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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 8 — Apr. 21, 2003
  • pp: 895–898
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Phase calculation based on curve fitting with a two-wavelength interferometer

Tetsuo Harimoto  »View Author Affiliations


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

Multiple-wavelength techniques are widely used in the measurement of distance and surface profiling [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]

2

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]

], since a synthetic wavelength generated from multiple wavelengths provides additional information for the measurement. In comparison with conventional one-wavelength interferometers, the synthetic wavelength can extend the unambiguous range of the interferometer [3

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

]. In the measurement of phase with a conventional one-wavelength interferometer such as a Michelson interferometer, the optical-path difference ΔL can be calculated by ΔL=(m+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]

], in which a phase shifter such as a nonlinear optical crystal or a scanning optical prism is necessary. Applying a voltage to the crystal or prism four times requires a relatively long time for the raw data measurement. In this paper, we propose a simple method to determine the phase 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.

Figure 1 shows an experimental setup of a two-wavelength interferometer based on a typical Michelson amplitude-division interferometer. The purpose of the system is to measure the temperature dependence of the refractive index of a nonlinear optical crystal for secondharmonic generation. Green and red lasers with wavelengths of 532 nm and 635 nm are used as sources, resulting in a synthetic wavelength of 3280 nm that extends the unambiguous range of the interferometer. The red laser beam is passed through the beam splitter BS0 and is overlapped with the green laser beam after the beam splitter BS1. The plate BS2 is used to compensate for the dispersion effect generated in BS1. A piezoelectric disk driven by an externally applied voltage is used to produce a phase shift in order to measure the phase Φ. The interference signal measured with the detector D can be analytically expressed by

I=I1+I2,
I1=I10[1+C1cos(2πΔLλ1)],
I2=I20[1+C2cos(2πΔLλ2)],
(1)

where λj (j=1, 2) is the wavelength, Ij is the maximum intensity with respect to the wavelength λj, and Cj is the contrast of the interference fringe. Equation (1) can be changed into the following form,

I=I10+I20+[I102C12+I202C22+2I10C1I20C2cos(2πΔLλs)]12cos(2πΔLλ1+φ0),
φ0=tan1{sin(2πΔLλs)[(I10C1)(I20C2)+cos(2πΔLλs)]},
λs=(1λ21λ1)1.
(2)

The root portion is the beat function due to the use of two wavelengths, and λs is the corresponding synthetic wavelength.

Fig. 1. Experimental setup. M: reflective mirror; BS0, BS1, BS2: beam splitters; A: aperture; D: detector; PZT: piezoelectric element; VS: variable mechanical stage; LD: red laser diode; Nd:YVO4: green laser; VND: variable neutral density filter.

According to wavelengths λ1, λ2, and λs, the optical-path difference ΔL is possibly expressed by

ΔL=(m1+Φ12π)λ1=(m2+Φ22π)λ2=(ms+Φs2π)Λ,
(3)

where mj (j=1. 2, s) is the integer and Φ j is the measured phase. The value of the integer mj cannot be determined from the phase measurement alone, so it is found by the method of excess fractions [4

4. Max Born and Emil Wolf, Principles of Optics (Cambridge University Press, 1999), p.324.

]. 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,

If=a+bcos(cx+d)+ecos(cxλ1λ2+f),
(4)

where 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 cc to cc 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,

Fig. 2. Experimental raw and theoretical curve-fitting results. Circle points and solid lines are respectively the experimental raw data and corresponding fitted result. The count points of the raw data and fitted results are 400.
If=a+[b2+e2+2becos(cxλ1λs+fd)]12cos(cx+φf),
φf=tan1{[bsind+esin(cxλ1λs+f)][bcosd+ecos(cxλ1λs+f)]}.
(5)

The phase Φ s is therefore calculated from the beat function that is the root portion of Eq. (4) as the following,

Φs=2π2πcxmaxλ1λs,
(6)

where 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 λ1, λ2 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.

In conclusion, we propose a simple method to calculate the phase measured by a twowavelength interferometer. In our experiment, the fitting coefficient 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. 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]

4.

Max Born and Emil Wolf, Principles of Optics (Cambridge University Press, 1999), p.324.

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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