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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 23 — Nov. 5, 2012
  • pp: 26027–26036
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Laser differential reflection-confocal focal-length measurement

Jiamiao Yang, Lirong Qiu, Weiqian Zhao, and Hualing Wu  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 26027-26036 (2012)
http://dx.doi.org/10.1364/OE.20.026027


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Abstract

A new laser differential reflection-confocal focal-length measurement (DRCFM) method is proposed for the high-accuracy measurement of the lens focal length. DRCFM uses weak light reflected from the lens last surface to determine the vertex position of this surface. Differential confocal technology is then used to identify precisely the lens focus and vertex of the lens last surface, thereby enabling the precise measurement of the lens focal length. Compared with existing measurement methods, DRCFM has high accuracy and strong anti-interference capability. Theoretical analyses and experimental results indicate that the DRCFM relative measurement error is less than 10 ppm.

© 2012 OSA

1. Introduction

2. DRCFM principle

Our group has previously established that the null point of the axial intensity response signal corresponds to the focus of the objective in a differential confocal system [17

17. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004). [CrossRef] [PubMed]

]. On this basis, DRCFM uses null points QA and QB of differential confocal response signals IA and IB to identify precisely the lens focus and vertex of the lens last surface, as shown in Fig. 1
Fig. 1 DRCFM principle. PBS is the polarized beam splitter, P is the one-fourth wave plate, Lc is the collimating lens, Lt is the test lens, R is the reflector, BS is the beam splitter, CCD1 and CCD2 are detectors, MO1 and MO2 are microscope objectives, M is the offset of the VPHs from the focus of Lc, and DMI is distance measurement interferometer.
. The distance between these two null positions is then determined to facilitate the high-accuracy measurement of the lens back focal length lF'.
lF=2|zAzB|,
(1)
where zA and zB are the position coordinates of reflector R corresponding to null points QA and QB, respectively.

When reflector R moves near position A along the optical axis of test lens Lt, the measurement beam reflected from R is again reflected by the polarized beam splitter (PBS) onto the beam splitter (BS). The two measurement beams split by the BS are received by the virtual pinholes (VPH1 and VPH2). The normalized intensity signals IA1(ν2,u, + uM) and IA2(ν2,u,-uM) received from CCD1 and CCD2 can be obtained by the Huygens–Fresnel diffraction integral formula.
IA1(ν2,u,+uM)=|0{01Pt(ρ)Pc(ρ)exp(iuρ22)J0(ν1ρ)ρdρ}×{01Pt(ρ)Pc(ρ)exp[iρ2(u2uM2)]02πJ0(ρν12+ν22+2ν1ν2cosθ)dθρdρ}ν1dν1|2,
(2)
IA2(ν2,u,uM)=|0{01Pt(ρ)Pc(ρ)exp(iuρ22)J0(ν1ρ)ρdρ}×{01Pt(ρ)Pc(ρ)exp[iρ2(u2+uM2)]02πJ0(ρν12+ν22+2ν1ν2cosθ)dθρdρ}ν1dν1|2,
(3)
where
{ν1=πλDftr1u=π2λzD2ft2and{ν2=πλDfcr2uM=π2λMD2fc2.
(4)
J0 is a zero-order Bessel function, ρ is the radial normalized radius of a pupil, θ is the angle of variable ρ in the polar coordinate, r1 is the radial coordinate on reflector R, v1 is the normalized coordinate of variable r1, z is the axial displacement between reflector R and position A, u is the normalized coordinate of variable z, r2 is the radial coordinate on the object plane of VPH1 or VPH2, v2 is the normalized coordinate of variable r2, M is the offset of the VPHs from the focus of collimating lens Lc, uM is the normalized offset of variable M, Pt(ρ) is the pupil function of test lens Lt, Pc(ρ) is the pupil function of collimating lens Lc, D is the effective aperture of Lc and Lt which is equal to the smaller aperture between Lc and Lt, D/fc' is the effective relative aperture of Lc, and D/ft' is the effective relative aperture of Lt. To ensure the full aperture measurement of Lt, the clear aperture of Lc should be no smaller than that of Lt.

When Pc(ρ) = 1 and Pt(ρ) = 1, the differential confocal response signal IA(u,uM) is obtained through the differential subtraction of IA1(0,u, + uM) and IA2(0,u,-uM).

IA(u,uM)=IA1(0,u,+uM)IA2(0,u,uM)=[sin(u/2uM/4)u/2uM/4]2[sin(u/2+uM/4)u/2+uM/4]2.
(5)

As shown in Fig. 1, null point QA of differential confocal response signal IA(u,uM) precisely corresponds to position A such that reflector R is exactly at the focus of Lt.

When reflector R moves near position B along the optical axis of Lt, a part of the measurement beam is reflected by the last surface of Lt. The overlap area of the measurement beam and the last surface of Lt is so small that the effect of the surface curvature on the differential confocal response signal is negligible. When the distance between reflector R and position B is z, the distance between the focusing point of the measurement beam and the vertex of the Lt last surface is 2z. Therefore, the differential confocal response signal IB(u,uM) obtained through the differential subtraction of IB1(0,u, + uM) and IB2(0,u,-uM) can be derived as follows.

IB(u,uM)=[sin(uuM/4)uuM/4]2[sin(u+uM/4)u+uM/4]2.
(6)

As shown in Fig. 1, null point QB of differential confocal response signal IB(u,uM) precisely corresponds to position B such that the measurement beam is focused on the vertex of the Lt last surface.

When the other parameters of Lt are given, the distance between the Lt second principle point and the vertex of the Lt last surface can be calculated by using ray tracing formulas. Then, the effective focal length of Lt can be indirectly measured. Generally, the distance between the Lt second principle point and the vertex of the Lt last surface is so small that the resulting measurement error is negligible. With a single lens as an example, the effective focal length can be calculated as follows.
ft'=lF+rb2n(r2r1)+(n1)b,
(7)
where r1 is the curvature radius of the Lt front surface, r2 is the curvature radius of the Lt back surface, n is the Lt refractive index, and b is the Lt thickness.

3. Effect analyses of key parameters

3.1 Focusing sensitivities

Sensitivities SA(0,uM) at null point QA and SB(0,uM) at null point QB can be obtained by differentiating Eqs. (5) and (6) on u, respectively.

SA(0,uM)=|IA(u,uM)u|u=0|=|2sinuM4(sinuM4uM4cosuM4)/(uM4)3|,
(8)
SB(0,uM)=|IB(u,uM)u|u=0|=|4sinuM4(sinuM4uM4cosuM4)/(uM4)3|.
(9)

Figure 2(a)
Fig. 2 Determination of the best offset of the VPHs: (a) focusing sensitivity curves, (b) differential confocal response signals.
shows that differential confocal response signals IA(u,uM) and IB(u,uM) have the largest sensitivities SAmax and SBmax at null points QA and QB when uM = 5.21.

SAmax=SA(0,5.21)=0.54andSBmax=SB(0,5.21)=1.08.
(10)

Figure 2(b) shows simulation curves IA(u,uM) and IB(u,uM) with uM = 5.21. The identification at the vertex of the Lt last surface is clearly much more sensitive than that at the Lt focus.

When uM = 5.21, DRCFM focusing errors σzA and σzB at the Lt focus and the vertex of the Lt last surface can be described as follows.
σzA=δIA(u,uM)SAmax2λπ(D/ft)2=2λ0.54πSNR(D/ft)2,
(11)
σzB=δIB(u,uM)SBmax2λπ(D/ft)2=2λ1.08πSNR(D/ft)2,
(12)
where δIA(u,uM) and δIB(u,uM) are the random noises of the differential confocal response signals caused by the CCDs, and SNR is the signal-to-noise ratio of the VPHs. Equations (11) and (12) show that the identification precision increases as relative aperture D/ft' increases.

3.2 Effect of axial alignment errors

As shown in Fig. 3
Fig. 3 Angles between DRCFM axes.
, axis m of the measurement beam, optical axis s of Lt, measurement axis p of the distance measurement interferometer (DMI), and optical axis q of reflector R should be coincident, whereas the deviation angles between them always exist in practice. Using the autocollimation method without Lt in the measurement path, the angle between axes m and q can be aligned such that the effect of this angle on the measurement can be negligible. If the angle between axes m and s is defined as α, and the angle between axes m and p is defined as β, the back focal length measurement error can be calculated using the following geometric relationship.

σaxialft(cosβcosα1).
(13)

Angle β is diminished through the careful alignment of axes p and m, and it should be adjusted only while establishing the system. When the focal length of Lc in the DRCFM system is 1000 mm, angle β can be reduced to less than 3 second. Angle α between axes m and s can be easily controlled within 2 minute by careful alignment.

3.3 Effect of two VPHs with different offsets

As shown in Fig. 4
Fig. 4 Light path schematics with different offsets.
, when the offsets of the two VPHs from the Lc focus are different, null points QA and QB of the differential confocal response signals deviate from positions A and B, respectively. Thus, the lF' measurements change.

Let the offsets of two VPHs from the Lc focus be M and -M + δM, their corresponding normalized offsets be uM and -uM + uδM, the offsets of QA from position A and of QB from position B in the measurement process be Δl1 and Δl2, and their normalized values be Δu1 and Δu2.

Then, the differential confocal response signals IA(u,uM) and IB(u,uM) are obtained according to Eqs. (5) and (6).

IA(u,uM)={sin(u/2uM/4)u/2uM/4}2{sin[u/2+(uMuδM)/4]u/2+(uMuδM)/4}2,
(14)
IB(u,uM)={sin(uuM/4)uuM/4}2{sin[u+(uMuδM)/4]u+(uMuδM)/4}2.
(15)

From Eqs. (14) and (15),

Δu1=uδM4andΔu2=uδM8.
(16)

From Eq. (4), the respective actual offsets of the corresponding positions QA and QB are

Δl1=14ft2fc2δMandΔl2=18ft2fc2δM.
(17)

Therefore, when the deviation of VPHs axial offset is δM, back focal length measurement error σoffset caused by δM can be obtained through Eq. (1).

σoffset=2(Δl1Δl2)=14ft2fc2δM.
(18)

Using a grating scale as an aided adjustment of the VPH offsets, offset error δM can be easily controlled within 10 μm, which is limited by both the position accuracy of the grating scale and the focusing accuracy of the Lc focus.

3.4 Distance measurement error

As shown in Fig. 1, the distance between positions A and B is measured by a single-frequency laser interferometer, and its measurement error is
σL=1×Lppm,
(19)
where L is the measurement distance.

3.5 Synthesis error

Considering the effect of the aforementioned errors on the measurement results of lF', the DRCFM total measurement error σlF' is

σlF=(lFLσL)2+(lFzAσzA)2+(lFzBσzB)2+σoffset2+σaxial2.
(20)

According to Eq. (1), the respective error transfer coefficients of σL, σzA, and σzB are

lFL=2,lFzA=2,andlFzB=2.
(21)

Using Eqs. (20) and (21), the DRCFM synthesis measurement error is

σlF=4σL2+4σzA2+4σzB2+σoffset2+σaxial2.
(22)

4. Experiments

4.1 Experimental setup

The experimental setup shown in Fig. 5
Fig. 5 Experimental setup. (1) He-Ne laser, (2) single-mode fiber, (3) VPH1, (4) VPH2, (5) beam splitter, (6) polarized beam splitter, (7) one-fourth wave plate, (8) aiming system, (9) beam splitter, (10) collimating lens, (11) test lens, (12) reflector, (13) air bearing slider, (14) XL-80 laser interferometer produced by Renishaw, and (15) air sensor of XL-80.
is established based on Fig. 1 to verify the validity of DRCFM.

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, a He-Ne laser with a wavelength of 632.8 nm is used as the light source. An achromatic lens (produced by LINOS Photonics GmbH & Co.) with a focal length of 1000 mm and a diameter of 100 mm is used as collimating lens Lc. An XL-80 laser interferometer (produced by Renishaw) is used as the distance measurement interferometer. A high-accuracy air bearing slider with a range of 1300 mm and a straightness of 0.1 μm is used as the motion rail. The CCDs used are OK-AM1100 with a pixel size of 8 μm. The SNR of the VPHs is 100:1, and the magnification of the VPH microscope objectives is 10 × . Test lens Lt is a cemented doublet lens with a diameter of 20 mm and a focal length of 200 mm, and its nominal back focal length lF' is 197.3 mm ( ± 2%).

4.2 Experiment results

As shown in Fig. 6
Fig. 6 Back focal length measurement curves.
, when reflector R moves across position A along the optical axis during measurement, DRCFM uses null point QA of differential confocal response signal IA(z) to determine precisely the focus of Lt. The position coordinate of reflector R corresponding to point QA is zA = −99.17363 mm.

Similarly, when reflector R moves across position B, DRCFM uses null point QB of differential confocal response signal IB(z) to determine precisely the vertex of the Lt last surface. The position coordinate of reflector R corresponding to point QB is zB = 0.17223 mm.

Therefore, the back focal length lF' of Lt is 2|zA - zB| = 198.6917 mm, and the repeatability achieved from ten measurements is σtest = 1.6 μm.

Using Eqs. (11)(13), (18), and (19), the errors existing in DRCFM are σL = 0.1 μm, σzA = 0.75 μm, σzB = 0.37 μm, σaxial = 0.03 μm, and σoffset = 0.1 μm. The system error obtained using Eq. (22) is

σlF=4σL2+4σzA2+4σzB2+σaxial2+σoffset2=4×0.12+4×0.752+4×0.372+0.032+0.12=1.7μm.
(23)

The relative error is

δ=σlFlF×100%=1.7198.6918×1000×100%0.00086%=8.6ppm.
(24)

Considering the environmental effects and some negligible errors, the relative error of DRCFM can be less than 10 ppm.

It can be concluded from the above error analyses that the focusing errors σzA and σzB have more significant effect on the measurement accuracy. So, the measurement accuracy increases as the relative aperture of test lens increases.

5. Conclusions

The proposed DRCFM uses the null points of differential confocal response signals to identify precisely the lens focus and vertex of the lens last surface. Consequently, the lens focal length can be measured with high accuracy. Theoretical analyses and experimental results show that the proposed approach has a relative error of less than 10 ppm and has the following advantages.

  • 1) It improves significantly the identification precision at the lens focus and vertex of the lens last surface because it has the best linearity and sensitivity at the null point of a differential confocal response signal.
  • 2) It can measure the lens spherical aberration in combination with annular pupil filtering technology.
  • 3) It has higher measurement accuracy and stronger anti-interference capability than existing approaches.

Therefore, DRCFM is a feasible method for the high-accuracy measurement of the lens focal length.

Acknowledgments

The authors gratefully acknowledge the support of the Major National Scientific Instrument and Equipment Development Project of China (No. 2011YQ040136), the National Science Foundation of China (No. 91123014, 60927012), and the Nature Science Association Foundation (No. 11076004).

References and links

1.

E. Keren, K. M. Kreske, and O. Kafri, “Universal method for determining the focal length of optical systems by moire deflectometry,” Appl. Opt. 27(8), 1383–1385 (1988). [CrossRef] [PubMed]

2.

C.-W. Chang and D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73(4), 257–262 (1989). [CrossRef]

3.

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]

4.

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. 31(28), 5984–5987 (1992). [CrossRef] [PubMed]

5.

F. Lei and L. K. Dang, “Measuring the focal length of optical systems by grating shearing interferometry,” Appl. Opt. 33(28), 6603–6608 (1994). [CrossRef] [PubMed]

6.

M. Thakur and C. Shakher, “Evaluation of the focal distance of lenses by white-light Lau phase interferometry,” Appl. Opt. 41(10), 1841–1845 (2002). [CrossRef] [PubMed]

7.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun. 248(4-6), 339–345 (2005). [CrossRef]

8.

S. Zhao, J. F. Wen, and P. S. Chung, “Simple focal-length measurement technique with a circular Dammann grating,” Appl. Opt. 46(1), 44–49 (2007). [CrossRef] [PubMed]

9.

Y. P. Kumar and S. Chatterjee, “Technique for the focal-length measurement of positive lenses using Fizeau interferometry,” Appl. Opt. 48(4), 730–736 (2009). [CrossRef] [PubMed]

10.

Y. Xiang, “Focus retrocollimated interferometry for focal-length measurements,” Appl. Opt. 41(19), 3886–3889 (2002). [CrossRef] [PubMed]

11.

I. K. Ilev, “Simple fiber-optic autocollimation method for determining the focal lengths of optical elements,” Opt. Lett. 20(6), 527–529 (1995). [CrossRef] [PubMed]

12.

D.-H. Kim, D. Shi, and I. K. Ilev, “Alternative method for measuring effective focal length of lenses using the front and back surface reflections from a reference plate,” Appl. Opt. 50(26), 5163–5168 (2011). [CrossRef] [PubMed]

13.

J.- Wu, J.- Chen, A.- Xu, X.-y. Gao, and S. Zhuang, “Focal length measurement based on Hartmann-Shack principle,” Optik (Stuttg.) 123(6), 485–488 (2012). [CrossRef]

14.

J. Wu, J. Chen, A. Xu, and X. Gao, “Uncollimated light beam illumination during the ocular aberration detection and its impact on the measurement accuracy by using Hartmann-Shack wavefront sensor,” Proc. SPIE 7508, 75080V, 75080V-12 (2009). [CrossRef]

15.

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.

16.

W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal ultra-long focal length measurement,” Opt. Express 17(22), 20051–20062 (2009). [CrossRef] [PubMed]

17.

W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004). [CrossRef] [PubMed]

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(180.1790) Microscopy : Confocal microscopy
(220.4840) Optical design and fabrication : Testing

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 24, 2012
Revised Manuscript: October 16, 2012
Manuscript Accepted: October 23, 2012
Published: November 2, 2012

Citation
Jiamiao Yang, Lirong Qiu, Weiqian Zhao, and Hualing Wu, "Laser differential reflection-confocal focal-length measurement," Opt. Express 20, 26027-26036 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-26027


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References

  1. E. Keren, K. M. Kreske, and O. Kafri, “Universal method for determining the focal length of optical systems by moire deflectometry,” Appl. Opt.27(8), 1383–1385 (1988). [CrossRef] [PubMed]
  2. C.-W. Chang and D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun.73(4), 257–262 (1989). [CrossRef]
  3. 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]
  4. K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt.31(28), 5984–5987 (1992). [CrossRef] [PubMed]
  5. F. Lei and L. K. Dang, “Measuring the focal length of optical systems by grating shearing interferometry,” Appl. Opt.33(28), 6603–6608 (1994). [CrossRef] [PubMed]
  6. M. Thakur and C. Shakher, “Evaluation of the focal distance of lenses by white-light Lau phase interferometry,” Appl. Opt.41(10), 1841–1845 (2002). [CrossRef] [PubMed]
  7. C. J. Tay, M. Thakur, L. Chen, and C. Shakher, “Measurement of focal length of lens using phase shifting Lau phase interferometry,” Opt. Commun.248(4-6), 339–345 (2005). [CrossRef]
  8. S. Zhao, J. F. Wen, and P. S. Chung, “Simple focal-length measurement technique with a circular Dammann grating,” Appl. Opt.46(1), 44–49 (2007). [CrossRef] [PubMed]
  9. Y. P. Kumar and S. Chatterjee, “Technique for the focal-length measurement of positive lenses using Fizeau interferometry,” Appl. Opt.48(4), 730–736 (2009). [CrossRef] [PubMed]
  10. Y. Xiang, “Focus retrocollimated interferometry for focal-length measurements,” Appl. Opt.41(19), 3886–3889 (2002). [CrossRef] [PubMed]
  11. I. K. Ilev, “Simple fiber-optic autocollimation method for determining the focal lengths of optical elements,” Opt. Lett.20(6), 527–529 (1995). [CrossRef] [PubMed]
  12. D.-H. Kim, D. Shi, and I. K. Ilev, “Alternative method for measuring effective focal length of lenses using the front and back surface reflections from a reference plate,” Appl. Opt.50(26), 5163–5168 (2011). [CrossRef] [PubMed]
  13. J.- Wu, J.- Chen, A.- Xu, X.-y. Gao, and S. Zhuang, “Focal length measurement based on Hartmann-Shack principle,” Optik (Stuttg.)123(6), 485–488 (2012). [CrossRef]
  14. J. Wu, J. Chen, A. Xu, and X. Gao, “Uncollimated light beam illumination during the ocular aberration detection and its impact on the measurement accuracy by using Hartmann-Shack wavefront sensor,” Proc. SPIE7508, 75080V, 75080V-12 (2009). [CrossRef]
  15. 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.
  16. W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal ultra-long focal length measurement,” Opt. Express17(22), 20051–20062 (2009). [CrossRef] [PubMed]
  17. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express12(21), 5013–5021 (2004). [CrossRef] [PubMed]

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