## Laser differential confocal ultra-long focal length measurement

Optics Express, Vol. 17, Issue 22, pp. 20051-20062 (2009)

http://dx.doi.org/10.1364/OE.17.020051

Acrobat PDF (815 KB)

### Abstract

A new laser differential confocal focal-length measurement method is proposed for the measurement of an ultra-long focal-length. The approach proposed uses the property of an axial intensity curve that the absolute zero precisely corresponds to the focus of the objective in a differential confocal focusing system (DCFS) to measure the variation in position of DCFS focus with and without a measured ultra-long focal-length lens (UFL), uses the distance between the two focuses to obtain the UFL focal-length, and thereby achieving the precise measurement of ultra-long focal-length. The method has a high focusing precision, a strong anti-interference capability and a short measurement light-path. The theoretical analyses and preliminary experimental results indicate that the relative measurement error is about 0.01% when the method is used for the measurement of back-focus-distance (BFD).

© 2009 OSA

## 1.Introduction

1. L. Glatt and O. Kafri, “Determination of the focal length of nonparaxial lenses by moire deflectometry,” Appl. Opt. **26**(13), 2507–2508 (1987). [CrossRef] [PubMed]

2. V. I. Meshcheryakov, M. I. Sinel’nikov, and O. K. Filippov, “Measuring the focal lengths of long-focus optical systems,” J. Opt. Technol. **66**, 458 (1999). [CrossRef]

3. Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. **24**(19), 3162–3166 (1985). [CrossRef] [PubMed]

5. K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. **31**(1), 75–79 (1992). [CrossRef] [PubMed]

6. 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]

7. 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 http://www.opticsinfobase.org/abstract.cfm?URI=OFT-2002-OWD8.

8. B. DeBoo and J. Sasian, “Precise focal-length measurement technique with a reflective Fresnel-zone hologram,” Appl. Opt. **42**(19), 3903–3909 (2003). [CrossRef] [PubMed]

7. 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 http://www.opticsinfobase.org/abstract.cfm?URI=OFT-2002-OWD8.

8. B. DeBoo and J. Sasian, “Precise focal-length measurement technique with a reflective Fresnel-zone hologram,” Appl. Opt. **42**(19), 3903–3909 (2003). [CrossRef] [PubMed]

## 2. Measurement principle

### 2.1 Differential confocal focusing principle

9. 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]

*R*is moved near the focus along the optical axis, intensity response

*I*

_{1}(

*v*,

*u*, +

*u*) received by detector 1 is [9

_{M}9. 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]

*J*

_{0}is a zero-order Bessel function,

*u*is the axial normalized offset of a pinhole,

_{M}*ρ*is a radial normalized radius,

*p*

_{1}(

*ρ*) is the pupil function of focusing system including RL,

*p*

_{2}(

*ρ*) is the pupil function of collecting system including Lens 2,

*u*and

*v*are axial and lateral normalized coordinates, respectively, and

*z*is the axial displacement of the object,

*r*is the radial coordinate of the objective,

*D*/

*f*is the relative aperture of objective RL used in a differential confocal focusing system (DCFS).

*I*

_{2}(

*v*,

*u*,-

*u*) received by detector 2 is [9

_{M}9. 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]

*I*(

*v*,

*u*,

*u*) obtained through the differential subtraction of

_{M}*I*

_{1}(

*v*,

*u*, +

*u*) and

_{M}*I*

_{2}(

*v*,

*u*,-

*u*) is:

_{M}*I*(0,

*u*,

*u*) obtained from Eq. (4) is:

_{M}*O*of differential axial response curve

*I*(0

*,u*,

*u*) precisely corresponds to DCFS focus and the precise focusing of a lens is achieved by exactly determining the position of the reflector corresponding to zero

_{M}*O*of

*I*(0,

*u*,

*u*).

_{M}**12**(21), 5013–5021 (2004). [CrossRef] [PubMed]

### 2.2 Differential confocal ultra-long focal-length focusing measurement principle

*d*between principal points of UFL and RL is simplified when RL is a plano-convex lens with coincided principal point and vertex.

*A*by RL without UFL in the light-path, and the differential response curve exactly corresponds to zero

*O*

_{1}when reflector

*R*passed through point

*A*along optical axis. The focus is changed from

*A*to

*B*when the UFL is combined with RL shown in Fig. 3 , and the differential response curve exactly corresponds to zero

*O*

_{2}when reflector

*R*passed through point

*B*along optical axis. Distance

*l = z*between focuses

_{A}-z_{B}*A*and

*B*can be obtained by measuring the movement displacement of reflector

*R*corresponding to two zeroes

*O*

_{1}and

*O*

_{2}of differential response curve.

*f'*, UFL focal-length

*f*

_{1}

*'*, RL focal-length

*f*

_{2}

*'*and distance

*d*between principal points of UFL and RL satisfy Eq. (6).

*l*is the distance between

*A*and

*B*,

*H*

_{2}

*H*

_{22}is the distance between the right principal points of combination Lens and RL.

*f*

_{1}

*'*obtained using Eq. (6) ~Eq. (8) can be written as:where

*d*=

*d*

_{0}+

*H*

_{12}

*V*

_{12},

*H*

_{12}

*V*

_{12}is the distance between the right principal point and vertex point of UFL,

*d*

_{0}is the distance between the vertex points of UFL and RL.

*f*

_{1}

*'*can be obtained by:where

*b*

_{1}is the UFL thickness,

*n*

_{1}is the refractive index, and

*r*

_{11}and

*r*

_{12}are the radii of curvature of the front and back surface.

## 3. Analyses of key parameters

### 3.1 Focusing sensitivity

*S*(0,0,

*u*) at absolute zero

_{M}*O*can be obtained by differentiating curve

*I*(0,

*u*,

*u*) on

_{M}*u*and is

*S*

_{max}at zero

*O*when

*u*= 5.21.

_{M}*u*and

_{M}*D*/

*f '*, the pinhole size and the lateral offset of the detector have an effect on the differential measurement sensitivity, but it can be eliminated by readjusting the optical path arrangement [10

10. W. Zhao, J. Tan, L. Qiu, and L. Zou, “A new laser heterodyne confocal probe for ultraprecision measurement of discontinuous contours,” Meas. Sci. Technol. **16**(2), 497–504 (2005). [CrossRef]

11. W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. **120**(1), 17–25 (2005). [CrossRef]

### 3.2 Measurement range of focal-length

*f*

_{1}

*'*is short, the combination lens focus

*B*will be far from point

*A*after UFL is introduced into the light-path. At the same time, the focusing intensity response curve will be out of shape and the focusing sensitivity will decrease obviously. If

*f*

_{1}

*'*> 2.5

*f*

_{2}

*'*and especially

*f*

_{1}

*'*→∞, the focus of the combination lens will be close to point

*A*and the focusing sensitivity is the best.

*f*

_{1}

*'*is limited by RL focal-length

*f*

_{2}

*'*and the best measurement range of

*f*

_{1}

*'*varies with

*f*

_{2}

*'*.

*u*satisfies Eq. (15) before UFL is inserted into the measurement light-path and DCFS focusing sensitivity is the best.

_{M}*f*

_{2}

*'*to

*f '*, and their ratio is expressed as:

*u*changes to

_{M}*u*´ when UFL is used in DCFS, and

_{M}*u*´ obtained from Eqs. (15) and (16) is

_{M}*f*

_{1}

*'*is equal to 1.5

*f*

_{2}

*'*, 2

*f*

_{2}

*'*, 2.5

*f*

_{2}

*'*, 3

*f*

_{2}

*'*, 5

*f*

_{2}

*'*, 10

*f*

_{2}

*'*,⋅⋅⋅, or ∞, respectively.

*f*

_{1}

*'*= 1.5

*f*

_{2}

*'*, the response curve has more peaks than the other curves and its slope near the point

*O*is reverse so that it cannot be used for focusing; when

*f*

_{1}

*'*≥2.5

*f*

_{2}

*'*, the differential focusing curves all pass through absolute zero

*O*although their sensitivity are different, and then, the focusing method using absolute zero

*O*on UFL focal-length measurement has a high precision. And therefore, it is that the measurement method proposed is especially fit for measurement of ultra-long focal-length.

### 3.3 Effect of two pinholes with different offsets

*l*will be changed.

*M*+

*δ*and

*M*, their corresponding normalized offsets are

*u*+

_{M}*u*and

_{δ}*u*, the offset of

_{M}*O*from the focus

_{1}*A*and

*O*from the focus

_{2}*B*in the measurement process are Δ

*l*and Δ

_{1}*l*, and their normalized amounts are

_{2}*u*

_{1}and

*u*

_{2}.

*l*

_{1}and Δ

*l*

_{2}of

*O*and

_{1}*O*are respectively:where

_{2}*f*

_{2}

*'*,

*f*and

_{Lens2}'*f '*are the focal-length of RL, Lens2 and combination lens respectively.

*f '*is

*δ*, the error of

*l*can be obtained from Eq. (21) and Eq. (22):

*δ*is not possible to be eliminate entirely but to be reduced largely in the adjustment of light path. The variation of

*l*is small because offset Δ

*l*and Δ

_{1}*l*is close in amount and the same in direction, so that they are counteracted in part. In a common DCFS, |Δ

_{2}*l|*<0.1

*δ*is obtained from Eq. (23) when

*f*=

_{Lens2}'*f*

_{2}

*'*and UFL focal-length satisfies

*f*

_{1}

*'*>5

*f*

_{2}

*'*and

*f*

_{1}

*'*>>d. Therefore, the effect of

*δ*on

*l*is smaller as the UFL focal-length

*f*

_{1}

*'*is longer.

## 4. Error analyses

*l*,

*f*

_{2}

*'*and

*d*

_{0}has an effect on measurement of UFL

*f*

_{BFD}

*'*, and the error propagation coefficient obtained by differentiating Eq. (10) on

*d*

_{0},

*l*and

*f*

_{2}

*'*is respectively:

*f*

_{BFD}

*'*/∂

*l*and

*f*

_{2}

*'*, ∂

*f*

_{BFD}

*'*/

*f*

_{2}

*'*and

*f*

_{2}

*'*are shown in Fig. 8 respectively when UFL focal-length is 10 m, 20 m or 30 m, respectively.

*l*is the largest in the error sources, and focusing error

*σ*has a direct effect on the measurement precision of focal-length. And

_{z}*σ*obtained from Eq. (14) is written aswhere

_{z}*SNR*is the signal noise ratio of the photoelectric detector.

*A*and

*B*is

*σ*, the measurement error of

_{L}*l*obtained through two focusing is:

*f*

_{2}

*'*can be obtained through measuring the BFD of RL by using the measurement principle as shown in Fig. 1, and the measurement error of

*f*

_{2}

*'*is:

*d*

_{0}is

*σ*, then UFL focal-length measurement error

_{d0}*σ*

_{f}_{BFD}

*can be obtained by considering the three error sources above, which are described in Eqs. (24)~(29):*

_{'}*l*and the effect of assembling error and vibration is neglected, the error is

*σ*≈0.2 μm,

_{L}*σ*≈1.5 μm,

_{RL}*σ*≈0.5 μm and

_{d0}*σ*≈0.2 μm when

_{z}*f*

_{BFD}

*'*≈10 m,

*D*= 200 mm,

*f*

_{2}

*'*= 1.5 m,

*d*

_{0}≈500 mm and

*SNR*= 200:1, and therefore, the relative measurement error obtained using Eq. (30) is less than 0.001%.

*σ*

_{f}_{2'}is the largest. And therefore, when the dual-frequency laser interferometer is used for calibrating the RL focal-length, the error is

*σ*≈0.5 μm,

_{l}*σ*

_{f}_{2´}≈50 μm and

*σ*≈5 μm, and UFL focal-length relative measurement error is

_{d0}*σ*

_{f}_{1´}= 0.007%.

## 5. Experiments

*Φ*= 20 mm,

*f*

_{BFD}

*'*= 988.73mm,

*f*

_{1}

*´*= 995.33 mm,

*b*

_{1}= 10mm,

*n*

_{1}= 1.5164,

*r*

_{11}= 513.99mm and

*r*

_{12}→∞, respectively. Both RL and colleting lens are

*Φ*= 20 mm and

*f*= 164.40 mm, and RL focal-length error σ

_{2}´*= 0.008 mm. And the distance between their vertexes is*

_{f2´}*d*

_{0}= 26.40 mm and σ

*= 0.005 mm.*

_{d0}*R*is moved along the RL optical axis, the focus

*A*of RL is precisely determined by the zero of differential confocal response signal in DCFS. The differential confocal focusing curve

*I*(

*z*) is shown in Fig. 9 , where

*I*

_{1}(

*z*) and

*I*

_{2}(

*z*) are the intensity signals received by two detectors,

*I*(

_{A}*z*) is the differential confocal focusing curve and the coordinate corresponding to the focus

*A*is 25.9998 mm.

*R*is moved along the optical axis, the focus

*B*is precisely determined by the zero of differential confocal focusing curve

*I*(

_{B}*z*), and the coordinate corresponding to the focus

*B*is 2.0117 mm.

*l*= 23.9881 mm, and the repetition measurement error of

*l*is

*σ*= 0.6 μm. BFD of the measured lens

_{l}*f*

_{BFD}

*´*obtained from Eq. (10) is:

*b*

_{1},

*n*

_{1},

*r*

_{11}and

*r*

_{12}, the focal-length of the measured lens

*f*

_{1}

*´*can be obtained from Eq. (11):

## 6. Conclusion

- 1) has a measurement light-path shorter than the measured focal-length so that the effect of environment on the measurement precision of the focal-length is small.
- 2) can reduce the focal-depth and improve the focusing sensitivity by using the pupil filtering technique and differential confocal technique.
- 3) can improve the environmental anti-interference capability by using the intensity modulation technique.

## Acknowledgment

## References and links

1. | L. Glatt and O. Kafri, “Determination of the focal length of nonparaxial lenses by moire deflectometry,” Appl. Opt. |

2. | V. I. Meshcheryakov, M. I. Sinel’nikov, and O. K. Filippov, “Measuring the focal lengths of long-focus optical systems,” J. Opt. Technol. |

3. | Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. |

4. | J. C. Bhattacharya and A. K. Aggarwal, “Measurement of the focal length of a collimating lens using the Talbot effect and the moiré technique,” Appl. Opt. |

5. | K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. |

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

7. | 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 http://www.opticsinfobase.org/abstract.cfm?URI=OFT-2002-OWD8. |

8. | B. DeBoo and J. Sasian, “Precise focal-length measurement technique with a reflective Fresnel-zone hologram,” Appl. Opt. |

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

10. | W. Zhao, J. Tan, L. Qiu, and L. Zou, “A new laser heterodyne confocal probe for ultraprecision measurement of discontinuous contours,” Meas. Sci. Technol. |

11. | W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3620) Instrumentation, measurement, and metrology : Lens system design

(180.1790) Microscopy : Confocal microscopy

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 7, 2009

Revised Manuscript: August 30, 2009

Manuscript Accepted: October 1, 2009

Published: October 20, 2009

**Citation**

Weiqian Zhao, Ruoduan Sun, Lirong Qiu, and Dingguo Sha, "Laser differential confocal ultra-long focal length measurement," Opt. Express **17**, 20051-20062 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-20051

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### References

- L. Glatt and O. Kafri, “Determination of the focal length of nonparaxial lenses by moire deflectometry,” Appl. Opt. 26(13), 2507–2508 (1987). [CrossRef] [PubMed]
- V. I. Meshcheryakov, M. I. Sinel’nikov, and O. K. Filippov, “Measuring the focal lengths of long-focus optical systems,” J. Opt. Technol. 66, 458 (1999). [CrossRef]
- Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162–3166 (1985). [CrossRef] [PubMed]
- J. C. Bhattacharya and A. K. Aggarwal, “Measurement of the focal length of a collimating lens using the Talbot effect and the moiré technique,” Appl. Opt. 30(31), 4479–4480 (1991). [CrossRef] [PubMed]
- K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. 31(1), 75–79 (1992). [CrossRef] [PubMed]
- 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]
- 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 http://www.opticsinfobase.org/abstract.cfm?URI=OFT-2002-OWD8 .
- B. DeBoo and J. Sasian, “Precise focal-length measurement technique with a reflective Fresnel-zone hologram,” Appl. Opt. 42(19), 3903–3909 (2003). [CrossRef] [PubMed]
- 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]
- W. Zhao, J. Tan, L. Qiu, and L. Zou, “A new laser heterodyne confocal probe for ultraprecision measurement of discontinuous contours,” Meas. Sci. Technol. 16(2), 497–504 (2005). [CrossRef]
- W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005). [CrossRef]

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