π 1 − p ρ; θ · po ρ; θ · po ρ;θ π 0 0 c 2 jρ2 2u−uM juρ2 2 411D∕f o 2 ρ2 8 · pc ρ;θ · e ρdρdθ ; (2)
Z Z 1 2π 1 p ρ; θ · po ρ; θ · po ρ; π θ I A u; uM π 0 0 c 2 2 2 jρ2 2uuM jλf 2 − o2 u Rρ πD ρdρdθ · pc ρ; π θ · e 2 Z Z 1 2π 1 − p ρ; θ · po ρ; θ · po ρ; π θ π 0 0 c 2 2 2 jρ2 2u−uM jλf 2 − o2 u Rρ 2 πD ρdρdθ ; (1) ·pc ρ; π θ · e
Laser differential confocal system
Lc
BS1
and
u
π D 2 z; 2λ f o
(3)
where ρ is the radial normalized radius of the pupil, θ is the angle of the variable ρ in the polar coordinate, z is the axial displacement between the test sphere T and null point position, u is the axial normalized optical coordinate of the variable z, uM is the normalized offset of the detectors from the focus of the collimating lens Lc, D is the diameter of the objective lens Lo, f o is the focal length of the objective lens Lo, pc ρ; θ is the pupil function of collimating lens Lc, and po ρ; θ is the pupil function of the objective lens Lo. In the scanning process, the position coordinates zA and zB of the null points OA and OB of the differential confocal intensity response curves corresponding to the cat’s eye position A and confocal position B are recorded by DMI. The ROC of test sphere T is calculated using Eq. (4). R zA − zB :
(4)
DCRMS is mainly composed of a laser differential confocal system, a distance measurement system and a control system, which all have effects on the ROC measurement results. Some effects are independently caused by one part, and some are caused
Lo
Distance measurement system Cat’s eye A Confocal B T
S
DMI
P Airy disk Detector 1
Slider
BS2 p1
zA
+M Detector 2 p2
Airy disk -M
Guide
zB
R
Control system I
I2(u,-uM) I1(u,+uM)
OA
OB
IA
IB
I(u,uM)=I1(u,+uM)-I2(u,-uM)
u u
Fig. 1. DCRMS principle. S, laser source; BS, beam splitter; Lc, collimating lens; Lo, objective lens; P, focus of Lo; T, test sphere; M, offset of detectors from focus of Lc. 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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3. Error Analyses
In DCRMS, the synthetic effects of the laser source offset Δs , the distance measurement error Δl, the angle γ of optical axes between the differential confocal system and the DMI, the position adjustment offset of the test sphere Δa , the slider motion error Δm, the test sphere figure Φρ; θ on the ROC measurement are expressed as R f R RDMI ; Δs ; Δl ; γ; Δa ; Δm ; Φρ; θ;
(5)
where, RDMI is the ROC measurement result obtained by the DMI. The effects of these factors on the ROC measurement accuracy are described in detail. A.
Laser Source Offset Δs
As shown in Fig. 2, when the axial offset of the laser source S from the focus Oc of collimating lens Lc is Δs , the distances from the convergence points p1 and p2 to focal plane 1 and 2 of the collimating lens Lc are Δzp1 and Δzp2 , respectively. The conjugate points of p1 and p2 on the objective lens Lo are point p0 , and the conjugate point of the laser source S on the objective lens Lo is S0 . Consequently, point p0 is the image of point S0 by the test sphere when the DCRMS moves to the null points near the cat’s eye position A or confocal position B. Δzp1 and Δzp2 are expressed as Δzp1 Δzp2 −Δs f c ∕f c 2Δs ;
(6)
where, f c is the focal length of the collimating lens Lc. The distance L between S0 and p0 is calculated using Eq. (7). L−
2Δs f 2o : f 2c
(7)
At the cat’s eye position A, the distance Lce between points S0 and p0 is obtained by ray tracing using Eq. (8). Z Lce
arctanD∕2f o 4−R−z 0
ce sinφ−αce cos αce tan φ dφ; sin2αce −φD∕2f o 2
(8)
and αce arcsin
R zce sin φ ; R
(9)
where φ is the divergence angle of the source point S0 , and zce is the distance between S0 and the test sphere T at the cat’s eye position A. At the confocal position B, the distance Lcf between S0 and p0 is calculated as follows: Z Lcf
arctanD∕2f o 4z 0
cf
Rsinφ αcf cos αcf tan φ dφ; sin2αcf φD∕2f o 2 (10)
and αcf arcsin
−zcf R · sin φ ; R
(11)
where zcf is the distance between S0 and the test sphere T at the confocal position B. Let Lce L;
(12)
Lcf L:
(13)
Then, zce is obtained by using Eqs. (7), (8) and (12), and zcf is obtained by using Eqs. (7), (10) and (13). The measurement deviation of ROC caused by laser source offset is expressed as ΔRs zce − zcf − R:
(14)
Using Eq. (14), the measurement deviation of ROC caused by the laser source offset Δs is shown in Fig. 3. It can be seen from Fig. 3 that the measurement deviation of ROC increases as the laser source offset Δs increases, and the measurement deviation of ROC increases as the test ROC decreases with the same Δs . In practice, the laser source offset is less than 0.7 mm. The maximum measurement deviations of ROC for the objective lens Lo with the different focal length are shown in Fig. 4. The measurement Measurement deviation of ROC (μm)
by an interaction effect of several parts. Therefore, the discovery and compensation of the effects on the ROC measurement are crucial in improving the measurement accuracy of the DCRMS.
R=-100mm R=-50mm R=-10mm R=10mm R=50mm R=100mm
fc=1000mm fo=150mm Laser source offset (mm)
Fig. 2. Effect of laser source offset Δs . 1080
APPLIED OPTICS / Vol. 54, No. 5 / 10 February 2015
Fig. 3. Effect of laser source offset on ROC measurements.
angle γ is less than 40 in the DCRMS, and then the ROC measurement uncertainty caused by angle γ is expressed as
Maximum deviation of ROC (µm)
0.5 0.4 0.3 0.2
σ 3 0.35 × 10−6 · R:
0.1 0
Adjustment Offset Δa of Test Sphere Position
D.
-0.1 fo=65mm fo=150mm fo=330mm fc=1000mm
-0.2 -0.3 -0.4 -0.5
-200 -150 -100 -50 0 50 100 150 200 ROC (mm)
Fig. 4. Maximum measurement deviation of ROC caused by laser source offset.
deviation of ROC decreases as the focal length of objective lens Lo decreases, and the maximum measurement deviation of ROC can be controlled within 0.1 μm for the appropriate objective lens. Then, the ROC measurement uncertainty caused by the laser source offset Δs is obtained as σ 1 0.05 μm: B.
(18)
As shown in Fig. 6, the optical axes of the test sphere T and the differential confocal system should be coincident. But in practice, the adjustment offset Δa of test sphere from the differential confocal system still exists, and the effect of the offset Δa on the ROC measurement is analyzed as follows. At the cat’s eye position A, the effect of a little offset Δa on Airy disk position may be ignored [8]. At the confocal position B, the offset dx; dy of the axes between the test sphere and the differential confocal system is obtained by recording the Airy disk center pixel position nx ; ny and calculating dx and dy using Eqs. (19) and (20), respectively.
An XL-80 laser interferometer produced by RENISHAW Company is used as DMI in the DCRMS, and the measurement accuracy of the machine is 0.5 ppm (k 2), then, the DMI measurement uncertainty is calculated using Eq. (16). 0.5 ppm · RDMI 0.25 × 10−6 · RDMI : 2
(19)
dy
ny · p · f o ; 2N · f c
(20)
where p is the pixel size of the CCD, and N is the magnification of the microscopy objective of the detector. Therefore, the adjustment offset Δa of test sphere position is calculated using Eq. (21).
(16) Δa
C. Angle γ of Optical Axes between Differential Confocal System and DMI
As shown in Fig. 5, the optical axes of the differential confocal system and DMI should be aligned in theory, whereas the angle γ between them always exists in practice. The measurement deviation of ROC caused by angle γ is obtained using Eq. (17) [5]. ΔRγ R1 − cos γ:
nx · p · f o ; 2N · f c
(15)
Distance Measurement Error Δl
σ2
dx
(17)
Practically, the angle γ can be adjusted through aligning the convergence points of DMI beam in the laser differential confocal system with the laser source S, when the objective lens Lo is not mounted, and the aligning accuracy is up to 1 mm. Hence, the
q dx2 dy2 :
(21)
The measurement deviation of ROC caused by Δa is obtained as Δa ΔRa R 1 − cos arctan : RDMI
(22)
The effect of Δa on the ROC measurement may be compensated by tracking the center position of the Airy disk. The compensated adjustment accuracy of the test sphere position is U dx U dy 8 × 10−6 · f o. Cat’s eye A R S RDMI Focal plane 1
Center
Confocal B T Δa T axis System axis nx n y
Focal plane 2 Image on focal plane
Fig. 5. Effect of angle of optical axes.
Fig. 6. Effect of adjustment offset of test sphere position. 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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ΔR
RDMI
xDMI ; yDMI , and the distance vector between the differential confocal system optical axis and the slider motion axis is xm ; ym , when the test sphere T mounted on the slider moves from the confocal position B to the cat’s eye position A, the measurement deviations of ROC caused by εx and εy are obtained, using Eqs. (26) and (27), respectively:
x, y
T DMI axis System axis
R
DMI
y
z
Motion axis Cat’s eye A
Confocal B
x
O
Fig. 7. Effects of translational errors δx and δy .
ΔRεx −yDMI · tan εx −
Hence, the ROC measurement uncertainty caused by Δa is: σ 4 3.2 × 10−11 · f 2o · R−1 DMI : E.
(23)
Motion Error Δm
As shown in Figs. 7–9, the slider motion error Δm consists of radial translation errors δx and δy in the x and y directions and rotation errors εx, εy and εz on x, y, and z axes during z motion, respectively. Their effects on the ROC measurement are described as follows: 1. Radial Translation Errors δx , δy As shown in Fig. 7, the test sphere T has radial translation δx and δy caused by the straightness of the slider when the test sphere T mounted on the slider moves from the confocal position B to the cat’s eye position A along the z axis. The measurement deviation of ROC caused by δx and δy is: ΔRδxy
q 8 < −R − R2 − δ2x δ2y ; q : −R R2 − δ2 δ2 ; x y
R0
:
(24)
The highly precise air-bearing slider used in DCRMS exhibits a traveling range of 1.3 m and the straightness of 1 μm. Then the ROC measurement uncertainty caused by the radial translation δx and δy is: σ 5 ≈ 10−6 · R−2 mmjR∶mm:
q R2 y2m 1−cos εx 2 −jRj ;
(26) q ΔRεy xDMI · tan εy R2 x2m 1 − cos εy 2 − jRj : (27) Figure 9 shows the effect of the rotation error εz on the ROC measurement. When the test sphere T mounted on the slider moves from the confocal B position to the cat’s eye A position, the radial deviation of the curvature center of the test sphere T caused by εz is: q (28) Δεz x2m y2m εz : Therefore, the measurement deviation of ROC caused by εz is: ΔRεz
p 2 − Δ2εz; −R − pR 2 −R R − Δ2εz ;
(29)
In the distance measurement system of DCRMS, xm 230 mm, and the alignment accuracy of ym, xDMI and yDMI is 2 mm. The motion rotation error of the highly precise air-bearing slider is below 500. Hence, the ROC measurement uncertainties caused by the rotation errors εx, εy and εz are obtained, respectively. σ 6 σ 7 ≈ 0.03 μm;
(30)
σ 8 ≈ 1.5 × 10−8 · R−2 mmjR∶mm:
(31)
(25)
2. Rotation Errors εx , εy and εz Figure 8 shows the effect of rotation errors εx and εy on the ROC measurement. Supposing that the distance vector between the DMI measurement axis and the differential confocal system optical axis is
F.
Test Sphere Figure Φρ; θ
Only several primary aberrations that exhibit distinct effects on the ROC measurement are considered, and the test sphere figure Φρ; θ is [15]: ΔR z
RDMI
DMI axis
T
Δz
APPLIED OPTICS / Vol. 54, No. 5 / 10 February 2015
y xm,ym
Motion axis Cat’s eye A
Fig. 8. Effects of rotation errors εx and εy .
DMI
R
System axis
1082
R0
z
z
O Confocal B
Fig. 9. Effects of rotation error εz.
x
Φρ; θ a40 ρ4 a31 ρ3 cos θ a22 ρ2 cos2 θ 2
a20 ρ a11 ρ cos θ;
R RDMI − ΔRs − ΔRγ − ΔRa − ΔRδxy
where a40 is the primary spherical aberration coefficient, a31 is the coma coefficient, a22 is the astigmatism coefficient, a20 is the curvature of field coefficient, and a11 is the distortion coefficient. The test sphere figure Φρ; θ can make the average ROC Ravg of the best-fit sphere deviate from the theoretical value r and also make the intensity response curves of DCRM change shape. The offset Δr of Ravg from r caused by Φρ; θ is [7]: Δr ≈
2π 2a40 a22 : λ
− ΔRεx − ΔRεy − ΔRεz − ΔRΦ :
(32)
(33)
(37)
Assuming the effects of the aforementioned errors on ROC measurements are independent, the total measurement uncertainty of the DCRMS is obtained by using Eq. (38). v u 9 uX (38) σ2: σ t R
i
i1
Substituting Eqs. (15), (16), (18), (23), (25), (30), (31) and (36) to Eq. (38), respectively, the total measurement uncertainty of DCRMS is superior to
The intensity response curve I B at the confocal position B with Φρ; θ is expressed as
σ R 0.067 μm 0.45 ppm · R:
Z Z jρ2 2uu 2 juρ2 1 2π 1 M 2 411D∕f o 2 ρ2 −j2kΦρ;θ 8 I B u; uM ; Φ e ρdρdθ ·e π 0 0 Z Z jρ2 2u−u 2 juρ2 1 2π 1 M 2 411D∕f o 2 ρ2 8 · e−j2kΦρ;θ ρdρdθ : − e π 0 0
Consequently, the null point offset of the intensity response curve is approximately equal to Δr [7], that is to say, the null point of the intensity response curve corresponds to the curvature center of bestfit sphere. Therefore, the effect of Φρ; θ on identifying confocal position B is negligible. At the cat’s eye position A, the overlap area of the measurement beam and the test sphere surface is extremely small, so that the overlap area can be considered as an ideal mirror with no aberration. Therefore, the measurement deviation of ROC is the distance between the overlap area and the best-fit sphere, thus, ΔRΦ −δfigure :
(35)
This deviation can be corrected precisely through the test sphere figure obtained by the phase shift interferometer. The figure measurement accuracy of the ZYGO GPI interferometer is up to λ∕20, and the ROC measurement uncertainty caused by the Φρ; θ after compensation is: σ9 G.
λ 0.016 μm: k · 20
The ROC measurement result after compensation is obtained by substituting Eqs. (14), (17), (22), (24), (26), (27), (29), and (35) to Eq. (5).
(34)
For k 2, the expanded uncertainty of the DCRMS is obtained by using Eq. (40). U kσ R 0.13 μm 0.9 ppm · R:
(40)
4. Experiments
In order to verify the measurement accuracy of DCRMS, the test sphere used is a high-precision standard ball with a diameter of 50 mm, which could be measured by a length measuring machine (LMM). And its nominal diameter measured by LMM is Φ 50.0527 mm, and the expanded uncertainty of the used LMM is U 0.08 0.4L μmk 2 L∶m. The ROC of the standard ball is repeatedly measured for 10 times using DCRMS. Table 1 shows the measurement results obtained. The pixels position corresponding to the Airy disk center at the confocal position B is nx ; ny 20; 15, then the measurement deviation of ROC is obtained using Eqs. (19)–(22). Table 1.
(36)
Errors Synthesis
(39)
Number 1 2 3 4 5
ROC Measurement Results
Result (mm)
Number
Result (mm)
25.02638 25.02641 25.02638 25.02635 25.02642
6 7 8 9 10
25.02654 25.02648 25.02658 25.02650 25.02642
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5. Conclusion
The error analysis and compensation model of DCRMS are established in this study, and the theoretical analyses and experiments indicate that: (1) The major error sources of DCRMS include laser source offset, distance measurement error, angle of optical axes between the differential confocal system and DMI, and rotational errors of the slider on axes x and y; (2) The expanded uncertainty of DCRMS is U 0.13 μm 0.9 ppm · Rk 2. Fig. 10. Standard ball configuration.
The established model for error analysis and compensation provides a crucial theoretical foundation to perform high-precision ROC measurement and optimize the DCRMS. The work was supported by the National Instrumentation Program of China (NIP, No. 2011YQ04013606) and the National Natural Science Foundation of China (No. 61327012). References
Fig. 11. ROC compensation.
measurement
results
obtained
ΔRa 4.5 × 10−2 nm:
through
(41)
The figure of the standard ball obtained by a ZYGO GPI interferometer is shown in Fig. 10, where PV 0.09λ and δfigure −0.019λ. And the figure measurement uses the same NA and fixed form with ROC measurement to make the test region the same approximately. Figure 11 shows the compensated ROC of the standard ball, where the mean value of the ROC is calculated as R 25.02644 mm:
(42)
The offset from the nominal ROC is ΔR R −
Φ 0.09 μm: 2
(43)
The expanded uncertainty of the DCRMS for this standard ball is obtained by Eq. (40): U 0.13 μm 0.9 ppm · R 0.15 μm:
(44)
Figure 11 shows that the nominal ROC is between the lower and upper limits of the measurement result obtained using DCRMS. Consequently, experimental result agreed well with the theoretical result. 1084
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