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Long distance real-time measurement of multi-points micro-vibration in region by digital holography
Optik 125 (2014) 2369–2373
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Long distance real-time measurement of multi-points micro-vibration in region by digital holography Lin Cong a,∗ , Wen Xiao a , Lu Rong b , Feng Pan a , Jianyi Li a , Fanjing Wang a , Zhaohai Zhang a a b
School of Instrumentation Science and Opto-electronics Engineering, Beihang University, 37 Xueyuan Street, Haidian District, Beijing 100191, China Institute of Information Photonics Technology, College of Applied Science, Beijing University of Technology, 100 Pingleyuan Road, Beijing 100124, China
a r t i c l e
a b s t r a c t
i n f o
Article history: Received 24 May 2013 Accepted 15 October 2013 Keywords: Digital holography Micro-vibration measurement Long distance detection Real-time detection
Many applications require micro-vibration measurement, especially multi-points detection at long distance in real-time. In this paper, a micro-vibration measurement approach based on digital holographic interferometry is proposed for middle-low frequency detection. It can be used to monitor irregular frequency/amplitude vibration in selected region over 10 m away simultaneously and synchronously. A series of experiments were conducted including real-time measurement of 300 Hz, 1 kHz, 2 kHz and 3 kHz constant frequency/amplitude periodic vibration, precision and frequency response tests with calibration of LDV, 1 kHz irregular amplitude vibration, irregular frequency/amplitude vibration as well as the real-time measurement and simultaneous display of multi-points vibration. The experimental results demonstrate the feasibility of the proposed method and reveal its unique advantages. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Great demand for micro-vibration (<3 kHz, /4–/100) measurement is required in many branches of engineering, i.e. aerospace [1], automotive [2], shipping [3] and MEMS (micro electro mechanical systems) [4]. The measurement approaches can be divided into point detection and area detection. The former category includes optical triangulation [5,6], moiré fringe [7], interference fringes [8,9], and laser Doppler vibration [10–15], assuming that any point in the detected area has the same vibration characteristics. While such assumption is not required for the latter category, i.e., holographic interferometry [16–22] and electronic speckle pattern interferometry (ESPI) [23–25]. Stetson and Powell demonstrated that for holographic recording architecture, interferograms indicating wave disturbances at different time can be generated for plane vibration measurement [16]. Khanna and Tonndorf proposed time-averaged holographic interferometry to detect absolute displacement amplitude at any point on the vibration plane through long time exposure [18]. However, this approach requires manual manipulation, and cannot investigate variant frequency vibration or real-time measurement. Santoyo developed ESPI, also known as TV holography, to visualize dynamic displacement of components from speckle patterns [23]. Doval et al. proposed double-exposure holographic interferometry
∗ Corresponding author. E-mail address: [email protected] (L. Cong). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.10.054
which extracts instantaneous displacement by comparing two consecutive holograms [22]. Due to limited processing speed, these methods [16–23] have been just applied for ultra-low frequency (<100 Hz) vibration measurement while not yet being expanded to middle-low frequency spectrum. Pedrini et al. and Fu proposed high-speed digital holographic interferometry with high speed camera for vibration measurement up to 1 kHz [26–28]. But the detected distance is less than 2 m. In this paper, we propose a real-time detection approach based on digital holographic interferometry for middle-low frequency micro-vibration measurement over 10 m. Vibration information is extracted directly and instantaneously from consecutive holograms using Fourier transform and filtering in frequency domain. Faster processing speed guarantees real-time vibration monitoring of selected area and dynamic measurement of irregular frequency/amplitude vibration. Using the high speed CMOS with smaller ROI (region of interest), the sampling rate of system is 48 kHz, and the highest detected vibration frequency is more than 2 kHz. Under the condition of long detected distance, a series of experiments have been conducted indicate: the detected range of frequency and precision of the proposed method. 2. Methods There are three specific requirements for the algorithm of realtime vibration demodulation. (1) The algorithm should be simple for faster processing speed. (2) Due to the long detection distance, the algorithm should be independent on the spatial resolution. (3)
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in which C0 and Cn are the Fourier transform of virtual image c0 and cn , fx and fy are the coordinates at frequency domain. 2.3. Drawing filtering mask In Eq. (1), A = [O02 (x, y, t) + R02 (x, y, t)] is the zero-order image, c(x, y, t)e−iϕr (x,y,t) is the virtual image and c ∗ (x, y, t)eiϕr (x,y,t) is the real image. Before vibration, a filtering mask TA can be drawn based on the first hologram to extract C0 and eliminate A and C* in F0 in the frequency domain. Assuming the optical setup and camera position do not change during the measurement, the same mask TA can be used to extract Cn in Fn . The inverse Fourier transform of C0 and Cn are expressed as: g0 (x, y) = F−1 [C0 (fx − f0x , fy − f0y )] = c0 (x, y)ei2(f0x x+f0y y) =
1 O0 (x, y)R0 (x, y)eiϕo0 (x,y) ei2(f0x x+f0y y) . 2
(4)
gn (x, y) = F−1 [Cn (fx − f0x , fy − f0y )] Fig. 1. Flow chart of the proposed method (FFT: fast Fourier transform, TA: filter task, IFFT: inverse Fast fourier transform, Div: divided, PD: phase demodulation, ID: instantaneous displacement).
For ease of use, it need not measure the power of object beam and reference beam before detection measurement. In this paper, we used a simplified vibration demodulation algorithm in which the vibration information is extracted from the holograms by transformation and filtering. The principle of the algorithm is illustrated in Fig. 1 and introduced below. 2.1. Recording digital holograms
1 O0 (x, y)R0 (x, y)eiϕon (x,y) ei2(f0x x+f0y y) . 2
2 I(x, y, t) = O(x, y, t) + R(x, y, t) = [O02 (x, y, t) + R02 (x, y, t)]
(1)
in which:
(5)
2.4. Calculating phase change The phase change ϕo due to vibration can be extracted by filtering O0 (x, y, t) and R0 (x, y, t) using Euler’s formula: G(x, y) =
gn (x, y) = ei[ϕon (x,y)−ϕo0 (x,y)] = eiϕo (x,y) g0 (x, y) = cos[ϕo (x, y)] + i sin[ϕo (x, y)],
The object beam O(x, y, t) = O0 (x, y, t)eiϕo (x,y,t) interferences with the reference beam R(x, y, t) = R0 (x, y, t)eiϕr (x,y,t) to form a hologram at the recording plane, in which O0 (x, y, t) and R0 (x, y, t) is the constant amplitude of the object beam and the reference beam, respectively. ϕr = 2(f0x x + f0y y) is the phase distribution of the reference beam, in which f0x and f0y are the carrier frequency at x and y direction. The hologram at the recording plane can be written as:
+c(x, y, t)e−iϕr (x,y,t) + c ∗ (x, y, t)eiϕr (x,y,t) ,
=
ϕo (x, y) = arctan
Im[G(x, y)] Re[G(x, y)]
.
(6)
(7)
2.5. Demodulation of instantaneous vibration displacement (IVD) If the vibration amplitude object is less than wavelength, i.e., ϕo (x, y) ∈ [−, ], IVD is L(x, y) =
ϕo (x, y). 4
(8)
2.6. Draw vibration curve
1 c = [O0 (x, y, t)R0 (x, y, t)eiϕo (x,y,t) ] 2
Selecting a random pixel in the measured region, the corresponding IVD can be calculated and output as a real-time vibration curve. It is noted that vibration information of any point within that region is restored in holograms, thus this approach is applicable for plane vibration detection.
1 c ∗ = [O0 (x, y, t)R0 (x, y, t)e−iϕo (x,y,t) ]. 2
2.2. Fast Fourier transform Assuming that the initial recording time is t0 , a random recorded time during the vibration detection is tn . The Fourier transform of the two corresponding holograms can be written as: At t0 : F0 (fx , fy ) = F[I0 (x, y)] = A(fx , fy ) + C0 (fx − f0x , fy − f0y ) + C0∗ (fx + f0x , fy + f0y ).
(2)
At tn : Fn (fx , fy ) = F[In (x, y)] = A(fx , fy ) + Cn (fx − f0x , fy − f0y ) + Cn∗ (fx + f0x , fy + f0y ),
(3)
3. Experiments and results 3.1. Experimental system The experimental setup, as depicted in Fig. 2, was based on a modified Michelson interferometer configuration. A doubledfrequency Nd:YAG laser with a power of 50 mW and a wavelength of 532 nm was used as a light source. A polarized beam splitter (PBS) was adopted to split and adjust the intensity ratio of the illumination beam to reference beam with the help of two attenuators (A1 and A2). Two half wave plates (HWP1 and HWP2) were used to obtain the same linear polarization state of the two beams, which were collimated as plane waves by beam expanders (BE1 and BE2
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pixel at 48 KFPS, and the exposure time was 18 s. The vibration information was extracted from the holograms by Eqs. (2)–(8) and output in real-time. Meanwhile, for clearly display the real-time vibration curve, a random pixel was chosen from reconstruction image as the output vibration signal, the amplitude–time signal was displayed in the window of demodulation software. Moreover, to demonstrate the feasibility of real-time detection by the proposed system, the oscilloscope (OS) received the input signal from SG and output signal which was converted as the voltage-time signal by Digital–Analog Change (DAC) from computer simultaneously for comparison. Fig. 2. Schematic layout of the experimental setup (M1, M2 and M3: mirror, PBS: polarizing beam splitter, A1 and A2: attenuator, HWP1 and HWP2: half-wave plate, BE1 and BE2: beam expander, BS: beam splitter, SG: signal generation, OS: oscilloscope, IL: imaging lens).
composed by spatial filters and collimating lens). The measured object was an actuator driven by the signal generator (SG) in the form of standard sinusoidal signal. The area of measured region was 40 mm × 40 mm. The measured actuator had normal diffusely reflecting surface. Since the detected distance was long and the exposure time of CMOS at 48 kHz FPS was too short, the weak scatter beam from the vibration surface strongly reduced the contrast of holograms. To gathered the weak scattered beam, a imaging lens (IL) was inserted downstream the object with 300 mm focus length. The object beam superposed with the reference beam by the beam splitter (BS) at the recording plane. A CMOS camera was used to record holograms with pixel array of 1280 × 1024, pixel pitch of 14 m × 14 m, 8 bit gray levels and 500 FPS (frame per second) sampling rate. The camera was connected with the data acquisition card via Camera Link. In order to demonstrate the feasibility of real-time detection by the proposed system, the oscilloscope (OS) received the input signal from SG and output signal from computer simultaneously. In the experiment, the detection distance from the vibration plane to the lens was 10 m. The amplitude of vibration plane has been calibrated by a laser Doppler velocimetry (LDV) whose precision is 0.1 nm. In order to increase the frame rate, the CMOS captured the holograms in the region of interest (ROI) of 80 × 32
3.2. Standard sinusoidal vibration detection The first experiment is standard sinusoidal vibration detection. The actuator was driven by a standard sinusoidal signal from SG at the frequency of 300 Hz, 1 kHz, 2 kHz and 3 kHz, respectively. The material of actuator’s surface is metal, so the vibration amplitude of every point in the surface can be regarded as consistent as about 50 nm under the monitor by LDV. The real-time vibration information in ROI was extracted from the holograms using the proposed approach. Fig. 3 illustrates the frequency comparison of input–output signal from OS. In every sub-figure, comparison of vibration curve in the object domain is on the left side, and comparison of frequency in the Fourier domain is on the right side. A real-time display software was developed to show the vibration information. Amplitude–time curve of the selected point in the ROI is shown in Fig. 4. It can be seen from Fig. 3 that the proposed system retrieves the frequency of standard sinusoidal vibration signal at the frequency from 300 Hz to 3 kHz very well. Fig. 4 shows that the amplitude of output waveform is close to the theoretical value. Both figures prove the effectiveness of real-time detection of micro-vibration. Following aspects should be considered for the proposed system. (1) There is a time delay between output signal and input signal by the processing time of data transmission, storage, reconstruction and retrieval. The delay is not constant, difficult to eliminate and depends on PC performance. But it is usually less than a vibration cycle, thus there is no obvious influence on real-time detection. (2) There is a fluctuation on output curve by the power instability of
Fig. 3. I/O contrast diagram of standard sinusoidal signals.
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Fig. 4. Retrieved amplitude–time curve.
Fig. 7. Test results of multi-points vibration.
Fig. 5. Precision and frequency response curve of the proposed system.
the reference beam. This random fluctuation is one of the measurement error sources affecting the stability of the proposed system. (3) The retrieved vibration amplitude at 3 kHz is smaller than the theoretical value. 3.3. System precision and frequency response test This experiment is to further test the precision of measured amplitude and the response ability at long distance. To observe the precision and frequency response of system, the vibratory amplitudes are hold in about 55 nm at every measured frequency through changing the voltage of driven signal under monitored by LDV. In Fig. 5, the curve a is the measured result by LDV, while curve b is the measured result by the proposed system at 48 kHz sampling rate. It can be seen that the system precision is good when the vibratory frequency from 100 Hz to 2600 Hz and the average error is about 4.05%. After 2600 Hz, the precision begins to descend quickly, which is possibly influenced by sampling rate. To prove it, we changed the sampling rate to 24 kHz and 12 kHz. The results are shown by curve c and d. The curve c illustrates that the precision decreases after 1.3 kHz at 24 kHz sampling rate and the precision
decreases after 700 Hz at 12 kHz sampling rate. The experimental results indicate that the frequency response rage of the system is limited by the sampling rate of system. To get the good precision, the sampling rate should be about 20 times over the vibration frequency. Except for the sampling rate, the precision was also influenced by the stability of light source. We assume that the power of illumination beam and reference beam are constant during the experiment. But in reality, the power of illumination has small fluctuation, which generates measurement error.
3.4. Random vibration detection In order to prove the feasibility to detect random vibration, two groups of experiment were conducted, the driving vibration sources of which were 1 kHz-frequency random-amplitude (amplitude: 30–60 nm) and random-frequency random-amplitude (amplitude: 30–60 nm, frequency: 500–2 kHz), respectively. The results shown in Fig. 6 prove that the proposed system can detect and extract these two random vibration signals at real-time. The extracted signals are basically as the same as the original signals. The result demonstrates the proposed system can be used to measure random vibration. When the amplitude or frequency of the
Fig. 6. Test results of random signals.
L. Cong et al. / Optik 125 (2014) 2369–2373
monitored object change, the system will synchronously detect and record this change. 3.5. Vibration of multi-points in the selected region detection The proposed system can measure multiple points synchronously in the selected vibration region. To prove this, we used headphone as the sample, since the amplitude of each point was significantly different. The driving vibration source was 1 kHz constant-voltage standard sinusoidal signal. Three different pixels were chosen and displayed on an oscilloscope as shown in Fig. 7. Fig. 7(a) is the photograph of the headphone with a dashed box marked with points A (close to the center), B (between point A and point C) and C (close to the edge). Fig. 7(b–d) illustrates the vibration detection curves of points A, B and C, respectively. For structural and material reasons, the vibration response on the headphone diaphragm surface is not uniform. The experimental results demonstrated that point A has the largest vibration amplitude, while point B and point C have smaller amplitude. 4. Conclusion In this paper, a real-time multi-points vibration detection approach based on digital holographic interferometry has been proposed for long distance (10 m) middle-low frequency (<3 kHz) micro-vibration (<100 nm) measurement. The experimental results demonstrate the effectiveness of the proposed method and reveal its unique advantages. (1) The measurement distance of microvibration is more than 10 m with imaging lens. (2) The method can synchronously measure and display the vibration of multiple points stimulated either by periodic signal or irregular one. (3) To increase the range of measured vibration frequency, smaller ROI and shorter exposure time was adopted with the expense of lower spatial resolution. (4) Unlike conventional reconstruction, vibration information can be determined directly and instantaneously from holograms using one time of Fourier transform and inverse transform. Faster processing speed guarantees real-time dynamic measurement. Unlike time-average approach, the proposed approach has only one manual step of filtering mask drawing before vibration and the whole vibration detection is automatic. In conclusion, this method overcomes the detected distance and the reconstruction speed limitation of digital holographic interferometry, so it may find wide applications in investigating real-time vibration phenomena. References [1] S.H. Zhang, H.L. Chen, Modeling and vibration analysis of a composite supporter for aerospace applications, Adv. Compos. Mater. 14 (2) (2005) 199–210.
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[2] L. Hinke, L. Pichler, H.J. Pradlwarter, B.R. Mace, T.P. Waters, Modeling of spatial variations in vibration analysis with application to an automotive windshield, Finite Elements Anal. Des. 47 (1) (2011) 55–62. [3] S.P. Singh, K. Saha, J. Singh, A.P.S. Sandhu, Measurement, analysis of vibration and temperature levels in global intermodal container shipments on truck, rail and ship, packaging, Technol. Sci. 25 (3) (2012) 149–160. [4] D. Teyssieux, S. Euphrasie, B. Cretin, MEMS in-plane motion/vibration measurement system based CCD camera, Measurement 44 (10) (2011) 2205–2216. [5] Z. Pawlowski, An objective test of vocal cord vibration using the photo-optic method, Otolaryngol. Pol. 18 (1964) 567–572. [6] R.G. Dorsch, G. Häusler, J.M. Herrmann, Laser triangulation: fundamental uncertainty distance measurement, Appl. Opt. 33 (7) (1994) 1306–1314. [7] A.M. Yeras, Moire technique and measurement of vibrations, Proc. SPIE 4419 (2001) 202–205. [8] O. Sasaki, T. Sato, Y. Taya, Vibration measuring system using interference fringes and a TV system, Appl. Opt. 19 (16) (1980). [9] G. He, X. Wang, Real-time micro-vibration measurement in sinusoidal phasemodulating interferometry, Optik 120 (3) (2009) 101–105. [10] T. Sato, Y. Nakatani, M. Ueda, Real time Doppler imaging system using forced vibration of the object, Appl. Opt. 16 (5) (1977) 1263–1267. [11] T. Sato, T. Kishimoto, K. Sasaki, Laser Doppler particle measuring system using forced vibration synchronized detection and power spectral analysis, Appl. Opt. 17 (2) (1978) 230–234. [12] O. Sasaki, T. Sato, T. Oda, Laser Doppler vibration measuring system using bispectral analysis, Appl. Opt. 19 (1) (1980) 151–153. [13] T. Yoshikawa, H. Kotera, K. Yoshida, D. Koyama, K. Nakamura, Y. Watanabe, Measurement of the resonant characteristics of a single bubble vibration by using a laser Doppler vibrometer, Jpn. J. Appl. Phys. 50 (7) (2011). [14] T. Matsumura, M. Esashi, H. Harada, S. Tanaka, Vibration mode observation of piezoelectric disk-type resonator by high-frequency laser Doppler vibrometer, IEE J. Trans. Electron. Information Syst. 95 (5) (2012) 33–41. [15] M. Guo, Y. Fu, P.B. Phua, Fiber-based multi-beam laser Doppler vibrometer for measuring transient vibrations, Proc. SPIE 7928 (2011). [16] K.A. Stetson, R.L. Powell, Interferometric hologram evaluation and real-time vibration analysis of diffuse objects, J. Opt. Soc. Am. 12 (55) (1965) 1694–1695. [17] J.W. Goodman, R.W. Lawrence, Digital image formulation from electronically detected holograms, Appl. Phys. Let. 11 (3) (1967) 77–79. [18] S.M. Khanna, J. Tonndorf, Tympanic membrane vibrations in cats studied by time-averaged holography, J. Opt. Soc. Am. 51 (6) (1972) 1904–1920. [19] K. Patorski, A. Jozwicka, A. Kalinowski, M. Pawlowski, Simplified time-averaged digital interferometry for vibration studies of microelements, Proc. SPIE 4933 (2003) 90–95. [20] G.G. Romero, E.E. Alanis, C.C. Martinez, L.G. Nallim, M.W. Toledo, L. Alvarez, Vibration analysis of perforated plates using time-average digital speckle pattern interferometry, Exp. Mech. 50 (7) (2010) 1013–1021. [21] F. Rao, X.F. Qian, Z.J. Peng, Vibration measurement based on time-average digital holography and the shearing interferometry, Proc. SPIE 8192 (2011). [22] A.F. Doval, C. Trillo, D. Cernadas, B.V. Dorrio, C. Lopez, J.L. Fernandez, M. PerezAmor, Measuring amplitude and phase of vibration with double-exposure stroboscopic TV holography, Interferometry Speckle Light: Theory Appl. (2000) 281–288. [23] F.M. Santoyo, M.C. Shellabear, J.R. Tyrer, Whole field in-plane vibration analysis using pulsed phase-stepped ESPI, Appl. Opt. 30 (7) (1991) 717–721. ˜ Analysis of mechanical [24] G. Ayala, A.L. Rivera, A. Dávila, G. Garnica, V.M. Castano, vibrations through speckle interferometry: a phase-space approach, Optik 121 (2010) 2028–2035. ˇ ´ P. Smíd, [25] M. Hrabovsky, P. Horváth, Z. Baˇca, Measurement of object vibrations using the theory of speckle pattern decorrelation, Optik 113 (2002) 117–120. [26] G. Pedrini, W. Osten, M.E. Gusev, High-speed digital holographic interferometry for vibration measurement, Appl. Opt. 45 (15) (2006) 3456–3462. [27] Y. Fu, G. Pedrini, W. Osten, Vibration measurement by temporal Fourier analyses of a digital hologram sequence, Appl. Opt. 46 (23) (2007) 5719–5727. [28] Y. Fu, H. Shi, H. Miao, Vibration measurement of a miniature component by high-speed image-plane digital holographic microscopy, Appl. Opt. 48 (11) (2009) 1990–1997.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Long distance real-time measurement of multi-points micro-vibration in region by digital holography Lin Cong a,∗ , Wen Xiao a , Lu Rong b , Feng Pan a , Jianyi Li a , Fanjing Wang a , Zhaohai Zhang a a b
School of Instrumentation Science and Opto-electronics Engineering, Beihang University, 37 Xueyuan Street, Haidian District, Beijing 100191, China Institute of Information Photonics Technology, College of Applied Science, Beijing University of Technology, 100 Pingleyuan Road, Beijing 100124, China
a r t i c l e
a b s t r a c t
i n f o
Article history: Received 24 May 2013 Accepted 15 October 2013 Keywords: Digital holography Micro-vibration measurement Long distance detection Real-time detection
Many applications require micro-vibration measurement, especially multi-points detection at long distance in real-time. In this paper, a micro-vibration measurement approach based on digital holographic interferometry is proposed for middle-low frequency detection. It can be used to monitor irregular frequency/amplitude vibration in selected region over 10 m away simultaneously and synchronously. A series of experiments were conducted including real-time measurement of 300 Hz, 1 kHz, 2 kHz and 3 kHz constant frequency/amplitude periodic vibration, precision and frequency response tests with calibration of LDV, 1 kHz irregular amplitude vibration, irregular frequency/amplitude vibration as well as the real-time measurement and simultaneous display of multi-points vibration. The experimental results demonstrate the feasibility of the proposed method and reveal its unique advantages. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Great demand for micro-vibration (<3 kHz, /4–/100) measurement is required in many branches of engineering, i.e. aerospace [1], automotive [2], shipping [3] and MEMS (micro electro mechanical systems) [4]. The measurement approaches can be divided into point detection and area detection. The former category includes optical triangulation [5,6], moiré fringe [7], interference fringes [8,9], and laser Doppler vibration [10–15], assuming that any point in the detected area has the same vibration characteristics. While such assumption is not required for the latter category, i.e., holographic interferometry [16–22] and electronic speckle pattern interferometry (ESPI) [23–25]. Stetson and Powell demonstrated that for holographic recording architecture, interferograms indicating wave disturbances at different time can be generated for plane vibration measurement [16]. Khanna and Tonndorf proposed time-averaged holographic interferometry to detect absolute displacement amplitude at any point on the vibration plane through long time exposure [18]. However, this approach requires manual manipulation, and cannot investigate variant frequency vibration or real-time measurement. Santoyo developed ESPI, also known as TV holography, to visualize dynamic displacement of components from speckle patterns [23]. Doval et al. proposed double-exposure holographic interferometry
∗ Corresponding author. E-mail address: [email protected] (L. Cong). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.10.054
which extracts instantaneous displacement by comparing two consecutive holograms [22]. Due to limited processing speed, these methods [16–23] have been just applied for ultra-low frequency (<100 Hz) vibration measurement while not yet being expanded to middle-low frequency spectrum. Pedrini et al. and Fu proposed high-speed digital holographic interferometry with high speed camera for vibration measurement up to 1 kHz [26–28]. But the detected distance is less than 2 m. In this paper, we propose a real-time detection approach based on digital holographic interferometry for middle-low frequency micro-vibration measurement over 10 m. Vibration information is extracted directly and instantaneously from consecutive holograms using Fourier transform and filtering in frequency domain. Faster processing speed guarantees real-time vibration monitoring of selected area and dynamic measurement of irregular frequency/amplitude vibration. Using the high speed CMOS with smaller ROI (region of interest), the sampling rate of system is 48 kHz, and the highest detected vibration frequency is more than 2 kHz. Under the condition of long detected distance, a series of experiments have been conducted indicate: the detected range of frequency and precision of the proposed method. 2. Methods There are three specific requirements for the algorithm of realtime vibration demodulation. (1) The algorithm should be simple for faster processing speed. (2) Due to the long detection distance, the algorithm should be independent on the spatial resolution. (3)
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in which C0 and Cn are the Fourier transform of virtual image c0 and cn , fx and fy are the coordinates at frequency domain. 2.3. Drawing filtering mask In Eq. (1), A = [O02 (x, y, t) + R02 (x, y, t)] is the zero-order image, c(x, y, t)e−iϕr (x,y,t) is the virtual image and c ∗ (x, y, t)eiϕr (x,y,t) is the real image. Before vibration, a filtering mask TA can be drawn based on the first hologram to extract C0 and eliminate A and C* in F0 in the frequency domain. Assuming the optical setup and camera position do not change during the measurement, the same mask TA can be used to extract Cn in Fn . The inverse Fourier transform of C0 and Cn are expressed as: g0 (x, y) = F−1 [C0 (fx − f0x , fy − f0y )] = c0 (x, y)ei2(f0x x+f0y y) =
1 O0 (x, y)R0 (x, y)eiϕo0 (x,y) ei2(f0x x+f0y y) . 2
(4)
gn (x, y) = F−1 [Cn (fx − f0x , fy − f0y )] Fig. 1. Flow chart of the proposed method (FFT: fast Fourier transform, TA: filter task, IFFT: inverse Fast fourier transform, Div: divided, PD: phase demodulation, ID: instantaneous displacement).
For ease of use, it need not measure the power of object beam and reference beam before detection measurement. In this paper, we used a simplified vibration demodulation algorithm in which the vibration information is extracted from the holograms by transformation and filtering. The principle of the algorithm is illustrated in Fig. 1 and introduced below. 2.1. Recording digital holograms
1 O0 (x, y)R0 (x, y)eiϕon (x,y) ei2(f0x x+f0y y) . 2
2 I(x, y, t) = O(x, y, t) + R(x, y, t) = [O02 (x, y, t) + R02 (x, y, t)]
(1)
in which:
(5)
2.4. Calculating phase change The phase change ϕo due to vibration can be extracted by filtering O0 (x, y, t) and R0 (x, y, t) using Euler’s formula: G(x, y) =
gn (x, y) = ei[ϕon (x,y)−ϕo0 (x,y)] = eiϕo (x,y) g0 (x, y) = cos[ϕo (x, y)] + i sin[ϕo (x, y)],
The object beam O(x, y, t) = O0 (x, y, t)eiϕo (x,y,t) interferences with the reference beam R(x, y, t) = R0 (x, y, t)eiϕr (x,y,t) to form a hologram at the recording plane, in which O0 (x, y, t) and R0 (x, y, t) is the constant amplitude of the object beam and the reference beam, respectively. ϕr = 2(f0x x + f0y y) is the phase distribution of the reference beam, in which f0x and f0y are the carrier frequency at x and y direction. The hologram at the recording plane can be written as:
+c(x, y, t)e−iϕr (x,y,t) + c ∗ (x, y, t)eiϕr (x,y,t) ,
=
ϕo (x, y) = arctan
Im[G(x, y)] Re[G(x, y)]
.
(6)
(7)
2.5. Demodulation of instantaneous vibration displacement (IVD) If the vibration amplitude object is less than wavelength, i.e., ϕo (x, y) ∈ [−, ], IVD is L(x, y) =
ϕo (x, y). 4
(8)
2.6. Draw vibration curve
1 c = [O0 (x, y, t)R0 (x, y, t)eiϕo (x,y,t) ] 2
Selecting a random pixel in the measured region, the corresponding IVD can be calculated and output as a real-time vibration curve. It is noted that vibration information of any point within that region is restored in holograms, thus this approach is applicable for plane vibration detection.
1 c ∗ = [O0 (x, y, t)R0 (x, y, t)e−iϕo (x,y,t) ]. 2
2.2. Fast Fourier transform Assuming that the initial recording time is t0 , a random recorded time during the vibration detection is tn . The Fourier transform of the two corresponding holograms can be written as: At t0 : F0 (fx , fy ) = F[I0 (x, y)] = A(fx , fy ) + C0 (fx − f0x , fy − f0y ) + C0∗ (fx + f0x , fy + f0y ).
(2)
At tn : Fn (fx , fy ) = F[In (x, y)] = A(fx , fy ) + Cn (fx − f0x , fy − f0y ) + Cn∗ (fx + f0x , fy + f0y ),
(3)
3. Experiments and results 3.1. Experimental system The experimental setup, as depicted in Fig. 2, was based on a modified Michelson interferometer configuration. A doubledfrequency Nd:YAG laser with a power of 50 mW and a wavelength of 532 nm was used as a light source. A polarized beam splitter (PBS) was adopted to split and adjust the intensity ratio of the illumination beam to reference beam with the help of two attenuators (A1 and A2). Two half wave plates (HWP1 and HWP2) were used to obtain the same linear polarization state of the two beams, which were collimated as plane waves by beam expanders (BE1 and BE2
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pixel at 48 KFPS, and the exposure time was 18 s. The vibration information was extracted from the holograms by Eqs. (2)–(8) and output in real-time. Meanwhile, for clearly display the real-time vibration curve, a random pixel was chosen from reconstruction image as the output vibration signal, the amplitude–time signal was displayed in the window of demodulation software. Moreover, to demonstrate the feasibility of real-time detection by the proposed system, the oscilloscope (OS) received the input signal from SG and output signal which was converted as the voltage-time signal by Digital–Analog Change (DAC) from computer simultaneously for comparison. Fig. 2. Schematic layout of the experimental setup (M1, M2 and M3: mirror, PBS: polarizing beam splitter, A1 and A2: attenuator, HWP1 and HWP2: half-wave plate, BE1 and BE2: beam expander, BS: beam splitter, SG: signal generation, OS: oscilloscope, IL: imaging lens).
composed by spatial filters and collimating lens). The measured object was an actuator driven by the signal generator (SG) in the form of standard sinusoidal signal. The area of measured region was 40 mm × 40 mm. The measured actuator had normal diffusely reflecting surface. Since the detected distance was long and the exposure time of CMOS at 48 kHz FPS was too short, the weak scatter beam from the vibration surface strongly reduced the contrast of holograms. To gathered the weak scattered beam, a imaging lens (IL) was inserted downstream the object with 300 mm focus length. The object beam superposed with the reference beam by the beam splitter (BS) at the recording plane. A CMOS camera was used to record holograms with pixel array of 1280 × 1024, pixel pitch of 14 m × 14 m, 8 bit gray levels and 500 FPS (frame per second) sampling rate. The camera was connected with the data acquisition card via Camera Link. In order to demonstrate the feasibility of real-time detection by the proposed system, the oscilloscope (OS) received the input signal from SG and output signal from computer simultaneously. In the experiment, the detection distance from the vibration plane to the lens was 10 m. The amplitude of vibration plane has been calibrated by a laser Doppler velocimetry (LDV) whose precision is 0.1 nm. In order to increase the frame rate, the CMOS captured the holograms in the region of interest (ROI) of 80 × 32
3.2. Standard sinusoidal vibration detection The first experiment is standard sinusoidal vibration detection. The actuator was driven by a standard sinusoidal signal from SG at the frequency of 300 Hz, 1 kHz, 2 kHz and 3 kHz, respectively. The material of actuator’s surface is metal, so the vibration amplitude of every point in the surface can be regarded as consistent as about 50 nm under the monitor by LDV. The real-time vibration information in ROI was extracted from the holograms using the proposed approach. Fig. 3 illustrates the frequency comparison of input–output signal from OS. In every sub-figure, comparison of vibration curve in the object domain is on the left side, and comparison of frequency in the Fourier domain is on the right side. A real-time display software was developed to show the vibration information. Amplitude–time curve of the selected point in the ROI is shown in Fig. 4. It can be seen from Fig. 3 that the proposed system retrieves the frequency of standard sinusoidal vibration signal at the frequency from 300 Hz to 3 kHz very well. Fig. 4 shows that the amplitude of output waveform is close to the theoretical value. Both figures prove the effectiveness of real-time detection of micro-vibration. Following aspects should be considered for the proposed system. (1) There is a time delay between output signal and input signal by the processing time of data transmission, storage, reconstruction and retrieval. The delay is not constant, difficult to eliminate and depends on PC performance. But it is usually less than a vibration cycle, thus there is no obvious influence on real-time detection. (2) There is a fluctuation on output curve by the power instability of
Fig. 3. I/O contrast diagram of standard sinusoidal signals.
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Fig. 4. Retrieved amplitude–time curve.
Fig. 7. Test results of multi-points vibration.
Fig. 5. Precision and frequency response curve of the proposed system.
the reference beam. This random fluctuation is one of the measurement error sources affecting the stability of the proposed system. (3) The retrieved vibration amplitude at 3 kHz is smaller than the theoretical value. 3.3. System precision and frequency response test This experiment is to further test the precision of measured amplitude and the response ability at long distance. To observe the precision and frequency response of system, the vibratory amplitudes are hold in about 55 nm at every measured frequency through changing the voltage of driven signal under monitored by LDV. In Fig. 5, the curve a is the measured result by LDV, while curve b is the measured result by the proposed system at 48 kHz sampling rate. It can be seen that the system precision is good when the vibratory frequency from 100 Hz to 2600 Hz and the average error is about 4.05%. After 2600 Hz, the precision begins to descend quickly, which is possibly influenced by sampling rate. To prove it, we changed the sampling rate to 24 kHz and 12 kHz. The results are shown by curve c and d. The curve c illustrates that the precision decreases after 1.3 kHz at 24 kHz sampling rate and the precision
decreases after 700 Hz at 12 kHz sampling rate. The experimental results indicate that the frequency response rage of the system is limited by the sampling rate of system. To get the good precision, the sampling rate should be about 20 times over the vibration frequency. Except for the sampling rate, the precision was also influenced by the stability of light source. We assume that the power of illumination beam and reference beam are constant during the experiment. But in reality, the power of illumination has small fluctuation, which generates measurement error.
3.4. Random vibration detection In order to prove the feasibility to detect random vibration, two groups of experiment were conducted, the driving vibration sources of which were 1 kHz-frequency random-amplitude (amplitude: 30–60 nm) and random-frequency random-amplitude (amplitude: 30–60 nm, frequency: 500–2 kHz), respectively. The results shown in Fig. 6 prove that the proposed system can detect and extract these two random vibration signals at real-time. The extracted signals are basically as the same as the original signals. The result demonstrates the proposed system can be used to measure random vibration. When the amplitude or frequency of the
Fig. 6. Test results of random signals.
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monitored object change, the system will synchronously detect and record this change. 3.5. Vibration of multi-points in the selected region detection The proposed system can measure multiple points synchronously in the selected vibration region. To prove this, we used headphone as the sample, since the amplitude of each point was significantly different. The driving vibration source was 1 kHz constant-voltage standard sinusoidal signal. Three different pixels were chosen and displayed on an oscilloscope as shown in Fig. 7. Fig. 7(a) is the photograph of the headphone with a dashed box marked with points A (close to the center), B (between point A and point C) and C (close to the edge). Fig. 7(b–d) illustrates the vibration detection curves of points A, B and C, respectively. For structural and material reasons, the vibration response on the headphone diaphragm surface is not uniform. The experimental results demonstrated that point A has the largest vibration amplitude, while point B and point C have smaller amplitude. 4. Conclusion In this paper, a real-time multi-points vibration detection approach based on digital holographic interferometry has been proposed for long distance (10 m) middle-low frequency (<3 kHz) micro-vibration (<100 nm) measurement. The experimental results demonstrate the effectiveness of the proposed method and reveal its unique advantages. (1) The measurement distance of microvibration is more than 10 m with imaging lens. (2) The method can synchronously measure and display the vibration of multiple points stimulated either by periodic signal or irregular one. (3) To increase the range of measured vibration frequency, smaller ROI and shorter exposure time was adopted with the expense of lower spatial resolution. (4) Unlike conventional reconstruction, vibration information can be determined directly and instantaneously from holograms using one time of Fourier transform and inverse transform. Faster processing speed guarantees real-time dynamic measurement. Unlike time-average approach, the proposed approach has only one manual step of filtering mask drawing before vibration and the whole vibration detection is automatic. In conclusion, this method overcomes the detected distance and the reconstruction speed limitation of digital holographic interferometry, so it may find wide applications in investigating real-time vibration phenomena. References [1] S.H. Zhang, H.L. Chen, Modeling and vibration analysis of a composite supporter for aerospace applications, Adv. Compos. Mater. 14 (2) (2005) 199–210.
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