Vibration detection by observation of speckle patterns Silvio Bianchi Dipartimento di Fisica, Università di Roma Sapienza, 00185 Rome, Italy ([email protected]) Received 12 November 2013; revised 12 January 2014; accepted 13 January 2014; posted 14 January 2014 (Doc. ID 201156); published 6 February 2014
When laser light illuminates a rough surface it is scattered into a speckle pattern that is strongly dependent on the surface geometry. Here, we show that it is possible to sense surface vibrations by measuring signal variations from a single pixel detector that collects a small portion of the scattered light. By carefully tuning the probing laser beam size and the detector’s aperture it was possible to record a good quality signal in the acoustic band. This approach eliminates the need for an interferometer and thus opens the door to the possibility of detecting vibrations at distances of few hundreds of meters. © 2014 Optical Society of America OCIS codes: (120.0280) Remote sensing and sensors; (120.7280) Vibration analysis. http://dx.doi.org/10.1364/AO.53.000931
1. Introduction
Since the advent of lasers, optical techniques have been increasingly adopted for measuring mechanical vibrations. Interferometry has been used to detect deformations of an object’s surface by measuring the optical path difference between a probe and a reference beam [1–4]. Using this approach, subwavelength accuracy can be achieved with a time resolution in a bandwidth well above the acoustic band [5]. A more powerful method is provided by laser Doppler vibrometry (LDV). In LDV, a probe beam is frequency shifted by a Bragg cell and then, after being reflected by the vibrating surface, it is subject to a Doppler frequency shift. The interference between the reflected probe beam and a reference beam is recorded by a photodetector so that the frequency shift, and thus the vibration velocity, can be retrieved [6,7]. Commercially available LDVs can detect velocities in the range of 0.02 μm∕s to 20 m∕s with a bandwidth up to 20 MHz. The above methods require an interferometer which, in addition to being sensitive to environment fluctuations, cannot probe objects at a distance greater than the laser coherence length. However, when a laser beam illuminates a target’s rough surface it is scattered into a speckle pattern 1559-128X/14/050931-06$15.00/0 © 2014 Optical Society of America
that is observable at any distance and varies substantially as the target shape changes. While in LDVs this phenomenon gives rise to undesired noise [8,9], an a priori knowledge of some properties of the speckle pattern can be used to measure the translation [10,11] or the rotation [12] of a target object. In [13,14] the authors used photo-EMF sensors, which are sensitive to speckle pattern intensity variations, to efficiently record the lateral vibrations of a speaker. However, photo-EMF sensors offer low sensitivity (especially in the frequency band below 1 kHz) which makes it difficult to probe a remote object. In [15] the intensity pattern on the photo-EMF sensor was created by the interference of a pulsed laser split into a probe and a reference beam. This last approach greatly improves the sensitivity of the technique but still relies on light interference. In this paper, we show that it is possible to sense vibrations by measuring only the output current of a photodiode in a noninterferometric setup. Unlike [13,14], we have a sensitive detector which is only capable of measuring the light intensity so, in order to retrieve a good signal, the probing laser beam size and the detector’s aperture must be correctly tuned. The bandwidth of photodiodes makes it possible to record vibrations in the acoustic band, including human speech. Due to the high sensitivity of semiconductor photodiodes, measurement requires only 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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a small portion of the scattered light, making it possible to apply the technique at distances greater than those achieved by long-range LDVs (typically tens of meters). A digital camera was also used for a more accurate and reliable measurement of low frequency vibrations. 2. Methods
Our experimental setup is shown in Fig. 1. A laser beam (Coherent Verdi G, λ 532 nm) illuminates a target membrane located at a distance L from the detector. Laser light is scattered by the membrane’s rough surface into a speckle field whose intensity pattern changes as the target surface deforms under the effect of sound waves generated by a PC speaker. Assuming that the vibration amplitude at is small enough, the detected power Pt can be expressed as a truncated Taylor expansion 1 Pat ≃ P0 P1 at P2 at2 ξt; 2
(1)
where Pn ∂n P∕∂an and ξ is a noise term. If the geometry of the system remains the same during measurement, Pn can be considered as constants, however, their values are random and they can only be estimated statistically. The constant term P0 scales with the portion of solid angle collected by the detector D2 ∕L2, where D is the detector aperture. To estimate the magnitude of the prefactors multiplying the signal a and the undesired terms an, the simple case is considered in which a vibration results in a proportional local tilt θ in the small region illuminated by the laser (see Fig. 1). On the detector plane, the local tilt of the surface causes a translation θL of the speckle pattern [12]. Since there is an equal chance of P increasing or decreasing as the target surface slightly changes shape, the statistical average of ∂n P∕∂an is zero. It can be
Fig. 1. Experimental setup. A laser beam strikes a vibrating membrane, and the scattered light forms a speckle pattern collected by a detector (camera or photodiode). W, laser spot size on the target membrane; L, detector–target distance; D, detector aperture; θ, local tilt of the membrane surface. 932
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shown (see Appendix A) that the root of the averaged square scales as follows: 1
D2 W n for D ≪ λL∕W; L2 λn DW n−1 for D ≫ λL∕W; ∼ Lλn−1
hP2n i2 ∼
(2)
where W is diameter of the laser beam on the target and λL∕W is the average speckle size [16]. In order to prove Eq. (2), a digital camera was used (Prosilica GC1280). The camera was positioned at a distance L 240 mm from a target membrane to acquire frames at 800 Hz for 2 s while the membrane vibrated at 150 Hz. Data was acquired with different frame sizes in order to establish a measurement of the signal as a function of D. Figure 2(a) shows the average amplitude of the first terms in Eq. (1). As described in Eq. (2), P1 , P2 , and P3 increase linearly or quadratically when D is, respectively, bigger or smaller than the speckle size. In the present case, the speckle size is about 100 μm while the values of D for which Pn change trend are between 90 and 60 μm. Figure 2(b) shows the power spectrum of Pt obtained for D 700 μm, note that the first harmonic peak is 102 higher than the second and third harmonic peaks. Even though the average ratio between the first and higher harmonics do not change much with D, if only a few speckles are collected by the detector, the values of Pn are subject to wide deviations from their average values. Figure 2(h) shows the standard deviations of the amplitudes of P1, P2 , and P3 divided by their average values. Only when D is more than 5 times greater than the speckle size, the plots show values below 10%. On the contrary, when D is small it may happen that higher order terms Pn become larger than P1 causing the appearance of large undesired harmonics as shown in Fig. 2(c), where the detector aperture is D 7 μm. The noise term ξ shows a white power spectrum whose amplitude as a function of D was fitted with the power law Dα, where α was found to be approximately 3∕2 [see Fig. 2(a)]. The desired signal is proportional to P1 and so the signal-to-noise ratio (SNR) scales as D or D−1 when D is, respectively, very small or very large compared to the speckle size. Figure 2(d) shows the squared ratio between the first harmonic amplitude and the noise amplitude [blue and black dots, respectively, in Fig. 2(a)] which is the SNR. The plot clearly underlines that the detector’s aperture must be comparable with the speckle size in order achieve optimal SNR. Figure 2(e) shows the dependence of the harmonic peak amplitudes as a function of the vibration amplitude a. It can be seen that, for small a, the peak amplitudes for the first, second, and third harmonics grow, respectively, as a, a2 , and a3 as expected from Eq. (1) but then, after this initial rise, they become comparable and start to slowly decrease. When
Fig. 2. (a) Amplitude of the terms in Eq. (1) plotted as a function of D: P0 (green), P1 (blue), P2 (red), P3 (purple), and ξ (black). Green and black lines plot D2 and D3∕2 , respectively, while blue lines highlight the behavior of P1 in the two regimes shown in Eq. (2). Power spectrum of Pt for D equal to (b) 700 μm and (c) 7 μm are shown in logarithmic scale. (d) Signal-to-noise ratio as a function of D calculated as the squared ratio between P1 and ξ. (e) Amplitudes of the first three harmonics and of ξ as a function of a. Power spectrum of Pt for (f) the lowest and (g) the highest value of vibration amplitude. (h) Standard deviation of P1 , P2 , and P3 divided their average values as a function of D. Blue, red, and purple arrows in (b), (c), (f), and (g) indicate the peaks corresponding to first, second, and third harmonic, respectively. Notice that, due to the limited acquisition frame rate, the harmonics above the second are aliased.
the vibration is small enough, the only relevant contribution to the first harmonic comes from the term linear in a while terms containing higher powers of a are suppressed, as shown in Fig. 2(f). On the other hand, when the vibration is large as in Fig. 2(g), higher harmonics clearly appear. Since in principle we have no knowledge of how the vibration changes the speckle pattern, to estimate sensitivity it is again assumed that the vibration results only in a local tilt θ of the target surface. The speckle displacement Lθ on the detector plane can be estimated by looking at the frames acquired by the camera so that, for each point in Fig. 2(e), its corresponding surface tilt can be established. Incidentally, the surface tilt is proportional to a that is expressed in arbitrary units. The lowest detectable tilt is estimated by extrapolating the curve of the signal amplitude [blue dots in Fig. 2(e)] until it equals the noise [black dots in Fig. 2(e)] while the highest detectable tilt is the value for which the undesired harmonics [red and purple dots in Fig. 2(e)] become comparable to the signal. In the same conditions as Fig. 2(e), where L 240 mm and D∕L ≃ 10−3 , tilts can be measured from 10−6 up to 10−2 radians. Equation (2) and the data shown in Fig. 2 give an idea of the parameters that influence the signal recorded by a detector. Given the target distance, the detector aperture D and the beam size W can easily be chosen using, respectively, a diaphragm and a couple of lenses, as shown in Fig. 1. The choice of these two parameters should be based on the following considerations: (i) signal-to-noise is maximized when the detector aperture D is comparable to speckle
size and (ii) a high D value prevents the appearance of large distortions in the signal. A good compromise between these two facts is to have D 5–6 times the speckle size, which means that a few tens of speckles must be collected. If the target is relatively close (i.e., small speckles) the minimum W value is chosen so that the detector aperture can be opened as much as possible. If instead the target is far away, D is maximized and W is set so that a few tens of speckles fit into the detector aperture. In order to test this technique, a target membrane was positioned 5 m away from a photodiode (Thorlabs PDA36A). The scattered light was collected and focused on the photodiode using a lens with an aperture set to 1 mm by a diaphragm. The photodiode output was sampled at audio frequencies (8 kHz) so that most audible vibrations could be recorded. Figure 3(a) shows a power spectrum related to a sinusoidal vibration of 1 s at 800 Hz, while in Fig. 3(b) vibrations from the experimenter’s voice are shown as a function of time (see also Media 1). In the latter plot, low frequency components were filtered out while background noise was reduced using the procedure explained in [17]. If the target was 250 m away, a standard two inch lens could be used to collect the same portion of scattered light and perform the same measurements. Up to this point P has been expressed only as a function of a vibration a. The case in which there are macroscopic changes in the position and/or shape of the target surface, occurring on timescales much longer than the period of the signal, are considered. The terms Pn in Eq. (1) are no longer constant in time but vary as the target surface changes so that 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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(a)
(b)
Fig. 3. (a) Power spectrum of a sinusoidal vibration of the membrane. (b) Output signal of the detector when target membrane vibration is caused by the experimenter’s voice (Media 1). Both measures were performed at a distance of 5 m and with a 1 mm detector aperture.
the resulting spectrum is given by the convolutions between an and Pn . A camera was used to record the signal produced by a target membrane vibrating at 100 Hz and at same time translating at a velocity in the order of a few centimeter/second. Figure 4 shows the spectrum of P when the target is fixed [Fig. 4(a)] and when it is translating [Figs. 4(b) and 4(c)]. The first harmonic peak is the convolution between a and P1 so it is low and broadened instead of being high and sharp. The broadening of the peak can be estimated with the following simple argument. If a surface is displaced by Δx, the speckle pattern on the detector plane is also translated by the same quantity [11]. If Δx is comparable or greater than D, the speckles collected by the detector are completely different, and so the value of P1 is also different. Calling v the velocity at which the target moves, the decorrelation time for
Fig. 4. (a) Power spectrum of P when the target position is fixed and when it is translating at a velocity of about (b) 2 cm∕s and (c) 4 cm∕s. The vibration peak at 100 Hz is broadened as the target speed increases. Such an effect can be overcome by tracking the speckle displacement whose power spectra are shown in (d) for a fixed target and in (e) and (f) for a target moving at about 2 and 4 cm∕s. 934
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P1 is about D∕v. The bandwidth of P1 is the inverse of the decorrelation time, and so the frequency broadening of the peak is D∕v−1 . Since data was acquired with a digital camera, it was possible to easily change D and it was verified that the peak width scales with D−1 as proposed above. The impossibility of correctly recording a vibration when a speckle pattern rapidly decorrelates is a major limitation for this technique. However, if it is only necessary to measure surface tilts due to vibration, any undesired effects can be eliminated by detecting speckle displacement. The displacement of a speckle intensity pattern I occurring in the time interval t; t Δt can be detected by looking at the correlation function ZZ cr; t Ir0 ; tIr0 r; t Δtdr0 ; (3) where r and r0 are position vectors located on the detector plane. The speckle displacement r¯ t is simply the value that maximizes cr; t. The power spectra of the displacement in Figs. 4(d)–4(f) are relative to the same data shown in Figs. 4(a)–4(c). The signal is now clearly recovered, and in addition, the spectrum does not exhibit any undesired harmonic. The time taken by the PC to analyze a 200 × 200 frame is 13 ms which limits the framerate to less than 100 Hz, but the frames can nevertheless be stored and successively analyzed so that the real limitation to the framerate is determined by the camera. The minimum detectable tilt is again limited by the SNR, which seems to be approximately the same as the previous technique described. On the other hand, the camera field of view limits the maximum detectable speckle displacement and thus the maximum detectable tilt. Under the same conditions described for the estimation of the range of the previous technique (L 240 mm, D∕L 10−3 ), it is possible to detect tilts from 10−6 to 10−3 radians. 3. Conclusions
Two methods for detecting surface vibrations are described. The first uses a single pixel detector and so provides a high sampling rate with a good sensitivity. Even though quantitative measurement of surface deformation is not possible, a signal exhibiting low distortion and low noise can be recorded. In the second approach, local surface tilts due to vibrations are detected using a digital camera. Undesired harmonics and signal degradation due to speckle decorrelation are eliminated but at the cost of a sampling rate limited by the framerate of digital cameras. Appendix A
In the Fraunhofer regime (L ≫ W 2 ∕λ), the field on the detector plane can be expressed as 1 ur L
ZZ
2πr · r0 0 iϕr dr0 ; exp i Lλ
(A1)
where r x; y is the position on the detector plane, while r0 is the position on the target surface plane. For the sake of simplicity, a square illuminated area is considered so that the integration domain is −W∕2; W∕2 for both x and y. The phase factor ϕ is a random variable which takes surface roughness into account. In the case considered here, ϕ is distributed in the interval 0; 2π and has the property hϕrϕr0 i δr − r0 , where h…i indicates the statistical average. Equation (A1) is a sum of plane waves of unitary amplitudes and random phases. The wavevectors of these plane waves k 2π∕Lλr have components, in both the x and y directions, uniformly distributed in −πW∕λL; πW∕λL. It is easy to believe that the speckle size is approximately Lλ∕W, which is the inverse of the field maximum spatial frequency. The speckle size is also the minimum distance between two points r1 and r2 for which ur1 and ur2 are uncorrelated. To estimate the magnitude ∂n I∕∂xn the root of its variance is calculated. First, Eq. (A1) must be multiplied by its complex conjugate to obtain the intensity then, bringing the derivative inside the integral, gives ∂n I ∂xn
Z
K12 · xˆ iK12 ·riΦ12 e dk1 dk2 ; 2kmax L2
(A2)
where
the intensity pattern Ix; y by Lθ in the x direction. The power recorded by the detector is the integral of Ix; y over an area D × D ZZ P Ix θL; ydxdy ZZ n X ∂ I n n θ L (A6) n dxdy: ∂x n0 Each term of the sum corresponds to those shown in Eq. (1) where a is replaced with θ. If the integration region is much smaller than the speckle size (i.e., D ≪ Lλ∕W), the terms inside the integral can be approximated as constants n 2 1 2 ∂ P D2 W n ∼ 2 n : n ∂θ L λ
On the contrary, if D ≫ Lλ∕W, the detector integrates the intensity given by a large number N of speckles whose average area is Lλ∕W2 . The integral in Eq. (A6) is replaced with a discrete sum of statistically independent terms
∂n P ∂θn
∂n I ∂xn
2 1 2
Z
K12 · xˆ K34 · xˆ 4k2max L4
1 2 × eiK12 K34 ·riΦ12 −iΦ34 d4 k :
(A3)
The integration now runs over four variables (each with two components) k1 , k2 , k3 , and k4 . The statistical average operator is now brought inside the integral and the exponential with the phase factors is replaced as follows: heiΦ12 −iΦ34 i δK12 δK34 δK14 δK23 :
(A4)
The first term annihilates the integral while integrating the second gives the following result: n 2 1 2 ∂ I knmax Wn ∼ : (A5) ∂xn L2 λn Ln2 The magnitude of the terms in Eq. (1) are now evaluated. A tilt of the target surface θ displaces
2 1 2
N X 1 N L 2 λ2 X ∂n I i ∂n I j 2 ≃L W 2 i0 j0 ∂xn ∂xn n
∼
K12 k1 − k2 ; Φ12 ϕk1 − ϕk2 ; kmax πW∕Lλ; kmax is the maximum value assumed by each component of k1 and k1 in the integral, notice also that the normalization term 2kmax −1 is introduced in the integral so that hIi L−2 . Squaring Eq. (A2) and taking the root of the average gives the following expression:
(A7)
DW n−1 : λn−1 L
(A8)
The above equation was derived from Eq. (A5) substituting N with DW2 ∕λL2 . References 1. C. Joenathan, B. Franze, P. Haible, and H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998). 2. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999). 3. G. Pedrini, W. Osten, and M. E. Gusev, “High-speed digital holographic interferometry for vibration measurement,” Appl. Opt. 45, 3456–3462 (2006). 4. I. Yamaguchi, A. Yamamoto, and S. Kuwamura, “Speckle decorrelation in surface profilometry by wavelength scanning interferometry,” Appl. Opt. 37, 6721–6728 (1998). 5. G. Smeets, “Laser interference microphone for ultrasonics and nonlinear acoustics,” J. Acoust. Soc. Am. 61, 872–875 (1977). 6. G. T. Feke and C. E. Riva, “Laser Doppler measurements of blood velocity in human retinal vessels,” J. Opt. Soc. Am. 68, 526–531 (1978). 7. T. A. Riener, A. C. Goding, and F. E. Talke, “Measurement of head/disk spacing modulation using a two channel fiber optic laser doppler vibrometer,” IEEE Trans. Magn. 24, 2745–2747 (1988). 8. S. J. Rothberg, “Numerical simulation of speckle noise in laser vibrometry,” Appl. Opt. 45, 4523–4533 (2006). 9. P. Martin and S. J. Rothberg, “Pseudo-vibration sensitivities for commercial laser vibrometers,” Mech. Syst. Signal Process. 25, 2753–2765 (2011). 10. J. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8, 38–44 (1997). 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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11. I. Yamaguchi, “Automatic measurement of in-plane translation by speckle correlation using a linear image sensor,” J. Phys. E 19, 944–949 (1986). 12. B. Rose, H. Imam, and S. G. Hanson, “Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement,” J. Opt. 29, 115–120 (1998). 13. N. A. Korneev and S. I. Stepanov, “Measurement of small lateral vibrations of speckle patterns using a non-steady-state photo-EMF in GaAs:Cr,” J. Mod. Opt. 38, 2153–2158 (1991). 14. N. Korneev and S. Stepanov, “Measurement of different components of vibrations in speckle referenceless configuration
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using adaptive photodetectors,” Opt. Commun. 115, 35–39 (1995). 15. P. Rodriguez, S. Trivedi, F. Jin, C. Wang, S. Stepanov, G. Elliott, J. F. Meyers, J. Lee, and J. Khurgin, “Pulsed-laser vibrometer using photoelectromotive-force sensors,” Appl. Phys. Lett. 83, 1893–1895 (2003). 16. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2010). 17. S. F. Boll, “Suppression of acoustic noise in speech using spectral subtraction,” IEEE Trans. Acoust., Speech, Signal Process. 27, 113–119 (1979).
When laser light illuminates a rough surface it is scattered into a speckle pattern that is strongly dependent on the surface geometry. Here, we show that it is possible to sense surface vibrations by measuring signal variations from a single pixel detector that collects a small portion of the scattered light. By carefully tuning the probing laser beam size and the detector’s aperture it was possible to record a good quality signal in the acoustic band. This approach eliminates the need for an interferometer and thus opens the door to the possibility of detecting vibrations at distances of few hundreds of meters. © 2014 Optical Society of America OCIS codes: (120.0280) Remote sensing and sensors; (120.7280) Vibration analysis. http://dx.doi.org/10.1364/AO.53.000931
1. Introduction
Since the advent of lasers, optical techniques have been increasingly adopted for measuring mechanical vibrations. Interferometry has been used to detect deformations of an object’s surface by measuring the optical path difference between a probe and a reference beam [1–4]. Using this approach, subwavelength accuracy can be achieved with a time resolution in a bandwidth well above the acoustic band [5]. A more powerful method is provided by laser Doppler vibrometry (LDV). In LDV, a probe beam is frequency shifted by a Bragg cell and then, after being reflected by the vibrating surface, it is subject to a Doppler frequency shift. The interference between the reflected probe beam and a reference beam is recorded by a photodetector so that the frequency shift, and thus the vibration velocity, can be retrieved [6,7]. Commercially available LDVs can detect velocities in the range of 0.02 μm∕s to 20 m∕s with a bandwidth up to 20 MHz. The above methods require an interferometer which, in addition to being sensitive to environment fluctuations, cannot probe objects at a distance greater than the laser coherence length. However, when a laser beam illuminates a target’s rough surface it is scattered into a speckle pattern 1559-128X/14/050931-06$15.00/0 © 2014 Optical Society of America
that is observable at any distance and varies substantially as the target shape changes. While in LDVs this phenomenon gives rise to undesired noise [8,9], an a priori knowledge of some properties of the speckle pattern can be used to measure the translation [10,11] or the rotation [12] of a target object. In [13,14] the authors used photo-EMF sensors, which are sensitive to speckle pattern intensity variations, to efficiently record the lateral vibrations of a speaker. However, photo-EMF sensors offer low sensitivity (especially in the frequency band below 1 kHz) which makes it difficult to probe a remote object. In [15] the intensity pattern on the photo-EMF sensor was created by the interference of a pulsed laser split into a probe and a reference beam. This last approach greatly improves the sensitivity of the technique but still relies on light interference. In this paper, we show that it is possible to sense vibrations by measuring only the output current of a photodiode in a noninterferometric setup. Unlike [13,14], we have a sensitive detector which is only capable of measuring the light intensity so, in order to retrieve a good signal, the probing laser beam size and the detector’s aperture must be correctly tuned. The bandwidth of photodiodes makes it possible to record vibrations in the acoustic band, including human speech. Due to the high sensitivity of semiconductor photodiodes, measurement requires only 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
931
a small portion of the scattered light, making it possible to apply the technique at distances greater than those achieved by long-range LDVs (typically tens of meters). A digital camera was also used for a more accurate and reliable measurement of low frequency vibrations. 2. Methods
Our experimental setup is shown in Fig. 1. A laser beam (Coherent Verdi G, λ 532 nm) illuminates a target membrane located at a distance L from the detector. Laser light is scattered by the membrane’s rough surface into a speckle field whose intensity pattern changes as the target surface deforms under the effect of sound waves generated by a PC speaker. Assuming that the vibration amplitude at is small enough, the detected power Pt can be expressed as a truncated Taylor expansion 1 Pat ≃ P0 P1 at P2 at2 ξt; 2
(1)
where Pn ∂n P∕∂an and ξ is a noise term. If the geometry of the system remains the same during measurement, Pn can be considered as constants, however, their values are random and they can only be estimated statistically. The constant term P0 scales with the portion of solid angle collected by the detector D2 ∕L2, where D is the detector aperture. To estimate the magnitude of the prefactors multiplying the signal a and the undesired terms an, the simple case is considered in which a vibration results in a proportional local tilt θ in the small region illuminated by the laser (see Fig. 1). On the detector plane, the local tilt of the surface causes a translation θL of the speckle pattern [12]. Since there is an equal chance of P increasing or decreasing as the target surface slightly changes shape, the statistical average of ∂n P∕∂an is zero. It can be
Fig. 1. Experimental setup. A laser beam strikes a vibrating membrane, and the scattered light forms a speckle pattern collected by a detector (camera or photodiode). W, laser spot size on the target membrane; L, detector–target distance; D, detector aperture; θ, local tilt of the membrane surface. 932
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shown (see Appendix A) that the root of the averaged square scales as follows: 1
D2 W n for D ≪ λL∕W; L2 λn DW n−1 for D ≫ λL∕W; ∼ Lλn−1
hP2n i2 ∼
(2)
where W is diameter of the laser beam on the target and λL∕W is the average speckle size [16]. In order to prove Eq. (2), a digital camera was used (Prosilica GC1280). The camera was positioned at a distance L 240 mm from a target membrane to acquire frames at 800 Hz for 2 s while the membrane vibrated at 150 Hz. Data was acquired with different frame sizes in order to establish a measurement of the signal as a function of D. Figure 2(a) shows the average amplitude of the first terms in Eq. (1). As described in Eq. (2), P1 , P2 , and P3 increase linearly or quadratically when D is, respectively, bigger or smaller than the speckle size. In the present case, the speckle size is about 100 μm while the values of D for which Pn change trend are between 90 and 60 μm. Figure 2(b) shows the power spectrum of Pt obtained for D 700 μm, note that the first harmonic peak is 102 higher than the second and third harmonic peaks. Even though the average ratio between the first and higher harmonics do not change much with D, if only a few speckles are collected by the detector, the values of Pn are subject to wide deviations from their average values. Figure 2(h) shows the standard deviations of the amplitudes of P1, P2 , and P3 divided by their average values. Only when D is more than 5 times greater than the speckle size, the plots show values below 10%. On the contrary, when D is small it may happen that higher order terms Pn become larger than P1 causing the appearance of large undesired harmonics as shown in Fig. 2(c), where the detector aperture is D 7 μm. The noise term ξ shows a white power spectrum whose amplitude as a function of D was fitted with the power law Dα, where α was found to be approximately 3∕2 [see Fig. 2(a)]. The desired signal is proportional to P1 and so the signal-to-noise ratio (SNR) scales as D or D−1 when D is, respectively, very small or very large compared to the speckle size. Figure 2(d) shows the squared ratio between the first harmonic amplitude and the noise amplitude [blue and black dots, respectively, in Fig. 2(a)] which is the SNR. The plot clearly underlines that the detector’s aperture must be comparable with the speckle size in order achieve optimal SNR. Figure 2(e) shows the dependence of the harmonic peak amplitudes as a function of the vibration amplitude a. It can be seen that, for small a, the peak amplitudes for the first, second, and third harmonics grow, respectively, as a, a2 , and a3 as expected from Eq. (1) but then, after this initial rise, they become comparable and start to slowly decrease. When
Fig. 2. (a) Amplitude of the terms in Eq. (1) plotted as a function of D: P0 (green), P1 (blue), P2 (red), P3 (purple), and ξ (black). Green and black lines plot D2 and D3∕2 , respectively, while blue lines highlight the behavior of P1 in the two regimes shown in Eq. (2). Power spectrum of Pt for D equal to (b) 700 μm and (c) 7 μm are shown in logarithmic scale. (d) Signal-to-noise ratio as a function of D calculated as the squared ratio between P1 and ξ. (e) Amplitudes of the first three harmonics and of ξ as a function of a. Power spectrum of Pt for (f) the lowest and (g) the highest value of vibration amplitude. (h) Standard deviation of P1 , P2 , and P3 divided their average values as a function of D. Blue, red, and purple arrows in (b), (c), (f), and (g) indicate the peaks corresponding to first, second, and third harmonic, respectively. Notice that, due to the limited acquisition frame rate, the harmonics above the second are aliased.
the vibration is small enough, the only relevant contribution to the first harmonic comes from the term linear in a while terms containing higher powers of a are suppressed, as shown in Fig. 2(f). On the other hand, when the vibration is large as in Fig. 2(g), higher harmonics clearly appear. Since in principle we have no knowledge of how the vibration changes the speckle pattern, to estimate sensitivity it is again assumed that the vibration results only in a local tilt θ of the target surface. The speckle displacement Lθ on the detector plane can be estimated by looking at the frames acquired by the camera so that, for each point in Fig. 2(e), its corresponding surface tilt can be established. Incidentally, the surface tilt is proportional to a that is expressed in arbitrary units. The lowest detectable tilt is estimated by extrapolating the curve of the signal amplitude [blue dots in Fig. 2(e)] until it equals the noise [black dots in Fig. 2(e)] while the highest detectable tilt is the value for which the undesired harmonics [red and purple dots in Fig. 2(e)] become comparable to the signal. In the same conditions as Fig. 2(e), where L 240 mm and D∕L ≃ 10−3 , tilts can be measured from 10−6 up to 10−2 radians. Equation (2) and the data shown in Fig. 2 give an idea of the parameters that influence the signal recorded by a detector. Given the target distance, the detector aperture D and the beam size W can easily be chosen using, respectively, a diaphragm and a couple of lenses, as shown in Fig. 1. The choice of these two parameters should be based on the following considerations: (i) signal-to-noise is maximized when the detector aperture D is comparable to speckle
size and (ii) a high D value prevents the appearance of large distortions in the signal. A good compromise between these two facts is to have D 5–6 times the speckle size, which means that a few tens of speckles must be collected. If the target is relatively close (i.e., small speckles) the minimum W value is chosen so that the detector aperture can be opened as much as possible. If instead the target is far away, D is maximized and W is set so that a few tens of speckles fit into the detector aperture. In order to test this technique, a target membrane was positioned 5 m away from a photodiode (Thorlabs PDA36A). The scattered light was collected and focused on the photodiode using a lens with an aperture set to 1 mm by a diaphragm. The photodiode output was sampled at audio frequencies (8 kHz) so that most audible vibrations could be recorded. Figure 3(a) shows a power spectrum related to a sinusoidal vibration of 1 s at 800 Hz, while in Fig. 3(b) vibrations from the experimenter’s voice are shown as a function of time (see also Media 1). In the latter plot, low frequency components were filtered out while background noise was reduced using the procedure explained in [17]. If the target was 250 m away, a standard two inch lens could be used to collect the same portion of scattered light and perform the same measurements. Up to this point P has been expressed only as a function of a vibration a. The case in which there are macroscopic changes in the position and/or shape of the target surface, occurring on timescales much longer than the period of the signal, are considered. The terms Pn in Eq. (1) are no longer constant in time but vary as the target surface changes so that 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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(a)
(b)
Fig. 3. (a) Power spectrum of a sinusoidal vibration of the membrane. (b) Output signal of the detector when target membrane vibration is caused by the experimenter’s voice (Media 1). Both measures were performed at a distance of 5 m and with a 1 mm detector aperture.
the resulting spectrum is given by the convolutions between an and Pn . A camera was used to record the signal produced by a target membrane vibrating at 100 Hz and at same time translating at a velocity in the order of a few centimeter/second. Figure 4 shows the spectrum of P when the target is fixed [Fig. 4(a)] and when it is translating [Figs. 4(b) and 4(c)]. The first harmonic peak is the convolution between a and P1 so it is low and broadened instead of being high and sharp. The broadening of the peak can be estimated with the following simple argument. If a surface is displaced by Δx, the speckle pattern on the detector plane is also translated by the same quantity [11]. If Δx is comparable or greater than D, the speckles collected by the detector are completely different, and so the value of P1 is also different. Calling v the velocity at which the target moves, the decorrelation time for
Fig. 4. (a) Power spectrum of P when the target position is fixed and when it is translating at a velocity of about (b) 2 cm∕s and (c) 4 cm∕s. The vibration peak at 100 Hz is broadened as the target speed increases. Such an effect can be overcome by tracking the speckle displacement whose power spectra are shown in (d) for a fixed target and in (e) and (f) for a target moving at about 2 and 4 cm∕s. 934
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P1 is about D∕v. The bandwidth of P1 is the inverse of the decorrelation time, and so the frequency broadening of the peak is D∕v−1 . Since data was acquired with a digital camera, it was possible to easily change D and it was verified that the peak width scales with D−1 as proposed above. The impossibility of correctly recording a vibration when a speckle pattern rapidly decorrelates is a major limitation for this technique. However, if it is only necessary to measure surface tilts due to vibration, any undesired effects can be eliminated by detecting speckle displacement. The displacement of a speckle intensity pattern I occurring in the time interval t; t Δt can be detected by looking at the correlation function ZZ cr; t Ir0 ; tIr0 r; t Δtdr0 ; (3) where r and r0 are position vectors located on the detector plane. The speckle displacement r¯ t is simply the value that maximizes cr; t. The power spectra of the displacement in Figs. 4(d)–4(f) are relative to the same data shown in Figs. 4(a)–4(c). The signal is now clearly recovered, and in addition, the spectrum does not exhibit any undesired harmonic. The time taken by the PC to analyze a 200 × 200 frame is 13 ms which limits the framerate to less than 100 Hz, but the frames can nevertheless be stored and successively analyzed so that the real limitation to the framerate is determined by the camera. The minimum detectable tilt is again limited by the SNR, which seems to be approximately the same as the previous technique described. On the other hand, the camera field of view limits the maximum detectable speckle displacement and thus the maximum detectable tilt. Under the same conditions described for the estimation of the range of the previous technique (L 240 mm, D∕L 10−3 ), it is possible to detect tilts from 10−6 to 10−3 radians. 3. Conclusions
Two methods for detecting surface vibrations are described. The first uses a single pixel detector and so provides a high sampling rate with a good sensitivity. Even though quantitative measurement of surface deformation is not possible, a signal exhibiting low distortion and low noise can be recorded. In the second approach, local surface tilts due to vibrations are detected using a digital camera. Undesired harmonics and signal degradation due to speckle decorrelation are eliminated but at the cost of a sampling rate limited by the framerate of digital cameras. Appendix A
In the Fraunhofer regime (L ≫ W 2 ∕λ), the field on the detector plane can be expressed as 1 ur L
ZZ
2πr · r0 0 iϕr dr0 ; exp i Lλ
(A1)
where r x; y is the position on the detector plane, while r0 is the position on the target surface plane. For the sake of simplicity, a square illuminated area is considered so that the integration domain is −W∕2; W∕2 for both x and y. The phase factor ϕ is a random variable which takes surface roughness into account. In the case considered here, ϕ is distributed in the interval 0; 2π and has the property hϕrϕr0 i δr − r0 , where h…i indicates the statistical average. Equation (A1) is a sum of plane waves of unitary amplitudes and random phases. The wavevectors of these plane waves k 2π∕Lλr have components, in both the x and y directions, uniformly distributed in −πW∕λL; πW∕λL. It is easy to believe that the speckle size is approximately Lλ∕W, which is the inverse of the field maximum spatial frequency. The speckle size is also the minimum distance between two points r1 and r2 for which ur1 and ur2 are uncorrelated. To estimate the magnitude ∂n I∕∂xn the root of its variance is calculated. First, Eq. (A1) must be multiplied by its complex conjugate to obtain the intensity then, bringing the derivative inside the integral, gives ∂n I ∂xn
Z
K12 · xˆ iK12 ·riΦ12 e dk1 dk2 ; 2kmax L2
(A2)
where
the intensity pattern Ix; y by Lθ in the x direction. The power recorded by the detector is the integral of Ix; y over an area D × D ZZ P Ix θL; ydxdy ZZ n X ∂ I n n θ L (A6) n dxdy: ∂x n0 Each term of the sum corresponds to those shown in Eq. (1) where a is replaced with θ. If the integration region is much smaller than the speckle size (i.e., D ≪ Lλ∕W), the terms inside the integral can be approximated as constants n 2 1 2 ∂ P D2 W n ∼ 2 n : n ∂θ L λ
On the contrary, if D ≫ Lλ∕W, the detector integrates the intensity given by a large number N of speckles whose average area is Lλ∕W2 . The integral in Eq. (A6) is replaced with a discrete sum of statistically independent terms
∂n P ∂θn
∂n I ∂xn
2 1 2
Z
K12 · xˆ K34 · xˆ 4k2max L4
1 2 × eiK12 K34 ·riΦ12 −iΦ34 d4 k :
(A3)
The integration now runs over four variables (each with two components) k1 , k2 , k3 , and k4 . The statistical average operator is now brought inside the integral and the exponential with the phase factors is replaced as follows: heiΦ12 −iΦ34 i δK12 δK34 δK14 δK23 :
(A4)
The first term annihilates the integral while integrating the second gives the following result: n 2 1 2 ∂ I knmax Wn ∼ : (A5) ∂xn L2 λn Ln2 The magnitude of the terms in Eq. (1) are now evaluated. A tilt of the target surface θ displaces
2 1 2
N X 1 N L 2 λ2 X ∂n I i ∂n I j 2 ≃L W 2 i0 j0 ∂xn ∂xn n
∼
K12 k1 − k2 ; Φ12 ϕk1 − ϕk2 ; kmax πW∕Lλ; kmax is the maximum value assumed by each component of k1 and k1 in the integral, notice also that the normalization term 2kmax −1 is introduced in the integral so that hIi L−2 . Squaring Eq. (A2) and taking the root of the average gives the following expression:
(A7)
DW n−1 : λn−1 L
(A8)
The above equation was derived from Eq. (A5) substituting N with DW2 ∕λL2 . References 1. C. Joenathan, B. Franze, P. Haible, and H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998). 2. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999). 3. G. Pedrini, W. Osten, and M. E. Gusev, “High-speed digital holographic interferometry for vibration measurement,” Appl. Opt. 45, 3456–3462 (2006). 4. I. Yamaguchi, A. Yamamoto, and S. Kuwamura, “Speckle decorrelation in surface profilometry by wavelength scanning interferometry,” Appl. Opt. 37, 6721–6728 (1998). 5. G. Smeets, “Laser interference microphone for ultrasonics and nonlinear acoustics,” J. Acoust. Soc. Am. 61, 872–875 (1977). 6. G. T. Feke and C. E. Riva, “Laser Doppler measurements of blood velocity in human retinal vessels,” J. Opt. Soc. Am. 68, 526–531 (1978). 7. T. A. Riener, A. C. Goding, and F. E. Talke, “Measurement of head/disk spacing modulation using a two channel fiber optic laser doppler vibrometer,” IEEE Trans. Magn. 24, 2745–2747 (1988). 8. S. J. Rothberg, “Numerical simulation of speckle noise in laser vibrometry,” Appl. Opt. 45, 4523–4533 (2006). 9. P. Martin and S. J. Rothberg, “Pseudo-vibration sensitivities for commercial laser vibrometers,” Mech. Syst. Signal Process. 25, 2753–2765 (2011). 10. J. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. 8, 38–44 (1997). 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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11. I. Yamaguchi, “Automatic measurement of in-plane translation by speckle correlation using a linear image sensor,” J. Phys. E 19, 944–949 (1986). 12. B. Rose, H. Imam, and S. G. Hanson, “Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement,” J. Opt. 29, 115–120 (1998). 13. N. A. Korneev and S. I. Stepanov, “Measurement of small lateral vibrations of speckle patterns using a non-steady-state photo-EMF in GaAs:Cr,” J. Mod. Opt. 38, 2153–2158 (1991). 14. N. Korneev and S. Stepanov, “Measurement of different components of vibrations in speckle referenceless configuration
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using adaptive photodetectors,” Opt. Commun. 115, 35–39 (1995). 15. P. Rodriguez, S. Trivedi, F. Jin, C. Wang, S. Stepanov, G. Elliott, J. F. Meyers, J. Lee, and J. Khurgin, “Pulsed-laser vibrometer using photoelectromotive-force sensors,” Appl. Phys. Lett. 83, 1893–1895 (2003). 16. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2010). 17. S. F. Boll, “Suppression of acoustic noise in speech using spectral subtraction,” IEEE Trans. Acoust., Speech, Signal Process. 27, 113–119 (1979).
