Displacement measurement using a wavelengthphase-shifting grating interferometer Ju-Yi Lee* and Geng-An Jiang Department of Mechanical Engineering, National Central University, 300 Jhongda Rd., Jhongli City, Taoyuan County 320, Taiwan *[email protected]

Abstract: A grating interferometer based on the wavelength-modulated phase-shifting method for displacement measurements is proposed. A laser beam with sequential phase shifting can be accomplished using a wavelength-modulated light passing through an unequal-path-length optical configuration. The optical phase of the moving grating is measured by the wavelength-modulated phase-shifting technique and the proposed timedomain quadrature detection method. The displacement of the grating is determined by the grating interferometry theorem with the measured phase variation. Experimental results reveal that the proposed method can detect a displacement up to a large distance of 1 mm and displacement variation down to the nanometer range. ©2013 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

C. M. Wu, “Heterodyne interferometric system with sub-nanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004). F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998). W. T. Estler, “High-accuracy displacement interferometry in air,” Appl. Opt. 24(6), 808–815 (1985). M. Nevièvre, E. Popov, B. Bojhkov, L. Tsonev, and S. Tonchev, “High-accuracy translation-rotation encoder with two gratings in a Littrow mount,” Appl. Opt. 38(1), 67–76 (1999). J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011). M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A Phys. 141(1), 217–223 (2008). S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009). K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009). K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Meas. 45(6), 1510–1514 (2012). K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009). A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992). S. Fourment, P. Arguel, J. L. Noullet, F. Lozes, S. Bonnefont, G. Sarrabayrouse, Y. Jourlin, J. Jay, and O. Parriaux, “A silicon integrated opto–electro–mechanical displacement sensor,” Sens. Actuators A Phys. 110(13), 294–300 (2004). C. F. Kao, C. C. Chang, and M. H. Lu, “Double-diffraction planar encoder by conjugate optics,” Opt. Eng. 44, 023063 (2005). J. Y. Lee and M. P. Lu, “Optical heterodyne grating shearing interferometry for long-range positioning applications,” Opt. Commun. 284(3), 857–862 (2011). C. F. Kao, S. H. Lu, H. M. Shen, and K. C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” J. Appl. Phys. 47, 1833–1837 (2008). C. C. Wu, C. C. Hsu, J. Y. Lee, Y. Z. Chen, and J. S. Yang, “Littrow-type self-aligned laser encoder with high tolerance using double diffractions,” Opt. Commun. 297, 89–97 (2013). A. Kimura, W. Gao, and L. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol. 21, 054005 (2010).

#195534 - $15.00 USD Received 9 Aug 2013; revised 27 Sep 2013; accepted 9 Oct 2013; published 18 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025553 | OPTICS EXPRESS 25553

18. A. Kimura, W. Gao, W. J. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012). 19. X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013). 20. C. C. Hsu, C. C. Wu, J. Y. Lee, H. Y. Chen, and H. F. Weng, “Reflection type heterodyne grating interferometry for in-plane displacement measurement,” Opt. Commun. 281(9), 2582–2589 (2008). 21. J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A Phys. 137(1), 185–191 (2007). 22. R. Onodera and Y. Ishii, “Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes,” Appl. Opt. 33(22), 5052–5061 (1994). 23. R. Onodera and Y. Ishii, “Two-wavelength laser-diode heterodyne interferometry with one phasemeter,” Opt. Lett. 20(24), 2502–2504 (1995). 24. J. Y. Lee, M. P. Lu, K. Y. Lin, and S. H. Huang, “Measurement of in-plane displacement by wavelengthmodulated heterodyne speckle interferometry,” Appl. Opt. 51(8), 1095–1100 (2012). 25. R. J. Moffat, “Describing the uncertainties in experimental results,” Exp. Therm. Fluid Sci. 1(1), 3–17 (1988).

1. Introduction Among the primary metrology parameters (dimension, mass, time, and frequency), the precision measurement of displacement plays an important role in modern technology. There is an increasing demand for nanometric measurement resolution in nanotechnology, semiconductors, precision manufacturing, photo-lithography, metrology instruments, highdensity mass data storage systems, etc. The optical interferometer is a typical measurement tool and has been widely used for precision measurement of displacement because it offers a high measurement resolution and a wide dynamic measurement range [1,2]. However, temperature, humidity, air pressure, and air flow in the environment must be controlled to maintain measurement accuracy [3,4]. Recently, common-optical-path heterodyne interferometers [5–10] integrated with surface plasmon resonance (SPR) [5,6] or total internal reflection (TIR) [7,8] have been developed for small displacement sensing. Their lens systems convert the displacement into an angle variation of the measurement beam. By detecting the optical phase variation of the measurement beam which passes through SPR or TIR, the displacement can be determined. Due to the common-optical-path configuration, these measurement systems can reduce environmental disturbance. However, the measurement range is only a few micrometers or less. In contrast, the grating interferometer is independent of the light source wavelength and provides better immunity against environmental disturbances such as variations in temperature, pressure, and humidity [4,11–14]. Different types of grating interferometers have been developed to measure displacement with high resolution. For example, Teimel [11] proposed a grating interferometer with polarization elements, and the displacement of the grating was determined by phase quadrature signals. Kao et al. [15] presented a diffractive laser encoder with a grating in the Littrow configuration. Kao’s laser encoder realized a maximum measurement error of 53 nm and repeatability within ± 20 nm. Wu et. al. [16] designed a Littrow-type self-aligned laser encoder with double diffractions. Due to the symmetric optical configuration, Wu’s laser encoder had high tolerance. These laser encoders for grating interferometers are based on phase quadrature detection. Although these encoders have high measurement resolution, there are many optical polarization components in the phase detection system, and the optical configurations are complex. Gao et. al. [17] measured the x-directional position and the z-directional out-of-straightness of a precision linear airbearing stage with a 2-degree-of-freedom linear encoder. Recently, they further developed the multi-degree-of-freedom (DOF) surface encoder [18,19] for the stage motion measurement. Their multi-DOF surface encoder is composed of a planar scale grating and a reference grating which is set in the optical sensor head. The diffracted beams from the scale and reference gratings mutually interfere to generate interference signals. The multi-DOF displacements can be determined by means of analyzing the phase variations of the interference signals. Besides, the surface encoder incorporates the laser autocollimators for angular sensing. Because of the well-designed mechanical structure, Gao’s multi-DOF surface encoder is compact and has high measurement resolution. However, it is not easy to compact

#195534 - $15.00 USD Received 9 Aug 2013; revised 27 Sep 2013; accepted 9 Oct 2013; published 18 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025553 | OPTICS EXPRESS 25554

the optical sensor head further, because there are many optical polarization components, such as polarizers and quarter-wave plates, and photodetectors in the displacement assembly. In this paper, we proposed a novel technique for the phase quadrature detection without any polarization components. This technique can be used to reduce the size of optical sensor head. Generally, heterodyne detection can effectively overcome the common problem of DC offset and amplitude variation of the homodyne grating interferometer. Hsu et. al. [20] developed a reflection-type heterodyne grating interferometer for in-plane displacement measurement with a resolution of 0.5 nm. In 2007, we proposed a heterodyne grating interferometer for measuring in-plane displacement with a resolution of 0.2 nm [21]. In the heterodyne device, an electro–optical modulator installed in the interferometer modulates the laser beams at different frequencies. Although the measurement resolution is high, the electro–optical modulator is quite expensive and bulky. Ishii et. al [22,23] conducted several studies on heterodyne interferometry with a frequency-modulated (or wavelength-modulated) laser diode for surface profile measurement. The injection current is continuously changed to introduce a time-varying phase difference between the two beams of an unbalanced Twyman– Green interferometer. By analyzing the time-varying interference fringes, the interference phase and surface profile can be determined. Following the thought of the wavelengthmodulated heterodyne detection, we presented a method of wavelength-modulated heterodyne speckle interferometry for in-plane displacement measurement [24]. Different from the electro–optical-modulation and acousto–optical-modulation methods, we combined an optical-path-difference configuration with wavelength modulation of a laser diode source by injection current modulation to achieve heterodyne detection. The displacement of the object can be determined by the speckle interferometry theorem with heterodyne phase detection. This previous work also demonstrated that the measurement system can detect a displacement variation down to nanometer scale with a measurement range of hundreds of micrometers. However, not only the wavelength, but also the light intensity of the laser diode is modulated by the injection current. The modulated intensity that causes the interference signal is not a pure sinusoid curve, and phase detection is difficult and inconvenient. In this study, we developed a wavelength-modulated phase-shifting method and a grating interferometer with double diffractions for displacement measurement. The principle used for this interferometry can be regarded as time-domain quadrature detection. Different from electro–optical or acousto–optical modulation, the phase shift of the light beam can be accomplished using a wavelength-modulated laser beam passing through an unequal-pathlength optical configuration. We developed a new phase-extraction algorithm to calculate the optical phase variation due to the Doppler shift from the moving grating. The displacement of the grating is determined by the grating interferometry theorem with the measured phase variation. From the experimental results, the measurement range of our system is up to millimeter scale. Considering the high-frequency noise, the measurement resolution of the system is about 2 nm. The feasibility is demonstrated. 2. Principle First, the double-diffraction interference system and the optical phase variation which results from the grating displacement are introduced in this section. Next, the wavelength-modulated technique for phase-resolution is described. 2.1 Double-diffraction interference system A schematic diagram of the double-diffraction interference system is shown in Fig. 1. For convenience, the + z axis is chosen to be along the direction of propagation, and the x axis is along the horizontal direction. A beam from the laser diode passes through the beam splitter BS and is incident onto the diffraction grating G. The laser beam is diffracted into the + 1stand −1st-order beams. According to Fourier optics analysis in our previous work [21], when the grating is displaced along x axis by an amount Δx, the optical phase in the + 1st- and −1storder beams increases and decreases, respectively, by φg = 2πΔx/Λ. Here Λ is the grating

#195534 - $15.00 USD Received 9 Aug 2013; revised 27 Sep 2013; accepted 9 Oct 2013; published 18 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025553 | OPTICS EXPRESS 25555

pitch. For convenience, we assume that the amplitude of the original laser beam is 1, then the amplitudes (E+1, E-1) of these two diffraction beams can be written as:  2π l ± iφ  .   λ 

E ±1 ∝ exp  i

±1

g

(1)

Here 2π/λ is the wave number, λ is the wavelength of the laser beam, and l+1 and l−1 are the optical paths of the + 1st- and −1st-order beams from the grating to mirrors M1 and M2, respectively. Then, these two diffraction beams are reflected from M1 and M2, and diffracted again by the grating G. These two double-diffracted beams can be expressed as:  2π 2l ± i 2φ  .   λ 

E ' ±1 ∝ exp  i

±1

g

(2)

Fig. 1. Schematic diagram of the wavelength phase-shifting grating interferometer. The diffraction beams can be reflected by (a) the mirrors or (b) the corner cube retro-reflectors. These reflected beams are diffracted by the grating G again, and interfere with each other. FG: Function Generator, LD: Laser Diode, BS: Beam Splitter, G: Grating, M: Mirror, C: Corner Cube Retro-reflectors, PZT: Piezoelectric actuators, PD: Photodetector, PC: Personal Computer.

These two double-diffracted beams propagate along the same optical path and interfere with each other. The intensity of the interference detected by the photodetector is: I ∝ E ' +1 + E ' −1

2

= 1 + cos ( 2πΔl λ + φ ) ,

(3)

where Δl = 2(l+1 − l−1) is the optical path difference of the two double-diffracted beams. φ = 2φg − (−2φg) = 4φg is the phase variation of the interference signal, which is 4 times the optical phase variation of the diffracted beams. The optical path difference Δl and the tunable wavelength of the laser diode are used to produce the phase shift for the measuring the phase variation φ. It is noticed that M1 and M2 can be replaced by the corner cube retro-reflectors C1 and C2 shown in Fig. 1(b). The optical configuration in Fig. 1(b) has better optical efficiency. From the above analysis, the relationship of the phase variation φ to the grating displacement Δx is given as: φ = 4φg = 8πΔx / Λ ,

(4a)

Δx = ( Λ / 8π ) ⋅ φ .

(4b)

or

It is obvious that the grating displacement Δx can be determined by measuring the phase variation φ of the interference signal.

#195534 - $15.00 USD Received 9 Aug 2013; revised 27 Sep 2013; accepted 9 Oct 2013; published 18 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025553 | OPTICS EXPRESS 25556

2.2 Wavelength modulation technique and quadrature method for phase detection In the present study, phase detection is based on the wavelength modulation technique and the quadrature method. When the LD is driven by an injection current signal S(t), the wavelength and the amplitude of the laser beam is a function of time. Considering the time-dependent injection current and the coherence length of the laser diode, the interference signal at the photodetector (Eq. (3)) can be rewritten as: I ( t ) ∝ S ( t ) ⋅ [1 + V cos ( 2πΔl λ ( t ) + φ )] ,

(5)

where V is the visibility of the interference signal. If the driving signal is a square waveform with the period T, then the LD emits two wavelengths (λ1 and λ2) sequentially in one period. The sequential interference signal can be expressed as: I 1 ∝ S1 ⋅ [1 + V cos ( 2πΔl λ1 + φ )] ,

0 < t < T 2,

(6a)

I 2 ∝ S 2 [1 + V cos ( 2πΔl λ2 + φ )] ,

T 2 < t < T,

(6b)

and

where S1 and S2 are the main intensities of the interference signals. Here we can select a suitable λ2 = λ1 + Δλ to make a π/2 phase difference between I2 and I1, that is: 

 2π



 λ1

I 2 ∝ S 2 ⋅ 1 + V cos 

Δl −

2π 2

λ1





 2π



 λ1

Δλ ⋅ Δl + φ   = S 2 1 + V sin 





Δl + φ   ,



(7)

where Δλ = (λ2 − λ1)

Displacement measurement using a wavelength-phase-shifting grating interferometer.

A grating interferometer based on the wavelength-modulated phase-shifting method for displacement measurements is proposed. A laser beam with sequenti...
2MB Sizes 0 Downloads 0 Views