Quantitative calibration of sound pressure in ultrasonic standing waves using the Schlieren method

Zheng Xu; Hao Chen; Xu Yan; Menglu Qian; Qian Cheng

doi:10.1364/OE.25.020401

1. Introduction

Ultrasound technology is widely used in various scientific fields, such as chemistry [1–4] and biology [5–7], and in industry [8,9]. In recent years, ultrasound standing waves (USW) at high frequency have been used for contactless handling of individual cells [10–13]. To gently manipulate the cell in high-precision, the measurement of ultrasonic power or intensity is important for the development and quality control of the USW device.

Radiation-force balance is commonly used to calibrate pressure distribution in ultrasound waves [14]. In this method, the net radiation pressure, integrated by a deflector or an absorber in the device, is measured. However, it is deficient in measuring ultrasonic fields at low power: the transducer should be carefully handled and the measurement is easily influenced by ambient vibrations. Scanning hydrophones are also widely used to determine the distribution of the pressure amplitude [15]. However, the measured and the real acoustic pressure distribution differ if the wavelength is comparable in size to or smaller than the hydrophone. This is caused by an interface that forms as a result of the different impedance values of the hydrophone needle and the liquid.

Schlieren imaging has been widely used to visualize ultrasound waves; it is based on the phase modulation of light produced by the acousto-optic effect. The Schlieren method has been applied in ultrasound transducer characterization and design [16], visualizing the propagation of ultrasound waves in complex structures [17–19], and the quantitative measurement of ultrasound pressure amplitudes [20–22]. Zanelli and Howard measured the relative sound pressure at the focal zone of a focused ultrasonic field; the value obtained using the Schlieren method agreed well with that obtained using a hydrophone [20]. Miyasaka et al. developed a tomography algorithm to reconstruct the ultrasound field recorded by the Schlieren method; the relative sound pressure in the axial and lateral directions agrees well with the values obtained using a hydrophone [21]. In 2016, Jiang et al. calibrated the sound pressure at the focal spot of a line-focused ultrasonic field by the Schlieren method. The sound pressure was quantitatively determined by measuring the intensity of diffracted light [22]. However, to our knowledge, almost all of these measurements were conducted on travelling waves. This is because the theory developed by Raman and Nath was derived for travelling waves [23]. Therefore, quantitative measurement of the pressure amplitude in standing waves using the Schlieren method should be developed for quality control of the USW device.

In this article, an equation for the intensity of light diffracted by a USW field is derived. Based on the derivation, a quantitative method to determine the sound pressure in a USW field is proposed. Experiments were conducted and the pressure amplitudes in the USW were calibrated using this theory.

2. Experimental apparatus

Schematic diagrams of the experimental setups used for Schlieren imaging and calibration are shown in Figs. 1(a) and 1(b), respectively. A laser beam (532 nm, DPGL-2150F, Photop, China) was expanded as a collimated light beam 75 mm in diameter using lenses L1 and L2, which were incident to the acoustic field. The acoustic field was generated in a transparent rectangular water tank made of quartz glass with dimensions of 120 × 120 × 200 mm³ (width × length × height). The rear surface of the tank was placed at the front focal plane of lens L3 and served as the object plane (OP). To visualize the acoustic field, a circular plate, which blocked zero-order diffraction light, was placed on the transform plane (TP). An intensified charge-coupled device (iCCD) camera (iStar DH734-18U-03, Andor, UK) was placed on the imaging plane (IP) to record the image, as illustrated in Fig. 1(a). To calibrate the acoustic pressure, an aperture was placed close to the rear wall of the tank. The size of the aperture was 6.5 × 1.3 mm² (height × width). The iCCD was placed at the TP, as shown in Fig. 1(b). The short exposure time of 1 μs ensured that the detector did not saturate. Moreover, to ensure accuracy of the results, we used the average value of three measurements.

Fig. 1 Schematic diagram of the experimental apparatus for (a) Schlieren imaging and (b) calibration.

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A planar wave field was generated by a commercial transducer (Panametrics V302 SU, Olympus, Japan). The radius and center frequency of the transducer were 12.7 mm and 1 MHz, respectively. To create the USW, an iron slab was placed at the bottom of the water tank. The distance between the transducer and the slab was 2.1 cm.

3. Theoretical calculation

The plane acoustic wave field propagating along the z axis can be written as

p = p_{s} \cos (ω_{a} t) \cos (k_{a} z)

where ω_a, k_a and p_s indicate the angular frequency, wavenumber and the pressure amplitude of the ultrasound wave, respectively. The complex amplitude of light can be expressed as [24]

U_{0} (z, t) = U_{a} e^{j ω_{r a y} t} e^{- j k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} \cos (ω_{a} t) \cos (k_{a} z + Δ)}

where U_a, ω_ray and k_ray are, respectively, the intensity, angular frequency and wavenumber of the light. The piezo-optic coefficient, mass density and speed of sound in the liquid are expressed as a_p, ρ₀ and c₀, respectively. The acousto-optic interaction length is expressed as L and Δ is the initial phase of the acoustic wave. Since

e^{j z \cos θ} = \sum_{n = - \infty}^{\infty} J_{n} (z) j^{n} e^{j n θ} = \sum_{n = - \infty}^{\infty} J_{n} (z) e^{j \frac{n π}{2}} e^{j n θ},

Equation (2) can be written as

U_{0} (z, t) = U_{a} e^{j ω_{r a y} t} \sum_{n = - \infty}^{\infty} J_{n} [k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} \cos (ω_{a} t)] e^{j \frac{n π}{2}} e^{j n k_{a} z + Δ} .

The complex amplitude, U_f, of the light on the TP can be expressed as

U_{f} (u, t) = U_{a} e^{j ω_{r a y} t} \sum_{n = - \infty}^{\infty} J_{n} [k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} \cos (ω_{a} t)] e^{j n Δ + j \frac{n π}{2}} δ (\frac{n}{λ_{a}} - \frac{u}{λ_{r a y} f_{2}}),

where u is the spatial coordinate in the TP. The wavelength of light and the focal length of lens L₂ are expressed as λ_ray and f₂ ( = 1 m), respectively. The wavelength of acoustic wave is expressed as λ_a and J_n indicates the nth-order Bessel function. The light intensity, I_f, on the TP can be expressed as

I_{f} (u, t) = \frac{U_{a}^{2}}{T} \int_{T} J_{n}^{2} [k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} \cos (ω_{a} t)] δ (\frac{n}{λ_{a}} - \frac{u}{λ_{r a y} f_{2}}) d t,

where T indicates the exposure time of the iCCD. If T is much shorter than the period of the acoustic wave, the sound pressure recorded by the iCCD can be written as

p (t_{0}) = p_{s} \cos (ω_{a} t_{0}) .

In this situation, the light intensity can be expressed as

I_{f} (u, t) = U_{a}^{'}^{2} J_{n}^{2} [k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p (t_{0})] δ (\frac{n}{λ_{a}} - \frac{u}{λ_{r a y} f_{2}}),

where

U_{a}^{'} = - \frac{e^{j k_{r a y} f_{2}}}{λ_{r a y} f_{2}} U_{a}

.

From Eq. (8), it can be seen that the light intensity is related to the sound pressure, p(t₀). To measure the sound pressure, the exposure duration has to be much shorter than the period of the acoustic wave; the real-time sound pressure can be measured by the light intensity. The pressure amplitude can be estimated from the maximum light intensity from multiple measurements. However, many images have to be captured to find the pressure peak. Furthermore, the exposure time must be very short because the sound pressure is averaged during exposure, which causes the low signal-to-noise ratio. To increase the signal-to-noise ratio, many images would have to be averaged, which is time consuming. Therefore, it is difficult to calibrate high-frequency USW using this method. Thus, a method to measure sound pressure for long exposure durations is very much needed.

Assuming s is the detection area of the iCCD, the intensity of the nth-order diffracted light ( $u = n \frac{λ_{r a y} f_{2}}{λ_{a}}$ ) is:

\begin{array}{l} I_{n} = \frac{U_{a}^{'}^{2}}{T} \int_{s} \int_{T} J_{n}^{2} [k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} \cos (ω_{a} t)] d t δ (\frac{n}{λ_{a}} - \frac{u}{λ_{r a y} f_{2}}) d s \\ = \frac{U_{a}^{'}^{2}}{T} \int_{T} J_{n}^{2} [k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} \cos (ω_{a} t)] d t . \end{array}

Since

J_{n}^{2} (z) = \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} (2 n + 2 m)!}{m! (2 n + m)! {[(n + m)!]}^{2}} {(\frac{z}{2})}^{2 n + 2 m},

Equation (9) becomes

I_{n} = \frac{U_{a}^{'}^{2}}{T} \int_{T} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} (2 n + 2 m)!}{m! (2 n + m)! {[(n + m)!]}^{2}} {[\frac{φ \cos (ω_{a} t)}{2}]}^{2 n + 2 m} d t,

where the argument, φ, is

φ = k_{r a y} \frac{a_{p}}{ρ_{0} c_{0}^{2}} L p_{s} .

The integral of the cosine function can be expressed in the following form

\int_{0}^{\frac{π}{2}} \cos^{2 n} (x) d x = \frac{Γ (n + \frac{1}{2}) \sqrt{π}}{2 Γ (n + 1)} .

In Eq. (13), the integral length is a quarter of one wave cycle. Moreover, the integral of cos²ⁿ(x) has the same value in the intervals [0, π/2], [π/2, π], [π, 3π/2] and [3π/2, 2π]. Substituting Eq. (13) into Eq. (11), the intensity of the nth-order diffracted light becomes

I_{n} = \frac{U_{a}^{'}^{2}}{T} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} (2 n + 2 m)!}{m! (2 n + m)! {[(n + m)!]}^{2} ω_{a}} \frac{Γ (n + m + \frac{1}{2}) \sqrt{π}}{Γ (n + m + 1)} {[\frac{φ}{2}]}^{2 n + 2 m} .

Because the exposure time can hardly be exactly one period, it is set to be far longer than one period to minimize the error caused by the cut-off of one period. Assuming the exposure time is $N T_{a} \pm Δ t$ , and because $Δ t$ is much smaller than $N T_{a}$ , the light intensity of the nth-order diffracted light can be expressed as

I_{n} \approx \frac{N U_{a}^{'}^{2}}{2 π} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} (2 n + 2 m)!}{m! (2 n + m)! {[(n + m)!]}^{2}} \frac{Γ (n + m + \frac{1}{2}) \sqrt{π}}{Γ (n + m + 1)} {[\frac{φ}{2}]}^{2 n + 2 m}

and the pressure amplitude of the standing-wave field can be calibrated by the measured intensity of the light fringe.

4. Simulation

Simulations were done to verify the theory. The sound propagation was simulated using finite element method software (COMSOL Multiphysics, COMSOL AB, Stockholm, Sweden); the boundary conditions are shown in Fig. 2(a). The transducer surface was set as the incident pressure field, the value of which changed depending on the experiment conditions. The surface of the iron slab was set as a hard boundary. Because the tank was much bigger than the radius of the transducer, the side boundary condition had little effect on the distribution of the ultrasonic field. Thus, the simulation was carried out in 2D axial symmetry and the right side of the boundary condition was set as plane wave radiation. The calculation area was rectangular (21 × 15 mm² (height × width)). The size of each triangular element used was set to less than the 1/12 of the wavelength in water at 1 MHz. The calculation area consisted of about 54,000 domain elements. A time-dependent solver was used with a time step of 0.1 μs (1/10 of the acoustic period).

Fig. 2 The (a) simulation model and (b–f) wave propagation in liquid at different times.

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5. Results and discussion

Figures 3(a) and 3(b) show, respectively, the experimental results and simulations of the light-fringe intensity distribution on the TP, where the color indicates the light intensity. The simulation results were obtained by the Fourier transformation of the data in the aperture taken from Fig. 2(f) at different ultrasonic pressure amplitudes. The light fringes in the centers of the images are the zero-order diffracted light. The positive and negative first- and second-order diffracted light is located on the upper and lower sides of the zero-order diffraction light, respectively. It is clear that the experimental results are in close agreement with the simulation. It can be seen in Fig. 3 that the intensity of the zero-order diffraction light decreases with the input voltage of the ultrasound transducer. At the same time, the intensity of the first-order and second-order light increases. The change in the intensity of the diffracted light is caused by the increase in the pressure amplitude of the ultrasound wave, which causes the light path in the liquid to increase. This increased light path causes the intensity of the light to shift to higher diffraction orders. It should be noted that the background colors in the experimental results and in the simulation are different, which indicates that the background is not an absolutely dark field in the experiment. In the following discussion, the background light field has been removed numerically.

Fig. 3 The (a) experimental results and (b) simulations of the light-fringe intensity distribution on the TP.

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Figure 4 shows the experimental results and the theoretical simulation of the intensities of the zero-order, first-order and second-order diffracted light with varying input voltage and φ. The experimental results were obtained from the data in Fig. 3(a).

Fig. 4 Experimental results (symbols) and theoretical simulations (curves) of the zero-order, first-order and second-order diffraction light intensity with varying input voltage and argument, φ. I₀, I₁ and I₂ represent the zero-, first- or second-order light intensity, respectively.

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The theoretical results were calculated from Eq. (15). It can be seen that the experimental results and the theoretical simulations have the same tendency. However, it is difficult to calibrate the sound pressure directly from the above results because the light intensity (U_a) is difficult to determine from the data. Furthermore, results indicate that the intensity of each order of the diffracted light is not monotonically increasing or decreasing with input voltage. This may cause one light intensity measurement to correspond to two or more sound pressures. Thus, it is difficult to determine the sound pressure from the individual intensities of the zero-, first- or second-order diffracted light. For the experiments, we repeated each measurement three times and the error bars in the figures represent the standard deviation from the mean. The main source of experimental error can be ascribed to instabilities in the experimental system; namely, in the photocurrent in the iCCD, the ambient temperature, and in the background light field (although all the experiments were conducted in a darkroom, the background was nonzero, as can be observed in Fig. 3(a)). For data processing, we numerically removed the average background light intensity, which introduces uncertainty, especially for low ultrasound input voltages. In all data, the error was less than 2%. However, there was discrepancy between the experimental and simulation results, which we ascribe to the ultrasound transducer, the output of which did not respond proportionally to the input voltage, especially at high input voltages.

To calibrate the sound pressure of the standing waves, the ratios of the light intensity (I₁/I₀ and I₂/I₁) were calculated and are shown in Fig. 5. In this way, the results are only related to φ and the sound pressure can be calculated from it using Eq. (12). It can be seen that, in the range from 0 to 88.8 V, I₁/I₀ increases with input voltage. Thus, the pressure amplitude in this range can be calibrated by I₁/I₀. In the range from 88.8 V to 134 V, the pressure amplitude can be calibrated by I₂/I₁. The error for I₂/I₁ at low input voltages is very large, and decreases with increasing input voltage. This is because I₁ and I₂ both start from zero, as can be observed in Fig. 4, resulting in a large error in their ratio. By increasing the input voltage from 0 to 60 V, both I₁ and I₂ increase and the error decreases. Because the sound pressure in the range of 0 to 88.8 V is calibrated using the ratio I₁/I₀, the large error in I₂/I₁ has no effect on the calibration.

Fig. 5 Experimental results for the ratio of the light intensity (I₁/I₀ and I₂/I₁) with varying input voltage of the ultrasound transducer.

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The calibration results are shown in Fig. 6. The results indicate that the pressure amplitude of the standing-wave field increases approximately linearly with the input voltage of the transducer. For input voltages greater than 70 V, the relationship between the input voltage and the pressure amplitude is not linear. This may be because of the decrease in the intensity of the zero-order light, which causes the noise signal to become more prominent. For input voltages greater than 88.8 V, it can be seen in Fig. 3(a) that the intensities of both the first- and second-order light are low; noise could play a role. The error in the measurements was less than 2%, which is comparable to the error associated with the conventional method. To reduce the noise in future work, we will try to control the background light field in our darkroom and increase the light intensity of the laser.

Fig. 6 The calibration results showing the relationship between the pressure amplitude and the input voltage. The red dashed line is the linear fit of experimental data.

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It should be mentioned that the calibration is based on φ. According to Eq. (12), φ is associated with the wavelength of the light, the acousto-optic interaction length and the piezo-optical coefficient of the liquid. Therefore, the calibration region may be different for different experimental parameters. It should also be mentioned that the higher sound pressure can be calibrated by the ratio of the intensity of the higher-order diffraction light. In future work, we will further investigate how to calibrate the pressure amplitude of non-planar ultrasonic waves.

6. Conclusion

In conclusion, we calculated the light intensity of the diffraction fringes induced by an USW field. Results demonstrated the pressure amplitude in an USW can be calibrated with the Schlieren method. Good agreement was found between the numerical results and the experimental findings, with the error in all measurements being less than 2%, which is comparable to the conventional methods, such as the radiation force balance measurement and hydrophone measurement. In contrast to these methods, our proposed method is applicable for measurement of high-frequency USW fields because the standing wave is not perturbed by an intrusive detector. Therefore, this work may enable the development of a new sound pressure calibration method for USW fields.

Funding Information

The National Natural Science Foundation of China (No. 11404245, 11374231, and 11674249), and the National Key Research and Development Plan (No. 2016YFA0100800 and 2012YQ150213).

References and links

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Quantitative calibration of sound pressure in ultrasonic standing waves using the Schlieren method

Abstract

1. Introduction

2. Experimental apparatus

3. Theoretical calculation

4. Simulation

5. Results and discussion

6. Conclusion

Funding Information

References and links

Cited By

Figures (6)

Equations (15)

Optics Express