Laser differential reflection-confocal focal-length measurement

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 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (180.1790) Confocal microscopy; (220.4840) Testing. 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). 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). 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). 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). 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). 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). 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). 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). 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). 10. Y. Xiang, “Focus retrocollimated interferometry for focal-length measurements,” Appl. Opt. 41(19), 3886–3889 (2002). 11. I. K. Ilev, “Simple fiber-optic autocollimation method for determining the focal lengths of optical elements,” Opt. Lett. 20(6), 527–529 (1995). 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). 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). 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). 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). #174943 $15.00 USD Received 24 Aug 2012; revised 16 Oct 2012; accepted 23 Oct 2012; published 2 Nov 2012 (C) 2012 OSA 5 November 2012 / Vol. 20, No. 23 / OPTICS EXPRESS 26027 17. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).


Introduction
The focal length is a key parameter that reflects performance of a lens.The focal length is related to the curvature radii of the lens surfaces, refractive index of the lens material, and lens thickness, among others.The actual focal length always deviates from the theoretical value as a result of manufacturing and assembly errors.Therefore, the precise measurement of the lens focal length is a core problem in the field of optical measurement.
Various methods for focal-length measurement have been proposed.Classical methods such as nodal slide and image magnification are based on the principles of geometry optics.Although these methods are easy to setup and manipulate, high accuracy is difficult to achieve because of subjectivity-related measurement errors.To improve the measurement accuracy, some novel methods based on the principle of physical optics have been presented.For example, moiré deflectometry [1] and Talbot interferometry [2][3][4] use a moiré fringe to determine the lens focal length.Their measurement accuracies (lower than 0.36%) are limited by the manual measurement of the inclination of the moiré fringes [3].Grating shearing interferometry measures the lens focal length by using the relative lateral shift between the undiffracted zero order and diffracted first order caused by the grating.The measurement accuracy (lower than 0.4%) is limited by the accuracy of charge-coupled device (CCD) measurements [5].Lau phase interferometry uses the slope of a phase map to determine the lens focal length [6,7].The measurement accuracy (lower than 0.2%) is influenced by the aberrations of an unwrapped phase map [7].The circular Dammann grating method uses the diffraction characteristics of a circular Dammann grating at the focal plane of a lens to locate the focus of the test lens.The identification accuracy of the focus (lower than 0.1%) is limited by the definition of a diffraction pattern [8].Phase-shift interferometry identifies the image point of the test lens using interference fringes, and then calculates the lens focal length by Gaussian or Newtonian equations [9,10].The measurement accuracy reaches 0.01% [10], which is the highest accuracy among those of existing measurement methods.Other proposed methods include the fiber-optic autocollimation method [11], reference plant method [12] and Hartmann-Shack method [13].However, their measurement accuracies need to be improved for engineering measurements.
The accuracies of existing focal-length measurement methods are lower than 0.01%, which cannot meet the demands of some sophisticated technologies [14,15].In the previous work, we have presented a laser differential confocal ultra-long focal length measurement with the relative error of about 0.01% [16], but it is not suitable for the short focal-length measurement.Therefore, a laser differential reflection-confocal focal-length measurement (DRCFM) method is proposed in this paper.DRCFM uses differential confocal intensity detection to identify the lens focus and vertex of the lens last surface with high accuracy.The method also has a strong anti-interference capability [17].Theoretical analyses and preliminary experiments show that the DRCFM relative measurement error is less than 10 ppm.Thus, DRCFM can enable the high-accuracy measurement of the lens focal length.

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].On this basis, DRCFM uses null points Q A and Q B of differential confocal response signals I A and I B to identify precisely the lens focus and vertex of the lens last surface, as shown in Fig. 1.The distance between these two null positions is then determined to facilitate the high-accuracy measurement of the lens back focal length l F '.

=2
, where z A and z B are the position coordinates of reflector R corresponding to null points Q A and Q B , respectively.When reflector R moves near position A along the optical axis of test lens L t , 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 I A1 (ν 2 ,u, + u M ) and I A2 (ν 2 ,u,-u M ) received from CCD1 and CCD2 can be obtained by the Huygens-Fresnel diffraction integral formula. where and .
J 0 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, r 1 is the radial coordinate on reflector R, v 1 is the normalized and L t which is equal to the smaller aperture between L c and L t , D/f c ' is the effective relative aperture of L c , and D/f t ' is the effective relative aperture of L t .To ensure the full aperture measurement of L t , the clear aperture of L c should be no smaller than that of L t .
When P c (ρ) = 1 and P t (ρ) = 1, the differential confocal response signal I A (u,u M ) is obtained through the differential subtraction of I A1 (0,u, + u M ) and I A2 (0,u,-u M ).

(
) ( ) As shown in Fig. 1, null point Q A of differential confocal response signal I A (u,u M ) precisely corresponds to position A such that reflector R is exactly at the focus of L t .
When reflector R moves near position B along the optical axis of L t , a part of the measurement beam is reflected by the last surface of L t .The overlap area of the measurement beam and the last surface of L t 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 L t last surface is 2z.Therefore, the differential confocal response signal I B (u,u M ) obtained through the differential subtraction of I B1 (0,u, + u M ) and I B2 (0,u,-u M ) can be derived as follows.
( ) ( ) As shown in Fig. 1, null point Q B of differential confocal response signal I B (u,u M ) precisely corresponds to position B such that the measurement beam is focused on the vertex of the L t last surface.
When the other parameters of L t are given, the distance between the L t second principle point and the vertex of the L t last surface can be calculated by using ray tracing formulas.Then, the effective focal length of L t can be indirectly measured.Generally, the distance between the L t second principle point and the vertex of the L t 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.
where r 1 is the curvature radius of the L t front surface, r 2 is the curvature radius of the L t back surface, n is the L t refractive index, and b is the L t thickness.

Focusing sensitivities
Sensitivities S A (0,u M ) at null point Q A and S B (0,u M ) at null point Q B can be obtained by differentiating Eqs. ( 5) and ( 6) on u, respectively.
( ) Therefore, when the deviation of VPHs axial offset is δ M , back focal length measurement error σ offset caused by δ M can be obtained through Eq. ( 1).
( ) Using a grating scale as an aided adjustment of the VPH offsets, offset error δ M 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 L c focus.

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 where L is the measurement distance.

Synthesis error
Considering the effect of the aforementioned errors on the measurement results of l F ', the DRCFM total measurement error σl F ' is .

Experimental setup
The experimental setup shown in Fig. 5

Experiment results
As shown in Fig. 6, when reflector R moves across position A along the optical axis during measurement, DRCFM uses null point Q A of differential confocal response signal I A (z) to determine precisely the focus of L t .The position coordinate of reflector R corresponding to point Q A is z A = −99.17363mm.Using Eqs. ( 11)-( 13), (18), and (19), the errors existing in DRCFM are σ L = 0.1 μm, σz A = 0.75 μm, σz B = 0.37 μm, σ axial = 0.03 μm, and σ offset = 0.1 μm.The system error obtained using Eq. ( 22) is The relative error is 1.7 100% 100% 0.00086% 8.6 ppm.198.6918 1000 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 σz A and σz B have more significant effect on the measurement accuracy.So, the measurement accuracy increases as the relative aperture of test lens increases.

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.

Fig. 1 .
Fig. 1.DRCFM principle.PBS is the polarized beam splitter, P is the one-fourth wave plate, L c is the collimating lens, L t 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 L c , and DMI is distance measurement interferometer.

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.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 L c .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 L t is a cemented doublet lens with a diameter of 20 mm and a focal length of 200 mm, and its nominal back focal length l F ' is 197.3 mm ( ± 2%).

Fig. 6 .
Fig. 6.Back focal length measurement curves.Similarly, when reflector R moves across position B, DRCFM uses null point Q B of differential confocal response signal I B (z) to determine precisely the vertex of the L t last surface.The position coordinate of reflector R corresponding to point Q B is z B = 0.17223 mm.Therefore, the back focal length l F ' of L t is 2|z A -z B | = 198.6917mm, and the repeatability achieved from ten measurements is σ test = 1.6 μm.Using Eqs.(11)-(13), (18), and (19), the errors existing in DRCFM are σ L = 0.1 μm, σz A = 0.75 μm, σz B = 0.37 μm, σ axial = 0.03 μm, and σ offset = 0.1 μm.The system error obtained using

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174943 -$15.00USD Received 24 Aug 2012; revised 16 Oct 2012; accepted 23 Oct 2012; published 2 Nov 2012 (C) 2012 OSA 5 November 2012 / Vol. 20, No. 23 / OPTICS EXPRESS 26035 1, z is the axial displacement between reflector R and position A, u is the normalized coordinate of variable z, r 2 is the radial coordinate on the object plane of VPH1 or VPH2, v 2 is the normalized coordinate of variable r 2 , M is the offset of the VPHs from the focus of collimating lens L c , u M is the normalized offset of variable M, P t (ρ) is the pupil function of test lens L t , P c (ρ) is the pupil function of collimating lens L c , D is the effective aperture of L c #174943 -$15.00USD Received 24 Aug 2012; revised 16 Oct 2012; accepted 23 Oct 2012; published 2 Nov 2012 (C) 2012 OSA coordinate of variable r