## Wide band interferometry for thickness measurement

Optics Express, Vol. 11, Issue 8, pp. 952-957 (2003)

http://dx.doi.org/10.1364/OE.11.000952

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### Abstract

In this work we present the concept of wide band interferometry as opposed to white-light interferometry to introduce a thickness measurement method that gains precision when the bandwidth is reduced to an adequate compromise in order to avoid the distortions arising from the material dispersion. The use of the widest possible band is a well established dogma when the highest resolution is desired in distance measurements with white-light interferometry. We will show that the dogma falls when thickness measurements must be carried out due to material dispersion. In fact the precise knowledge of the frequency dependence of the refractive index is essential for adequate thickness retrieval from the optical experiments. The device we present is also useful to obtain the group refractive index that is necessary to calculate the absolute thickness value. As an example, we show the spreading of a silicone oil on a reference surface in real time.

© 2003 Optical Society of America

## 1. Introduction

1. A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical scanning interferometry,” Appl. Opt. **39**, 2107–2115 (2000) [CrossRef]

2. P. de Groot, “Derivation algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. **34**, 4723–4730 (1995) [CrossRef]

3. L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. **33**, 7334–7338 (1994) [CrossRef] [PubMed]

5. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. **32**, 438–3441 (1993) [CrossRef]

6. P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in whitelight phase-shifting interferometry,” J. Mod. Opt. **44**, 519–534 (1997) [CrossRef]

7. A. Dobroiu, H. Sakai, H. Ootaki, M. Sato, and N. Tanno, “Coaxial Mireau interferometer”, Opt. Lett. **27**, 1153–1155 (2002) [CrossRef]

8. D. Kim, S. Kim, H. J. Kong, and Y. Lee, “Measurement of the thickness profile of a transparent thin film deposited upon a pattern structure with an acusto-optic tunable filter,” Opt. Lett. **27**, 1893–1895 (2002) [CrossRef]

9. J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. **19**, 995–997 (1994) [CrossRef] [PubMed]

10. J. E. Calatroni, P. Sandoz, and Gilbert Tribillon, “Surface profiling by means of double spectral modulation,” Appl. Opt. **32**, 30–36 (1993) [CrossRef] [PubMed]

## 2. Wide band interferometry

*I*

_{ref}and

*I*

_{surf}are the intensities from the reference and surface arms respectively,

*k*is the wave vector and

*Δl*is the OPD. If the interferogram is dispersed using a spectrometer, the intensity profile can be described by the spectrum of the light source modulated by a cosine function with a frequency that corresponds to the OPD. Consequently, a Fourier-transform analysis and the adequate filters enable to evaluate

*Δl*without using any kind of phase unwrapping algorithms [9

9. J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. **19**, 995–997 (1994) [CrossRef] [PubMed]

*I*

_{int}in a complex form and a gaussian spectrum for the light source, with a bandwidth

*Δk*, Eq. (1) becomes

*c*is the speed of light in vacuum,

*n*(

*k*) is the refractive index,

*d*is the thickness of the sample and

*k*

_{0}is the central wave vector of the light source.

*k*it is possible to calculate the Fourier transform of the intensity analytically

*n*(

*k*) is expanded up to higher orders there is no analytical solution and in order to evaluate how the dispersion affects the accuracy of the measurements we have numerically calculated the Fourier transform. For an optical glass sample it is possible to use the Sellmeier coefficients for different bandwidths and obtain the thickness.

^{-3}and in the case of wide band less than 1×10

^{-5}.

## 3. Experimental results

*n*

_{g}=1.440±0.001. The error of this magnitude is obtained using the standard deviation of each of the distance measurements which arises from the noise in the photodiode array.

_{0}measuring the voltage of the central pixel.

*s*provides an expression to calculate the refractive index and its first order dispersion.

*n*

_{0}=1.40±0.01 and

*dn*/

*dλ*=-0.0497±0.0005µm

^{-1}, where the values of the errors are determined by the standard deviation of the linear fit performed in Fig. 5. In this situation the main contribution to the deviation value arises from the error in the determination of the interference fringes maxima.

## 4. Conclusions

^{-3}. The magnitude of the error is originated in the low quality photodiode array that was employed. The accuracy obtained is in the order of the theoretical limits imposed by the index dispersion for low dispersive optical glass for white-light interferometry. Furthermore, the precision obtained using this method represents 300 times smaller than the coherence length of the light source. In other words, the coherence length is 30µm while, the absolute error of the measurements is 100nm and it is computed from the statistical distribution of the modulation frequency and it does not scale with the thickness of the sample.

## References and Links

1. | A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical scanning interferometry,” Appl. Opt. |

2. | P. de Groot, “Derivation algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. |

3. | L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. |

4. | M. Hart, D. G. Vass, and M. L. Begbie, “Fast surface profiling by spectral analysis of white-light interferograms with Fourier transform spectroscopy,” Appl. Opt. |

5. | P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. |

6. | P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilometry by fringe-order identification in whitelight phase-shifting interferometry,” J. Mod. Opt. |

7. | A. Dobroiu, H. Sakai, H. Ootaki, M. Sato, and N. Tanno, “Coaxial Mireau interferometer”, Opt. Lett. |

8. | D. Kim, S. Kim, H. J. Kong, and Y. Lee, “Measurement of the thickness profile of a transparent thin film deposited upon a pattern structure with an acusto-optic tunable filter,” Opt. Lett. |

9. | J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. |

10. | J. E. Calatroni, P. Sandoz, and Gilbert Tribillon, “Surface profiling by means of double spectral modulation,” Appl. Opt. |

11. | G. P. Agrawal, |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(220.4840) Optical design and fabrication : Testing

(240.0310) Optics at surfaces : Thin films

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 18, 2003

Revised Manuscript: April 14, 2003

Published: April 21, 2003

**Citation**

Santiago Costantino, Oscar Martinez, and Jorge Torga, "Wide band interferometry for thickness measurement," Opt. Express **11**, 952-957 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-8-952

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### References

- A. Harasaki, J. Schmit and J. C. Wyant, "Improved vertical scanning interferometry,�?? Appl. Opt. 39, 2107-2115 (2000) [CrossRef]
- P. de Groot, �??Derivation algorithms for phase-shifting interferometry using the concept of a data-sampling window,�?? Appl. Opt. 34, 4723-4730 (1995 [CrossRef]
- L. Deck and P. de Groot, �??High-speed noncontact profiler based on scanning white-light interferometry,�?? Appl. Opt. 33, 7334-7338 (1994) [CrossRef] [PubMed]
- M. Hart, D. G. Vass and M. L. Begbie, �??Fast surface profiling by spectral analysis of white-light interferograms with Fourier transform spectroscopy,�?? Appl. Opt. 37, 1764-1769 (1998) [CrossRef]
- P. J. Caber, �??Interferometric profiler for rough surfaces,�?? Appl. Opt. 32, 438-3441 (1993) [CrossRef]
- P. Sandoz, R. Devillers and A. Plata, �??Unambiguous profilometry by fringe-order identification in whitelight phase-shifting interferometry,�?? J. Mod. Opt. 44, 519-534 (1997) [CrossRef]
- A. Dobroiu, H. Sakai, H. Ootaki, M. Sato and N. Tanno, �??Coaxial Mireau interferometer,�?? Opt. Lett. 27, 1153-1155 (2002) [CrossRef]
- D. Kim, S. Kim, H. J. Kong and Y. Lee, "Measurement of the thickness profile of a transparent thin film deposited upon a pattern structure with an acusto-optic tunable filter," Opt. Lett. 27, 1893-1895 (2002) [CrossRef]
- J. Schwider and L. Zhou, �??Dispersive interferometric profilometer,�?? Opt. Lett. 19, 995-997 (1994) [CrossRef] [PubMed]
- J. E. Calatroni, P. Sandoz and Gilbert Tribillon, �??Surface profiling by means of double spectral modulation,�?? Appl. Opt. 32, 30-36 (1993) [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear fiber optics, (Academic Press, 1989), Chap. 3.

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