## Phase calculation based on curve fitting with a two-wavelength interferometer

Optics Express, Vol. 11, Issue 8, pp. 895-898 (2003)

http://dx.doi.org/10.1364/OE.11.000895

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### Abstract

We propose a simple method to calculate the phase measured by a two-wavelength interferometer. Our experiment showed that the fitting coefficient with respect to the phase was obtained with a final accuracy of 7.9×10^{-5}.

© 2003 Optical Society of America

1. Hirokazu Matsumoto and Lijiang Zeng, “Two-color interferometer for surface characterization using two frequency doublers and a four-phase-step method,” Appl. Opt. **35**, 2179–2181 (1996). [CrossRef] [PubMed]

2. Hirokazu Matsumoto, Yucong Zhu, Shigeo Iwasaki, and Tadanao O’ishi, “Measurement of the changes in air refractive index and distance by means of a two-color interferometer,” Appl. Opt. **31**, 4522–4526 (1992). [CrossRef] [PubMed]

3. Peter J. de Groot. “Extending the unambiguous range of two-color interferometers,” Appl. Opt. **33**, 5948–5953 (1994). [CrossRef] [PubMed]

**Φ**/2π)λ, where m is an integer, λ is the wavelength, and

**Φ**is the measured phase (0<

**Φ**<2π). The phase

**Φ**is generally calculated by a four-step phase-shifting technique [1

1. Hirokazu Matsumoto and Lijiang Zeng, “Two-color interferometer for surface characterization using two frequency doublers and a four-phase-step method,” Appl. Opt. **35**, 2179–2181 (1996). [CrossRef] [PubMed]

**Φ**with a conventional curve-fitting technique. The phase-variation history is involved throughout the curve-fitting process so that a higher accuracy for the phase measurement is achieved.

**Φ**. The interference signal measured with the detector D can be analytically expressed by

_{j}(j=1, 2) is the wavelength, I

_{j}is the maximum intensity with respect to the wavelength λ

_{j}, and C

_{j}is the contrast of the interference fringe. Equation (1) can be changed into the following form,

_{s}is the corresponding synthetic wavelength.

_{1}, λ

_{2}, and λ

_{s}, the optical-path difference ΔL is possibly expressed by

_{j}(j=1. 2, s) is the integer and

**Φ**

_{j}is the measured phase. The value of the integer m

_{j}cannot be determined from the phase measurement alone, so it is found by the method of excess fractions [4]. The phase

**Φ**

_{s}due to the synthetic wavelength can be calculated by a curve-fitting technique. For this purpose, the interference fringes derived in Eq. (1) are fit with the following,

*a*,

*b*,

*c*,

*d*,

*e*, and

*f*are unknown coefficients which will be determined by fitting the raw experimental data,

*x*is a count number of the sampling raw data. We want to find values for the coefficients such that the function matches the raw data as well as possible. The best values of the coefficients are the ones that minimize the value of the standard deviation. However, the best values corresponding to the coefficients are not easily and accurately found due to the complicated form of the function in Eq. (3) in comparison with the conventional sine function. Here we propose a simple method to find the coefficients with high accuracy using the following four steps: First, the initial value of the coefficient

*a*is approximately determined by averaging the experimental raw data; Second, the initial value of the coefficient

*c*is approximately found by fitting the raw data with a sine function; Third, coefficients

*b*and

*e*are assumed as the half of the maximum value of the raw data, and coefficients

*d*and

*f*are assumed to be zero; Finally, by changing coefficient

*c*from

*c*-Δ

*c*to

*c*+Δ

*c*and calculating the minimum standard deviation, we can find the final

*c*and the other coefficients. The above steps can be automatically achieved after the raw data were input. The function in Eq. (3) can be expressed by a beat function with form of the following,

**Φ**

_{s}is therefore calculated from the beat function that is the root portion of Eq. (4) as the following,

*x*

_{max}is the count number with respect to the first maximum peak of the interference fringe. Therefore the phase is only determined by the initial wavelengths λ

_{1}, λ

_{2}and the fitting coefficient

*c*. Figure 2 is the fitted result for the experimental raw data, in which the coefficient

*c*is well determined within a standard-deviation accuracy of 0.000079. The other coefficients

*a*,

*b*,

*d*,

*e*, and

*f*are fitted with accuracies of 0.0072, 0.01, 0.053, 0.01, and 0.062. The repeatability of the standard deviation for the fitting coefficient

*c*is approximately 0.000079±000005 with respect to the count range shown in Fig. 2.

*c*that primarily determines the phase is obtained with an accuracy of <10

^{-4}.

## References and links

1. | Hirokazu Matsumoto and Lijiang Zeng, “Two-color interferometer for surface characterization using two frequency doublers and a four-phase-step method,” Appl. Opt. |

2. | Hirokazu Matsumoto, Yucong Zhu, Shigeo Iwasaki, and Tadanao O’ishi, “Measurement of the changes in air refractive index and distance by means of a two-color interferometer,” Appl. Opt. |

3. | Peter J. de Groot. “Extending the unambiguous range of two-color interferometers,” Appl. Opt. |

4. | Max Born and Emil Wolf, |

**OCIS Codes**

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(260.3160) Physical optics : Interference

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 31, 2003

Revised Manuscript: April 9, 2003

Published: April 21, 2003

**Citation**

Tetsuo Harimoto, "Phase calculation based on curve fitting with a two-wavelength interferometer," Opt. Express **11**, 895-898 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-8-895

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### References

- Hirokazu Matsumoto and Lijiang Zeng, �??Two-color interferometer for surface characterization using two frequency doublers and a four-phase-step method,�?? Appl. Opt. 35, 2179-2181 (1996). [CrossRef] [PubMed]
- Hirokazu Matsumoto, Yucong Zhu, Shigeo Iwasaki, Tadanao O�??ishi, �??Measurement of the changes in air refractive index and distance by means of a two-color interferometer,�?? Appl. Opt. 31, 4522-4526 (1992). [CrossRef] [PubMed]
- Peter J. de Groot. �??Extending the unambiguous range of two-color interferometers,�?? Appl. Opt. 33, 5948-5953 (1994). [CrossRef] [PubMed]
- Max Born and Emil Wolf, Principles of Optics (Cambridge University Press, 1999), p.324.

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