Characteristics of vibration frequency measurement based on sound field imaging by digital holography

Sudheesh K. Rajput; Osamu Matoba; Yasuhiro Awatsuji

doi:10.1364/OSAC.1.000200

1. Introduction

In digital holography (DH), the optical field is recorded as a digital hologram by an image sensor and the propagation of optical fields is described by diffraction theory, which allows numerical reconstruction of the two- and three-dimensional images [1,2]. The DH has been researched in several applications because of its significant advantages such as the ability to acquire holograms rapidly, availability of complete amplitude and phase information of the optical field, and suitability of the image processing techniques for recorded holograms [1,2]. The applications of DH include microscopy, optical security, vibration analysis, and sound field imaging [3–15]. The recent and new application of DH is a sound field imaging where acoustical quantities are acquired by optical means, which is of growing interest because of its contactless nature. It has been used to image a spatial and temporal distribution of sound field by measuring phase or amplitude modulation of light caused by sound field [7–9]. The sound field imaging technique has also been used successfully to visualize the optical voice [7].

The vibration measurement and analysis is also an important application area of DH where off-axis and inline optical setups have been utilized [10–28]. It has been used in many scientific fields, like acoustics, entertainment devices, security, and surveillance [10–28]. The DH based techniques have been considered as novel approach to measure several parameters of the vibration in real time [10–25]. These techniques have been preferred over other measurement techniques for vibration analysis because of their non-contact nature. In DH, the hologram has been recorded which is the interference pattern of light reflected from vibrating object and stationary reference beam. Therefore, any change in the interference pattern caused by the vibrating object can be visualized and analyzed [10–25]. Because of the rapid development of high speed image sensor, it is possible to record a sequence of holograms of the object with high frame rate as a function of the time [20–22]. This capability of high speed DH helps to analyze the motion of object by getting phase difference and by Fourier transform analysis in the time domain. For complete vibration study, the frame rate should be two times higher than the vibration frequency as it can be understood from the Nyquist-Shannon theorem.

The simplified processing of the hologram data has been performed by the concept of time-averaged holography using digital image sensor. In the time averaged holography, there is no limit in vibration frequency and do not requires expensive fast image sensor [16]. Further, this technique has been extended for the detection of the optical signal at the vibration sideband frequency [17]. In sideband holography based techniques, the frequency corresponding to reference beam is tuned in order to select the sideband. This tuning has been successfully implemented by the heterodyne holography technique [18], which are able to measure vibration parameters of higher frequencies [18]. These techniques record the holographic signal over a large number of vibration periods, which are not sensitive to the phase of the vibration and hence unable to instantaneous measurement of some parameters. It has also been found that in most of the vibration analysis and measurement applications, the characteristics such as amplitude, displacement, velocity, and acceleration of object have been determined [10–23].

Other category of vibration measurement is the image-based methods, which are also attracting researcher’s community and becoming a reliable alternative to non-contact measurement of moving and vibrating objects [26–29]. These methods are based on object recognition and tracking through digital image correlation [26,27], sub-pixel method [28], and local multi-threshold technique [29]. In local multi-threshold technique, frames of light intensity variation due to object vibration are recorded. Then the analysis is done at different threshold levels of binary information and number of white pixels inside that region is tracked in order to obtain the frequency of the vibration movement. Usually in vibration analysis, the amplitude of vibration is measured. There are only few optical schemes, which determine the frequency of vibration [25,26,29]. Also these methods use imaging of vibrating objects instead of imaging of vibration field. In the cases where imaging of vibrating objects is difficult and hence it may not be possible to analyze complete vibration. Therefore, the new methods should be developed to remove this problem. Frequency estimation of signal is a fundamental problem in speech processing, electric power system, biomedical signal processing, microelectromechanical system, and telecommunications [30]. And developing the quantitative methods for detecting and identifying unknown vibration frequency is also challenging task [25]. Therefore, for the testing of scientific vibrating objects, more accurate and quantitative frequency measurement methods need to be developed.

In this paper, we present the characteristics of DH based sound field imaging on frequency measurement of vibrating objects where phase change characteristics of a medium due to sound wave propagation is utilized. The measurement capability of vibration frequency using DH for different objects and quantitative characteristics of the sound power are presented. In this method, an object is placed near to one arm of off-axis DH and vibrated perpendicular to beam. The interference patterns of it with a reference wave are recorded with a high-speed image sensor in digital holograms as a function of time. After using inverse Fresnel propagation reconstruction and some data processing techniques, the spatial and temporal distributions are recovered. The Fourier transform of these distributions can determine the frequency of vibrating object. Our method is based on the imaging of vibration field instead of imaging of vibrating object and it is also a non-contact method. So, it is possible to record the information of vibration without disturbing the field. This system may be very useful for testing the unknown frequency of objects in addition to standard objects. Here, the standard object is defined as the object, which can produce the sound wave with specific frequency. We present experimental results from hundreds Hz to tens of kHz that covers audible range. We also present comparative study with microphone recording and analysis to check the strength and stability of scheme.

2. Frequency measurement system

2.1 Sound field recording by DH

We start this Sec. with sound field imaging using DH. The basic principle behind this imaging technique is to measure phase change over time, which is caused by variation in refractive index of medium due to sound field [7–9,31]. Therefore, the propagation of sound field modulates a refractive index distribution. According to theory of sound wave propagation described in Ref. 31, the refractive-index change Δn in propagation medium is described by

Δ n = \frac{(n - 1)}{γ} \frac{P}{P_{0}},

where n is the refractive index of the propagation medium at atmospheric pressure P₀, P is the increased pressure due to the sound waves, and γ is the specific heat ratio. Equation (1) shows the linear relationship between the change in refractive index and sound pressure induced by sound wave. To the optics point of view, if the light passes through the region where the change in refractive index occurs, the phase will also change. This phase change can be measured by some optical means such as DH [7–9].

In DH based sound field imaging schemes, an object wave passes through the region where the sound wave is propagating; the phase retardation is occurred in the object wave [7]. The phase retarded object wave interferes with a reference wave, which can be recorded as an interference pattern. This temporal phase retardation can be measured using DH and some data processing techniques [7–9]. Here, the vibrating objects whose frequencies are to be determined are placed near to an object arm of off-axis DH setup as shown in Fig. 1(a)

Fig. 1 (a) Optical setup for frequency measurement for vibrating objects based on off-axis DH and (b) diagram to show directions of sound propagation and laser beam. BS: Beam splitter, OM: Object mirror, RM: Reference mirror, HSC: High speed camera, and BE: Beam expander.

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. In this Figure, the laser light after passing through beam splitter (BS) is divided into object and reference beams and expanded by beam expander (BE). Two mirrors namely, object mirror (OM) and reference mirror (RM) have also been used, corresponding to object and reference beams, respectively. The sound waves originated from the vibration source propagate perpendicular to the direction of object beam. The direction of vibrating sound propagation is explained through Fig. 1(b). The propagation directions of the object light and the sound wave are z and y axes, respectively. The interference pattern between phase retarded vibrating object wave and reference wave are recorded in holograms as a function of time using high speed image sensor by off-axis DH [7]. Mathematically, the recording of holograms, h(x,y) can be explained as follows:

\begin{array}{l} h (x, y) = {| o (x, y) |}^{2} + {| r (x, y) |}^{2} + o^{*} (x, y) r (x, y) \\ + o (x, y) r^{*} (x, y), \end{array}

where o and r denote object and reference optical fields, respectively. The symbol ‘∗’ denotes the complex conjugate. In Eq. (2),

\begin{array}{l} o (x, y) = | o (x, y) | e x p (i p (x, y)) \\ = | o (x, y) | e x p (i \frac{2 π Δ n (x, y) d}{λ}), \end{array}

where p(x,y), d, and λ are the phase image of the sound wave, the effective interaction length between the object light and the sound wave, and wavelength of light, respectively.

2.2 Hologram reconstruction and processing

Now the object wave o(x,y) which has the information of vibrating objects, is reconstructed from holograms using angular spectrum method. This is basically numerical calculation of free space propagation [1,2] which can reconstruct amplitude and phase of the objects. The complex amplitude of the object wave field extracted by off-axis holography at the hologram plane is h’(x,y;0) and its FT is a(ξ,ρ;0). The angular spectrum of reconstructed wave at perpendicular distance z, a(ξ,ρ;z) can be calculated from a(ξ,ρ;0) as

a (ξ, ρ, z) = a (ξ, ρ, 0) \times \exp [i \frac{2 π z}{λ} \sqrt{1 - {(λ ξ)}^{2} - {(λ ρ)}^{2}}] .

From these spectra, input complex distribution, b(x’,y’), at the object plane where the sound wave is propagating can be obtained by taking inverse FT at particular value of z.

\begin{array}{l} b (x^{'}, y^{'}) = I F T {a (ξ, ρ, z)} \\ = I F T {a (ξ, ρ, 0) \times \exp [i \frac{2 π z}{λ} \sqrt{1 - {(λ ξ)}^{2} - {(λ ρ)}^{2}}]} . \end{array}

The phase image, p(x’,y’) can be calculated by taking angle of obtained complex function.

p (x^{'}, y^{'}) = a n g l e [b (x^{'}, y^{'})] .

Now, one point from each reconstructed phase at same pixel, r_p = (x_p,y_p), is extracted as a sequence of arrays, t_p(α). These phase values with respect to time is basically optical sound information. Finally, Fourier transform of phase values gives the frequency response (temporal phase distributions), T_p(β) which can determine the frequencies of vibrating objects. It can be explained as follows:

T_{p} (β) = F T [t_{p} (α)] .

From this distribution, the fundamental frequency of a vibrating object can be estimated corresponding to highest amplitude. Inverse Fresnel propagation and frequency measurement procedure has also been explained through block diagram as shown in Fig. 2

Fig. 2 Block diagram for reconstruction and detection of frequency. IFP: Inverse Fresnel propagation, DP: Data processing, and FT: Fourier transform.

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.

3. Experiment results

For the frequency measurement method, an experiment based on off-axis digital holographic setup has been carried out to check the vibration frequencies of tuning fork and loudspeaker diaphragm as the standard objects. In this experiment, a green laser of wavelength 532 nm is used as a light source. Two image sensors with frame rates of 2,000 fps (frames per second) and 36,000 fps have been used to check frequency estimation capability in lower and higher audible ranges. The maximum frequencies to be recorded by the two image sensors are 1,000 Hz and 18,000 Hz, respectively.

3.1 Results for lower audible range

For the image sensor with 2,000 fps to check the lower audible range frequencies, the holograms of size 512 × 512 pixels with a pixel pitch of 16 μm have been recorded. The size of observation field is 8.192 mm × 8.192 mm. The same size of hologram is processed. For the hologram recording in our experiment, the angle between object wave and reference wave is 0.007 radian. We present some results to check fundamental frequencies of tuning fork and loudspeaker. For tuning fork, the frequency of 528 Hz has been used for testing. For loudspeaker membrane, two frequencies of 300 Hz and 500 Hz have also been tested. It is assumed that we do not know the frequencies of vibrating objects to be determined. By using our method, frequency accuracy can be determined by the inverse number of the measurement time length. The frequency interval, Δf, is determined as Δf = 1/NΔt where N is the number of sampling and Δt is the sampling interval. In this experiment, Δf is 2 Hz when N = 1000 and Δt = 0.5 ms (2,000 fps).

Results to determine the frequency of tuning fork of 528 Hz are shown in Fig. 3

Fig. 3 Results to determine the frequencies of tuning fork. (a) One of the recorded hologram for tuning fork of 528 Hz, (b) reconstructed phase image from hologram, (c) plot of phase values against time, and (d) detected frequency at 528 Hz of tuning fork. The image size in Fig. 3(a) and 3(b) is 8.192 mm × 8.192 mm.

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. One of the recorded holograms and reconstructed phase images for 528 Hz tuning fork are shown Figs. 3(a) and 3(b), respectively. Figure 3(c) shows the plot of phase values against time and Fig. 3(d) shows the detected frequency at 528 Hz of tuning fork. The plot of phase vs time is obtained by selecting one point from each reconstructed phase image corresponding to same pixel and plotted against time. And the vibration frequency is detected by following Eq. (7) as discussed in Sec. 2. It should also be noticed that the subtraction of the local average is implemented to remove background phase and the one-dimensional phase unwrapping is used to avoid the phase jump. In Fig. 3(d), second order frequency is observed at 944 Hz. This is the wraparound frequency of 1,056 Hz because the maximum frequency is 1,000 Hz. However, the highest peak is obtained at the 528 Hz.

Results to determine the fundamental frequency of loudspeaker at 300 Hz and 500 Hz are shown in Fig. 4 and 5

Fig. 4 Results to determine the fundamental frequency of loudspeaker diaphragm at 300 Hz. (a) One of the recorded hologram, (b) reconstructed phase image from hologram, (c) plot of phase values against time, and (d) detected frequency at 300 Hz of loudspeaker diaphragm (membrane).

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Fig. 5 Results to determine the fundamental frequency of loudspeaker diaphragm at 500 Hz. (a) One of the recorded hologram, (b) reconstructed phase image from hologram, (c) plot of phase values against time, and (d) detected frequency at 500 Hz of loudspeaker diaphragm (membrane), and (e) detected frequency at 500 Hz with 100 frames.

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, respectively. Loudspeaker is forced to vibrate at some selective frequencies using tone generator software. Figure 4(a) shows the one of recorded hologram for 300 Hz. Figure 4(b) shows the reconstructed phase image from hologram. Figure 4(c) shows the plot of phase values against time. Figure 4(d) shows the detected frequency at 300 Hz of loudspeaker. In Fig. 4(d), there are several peaks. Peaks at 600 Hz and 900 Hz are second and third order frequencies. Peak at 800 Hz is the wraparound frequency of fourth order frequency at 1,200 Hz. Highest peak is obtained at the correct frequency of 300 Hz.

Results to determine the frequencies of loudspeaker of 500 Hz are shown in Fig. 5. One of the recorded hologram and reconstructed phase image for 500 Hz of loudspeaker are shown in Figs. 5(a) and 5(b), respectively. Figure 5(c) shows the plot of phase values against time for 500 Hz. Figure 5(f) shows the detected frequency at 500 Hz of loudspeaker. To check the computational load, we have also performed processing using less number of frames. By using less number of frames in the processing, it can reduce computational load as well as give the possibility of implementing the scheme in real time. For example, processing 100 frames instead of 4,000 frames reduce the time by factor of 1/20. The detected frequency at 500 Hz with 100 frames is shown in Fig. 5(e). To get plots of Figs. 3(c), 4(c), and 5(c), the point of pixel at (244, 186) is selected from respective phase images.

3.2 Results for higher audible range

For the image sensor with 36,000 fps to check the higher audible range frequencies, the holograms of size 384 × 384 pixels with pixel pitch of 20 μm have been recorded. The size of observation field is 7.680 mm × 7.680 mm. A higher frequency of 12,000 Hz has been tested. In this case, Δf is 4 Hz when N = 9000 and Δt = 0.278 ms. Results to estimate frequency of 12,000 Hz, are shown in Fig. 6

Fig. 6 Results to determine the fundamental frequency of loudspeaker diaphragm at 12,000 Hz when point (244, 186) is selected. (a) Plot of phase values against time and (b) detected frequency at 12,000 Hz of loudspeaker diaphragm (membrane).

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. Figure 6(a) shows the plot of phase values against time. To obtain this phase plot, point (244, 186) is selected from reconstructed phase images. Figure 6(b) shows the detected frequency at 12,000 Hz of loudspeaker diaphragm (membrane) which can obtained by taking FT of phase values. Further to check the capability of scheme at different pixel values from phase image, results with average of some points have also been presented. Figure 7(a)

Fig. 7 (a) The detected frequency at 12,000 Hz of loudspeaker diaphragm (membrane) when average of points (10, 210), (184, 126) and (244, 186) have been used and (b) detected frequency at 12,000 Hz of loudspeaker diaphragm (membrane) when average of all points are used.

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shows the detected frequency at 12,000 Hz of loudspeaker diaphragm (membrane) when average of points (10, 210), (184, 126) and (244, 186) have been used. Figure 7(b) shows the detected frequency at 12,000 Hz of loudspeaker diaphragm (membrane) when average of all points are used. From these results, it can be noticed that peak corresponding to detected frequency can be clearly seen irrespective of points in phase images. These results support the detection capability of scheme by selecting any points of spectrum (in phase image). Here, we successfully demonstrated detection of frequencies up to few kHz. However, it is very straightforward to measure higher frequencies of most of the vibrating objects up to few MHz using available very high-speed image sensor [32].

3.3 Sound power evaluation and comparison with microphone

The sensitivity of the proposed scheme is evaluated by changing the amplitude of the audio data to be applied to loudspeaker by multiplying the coefficient from 0 to 1 to the original sinusoidal data. Further, the stability is also analyzed by performing several experiments at different times. To evaluate the strength of scheme, we compared the results with microphone recording which are shown in Fig. 8

Fig. 8 (a) Comparative plot of normalized peak signal as a function of the amplitude coefficient of sinusoidal data at 12,000 Hz for DH and microphone measurements.

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. This figure shows the scatter plot for DH and microphone by using five amplitude coefficients of sinusoidal data at 12,000 Hz applied to a loudspeaker. From these results, it can be confirmed that, there is similar behavior between microphone recording and DH recording for frequency measurement capability.

To check effectiveness and stability of scheme, we have carried out five measurements. Then analysis is done by calculating mean and standard deviation. Figure 9

Fig. 9 Stability measurement. Averages and standard deviations of normalized amplitude of the peak signal obtained by DH as a function of the amplitude coefficient of sinusoidal data at 12,000 Hz.

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shows the plots between amplitude coefficient of sinusoidal data applied to the loudspeaker and the amplitude of peak signal obtained by DH at 12,000 Hz. From this plot, it can be seen that there is linear relationship between normalized peak of measured signal and amplitude coefficient of applied sound data, which support the validity of Eq. (1).

3.4 Discussion

Now we discuss some points regarding implementation and capability of scheme with some comparative study and analysis. It should also be noticed that the vibrating objects should not be placed in such a way to block the object beam completely. It should be placed near to beam in such a way that the vibration propagates perpendicular to beam propagation. To minimize the effect of vibration induced by loudspeaker, the speaker is placed and tied over properly fixed base on optical table with both sided tape. The oscillation of the optical phase is observed at the point where the sound wave passes through an object optical wave. And after the object optical wave up to image sensor position which have information of vibration sound should have same frequency. This has been checked by reconstructing phase images with distances from 0.0001 mm to 250 mm when the vibrating object is placed at distance of 250 mm. All the distances in this range give similar results. Therefore, any distance between 0.0001 mm to 250 mm can be used for frequency measurement.

We have confirmed the detecting capability of our proposed system with some standard vibrating objects up to frequencies of tens kHz. We can safely claim that our system can be used for detecting frequencies of most of the vibrating objects with unknown frequency. From this point of view, most of the scientific and industrial vibrating device can be calibrated with our proposed method. From Fig. 4(d), it can also be seen that the frequencies corresponding to second harmonics are also detected where measured frequency range is below 500 Hz. This is because the speed of the used image sensor is 2,000 fps, which can detect frequencies below 1,000 Hz, and second harmonics will be out of range. The wraparound signal is also detected. Harmonic and wraparound signal, however, are much weaker than the original one.

From these results, it can be confirmed that the frequencies of scientific and other standard objects can be determined very clearly. So, we can claim that the unknown vibration frequencies of most of the vibrating objects can also be determined with high accuracy. In our experiment, we presented two standard vibrating objects namely, tuning fork and loudspeaker. However, some more instruments or objects whose frequencies are to be determined can also be tested straight forward. At present, the available high-speed image sensors have frame rates up to Mega frames per second [32]. So, the frequencies up to half of frame rates of image sensor can be detected with our system. Future scope could also be to detect up to ultrasound frequency by using very high speed and ultrahigh speed image sensor [32] or heterodyne technique. Our scheme may also be suitable for verifying ultrahigh acoustical frequency generators.

4. Conclusions

We have presented the characteristics of indirect vibration sound measurement system based on off-axis DH by evaluating the measurement capability of vibration frequency for different objects and quantitative characteristics of the sound power. The system is implemented using off-axis DH where holograms are recorded with an image sensor as a function of time. Phase image reconstruction after numerical calculation of inverse Fresnel propagation and some acousto-optic data processing techniques, the frequency can be calculated. With this system, it is easy to determine the frequency of the standard and other vibrating objects with high accuracy. Our method is based on imaging of vibration field rather than imaging of vibrating objects. This important point makes the scheme suitable where imaging of vibrating objects is difficult or not possible. And it may be very useful where the external measurement device is difficult or impossible to use. The method is quantitative and has been demonstrated for hundreds Hz to tens of kHz of vibrating objects that covers audible range.

Funding

Hoso Bunka Foundation and JSPS Postdoctoral research fellow grant (17F17369); Science and Engineering Research Board (SERB), Government of India ( SB/OS/PDF-117/2015-16).

Acknowledgment

Authors acknowledge the Science and Engineering Research Board (SERB), Government of India, through award No. SB/OS/PDF-117/2015-16.

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Characteristics of vibration frequency measurement based on sound field imaging by digital holography

Abstract

1. Introduction

2. Frequency measurement system

2.1 Sound field recording by DH

2.2 Hologram reconstruction and processing

3. Experiment results

3.1 Results for lower audible range

3.2 Results for higher audible range

3.3 Sound power evaluation and comparison with microphone

3.4 Discussion

4. Conclusions

Funding

Acknowledgment

References

Cited By

Figures (9)

Equations (7)

OSA Continuum