## Laser differential reflection-confocal focal-length measurement |

Optics Express, Vol. 20, Issue 23, pp. 26027-26036 (2012)

http://dx.doi.org/10.1364/OE.20.026027

Acrobat PDF (2845 KB)

### 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

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]

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]

## 2. DRCFM principle

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]

*Q*and

_{A}*Q*of differential confocal response signals

_{B}*I*and

_{A}*I*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

_{B}*l*'.where

_{F}*z*and

_{A}*z*are the position coordinates of reflector R corresponding to null points

_{B}*Q*and

_{A}*Q*, respectively.

_{B}*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*

_{A}_{1}(

*ν*

_{2},

*u*, +

*u*) and

_{M}*I*

_{A}_{2}(

*ν*

_{2},

*u*,-

*u*) received from CCD1 and CCD2 can be obtained by the Huygens–Fresnel diffraction integral formula.

_{M}*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 coordinate of variable

*r*

_{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*is the normalized offset of variable

_{M}*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}and L

_{t}which is equal to the smaller aperture between L

_{c}and L

_{t},

*D*/

*f*is the effective relative aperture of L

_{c}'_{c}, and

*D*/

*f*is the effective relative aperture of L

_{t}'_{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}.

*P*(

_{c}*ρ*) = 1 and

*P*(

_{t}*ρ*) = 1, the differential confocal response signal

*I*(

_{A}*u*,

*u*) is obtained through the differential subtraction of

_{M}*I*

_{A}_{1}(0,

*u*, +

*u*) and

_{M}*I*

_{A}_{2}(0,

*u*,-

*u*).

_{M}*Q*of differential confocal response signal

_{A}*I*(

_{A}*u*,

*u*) precisely corresponds to position

_{M}*A*such that reflector R is exactly at the focus of L

_{t}.

*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 2

*z*. Therefore, the differential confocal response signal

*I*(

_{B}*u*,

*u*) obtained through the differential subtraction of

_{M}*I*

_{B}_{1}(0,

*u*, +

*u*) and

_{M}*I*

_{B}_{2}(0,

*u*,-

*u*) can be derived as follows.

_{M}*Q*of differential confocal response signal

_{B}*I*(

_{B}*u*,

*u*) precisely corresponds to position

_{M}*B*such that the measurement beam is focused on the vertex of the L

_{t}last surface.

_{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.

## 3. Effect analyses of key parameters

### 3.1 Focusing sensitivities

*S*(0,

_{A}*u*) at null point

_{M}*Q*and

_{A}*S*(0,

_{B}*u*) at null point

_{M}*Q*can be obtained by differentiating Eqs. (5) and (6) on

_{B}*u*, respectively.

*I*(

_{A}*u*,

*u*) and

_{M}*I*(

_{B}*u*,

*u*) have the largest sensitivities

_{M}*S*

_{A}_{max}and

*S*

_{B}_{max}at null points

*Q*and

_{A}*Q*when

_{B}*u*= 5.21.

_{M}*I*(

_{A}*u*,

*u*) and

_{M}*I*(

_{B}*u*,

*u*) with

_{M}*u*= 5.21. The identification at the vertex of the L

_{M}_{t}last surface is clearly much more sensitive than that at the L

_{t}focus.

*u*= 5.21, DRCFM focusing errors

_{M}*σz*and

_{A}*σz*at the L

_{B}_{t}focus and the vertex of the L

_{t}last surface can be described as follows. where

*δI*(

_{A}*u*,

*u*) and

_{M}*δI*(

_{B}*u*,

*u*) are the random noises of the differential confocal response signals caused by the CCDs, and

_{M}*SNR*is the signal-to-noise ratio of the VPHs. Equations (11) and (12) show that the identification precision increases as relative aperture

*D*/

*f*increases.

_{t}'### 3.2 Effect of axial alignment errors

*m*of the measurement beam, optical axis

*s*of L

_{t}, 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 L

_{t}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.

*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 L

_{c}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

_{c}focus are different, null points

*Q*and

_{A}*Q*of the differential confocal response signals deviate from positions

_{B}*A*and

*B*, respectively. Thus, the

*l*' measurements change.

_{F}_{c}focus be

*M*and -

*M*+

*δ*, their corresponding normalized offsets be

_{M}*u*and -

_{M}*u*+

_{M}*u*, the offsets of

_{δM}*Q*from position

_{A}*A*and of

*Q*from position

_{B}*B*in the measurement process be Δ

*l*

_{1}and Δ

*l*

_{2}, and their normalized values be Δ

*u*

_{1}and Δ

*u*

_{2}.

*I*(

_{A}*u*,

*u*) and

_{M}*I*(

_{B}*u*,

*u*) are obtained according to Eqs. (5) and (6).

_{M}*δ*, back focal length measurement error

_{M}*σ*caused by

_{offset}*δ*can be obtained through Eq. (1).

_{M}*δ*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

_{M}_{c}focus.

### 3.4 Distance measurement error

*A*and

*B*is measured by a single-frequency laser interferometer, and its measurement error iswhere

*L*is the measurement distance.

### 3.5 Synthesis error

*l*', the DRCFM total measurement error

_{F}*σl*' is

_{F}## 4. Experiments

### 4.1 Experimental setup

_{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*' is 197.3 mm ( ± 2%).

_{F}### 4.2 Experiment results

*A*along the optical axis during measurement, DRCFM uses null point

*Q*of differential confocal response signal

_{A}*I*(

_{A}*z*) to determine precisely the focus of L

_{t}. The position coordinate of reflector R corresponding to point

*Q*is

_{A}*z*= −99.17363 mm.

_{A}*B*, DRCFM uses null point

*Q*of differential confocal response signal

_{B}*I*(

_{B}*z*) to determine precisely the vertex of the L

_{t}last surface. The position coordinate of reflector R corresponding to point

*Q*is

_{B}*z*= 0.17223 mm.

_{B}*l*' of L

_{F}_{t}is 2|

*z*-

_{A}*z*| = 198.6917 mm, and the repeatability achieved from ten measurements is

_{B}*σ*= 1.6 μm.

_{test}*σ*= 0.1 μm,

_{L}*σz*= 0.75 μm,

_{A}*σz*= 0.37 μm,

_{B}*σ*= 0.03 μm, and

_{axial}*σ*= 0.1 μm. The system error obtained using Eq. (22) is

_{offset}*σz*and

_{A}*σz*have more significant effect on the measurement accuracy. So, the measurement accuracy increases as the relative aperture of test lens increases.

_{B}## 5. Conclusions

- 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.

## Acknowledgments

## 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. |

2. | C.-W. Chang and D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. |

3. | P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, “Measurement of focal length with phase-shifting Talbot interferometry,” Appl. Opt. |

4. | K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. |

5. | F. Lei and L. K. Dang, “Measuring the focal length of optical systems by grating shearing interferometry,” Appl. Opt. |

6. | M. Thakur and C. Shakher, “Evaluation of the focal distance of lenses by white-light Lau phase interferometry,” Appl. Opt. |

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. |

8. | S. Zhao, J. F. Wen, and P. S. Chung, “Simple focal-length measurement technique with a circular Dammann grating,” Appl. Opt. |

9. | Y. P. Kumar and S. Chatterjee, “Technique for the focal-length measurement of positive lenses using Fizeau interferometry,” Appl. Opt. |

10. | Y. Xiang, “Focus retrocollimated interferometry for focal-length measurements,” Appl. Opt. |

11. | I. K. Ilev, “Simple fiber-optic autocollimation method for determining the focal lengths of optical elements,” Opt. Lett. |

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. |

13. | J.- Wu, J.- Chen, A.- Xu, X.-y. Gao, and S. Zhuang, “Focal length measurement based on Hartmann-Shack principle,” Optik (Stuttg.) |

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 |

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. | W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express |

**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

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