Characterization of Sound Fields Generated by Ultrasonic Transducers

Abstract

The metrological characterization of sound fields represents an important step in the design and optimization of ultrasonic transducers. In this chapter, we will concentrate on the so-called light refractive tomography (LRT), which is an optical-based measurement principle. It allows noninvasive, spatially as well as temporally resolved acquisition of both, sound fields in fluids and mechanical waves in optical transparent solids. Before the history and fundamentals (e.g., tomographic reconstruction) of LRT are studied in Sects. 8.2 and 8.3, we will discuss conventional measurement principles (e.g., hydrophones) for such measuring tasks. Section 8.4 addresses the application of LRT for investigating sound fields in water. For instance, the disturbed sound field due to a capsule hydrophone will be quantified. In Sect. 8.5, LRT results for airborne ultrasound are shown and verified through microphone measurements. Finally, LRT will be exploited to quantitatively acquire the propagation of mechanical waves in optically transparent solids, which is currently impossible by means of conventional measurement principles.

The metrological characterization of sound fields represents an important step in the design and optimization of ultrasonic transducers. In this chapter, we will concentrate on the so-called light refractive tomography (LRT), which is an optical-based measurement principle. It allows noninvasive, spatially as well as temporally resolved acquisition of both, sound fields in fluids and mechanical waves in optical transparent solids. Before the history and fundamentals (e.g., tomographic reconstruction) of LRT are studied in Sects. 8.2 and 8.3, we will discuss conventional measurement principles (e.g., hydrophones) for such measuring tasks. Section 8.4 addresses the application of LRT for investigating sound fields in water. For instance, the disturbed sound field due to a capsule hydrophone will be quantified. In Sect. 8.5, LRT results for airborne ultrasound are shown and verified through microphone measurements. Finally, LRT will be exploited to quantitatively acquire the propagation of mechanical waves in optically transparent solids, which is currently impossible by means of conventional measurement principles.

8.1 Conventional Measurement Principles

In this section, let us briefly describe conventional measurement principles for analyzing sound fields in fluids as well as mechanical waves in optically transparent solids. The measurements principles are categorized into five groups: (i) hydrophones, (ii) microphones, (iii) pellicle-based optical interferometry, (iv) Schlieren optical method, and (v) light diffraction tomography. At the end, the different measurement principles are compared regarding important requirements (e.g., spatially resolved results) in practical applications.

8.1.1 Hydrophones

Sound fields in water and water-like liquids are frequently analyzed by means of so-called hydrophones [2, 26]. Since this measurement device has always to be immersed in the liquid, the incident sound waves will be reflected as well as diffracted at the hydrophone body. Therefore, the underlying measurement principle is invasive (see Sect. 8.4.4 and [21]). Hydrophones are usually based on piezoelectric materials such as thin piezoceramics or PVDF  (polyvinylidene fluoride) foils. Piezoceramic materials provide higher coupling factors for converting mechanical into electrical energy, but due to its low acoustic impedance, PVDF is much better suited for water (see Sect. 3.6.3). Consequently, PVDF does not require a $\lambda _{\mathrm {aco}}/4$ layer for matching acoustic impedances of piezoelectric material and water. As a result, PVDF hydrophones feature higher measurement bandwidths than those exploiting piezoceramics and, thus, PVDF hydrophones are more often used in practical applications.

In general, we can distinguish between three different types of piezoelectric hydrophones, namely (i) needle, (ii) capsule, and (iii) membrane hydrophones. As the name already suggests, needle hydrophones have the form of a needle. A piezoelectric material with a typical effective diameter of $\ll 1\,\text {mm}$ is directly located on the needle tip. Depending on the utilized piezoelectric material and without a preamplifier, these hydrophones offer currently nominal sensitivities ranging from 12 to $1200\,\text {nV Pa}^{-1}$ (i.e., $-278\,\text { to }\,{-}238\,\text {dB}$ re $1\,\text {V}\,\upmu \text {Pa}^{-1}$) and provide measurement bandwidths of $1{-}20\,\text {MHz}$ [30]. A larger effective diameter of the piezoelectric material yields a better hydrophone sensitivity but reduces the acceptance angle for incident sound pressure waves. For example, a needle hydrophone with the nominal sensitivity $1200\,\text {nV Pa}^{-1}$ exhibits only an acceptance angle of $15^{\circ }$ at the sound frequency $5\,\text {MHz}$. Besides, a large effective diameter is crucial because sound pressure values are averaged over the hydrophone’s active area. Note that this fact refers to all three types of piezoelectric hydrophones and may pose especially problems in the near field of ultrasonic transducers, where sound fields exhibit high spatial frequencies (cf. Fig. 8.10).

The second type of hydrophones, so-called capsule hydrophones, looks like a projectile and uses PVDF as piezoelectric material (see Fig. 8.1a). The designs as well as specifications of capsule and needle hydrophones are quite similar, but the sensitivity of capsule hydrophones does not depend so strongly on sound frequency. Although its special geometry allows a solid construction even for small PVDF diameters, there occur only minor reflections as well as diffractions of the incident sound waves at the hydrophone body.

Fig. 8.1

Illustration of a capsule hydrophone and b membrane hydrophone [2]

Membrane hydrophones  (see Fig. 8.1b), which represent the last hydrophone type, consist of an acoustically transparent PVDF membrane (thickness of a single PVDF foil ${<}30\,\upmu \text {m}$; diameter $100\,\text {mm}$). Each side of the membrane is covered by electrodes in a manner that a small area at the membrane’s center with a typical diameter ${<}1\,\text {mm}$ gets piezoelectric after poling [3, 50]. In doing so, this area can be used to convert incident sound pressures waves into corresponding electrical output signals. Compared to the other two types, membrane hydrophones provide a larger measurement bandwidth ranging from 1 up to $50\,\text {MHz}$ and more. For this reason, such hydrophones are perfectly suited for undistorted acquisition of pulse-shaped ultrasonic waves in water. Continuous sound waves may, however, lead to standing mechanical waves inside the thin membrane that distort sound field as well as electrical output signals [21]. Membrane hydrophones are also rather sensitive to the angle of incident sound pressure waves, which results in a small acceptance angle of typically ${<}30^{\circ }$. Apart from piezoelectric hydrophones, fiber-optic hydrophones are sometimes utilized for characterizing sound fields in water since they enable sound pressure measurements far above $10\,\text {MPa}$ [45].

8.1.2 Microphones

Devices for acquiring sound fields in air are called microphones. For precise as well as accurate measurements of sound pressure values and levels, one commonly utilizes electrostatic capacitor-based microphones such as condenser and electret microphones [26]. While condenser microphones require an external voltage supply $U_{\mathrm {bias}}$ for polarization, electret microphones exploit a permanently charged material. Depending on the condenser microphone, the values for $U_{\mathrm {bias}}$ lie in the range $20{-}200\,\text {V}$. Figure 8.2a depicts the schematic structure of a condenser microphone. A moveable as well as mechanically prestressed circular membrane that oscillates with the incident sound waves serves as one plate of the capacitor. The membrane is commonly made of nickel or duraluminum of ${\approx }10\,\upmu \text {m}$ thickness. For special measurement applications, a metallized polymer foil of less thickness is used instead. The air gap between membrane and backing electrode equals typically $30\,\upmu \text {m}$. To increase the amplitude of membrane oscillations, the backing electrode oftentimes contains small holes, which increase the air volume within the microphone but barely alter its capacitance $C_{\mathrm {mic}}$. For the air gap $s_0$ in equilibrium state (i.e., without sound pressure wave) and the active area $A_{\mathrm {mic}}$ of the microphone, $C_{\mathrm {mic}}$ becomes

$$\begin{aligned} C_{\mathrm {mic}} \!\left( s_{\sim } \right) = \frac{\varepsilon _0 A_{\mathrm {mic}}}{s_0 + s_{\sim }} \;, \end{aligned}$$

(8.1)

whereby $\varepsilon _0$ stands for the electric permittivity of air and $s_{\sim }$ represents the membrane deflection¹ due to incident sound pressure waves. If the input resistance R of the preamplifier (see Fig. 8.2a) fulfills the condition $R \gg (2\pi f C_{\mathrm {mic}})^{-1}$ (sound frequency f), the electric charges $Q_0$ on the capacitor plates will remain constant. Under additional assumption of small membrane deflections (i.e., $s_{\sim }\ll s_0$), the electric output voltage $u_{\mathrm {mic}} \!\left( s_{\sim } \right)$ of the condenser microphones simplifies to

$$\begin{aligned} u_{\mathrm {mic}} \!\left( s_{\sim } \right) = \frac{Q_0}{C_{\mathrm {mic}}\!\left( s_{\sim } \right) } - U_{\mathrm {bias}} = \frac{Q_0\, \!\left( s_0 + s_{\sim } \right) }{\varepsilon _0 A_{\mathrm {mic}}} - \underbrace{\frac{Q_0 s_0}{\varepsilon _0 A_{\mathrm {mic}}}}_{U_{\mathrm {bias}}} \approx E_0 s_{\sim } \end{aligned}$$

(8.2)

with the (constant) electric field intensity $E_0=Q_0 (\varepsilon _0 A_{\mathrm {mic}})^{-1}$ in the air gap. Therefore, $u_{\mathrm {mic}} \!\left( s_{\sim } \right)$ depends on the incident sound pressure wave.

Fig. 8.2

Illustration of a condenser microphone and b electret microphone; air gap $s=s_0 + s_{\sim }$ [26]

In 1962, Sessler and West [42] invented the so-called electret microphones. Contrary to condenser microphones, the electric field within the air gap results from a permanently charged dielectric foil (electret) that is located between circular membrane and backing electrode. The dielectric foil is a fluoropolymer (e.g., PVDF) of $d_{\mathrm {foil}}=6$ to $25\,\upmu \text {m}$ thickness, which is metallized at one side. By means of corona discharge, the fluoropolymer gets negatively charged at the other side. Figure 8.2b shows an embodiment of an electret microphone for the electret being located at the backing electrode. In the equilibrium state, the electric field intensity $E_0$ in the air gap of thickness $s_0$ computes as

$$\begin{aligned} E_0 = \frac{\sigma _{\mathrm {cor}} \, d_{\mathrm {foil}} }{\varepsilon _0 \!\left( d_{\mathrm {foil}} + \varepsilon _{\mathrm {r}} s_0 \right) }\;. \end{aligned}$$

(8.3)

Here, $\sigma _{\mathrm {cor}}$ expresses the electric surface charge on the electret and $\varepsilon _{\mathrm {r}}$ its relative electric permittivity, respectively. Just as for condenser microphones, the electric output signals of electret microphones due to incident sound pressure waves can be approximated by the simple relation $u_{\mathrm {mic}}\!\left( s_{\sim } \right) \approx E_0 s_{\sim }$. Although such microphones do not need an external voltage supply, their performance (e.g., sensitivity) is comparable or even better than that of conventional condenser microphones. Under normal operating conditions, the permanently charged fluoropolymer almost entirely retains its electric surface charge [41]. The sensitivity of electret microphones to incident sound pressure waves is only reduced by less than $1\,\text {dB}$ per year.

For measurement applications, condenser and electret microphones are commercially available in different sizes of the circular membrane. The membrane diameter commonly ranges from $1/8\,\text {inch}$ to $1\,\text {inch}$. In fact, large membranes provide high microphone sensitivities for incident sound pressure waves, but they are not suitable for the acquisition of ultrasonic waves. Similar to hydrophones, this can be ascribed to the fact that the electric microphone output relates to the averaged deflection of the membrane. When the wavelength $\lambda _{\mathrm {aco}}$ of the sound waves is close to the membrane diameter, averaging over the membrane surface will lead to remarkable deviations between acquired and actually occurring sound pressure values. The membrane size affects, moreover, the acceptance angle for incident sound pressure waves. If the membrane size is large compared to $\lambda _{\mathrm {aco}}$, the acceptance angle will be small. Those are the reasons why one has to carefully select the utilized microphone. While $1\,\text {inch}$ condenser microphones offer currently sensitivities up to $100\,\text {mVPa}^{-1}$ and can be used in the frequency range $10\,\text {Hz}{-}10\,\text {kHz}$, $1/8\,\text {inch}$ versions provide only sensitivities of $1\,\text {mVPa}^{-1}$ [9]. Due to its small size, a $1/8\,\text {inch}$ condenser microphone is, however, suitable for reliable sound pressure measurements up to $140\,\text {kHz}$.

8.1.3 Pellicle-Based Optical Interferometry

The pellicle-based optical interferometry exploits particle displacements that are caused by propagating sound pressure waves. In 1988, Bacon [4] suggested this measurement approach for primary calibration of hydrophones. He utilized a thin gold-coated pellicle of $3\,\upmu \text {m}$ thickness, which was immersed in water. Owing to its low thickness, the pellicle is acoustically transparent but optically opaque. It should, therefore, be able to follow sound pressure waves passing through. If the pellicle movement is acquired by an appropriate device such as a Michelson interferometer (see Fig. 8.3), one can deduce sound pressure values with respect to time. In the presented implementation, the measurement uncertainty varies from 2.3 to $6.6\%$ for the frequency range $0.5{-}15\,\text {MHz}$ of the investigated sound field.

Fig. 8.3

Illustration of pellicle-based optical interferometry [4]; ultrasonic transducer UT

Koch and Molkenstruck [25] enhanced the experimental arrangement by mounting the pellicle on the water surface to extend the upper frequency limit to $70\,\text {MHz}$. Because of its high accuracy, this enhanced pellicle-based optical interferometry has become the standard for primary calibration of hydrophones in several countries, e.g., Germany [24]. Nevertheless, the measurement approach exhibits several limitations. Even though the thin pellicle is nearly nonperturbing, it has to be immersed in the sound propagation medium and, thus, the approach is, strictly speaking, invasive. Moreover, the output of the interferometer strongly depends on the incidence angle of the sound pressure waves on the pellicle. When the sound waves do not impinge perpendicular to the pellicle surface, the determined sound pressure amplitudes seem to be smaller than they actually are. On these grounds, pellicle-based optical interferometry should be only applied for primary calibration of hydrophones.

8.1.4 Schlieren Optical Methods

Schlieren optical methods exploit interactions between electromagnetic waves and acoustic waves in a sound field that is present in an optically transparent medium, i.e., fluid or solid [43, 54]. As illustrated in Fig. 8.4, a collimated electromagnetic wave (e.g., laser beam) propagates through the investigated sound field. Since sound pressure waves cause local variations of the density in the sound propagation medium, its optical refractive index also varies locally. This fact leads to a phase grating for the electromagnetic waves, which, therefore, get diffracted into different orders. In other words, the diffracted electromagnetic waves contain information about the sound field. After passing the sound propagation medium, the electromagnetic waves are focused by a lens. With a view to isolating the high-order diffractions of those waves, the zeroth diffraction order is removed by placing an optical stop as spatial filter at the focal plane of the lens. For instance, the necessary spatial filtering can be realized through a digital micromirror device, which allows applying different filters (e.g., knife-edge or low-pass filter) sequentially [49]. The remaining part (i.e., high-order diffractions) of the electromagnetic waves that contains sound field information is finally captured with an appropriate camera (Fig. 8.4).

Fig. 8.4

Illustration of Schlieren optical method [43]; ultrasonic transducer UT

Compared to hydrophones in water and microphones in air, Schlieren optical methods are noninvasive and provide two-dimensional sound field information in real time. Each pixel in a Schlieren image is, however, proportional to the integration of the acoustic intensity along the path of the corresponding electromagnetic waves [39]. As a result, Schlieren optical methods do not yield spatially resolved sound pressure values even if one applies tomographic imaging in addition. Instead, sound power distributions and normalized sound pressure values are commonly reconstructed (e.g., [28, 53]). To sum up, it can be stated that Schlieren optical methods are excellently suited for visualizing sound fields in optically transparent media, but those methods currently do not deliver absolute values for the sound pressure.

8.1.5 Light Diffraction Tomography

In 1984, Reibold and Molkenstruck [37] presented the so-called light diffraction tomography, which also exploits interactions between electromagnetic waves and acoustic waves. Contrary to Schlieren optical methods, this noninvasive measurement approach provides absolute values for the sound pressure. Figure 8.5 depicts the experimental setup of light diffraction tomography that mainly differs in two points from setups of Schlieren optical methods: (i) The optical stop is replaced by a slit aperture and (ii) the camera is replaced by a combination of pinhole and photodiode. With the aid of an appropriately shaped slit aperture, the zeroth and first positive or negative diffraction orders are only allowed to pass through. Their spatial intensity and phase distributions are acquired by moving pinhole and photodiode in parallel to the xy-plane. By repeating these measurements for different projection angles (e.g., through rotating sound source), tomographic reconstruction leads to spatially resolved information about the sound field. Enhanced experimental setups enable sound pressure measurements up to sound frequencies of $10\,\text {MHz}$ with uncertainties ${<}10\%$ [1]. However, light diffraction tomography does not deliver temporally resolved results and, therefore, can be utilized solely in case of continuous harmonic or standing sound pressure waves [36, 37].

Fig. 8.5

Illustration of light diffraction tomography [1]; ultrasonic transducer UT

8.1.6 Comparison

Finally, let us compare the mentioned measurement principles with regard to important requirements in practical applications (see Table 8.1). Actually, we desire a highly precise measurement approach that is noninvasive and provides temporally as well as spatially resolved absolute values of sound pressure. The utilized approach should not be limited to measurements in fluids but should also enable investigations of mechanical waves in solids. Moreover, since waves may propagate in different directions (e.g., reflections), the measurement approach is desired to be omnidirectional, i.e., equal sensitivity in all directions.

Table 8.1 Comparison of conventional measurement principles for analyzing sound fields in fluids and mechanical waves in solids

Even though each of those conventional measurement principles offers significant benefits, none of them fulfills all requirements in Table 8.1. Consequently, there is a great demand for alternative measurement approaches that can cope with the listed requirements. The remaining part of this chapter is dedicated to light refractive tomography, which represents such an approach.

8.2 History of Light Refractive Tomography

Just as Schlieren optical methods and light diffraction tomography (see Sects. 8.1.4 and 8.1.5), LRT exploits interactions between electromagnetic waves and acoustic waves, i.e., sound waves. Jia et al. [22] firstly measured variations of the optical refractive index in water and in air due to propagating sound waves. They utilized a heterodyne interferometer, whose output signal is directly proportional to the integral of these variations along the emitted laser beam. By assuming plane sound waves, the solution of the underlying integral equation is considerably simplified. In doing so, one can directly relate the interferometer output to sound pressure values. To get rid of this assumption, Matar et al. [27] additionally applied tomographic imaging, which enables spatially resolved reconstruction of the investigated sound field. Harvey and Gachagan [18] replaced the heterodyne interferometer with a commercial single-point laser Doppler vibrometer and, thus, reduced complexity of the experimental setup. Zipser and Franke [55] used a scanning vibrometer instead to lower measuring time. However, they exclusively concentrated on visualization of sound propagation in various practical applications but did not intend quantitative reconstruction of spatially resolved sound pressure. In summary, several researchers developed and utilized LRT for analyzing sound fields in water and in air. Nevertheless, one can barely find quantitative verifications of LRT results through conventional measurement principles such as hydrophones and microphones.

In 2006, Bahr started to research on LRT at the Chair of Sensor Technology (Friedrich-Alexander-University Erlangen-Nuremberg). Together with Lerch, he figured out that the filtered back projection algorithm provides the most reliable results for reconstructing spatially resolved sound fields [5]. They acquired sound fields of rotationally symmetric ultrasonic transducers operating in water and visualized mechanical waves in a PMMA block. Chen et al. (e.g., [12,13,14]) extended both the experimental setup and the reconstruction approach to investigate sound fields of arbitrarily shaped ultrasonic transducers as well as mechanical waves in optically transparent solids. In the following, fundamentals of LRT and important steps toward the extended version will be detailed.

8.3 Fundamentals of Light Refractive Tomography

In this section, the most important fundamentals of LRT will be given. We start with the underlying measurement principle and specify physical quantities (e.g., sound pressure) that can be determined in sound propagation media, i.e., fluids and solids. Section 8.3.2 details tomographic imaging, which allows spatially as well as temporally resolved reconstruction of the physical quantities through LRT measurements. In Sect. 8.3.3, the measurement procedure will be explained. Furthermore, the measurement setup will be presented that was realized at the Chair of Sensor Technology. Afterward, decisive parameters (e.g., number of projections) for LRT measurements are theoretically determined as well as optimized. Section 8.3.5 deals with sources for measurement deviations such as placement errors. Finally, we will discuss the range of sound frequencies, which can be acquired by means of LRT.

8.3.1 Measurement Principle

As stated above, LRT exploits interactions between electromagnetic waves and sound waves. Such interactions arise in each optically transparent medium (e.g., water) through which sound waves propagate [43]. A sound pressure wave causes changes of the density  $\varrho \!\left( x,y,z,t \right)$ in the propagation medium that depend on both space (coordinates x, y and z) and time t. Owing to those changes, the optical refractive index $n\!\left( x,y,z,t \right)$ of the propagation medium also varies with respect to space and time. In a homogeneous medium, this fact leads to the deviation $\varDelta n\!\left( x,y,z,t \right)$ of the optical refractive index from its value $n_0$ in the equilibrium state, i.e., without sound wave. Generally speaking, $n\!\left( x,y,z,t \right)$ will rise when the sound pressure $p_{\sim }\!\left( x,y,z,t \right)$ and, consequently, $\varrho \!\left( x,y,z,t \right)$ increase.

Let us consider a laser beam propagating in y-direction through the sound field of an ultrasound source (see Fig. 8.6). In case of LRT, the laser beam is usually reflected back to the laser source by means of an optical reflector (e.g., [5, 14]). Since $n\!\left( x,y,z,t \right)$ varies along the laser beam due to the sound waves, the optical path length changes. Under the assumption that electromagnetic waves propagate much faster than sound waves (i.e., wave propagation velocity $c_{\mathrm {em}}\gg c_{\mathrm {aco}}$), which is always fulfilled, the optical path difference $\varDelta L$ along the laser beam becomes

$$\begin{aligned} \varDelta L \!\left( x,z,t \right) = 2 \int \varDelta n\!\left( x,y,z,t \right) \mathrm {d}y \;. \end{aligned}$$

(8.4)

Therefore, the optical reflector undergoes the virtual displacement $\varDelta L \!\left( x,z,t \right)$ in y-direction although laser source and reflector exhibit a constant geometric distance. If $\varDelta L \!\left( x,z,t \right)$ is acquired with an appropriate measurement device (e.g., laser Doppler vibrometer) and tomographic imaging (see Sect. 8.3.2) is applied in addition, one can reconstruct spatially as well as temporally resolved refractive index changes $\varDelta n\!\left( x,y,z,t \right)$.

Fig. 8.6

Schematic representation of LRT principle; laser beam of laser sources (e.g., laser Doppler vibrometer) propagates in y-direction through sound field of ultrasound source; laser beam is reflected back to laser source by optical reflector

Sound Pressure in Fluids

The refractive index change $\varDelta n\!\left( x,y,z,t \right)$ can be utilized to calculate the sound pressure $p_{\sim } \!\left( x,y,z,t \right)$ in optically transparent fluids, i.e., gases and liquids. According to the so-called piezo-optic effect  [48], $\varDelta n\!\left( x,y,z,t \right)$ is directly proportional to $p_{\sim } \!\left( x,y,z,t \right)$, which is demonstrated by the relation

$$\begin{aligned} \varDelta n \!\left( x,y,z,t \right) = \!\left( \frac{\partial n}{\partial p} \right) _{\mathrm {\!S}} \cdot p_{\sim } \!\left( x,y,z,t \right) \end{aligned}$$

(8.5)

with the piezo-optic coefficient  $(\partial n / \partial p)_{\mathrm {S}}$ (index S for adiabatic conditions). For the sound pressure amplitudes $\hat{p}_{\sim }$ commonly occurring in acoustic wave propagation, this coefficient remains nearly constant [40, 51]. We are able to reconstruct the spatially as well as temporally resolved sound pressure $p_{\sim } \!\left( x,y,z,t \right)$ with the aid of LRT.

Dilatation in Solids

While solely longitudinal waves propagate in nonviscous fluids, there additionally exist mechanical transverse waves in solid media (see Sect. 2.2). That is why the full description of propagating mechanical waves in solids demands tensor quantities (e.g., mechanical strain $\mathbf {S}$), which cannot be uniquely reconstructed from the scalar quantity refractive index change $\varDelta n\!\left( x,y,z,t \right)$. Nevertheless, we are able to determine density changes $\varDelta \varrho \!\left( x,y,z,t \right)$ in a homogenous optically transparent solid from $\varDelta n\!\left( x,y,z,t \right)$. By assuming constant electric polarizability of atoms or molecules within the medium, Maxwell’s equations yield the so-called Lorentz–Lorenz equation  [7]

$$\begin{aligned} R_{\mathrm {LL}} = \frac{n^2 \!\left( x,y,z,t \right) - 1}{n^2 \!\left( x,y,z,t \right) + 2} \cdot \frac{1}{\varrho \!\left( x,y,z,t \right) } = \mathrm {const.} \end{aligned}$$

(8.6)

with

$$\begin{aligned} \left\{ \begin{array}{llll} n\!\left( x,y,z,t \right) &{} = n_0 &{}+ &{}\varDelta n\!\left( x,y,z,t \right) \\[1ex] \varrho \!\left( x,y,z,t \right) &{}= \varrho _0 &{}+ &{}\varDelta \varrho \!\left( x,y,z,t \right) \end{array} \right. \;. \end{aligned}$$

Here, $n_0$ and $\varrho _0$ stand for optical refractive index and density of the solid in the equilibrium state, respectively. The expression $R_{\mathrm {LL}}$ is the Lorentz–Lorenz specific refraction of the medium changing only by ${<}1\%$ even under extreme variations of temperature and pressure [6]. One may utilize $\varDelta \varrho \!\left( x,y,z,t \right)$ of the solid to additionally calculate its relative volume change $\varDelta V/V_0$, which is referred to as dilatation  $\delta _{\mathrm {dil}}$ [26]. Since propagation of mechanical waves in solids is commonly accompanied by extremely small relative volume changes (i.e., $V_0\gg \varDelta V$), $\delta _{\mathrm {dil}}$ takes the form

$$\begin{aligned} \delta _{\mathrm {dil}} = \frac{\varDelta V}{V_0} \approx \frac{\varDelta V}{V_0 + \varDelta V} = - \frac{\varDelta \varrho }{\varrho _0} \;. \end{aligned}$$

(8.7)

Therefore, $\varDelta n\!\left( x,y,z,t \right)$ leads to the spatially as well as temporally resolved dilatation $\delta _{\mathrm {dil}}\!\left( x,y,z,t \right)$ in an optically transparent isotropic solid (e.g., [11, 12]). Note that only longitudinal waves alter the medium volume and, consequently, the dilatation, which is also shown in (normal strains $S_{ii}$)

$$\begin{aligned} \delta _{\mathrm {dil}} = S_{\mathrm {xx}} + S_{\mathrm {yy}} + S_{\mathrm {zz}} \;. \end{aligned}$$

(8.8)

For this reason, LRT is restricted to the acquisition of mechanical longitudinal waves in optically transparent solids.

8.3.2 Tomographic Imaging

To reconstruct spatially resolved refractive index changes $\varDelta n \!\left( x,y,z,t \right)$ from the optical path differences $\varDelta L\!\left( x,z,t \right)$, tomographic imaging has to be applied. The basis for tomographic imaging is the so-called Radon transform published by Radon in 1917 [35]. He suggested a mathematical formulation enabling reconstruction of a function from its projections. The filtered back projection  (FBP) represents the best-known reconstruction approach in tomographic imaging and is oftentimes utilized in medical examinations as well as nondestructive testing (e.g., [10, 20, 23]). Below, the fundamentals of tomographic imaging are briefly outlined. This includes the Fourier slice theorem and the reconstruction procedure for parallel projections through FBP algorithms.

Fourier Slice Theorem

The Fourier slice theorem is a fundamental principle in tomographic imaging because it links object projections in the spatial domain and distributions in the spatial frequency domain. Figure 8.7 illustrates a graphical interpretation of this theorem. In order to explain the mathematical background, let us introduce the two-dimensional (2-D) object function $f\!\left( x,y \right)$ defined by the Cartesian coordinates xy and its 2-D Fourier transform $F\!\left( u,v \right)$ in the spatial frequency domain

$$\begin{aligned} F\!\left( u,v \right) = \int \limits _{-\infty }^{+\infty } \int \limits _{-\infty }^{+\infty } f\!\left( x,y \right) \mathrm {e}^{-\mathrm {j}2 \pi (ux +vy)} \mathrm {d}x \mathrm {d}y \;. \end{aligned}$$

(8.9)

Fig. 8.7

Graphical representation of Fourier slice theorem for 2-D object function $f\!\left( x,y \right)$ and its 2-D Fourier transform $F\!\left( u,v \right)$; 1-D Fourier transform $\mathfrak {O}_{\varTheta } \!\left( \nu \right)$ of projections $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ in spatial domain is equal to distribution along corresponding radial lines in spatial frequency domain; step size $\varDelta \xi$ in $\xi$-direction

Here, the arguments u and v stand for spatial frequencies, respectively. The projection²  $\mathfrak {o}_{\varTheta } \!\left( x \right)$ of $f\!\left( x,y \right)$ at x along the y-axis results in

$$\begin{aligned} \mathfrak {o} \!\left( x \right) = \int \limits _{-\infty }^{+\infty } f\!\left( x,y \right) \mathrm {d}y\;. \end{aligned}$$

(8.10)

Without limiting the generality, we are able to transform the object function to the coordinate system $(\xi ,\eta )$, which should represent a rotated version of (x, y) with the rotation angle $\varTheta$; i.e., the underlying coordinate transform reads as

$$\begin{aligned} \!\left[ \begin{array}{c} \xi \\ \eta \end{array} \right] = \!\left[ \begin{array}{c@{~~}c} \cos \varTheta &{} \sin \varTheta \\ - \sin \varTheta &{} \cos \varTheta \end{array} \right] \!\left[ \begin{array}{c} x\\ y \end{array} \right] \;. \end{aligned}$$

(8.11)

For the rotated coordinate system, the projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ at $\xi$ along the $\eta$-axis becomes (see Fig. 8.7)

$$\begin{aligned} \mathfrak {o}_{\varTheta } \!\left( \xi \right) = \int \limits _{-\infty }^{+\infty } f\!\left( \xi ,\eta \right) \mathrm {d}\eta \end{aligned}$$

(8.12)

with its one-dimensional (1-D) Fourier transform

$$\begin{aligned} \mathfrak {O}_{\varTheta } \!\left( \nu \right)&= \int \limits _{-\infty }^{+\infty } \mathfrak {o}_{\varTheta }\!\left( \xi \right) \mathrm {e}^{-\mathrm {j}2 \pi \nu \xi } \mathrm {d}\xi \nonumber \\&= \int \limits _{-\infty }^{+\infty } \!\left[ \int \limits _{-\infty }^{+\infty } f\!\left( \xi ,\eta \right) \mathrm {d}\eta \right] \mathrm {e}^{-\mathrm {j}2 \pi \nu \xi } \mathrm {d}\xi \nonumber \\&= \int \limits _{-\infty }^{+\infty } \int \limits _{-\infty }^{+\infty } f\!\left( \xi ,\eta \right) \mathrm {e}^{-\mathrm {j}2 \pi \nu \xi } \mathrm {d}\eta \mathrm {d}\xi \;. \end{aligned}$$

(8.13)

The expression $\nu$ denotes the spatial frequency and is a rotated version of u. Now, one can apply the coordinate transform (8.11) leading to the mathematical relation

$$\begin{aligned} \mathfrak {O}_{\varTheta } \!\left( \nu \right)&= \int \limits _{-\infty }^{+\infty } \int \limits _{-\infty }^{+\infty } f\!\left( x,y \right) \mathrm {e}^{-\mathrm {j}2 \pi (x \nu \cos \varTheta + y \nu \sin \varTheta ) } \mathrm {d}x \mathrm {d}y \nonumber \\&= F \!\left( \nu \cos \varTheta , \nu \sin \varTheta \right) \;. \end{aligned}$$

(8.14)

Because $u=\nu \cos \varTheta$ and $v=\nu \sin \varTheta$, the 1-D Fourier transform $\mathfrak {O}_{\varTheta } \!\left( \nu \right)$ of the projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ at angle $\varTheta$ is equal to the linear intersection of the 2-D Fourier transform $F\!\left( u,v \right)$ at angle $\varTheta$, which results from the object function $f\!\left( x,y \right)$. This fact is commonly named Fourier slice theorem [10, 23].

Reconstruction for Parallel Projections

The general aim of tomographic imaging is to reconstruct object functions $f\!\left( x,y \right)$ from object projections. In LRT measurements, the refractive index change $\varDelta n\!\left( x,y,z,t \right)$ represents the object function and the optical path difference $\varDelta L\!\left( x,z,t \right)$ its projection. Due to the fact that $\varDelta L\!\left( x,z,t \right)$ is acquired solely in y-direction (see Fig. 8.6), let us concentrate hereafter on parallel projections; i.e., $f\!\left( x,y \right)$ should be projected under different angles $\varTheta$ in parallel yielding the projections $\mathfrak {o}_{\varTheta }\!\left( \xi \right)$ (see (8.12)). According to the aforementioned Fourier slice theorem, one can determine the object information $F\!\left( u,v \right)$ in the whole spatial frequency domain if such projections are available for a sufficient number of projection angels $\varTheta$. By performing the 2-D inverse Fourier transform

$$\begin{aligned} f\!\left( x,y \right) = \int \limits _{-\infty }^{+\infty } \int \limits _{-\infty }^{+\infty } F\!\left( u,v \right) \mathrm {e}^{\mathrm {j}2 \pi (ux +vy)} \mathrm {d}u \mathrm {d}v \;, \end{aligned}$$

(8.15)

we finally obtain the desired quantity $f\!\left( x,y \right)$. However, regarding data acquisition in practical applications (e.g., LRT measurements), the object information in the spatial frequency domain is given rather in polar coordinates $(\nu ,\varTheta )$ than in spatial frequencies (u, v). It makes, therefore, sense to rewrite $F\!\left( u,v \right)$ as $F\!\left( \nu ,\varTheta \right) \equiv F\!\left( \nu \cos \varTheta ,\nu \sin \varTheta \right)$ through conducting the substitutions

$$\begin{aligned} u=\nu \cos \varTheta \qquad \text {and} \qquad v=\nu \sin \varTheta \;. \end{aligned}$$

(8.16)

In doing so, (8.15) takes the form

$$\begin{aligned} f\!\left( x,y \right)&= \int \limits _{0}^{2\pi } \int \limits _{0}^{+\infty } F\!\left( \nu ,\varTheta \right) \mathrm {e}^{\mathrm {j}2 \pi \nu (x \cos \varTheta +y \sin \varTheta )} \nu \mathrm {d}\nu \mathrm {d}\varTheta \nonumber \\&= \int \limits _{0}^{\pi } \int \limits _{-\infty }^{+\infty } F\!\left( \nu ,\varTheta \right) \mathrm {e}^{\mathrm {j}2 \pi \nu (x \cos \varTheta +y \sin \varTheta )} \left| \nu \right| \mathrm {d}\nu \mathrm {d}\varTheta \;. \end{aligned}$$

(8.17)

Since $F\!\left( \nu ,\varTheta \right)$ represents the linear intersection of $F\!\left( u,v \right)$ at angle $\varTheta$, $F\!\left( \nu ,\varTheta \right)$ can be replaced by the 1-D Fourier transform $\mathfrak {O}_{\varTheta } \!\left( \nu \right)$ of the projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$, which leads to

$$\begin{aligned} f\!\left( x,y \right) = \int \limits _{0}^{\pi } \mathfrak {T}_{\varTheta }\!\left( x \cos \varTheta + y \sin \varTheta \right) \mathrm {d}\varTheta \end{aligned}$$

(8.18)

with

$$\begin{aligned} \mathfrak {T}_{\varTheta } \!\left( r \right) = \int \limits _{-\infty }^{+\infty } \mathfrak {O}_{\varTheta }\!\left( \nu \right) \left| \nu \right| \mathrm {e}^{\mathrm {j}2 \pi \nu r} \mathrm {d}\nu \;. \end{aligned}$$

(8.19)

As a result, we will be able to reconstruct $f\!\left( x,y \right)$ if $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ and, consequently, $\mathfrak {O}_{\varTheta } \!\left( \nu \right)$ are known. This forms the basis for FBP algorithms [10, 23].

In practical applications of tomographic imaging, spatial and temporal sampling rates are actually limited due to data acquisition and measuring time. One has to cope with a limited number of spatial sampling points in both $\xi$-direction and $\varTheta$-direction for FBP meaning that discrete formulations of (8.18) and (8.19) are indispensable. To explain the discrete formulations, we consider $N_{\mathrm {proj}}$ projections of $f\!\left( x,y \right)$ exhibiting the angular increment $\varDelta \varTheta = \pi / N_{\mathrm {proj}}$. Each projection is assumed to comprise $N_{\mathrm {ray}}$ sampling points in $\xi$-direction with equidistant step size $\varDelta \xi$ (see Fig. 8.7). Under these assumptions, (8.18) gets modified to

$$\begin{aligned} f\!\left( x,y \right) = \frac{\pi }{N_{\mathrm {proj}}} \sum \limits _{i=1}^{N_{\mathrm {proj}}} \mathfrak {T}_{\varTheta _i} \!\left( x \cos \varTheta _i + y \sin \varTheta _i \right) \end{aligned}$$

(8.20)

with

$$\begin{aligned} \varTheta _i = \frac{\!\left( i-1 \right) \pi }{N_{\mathrm {proj}}} \quad \forall ~i=1,\ldots , N_{\mathrm {proj}} \end{aligned}$$

and (8.19) becomes

$$\begin{aligned} \mathfrak {T}_{\varTheta _i} \!\left( m \varDelta \xi \right) =&\varDelta \xi \cdot \mathsf {IDFT} \Big \{ \mathsf {DFT} \big \{ \mathfrak {o}_{\varTheta _i} \!\left( m \varDelta \xi \right) \!\big \} \cdot \mathsf {DFT} \big \{h_{\mathrm {ck}} \!\left( m \varDelta \xi \right) \!\big \} \cdot \text {window} \Big \} \nonumber \\&\forall ~ m = -\frac{N_{\mathrm {ray}}}{2}, \ldots , \frac{N_{\mathrm {ray}}}{2} - 1 \;. \end{aligned}$$

(8.21)

Here, $m\varDelta \xi$ denotes the mth sampling point in $\xi$-direction and $\varTheta _i$ is the angle of the ith projection. The operators $\mathsf {DFT}\!\left\{ \cdot \right\}$ and $\mathsf {IDFT}\!\left\{ \cdot \right\}$ in (8.21) stand for the 1-D discrete Fourier transform and 1-D inverse discrete Fourier transform, respectively. Instead of $\left| \nu \right|$ in (8.19), we use the 1-D discrete Fourier transform of an appropriate convolution kernel $h_{\mathrm {ck}}\!\left( m\varDelta \xi \right)$ that should serve as an additional filter to suppress noise in measurement data. For the implemented LRT setup, the so-called Ram-Lak kernel turned out to be a good choice because it is rather simple and provides excellent reconstruction results (e.g., [5, 11]). The Ram-Lak kernel is mathematically defined as

$$\begin{aligned} h_{\mathrm {ck}} \!\left( m\varDelta \xi \right) = {\left\{ \begin{array}{ll} \!\left( 2\varDelta \xi \right) ^{-2} &{} \quad \mathrm {for} \quad m=0 \\ 0 &{} \quad \mathrm {for} \quad m~\text {is even} \\ -\!\left( m \pi \varDelta \xi \right) ^{-2} &{} \quad \mathrm {for} \quad m~\text {is odd} \end{array}\right. } \;. \end{aligned}$$

(8.22)

Besides, (8.21) contains a window function, which is not necessarily required for reconstruction purpose but can significantly improve the imaging quality.

8.3.3 Measurement Procedure and Realized Setup

According to the previous subsections, one has to project the investigated sound field under different angles $\varTheta$ to reconstruct spatially and temporally resolved field quantities by means of LRT. Basically, there exist two possibilities for this task, namely (i) simultaneous rotation of laser source as well as optical reflector and (ii) rotation of ultrasound source.³ The first possibility is oftentimes applied for tomographic imaging principles in medical examinations and nondestructive testing (e.g., X-ray computed tomography [10]). While the investigated object retains its position and orientation, the measuring components rotate around the object. However, in case of LRT, the simultaneous rotation of laser source and optical reflector constitutes various problems. For example, it imposes high technical demands to precisely rotate both devices around a water tank that is necessary for acquiring sound fields in water. Besides, an additional optical path difference along the laser beam may occur during rotation due to optical refraction at the interfaces of different media (e.g., water tank and water) when the laser beam does not impinge orthogonal to those interfaces. The second possibility (i.e., rotation of ultrasound source) should, thus, be preferred for practical implementation of LRT.

Figure 8.8 illustrates an appropriate measurement procedure for LRT in order to obtain the spatially and temporally resolved refractive index change $\varDelta n\!\left( x,y,z,t \right)$ in a single xy-plane. Note that the procedure will only work if the ultrasound source is periodically excited with the same signal. Taking into account the reconstruction of $\varDelta n\!\left( x,y,z,t \right)$, the entire measurement process consists of three main steps:

• 1. The laser Doppler vibrometer (LDV), which emits a laser beam in y-direction, is moved in parallel to the x-axis with step size $\varDelta \xi$. Hence, the z-distance between ultrasound source and LDV remains constant. The optical reflector being aligned in parallel to the xz-plane reflects the laser beam back to the LDV. At each LDV position, the optical path difference $\varDelta L\!\left( x,z,t \right)$ arising from the sound field is acquired and transferred to an evaluation unit.

• 2. In step 2, the ultrasound source is rotated around an axis parallel to the z-axis by the angle increment $\varDelta \varTheta$. Afterward, step 1 is conducted again, i.e., LDV signals are acquired and transferred at each LDV position. The sequence of step 1 and step 2 is repeated until the angular range $[0,180^{\circ }-\varDelta \varTheta ]$ is completely covered. At this point, it should be mentioned that further angular steps (e.g., $180^{\circ }$ and $180^{\circ }+\varDelta \varTheta$) do not provide extra information because $\varDelta L\!\left( x,z,t \right)$ is independent of the direction in which the laser beam passes the sound field.

• 3. The stored LDV signals represent projections of the sound field under different angles. By combining these signals through FBP, one can finally reconstruct the refractive index change $\varDelta n\!\left( x,y,z,t \right)$ in the investigated xy-plane.

When three-dimensional (3-D) information of the sound field is desired, several xy-planes will be required which means that the z-distance between ultrasound source and LDV needs to be altered. As a matter of course, step 1 to step 3 have to be performed for each xy-plane. In doing so, we are able to analyze sound fields of nearly arbitrarily shaped ultrasound sources in three spatial dimensions with respect to time.

Fig. 8.8

Measurement procedure for LRT to obtain refractive index change $\varDelta n\!\left( x,y,z,t \right)$ in xy-plane (cf. Fig. 8.6); 1: scanning of laser Doppler vibrometer (LDV) along x-direction with step size $\varDelta \xi$; 2: rotation of ultrasound source around its axis (i.e., parallel to z-axis) by angle increment $\varDelta \varTheta$

In Fig. 8.9, one can see the experimental arrangement of LRT that was established at the Chair of Sensor Technology [11]. A differential LDV (Polytec OFV 512 [32]) containing two sensor heads serves as instrument to measure the optical path difference $\varDelta L\!\left( x,z,t \right)$ along the laser beam. One of its fiber-optic sensor heads is mounted on a linear positioning system comprising three translation axes (Physik Instrumente M-531.DG [31]), which enable precise movements in xyz-direction. The other sensor head is mirrored and, therefore, does not contribute to the measurement procedure. To optimally reflect the laser beam back to the LDV, the optical reflector (glass plate coated with chrome) is placed onto an adjustable base. The analog output signal of the LDV is, depending on the applied decoder, either directly proportional to $\varDelta L\!\left( x,z,t \right)$ or to the resulting velocity. With a view to rotating the investigated ultrasonic transducer that represents the ultrasound source, a rotation unit (Physik Instrumente M-037.DG [31]) is connected to a gear via a timing belt. The gear directly rotates a cylindrical mount in which the ultrasonic transducer is fixed.

Fig. 8.9

Realized experimental setup of LRT containing linear positioning system to move differential LDV and rotation unit to rotate ultrasonic transducer, which is fixed in cylindrical mount [11]

A single substep of the LRT setup in Fig. 8.9 including LDV movement, waiting time as well as data acquisition and data transfer takes approximately $0.7\,\text {s}$. Let us assume that one xy-plane requires 5000 substeps. Therewith, the entire measurement procedure (step 1 and step 2) takes approximately one hour. In contrast, the reconstruction of the spatially as well as temporally resolved refractive index change through the FBP algorithm in step 3 takes only a few minutes on a commercial PC.

8.3.4 Decisive Parameters for LRT Measurements

Below, the decisive parameters (e.g., number of projections) for LRT measurements are determined from the theoretical point of view. Subsequently, we will optimize these parameters with regard to a short measuring time as well as reasonable reconstruction results. Finally, an appropriate window function is given which helps to filter images during the reconstruction stage.

Theoretical Determination of Measurement Parameters

For reliable investigations of sound fields by means of LRT, we have to fulfill the Nyquist sampling theorem in time domain and spatial domain [11, 14]. The sampling rate in both domains needs, thus, to be more than twice as fast as the highest frequency components of the signals. In the time domain, this can be simply guaranteed with conventional digital storage oscilloscopes (e.g., Tektronix TDS 3054 [46]). However, since most time in LRT measurements is spent for positioning tasks, let us take a closer look at the spatial domain. This domain is defined by the scanning area of the LDV as well as the number of sampling points $N_{\mathrm {ray}}$ along a single projection and the number of projections $N_{\mathrm {proj}}$.

In order to theoretically determine the scanning area, $N_{\mathrm {ray}}$ and $N_{\mathrm {proj}}$, we consider the sound field of a piston-type ultrasonic transducer in water (sound velocity $c_{\mathrm {aco}}=1480\,\text {m s}^{-1}$). The active circular area of the transducer featuring the radius $R_{\mathrm {T}}=6.35\,\text {mm}$ is assumed to oscillate uniformly at a frequency of $f=1\,\text {MHz}$. The axisymmetric sound pressure field $p_{\sim }\!\left( x,z,t \right)$ was calculated through FE simulations, whereby absorbing boundary conditions suppressed unwanted reflections at the boundaries of the computational domain (see Sect. 4.4). Due to the fact that measurements are always affected by noise, white Gaussian noise was added to the simulated sound pressure field so that the signal-to-noise ratio  (SNR) amounts $30\,\text {dB}$. Such SNR value can be easily reached in practical experiments by performing signal averaging. Figure 8.10a shows the resulting sound pressure distribution $p_{\sim }\!\left( x,z \right)$ normalized to its maximum $\left| p_{\sim }\!\left( x,z \right) \right| _{\mathrm {max}}$ and at an arbitrary instant of time. As can be clearly seen, the sound field in the computational domain concentrates within a small area, whose geometric distance from the rotation axis (i.e., z-axis) corresponds approximately to $R_{\mathrm {T}}$. It seems only natural that in regions of low sound pressure amplitudes, the optical path difference $\varDelta L\!\left( x,z,t \right)$ caused by the propagating acoustic wave is also small. Hence, one can restrict the acquisition of LDV signals to areas, where remarkable sound energy arises. For the considered piston-type ultrasonic transducer, let us take into account $\varDelta L\!\left( x,z,t \right)$ up to a distance of $20\,\text {mm}$ from the z-axis, which is $\approx 3R_{\mathrm {T}}$. Consequently, the LDV has to be moved for each projection along a line of $40\,\text {mm}$ in x-direction representing the scanning area.

Fig. 8.10

a Snapshot of simulated sound pressure field $p_{\sim } \!\left( x,z \right)$ in spatial domain of piston-type transducer (radius $R_{\mathrm {T}}=6.35\,\text {mm}$; excitation frequency $f=1\,\text {MHz}$) located at $z=0\,\text {mm}$; normalized to maximum $\left| p_{\sim }\!\left( x,z \right) \right| _{\mathrm {max}}$; b resulting normalized distribution of each horizontal line from a in spatial frequency domain with spatial frequency $\nu$

In a next step, the necessary number of sampling points $N_{\mathrm {ray}}$ in the previously identified scanning area will be determined. The excitation frequency $f=1\,\text {MHz}$ leads to the wavelength $\lambda _{\mathrm {aco}}=c_{\mathrm {aco}} / f =1.48\,\text {mm}$ and, thus, to the spatial frequency $\nu _{\mathrm {max}}=\lambda _{\mathrm {aco}}^{-1}=676\,\text {m}^{-1}$, which represents the highest spatial frequency that is possible. This can also be observed in Fig. 8.10b depicting the actual distribution of spatial frequencies $\nu$ in x-direction for the simulated sound field. Thereby, each horizontal line denotes the 1-D Fourier transform of the corresponding horizontal line in Fig. 8.10a. According to Nyquist sampling theorem for the spatial domain, the minimum spatial sampling rate $\nu _{\mathrm {samp}}$ in radial direction becomes

$$\begin{aligned} \nu _{\mathrm {samp}} > 2\nu _{\mathrm {max}}=2\lambda _{\mathrm {aco}}^{-1}=1351\,\text {m}^{-1} \end{aligned}$$

(8.23)

meaning that $\varDelta \xi < 0.74\,\text {mm}$ ($\widehat{=} 1/2 \nu _{\mathrm {max}}$) has to be fulfilled for the distance of two neighboring LDV positions. As a result, the scanning area of $40\,\text {mm}$ requires at least $N_{\mathrm {ray}}=55$ sampling points for each projection.

A further decisive parameter in LRT measurements is the number of projections $N_{\mathrm {proj}}$. Just as for the radial direction, the sound field of ultrasonic transducers may also exhibit in tangential direction spatial frequencies up to $\nu _{\mathrm {max}}$. The sampling rate in tangential direction should therefore be equal to the sampling rate in radial direction. Although this constitutes the worst-case scenario that does commonly not occur, we will determine $N_{\mathrm {proj}}$ for such case. Without limiting the generality, the projections are assumed to be evenly distributed with the angular increment $\varDelta \varTheta$, which yields [11, 23]

$$\begin{aligned} \nu _{\mathrm {max}} \varDelta \varTheta = \nu _{\mathrm {max}} \frac{\pi }{N_{\mathrm {proj}}} \;. \end{aligned}$$

(8.24)

Because the maximum spatial frequency $\nu _{\mathrm {max}}$ also influences the number of sampling points $N_{\mathrm {ray}}$ in radial direction, one can deduce from $\varDelta \varTheta \approx 2 / N_{\mathrm {ray}}$ the relation

$$\begin{aligned} N_{\mathrm {proj}} \approx \frac{\pi }{2} N_{\mathrm {ray}}\; , \end{aligned}$$

(8.25)

which links the sampling points in both directions. For the considered sound field of the piston-type transducer, this results in $N_{\mathrm {proj}}=86$ projections.

Optimization of Measurement Parameters

As the previous theoretical determination suggests, a sampling interval of $\varDelta \xi = 0.7 \,\text {mm}$ in radial direction should be sufficient to reconstruct the aimed values in a xy-plane unambiguously. However, in real measurements, $\varDelta \xi$ has to be much smaller than the Nyquist rate. This can be attributed to the following facts:

• Each measurement is contaminated with noise. In order to avoid aliasing effects, $\varDelta \xi$ has, therefore, to be remarkably reduced.

• Within the framework of tomographic reconstruction, an appropriate window function (see (8.21)) is commonly applied for filtering images. Such filters are difficult to be implemented if $\varDelta \xi$ is close to the Nyquist rate.

• Sampling intervals, which should be theoretically sufficient, result in visually rough images. Smoothing can be achieved by means of various interpolation algorithms but may, on the other hand, hide important information in the reconstructions.

For these reasons, the sampling interval $\varDelta \xi$ has to be significantly reduced. A value of $\varDelta \xi =0.2\,\text {mm}$ turned out to be excellent choice in case of the considered sound field since it leads to a good compromise between measuring time and accuracy of measurements [11]. Consequently, the number of sampling points in radial directions increases to $N_{\mathrm {ray}}=201$, which almost quadruples the entire measuring time compared to $N_{\mathrm {ray}}=55$.

Now, let us apply the specified scanning area (i.e., $40\,\text {mm}$) as well as the determined values $N_{\mathrm {ray}}=201$ and $N_{\mathrm {proj}}=86$ to emulate LRT measurements for the modeled ultrasonic transducer. In doing so, the optical path difference $\varDelta L\!\left( x,z,t \right)$ from the sound pressure field in Fig. 8.10a is calculated through (8.4) and (8.5). Figure 8.11a and b show original normalized sound pressure curves with respect to x as well as reconstructed ones for the xy-planes at the axial distances $z=7.5\,\text {mm}$ and $z=25.0\,\text {mm}$, respectively. As the results illustrate, the reconstructed values coincide very well with the original sound pressure curves. The normalized relative deviation $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) of the reconstruction is always smaller than $1.5\%$ in both xy-planes (see Fig. 8.11c and d).

Fig. 8.11

a and b Comparison of original sound pressure distribution $p_{\sim }\!\left( x,z \right)$ and reconstruction with respect to x-position at axial distances $z=7.5\,\text {mm}$ and $z=25.0\,\text {mm}$, respectively (see Fig. 8.10a); normalized to maximum $\left| p_{\sim } \!\left( x,z \right) \right| _{\mathrm {max}}$; c and d normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ between reconstruction results and originals

Even though the selected scanning area and sampling parameters lead to convincing reconstruction results, it is desired to reduce measuring time in LRT measurements. Actually, this will constitute a crucial point especially when many xy-planes of the sound field should be investigated. While a decrease of the scanning area is not recommended, one is able to reduce $N_{\mathrm {ray}}$ as well as $N_{\mathrm {proj}}$. Let us start with the idea behind reducing $N_{\mathrm {ray}}$. Owing to the high concentration of ultrasonic energy nearby the rotation axis of the ultrasonic transducer (see Fig. 8.10a), nonequidistant sampling seems to be a proper way for decreasing $N_{\mathrm {ray}}$ [11, 15]. Regions of high energy demand fine sampling, but we can reduce the sampling rate outside such regions, which means skipping of sampling points. The skipped sampling points have to be filled up through suitable interpolation approaches like cubic spline interpolation [47]. For the considered sound field, the sampling intervals $\varDelta \xi =0.2\,\text {mm}$ for $x\le 6.4\,\text {mm}$ (i.e., ${\approx }R_{\mathrm {T}}$) and $\varDelta \xi = 0.8 \,\text {mm}$ beyond in the region $6.4<x\le 20.0 \,\text {mm}$ are a good choice. Therewith, the number of sampling points $N_{\mathrm {ray}}$ in radial direction decreases from 201 to 99 and, thus, measuring time will be halved.

To compare equidistant sampling and nonequidistant sampling, the simulated sound field was completely reconstructed in the region $x \times z = [0,20\,\text {mm}] \times [0,33\,\text {mm}]$ by individually computing sound pressure curves in each xy-plane. Figure 8.12a and b depict the normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between reconstruction results and original sound pressure field. Although $\left| \epsilon _{\mathrm {r}} \right|$ slightly increases in the peripheral (i.e., $x>6.4\,\text {mm}$) for nonequidistant sampling, the maximum relative deviation for both sampling methods stays below $5\%$. On this account, nonequidistant sampling is a great opportunity to reduce measuring time in LRT measurements, particularly if the main emphasis lies on central areas of sound fields.

Fig. 8.12

Normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ between reconstruction results and original sound pressure field (see Fig. 8.10a) for a equidistant sampling and b nonequidistant sampling in x-direction

Besides the parameter $N_{\mathrm {ray}}$, we may also reduce the number of projections $N_{\mathrm {proj}}$ in LRT measurements. To optimize $N_{\mathrm {proj}}$, let us take a closer at the influence of that number on reconstruction results. For this task, LRT measurements were emulated again on the basis of simulated sound field (see Fig. 8.10a). The tomographic reconstruction was conducted with different amounts of projections $N_{\mathrm {proj}}$ ranging from 5 to 200 projections in steps of 5. Figure 8.13a and b display maximum relative deviations and mean relative deviations as function of $N_{\mathrm {proj}}$ for the entire reconstruction result, i.e., in the region $x\times z = [0,20\,\text {mm}] \times [0,33\,\text {mm}]$. Both deviations drop quickly at the beginning for increasing number of projections, while they remain almost constant for $N_{\mathrm {proj}}\ge 45$. Hence, it is reasonable to choose 50 projections in LRT measurements instead of $N_{\mathrm {proj}}=86$. Apart from that, if one is exclusively interested in the central area of the sound field (here $x<6.4\,\text {mm}$), a much lower number of projections might be sufficient. This follows from the sampling point density, which is always higher close to the xy-plane’s center than in its periphery [10]. For the considered sound field, $N_{\mathrm {proj}}=15$ already yields reconstruction results, whose maximum and mean relative deviations are in the range of $6\%$ and $1\%$ (see Fig. 8.13a and b), respectively. In other words, when rough information is desired only in LRT measurements, we can utilize a rather small number of projections and, therefore, get rid of long measuring times.

Fig. 8.13

a and b Maximum and mean of normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between reconstruction results and originals with respect to number of projections $N_{\mathrm {proj}}$ for entire sound pressure field (see Fig. 8.10a) and within central area (i.e., $x<6.4\,\text {mm}$)

Window Function for Tomographic Imaging

One can suppress noise in tomographic imaging by means of an appropriate window function (see (8.21)). For example, the Hann window is oftentimes used in X-ray computed tomography because the spatial frequency components of the target spread in a large spatial frequency range [23]. However, this window may either damp wanted signals or insufficiently removes noise in LRT measurements. That is why we apply the so-called Turkey window, which combines rectangular and Hann window: Signal components with spatial frequencies $<\nu _{\mathrm {max}}~(\widehat{=} \lambda _{\mathrm {aco}}^{-1})$ are not altered due to the rectangular window, while high-frequency components can be strongly damped through the Hann window [11, 14]. In doing so, we protect signals featuring a reasonable SNR and remove high-frequency noise without raising unwanted ringing effects in the spatial domain. Numerical studies revealed the transition band $[\nu _{\mathrm {max}},3\nu _{\mathrm {max}}]$ connecting passband and stopband as a proper choice for the Turkey window. Indeed, such transition band requires oversampling in the spatial domain; i.e., $\nu _{\mathrm {samp}}>6 \nu _{\mathrm {max}}$ has to be fulfilled. For the considered sound field, the minimum sampling rate results in $\nu _{\mathrm {samp}} > 4054\,\text {m}^{-1}$, which yields the sampling interval $\varDelta \xi < 0.25\,\text {mm}$. Note that the Turkey window was already applied in the previous reconstruction procedures (see Figs. 8.11, 8.12 and 8.13).

8.3.5 Sources for Measurement Deviations

For reconstructing spatially as well as temporally resolved quantities by means of LRT, projections from different angles have to be combined. All projections contribute to the final results. Owing to this fact, the result quality actually depends on the measurement accuracy over the whole cross section. Minor imperfections in the realized LRT setup accumulate and may cause substantial measurement deviations followed by completely distorted images. That is why we need to take care about potential sources for such measurement deviations. Here, two different types of sources are discussed in detail: (i) placement errors arising from misalignments of LRT components and (ii) optical errors originating from the nonideal laser beam of the LDV.

Placement Errors

The reliable reconstruction in LRT measurements demands specific knowledge of the sampling positions along a single projection and the projections angles. Therefore, one should utilize precise linear positioning systems as well as rotation units. Such components do not, however, ensure highly accurate reconstruction results because their geometric alignment is a further decisive point in LRT measurements. In order to study effects of misalignments, let us take a look at the geometric orientation of a LDV scanning plane relative to the cylindrical mount, which contains the ultrasound source. The Cartesian coordinate system xyz belongs to the scanning plane, whereas the front surface of the cylindrical mount represents the origin of the Cartesian coordinate system $x_{\mathrm {c}}y_{\mathrm {c}}z_{\mathrm {c}}$ (see Fig. 8.14). If all components of the LRT setup are perfectly aligned, the z-axes (i.e., symmetry axes) of both coordinate systems will coincide. In practical setups, there always arise deviations that can be understood as geometric uncertainties and cause systematic errors in measurements. For the sake of simplicity, the laser beam of the LDV is here assumed to propagate in y-direction, which can be easily achieved in the realized setup. We then have to consider only three parameters defining relevant deviations of both coordinate systems in LRT measurements. Guided by nautical terms, the three parameters are named (i) sway distance, (ii) yaw angle, and (iii) pitch angle. Under the assumption that the coordinate system xyz of the scanning plane is spatially fixed, they can be interpreted as follows (see Fig. 8.14):

• The sway distance $\varDelta x_{\mathrm {c}}$ stands for the horizontal distance between supposed and actual symmetry axes of the cylindrical mount.

• The yaw angle $\varPhi _{\mathrm {c}}$ indicates the angle around $y_{\mathrm {c}}$ by which the front surface of the cylindrical mount is rotated away from the xy-plane.

• The pitch angle $\varTheta _{\mathrm {c}}$ indicates the angle around $x_{\mathrm {c}}$ by which the front surface of the cylindrical mount is rotated away from the xy-plane.

Fig. 8.14

Illustration of sway distance $\varDelta x_{\mathrm {c}}$, yaw angle $\varPhi _{\mathrm {c}}$, and pitch angle $\varTheta _{\mathrm {c}}$ representing geometric uncertainties in LRT setups [14]; Cartesian coordinate system $x_{\mathrm {c}}y_{\mathrm {c}}z_{\mathrm {c}}$ of cylindrical mount; Cartesian coordinate system xyz of scanning plane

To rate the impacts of these parameters on LRT measurements, the simulated sound pressure field of Fig. 8.10a was used. Depending on the parameter and its value, the original sound field has to be slightly shifted and rotated [14]. On the basis of the modified sound pressure field, LDV signals were emulated in the scanning planes leading to distorted projections, which served as input for the FBP algorithm. The reconstruction results can subsequently be compared to the original sound field. Table 8.2 contains maxima of the normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between reconstruction results and original sound pressure curves at the axial distances $z=3.0\,\text {mm}$, $z=7.5\,\text {mm}$ as well as $z=25.0\,\text {mm}$. The parameters $\varDelta x_{\mathrm {c}}$, $\varPhi _{\mathrm {c}}$, and $\varTheta _{\mathrm {c}}$ were varied separately. As the table entries demonstrate, small sway distances $x_{\mathrm {c}}$ induce large deviations between reconstruction results and original sound pressure curves, especially in the near field of the transducer (i.e., $z=3.0\,\text {mm}$). This is due to the geometric shift of the projections against each other, which have to be combined in the reconstruction stage and, therefore, yield blurred images. Although the maxima of $\left| \epsilon _{\mathrm {r}} \right|$ exhibit for the chosen yaw angles $\varPhi _{\mathrm {c}}$ and pitch angles $\varTheta _{\mathrm {c}}$ the same value range as for $\varDelta x_{\mathrm {c}}$, both parameters are actually not that critical for tomographic reconstruction. Instead of shifting projections against each other, $\varPhi _{\mathrm {c}}$ and $\varTheta _{\mathrm {c}}$ exclusively tilt the cylindrical mount. Consequently, a slightly different cross section of the sound field is projected and reconstructed in LRT measurements. Nevertheless, depending on the sound field, this may also cause high values for $\left| \epsilon _{\mathrm {r}} \right|$ because the projected cross section does not coincide with the supposed one.

Table 8.2 Maxima of normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between reconstruction results and original sound pressure field (see Fig. 8.10a) at three xy-planes with different axial distances from transducer surface; $\varDelta x_{\mathrm {c}}$, $\varPhi _{\mathrm {c}}$ as well as $\varTheta _{\mathrm {c}}$ represent geometric uncertainties in LRT setups

The previous investigations demonstrated that the LRT components need to be precisely adjusted. For the dedicated LRT setup (see Fig. 8.9), the alignment procedure was based on the intensity of the reflected LDV laser beam [11]. When the emitted laser beam is partially blocked, the intensity of the reflected beam will be decreased. In this way, one can detect edges of the cylindrical mount by moving the LDV in the xz-plane. Through corrections of component alignment and intensity measurements, it is possible to guarantee a sway distance $\left| {\varDelta }x_{\mathrm {c}} \right| <0.01\,\text {mm}$, a yaw angle $\left| \varPhi _{\mathrm {c}} \right| <0.02^{\circ }$, and a pitch angle $\left| \varTheta _{\mathrm {c}} \right| <0.40^{\circ }$. The remaining uncertainties in geometric component alignment induce only small deviations between reconstruction results and original sound pressure curves for the considered sound field (cf. Table 8.2).

Optical Errors

Besides placement errors, the nonzero spot size of the laser beam is a crucial point in LRT measurements. The helium–neon laser of the utilized LDV (Polytec OFV 512 [32]) emits a laser beam, whose beam profile is very close to an ideal Gaussian beam  [44]. In order to describe beam properties (e.g., divergence) of the LDV, we can, therefore, apply fundamental relations that are valid for ideal Gaussian beams. For such a Gaussian beam, the spot size $w_{\mathrm {em}}\!\left( \zeta \right)$ at which the beam intensity has decreased to $1/\mathrm {e}^{2}$ times its value at the center becomes (see Fig. 8.15) [34, 48]

$$\begin{aligned} w_{\mathrm {em}}\!\left( \zeta \right) = w_0 \sqrt{1+\!\left( \frac{\zeta }{\zeta _0} \right) ^2} \;. \end{aligned}$$

(8.26)

The expression $\zeta$ stands for the axial position along the laser beam, and $\zeta _0$ is the so-called Rayleigh range that computes as

$$\begin{aligned} \zeta _0 = \frac{w_0^2 \pi }{\lambda _{\mathrm {em}}} \;. \end{aligned}$$

(8.27)

The minimum spot size $w_0$ of the laser beam, which is named beam waist, appears at $\zeta =0$. While a helium–neon laser emits in air laser beams at a wavelength of $\lambda _{\mathrm {em}}=632.8\,\text {nm}$, the wavelength changes to $475.8\,\text {nm}$ in water since its optical refractive index is $n_0=1.33$.

Actually, $w_{\mathrm {em}}\!\left( \zeta \right)$ should be in LRT measurements as small as possible throughout the entire sound field of the ultrasound source. A low divergence of the laser beam is, however, accompanied by a large value of $w_0$ (cf. (8.26)). Hence, one has to find a compromise between beam waist extension and divergence so that the spot size $w_{\mathrm {em}}\!\left( \zeta \right)$ of the laser beam stays sufficiently small in the investigated sound field. For the realized LRT setup, a small beam waist extension of the LDV turned out to be a good choice. As the laser beam propagates through the sound field twice, it makes sense to position the beam waist directly onto the surface of the optical reflector. The ultrasound source should be nearby the reflector in LRT measurements to achieve a tight laser beam in the sound field. However, when the reflector is located too close to the ultrasound source, the sound field will be strongly disturbed by the reflector. It is for this reason rather important to consider the investigated sound field for selecting an appropriate distance between ultrasound source and optical reflector.

Fig. 8.15

Increase of spot size $w_{\mathrm {em}}\!\left( \zeta \right)$ of laser beam along axial distance $\zeta$ from beam waist $w_0$ at $\zeta =0$; Rayleigh range $\zeta _0$

Now, let us evaluate optical errors arising in case of the dedicated LRT setup if we measure the sound field of Fig. 8.10a. In doing so, the spot size $w_{\mathrm {em}}\!\left( \zeta \right)$ of the laser beam and its normalized transverse energy distribution are required. Because the sound pressure almost completely vanishes at $x=30\,\text {mm}$, this distance represents a good choice for the y-spacing of ultrasound source and optical reflector. Related to the symmetry axis of cylindrical mount and, consequently, to the sound field distribution, the distance needs to be considered once more, which yields the range $\zeta =[0,60\,\text {mm}]$ for relevant axial positions along the laser beam. Taking into account the beam waist $w_0=94\,\upmu \text {m}$ of the emitted laser beam as well as its wavelength $\lambda _{\mathrm {em}}=475.8\,\text {nm}$ in water, $w_{\mathrm {em}}\!\left( \zeta \right)$ exhibits the extrema (see (8.26))

$$\begin{aligned} w_{\mathrm {em}}\!\left( \zeta \right) = {\left\{ \begin{array}{ll} 94\,\upmu \text {m} &{} \quad \mathrm {at} \quad \zeta =0\,\text {mm} \\ 135\,\upmu \text {m} &{} \quad \mathrm {at} \quad \zeta =60\,\text {mm} \end{array}\right. } \;. \end{aligned}$$

(8.28)

The normalized transverse distribution $E_{\mathrm {n}}\!\left( x \right)$ of the electric field magnitude for a Gaussian laser beam is given by [19]

$$\begin{aligned} E_{\mathrm {n}}\!\left( x \right) = \mathrm {e}^{-x^2 \, \!\left( 2\sigma _{\mathrm {em}}^2 \right) ^{-1}} \;. \end{aligned}$$

(8.29)

As $w_{\mathrm {em}}\!\left( \zeta \right)$ specifies the distance at which the beam intensity ($\propto E_{\mathrm {n}}^2$) has decreased to $1/\mathrm {e}^{2}$ times its maximum, a single unknown remains in (8.29) that can be determined according to

$$\begin{aligned} E_{\mathrm {n}}^2\!\left( w_{\mathrm {em}} \right)&= \mathrm {e}^{-2w_{\mathrm {em}}^2 \, \!\left( 2\sigma _{\mathrm {em}}^2 \right) ^{-1}} \equiv E_{\mathrm {n}}^2\!\left( 0 \right) \cdot \mathrm {e}^{-2} = \mathrm {e}^{-2} \nonumber \\ \Rightarrow&\quad \sigma _{\mathrm {em}} = \frac{w_{\mathrm {em}}}{\sqrt{2}} \;. \end{aligned}$$

(8.30)

Thus, it is possible to calculate $E_{\mathrm {n}}\!\left( x \right)$ for both spot sizes (8.28) of the laser beam and the normalized spatial Fourier transform $A_{\mathrm {n}}\!\left( \nu \right) \propto \mathcal {F}\!\left\{ E_{\mathrm {n}}\!\left( x \right) \right\}$ (1-D; spatial frequency $\nu$), respectively. The beam waist represents the best case, while $w_{\mathrm {em}}\!\left( \zeta \right)$ at $\zeta =60\,\text {mm}$ is the worst case. The resulting relations for the beam profiles read as⁴

$$\begin{aligned} \left. \begin{array}{lll} E_{\mathrm {n,best}}\!\left( x \right) &{} = &{} \mathrm {e}^{-x^2 \cdot 1.13 \cdot 10^8 \,\text {m}^{-2}} \\[1ex] E_{\mathrm {n,worst}}\!\left( x \right) &{} = &{} \mathrm {e}^{-x^2 \cdot 5.50 \cdot 10^7 \,\text {m}^{-2}} \end{array} \right\} \quad \underrightarrow{\mathcal {F}} \quad \left\{ \begin{array}{lll} A_{\mathrm {n,best}}\!\left( \nu \right) &{} = &{} \mathrm {e}^{-\nu ^2 \cdot 8.72 \cdot 10^{-8} \,\text {m}^{2}} \\[1ex] A_{\mathrm {n,worst}}\!\left( \nu \right) &{} = &{} \mathrm {e}^{-\nu ^2 \cdot 1.79 \cdot 10^{-7} \,\text {m}^{2}} \end{array} \right. \;. \end{aligned}$$

In the spatial frequency domain, LRT measurements can be understood as multiplication of ideal projections $\mathfrak {O}_{\varTheta } \!\left( \nu \right)$ (see (8.13)) with the beam profile $A_{\mathrm {n}}\!\left( \nu \right)$. It becomes clear from Fig. 8.16 depicting $A_{\mathrm {n}}\!\left( \nu \right)$ for the best and worst cases that the nonzero spot size of the laser beam corresponds to a spatial low-pass filter. The larger the spot size $w_{\mathrm {em}}\!\left( \zeta \right)$, the lower will be the spatial cutoff frequency $\nu _{\mathrm {lp}}$ of the low-pass filter and, therefore, ideal projections will be more strongly affected. For the considered sound field and realized setup, the maximum spatial frequency $\nu _{\mathrm {max}}=676\,\text {m}^{-1}$ is attenuated by ${\approx }8\%$ (worst case) in LRT measurements.

Fig. 8.16

Magnitude of Fourier transform $A_{\mathrm {n}}\!\left( \nu \right)$ in spatial frequency domain; best case $w_{\mathrm {em}}=94\,\upmu \text {m}$ and worst case $w_{\mathrm {em}}=135\,\upmu \text {m}$ refer to LDV beam profiles for realized LRT setup; hydrophone ($R_{\mathrm {HY}}=0.2\,\text {mm}$) is assumed to be uniformly sensitive across its active surface

At this point, it should be mentioned that one is concerned with similar effects in microphone and hydrophone measurements. The capsule hydrophone Onda HGL-400, which is used for comparative measurements in Sect. 8.4, features the radius $R_{\mathrm {HY}}=0.2\,\text {mm}$. Under the assumption of uniform sensitivity across the hydrophone’s active surface, the normalized sensitivity $a_{\mathrm {HY}}\!\left( x \right)$ in the spatial domain takes the form

$$\begin{aligned} a_{\mathrm {HY}}\!\left( x \right) = {\left\{ \begin{array}{ll} 1 &{} \quad \mathrm {for} \quad \left| x \right| \le R_{\mathrm {HY}} \\ 0 &{} \quad \mathrm {elsewhere} \end{array}\right. } \end{aligned}$$

(8.31)

yielding the normalized Fourier transform [8]

$$\begin{aligned} A_{\mathrm {n}}\!\left( \nu \right) = \frac{\sin \!\left( 2 \pi \nu R_{\mathrm {HY}} \right) }{2 \pi \nu R_{\mathrm {HY}}} = \text {sinc}\!\left( 2 \pi \nu R_{\mathrm {HY}} \right) = \text {sinc}\!\left( \nu \cdot 1.26 \cdot 10^{-3} \,\text {m} \right) \end{aligned}$$

(8.32)

in the spatial frequency domain. Figure 8.16 reveals that high-frequency components are more attenuated in hydrophone measurements due to averaging over the active surface than in LRT measurements, even for the worst case of the laser beam profile. In the far field of ultrasound sources, spatial frequencies in radial direction are, however, much lower than in propagation direction, i.e., axial direction. That is the reason why we are able to characterize also high-frequency sound fields through hydrophones, e.g., up to $20\,\text {MHz}$ (i.e., $\nu _{\mathrm {max}}=1.35\cdot 10^4\,\text {m}^{-1}$) with the capsule hydrophone Onda HGL-400. Despite this fact, measurement data provided by typical hydrophones in transducer near fields is only useful to a limited extent.

8.3.6 Measurable Sound Frequency Range

Similar to conventional measurement principles (e.g., hydrophones) for sound field analysis, LRT has certain limits concerning measurable sound frequencies. The maximum measurable sound frequency $f_{\mathrm {max}}$ is mainly determined by the laser beam profile, whereas the spatial extension of the optical reflector defines the minimum measurable sound frequency $f_{\mathrm {min}}$. Below, a theoretical analysis for both frequency limits is conducted.

Maximum Measurable Sound Frequency

As discussed in Sect. 8.3.5, the nonzero spatial extension of the LDV laser beam leads to a low-pass filter in LRT measurements. When its spatial cutoff frequency $\nu _{\mathrm {lp}}$ is below decisive frequency components of the investigated sound field, those components will be attenuated and the measured sound pressure amplitude seems then to be decreased. On account of this fact, the spot size $w_{\mathrm {em}}\!\left( \zeta \right)$ of the laser beam throughout the sound field determines the maximum measurable frequency $f_{\mathrm {max}}$ of sound. In the case that an attenuation of $3\,\text {dB}$ is acceptable in LRT measurements, $f_{\mathrm {max}}$ directly relates to $\nu _{\mathrm {lp}}$. For the dedicated LRT setup and an assumed radial field extension of $30\,\text {mm}$, the largest laser spot size $w_{\mathrm {em}}\!\left( \zeta =60\,\text {mm} \right) =135\,\upmu \text {m}$ (worst case in Fig. 8.16) yields

$$\begin{aligned} f_{\mathrm {max}} = c_{\mathrm {aco}} \cdot \nu _{\mathrm {lp}} = 1480\,\text {m s}^{-1} \cdot 1390\,\text {m}^{-1} = 2.1\,\text {MHz} \end{aligned}$$

(8.33)

representing the maximum measurable frequency in water. To increase $f_{\mathrm {max}}$, one may think about reducing $w_{\mathrm {em}}\!\left( \zeta \right)$ throughout the sound field by raising the frequency of the laser light, which means reducing its wavelength $\lambda _{\mathrm {em}}$. Major changes of $\lambda _{\mathrm {em}}$ are, however, hardly possible since commercial laser Doppler vibrometers (e.g., from the company Polytec GmbH [32]) usually work at fixed wavelengths within the visible range. Nevertheless, if an ultrasound source emits a strongly focused field, we may reliably analyze higher frequencies of sound with LRT. In such situation, the z-spacing between ultrasound source and optical reflector can be reduced. Owing to the small radial extension of the sound field, the beam waist $w_0$ can be decreased by increasing beam divergence. Anyway, LRT should not be used for investigations of sound fields in water with frequencies $f_{\mathrm {max}}>5\,\text {MHz}$.

Minimum Measurable Sound Frequency

In tomographic imaging, it is of utmost importance to acquire projections of the whole object under test. The object has, thus, to exhibit spatial extensions, which can be completely covered by the projections. Due to the fact that there do not exist clear boundaries for sound fields, an alternative criterion is needed for determining the scanning area (i.e., in x-direction) in LRT measurements. The directivity pattern of sound fields contains a dominant main lobe and several side lobes. With a view to achieving reliable LRT results, it must be ensured that at least the main lobe is entirely inside the scanning area [11]. Consequently, we are able to estimate for an ultrasound source the minimum scanning area, which should be exceeded in LRT measurements.

Let us take a look at the directivity pattern of a piston-type ultrasonic transducer with the radius $R_{\mathrm {T}}$. In case of a harmonically excitation, the sound pressure amplitude $\hat{p}_{\sim }\!\left( r,\theta \right)$ at an arbitrary position in the far field is proportional to (wave number k; cf. (7.49, p. 280))

$$\begin{aligned} \hat{p}_{\sim }\!\left( r,\theta \right) \propto \frac{J_1 \!\left( k R_{\mathrm {T}} \sin \theta \right) }{k R_{\mathrm {T}} \sin \theta }\;. \end{aligned}$$

(8.34)

Here, r stands for the geometric distance from transducer center and $\varTheta$ is the angle between connecting line and z-axis (see Fig. 8.17). $J_1 \!\left( \cdot \right)$ represents the Bessel function of the first kind and order 1. According to (8.34), $\hat{p}_{\sim }\!\left( r,\theta \right)$ will be zero if

$$\begin{aligned} \frac{J_1 \!\left( k R_{\mathrm {T}} \sin \theta \right) }{k R_{\mathrm {T}} \sin \theta } = 0 \;, \end{aligned}$$

(8.35)

which is fulfilled for the first time at $k R_{\mathrm {T}} \sin \theta = 3.83\,\text {rad}$. Because this zero point defines the main lobe, one can calculate its diameter $D_{\mathrm {main}}$ at a given z-position in the far field through

$$\begin{aligned} D_{\mathrm {main}} \!\left( z \right) = 2 z \tan \varphi = 2z \tan \!\left[ \arcsin \!\left( \frac{3.83\,\text {rad}}{k R_{\mathrm {T}}} \right) \right] \;. \end{aligned}$$

(8.36)

Fig. 8.17

Geometric quantities for piston-type transducer (radius $R_{\mathrm {T}}$) to compute sound pressure amplitude $\hat{p}_{\sim }\!\left( r,\theta \right)$ at arbitrary position given by distance r and angle $\theta$; $r \sin \theta = z \tan \theta$

The scanning area in LRT measurements should cover at least the spatial extension of the main lobe, i.e., $D_{\mathrm {main}}$. Note that this fact does not only refer to the linear positioning system but also to the optical reflector, which features an edge length of $l_{\mathrm {opt}}=100\,\text {mm}$ in the realized LRT setup. The condition $D_{\mathrm {main}}\le l_{\mathrm {opt}}$ and (8.36) lead to the minimum wave number  $k_{\mathrm {min}}$, which can be just measured by means of LRT. For instance, at an axial distance $z=50\,\text {mm}$ of a harmonically excited ultrasonic transducer (piston-type; radius $R_{\mathrm {T}}=6.35\,\text {mm}$) that operates in water, one obtains $k_{\mathrm {min}}=853\,\text {rad m}^{-1}$. The minimum sound frequency $f_{\mathrm {min}}$ then results from

$$\begin{aligned} f_{\mathrm {min}} = \frac{k_{\mathrm {min}} c_{\mathrm {aco}}}{2\pi } = \frac{853\,\text {rad m}^{-1} \cdot 1480\,\text {m s}^{-1}}{2\pi } = 201\,\text {kHz}\;. \end{aligned}$$

(8.37)

When lower sound frequencies should be acquired, the reflector position would need to be changed during measurements or, alternatively, a larger optical reflector has to be used.

Additionally, there exists another frequency limitation that exclusively refers to the ultrasound source. The $\arcsin \!\left( \cdot \right)$ function in (8.36) demands an argument fulfilling $kR_{\mathrm {T}}\ge 3.83\,\text {rad}$. However, if this requirement is not fulfilled, the ultrasound source will not generate a sound field with pronounced side lobes. Instead, a wide main lobe will be emitted whose spatial extension is, strictly speaking, not limited. It is, therefore, complicated to analyze such sound fields through LRT. For the considered piston-type ultrasonic transducer operating in water, the critical wave number becomes $k=603\,\text {rad m}^{-1}$ yielding the minimum sound frequency $f_{\mathrm {min}}=142\,\text {kHz}$ (see (8.37)) that can be measured in a reliable way.

8.4 Sound Fields in Water

In Sect. 8.3, we have studied the fundamentals of LRT including measurement principle, tomographic reconstruction, realized experimental setup as well as choosing decisive measurement parameters. Now, these fundamentals are applied to actual LRT measurements of sound fields in water, which is the most common propagation medium for ultrasonic waves. The piezo-optic coefficient  $(\partial n / \partial p)_{\mathrm {S}}$ is assumed to be (e.g., [11, 15])

$$\begin{aligned} \!\left( \frac{\partial n}{\partial p} \right) _{\mathrm {\!S}} = 1.473 \cdot 10^{-10} \,\text {Pa}^{-1} \end{aligned}$$

(8.38)

for the wavelength $\lambda _{\mathrm {em}}=475.8\,\text {nm}$ of electromagnetic waves (i.e., laser beam of LDV) in water and the water temperature $20\,^{\circ }\text {C}$.

Firstly, the sound pressure field of a piston-type ultrasonic transducer is analyzed in selected cross sections (see Sect. 8.4.1). The reconstructed sound pressure amplitudes will be compared to results of conventional hydrophone measurements. Afterward, we perform the same comparison for a cylindrically focused ultrasonic transducer. In Sect. 8.4.3, acceleration of the time-consuming measurement procedure is discussed. Finally, the disturbance of the sound field due to an immersed hydrophone will be quantified by means of LRT, which is hardly possible with other measurement approaches.

8.4.1 Piston-Type Ultrasonic Transducer

At the beginning, let us investigate the sound field of a piston-type ultrasonic transducer (Olympus V306-SU [29]) that was immersed in water. The ultrasonic transducer with a radius of $R_{\mathrm {T}}=6.35\,\text {mm}$ was excited by a sinusoidal burst signal of 40 cycles at $f=1\,\text {MHz}$ and fixed in the cylindrical mount of the realized LRT setup (see Fig. 8.9). For the chosen transducer excitation, the near-field length equals $N_{\mathrm {near}}=27\,\text {mm}$. Several cross sections (i.e., in parallel to the xy-plane) of the arising sound field were acquired at different axial distances z, whereby $z=0\,\text {mm}$ relates to the transducer front [14]. Figure 8.18 depicts the averaged output signal of the differential LDV at $(x,z)=(0,32.8\,\text {mm})$, which represents data at a single point of a projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ with respect to time t. By means of averaging 16 output signals directly within the utilized digital storage oscilloscopes Tektronix TDS 3054, the SNR exceeds $30\,\text {dB}$ [11].

Fig. 8.18

Averaged output (mean of 16 signals) of LDV at position $(x,z)=(0,32.8\,\text {mm})$ representing single point of projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ for tomographic reconstruction; resulting $\text {SNR}\approx 34\,\text {dB}$; piston-type ultrasonic transducer Olympus V306-SU; excitation frequency $f=1\,\text {MHz}$

In Sect. 8.3.4, we determined decisive parameters for LRT measurements from the theoretical point of view. Thereby, a piston-type transducer was considered featuring the same radius $R_{\mathrm {T}}$ and excitation frequency as the investigated one. Here, exactly those parameters were applied to acquire sound pressure fields $p_{\sim }\!\left( x,y,t \right)$ in the cross sections $z=32.8\,\text {mm}$ (far field) as well as $z=8.4\,\text {mm}$ (near field). This refers to the scanning area but also to the number of projections $N_{\mathrm {proj}}$ and nonequidistant sampling between two neighboring LDV positions. Table 8.3 summarizes the most important parameters for the conducted LRT measurements. After recording the LDV output signals at the selected scanning positions, tomographic reconstruction was performed for each instant of time individually.

Fig. 8.19

a and c LRT measurements for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in cross sections at $z=32.8\,\text {mm}$ and $z=8.4\,\text {mm}$, respectively; b and d corresponding hydrophone (HYD) measurements; piston-type ultrasonic transducer Olympus V306-SU; excitation frequency $f=1\,\text {MHz}$

Table 8.3 Decisive parameters for LRT measurement of sound field arising from piston-type ultrasonic transducer

Figure 8.19a and c show the reconstructed sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in the cross sections⁵ at $z=32.8\,\text {mm}$ and $z=8.4\,\text {mm}$, respectively. As expected from piston-type ultrasonic transducers, the sound pressure amplitudes are rotationally symmetric distributed in a cross section. Compared to the near field at $z=8.4\,\text {mm}$, $\hat{p}_{\sim }\!\left( x,y \right)$ decreases in the far field at $z=32.8\,\text {mm}$, which coincides with the theory of sound propagation.

Fig. 8.20

a and b Comparison of LRT and hydrophone (HYD) measurements for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ along x-axis in cross sections at $z=32.8\,\text {mm}$ and $z=8.4\,\text {mm}$, respectively; c and d relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) along x-axis between LRT and hydrophone measurements normalized to maximum of hydrophone output; piston-type ultrasonic transducer Olympus V306-SU

To verify the LRT results, hydrophone measurements were additionally carried out for both cross sections. Guided by the spatial resolution in LRT measurements, the utilized capsule hydrophone (Onda HGL-0400 [30]) was moved with the step size $0.2\,\text {mm}$ in x- and y-direction. Because of the fine spatial resolution and a waiting time of $2\,\text {s}$ that is required for mechanical vibrations to settle down after each hydrophone movement, such measurement takes approximately $12\,\text {h}$. Figure 8.19b and d display the obtained results $\hat{p}_{\sim }\!\left( x,y \right)$ from hydrophone measurements. As the comparison clearly reveals, the results coincide very well with the corresponding LRT measurements. This also becomes obvious if we take a look at the acquired sound pressure amplitudes along the x-axis (see Fig. 8.20a and b) and the normalized relative deviations $\epsilon _{\mathrm {r}}$ (see Fig. 8.20c and d) between the different measurement approaches. While $\left| \epsilon _{\mathrm {r}} \right|$ is always smaller than $5.4\%$ in the far field (i.e., $z=32.8\,\text {mm}$), there occur, however, relative deviations up to $12.3\%$ in the near field (i.e., $z=8.4\,\text {mm}$). The pronounced deviations between LRT and hydrophone measurements in the near field can be mainly ascribed to three points. Firstly, the hydrophone is calibrated for ultrasound measurements in the far field, where the spatial frequencies $\nu$ in x- and y-direction are much smaller than in the near field (cf. Fig. 8.10b). As a second point, one has to keep in mind that a perfect alignment of the system components does not avoid spatial deviations of the coordinate systems for LRT and hydrophone measurements. Therefore, we always compare slightly different cross sections, which can be especially crucial in the near field of an ultrasonic transducer since sound fields are subject to strong fluctuations there. Lastly, hydrophone measurements are invasive and, thus, affect the analyzed sound field due to reflections of propagating sound waves, which is again critical in the near field. Nevertheless, the comparison of hydrophone and LRT measurements definitely proves that LRT is a reliable approach for investigating rotationally symmetric sound fields in water.

8.4.2 Cylindrically Focused Ultrasonic Transducer

LRT measurements usually concentrate on the inspection of rotationally symmetric sound fields (e.g., [5]). The dedicated LRT setup enables, however, the investigation of sound fields arising from nearly arbitrarily shaped sound sources. With a view to demonstrating this fact, a cylindrically focused ultrasonic transducer (Olympus V306-SU-CF1.00N [29]) exhibiting the focal length $25.4\,\text {mm}$ was utilized to generate nonaxisymmetric sound fields in water [14]. Again, the transducer was excited by a sinusoidal burst of 40 cycles at $f=1\,\text {MHz}$. In contrast to the previous investigations, fairly conservative measurement parameters were chosen here. The radial scanning area was extended from 40 to $60\,\text {mm}$ (i.e., $x\in [-30,30]\,\text {mm}$), and nonequidistant sampling between two neighboring LDV positions was abandoned. Consequently, the measuring time for a single cross section increases from $2\,\text {h}$ to $6\,\text {h}$. Table 8.4 summarizes the most important parameters for the conducted LRT measurements.

Table 8.4 Decisive parameters for LRT measurement of sound field arising from cylindrically focused ultrasonic transducer

Figure 8.21a depicts the reconstructed sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in the cross section at $z=25.4\,\text {mm}$ representing the focal plane of the investigated cylindrically focused transducer. While the sound field is strongly focused in one direction, there hardly occurs focusing perpendicular to it, which is typical for such transducer shape.

Fig. 8.21

a LRT measurement for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in cross section at $z=25.4\,\text {mm}$; b corresponding hydrophone (HYD) measurement; cylindrically focused ultrasonic transducer Olympus V306-SU-CF1.00N; excitation frequency $f=1\,\text {MHz}$

Fig. 8.22

a and b Comparison of LRT and hydrophone (HYD) measurements for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ along x-axis and along y-axis in cross section at $z=25.4\,\text {mm}$, respectively; c and d relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between LRT and hydrophone measurements normalized to maximum of hydrophone output; cylindrically focused ultrasonic transducer Olympus V306-SU-CF1.00N

To verify the LRT results, hydrophone measurements were carried out in the same cross section (see Fig. 8.21b), whereby the hydrophone was moved with the step size $0.2\,\text {mm}$ in x- and y-direction. Both images show a fairly good agreement and have many fine details in common. This fact is also demonstrated in the sound pressure amplitudes along the x-axis (see Fig. 8.22a) and y-axis (see Fig. 8.22b), respectively. As displayed in Fig. 8.22c and d, the normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between LRT and hydrophone measurements along these axes are always smaller than $3.4\%$. Hence, we can state that in addition to rotationally symmetric sound fields, the presented LRT measurement approach is applicable for precise measurements of nonaxisymmetric sound fields in water.

8.4.3 Acceleration of Measurement Process

So far, the sound fields of piston-type and cylindrically focused ultrasonic transducers were projected under 100 angles per cross section to reconstruct field information (e.g., sound pressure amplitudes) through LRT measurements. The procedure comprising measurement and reconstruction takes a few hours for a single cross section. When sound propagation should be investigated with respect to space and time, we would need sound information in numerous cross sections [14]. It is, therefore, of utmost importance to accelerate LRT measurements. In the following, we will discuss two possibilities for acceleration: (i) reducing the number of projections and (ii) assumption-based reconstruction for rotationally symmetric ultrasonic transducers.

Reduction of the Number of Projections

A proper way to lower measuring time in LRT is reducing the number of utilized projections $N_{\mathrm {proj}}$. The theoretical investigations in Sect. 8.3.4 indicated that $N_{\mathrm {proj}}=50$ should be sufficient to obtain reliable reconstruction results. Let us verify this value by means of LRT measurements for the piston-type ultrasonic transducer Olympus V306-SU. As an example, the sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in the cross section at $z=32.8\,\text {mm}$ are considered. Figure 8.19a depicts the reconstruction result for $N_{\mathrm {proj}}=100$ acquired projections. Here, this value is reduced by picking out every Mth projection and ignoring the remaining ones for tomographic reconstruction. M was selected to be [2, 4, 5, 10, 20, 50] yielding $N_{\mathrm {proj}}\in [50,25,20,10,5,2]$ projections. In doing so, the reconstruction results for reduced numbers of projections can be compared to the full reconstruction, which follows from $N_{\mathrm {pro}}=100$.

Figure 8.23a and b illustrate maximum relative deviations and mean relative deviations from full reconstruction as functions of $N_{\mathrm {proj}}$. Both deviations decrease with rising amount of projections for the considered cross section, i.e., $x\times y = [-10\,\text {mm},10\,\text {mm}]\times [-10\,\text {mm},10\,\text {mm}]$. The same holds in the central area of the sound field, which is here defined as $\left| x,y \right| <6.4\,\text {mm}$. However, since the deviations are smaller in the central area, reconstruction results in the periphery (i.e., $\left| x,y \right| \ge 6.4\,\text {mm}$) of the sound field suffer more from reducing $N_{\mathrm {proj}}$. In accordance with the theoretical investigations, $N_{\mathrm {proj}}=50$ leads in any case to precise LRT results and, therefore, constitutes a good compromise between measuring time and measurement accuracy. But, if solely information from the central area is desired, we will be able to additionally reduce the number of projections, e.g., $N_{\mathrm {proj}}=20$ or even less.

Fig. 8.23

a and b Maximum and mean of normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between reconstruction results and full reconstruction ($N_{\mathrm {proj}}=100$; see Fig. 8.19a) versus number of projections $N_{\mathrm {proj}}$ for entire sound pressure field (i.e., $\left| x,y \right| \le 10 \,\text {mm}$) and within central area (i.e., $\left| x,y \right| <6.4\,\text {mm}$), respectively

Axisymmetric Assumption

The realized LRT setup combined with tomographic reconstruction allows spatially and temporally resolved acquisition of sound fields generated by nearly arbitrarily shaped ultrasonic transducers. This is achieved by projecting the investigated sound field under a sufficient number of angles through a LDV. Here, let us study reconstruction results in cases where projections under different angles are not available; i.e., there exists only one projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ of the sound field. For instance, such situation will be present when the sound source cannot be rotated or measuring time of LRT should be minimal. As a result, one has to reconstruct the entire sound field from a single projection, which means that an axisymmetric field has to be assumed. Such assumption will only make sense, however, if the sound source is of rotationally symmetric shape, e.g., piston-type or spherically focused.

As a first example, the sound field generated by the piston-type ultrasonic transducer Olympus V306-SU is considered. Again, we take a look at the sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in the cross sections $z=32.8\,\text {mm}$ (far field) and $z=8.4\,\text {mm}$ (near field), which are shown in Fig. 8.19a and c for full reconstruction, i.e., $N_{\mathrm {proj}}=100$. To emulate the so-called assumption-based reconstruction that uses only one acquired projection, a single projection was picked out and replicated hundred times yielding $N_{\mathrm {proj}}=100$ projections. These identical projections served then as input for tomographic reconstruction. Figure 8.24a and b display the results of the assumption-based reconstruction at $z=32.8\,\text {mm}$ and $z=8.4\,\text {mm}$, respectively. By comparing assumption-based and full reconstructions (see Fig. 8.19a and c), it becomes apparent that the spatial distributions of $\hat{p}_{\sim }\!\left( x,y \right)$ as well as their absolute values agree very well, especially in the far field. This can be also seen from the sound pressure amplitudes along the x-axis (see Fig. 8.24c and d) and the resulting normalized relative deviations $\epsilon _{\mathrm {r}}$ (see Fig. 8.24e and f) between the different reconstruction approaches. Hence, the question arises why one should acquire projections under different angles, which is indeed a time-consuming procedure.

Fig. 8.24

a and b LRT results for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in cross sections at $z=32.8\,\text {mm}$ and $z=8.4\,\text {mm}$ for assumption-based reconstruction, respectively; c and d $\hat{p}_{\sim }\!\left( x,y \right)$ along x-axis in cross sections at $z=32.8\,\text {mm}$ and $z=8.4\,\text {mm}$ for full and assumption-based reconstruction; (e) and (f) normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between both reconstruction approaches; piston-type ultrasonic transducer Olympus V306-SU; excitation frequency $f=1\,\text {MHz}$

In order to answer the aforementioned question, let us investigate the sound field generated by the piston-type ultrasonic transducer Krautkramer Benchmark ISS 3.5 [17]. The transducer was excited again by a sinusoidal burst of 40 cycles at $f=1\,\text {MHz}$. Figure 8.25a depicts the resulting sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ at the cross section $z=6.5\,\text {mm}$ (near field) for full reconstruction; i.e., $N_{\mathrm {proj}}=100$ independent projections were acquired under 100 angles. Although the ultrasonic transducer features a rotationally symmetric shape, the spatial distribution of $\hat{p}_{\sim }\!\left( x,y \right)$ is by no means axisymmetric, which can be attributed to partial damages of the transducer’s matching layer. In a next step, assumption-based reconstruction was performed for the same cross section by picking out a single projection and replicating it so that $N_{\mathrm {proj}}=100$ identical projections are available. Not surprisingly, the assumption-based reconstruction (see Fig. 8.25b) completely differs from full reconstruction regarding spatial distribution as well as absolute values. It can, therefore, be concluded that even though ultrasonic transducers are of rotationally symmetric shape, assumption-based reconstruction may cause remarkable deviations in LRT measurements of sound fields.

Fig. 8.25

LRT measurement for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in cross section at $z=6.5\,\text {mm}$ for a full reconstruction and b assumption-based reconstruction; piston-type ultrasonic transducer Krautkramer Benchmark ISS 3.5; excitation frequency $f=1\,\text {MHz}$

According to the previous discussions, we desire a simple criterion answering the question whether assumption-based reconstruction provides reliable results in LRT measurements. Such a criterion follows from a single sound field projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ along a line, which is acquired by the LDV [11]. If that projection is symmetrical with respect to the rotation axis (i.e., z-axis), assumption-based reconstruction may be permitted. Figure 8.26 contains single projection magnitudes along the x-axis for both piston-type transducers. In contrast to the Olympus transducer, the projection for the Krautkramer transducer is completely asymmetrical around the rotation axis, which gets also clear by comparing mirrored projections from the left side (i.e., $x\in [-20\,\text {mm},0]$) with the original ones from the right side. It, thus, seems only natural that we cannot apply assumption-based reconstruction in LRT measurements for the Krautkramer transducer. However, the simple criterion is necessary but not sufficient for achieving reliable results with assumption-based reconstruction because the comparison of a single projection may lead to a wrong conclusion. For this reason, full reconstruction should be preferred even for rotationally symmetric transducer shapes, especially when precise information of the investigated sound fields is demanded.

Fig. 8.26

Projections $\left| \mathfrak {o}_{\varTheta } \!\left( x \right) \right|$ (magnitude) of sound field along x-axis for a Olympus transducer V306-SU and b Krautkramer transducer Benchmark ISS 3.5; dashed lines represent mirrored version from left side, i.e., $x\in [-20\,\text {mm},0]$

8.4.4 Disturbed Sound Field due to Hydrophones

To verify LRT results in water, we compared them so far to hydrophone measurements, whereby the hydrophone had to be placed directly within the investigated sound field. Due to the different acoustic properties of hydrophone and the surrounding water, there occur, of course, reflections as well as diffractions of incident sound waves at the interface of water and hydrophone. Therefore, the sound field is influenced by the presence of the hydrophone, which especially constitutes a problem close to the transducer surface and medium boundaries. In the following, the disturbed sound field should be quantified. The applied measurement approach has to meet mainly three requirements:

• With a view to avoiding further disturbances of the sound field, the measurement has to be noninvasive and nonreactive.

• Spatially and temporally resolved as well as absolute measurement results should be provided.

• The measurement approach has to feature omnidirectional sensitivity because incident and reflected sound waves propagate in different directions.

In accordance with these requirements, LRT is an outstanding candidate for the quantitative analysis of sound reflections and diffractions at the hydrophone surface [11, 15].

Figure 8.27 shows the relevant part of the experimental setup comprising cylindrical mount, piston-type ultrasonic transducer Olympus V306-SU, optical reflector and capsule hydrophone Onda HGL-0400 as well as its preamplifier Onda AH-2010. The rotationally symmetric hydrophone was placed at the axial distance $z_{\mathrm {H}}=27.0\,\text {mm}$ in front of the transducer. Since LRT measurements demand sound field projections under different angles and the realized setup allows only transducer rotation, the rotation axes of both cylindrical mount and hydrophone have to be aligned. The horizontal hydrophone alignments in the xz-plane were based again on the intensity of the reflected LDV laser beam (see Sect. 8.3.4). For vertical hydrophone alignments in the yz-plane, we analyzed pictures from a SLR camera (Nikon D80; 10.2 megapixels) that was located about $5\,\text {m}$ away from the LRT setup [15]. In doing so, it is possible to ensure geometrical deviations of both rotation axes ${<}5\,\upmu \text {m}$ in x-direction and ${<}50\,\upmu \text {m}$ in y-direction, respectively.

Fig. 8.27

Relevant part of realized LRT setup to investigate disturbance of sound field at capsule hydrophone Onda HGL-0400 [15]; ultrasonic transducer UT

The ultrasonic transducer was excited by a sinusoidal pulse of 8 cycles at $f=1\,\text {MHz}$. With the aid of LRT, the transient sound field was acquired in 134 cross sections (i.e., xy-planes), which were equidistantly distributed between transducer front and hydrophone tip (see Table 8.5 for measurement parameters). After measurement and tomographic reconstruction, the entire sound field at a selected instant of time follows from assembling sound pressure values of all cross sections for that instant. Beyond the hydrophone tip (i.e., $z\ge 27.0\,\text {mm}$), LRT does not provide, however, absolute information about the sound field because the hydrophone blocks the LDV laser beam and this data is then missing for tomographic reconstruction. Nevertheless, each point of a projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ is proportional to the integral of the sound pressure along the laser beam and, consequently, contains certain information about sound propagation. Instead of reconstructing spatially resolved sound pressure values for $z\ge 27.0\,\text {mm}$, sound propagation was only visualized there through recording a single projection in every cross section [55].

Table 8.5 Decisive parameters for LRT measurement of disturbed sound field between transducer front and hydrophone tip

Fig. 8.28

Sound pressure fields in xz-plane (i.e., $y=0$) at four instants of time after starting emitting sinusoidal burst (8 cycles at $1\,\text {MHz}$); reconstruction results for $z<27.0\,\text {mm}$; visualizations for $z\ge 27.0\,\text {mm}$; a transducer finished emitting burst signal; b sound waves almost reached hydrophone; c constructive interference of incident and reflected sound waves; d all wave fronts passed hydrophone tip at $z_{\mathrm {H}}=27.0\,\text {mm}$; piston-type ultrasonic transducer Olympus V306-SU

In Fig. 8.28, one can see the measured sound pressure fields in the xz-plane (i.e., $y=0$) at four representative instants of time after starting pulse emission, namely 12.80, 19.92, 24.64, and $28.44\,\upmu \text {s}$. Thereby, reconstructions between transducer front and hydrophone are combined with visualizations beyond hydrophone tip. To achieve meaningful images, visualizations were rescaled so that the absolute values of reconstructed sound pressure and visualized projections coincide nearby $z=27.0\,\text {mm}$. At the first two instants of time $t=12.80$ and $t=19.92\,\upmu \text {s}$ (see Fig. 8.28a and b), the sound pressure waves have not reached the hydrophone tip, which means that there do not emerge disturbances of the sound field. The already existing constructive and destructive interference patterns stem from the beam characteristic of the piston-type ultrasonic transducer. In Fig. 8.28c, constructive interference of incident and reflected sound pressure waves arises since exactly four wave fronts passed the hydrophone tip. At the last instant of time $t=28.44\,\upmu \text {s}$, all wave fronts have passed the hydrophone tip and the expected reflections propagate in negative z-direction toward to transducer. When the hydrophone is placed close to the transducer surfaces or medium boundaries, exactly such reflected waves can cause remarkably deviations in hydrophone measurements.

For better interpretation, Fig. 8.29 depicts a binarized version of Fig. 8.28d; i.e., negative and positive sound pressure values are shown in black-and-white color, respectively. It can be clearly observed that the reflected sound waves appear as eight concentric white circular regions in front of the hydrophone tip (i.e, $z<27.0\,\text {mm}$). Besides, sound waves propagating in positive z-direction are still present for $z<27.0\,\text {mm}$, which is a consequence of the transducer’s beam characteristic. At this point, it should be mentioned again that the cross sections were individually reconstructed between transducer front and hydrophone tip. The spatial continuity and symmetry in the resulting images, thus, proves once more applicability of LRT even for challenging measurement tasks.

Fig. 8.29

Binarized version of sound pressure field at $t=28.44\,\upmu \text {s}$ (see Fig. 8.28d); positive and negative sound pressure values in black and white color; respectively

In addition to the spatial investigations of the disturbed sound field, let us now regard the reconstructed time-dependent values $p_{\sim }\!\left( t \right)$ at selected positions on the z-axis, i.e., $(x,y)=(0,0)$. Figure 8.30a displays $p_{\sim }\!\left( t \right)$ at $z=24.6\,\text {mm}$, which means $2.4\,\text {mm}$ in front of the hydrophone tip. One can distinguish between three time periods within the sound pressure curve: (i) exclusively waves propagating in positive z-direction; (ii) interference between waves propagating in positive and negative z-direction; (iii) exclusively waves propagating in negative z-direction. In Fig. 8.30a, the time periods (i), (ii), and (iii) cover approximately the intervals $[18,22]\,\upmu \text {s}$, $[22,27]\,\upmu \text {s}$ and $[27,31]\,\upmu \text {s}$, respectively. Figure 8.30b shows $p_{\sim }\!\left( t \right)$ for the axial position $z=26.6\,\text {mm}$, i.e., very close to the hydrophone tip. Contrary to $z=24.6\,\text {mm}$ where constructive interference is present in the sound pressure curve, time period (ii) is dominating for $z=26.6\,\text {mm}$ and refers to destructive interference. For these reasons, it can be stated that sound pressure amplitudes in the disturbed sound field vary greatly close to the hydrophone tip [15].

Fig. 8.30

Time-dependent sound pressure values $p_{\sim }\!\left( t \right)$ at selected positions on z-axis; a partly constructive interference of incident and reflected sound waves at $z=24.6\,\text {mm}$; b dominating destructive interference at $z=26.6\,\text {mm}$; c hydrophone output at $z_{\mathrm {H}}=27.0\,\text {mm}$; d LRT result at $z=27.0\,\text {mm}$ in the absence of hydrophone

Finally, the hydrophone output is compared with the LRT result at the position of the hydrophone tip (see Fig. 8.30c and d). Note that for the LRT measurement, the hydrophone was removed from the water path and, consequently, there did not exist sources for disturbing the sound field anymore. The acquired sound pressure curves $p_{\sim }\!\left( t \right)$ of both measurement approaches coincide very well regarding time behavior and absolute values. Related to the hydrophone output, the relative deviation $\epsilon _{\mathrm {r}}$ of the amplitudes becomes

$$\begin{aligned} \epsilon _{\mathrm {r}} = \frac{\hat{p}_{\mathrm {L\sim }} - \hat{p}_{\mathrm {H\sim }}}{ \hat{p}_{\mathrm {H\sim }} } = \frac{10.6\,\text {kPa} - 9.9\,\text {kPa} }{ 9.9\,\text {kPa} } = 7.1\% \;. \end{aligned}$$

(8.39)

The expressions $\hat{p}_{\mathrm {H \sim }}$ and $\hat{p}_{\mathrm {L\sim }}$ stand for the determined values of hydrophone and LRT measurements, respectively. Taking into account the uncertainty of $10.4\%$ in hydrophone measurements demonstrates once again the reliability of LRT results[33].

From the presented results in Sect. 8.4, it can be concluded that LRT is an excellent measurement approach providing spatially as well as temporally resolved data for sound fields in water. This also applies to situations where other measurement approaches fail, e.g., hydrophone for disturbed sound fields.

8.5 Sound Fields in Air

Air serves in various applications as propagation medium for ultrasonic waves. For example, airborne ultrasound is oftentimes employed in distance measurements such as parking sensors (see Sect. 7.6.1). Here, we will prove the applicability of LRT for quantitative measurements of airborne ultrasound. In Sect. 8.5.1, the piezo-optic coefficient $(\partial n / \partial p)_{\mathrm {S}}$ in air is derived. Section 8.5.2 deals with the slightly modified LRT setup, which additionally contains foam to avoid disturbing reflections of ultrasound. Finally, we discuss reconstructed sound pressure amplitudes for an air-coupled ultrasonic transducer that operates at a frequency of $f=40\,\text {kHz}$. The results are verified by microphone measurements.

8.5.1 Piezo-optic Coefficient in Air

With a view to reconstructing spatially as well as temporally resolved sound pressure values for airborne ultrasound, LRT measurements require knowledge of the piezo-optic coefficient $(\partial n / \partial p)_{\mathrm {S}}$ in air. Let us deduce this coefficient for an ideal gas, which constitutes a good approximation of air. The adiabatic state equation for an ideal gas is defined as (cf. (2.108, p. xx))

$$\begin{aligned} \frac{p_0 + \varDelta p}{p_0} = \!\left( \frac{\varrho _0 + \varDelta \varrho }{\varrho _0} \right) ^{\kappa } \end{aligned}$$

(8.40)

with the pressure $p_0$ and the density $\varrho _0$ in the equilibrium state, respectively. $\varDelta p$ and $\varDelta \varrho$ stand for slight fluctuations around the equilibrium state, and $\kappa$ is the adiabatic exponent. By utilizing the so-called Gladstone–Dale relation for gases [55]

$$\begin{aligned} K_{\mathrm {G}} \underbrace{\!\left( \varrho _0 + \varDelta \varrho \right) }_{\varrho }= \underbrace{n_0 + \varDelta n}_{n} - 1 \end{aligned}$$

(8.41)

where $K_{\mathrm {G}}$ denotes the Gladstone–Dale constant, (8.40) becomes

$$\begin{aligned} \frac{\varDelta p}{p_0} = \!\left( \frac{n_0 + \varDelta n - 1 }{n_0 - 1} \right) ^{\kappa } -1 = \!\left( 1 + \frac{\varDelta n}{n_0 - 1} \right) ^{\kappa } - 1 \;. \end{aligned}$$

(8.42)

Again, the expressions $n_0$ and $\varDelta n$ represent the optical refractive index of the gas in equilibrium state and its fluctuation, respectively. Owing to the fact that $\varDelta n \ll n_0$ is usually fulfilled for airborne ultrasound, we can introduce the Taylor approximation $(1+x)^k\approx 1+kx$. Therewith, (8.42) simplifies to

$$\begin{aligned} \frac{\varDelta p}{p_0} \approx 1 + \kappa \frac{\varDelta n}{n_0 - 1} - 1 = \frac{\kappa \varDelta n}{n_0 - 1}\;. \end{aligned}$$

(8.43)

Rewriting this equation finally yields the piezo-optic coefficient of an ideal gas

$$\begin{aligned} \!\left( \frac{\partial n}{\partial p} \right) _{\mathrm {\!S}} = \frac{\varDelta n}{\varDelta p}\approx \frac{n_0 - 1}{\kappa p_0} \;, \end{aligned}$$

(8.44)

which can be used to link refractive index changes $\varDelta n$ to sound pressure values $p_{\sim }$ ($\widehat{=}\varDelta p$) when the quantities $n_0$, $p_0$ as well as $\kappa$ are known.

In LRT measurements of airborne ultrasound, we suppose the following conditions and parameters for air [11, 38]:

• wavelength of laser beam $\lambda _{\mathrm {em}}=632.8\,\text {nm}$

• air temperature $20\,^{\circ }\text {C}$

• adiabatic exponent $\kappa =1.4$ at $20\,^{\circ }\text {C}$

• static air pressure $p_0=101.325\,\text {kPa}$

• relative humidity $40\%$

• carbon dioxide content $0.045\%$.

According to the empirical formula in [16], these values lead to the optical refractive index $n_0=1.000271$ in the equilibrium state and, consequently, to the piezo-optic coefficient

$$\begin{aligned} \!\left( \frac{\partial n}{\partial p} \right) _{\mathrm {\!S}} \approx \frac{n_0 - 1}{\kappa p_0} = \frac{1.000271 - 1}{1.4 \cdot 101325\,\text {Pa}} = 1.91 \cdot 10^{-9} \,\text {Pa}^{-1} \end{aligned}$$

(8.45)

in air. Actually, this coefficient is subject to certain fluctuations in practical situations. For instance, if the air temperature increases by $1\,^{\circ }\text {C}$, $n_0$ will decrease by $1.1\cdot 10^{-6}$ and $(\partial n / \partial p)_{\mathrm {S}}$ will be reduced by $0.4\%$ (e.g., [52]). It can be stated, however, that under normal ambient conditions, the piezo-optic coefficient in air exhibits a maximum uncertainty ${<}20\%$. Compared to water, whose piezo-optic coefficient amounts $\approx 10^{-10}\,\text {Pa}^{-1}$ (see (8.38)), the value in air is more than ten times larger.

8.5.2 Experimental Setup

The realized LRT setup for investigating airborne ultrasound is similar to the experimental arrangement that we utilized in water. This refers to the differential LDV, the linear positioning system, and rotation unit but also to the cylindrical mount as well as optical reflector (see Fig. 8.31). In order to avoid disturbing reflections of ultrasound, several components of the experimental setup have to be additionally surrounded and lined by a foam, which absorbs sound waves and, therefore, reduces echoes.

Fig. 8.31

Relevant part of realized LRT setup to investigate airborne ultrasound [38]; foam avoids disturbing reflections of sound waves; ultrasonic transducer UT

In fact, there arise significant differences between sound propagation in water and in air. On the one hand, attenuation of ultrasonic waves in air is much higher than in water (cf. Table 2.8 on p. xx). The acoustic impedance $Z_{\mathrm {aco}}$ of air is much lower than that of water on the other hand. While the acoustic impedance of water amounts $1.48\cdot 10^6 \,\text {N s m}^{-3}$, air exhibits a value of $413.5 \,\text {N s m}^{-3}$ at $20\,^{\circ }\text {C}$. For these reasons, both the frequencies of airborne ultrasound and the resulting sound pressure values are usually comparatively small. In the present case, the air-coupled ultrasonic transducer Sanwa SCS-401T with a radius of $R_{\mathrm {T}}=6.5\,\text {mm}$ served as sound source. The piston-type transducer was fixed again in the cylindrical mount. The transducer was excited by a sinusoidal burst signal of 48 cycles at $f=40\,\text {kHz}$, which is a typical frequency for airborne ultrasound in practical applications, e.g., parking sensors. To achieve a satisfactory SNR in LRT measurements, the excitation voltage was chosen to be $24\,\text {V}_{\mathrm {pp}}$ (peak-to-peak). Figure 8.32 shows the averaged output signal (mean of 64 signals) of the differential LDV at $(x,z)=(0,20\,\text {mm})$, which represents data at a single point of a projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ with respect to time t. For such excitation voltage, the SNR exceeds $30\,\text {dB}$ and, thus, is sufficient for LRT measurements. However, it should be noted that an appropriate microphone (e.g., 1 / 4 inch microphone from Brüel & Kjær) usually provides considerably higher SNR values [11, 38].

Fig. 8.32

Averaged output (mean of 64 signals) of LDV at position $(x,z)=(0,20\,\text {mm})$ representing single point of projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ for tomographic reconstruction; transducer excitation started at $0.5\,\text {ms}$; resulting $\text {SNR}\approx 31\,\text {dB}$; piston-type ultrasonic transducer Sanwa SCS-401T; excitation frequency $f=40\,\text {kHz}$

8.5.3 Results for Piston-Type Ultrasonic Transducer

By means of LRT, we acquired sound pressure fields $p_{\sim }\!\left( x,y,t \right)$ of the air-coupled transducer in three different cross sections. The cross sections were located at the axial distances $z=5\,\text {mm}$, $z=20\,\text {mm}$ and $z=90\,\text {mm}$ from the transducer front. Compared to the LRT experiments in water, the scanning area in x-direction has been extended because wavelength $\lambda _{\mathrm {aco}}$ of the sound waves and, consequently, the main lobe’s diameter $D_{\mathrm {main}}$ of the directivity pattern increases (see Sect. 8.3.6). Moreover, the higher value of $\lambda _{\mathrm {aco}}$ enables increasing the sampling interval $\varDelta \xi$ between two neighboring LDV positions as well as reducing the number of projections $N_{\mathrm {proj}}$, which is necessary for tomographic reconstruction. Table 8.6 summarizes the most important parameters for the conducted LRT measurements.

Table 8.6 Decisive parameters for LRT measurement of sound field in air arising from piston-type ultrasonic transducer

Figure 8.33a and c presents the reconstructed sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ of the piston-type transducer in the cross sections⁶ at $z=5\,\text {mm}$ and $z=20\,\text {mm}$, respectively. The sound pressure amplitudes are rotationally symmetric distributed in both cross sections. For the chosen transducer excitation $f=40\,\text {kHz}$, the near-field length equals $N_{\mathrm {near}}=2.8\,\text {mm}$. Hence, the maximum of $\hat{p}_{\sim }\!\left( x,y \right)$ should be larger at $z=5\,\text {mm}$ than at $z=20\,\text {mm}$, which is also confirmed in the LRT measurements.

Fig. 8.33

a and c LRT measurements for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in cross sections at $z=5\,\text {mm}$ and $z=20\,\text {mm}$, respectively; b and d corresponding microphone (MIC) measurements; piston-type ultrasonic transducer Sanwa SCS-401T; excitation frequency $f=40\,\text {kHz}$

With a view to verifying LRT results, we performed microphone measurements in addition [11, 38]. The utilized 1/4 inch condenser microphone (Brüel & Kjær; type 4939 [9]), whose output response to sound waves is almost constant up to $100\,\text {kHz}$, was moved with the step size $1.0\,\text {mm}$ in x- and y-direction. Overall, this procedure takes $2.5\,\text {h}$ and, thus, much more time than corresponding LRT measurements (see Table 8.6). In Fig. 8.33b and d, one can see the obtained amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ from microphone measurements. The comparison reveals that LRT and microphone results coincide very well in both cross sections. This is also demonstrated by the acquired sound pressure amplitudes along the x-axis (see Fig. 8.34a and b) and the normalized relative deviations $\epsilon _{\mathrm {r}}$ (see Fig. 8.34c and d) between the different measurement approaches. The results exhibit a maximum difference of 11.7 and $8.9\%$ at $z=5\,\text {mm}$ and $z=20\,\text {mm}$, respectively. From there, LRT seems to be a suitable approach to replace conventional microphone measurements, especially when we desire sound pressure information in the whole cross section.

Fig. 8.34

a and b Comparison of LRT and microphone (MIC) measurements for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ along x-axis in cross sections at $z=5$ and $z=20\,\text {mm}$, respectively; c and d relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) along x-axis between LRT and microphone measurements normalized to maximum of microphone output; piston-type ultrasonic transducer Sanwa SCS-401T

Apart from the axial distances 5 and $20\,\text {mm}$, the sound field was investigated in the cross section at $z=90\,\text {mm}$. Figure 8.35a and b display the achieved LRT and microphone results for the sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,t \right)$. In contrast to the other cross sections, there emerge enormous deviations between the different measurement approaches, which gets also obvious in the acquired values along the x-axis (see Fig. 8.35c). The reason for the large deviations lies in the spatial main lobe extension of the sound field. At the axial distance $z=90\,\text {mm}$, the main lobe has a diameter of $D_{\mathrm {main}}=244\,\text {mm}$ (see (8.36)), which, therefore, remarkably exceeds the geometric dimensions $l_{\mathrm {opt}}=100\,\text {mm}$ of the utilized optical reflector. Due to this fact, the main lobe cannot be completely covered by means of a single reflector position in LRT measurements. The projections $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ for tomographic reconstruction may exhibit relatively large values at the boundaries of the scanning area (i.e., $x=\pm 45\,\text {mm}$), which is proven in Fig. 8.35d.

Fig. 8.35

a and b LRT and microphone (MIC) measurements for sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in cross section at $z=90\,\text {mm}$, respectively; c comparison of $\hat{p}_{\sim }\!\left( x,y \right)$ along x-axis; d magnitude $\left| \mathfrak {o}_{\varTheta } \!\left( x \right) \right|$ of single projection; piston-type ultrasonic transducer Sanwa SCS-401T; excitation frequency $f=40\,\text {kHz}$

An insufficient scanning area has mainly two consequences for LRT results [11]. Firstly, nonzero projections at the scanning boundaries can be understood as spectral leakage in the spatial frequency domain. Such spectral leakage leads to spatial oscillations in the reconstructed images (see, e.g., Fig. 8.35a). As a second consequence, the energies of reconstructed and actual sound pressure field may differ in a cross section. Let us explain this circumstance through Fig. 8.36, which illustrates an insufficient scanning area in LRT measurements for a rotationally symmetric sound field. Within the scanning area, the entire sound information along the LDV beam contributes to the measurement. However, sound information is not available beyond this scanning area and, thus, sound energy and sound pressure values are set to zero there, regardless of their actual values. In other words, we are mixing up two situations differing in sound energy during tomographic reconstruction. It seems only natural that in case of an insufficient scanning area, LRT measurements are then accompanied by large deviations.

Fig. 8.36

Insufficient scanning area in LDV measurements for rotationally symmetric sound field [38]; extension of actual sound field remarkably exceeds reconstructed one

To sum up, LRT is also applicable for acquiring sound pressure values of airborne ultrasound. Nevertheless, this technique should be restricted to applications where sound pressure amplitudes allow a satisfactory SNR in the LDV outputs and the spatial extension of the sound field can be completely covered.

8.6 Mechanical Waves in Optically Transparent Solids

So far, LRT was exclusively applied to analyze sound propagation in fluids such as water and air. Apart from fluids, solid media are often involved in sound as well as ultrasound applications (e.g., nondestructive testing). At the boundary of a fluid and a solid, sound pressure waves propagating in the fluid get converted to mechanical waves, which propagate in the solid. Conventional measurement approaches (see Sect. 8.1) do not, however, allow quantitative investigations of mechanical waves within the solid. For instance, the Schlieren optical method can be utilized to visualize mechanical waves in optically transparent solids, but this method does not provide absolute information about physical quantities (e.g., mechanical stress) describing the waves. In this section, we will prove the applicability of LRT to determine such quantities in optically transparent solids with respect to both time and space.

As a starting point, the mechanical normal stress in an isotropic solid will be deduced from dilatation. Section 8.6.2 deals with the experimental setup that was used to excite mechanical waves in optically transparent solids. Finally, we present selected LRT results, which will be verified by numerical simulations and characteristic parameters (e.g., reflection coefficient).

8.6.1 Normal Stress in Isotropic Solids

As already discussed in Sect. 8.3.1, LRT enables acquisition of mechanical longitudinal waves in optically transparent solids. In doing so, we reconstruct the spatially as well as temporally resolved dilatation  $\delta _{\mathrm {dil}}$, which corresponds to the sum of normal strains $S_{ii}$ within the solid medium (cf. (8.8)). This quantity is, however, not common for describing propagation of mechanical waves. Besides, another physical quantity is desired instead of dilatation to compare sound fields in fluids with mechanical waves in solids. The mechanical stress tensor $\left[ {\mathbf {T}} \right]$ seems to be such a quantity since it shares the same physical unit of measurement as the sound pressure $p_{\sim }$, i.e., $\text {N m}^{-2} \,\widehat{=} \,\text {Pa}$.

Let us deduce a normal component $T_{ii}$ of the mechanical stress tensor from $\delta _{\mathrm {dil}}$. For the propagating mechanical waves, the normal stress $T_{\mathrm {zz}}$ in z-direction is assumed to be dominant in the following. According to Hooke’s law  (see Sect. 2.2.3), $T_{\mathrm {zz}}$ in an isotropic and homogeneous solid is given by

$$\begin{aligned} T_{\mathrm {zz}} = \lambda _{\mathrm {L}} \!\left( S_{\mathrm {xx}} + S_{\mathrm {yy}} + S_{\mathrm {zz}} \right) + 2 \mu _{\mathrm {L}} S_{\mathrm {zz}} \end{aligned}$$

(8.46)

with the Lamé parameters $\lambda _{\mathrm {L}}$ and $\mu _{\mathrm {L}}$. Expressing these parameters with Young’s modulus $E_{\mathrm {M}}$ as well as Poisson’s ratio $\nu _{\mathrm {P}}$ and inserting (8.8) leads to

$$\begin{aligned} T_{\mathrm {zz}}&= \frac{E_{\mathrm {M}}}{\!\left( 1 + \nu _{\mathrm {P}} \right) \!\left( 1 - 2 \nu _{\mathrm {P}} \right) } \!\left[ \nu _{\mathrm {P}} S_{\mathrm {xx}} + \nu _{\mathrm {P}} S_{\mathrm {yy}} + \!\left( 1-\nu _{\mathrm {P}} \right) S_{\mathrm {zz}} \right] \nonumber \\&= \frac{E_{\mathrm {M}} \!\left( 1-\nu _{\mathrm {P}} \right) }{\!\left( 1+\nu _{\mathrm {P}} \right) \!\left( 1 - 2 \nu _{\mathrm {P}} \right) } \delta _{\mathrm {dil}} + \frac{E_{\mathrm {M}} \!\left( 2\nu _{\mathrm {P}} -1 \right) }{\!\left( 1+\nu _{\mathrm {P}} \right) \!\left( 1 - 2\nu _{\mathrm {P}} \right) } \!\left( S_{\mathrm {xx}} + S_{\mathrm {yy}} \right) \;. \end{aligned}$$

(8.47)

Actually, LRT measurements in optically transparent solids provide exclusively values for $\delta _{\mathrm {dil}}$. In other words, information about the normal strains $S_{\mathrm {xx}}$ and $S_{\mathrm {yy}}$ is not available [11, 12]. Owing to this fact, we approximate $T_{\mathrm {zz}}$ by neglecting the second term in (8.47), i.e.,

$$\begin{aligned} T_{\mathrm {zz}} \approx k_{\mathrm {m}} \delta _{\mathrm {dil}} \qquad \text {with} \qquad k_{\mathrm {m}} = \frac{E_{\mathrm {M}} \!\left( 1-\nu _{\mathrm {P}} \right) }{\!\left( 1+\nu _{\mathrm {P}} \right) \!\left( 1 - 2 \nu _{\mathrm {P}} \right) } \;. \end{aligned}$$

(8.48)

The expression $k_{\mathrm {m}}$ stands for a material-dependent constant of the solid. Therewith, the relative deviation $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) of the approximation becomes

$$\begin{aligned} \left| \epsilon _{\mathrm {r}} \right|&= \left| \frac{k_{\mathrm {m}} \delta _{\mathrm {dil}} - T_{\mathrm {zz}}}{T_{\mathrm {zz}}} \right| \nonumber \\&= \left| \frac{-\!\left( 2\nu _{\mathrm {P}} -1 \right) \!\left( S_{\mathrm {xx}} + S_{\mathrm {yy}} \right) }{\!\left( 1- \nu _{\mathrm {P}} \right) \delta _{\mathrm {dil}} + \!\left( 2\nu _{\mathrm {P}} -1 \right) \!\left( S_{\mathrm {xx}} + S_{\mathrm {yy}} \right) } \right| \nonumber \\&= \left| 1 + \frac{1-\nu _{\mathrm {P}}}{2\nu _{\mathrm {P}} - 1 } \!\left[ 1 + \frac{S_{\mathrm {zz}}}{S_{\mathrm {xx}} + S_{\mathrm {yy}}} \right] \right| ^{-1} \end{aligned}$$

(8.49)

and, thus, strongly depends on Poisson’s ratio of the solid. As (8.49) indicates, one can approximate a normal component of the stress tensor (e.g., $T_{\mathrm {zz}}$) with $k_{\mathrm {m}}\delta _{\mathrm {dil}}$ very well when the normal strain in this direction dominates, e.g., $S_{\mathrm {zz}}\gg S_{\mathrm {xx}} + S_{\mathrm {yy}}$. A sound pressure wave impinging perpendicular to an interface of fluid and solid causes such situation in the solid. By means of numerical simulations, this fact was proven in [12]. Summing up, it can be stated that LRT measurements should be applicable for reliably estimating normal stresses due to mechanical longitudinal waves, which propagate in an optically transparent solid.

8.6.2 Experimental Setup

The realized LRT setup for analyzing mechanical waves in solids is identical to the one, which was employed in water (cf. Fig. 8.9). As a sound source, we utilized either the piston-type ultrasonic transducer V306-SU or the cylindrically focused ultrasonic transducer V306-SU-CF1.00IN, both from the company Olympus Corporation [29] and optimized for generating sound fields in water. A PMMA (poly(methyl methacrylate)) block served as optically transparent solid in which the propagation of mechanical longitudinal waves should be investigated through LRT. This PMMA block with a geometric dimension of $160\,\text {mm} \times 60 \,\text {mm} \times l_{\mathrm {B}}$ (block length $l_{\mathrm {B}}$ in z-direction) was placed directly onto the optical reflector at the axial distance $z_{\mathrm {B}}$ from the transducer front (see Fig. 8.37). Both the PMMA block and the cylindrical mount containing the ultrasonic transducer were immersed in water.

Fig. 8.37

Relevant part of realized LRT setup to investigate mechanical waves in optically transparent PMMA block [12]; geometric block dimension $160\,\text {mm} \times 60\,\text {mm} \times l_{\mathrm {B}}$; axial distance $z_{\mathrm {B}}$ of block surface from transducer front; ultrasonic transducer UT

To reconstruct spatially as well as temporally resolved quantities (e.g., $\delta _{\mathrm {dil}}$), LRT measurements demand projections under different angles, which are acquired by the LDV. Note that this fact does not only refer to sound fields in fluids but also to mechanical waves in optically transparent solids. The dedicated experimental setup allows solely rotations of the ultrasonic transducer. It is, therefore, of utmost importance that such rotations do not alter sound fields as well as mechanical waves. Consequently, the PMMA block has to be aligned with respect to the cylindrical mount so that the block surface facing the transducer is in parallel to the scanning planes. Again, this alignment was conducted by evaluating the intensity of the reflected LDV beam (see Sect. 8.3.5). The PMMA block can then be treated as axisymmetric in LRT measurements.

8.6.3 Results for Different Ultrasonic Transducers

Two experiments will be presented in order to demonstrate applicability of LRT for quantitative measurements in the optically transparent solids. While one experiment exclusively deals with stress amplitudes in a cross section inside a PMMA block, the second experiment concentrates on transient field quantities for a wave propagating throughout water and PMMA [11, 12].

Mechanical Waves within a PMMA Block

In the first experiment, the mechanical longitudinal waves within a PMMA block (length $l_{\mathrm {B}}=45\,\text {mm}$) were investigated through LRT. The block was placed at the axial distance $z_{\mathrm {B}}=11\,\text {mm}$ from the transducer front. The ultrasonic transducer (piston-type or cylindrically focused) was excited by a sinusoidal burst signal of 12 cycles at $f=1\,\text {MHz}$. At the interface water/PMMA, the generated sound pressure waves in water get reflected as well as converted to mechanical waves in the solid. Figure 8.38 depicts the averaged output signal (mean of 16 signals) of the differential LDV at $(x,z)=(0,25.4\,\text {mm})$, which represents data at a single point of a projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ with respect to time t. Note that this position is within the PMMA block. In the acquired LDV signal, one can clearly see the incident mechanical waves as well as the reflections due to the interface of PMMA and water. Just as in water (cf. Fig. 8.18), the SNR of the LDV signal in the PMMA block exceeds $30\,\text {dB}$.

Fig. 8.38

Averaged LDV output (mean of 16 signals) within PMMA block at position $(x,z)=(0,25.4\,\text {mm})$ representing single point of projection $\mathfrak {o}_{\varTheta } \!\left( \xi \right)$ for tomographic reconstruction; incident mechanical waves as well as reflections at interface PMMA/water are present; resulting $\text {SNR}\approx 37\,\text {dB}$; piston-type ultrasonic transducer Olympus V306-SU; excitation frequency $f=1\,\text {MHz}$

LRT was used to reconstruct amplitudes $\hat{T}_{\mathrm {zz}}\!\left( x,y \right)$ of the mechanical stress in a cross section at the axial distance $z=25.4\,\text {mm}$, i.e., directly within the PMMA block. The most important parameters for the conducted LRT measurements can be found in Table 8.4. Figure 8.39a and b show the obtained results⁷ for the piston-type and cylindrically focused transducer, respectively. The piston-type transducer produces an axisymmetric distribution of $\hat{T}_{\mathrm {zz}}\!\left( x,y \right)$ in the PMMA block. In contrast, the cylindrically focused transducer causes a distribution that is slightly focused in one direction. However, compared to the distribution of sound pressure amplitudes $\hat{p}_{\sim }\!\left( x,y \right)$ in water at $z=25.4\,\text {mm}$ (cf. Fig. 8.21), focusing is less pronounced in the PMMA block. This can be ascribed to the fact that the cylindrically focused transducer is designed for operating in water and, thus, the focal length $25.4\,\text {mm}$ also relates to water.

Fig. 8.39

LRT results for stress amplitudes $\hat{T}_{\mathrm {zz}}$ within PMMA block in cross section at $z=25.4\,\text {mm}$; a piston-type ultrasonic transducer Olympus V306-SU; b cylindrically focused ultrasonic transducer Olympus V306-SU-CF1.00IN; excitation frequency $f=1\,\text {MHz}$

Up to now, conventional measurement approaches (e.g., Schlieren optical method) do not allow quantitative verification of the LRT results in PMMA. For this reason, let us compare the LRT results for the piston-type transducer with FE simulations [12]. In doing so, the ultrasonic transducer was modeled as a circular area of radius $R_{\mathrm {T}}=6.35\,\text {mm}$ oscillating uniformly with the applied excitation signal in the experiments, i.e., 12 cycles at $f=1\,\text {MHz}$. The decisive material properties of the PMMA block were identified by measuring the wave propagation velocities of mechanical longitudinal and transverse waves in the block (see Sect. 5.1.2). These measurements lead to Young’s modulus $E_{\mathrm {m}}=5.98\,\text {GPa}$ and Poisson’s ratio $\nu _{\mathrm {P}}=0.33$.

Figure 8.40a compares measured (i.e., LRT results) and simulated amplitudes $\hat{T}_{\mathrm {zz}}$ of the mechanical normal stress along the x-axis at $z=25.4\,\text {mm}$. Thereby, the simulation results have been scaled so that measured and simulated quantities feature the same energy within the PMMA block. The distribution of both quantities along the x-axis coincides very well, which is also demonstrated by their normalized relative deviation $\left| \epsilon _{\mathrm {r}} \right|$ (see Fig. 8.40b). Besides the mechanical normal stress $\hat{T}_{\mathrm {zz}}$, Fig. 8.40a contains the simulated dilatation $\hat{\delta }_{\mathrm {dil;sim}}$ that was scaled with the material-dependent constant $k_{\mathrm {m}}$. As discussed above, LRT measurements within optically transparent solids exploit the approximation of $T_{ii}$ by this quantity (see (8.48)). Because the normalized relative deviation of $\hat{T}_{\mathrm {zz;sim}}$ and $k_{\mathrm {m}}\hat{\delta }_{\mathrm {dil;sim}}$ is always smaller than $2\%$ (see Fig. 8.40b), it can be stated that the applied approximation yields reliable LRT results for the normal stress, which is caused by mechanical longitudinal waves.

Fig. 8.40

a Comparison of measured stress amplitudes $\hat{T}_{\mathrm {zz;meas}}$, simulated stress amplitudes $\hat{T}_{\mathrm {zz;sim}}$ as well as scaled dilatations $k_{\mathrm {m}} \hat{\delta }_{\mathrm {dil;sim}}$ within PMMA block along x-axis at $z=25.4\,\text {mm}$; b normalized relative deviations $\left| \epsilon _{\mathrm {r}} \right|$ (magnitude) between measured and simulated stress amplitudes as well as between simulated stress amplitudes and scaled version of simulated dilatations; piston-type ultrasonic transducer Olympus V306-SU

Fields Throughout Water and PMMA

In the second experiment, wave propagation phenomena throughout water and PMMA were analyzed by means of LRT. For this purpose, a PMMA block with the length $l_{\mathrm {B}}=22\,\text {mm}$ was placed onto the optical reflector at the axial distance $z_{\mathrm {B}}=24.4\,\text {mm}$ from the transducer front (see Fig. 8.37). To avoid overlaps of multiple reflections, the piston-type ultrasonic transducer Olympus V306-SU was excited by a sinusoidal burst signal consisting only of 8 cycles. Since this experiment required the acquisition of numerous cross sections in water as well as PMMA, we applied nonequidistant sampling between two neighboring LDV positions. Table 8.7 summarizes the most important parameters for the conducted LRT measurements.

Table 8.7 Decisive parameters for LRT measurement of sound pressure waves in water and mechanical waves in PMMA block

The temporally resolved LRT results in the cross sections were assembled together yielding transient as well as spatially resolved information about wave propagation throughout water and PMMA. With a view to achieving visually continuous images, the spatial resolution in z-direction has been additionally doubled by cubic spline interpolation, i.e., from $\varDelta z= 0.4\,\text {mm}$ to $\varDelta z = 0.2\,\text {mm}$. Figures 8.41a–c display the assembled data in the xz-plane (i.e., $y=0$) for three different instants of time t after starting pulse emission, namely 18.2, 25.4, and $32.0\,\upmu \text {s}$. Because sound pressure $p_{\sim }$ and mechanical stress $T_{\mathrm {zz}}$ have the same physical unit, they can be represented by a single color map in the images. Before discussing the LRT results, it should be pointed out that there is information missing in several cross sections ($z\in [24.4,27.0] \cup [43.6,46.2]\,\text {mm}$) at the left and right sides of the PMMA block. The reason for this was the mechanical processing of the PMMA block. Cutting and milling alters the material density near the surfaces of the block and, consequently, the optical refractive index n changes permanently. Owing to these changes, the laser beam of the LDV gets deflected and is not reflected back to its sensor head anymore. Hence, projections are not available there and the field information cannot be reconstructed in LRT measurements.

Fig. 8.41

Acquired sound pressure fields $p_{\sim }$ and mechanical fields $T_{\mathrm {zz}}$ in xz-plane (i.e., $y=0$) throughout water and PMMA at three instants of time after starting emitting sinusoidal burst (8 cycles at $1\,\text {MHz}$); a sound pressure waves almost reached PMMA block; b mechanical waves reached right side of block; c mechanical waves got converted to sound pressure waves; no reconstructions at left and right block sides due to missing LDV signals; d normalized result of numerical simulation at $t=32.0\,\upmu \text {s}$; piston-type ultrasonic transducer Olympus V306-SU

At $t=18.2\,\upmu \text {s}$ (see Fig. 8.41a), the sound pressure waves in water have not reached the PMMA block and, therefore, neither sound reflections at the interface water/PMMA nor mechanical waves in the PMMA block arise. However, one can observe constructive and destructive interference patterns, which stem from the beam characteristic of the piston-type ultrasonic transducer (cf. Fig. 8.28). In Fig. 8.41b ($t=25.4\,\upmu \text {s}$), almost the whole wave front has passed the block surface facing the transducer. The reflected sound pressure waves propagate back to the transducer, i.e., in negative z-direction. Besides, the first wave front of mechanical waves resulting from converted sound pressure waves was about to reach the right side of the PMMA block. As the comparison of sound pressure waves before the block and longitudinal mechanical waves in PMMA reveals, the later ones exhibit a greater wavelength. This is also expected from the theoretical point of view since the wave propagation velocity in PMMA is higher than in water. Moreover, the energy of mechanical waves is distributed in a larger area in lateral direction, i.e., in x- and y-direction. The same behavior can be seen in Fig. 8.41c ($t=32.0\,\upmu \text {s}$), where most of the mechanical waves have reached the right block side and have been partially reflected as well as converted back to sound pressure waves again.

For qualitative comparison, a transient FE simulation was carried out in addition. Therewith, it is possible to predict the sound pressure field in water and the mechanical waves in PMMA. Figure 8.41d illustrates the normalized simulation result at the instant of time $t=32.0\,\upmu \text {s}$ (cf. Fig. 8.42c). Apart from the missing data at the border area of the PMMA block, simulations and LRT results share many fine details (e.g., waveforms), which demonstrates reliability of LRT measurements in optically transparent solids.

Now, let us regard the reconstructed time-dependent variations $p_{\sim }\!\left( t \right)$ as well as $T_{\mathrm {zz}}\!\left( t \right)$ (see Fig. 8.42) at three selected positions on the z-axis, i.e., $(x,y)=(0,0)$. The positions are located (i) in water at $z=15.4\,\text {mm}$, i.e., between transducer and PMMA block, (ii) in the PMMA block at $z=33.6\,\text {mm}$, and (iii) in water at $z=46.8\,\text {mm}$, i.e., behind the PMMA block. In Fig. 8.42a, one can clearly observe the emitted sound pressure wave (group 1), which was firstly reflected by the left side of the PMMA block. Subsequently, these reflected waves got reflected by the transducer front again leading to a sound pressure wave (group 2) propagating in positive z-direction toward the PMMA block. Furthermore, several other wave groups are present that result from multiple reflections between medium boundaries. Both wave group 1 and wave group 2 also exist in the PMMA block (see Fig. 8.42b), i.e., in the time-dependent variation $T_{\mathrm {zz}}\!\left( t \right)$. However, because the second position is further away from the transducer front than the first one, the dominant wave groups are shifted in time. An additional shift occurs at $z=46.8\,\text {mm}$ (see Fig. 8.42c), which is located in water behind the PMMA block. Overall, the wave groups at the three positions are of similar shape with the exception that they appear upside-down at the third position. Note that this is a consequence of the continuity condition at a solid–fluid interface (see (4.104, p. xx)).

Fig. 8.42

Time-dependent sound pressure $p_{\sim }\!\left( t \right)$ and mechanical stress $T_{\mathrm {zz}}\!\left( t \right)$ at selected positions on z-axis; a in water between transducer front and PMMA block at $z=15.4\,\text {mm}$; b within PMMA block at $z=33.6\,\text {mm}$; c in water behind PMMA block at $z=46.8\,\text {mm}$; wave group 1 emitted by ultrasonic transducer; wave group 2 reflected by transducer front

8.6.4 Verification of Experimental Results

Finally, the previous LRT results for the optically transparent PMMA block should be quantitatively verified. For this purpose, we determine the reflection coefficient at the interface water/PMMA as well as the amplitude ratio for dominating wave groups.

Reflection Coefficient

According to the LRT results in Fig. 8.42a, the reflected sound pressure waves feature the amplitude $\hat{p}_{\mathrm {LRT}}\!\left( z=15.4\,\text {mm} \right) = 3.84\,\text {kPa}$. Due to the fact that the PMMA block is located at $z_{\mathrm {B}}=24.4\,\text {mm}$, the reflected sound pressure waves propagate altogether $33.4\,\text {mm}$ there⁸ (see Fig. 8.43). In the absence of PMMA block, a hydrophone measurement yielded the sound pressure amplitude $\hat{p}_{\mathrm {HYD}}\!\left( z=33.4\,\text {mm} \right) = 10.78\,\text {kPa}$ at the axial distance $z=33.4\,\text {mm}$ from the transducer front. We can utilize these two amplitudes to estimate the reflection coefficient $r_p$ for an incident sound pressure wave at the interface water/PMMA, i.e.,

$$\begin{aligned} r_p = \frac{\hat{p}_{\mathrm {LRT}} \!\left( z=15.4\,\text {mm} \right) }{\hat{p}_{\mathrm {HYD}} \!\left( z=33.4\,\text {mm} \right) } = \frac{3.84\,\text {kPa}}{10.78\,\text {kPa}} = 0.356 \;. \end{aligned}$$

(8.50)

From the theoretical point of view, the reflection coefficient $r'_p$ in case of plane sound waves results in (see (2.139, p. xx))

$$\begin{aligned} r'_p&= \frac{Z_{\mathrm {PMMA}} - Z_{\mathrm {water}}}{Z_{\mathrm {PMMA}} + Z_{\mathrm {water}}} \nonumber \\&= \frac{3.26 \cdot 10^6 \,\text {N s m}^{-3} - 1.48 \cdot 10^6 \,\text {N s m}^{-3}}{3.26 \cdot 10^6 \,\text {N s m}^{-3} + 1.48 \cdot 10^6 \,\text {N s m}^{-3}} = 0.376 \;. \end{aligned}$$

(8.51)

The expressions $Z_{\mathrm {PMMA}}$ and $Z_{\mathrm {water}}$ stand for the acoustic impedance of PMMA and water, respectively. Although two completely different approaches were applied to determine the reflection coefficient at the interface water/PMMA, both values coincide very well and exhibit a relative deviation of only $-5.1\%$ (rel. to $r'_p$).

Fig. 8.43

Illustration of wave propagation throughout water and PMMA block; time-dependent curves of sound pressure $p_{\sim }\!\left( t \right)$ and mechanical stress $T_{\mathrm {zz}} \!\left( t \right)$ are shown in Fig. 8.42a–c

Amplitude Ratio

As the time-dependent curves in Fig. 8.42 indicate, there exist two dominant wave groups (group 1 and group 2), which originate from the emitted sinusoidal burst and the reflection by the transducer front. Let us evaluate the amplitude ratios of the wave groups in the PMMA block as well as in water behind the block (see Fig. 8.43). In the PMMA block at $z=33.6\,\text {mm}$, the amplitude ratio $\alpha _T$ is defined by the amplitudes of the mechanical normal stress, i.e.,

$$\begin{aligned} \alpha _T = \frac{\hat{T}_{\mathrm {zz;1}} - \hat{T}_{\mathrm {zz;2}}}{\hat{T}_{\mathrm {zz;1}}} = \frac{11.10\,\text {kPa} - 2.60\,\text {kPa}}{11.10\,\text {kPa}} = 0.766 \end{aligned}$$

(8.52)

with $\hat{T}_{\mathrm {zz;1}}$ and $\hat{T}_{\mathrm {zz;2}}$ representing the stress amplitudes of incident and reflected waves, respectively. In water at $z=46.8\,\text {mm}$, the amplitude ratio $\alpha '_T$ of the corresponding sound pressure waves results in

$$\begin{aligned} \alpha '_T = \frac{\hat{p}_{\sim ;1} - \hat{p}_{\sim ;2}}{\hat{p}_{\sim ;1}} = \frac{5.28\,\text {kPa} - 1.18\,\text {kPa}}{5.28\,\text {kPa}} = 0.777 \;. \end{aligned}$$

(8.53)

Since the relative deviation of the amplitude ratios is only $-1.3\%$ (rel. to $\alpha '_T$), it can be stated once more that LRT measurements lead to reliable information about waves, which propagate throughout water and PMMA.

To sum up, LRT also enables spatially as well as temporally resolved measurements of mechanical waves in optically transparent solids. In such media, mechanical longitudinal waves locally alter the dilatation, which can be acquired through LRT. By additionally introducing a material-dependent approximation, LRT measurements provide absolute values for the mechanical normal stresses within the solid.