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Abstract
A method for measuring a surface slope distribution of a capillary wave is proposed. The method uses an optical imaging system that can capture a one-shot image of a light-reflectance direction field in a two-dimensional image plane. A dispersion relation between the wavelength and frequency of the capillary wave is shown to be obtainable by the imaging system, which agrees well with the theoretical prediction.
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1. INTRODUCTION
Capillary waves are waves that travel along an interface between two fluids, where the dynamics and phase velocity are mainly affected by surface tension. These waves, also known as ripples, are widely observed in nature. Typically, the wavelength of capillary waves on water is less than a few centimeters, and the phase speed exceeds one-tenth of a few meters per second.
Capillary waves can be used to study the surface properties of materials such as surface tension [1–4]. Laser-induced surface capillary waves are shown to enhance material removal using laser processing [5] or to create a new surface structure [6]. Knowledge of capillary wave behavior is crucial in the design of ships and other watercraft. The size and shape of capillary waves can have a significant impact on the drag experienced by a ship which, in turn, affects its speed and fuel efficiency [7]. In biomedical engineering, the capillary waves can be used to study the behavior of cells and tissues [8–10]. Optical coherence tomography is a powerful tool to measure the surface slope distributions of capillary waves [11]. A scanning beam directed by galvano mirrors is used to capture the displacement distribution of a capillary wave within a specific field of view. A time delay effect caused by the scanning, however, sometimes makes it difficult to capture fast-propagating waves in a two-dimensional field of view.
A two-dimensional distribution of a surface reflectance direction field can be captured using an optical imaging system employing multicolor filters [12–15]. A method for measuring a surface slope distribution of a capillary wave is therefore proposed here using one-shot reflectance direction field imaging. The surface reflectance direction can be described by the bidirectional reflectance distribution function (BRDF) [16]. Thus, the imaging system in this paper is called a one-shot BRDF imaging system or, for brevity, a one-shot BRDF.
The remainder of this paper is organized as follows. First, the basic structure of the one-shot BRDF imaging system, which can capture a light-reflectance direction field of a capillary wave in a two-dimensional image plane, is described. Secondly, theoretical equations for a capillary wave are described, namely, the dispersion relationship between the wavelength and frequency, and the surface slope angle obtainable by the one-shot BRDF imaging system. Thirdly, the experimental results for capillary waves generated by a transducer are described. Surface reflectance direction fields for the capillary waves in the two-dimensional image plane are experimentally captured by the one-shot BRDF. The surface slope angle can then be calculated from the reflectance direction. The dispersion relationship between the wavelength and frequency can also be obtained, which is compared with that of the theoretical prediction. Lastly, discussions and conclusions are presented.
2. ONE-SHOT BRDF IMAGING SYSTEM FOR MEASURING CAPILLARY WAVES
Figure 1 shows a schematic cross-sectional view of the one-shot BRDF. The one-shot BRDF consists mainly of an illumination optical system and an imaging optical system. In addition to the optical system, a transducer is used to produce an acoustic radiation force at an interface between water and air.
Fig. 1. Schematic cross-sectional view of the one-shot BRDF imaging system (one-shot BRDF). The one-shot BRDF consists mainly of an illumination optical system and an imaging optical system. In addition to the optical system, a transducer is used to produce the acoustic radiation force at an interface between water and air. The imaging optical system has an imaging lens and a stripe-pattern multicolor filter that is placed at the focal plane of the imaging lens. The optical axis of the imaging lens is set to the $z$ axis in a global Cartesian coordinate system. The multicolor filter is set parallel to the $xy$ plane and has translational symmetry in the $y$ axis. The coordinate origin $O$ is in the multicolor filter.
The illumination optical system has an LED, a pinhole, and a collimator lens that can convert the diverging light rays emitted from the LED to collimated light rays. The collimated light rays are reflected by a beam splitter and travel toward a water surface. The imaging optical system has an imaging lens and a stripe-pattern multicolor filter that consists of multiple regions with different color filters. The multicolor filter is placed at the focal plane of the imaging lens at a focal length $f$ from a principal plane of the imaging lens. The optical axis of the imaging lens is set to the $z$ axis in a global Cartesian coordinate system. The multicolor filter is set parallel to the $xy$ plane and has translational symmetry in the $y$ direction. The coordinate origin $O$ is in the multicolor filter. The light rays reflected from the water surface pass through the beam splitter and will be imaged on an image sensor passing through the multicolor filter. In this way, a light ray reflected by an object point on the water surface is imaged onto an image point on the image sensor with its color selected depending on its direction.
As shown in Fig. 1, a position vector ${\boldsymbol r}$ represents a point where a light ray with an angle $\theta$ to the optical axis passes through the multicolor filter placed at the focal plane. The ${\boldsymbol r}$, which is projected on $xy$ plane, can be derived based on the geometrical optics with an azimuth angle $\phi$ to the $x$ direction and the focal length $f$ as
A two-dimensional angle vector ${\boldsymbol \theta}$ having two components of ${\theta _x}$ and ${\theta _y}$ is here defined as
Under the assumption of the paraxial approximation, the two-dimensional angle vector ${\boldsymbol \theta}$ can be written using the position vector ${\boldsymbol r}$ in the two-dimensional representation as
3. THEORETICAL EQUATIONS FOR CAPILLARY WAVES
A. Dispersion Relation between the Wavelength and Frequency
A dispersion relation between the wavelength and frequency for a capillary wave is an important feature used to characterize the wave. The dispersion relation can be theoretically derived based on the elastic theory as follows.
A capillary wave in the ($x$, $y$, $z$) Cartesian coordinate system is assumed to be propagating along the $x$ axis. A transverse surface displacement $h$ of the capillary wave can be written with an amplitude ${a_0}$ as
where $k$ denotes a wavenumber, and $\omega$ denotes an angular frequency. The wavenumber can be written using a wavelength $\Lambda$ of the capillary wave asA dispersion relation between the wavelength and frequency can be written based on the elastic theory as [11]
where $\sigma$ denotes a surface tension, and $\rho$ denotes a density of fluid.B. Surface Slope Angle Obtainable by the One-Shot BRDF Imaging System
The direction of a light ray reflected from the surface of a capillary wave can be described by the direction of the normal vector on the surface. Once the direction of the light ray is obtained by using the one-shot BRDF imaging system, the direction of the surface normal can then be determined. The surface slope angle, which is determined by the surface normal, can thus be derived from the light ray direction as follows.
A surface normal vector, ${\boldsymbol n}$, on the capillary wave propagating along the $x$ axis can be derived from the transverse surface displacement $h$, which can be written as [14]
A light ray direction, ${{\boldsymbol e}_r}$, reflected from the capillary wave surface, can be derived based on the geometrical optics, assuming that an incident light ray has a direction of ${{\boldsymbol e}_i}$ and undergoes regular reflection as
The incident light ray direction is here set parallel to the $z$ axis as
Using Eqs. (7)–(9), the reflected light ray direction, ${{\boldsymbol e}_r}$, can be written as
Inserting Eq. (4) into Eq. (10), the following equation can be derived as
The amplitude of the displacement is here assumed to be small enough in comparison with the wavelength. Equation (11) can thus be approximated as
The point ${\boldsymbol r}$, where the reflected light ray will pass through in the multicolor filter of the one-shot BRDF, can then be written in a three-dimensional representation as
The reflectance direction component, ${\theta _x}$, that is obtainable by the one-shot BRDF imaging system can thus be written using Eqs. (3) and (13) as
Equation (14) can also be rewritten as
This equation indicates that the surface slope of the capillary wave is obtainable from the reflectance direction component captured by the one-shot BRDF. The surface displacement $h$ in Eq. (15) can be considered as a scalar potential, from which the two-dimensional angle vector ${\boldsymbol \theta}$ can be derived [17,18]. Assuming that the surface displacement is sufficiently small in comparison with the wavelength, an angle of the surface slope, $\Theta$, can thus be derived from the reflectance direction component, ${\theta _x}$, as
Equation (16) indicates that the surface slope angle can be obtainable by the one-shot BRDF imaging system.
4. EXPERIMENT
Figure 2 shows a perspective view of a prototype of the one-shot BRDF imaging system. An imaging lens has a focal length of 200 mm (Nikon, NIKKOR). A collimator lens has a focal length of 200 mm, which converts the light rays emitted from an LED into parallel light rays. The divergent angle of the parallel light rays is set to 0.057° using a pinhole with a diameter of 0.4 mm. An image sensor has RGB color channels. A multicolor filter is set up to have 22 steps of hue variation from blue to red with a distance of 11.5 mm.
Fig. 2. Perspective view of a prototype of the one-shot BRDF imaging system. A collimator lens has a focal length of 200 mm, which converts the light rays emitted from an LED into parallel light rays. The divergent angle of the parallel light rays is set to 0.057° using a pinhole with a diameter of 0.4 mm. An image sensor has RGB color channels. A multicolor filter is set up to have 22 steps of hue variation from blue to red with a distance of 11.5 mm.
The capillary wave is formed by a transducer at an interface between water in a tank and air. The water has a depth of about 40 mm. The transducer used to apply sound pressure to the water surface is an airborne ultrasonic transducer (FUS-200A, Fuji Ceramics) with a resonance frequency of 200 kHz. The transducer is mounted facing the water surface with a gap using a flexible arm. A double sideband suppressed carrier amplitude modulation signal with a specified modulation frequency and a carrier frequency of 200 kHz is generated from an arbitrary waveform generator (33621A, Keysight) and fed into a power amplifier (HSA 4014, NF). The amplified signal of 10 modulation cycles is used to drive the transducer. The modulation frequency can be practically set in a wide range of less than about 10 kHz.
Fig. 3. Unprocessed images captured by the one-shot BRDF for the capillary waves with frequencies of 30, 60, and 90 Hz. The capillary wave propagates along the x direction.
5. RESULTS
A. Surface Slope Angle Distribution Obtained Using the One-Shot BRDF
The surface slope angle distribution of a capillary wave in the $xy$ plane can be calculated from a light ray direction field using Eq. (16), where the ray direction field is obtainable using the one-shot BRDF.
Figure 3 shows unprocessed images captured by the one-shot BRDF for the capillary waves with frequencies of 30, 60, and 90 Hz. The capillary wave propagates along the $x$ direction. It can be found from these images that the wavelengths of the capillary waves decrease as the frequencies increase.
As an alternative representation of the RGB color model, hue, saturation, and value are used for color mapping of light ray directions [19]. Figure 4 shows the color-direction relationship between the hue and the reflectance direction component ${\theta _x}$ in unit of degrees. The relationship was obtained by fitting a polynomial function of up to the fifth order to the experimental data obtained from the surface reflection of a glass plate tilted at several angles using a goniometer. The horizontal axis denotes the hue, and the vertical axis denotes the reflectance direction component ${\theta _x}$. The solid line indicates the color-direction relationship.
Fig. 4. Color-direction relationship between the hue and the reflectance direction component ${\theta _x}$ in unit of degrees. The relationship was obtained by fitting a polynomial function of up to the fifth order to the experimental data obtained from the surface reflection of a glass plate tilted at several angles using a goniometer. The horizontal axis denotes the hue, and the vertical axis denotes the reflectance direction component ${\theta _x}$.
A two-dimensional field of the reflectance direction component, ${\theta _x}$, can be obtained from the image captured by the one-shot BRDF using the color-direction relationship, as shown in Fig. 4. Once the reflectance field of ${\theta _x}$ is obtained, the surface slope angle distribution of the capillary wave is calculable using Eq. (16). Figure 5 shows images that represent the calculated surface slope angle, $\Theta$, for the respective capillary waves in the $xy$ plane with frequencies of 30, 60, and 90 Hz. Each capillary wave propagates along the $x$ direction. The color contour indicates the surface slope angle, $\Theta$, in unit of degrees.
Fig. 5. Calculated surface slope angle, $\Theta$, for the respective capillary waves in the $xy$ plane with frequencies of 30, 60, and 90 Hz. Each capillary wave propagates along the $x$ direction. The color contour indicates the surface slope angle, $\Theta$, in unit of degrees.
B. Comparison of the Measured Dispersion Relation with the Theoretical Prediction
The wavelength of the capillary wave is obtainable from the surface slope angle distribution, as shown in Fig. 5. Figure 6 shows a plot of the surface slope angle distribution that is projected onto a plane orthogonal to the $y$ axis for a capillary wave with a frequency of 90 Hz as an example. In this plot, a dot indicates a slope angle at each $x$, and a solid line indicates an averaged slope angle with respect to $x$. The wavelength is calculated by averaging the spacing between the positions where the solid line reaches its maximum or minimum values.
Fig. 6. Plot of the surface slope angle distribution that is projected onto a plane orthogonal to the $y$ axis for a capillary wave with a frequency of 90 Hz as an example.
Figure 7 shows the dispersion relation between the wavelength and frequency for the capillary wave. The horizontal axis denotes the wavelength, and the vertical axis denotes the frequency. The circle marks indicate experimental data, and a solid line indicates the theoretical prediction. The theoretical prediction is calculated by using Eq. (6) with the parameters of $\rho = {1000}\;{{\rm kg/m}^3}$ and $\sigma = {0.074}\;{\rm N/m}$. The experimental data are obtained using the one-shot BRDF for the capillary waves with the frequencies of 30, 40, 50, 60, 70, 80, 90, 100, and 110 Hz. The experimental results agree well with the theoretical prediction.
Fig. 7. Dispersion relation between the wavelength and frequency. The horizontal axis denotes the wavelength, and the vertical axis denotes the frequency. The circle marks indicate experimental data, and a solid line indicates the theoretical prediction. The experimental data are obtained using the one-shot BRDF optical system for the capillary waves with the frequencies of 30, 40, 50, 60, 70, 80, 90, 100, and 110 Hz. The experimental results agree well with the theoretical prediction.
6. DISCUSSION
At the leading edge of a capillary wave, the amplitude and wavelength are more likely to become unstable. The wavelength measurement, however, was carried out at the leading edge of the wave because a measurement carried out at the rear of the wave, as a subsequent wave after the leading edge, would result in the leading edge reaching the wall of the tank and being reflect back. When the reflected wave is mixed in the measurement, the calculation of the wavelength becomes somewhat complicated, making accurate measurement difficult. Therefore, the wavelength measurement is carried out at the leading edge of the wave. This might lead to the deviations in the dispersion relation between the theoretical prediction and the measured value, as shown in Fig. 7.
The velocity of a capillary wave increases with an increase in frequency. Therefore, if the frequency is increased, the exposure time for imaging with the one-shot BRDF should be set shorter. Otherwise, the imaging would be blurred. Achieving a shorter exposure time, however, requires the use of higher intensity illumination.
An instant measurement of a capillary wave in a two-dimensional field of view can be carried out using the proposed method. This enables real-time observation of the behavior of the capillary wave during its propagation. In addition, using the one-shot BRDF, it is possible to measure the three-dimensional surface of the capillary wave at an instant [14]. This also allows the amplitude distribution of the capillary wave in the two-dimensional field of view to be measured in real time during its propagation.
7. CONCLUSION
A measurement method to obtain a surface slope distribution of a capillary wave is proposed. The method uses the optical imaging system of the one-shot BRDF that can capture a one-shot image of the light-reflectance direction field in a two-dimensional image plane.
A theoretical equation for a surface slope angle of a capillary wave is derived as represented in Eq. (16). The equation shows that a distribution of the surface slope angle in a two-dimensional plane can be obtained using an image captured by the one-shot BRDF. Furthermore, a dispersion relationship between the wavelength and frequency can also be obtained from the captured images. This dispersion relationship is measured experimentally and agrees well with the theoretical prediction.
An instantaneous measurement of a capillary wave in a two-dimensional field of view can be carried out using the proposed method. This method enables real-time observation of the behavior of a capillary wave during its propagation.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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