Homodyne laser vibrometer capable of detecting ... - OSA Publishing

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Research Article

Vol. 54, No. 34 / December 1 2015 / Applied Optics

Homodyne laser vibrometer capable of detecting nanometer displacements accurately by using optical shutters JINGHAO ZHU, PENGCHENG HU,*

AND

JIUBIN TAN

D-403 Science Park, Harbin Institute of Technology, 2 Yikuang Street, Harbin, 150080, China *Corresponding author: [email protected] Received 18 August 2015; revised 1 November 2015; accepted 5 November 2015; posted 5 November 2015 (Doc. ID 248205); published 30 November 2015

This paper describes a homodyne laser vibrometer with optical shutters. The parameters that define the nonlinearity of the quadrature signals in a vibrometer can be pre-extracted before the measurement, and can then be used to compensate for nonlinear errors, such as unequal AC amplitudes and DC offsets. The experimental results indicated that the homodyne laser vibrometer developed has the ability to accurately detect the vibration state of the object to be measured, even when the amplitude is ≤λ∕4. The displacement residual error can be reduced to a value under 0.9 nm. © 2015 Optical Society of America OCIS codes: (000.2170) Equipment and techniques; (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry; (120.7280) Vibration analysis. http://dx.doi.org/10.1364/AO.54.010196

1. INTRODUCTION With the rapid development of scientific research and industrial processing, a great demand for accurate vibration measurements with subfringe amplitude has arisen in many interesting areas, such as ultrasonic testing, engine vibration analysis, and muffler vibration analysis [1–3]. By offering high sensitivity, noncontact measurements, and wide frequency response, the laser interference vibration method has become a common technique for vibration measurements [1]. The two-detector homodyne interference structure is a widely used configuration of the homodyne laser vibrometer, which is easier to implement—with a minimum number of optical components—than the four-detector configuration [4]. For a two-detector homodyne laser vibrometer, nonlinear errors, especially DC offset, are unavoidable due to nonlinearity inherent in the periodic interference process. By using ellipsefitting algorithms [5,6] or gain/offset correction methods based on peak value extraction [7], the nonlinear error of the quadrature signal can be identified and well corrected. However, when the vibration amplitude is less than λ∕4, the Lissajous trace will become an incomplete ellipse, and thus the identification of ellipse parameters will be unachievable and nonlinear errors cannot be corrected [8,9]. Požar et al. [10] explored an enhanced ellipse-fitting method to operate the interferometer with nanometric accuracy provided that there are sufficient data to define at least a quarter-arc of the ellipse. However, the vibration measurement for a very small amplitude is still a

challenging issue when the vibration displacement is smaller than a quarter-arc because of the difficulty of acquiring the nonlinearity parameters in a short time for a high-speed vibration measurement. In this study, after analyzing the form of the quadrature interference signals in a homodyne laser vibrometer, an enhanced homodyne laser vibrometer with optical shutters is designed to detect the vibration of an object, even when the vibration amplitude is much less than λ∕4. The nonlinear error parameters can be pre-extracted before the vibration measurement and then be used effectively to compensate the nonlinear error during the vibration measurement. 2. HOMODYNE LASER VIBROMETER WITH OPTICAL SHUTTERS A. Principles of the Homodyne Laser Vibrometer

The schematic of a homodyne laser vibrometer with optical shutters is shown in Fig. 1. A 45° linearly polarized beam from a He–Ne laser is split into two beams by a nonpolarizing beam splitter (NPBS), i.e., a measurement beam and a reference beam. The reference beam passes through an octadic-wave plate (OWP), is reflected by the reference mirror (RM), and then passes through the OWP a second time, thereby being converted into a circularly polarized beam; thus, the interference signals on the photodiodes are shifted by 90°. The measurement beam is reflected from the polished surface of a vibrating target with linear polarization.

1559-128X/15/3410196-04$15/0$15.00 © 2015 Optical Society of America

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where Dx and Dy are the DC offsets, Ax and Ay are the AC amplitudes, and δ is the lack of quadrature. If the vibration amplitude is larger than λ∕4, the nonlinear error parameters of the quadrature signal can be obtained and then refined by using ellipse-fitting algorithms or a gain/offset correction method based on peak-value extraction. However, when the vibration amplitude is less than λ∕4, the Lissajous trace will become an incomplete ellipse, and thus the ellipse parameters cannot be identified and the nonlinear errors cannot be corrected in this case. B. Nonlinearity Compensation Based on Using Optical Shutters

Fig. 1. Schematic of a homodyne laser vibrometer with optical shutters. RM, reference mirror; WP, Wollaston prism; OPW, octadicwave plate; NPBS, nonpolarizing beam splitter; PD, photodiode.

The electric field vectors of the beams reflected from the RM and the vibrating surface are expressed as follows: E m t E m0 cosωt φm t;

(1)

E r t E r0 cosωt φr ;

(2)

where E m0 and E r0 are the amplitudes of E m and E r , respectively, and ω is the laser frequency. These two beams recombine at the NPBS. After passing through the Wollaston prism, the reference beam and the measurement beam will be separated into two linearly polarized beams with a phase difference of 90° between them, and will then interfere with each other on photoelectric detectors PD1 and PD2 . Under ideal conditions, the output voltage of the both PDs can be expressed as [10] I (3) ux α 0 1 cosΔφ; 4 I0 1 sinΔφ; (4) 4 where I 0 stands for the laser output intensity and α denotes the photoelectric conversion efficiency of the detector. After coherent signal processing, the resulting phase difference Δφ φm − φr . The quantity Δφ depends on the displacement of the target (Δφ 4πΔL∕λ, where λ is the wavelength of the interferometric laser in air), and can be determined using uy − αI 0 ∕4 mπ; (5) Δφ arctan ux − αI 0 ∕4 uy α

where the integer m is the output value of the bidirectional counter when the phase has a sudden jump. However, due to imperfections in the optics used, or misalignment of the axes of the optics, the generalized signals commonly contain nonlinearities, which can be expressed as ux Dx Ax cosΔφ;

(6)

uy Dy Ay sinΔφ δ;

(7)

The lack of quadrature, δ, is caused by the imperfections or misalignment of the OWP and NPBS. And it is always fixed during the displacement measurement. Consequently, it can be corrected easily by using a passive method based on adjusting the rotation angles of a Wollaston prism (WP) in the interferometer [11,12]. Because the vibration measurement is in general finished in a short time, the change in DC offsets and AC amplitudes caused by the fluctuation of light intensity or the temperature drift can be ignored during the vibration measurement. In order to compensate for the DC offsets and unequal AC amplitudes, two optical shutters (SR, SM) are placed in the optical paths of the interferometer. As shown in Fig. 1, one is located in the reference arm between the OWP and the RM; the other one is located in the measuring arm between the NPBS and the vibrating target. When shutter SR is open and SM is closed, the voltage signals of the PDs that only contain reference signals can be expressed as uxR αjE r0x cosωr t φrk j2 ;

(8)

uyR αjE r0y cosωr t φrk j2 :

(9)

The average of the square of the cosine during one laser frequency cycle is 1/2, so the final output photocurrent can be expressed as 1 (10) uxR αE 2r0x ; 2 1 uyR αE 2r0y : (11) 2 By opening shutter SM and closing SR, the voltage signals from the detectors can be expressed as 1 (12) uxM αE 2m0x ; 2 1 (13) uyM αE 2m0y : 2 When both shutters are open, the interference voltage signal output from each measuring channel can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux αjE rx E mx j2 uxM uxR 2 uxM uxR cosΔφ; (14) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi uy αjE ry E my j2 uyM uyR 2 uyM uyR cosΔφ: (15)

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In Eqs. (14) and (15), the first and second terms are unwanted DC offsets, and the third terms are the AC voltage signals, whose amplitudes are in general unequal. To correct errors due to the DC offsets and the unequal amplitudes, Eqs. (11) and (12) can be reconstructed as u − uxM − uxR cosΔφ; (16) ux0 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 uxM uxR uy0

uy − uyM − uyR pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosΔφ: 2 uyM uyR

(17)

The phase difference φm then can be obtained accurately according to 0 uy 0 (18) Δφ arctan 0 mπ: ux Equations (16) and (17) demonstrate that the corrected quadrature signals have become equal amplitudes without the DC offsets as a result of reconstructing the signals. The ellipse is thus transformed to a perfect circle. 3. EXPERIMENTAL WORK AND RESULTS A. Setup

In order to verify the performance of the proposed homodyne laser vibrometer with optical shutters, experiments were conducted with high-quality components using the configuration shown in Fig. 1. The vibrating mirror target was attached to a one-axis piezoelectric flexure stage (P-753.2CD, Physik Instrument, Germany), which was controlled by a high-speed stage position controller (PIE-709.CP, Physik Instrument, Germany). The stage was driven open-loop by a 1 kHz triangular voltage signal within the range of 0–2 V, causing a corresponding displacement of approximately 0–400 nm. Lissajous graphs and unwrapped displacements were displayed by LabVIEW, which was also used to perform the data processing. Before the measurement, the incoming polarization plane and the wave plate have to be iteratively rotated in order to achieve the constant sensitivity and the quadrature of the two-channel signals [12]. Consequently, the lack of the quadrature, δ, has been changed. After the data acquisition and lowpass filtering, signal processing was implemented as previously described in Section 2. To assess the performance of the homodyne laser vibrometer, the Heydemann ellipse-fitting method was used as the reference algorithm, whose ellipse parameters, fDx ; Dy ; Ax ; Ay ; δg, were obtained using a set of data with a larger vibration amplitude u0 200 nm (more than a full ellipse) just before the data shown in Fig. 2 were collected.

Fig. 2. Lissajous patterns of the measured signal without (small arcs) and with shutters (large arcs), with corresponding vibration amplitudes for three sections of arc: 10, 20, and 30 nm.

vibrometer, even if the whole fringe could not be detected on each photodiode. The unwrapped displacement information of the vibration is shown in Fig. 3. The red line is the measured data without using optical shutters. The black line is the corresponding corrected displacement that results from using the Heydemann ellipse-fitting method according to the obtained ellipse parameters fDx ; Dy ; Ax ; Ay ; δg, which is considered the reference displacement. The amplitude of the vibration measured without using shutters [according to Eq. (5)] has a large deviation from

B. Results and Discussion

As shown in Fig. 2, the small arc describes the Lissajous trace of the signal processed according to Eq. (5) and is a portion of an ellipse. In this case, the displacement of the vibrating target could not be measured accurately due to existing of DC offset. The big arc is the reconstructed signal measured with optical shutters, whose center is fixed at the zero point, and the amplitudes of the signals are normalized to approximately 1 V. The result indicates that the offset error and the unequal amplitude error can still be largely removed by using this laser

Fig. 3. Unwrapped displacements. Vibration amplitude of (a) 10 nm, (b) 20 nm, and (c) 30 nm.

Research Article

Fig. 4. Displacement residual amplitude (solid lines) and corresponding relative residual amplitude (dotted lines) for various lengths of the ellipse arc.

the reference displacement. In contrast, the amplitude of the vibration measured using shutters (green line) is almost equal to that of the reference value. Figure 4 presents the displacement residual amplitudes for various lengths of the ellipse arc measured with the developed homodyne laser vibrometer. The vibration amplitude was increased discretely from 5 nm (corresponding to almost 1/32 of a fringe) to 160 nm (corresponding to almost a full fringe). The displacement residual amplitude is defined as the peak value of the difference between the measured displacement and the reference displacement. As shown in Fig. 4, the displacement residual amplitude increased as the vibration amplitude increased, and became flat when the vibration amplitude exceeded 1/2 of a fringe, because the increase in the vibration amplitude would result in an increase in the nonlinear error within a full circle. The relative residual amplitude shown in Fig. 4 is the deviation of the displacement residual amplitude from the vibration amplitude, which decreases with increasing ellipse arc to demonstrate the variation of the displacement error. The displacement residual amplitude with correction was reduced from 50 to 0.9 nm; this residual error can be caused by laser power electrical noise and/or the instability of the refractive index of air. The amplitude error of 0.9 nm is small enough, especially for high-speed vibration measurements.

4. CONCLUSION In summary, this paper describes a homodyne laser vibrometer with optical shutters, based on quadrature detection and uniform fringe subdivision. Unequal AC amplitudes and DC offsets can be corrected by using the optical shutters and reconstructing the quadrature signals using the detected photoelectric signals. The experimental results show that the displacement can still be measured accurately when the vibration has tiny amplitudes (smaller than λ∕4) by using the developed homodyne laser vibrometer. This system can achieve accurate vibration measurements with a displacement residual error of less than 0.9 nm. When the vibration amplitude is larger than λ∕4, ellipse-fitting algorithms can be applied to improve the accuracy of the vibration amplitude measurement.


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It should be considered that light intensity, polarization state, or contrast might change with the time. However, a commercial He–Ne laser applied in vibration measurement normally has an intensity stability of 0.2% (over 1 h) [13]. Moreover, a vibration measurement of target can generally be accomplished in a shorter time period than that of the laser intensity fluctuations. Consequently, the effect caused by drift factors is small enough to be ignored during the vibration measurement. As a result, the nonlinearity parameters of the homodyne laser vibrometer can be regarded as almost time independent. Compared with the fluctuation of intensity for one target during one vibration measurement, the change in intensity caused by changing the measurement target is greater. In this case, in order to ensure the effectiveness of the nonlinearity compensation, the optical shutters of the homodyne laser vibrometer will be switched again to update the nonlinearity parameters. This vibrometer also proved to be a useful tool for measuring extremely low-amplitude vibration and for the calibration of vibration and shock transducers in the sub-fringestroke range. Funding. Fundamental Research Funds for the Central Universities (HIT.NSRIF.20168); National Natural Science Foundation of China (NSFC) (51105114). REFERENCES 1. T. AbdElrehim and M. H. A. Raouf, “Detection of ultrasonic signal using polarized homodyne interferometer with avalanche detector and electrical filter,” MAPAN J. Metrol. Soc. 29, 1–8 (2014). 2. M.-A. Beeck and W. Hentschel, “Laser metrology—a diagnostic tool in automotive development processes,” Opt. Lasers Eng. 34, 101–120 (2000). 3. P. Castellini, M. Martarelli, and E. P. Tomasini, “Laser Doppler vibrometry: development of advanced solutions answering to technology’s needs,” Mech. Syst. Signal Process. 20, 1265–1285 (2006). 4. E. Tomasini, L. Zhang, and R. Kumme, “Investigation of a homodyne and a heterodyne laser interferometer for dynamic force measurement,” Proc. SPIE 5503, 608–615 (2004). 5. P. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981). 6. C. M. Wu, C. S. Su, and G. S. Peng, “Correction of nonlinearity in onefrequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996). 7. G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15, 444–450 (2004). 8. Y. Li and R. Baets, “Homodyne laser Doppler vibrometer on siliconon-insulator with integrated 90 degree optical hybrids,” Opt Express 21, 13342–13350 (2013). 9. Q. Sun, W. Wabinski, and T. Bruns, “Investigation of primary vibration calibration at high frequencies using the homodyne quadrature sineapproximation method: problems and solutions,” Meas. Sci. Technol. 17, 2197–2205 (2006). 10. T. Požar, P. Gregorčič, and J. Možina, “A precise and wide-dynamicrange displacement-measuring homodyne quadrature laser interferometer,” Appl. Phys. B 105, 575–582 (2011). 11. J. Ahn, J.-A. Kim, C.-S. Kang, J. W. Kim, and S. Kim, “A passive method to compensate nonlinearity in a homodyne interferometer,” Opt. Express 17, 23299–23308 (2009). 12. T. Požar, P. Gregorčič, and J. Možina, “Optimization of displacementmeasuring quadrature interferometers considering the real properties of optical components,” Appl. Opt. 50, 1210–1219 (2011). 13. S. M. Goldwasser, “Commercial stabilized HeNe lasers,” http://www .repairfaq.org/sam/laserhst.htm#hstlfs220.