Three-color laser-diode interferometer

Three-color laser-diode interferometer Peter de Groot

The combined optical spectrum of a pair of multimode laser diodes is composed of a large number of welldefined wavelengths. This work reports the use of three of these wavelengths in a phase-modulated interferometer to measure absolute distance over 360-JLm intervals with a resolution of 0.5 nm. The laboratory demonstration system is composed of a three-wavelength source coupled by single-mode fiber to a compact interferometric probe. This system has been used for displacement measurement and profiling of optical surfaces. Key words: Interferometry, metrology, laser diode.

1.

Introduction

In 1895, Michelson and Benoit reported on the calibration of the wavelength of the red cadmium line (643.8 nm) with respect to the international prototype meter.I One of the purposes of their experiments was to provide a portable reference for length measurement based on the platinum-iridium alloy bar stored at the International Bureau of Weights and Measures in France. The number of wavelengths contained in the prototype meter was found by using multiple-color interferometry and the method of excess fractions. 2 The same multiple-color interferometric techniques could then be applied anywhere in the world to the inverse task of measuring mechanical lengths with respect to the red cadmium line. Although the way in which we define fundamental units of measure has changed radically since 1895, the importance of multiple-color interferometry in length measurement has not diminished. Modern versions of these instruments include those based on specialized CO 2 lasers,3 combinations of Ar and He-Ne lasers,4 and semiconductor laser diodes. 5- 11 The applications include ranging,3 metrology for optics manufacture,4,10,11 optical inspection of electronic circuits,6 and non contact gauging of machined parts. 8

This paper reports a unique embodiment of a threecolor interferometer for absolute distance measurement. The light source is composed of two multimode laser diodes, which generate a number of different colors for use in removing the 271" ambiguity of the interferometric phase measurement. The instrument provides absolute distance information with subnanometer resolution over a 360-JLm interval, making it a useful tool for small distance measurement and profilometry. 2.

Theory

When monochromatic light is made to interfere with itself in a two-beam interferometer, the output intensityas measured by a square-law detector is proportional to a function h: h(L,'A)

= cos2(27rL/'A),

where L is the one-way optical path difference, including the refractive index, and A is the vacuum wavelength. For convenience, we define a real number m, referred to here as the fringe number, such that L

= m'A/2,

(2) 2

h(m) = cos (m7r).

When this research was performed the author was with the Research Department, Hughes Danbury Optical Systems, 100 Wooster Heights Road, Danbury, Connecticut 06810. He is currently with Boeing High Technology Center, P.O. Box 3999, MS 7J-27, Seattle, Washington 98124. Received 17 September 1990. 0003-6935/91/253612-05$05.00/0. ©

1991 Optical Society of America.

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APPLIED OPTICS / Vol. 30, No. 25 / 1 September 1991

(1)

(3)

Because h is a periodic function, the integer part of fringe number m cannot be determined by inverting Eq. (3). Interferometry normally provides only the fractional part f(m) of the fringe number, with the consequence that only changes in the length L can be measured directly. This integer fringe-number ambiguity limits the usefulness of interferometry in many applications. The purpose of multiple-color interferometry is to measure the integer part of m, so that the entire length L can be measured directly in terms of

vacuum wavelen.gth Awith great precision.

Analytical procedures for determining lengths from multiple-color interferometry exist in a variety of forms. 2,6,9,12 Here we use the concept of synthetic wavelengths, corresponding to differences in phase measurements for pairs of colors in the interferometer. 6,!0 Consider three optical wavelengths ~q, A2, and A3. There are three possible synthetic wavelengths defined by (4)

Note that a synthetic wavelength can be made much larger than a visible wavelength by choosing appropriate pairs of wavelengths Ai, Aj. The corresponding synthetic fringe numbers Mij are obtained from the differences in optical fringe numbers mi and mj: (5)

It is easy to verify that length L can be calculated from a synthetic wavelength measurement using (6)

The larger the synthetic wavelength, the greater the range of distances L that can be accommodated without possibility of error because of integer ambiguity in the value of Mij. On the other hand, the precision in the measurement of L is best for small synthetic wavelengths. For precision multiple-color interferometry, synthetic wavelengths of different sizes are used to reduce progressively the uncertainty in the measurement of L. In the three-color interferometer described in this paper, we use two synthetic wavelengths, A12 = 720 ~m and A13 = 20 ~m, and a final optical wavelength Al = 0.785 ~m. The procedure for measuring a distance L with interferometer resolution begins as follows. Assuming thatL is less than A12/4, the integer part of the synthetic fringe number M12 is zero, and (7)

The fractional parts !(mi) of the interferometric fringe numbers mi are obtained by inverting Eq. (1) or by performing some equivalent phase-detection algorithm. The next step is to use the shorter synthetic wavelength A13 to increase the precision in the measurement. To calculate M13 after M12 is known, we note that M 12A 12/A13 is an approximation toM13 that is free of integer ambiguity. Now since the integer part of M13 is obtained by subtracting the fractional part !(M13 ), the following equation can be used to calculate M13 without ambiguity: M13 = {(MI3) +

J

I[ M:~12 - {(MI3)

(8)

where !(MI3 ) = !(m l )

-

!(m3)'

(9)

and the function I[a] appearing in Eq. (8) yields the integer nearest to argument a. This calculation is correct as long as the estimate for the integer value of

M 13 that appears as the argument of function I is not in error by more than ± %. When M13 has been obtained from Eq. (8)', the optical fringe number ml can be calculated from a similar equation: (10)

The final step is to use Eq. (2) for distance L. The measurement is therefore a three-step process, wherein M12 is used to remove the integer fringe-number ambiguity in the calculation of M 13 , and M13 is used in the calculation of mI. The result is interferometric accuracy, but without the integer fringe-number ambiguity of conventional single-color interferometry. Since the largest synthetic wavelength is 720 ~m, the instrument can measure distance L absolutely over a 180-~m range about zero. When L is equal to zero there is no optical path difference between the reference and object beams in the interferometer optics. For values of L outside this ±180-~m range, the measurement is relative, with an ambiguity interval of 360 ~m. For all the intended applications of the present instrument, there is sufficient a priori knowledge to remove this ambiguity. 3. Error Sources and Selection of Synthetic Wavelengths

The fringe-number calculations in Eqs. (8) and (10) require well-calibrated wavelengths and an accurate phase-demodulation algorithm to determine the fractional parts of the optical fringe numbers. Ultimately, these considerations place a limit on the synthetic wavelengths that may be used. The following analysis quantifies the limits on the synthetic wavelengths given normal phase-demodulation and calibration errors. For Eq. (8) we require that the absolute value of the error in the initial estimate of the integer part of M12 be less than 1/2:

I{M~::I2

_{(M )]I < I3

1/2,

(11)

where the 0 denotes error. Expanding inequality (11), we obtain

(12)

We now denote the maximum absolute value of an error by fl.. The following identifications are appropriate for un correlated errors: 11m

= lom1lmax = lom 21max = lom3lmax , X I1A I2 / Al2 = loA l2 / Al2lmax, I1AI3/Al3 = IOAI3/Al3lmax.

(13)

The maximum absolute value is the largest error that can be reasonably expected. Since both random and systematic errors are present, the relationships between the fl. values (maximum absolute error) and conventional statistical parameters cannot be rigorously derived. In what follows, the fl. values are han1 September 1991 I Vol. 30, No. 25 I APPLIED OPTICS

3613

dIed algebraically in the same manner as uncertainties; however, it should be clear that, if the instrument is to operate reliably, these uncertainties must be taken to be much larger than the relevant standard deviations. Substituting the relationships in Eqs. (13) into inequality (12) and rearranging, we obtain the following restriction on the synthetic wavelength A12: A12

< _1_ [A 13 _ !=. (AA12 + AA 13 )]. Am

4

2

A12

A13

~1

/

\

(14)

A similar calculation can be performed for A13: A < _1_ [~ _ !=. (AA 13 13 Am 4 2 A13

+ AA)] + ~ A

2'

780

where tlX/X = loXtiAllmax. The implementation considered in Section 4 provides an example of the use of these equations. It was found that the fractional fringe is measured with an rms accuracy of 0.125%. In selecting the synthetic wavelengths, the tlm chosen for use in inequalities (14) and (15) was three times the standard deviation. This reduces the probability of an accidental miscalculation of the integer part of the fringe numbers caused by a random fluctuation in tlm to