## Detection method of nonlinearity errors by statistical signal analysis in heterodyne Michelson interferometer

Optics Express, Vol. 18, Issue 6, pp. 5831-5839 (2010)

http://dx.doi.org/10.1364/OE.18.005831

Acrobat PDF (200 KB)

### Abstract

Periodic nonlinearity that ranges from tens of nanometers to a few nanometers in heterodyne interferometer limits its use in high accuracy measurement. A novel method is studied to detect the nonlinearity errors based on the electrical subdivision and the analysis method of statistical signal in heterodyne Michelson interferometer. Under the movement of micropositioning platform with the uniform velocity, the method can detect the nonlinearity errors by using the regression analysis and Jackknife estimation. Based on the analysis of the simulations, the method can estimate the influence of nonlinearity errors and other noises for the dimensions measurement in heterodyne Michelson interferometer.

© 2010 OSA

## 1. Introduction

1. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. **14**(2), 91–98 (1992). [CrossRef]

2. C. M. Wu and C. S. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. **7**(1), 62–68 (1996). [CrossRef]

1. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. **14**(2), 91–98 (1992). [CrossRef]

7. A. Rosenbluth and N. Bobroff, “Optical source of nonlinearity of heterodyne interferometers,” Precis. Eng. **12**(1), 7–11 (1990). [CrossRef]

10. W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. **30**(3), 337–346 (2006). [CrossRef]

1. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. **14**(2), 91–98 (1992). [CrossRef]

8. A. Yacoot and M. J. Downs, “The use of X-ray interferometry to investigate the linearity of NPL differential plane mirror optical interferometer,” Meas. Sci. Technol. **11**(8), 1126–1130 (2000). [CrossRef]

9. V. Badami, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. **24**(1), 41–49 (2000). [CrossRef]

23. T. L. Schmitz and H. S. Kim, “Monte Carlo evaluation of periodic error uncertainty,” Precis. Eng. **31**(3), 251–259 (2007). [CrossRef]

**14**(2), 91–98 (1992). [CrossRef]

10. W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. **30**(3), 337–346 (2006). [CrossRef]

16. J. Lawall and E. Kessler, “Michelson interferometry with 10pm accuracy,” Rev. Sci. Instrum. **71**(7), 2669–2676 (2000). [CrossRef]

28. K. N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express **18**(2), 1159–1165 (2010). [CrossRef] [PubMed]

9. V. Badami, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. **24**(1), 41–49 (2000). [CrossRef]

23. T. L. Schmitz and H. S. Kim, “Monte Carlo evaluation of periodic error uncertainty,” Precis. Eng. **31**(3), 251–259 (2007). [CrossRef]

27. T. L. Schmitz, L. Houck III, D. Chu, and L. Kalem, “Bench-top setup for validation of real time, digital periodic error correction,” Precis. Eng. **30**(3), 306–313 (2006). [CrossRef]

25. T. L. Schmitz, D. Chu, and L. Houck III, “First-order periodic error correction: validation for constant and nonconstant velocities with variable error magnitudes,” Meas. Sci. Technol. **17**(12), 3195–3203 (2006). [CrossRef]

26. T. L. Schmitz, D. Chu, and H. S. Kim, “First and second order periodic error measurement for non-constant velocity motions,” Precis. Eng. **33**(4), 353–361 (2009). [CrossRef]

17. B. Efron, “Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods,” Biometrika **68**(3), 589–599 (1981). [CrossRef]

## 2. The electrical subdivision algorithm

22. S. H. Lu, C. I. Chiueh, and C. C. Lee, “Differential wavelength-scanning heterodyne interferometer for measuring large step height,” Appl. Opt. **41**(28), 5866–5871 (2002). [CrossRef] [PubMed]

29. J. Flügge, Ch. Weichert, H. Hu, R. Köning, H. Bosse, A. Wiegmann, M. Schulz, C. Elster, and R. D. Geckeler, “Interferometry at the PTB Nanometer Comparator: design, status and development,” Proc. SPIE **7133**, 713346 (2008). [CrossRef]

*L*is the displacement of the micropositioning platform,

*λ*is the wavelength of laser, and Δ

*f*is the Doppler frequency shift.

*φ*to calculate

*L*:where Δ

*φ*is the phase shift that is caused by Doppler frequency shift.

*I*,

_{m1}*I*and

_{m2}*I*by photodiodes D

_{r}_{M1}, D

_{M2}and D

_{R}in the scheme under the ideal conditions: where

*I*and

_{m0}*I*are the amplitude of signals,

_{r0}*f*and

_{1}*f*are the frequencies of incident beams,

_{2}*φ*and

_{m0}*φ*are the initial and constant phase shift, Δ

_{r0}*φ*is the phase shift that is caused by Doppler frequency shift.

*I*and

_{mr1}*I*:

_{mr2}*I*and

_{mr1}*I*pass by the low-pass filter, we can obtain: where

_{mr2}*φ*.

_{mr0}= φ_{m0}–φ_{r0}*φ*as follows:

*φ*as follows:where

_{mr0}*I*and

_{1s}*I*are the value of

_{2s}*I*and

_{1}*I*when the micropositioning platform stops.

_{2}*I*,

_{1}*I*. The FPGA receives the sampled signals and outputs Δ

_{2}*φ*to PC by Eq. (10) and Eq. (11). In contrast to

*f*, Δ

_{1}-f_{2}*f*is small if the speed of the micropositioning platform is not fast. So we can obtain more sampling points in a period by ADC. In other words, the method can get higher resolution than sampling

*I*and

_{m}*I*directly.

_{r}## 3. The nonlinearity of heterodyne interferometers

*a*,

*b*,

*c*,

*d*,

*θ*, and

_{a}*θ*are the influence factors that are determined by the incident beams with all possible polarizing imperfections and the attenuation and the phase shift while the beams travel through the diverse optical parts.

_{c}*b/a*and

*d/c*are the frequency mix ratios in two interferometer arms.

*θ*and

_{a}*θ*are the initial phase shifts of frequency

_{c}*f*and

_{1}*f*respectively. The derivation process of the Eq. (12) and the detailed definition of the influence factors are described in Ref.5 and Ref.10.

_{2}## 4. Detection of nonlinearity errors

### 4.1 Phase shift under uniform velocity of the micropositioning platform

*f*is the frequency shift,

*v*is the instant velocity of the micropositioning platform, c is the velocity of light, and f is the frequency of beam, respectively.

*φ*is a constant value in a constant period of time when the micropositioning platform moves with the uniform velocity. So we can get:where

*k*is the parameter that is defined by the velocity

*v*of the micropositioning platform and the frequency

*f*of the beam as follows:

*φ*under the uniform velocity of the micropositioning platform as follows:

_{m}*γ*is smooth, but also the probability density function of the nonlinearity errors

*γ*is not normal distribution. Therefore, we cannot use the traditional regression analysis to estimate

*k*of Eq. (19) because the traditional regression analysis requires that the distribution of error is normal distribution. Therefore, we use Jackknife method to estimate

*k*because the method has not the limitation of error distribution that is normal distribution, and has the excellent estimation performance.

### 4.2 Jackknife method for detection of nonlinearity errors

*n*is the time of sampling point

_{i}*i*and is determined by the sampling period of the ADC.

*k*:

*n*and Δ

_{j}*φ*where

_{mj}*j*≦

*N*in the series and obtain a new series whose name is series

*j*and the length of series

*j*is

*N-1*. We also use least square method to estimate

*k*in series

^{(j)}*j*. We repeat the above method for

*N*times and get a new series of

*k*where

^{(j)}*j*= 1,2,3,…,

*N*. Therefore, we can obtain the unbiased estimators of

*k*as follows:

*k*as follows:

*k*, we can use the unbiased estimator to fit the image of phase shift that does not include the nonlinearity errors. By comparing the fitting phase shift with the measured phase shift, we can achieve the detection of the nonlinearity errors. The estimated nonlinearity errors are calculated as follows:

## 5. Simulations

*v*is the velocity of the micropositioning platform, and other parameters

*a*,

*b*,

*c*,

*d*,

*θ*, and

_{a}*θ*are the influence factors of nonlinearity errors, respectively.

_{c}### 5.1 Simulation 1

*n*is 0.00001. Then we use Jackknife method to estimate

*k*. The results of the simulation are shown in Table 2 . The function image of Δ

*φ*, estimated

_{m}*kn*and nonlinearity errors in group 1 is shown in Fig. 3 .

### 5.2 Simulation 2

^{−6}

*k*. Although the electrical signals from the photosensors pass the filters and other electrical elements in order to remove the noises, we cannot avoid the condition that the electrical signals carry the tiny noises. So, we can get the measured value of phase shift Δ

*φ*as follows:where

_{me}*η*is the tiny electrical noise that is often defined as a white Gaussian noise.

*n*is 0.00001. Then we use Jackknife method to estimate

*k*. The results of the simulation are shown in Table 3 . The function image of Δ

*φ*, estimated

_{me}*kn*and nonlinearity errors in group 1 is shown in Fig. 4 .

### 5.3 Simulation 3

*v*and variance 10

^{−4}

*v*. We can get the measured value of phase shift Δ

*φ*as follows:where

_{mes}*ε*is the bias value that is caused by the nonuniform velocity movement of the micropositioning platform.

*n*is 0.00001. Then we use Jackknife method to estimate

*k*. The results of the simulation are shown in Table 4 . The function image of Δ

*φ*, estimated

_{mes}*kn*and nonlinearity errors in group 1 is shown in Fig. 5 .

## 6. Conclusion

## Acknowledgements

## References and links

1. | W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. |

2. | C. M. Wu and C. S. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. |

3. | R. C. Quenelle, “Nonlinearity in interferometric measurement,” Hewlett Packard J. |

4. | N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. |

5. | W. Hou and X. Zhao, “The drift of the nonlinearity of heterodyne interferometers,” Precis. Eng. |

6. | J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. |

7. | A. Rosenbluth and N. Bobroff, “Optical source of nonlinearity of heterodyne interferometers,” Precis. Eng. |

8. | A. Yacoot and M. J. Downs, “The use of X-ray interferometry to investigate the linearity of NPL differential plane mirror optical interferometer,” Meas. Sci. Technol. |

9. | V. Badami, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. |

10. | W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. |

11. | W. Hou, Y. Zhang, and H. Hu, “A simple technique for eliminating the nonlinearity of a heterodyne interferometer,” Meas. Sci. Technol. |

12. | C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. |

13. | O. P. Lay and S. Dubovitsky, “Polarization compensation: a passive approach to a reducing heterodyne interferometer nonlinearity,” Opt. Lett. |

14. | S. Dubovitsky, O. P. Lay, and D. J. Seidel, “Elimination of heterodyne interferometer nonlinearity by carrier phase modulation,” Opt. Lett. |

15. | C. M. Wu, “Periodic nonlinearity resulting from ghost reflections in heterodyne interometry,” Opt. Commun. |

16. | J. Lawall and E. Kessler, “Michelson interferometry with 10pm accuracy,” Rev. Sci. Instrum. |

17. | B. Efron, “Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods,” Biometrika |

18. | S. Mori, T. Akatsu, and C. Miyazaki, “Laser measurement system for precise and fast positioning,” Opt. Eng. |

19. | G. E. Sommargren, “A new laser measurement system for precision metrology,” Precis. Eng. |

20. | S. Hosoe, “Laser interferometric system for displacement measurement with high precision,” Nanotechnology |

21. | N. Hagiwara, Y. Nishitani, M. Yanase, and T. Saegusa, “A phase encoding method for improving the resolution and reliability of laser interferometers,” IEEE Trans. Instrum. Meas. |

22. | S. H. Lu, C. I. Chiueh, and C. C. Lee, “Differential wavelength-scanning heterodyne interferometer for measuring large step height,” Appl. Opt. |

23. | T. L. Schmitz and H. S. Kim, “Monte Carlo evaluation of periodic error uncertainty,” Precis. Eng. |

24. | D. Chu, and A. Ray, “Nonlinearity measurement and correction of metrology data from an interferometer system,” Proc. of 4th Euspen Int. Conf., 300–301 (2004). |

25. | T. L. Schmitz, D. Chu, and L. Houck III, “First-order periodic error correction: validation for constant and nonconstant velocities with variable error magnitudes,” Meas. Sci. Technol. |

26. | T. L. Schmitz, D. Chu, and H. S. Kim, “First and second order periodic error measurement for non-constant velocity motions,” Precis. Eng. |

27. | T. L. Schmitz, L. Houck III, D. Chu, and L. Kalem, “Bench-top setup for validation of real time, digital periodic error correction,” Precis. Eng. |

28. | K. N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express |

29. | J. Flügge, Ch. Weichert, H. Hu, R. Köning, H. Bosse, A. Wiegmann, M. Schulz, C. Elster, and R. D. Geckeler, “Interferometry at the PTB Nanometer Comparator: design, status and development,” Proc. SPIE |

**OCIS Codes**

(120.1880) Instrumentation, measurement, and metrology : Detection

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 17, 2009

Revised Manuscript: February 12, 2010

Manuscript Accepted: February 24, 2010

Published: March 9, 2010

**Citation**

Juju Hu, Haijiang Hu, and Yinghua Ji, "Detection method of nonlinearity errors by statistical signal analysis in heterodyne Michelson interferometer," Opt. Express **18**, 5831-5839 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5831

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### References

- W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14(2), 91–98 (1992). [CrossRef]
- C. M. Wu and C. S. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996). [CrossRef]
- R. C. Quenelle, “Nonlinearity in interferometric measurement,” Hewlett Packard J. 34, 3–13 (1983).
- N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. 26(13), 2676–2682 (1987). [CrossRef] [PubMed]
- W. Hou and X. Zhao, “The drift of the nonlinearity of heterodyne interferometers,” Precis. Eng. 16(1), 25–35 (1994). [CrossRef]
- J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4(10), 1173–1176 (1993). [CrossRef]
- A. Rosenbluth and N. Bobroff, “Optical source of nonlinearity of heterodyne interferometers,” Precis. Eng. 12(1), 7–11 (1990). [CrossRef]
- A. Yacoot and M. J. Downs, “The use of X-ray interferometry to investigate the linearity of NPL differential plane mirror optical interferometer,” Meas. Sci. Technol. 11(8), 1126–1130 (2000). [CrossRef]
- V. Badami, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24(1), 41–49 (2000). [CrossRef]
- W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006). [CrossRef]
- W. Hou, Y. Zhang, and H. Hu, “A simple technique for eliminating the nonlinearity of a heterodyne interferometer,” Meas. Sci. Technol. 20(10), 105303 (2009). [CrossRef]
- C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. 38(19), 4089–4094 (1999). [CrossRef]
- O. P. Lay and S. Dubovitsky, “Polarization compensation: a passive approach to a reducing heterodyne interferometer nonlinearity,” Opt. Lett. 27(10), 797–799 (2002). [CrossRef]
- S. Dubovitsky, O. P. Lay, and D. J. Seidel, “Elimination of heterodyne interferometer nonlinearity by carrier phase modulation,” Opt. Lett. 27(8), 619–621 (2002). [CrossRef]
- C. M. Wu, “Periodic nonlinearity resulting from ghost reflections in heterodyne interometry,” Opt. Commun. 215(1-3), 17–23 (2003). [CrossRef]
- J. Lawall and E. Kessler, “Michelson interferometry with 10pm accuracy,” Rev. Sci. Instrum. 71(7), 2669–2676 (2000). [CrossRef]
- B. Efron, “Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods,” Biometrika 68(3), 589–599 (1981). [CrossRef]
- S. Mori, T. Akatsu, and C. Miyazaki, “Laser measurement system for precise and fast positioning,” Opt. Eng. 27, 823–829 (1988).
- G. E. Sommargren, “A new laser measurement system for precision metrology,” Precis. Eng. 9(4), 179–184 (1987). [CrossRef]
- S. Hosoe, “Laser interferometric system for displacement measurement with high precision,” Nanotechnology 2(2), 88–95 (1991). [CrossRef]
- N. Hagiwara, Y. Nishitani, M. Yanase, and T. Saegusa, “A phase encoding method for improving the resolution and reliability of laser interferometers,” IEEE Trans. Instrum. Meas. 38(2), 548–551 (1989). [CrossRef]
- S. H. Lu, C. I. Chiueh, and C. C. Lee, “Differential wavelength-scanning heterodyne interferometer for measuring large step height,” Appl. Opt. 41(28), 5866–5871 (2002). [CrossRef] [PubMed]
- T. L. Schmitz and H. S. Kim, “Monte Carlo evaluation of periodic error uncertainty,” Precis. Eng. 31(3), 251–259 (2007). [CrossRef]
- D. Chu, and A. Ray, “Nonlinearity measurement and correction of metrology data from an interferometer system,” Proc. of 4th Euspen Int. Conf., 300–301 (2004).
- T. L. Schmitz, D. Chu, and L. Houck, “First-order periodic error correction: validation for constant and nonconstant velocities with variable error magnitudes,” Meas. Sci. Technol. 17(12), 3195–3203 (2006). [CrossRef]
- T. L. Schmitz, D. Chu, and H. S. Kim, “First and second order periodic error measurement for non-constant velocity motions,” Precis. Eng. 33(4), 353–361 (2009). [CrossRef]
- T. L. Schmitz, L. Houck, D. Chu, and L. Kalem, “Bench-top setup for validation of real time, digital periodic error correction,” Precis. Eng. 30(3), 306–313 (2006). [CrossRef]
- K. N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express 18(2), 1159–1165 (2010). [CrossRef] [PubMed]
- J. Flügge, Ch. Weichert, H. Hu, R. Köning, H. Bosse, A. Wiegmann, M. Schulz, C. Elster, and R. D. Geckeler, “Interferometry at the PTB Nanometer Comparator: design, status and development,” Proc. SPIE 7133, 713346 (2008). [CrossRef]

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