## Heterodyne moiré surface profilometry |

Optics Express, Vol. 22, Issue 3, pp. 2845-2852 (2014)

http://dx.doi.org/10.1364/OE.22.002845

Acrobat PDF (1448 KB)

### Abstract

In this study, a novel moiré fringe analysis technique is proposed for measuring the surface profile of an object. After applying a relative displacement between two gratings at a constant velocity, every pixel of CMOS camera can capture a heterodyne moiré signal. The precise phase distribution of the moiré fringes can be extracted using a one-dimensional fast Fourier transform (FFT) analysis on every pixel, simultaneously filtering the harmonic noise of the moiré fringes. Finally, the surface profile of the tested objected can be generated by substituting the phase distribution into the relevant equation. The findings demonstrate the feasibility of this measuring method, and the measurement error was approximately 4.3 μm. The proposed method exhibits the merits of the Talbot effect, projection moiré method, FFT analysis, and heterodyne interferometry.

© 2014 Optical Society of America

## 1. Introduction

## 2. Principle

*z*-axis is assigned as the observation axis of the CMOS camera C, and the

*y*-axis is set perpendicular to the paper plane. A laser light beam (wavelength

*λ*) passes through a beam expander to form an expanded and collimated light beam. Subsequently, the light beam impinges on linear grating G

_{1}at projection angle

*α*forming a self-image of grating G

_{1}and projecting the first self-image on the test surface. The first self-image distance

*Z*

_{1}can be expressed as follows [19

19. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. **129**(3-4), 167–172 (1996). [CrossRef]

*p*is the pitch of grating G

_{1}. The projected fringes are deformed because of the surface profile; this can be expressed as follows:where

*γ*is the visibility of the deformed fringes,

*z*(

*x*,

*y*) is the height distribution of the tested surface, and

*ϕ*

_{1}is the initial phase of grating G

_{1}. The deformed fringes are imaged at 1 × magnification on reference grating G

_{2}with a grating pitch of

*p*to form the moiré fringes; the image was captured using a CMOS camera C. The captured fringes can be expressed as follows:where

*q*

_{1}(−

*x*, −

*y*) denotes the imaging fringes of

*q*

_{1}(

*x*,

*y*) by using the imaging lens L

_{2};

*W*(

*x*,

*y*) is the transmission function of grating G

_{2}; and

*ϕ*

_{2}is the initial phase of grating G

_{2}. Equation (3) can be written as follows:

*ϕ*(

*x*,

*y*) is the phase distribution of the moiré fringes. In Eq. (4), the second, third, and fourth terms are the harmonic noise, and the last term is the moiré fringe, which is primarily relative to the height distribution of the tested surface. To obtain the phase distribution

*ϕ*, gratings G

_{1}and G

_{2}are moved at a constant speed,

*v*along the −

*x*-direction by using the motorized translation stages M

_{1}and M

_{2}, respectively. Thus, every pixel of the CMOS camera C can receive the time-varying signal and the moiré fringes can be expressed as follows:where

*f*

_{0}=

*v*/

*p*is the frequency introduced by the grating movement. Equation (6) can be rewritten as:

*x*-direction and

*f*= 2

_{h}*f*

_{0}is the heterodyne moiré frequency. For convenience, Eq. (7) can be rewritten using the Euler formula as follows [10

10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

*Q*,

*A*,

*B*,

*C*, and

*D*are the Fourier transform of

*q*

_{3},

*a*,

*b*,

*c*, and

*d*. Subsequently,

*D*(

*f*−

*f*,

_{h}*x*,

*y*) is shifted to

*D*(

*f*,

*x*,

*y*) and processed using the inverse Fourier transform to obtain

*d*(

*t*,

*x*,

*y*). Calculating a logarithm of

*d*(

*t*,

*x*,

*y*) yields:Extracting the imaginary part of Eq. (14) allows generating

*ϕ*+

*ϕ*

_{1}−

*ϕ*

_{2}. To generate phase distribution

*ϕ*(

*x*,

*y*), this procedure was applied to every pixel of camera C, and the initial phase difference

*ϕ*

_{1}−

*ϕ*

_{2}was neglected because it did not influence the relative height distribution of the surface measurement. Considering the image reversal that results from the imaging lens, Eq. (5) can be rewritten as follows:Finally, substituting

*ϕ*(

*x*,

*y*) into Eq. (15) and considering the image reversal that resulted from the imaging lens, the surface profile of the tested object can be reconstructed.

## 3. Experimental results and discussions

*f*= 20 Hz (

_{h}*v*= 0.2822 mm/s), and a CMOS camera (Basler /A504k) with an 8-bit gray level and 1280 × 1024 image resolution. For convenience, the projection angle was set at 30°, the frame rate of the CMOS camera was set at

*f*= 120 fps, the exposure time was set at

_{s}*t*= 8 ms, and the total recording time was set at

_{e}*T*= 1 s to record the heterodyne moiré signals at various times. Figures 2(b)-2(d) display the experimental results. Figure 2(b) is a recorded moiré image, showing that grating noise spread throughout the image. Despite the nonuniform light intensity distribution, nonuniform scattering property of the coin surface, and the height distribution of the coin, which is lower than that one period of the moiré fringe can represent, the phase distribution of the moiré fringes were extracted by using the proposed fringe analysis method [Fig. 2(c)]. The surface profile of the coin was reconstructed using Eq. (15), as shown in Fig. 2(d). To prove the accuracy of the proposed method, the contour from A to A′ [Fig. 2(a)] was measured using a stylus profilometer (KLA-Tencor/Alpha Step D-100) toserve as reference data. Figure 3 shows the contour lines from the experimental and reference data, indicating that the experimental and reference data exhibited the same height trends; this proves the feasibility of the proposed method.

*p*is the grating pitch error, Δ

*α*is the projection angle error, and Δ

*ϕ*is the phase error. The grating pitch error resulted from fabrication error. Because the gratings in the experiment were produced by plating chromium on a glass plate using a mask laser writer, the grating pitch error Δ

*p*was approximately 0.1 μm. The projection angle error resulted from axis alignment error and the resolution of the rotary stage. When the axis alignment error was 0.05° and the resolution of the rotary stage was 10 arcmin, the projection angle error Δ

*α*was approximately 0.22°. The phase error of the proposed method primarily resulted from the sampling error. When heterodyne moiré signals are captured using a CMOS camera, using an exposure and integration process, the recorded intensity of one pixel at the

*k*th sampling point can be expressed as follows:

*t*= 1/

_{s}*f*is the frame period. The intensity is subsequently quantized in gray-level units as:where

_{s}*n*denotes the number of the gray level. After substituting the experimental conditions in Eqs. (17) and (18), the captured heterodyne moiré signal of one pixel can be simulated and analyzed using FFT. The phase error Δ

*ϕ*can be obtained by comparing the calculated phases of FFT and set

*ϕ*when the relative phase measurement of the surface profilometry is considered and the light intensity error resulting from the stability of the light source and grating vibration is 1%. The phase error Δ

*ϕ*of the proposed method can be estimated as approximately 1.53°. Substituting the experimental conditions, Δ

*p*, Δ

*α*, and Δ

*ϕ*into Eq. (16) yielded a height error Δ

*z*of approximately 4.3 μm.

*d*(

*x*,

*y*) in Eq. (12). It is obvious that if visibility is zero, the heterodyne frequency component and the phase of this component certainly cannot be extracted by FFT. Therefore, the suggested minimum visibility is about 0.4 because the height error in this condition reaches 6.67 μm. The lower visibility causes much larger height error and results in a disappointing result. Besides, the angular misalignment between the directions of the projected fringes and the grating line of G

_{2}can reach 0.01 arcmin in our experiment. The resultant phase variation caused by this small angle in whole image can be estimated merely within 1.7 × 10

^{−5}degree [4

4. J. Dhanotia and S. Prakash, “Automated collimation testing by incorporating the Fourier transform method in Talbot interferometry,” Appl. Opt. **50**(10), 1446–1452 (2011). [CrossRef] [PubMed]

20. D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacement,” J. Opt. **27**(1), 19–23 (1996). [CrossRef]

10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

16. J. A. N. Buytaert and J. J. J. Dirckx, “Moiré profilometry using liquid crystals for projection and demodulation,” Opt. Express **16**(1), 179–193 (2008). [CrossRef] [PubMed]

17. J. J. J. Dirckx, J. A. N. Buytaert, and S. A. M. Van Der Jeught, “Implementation of phase-shifting moire profilometry on a low-cost commercial data projector,” Opt. Lasers Eng. **48**(2), 244–250 (2010). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | H. Takasaki, “Moiré topography from its birth to practical application,” Opt. Lasers Eng. |

2. | J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, “Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moire topograms,” Appl. Opt. |

3. | A. A. Mudassar and S. Butt, “Self-imaging-based laser collimation testing technique,” Appl. Opt. |

4. | J. Dhanotia and S. Prakash, “Automated collimation testing by incorporating the Fourier transform method in Talbot interferometry,” Appl. Opt. |

5. | J. Y. Lee, Y. H. Wang, L. J. Lai, Y. J. Lin, and Y. H. Chang, “Development of an auto-focus system based on the moiré method,” Measurement |

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

7. | M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform,” Opt. Commun. |

8. | M. Ramulu, P. Labossiere, and T. Greenwell, “Elastic–plastic stress/strain response of friction stir-welded titanium butt joints using moiré interferometry,” Opt. Lasers Eng. |

9. | K. S. Lee, C. J. Tang, H. C. Chen, and C. C. Lee, “Measurement of stress in aluminum film coated on a flexible substrate by the shadow moiré method,” Appl. Opt. |

10. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

11. | Y. M. He, C. J. Tay, and H. M. Shang, “Deformation and profile measurement using the digital projection grating method,” Opt. Lasers Eng. |

12. | S. Mirza and C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng. |

13. | C. Quan, Y. Fu, and C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. |

14. | Y. B. Choi and S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. |

15. | J. A. N. Buytaert and J. J. J. Dirckx, “Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation,” J. Opt. Soc. Am. A |

16. | J. A. N. Buytaert and J. J. J. Dirckx, “Moiré profilometry using liquid crystals for projection and demodulation,” Opt. Express |

17. | J. J. J. Dirckx, J. A. N. Buytaert, and S. A. M. Van Der Jeught, “Implementation of phase-shifting moire profilometry on a low-cost commercial data projector,” Opt. Lasers Eng. |

18. | J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. |

19. | M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. |

20. | D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacement,” J. Opt. |

**OCIS Codes**

(040.2840) Detectors : Heterodyne

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(120.4120) Instrumentation, measurement, and metrology : Moire' techniques

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 24, 2013

Revised Manuscript: January 20, 2014

Manuscript Accepted: January 29, 2014

Published: January 31, 2014

**Citation**

Wei-Yao Chang, Fan-Hsi Hsu, Kun-Huang Chen, Jing-Heng Chen, and Ken Y. Hsu, "Heterodyne moiré surface profilometry," Opt. Express **22**, 2845-2852 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2845

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### References

- H. Takasaki, “Moiré topography from its birth to practical application,” Opt. Lasers Eng. 3(1), 3–14 (1982). [CrossRef]
- J. J. J. Dirckx, W. F. Decraemer, G. Dielis, “Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moire topograms,” Appl. Opt. 27(6), 1164–1169 (1988). [CrossRef] [PubMed]
- A. A. Mudassar, S. Butt, “Self-imaging-based laser collimation testing technique,” Appl. Opt. 49(31), 6057–6062 (2010). [CrossRef]
- J. Dhanotia, S. Prakash, “Automated collimation testing by incorporating the Fourier transform method in Talbot interferometry,” Appl. Opt. 50(10), 1446–1452 (2011). [CrossRef] [PubMed]
- J. Y. Lee, Y. H. Wang, L. J. Lai, Y. J. Lin, Y. H. Chang, “Development of an auto-focus system based on the moiré method,” Measurement 44(10), 1793–1800 (2011). [CrossRef]
- Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162–3166 (1985). [CrossRef] [PubMed]
- M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform,” Opt. Commun. 136(5-6), 370–374 (1997). [CrossRef]
- M. Ramulu, P. Labossiere, T. Greenwell, “Elastic–plastic stress/strain response of friction stir-welded titanium butt joints using moiré interferometry,” Opt. Lasers Eng. 48(3), 385–392 (2010). [CrossRef]
- K. S. Lee, C. J. Tang, H. C. Chen, C. C. Lee, “Measurement of stress in aluminum film coated on a flexible substrate by the shadow moiré method,” Appl. Opt. 47(13), C315–C318 (2008). [CrossRef] [PubMed]
- M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
- Y. M. He, C. J. Tay, H. M. Shang, “Deformation and profile measurement using the digital projection grating method,” Opt. Lasers Eng. 30(5), 367–377 (1998). [CrossRef]
- S. Mirza, C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng. 44(1), 013601 (2005). [CrossRef]
- C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230(1-3), 23–33 (2004). [CrossRef]
- Y. B. Choi, S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37(3), 1005–1010 (1998). [CrossRef]
- J. A. N. Buytaert, J. J. J. Dirckx, “Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation,” J. Opt. Soc. Am. A 24(7), 2003–2013 (2007). [CrossRef] [PubMed]
- J. A. N. Buytaert, J. J. J. Dirckx, “Moiré profilometry using liquid crystals for projection and demodulation,” Opt. Express 16(1), 179–193 (2008). [CrossRef] [PubMed]
- J. J. J. Dirckx, J. A. N. Buytaert, S. A. M. Van Der Jeught, “Implementation of phase-shifting moire profilometry on a low-cost commercial data projector,” Opt. Lasers Eng. 48(2), 244–250 (2010). [CrossRef]
- J. A. N. Buytaert, J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. 40(3), 114–131 (2011). [CrossRef]
- M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129(3-4), 167–172 (1996). [CrossRef]
- D. C. Su, M. H. Chiu, C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacement,” J. Opt. 27(1), 19–23 (1996). [CrossRef]

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