## Abstract

Quantitatively measuring a dynamic liquid surface often presents a challenge due to high transparency, fluidity and specular reflection. Here, a novel Transmission-Lattice based Geometric Phase Analysis (TLGPA) method is introduced. In this method, a special lattice is placed underneath a liquid to be tested and, when viewed from above, the phase of the transmission-lattice image is modulated by the deformation of the liquid surface. Combining this with multi-directional Newton iteration algorithms, the dynamic deformation field of the liquid surface can be calculated from the phase variation of a series of transmission-lattice images captured at different moments. The developed method has the advantage of strong self-adaption ability to initial lattice rotational errors and this is discussed in detail. Dynamic 3D ripples formation and propagation was investigated and the results obtained demonstrated the feasibility of the method.

© 2014 Optical Society of America

## 1. Introduction

A liquid free surface can experience a wide spectrum of motions and form a general liquid sloshing which may affect structures such as fuel tanks, heavy ships, large dams and elevated water towers under ground motion etc.; this has always been an area of intense study in scientific and engineering fields [1–6]. For example, in some liquid propellant rockets or spacecraft, the 3D sloshing and control of fuel is very important for keeping such systems stable [2, 3]. Alternatively, other areas of important study in recent years include how the legs of some insects living on the water surface, e.g. Striders, can propel their body across the water surface [4–6]. The most fundamental way to experimentally resolve these problems is to measure the 3D shape of the water surface created by internal turbulence or external fluctuations. However, a great many conventional optical methods have failed in this measurement [7–11], mainly because of three main issues relating to transparent liquid surfaces; specular reflection, high transparency and fluidity.

The Fringe Reflection Technique (FRT) is one of the main practical methods to measure specular reflective surfaces [12–16]. Based on this technique, Tang and Su et al. [14, 15] developed an advanced Phase Measuring Deflectometry (PMD) method to measure the 3D shape of an aspheric mirror with the assistance of a Liquid Crystal Display (LCD). Huang et al. [16] improved the FRT by developing monoscopic fringe reflectometry for sensing water wave variations. However, this category of methods (i.e. FRT) contains some insurmountable limitations meaning that they are not suitable for measuring specular surfaces with large curvatures [16]. Other techniques for the deformation measurement of a liquid surface are mainly based on traditional laser interferometry, which includes holographic shearing interferometry [17] and holographic interferometry [18]. The sensitivity of the two methods can be at the sub-micron level, but the measuring range is within the sub-millimeter or micron scale and is limited by the fringe resolution.

Liu and Huang et al. [19, 20] first proposed the Fringe Transmission Technique (FTT) for measuring liquid surface deformation by analyzing the virtual image of transmission fringes, whereas the center line extracting method for fringe processing is a time-consuming process that has low precision [19]. A unidirectional iterative algorithm will lead to the accumulation of a calculation error and there can also be difficulty in aligning the fringe; a small rotational error or tilt angle inevitably exists between the principal direction of the pre-set fringe and the horizontal (or the vertical), thus calculation errors will be introduced that are difficult to eliminate. In general, this method is generally used for measuring the morphology of a static liquid surface and is not suitable for dynamic liquid surfaces.

A novel Transmission-Lattice based Geometric Phase Analysis (TLGPA) method is introduced for measuring the 3D dynamic deformation of a liquid surface. The principle of the method is based on placing a periodic lattice underneath a transparent liquid to be tested. When viewed from above, the phase of the lattice virtual image is modulated by the deformation of the liquid surface. Combining this with multi-directional Newton iteration algorithms, the dynamic deformation field of the liquid surface can be calculated from the phase variation of a series of transmission-lattice images captured at different moments. The complete relationship between the variation of transmitted light and liquid surface deformation was theoretically and systematically investigated. With this, it was possible to implement an automatic measurement by phase analysis, instead of semi-automatic processing by extracting center lines. This, to a large extent, saves calculation time and is extremely suitable for dynamic measurements. Furthermore, the method has a higher measurement precision due to its self-adaption in lattice rotational errors.

## 2. Mathematical basis for the deformation measurement of a liquid surface

A glance at the surface of a pond reveals one of the more delightful images of summer: the shimmering ripples with graceful strokes made by a tossed pebble. A leaf laying on the water moves, fish darts away, and several pretty large logs may still be seen lying on the bottom, where, owing to the undulation of the surface, they look like huge water snakes in motion. Such bizarre optical illusions take place due to the fluctuations of water surface change the propagation direction of the transmitted light resulting in a translocation and/or deformation of a virtual image of the submerged object. Here, the submerged object is replaced by a periodic lattice pattern. As shown in Fig. 1(a), when observed from directly above, a deformed virtual image of the periodic lattice pattern will be seen when a 3D ripple occurs.

The degree of surface deformation and degree of transformation of the virtual image are mathematical related. For example, point A transfers to A′ (A→A′, see Fig. 1(b)) when the water surface slopes, whereas the virtual image A′ is on the reverse extension line of the refracted ray and on the image plane as observed. So for right triangle (RT) AN* _{i}*A′, the geometric relationship is as follows:

*h*(

*x*,

*y*,

*t*) is the out-of-plane displacement field of the water surface at time

*t*;

*S*(

*x*,

*y*,

*t*) is the in-plane displacement of the virtual image of the underwater lattice at time

*t*, which is the distortion deformation of the transmission-lattice image; and ∠AN

*A′ =*

_{i}*ϕ*–

*θ*(see magnified view of a small portion section curve in Fig. 1(c)).

When a refracted ray is transmitted out of the water surface at an arbitrary position N* _{i}*, the Snell’s law applies:

*n*

_{w}and

*n*

_{a}(usually = 1) are the refractive indexes of water and air;

*θ*and

*ϕ*are the incidence and emergence angles making with the surface normal, respectively.

From Fig. 1(c) it can be seen that in right triangle (RT) N* _{i}*P

*N*

_{i}

_{i}_{-1}the mathematical relations between

*h*and

*ϕ*can be derived using the definition of the integration:

*y*and

*t*constant, vary

*x*, let

*h*and

*S*be functions of

*x*,

*y*and

*t*.

This is the mathematical basis of the method for measuring liquid surface deformation, which reveals that the 3D deformation measurement, i.e. *h*(*x*, *y*, *t*), can be changed into the in-plane displacement measurement of a virtual image of a transmission-lattice, i.e. *S*(*x*, *y*, *t*).

## 3. Method for the dynamic 3D measurement of a liquid surface

#### 3.1 Transmission-lattice based geometric phase analysis for displacement measurements

A schematic diagram of the experimental layout of this method is shown in Fig. 2. A physical periodic lattice pattern is placed underneath the water, and the transmitted light from the periodic lattice passes through the deformed liquid surface with its refracted light forming a distorted virtual image viewed by the high speed camera. For this distorted virtual image covered with dense periodic lattice, the phase of each point is modulated. The phase map of the lattice image is directly related to the in-plane displacement field, *S*(*x*, *y*, *t*) [21], which as well suggests that it is related to the 3D deformation of liquid surface, *h*(*x*, *y*, *t*).

As discussed in Section 2, the morphology of 3D ripples on the liquid surface can be measured through calculating the in-plane displacement or phase of the transmission-lattice image and, here, the GPA technique is used [21–24]. This technique is applied by centering a small aperture on a strong reflection (i.e. selecting peak values) in the Fast Fourier Transform (FFT) of a lattice image and then performing an Inverse Fast Fourier Transform (IFFT). The phase component of the resultant complex image gives information about the local displacement of the transmission-lattice. The intensity in a lattice image, *I*(** r**), is written as [21]:

*I*(

**) is the intensity in the image at position**

*r***, and**

*r***is the reciprocal lattice vectors describing the undistorted lattice. The local Fourier components are obtained by filtering in the Fourier space, and have amplitude and phase [22]:**

*g**A*(

_{g}**) describes the local contrast of the lattice image and the phase component.**

*r**P*(

_{g}**) describes their position change, which is related to the displacement field**

*r***by the following [23]:By combining the phase information from the lattice image, the displacement vector**

*S*(*r*)**can be calculated from the unwrapped phase images as [24]:Where**

*S*(*r*)**is the original lattice pitch in real space, here $a={g}^{-1}$ .**

*a*In the orthogonal coordinate system, the complete equations for liquid surface deformation directly from the phase or displacement of the transmission-lattice images are as given as follows:

*P*

_{g}_{1}and

*P*

_{g}_{2}are the two phase fields along the directions of

**and**

*g*_{1}**.**

*g*_{2}

*a*_{1}

*and*

_{x}

*a*_{2}

*are the component of*

_{x}

*a*_{1},

*a*_{2}in the

*x*direction. It is noted that in Eq. (9), the Newton iteration to be performed is along the positive

*x*direction. The calculations along other directions using the Newton iteration are similar to that of Eq. (9).

A calculation process for a liquid surface morphology using the TLGPA method is shown in Fig. 3. A simulated transmitted-lattice image (shown in Fig. 3(a)) is a distorted lattice implying liquid surface deformation. Figure 3(b) is the diffraction spectrum after performing FFT of Fig. 3(a). By centering a small aperture around a strong peak value in the diffraction spectrum of the lattice image and then performing IFFT, the raw phase image was easily calculated from the resulting complex image. The unwrapped phase image was obtained from the raw phase image (Figs. 3(c) and 3(d)) by subtracting the reference phase wherever necessary. Figures 3(e) and 3(f) are the calculated displacement fields in the *x* and *y* directions. Eventually, the whole height distribution of the simulated liquid surface can be calculated point by point using the Newton iteration algorithm along the *x* (or *y*) direction with the U or/and V displacement field (Figs. 3(e) and 3(f)), as shown in Fig. 3(g).

#### 3.2 Sensitivity and error analysis

From Eq. (9), it is known that the 3D shape of a liquid surface, *h*(*x*, *y*, *t*), is directly related to the displacement or phase, *S*(*x*, *y*, *t*), so that the sensitivity of the height measurement is directly dependable on the sensitivity of the in-plane displacement measurement. The GPA technique is a phase analysis method with a displacement measurement sensitivity of ~1% of the lattice pitch used [21]. If a lattice with a period of 1 mm is used then the displacement measurement sensitivity can reach up to 10 μm. This means that for a liquid depth of 20 mm and a calculation step of 1 pixel (supposing that 1 pixel = 0.1 mm as to common image apparatus), the sensitivity of the shape measurement is about 0.2 μm. In real experiments, if the lattice frequency is chosen properly, then the sensitivity can be further increased. In addition, increasing the initial liquid depth (H) can make the displacement *S*(*x*, *y*, *t*) larger (see Fig. 1(b)), which is equivalent to improving the sensitivity.

It should be noted that there is always an inevitable small rotational error or tilt between the *x* (or *y*) direction and the principal directions of the lattice, such that the Fringe Center Method (FCM) may result in some errors when calculating the fringe deformation as well as 3D shape measurement of liquid surface when ignoring the rotation. This problem can be resolved by the developed TLGPA method. As shown in Fig. 4, the rotation angle *α* (or *β*) between the *x* (or *y*) direction and the principal directions of lattice can be automatically calculated when the diffraction points have been selected. The precise calculation of the angle ** α** is based on the coordinate relationship between the center point of the selected diffraction orders and the center point of the whole lattice. Hence, the displacement distribution in

*x*direction,

*S*can be corrected automatically using:

_{x,}*S*

_{1}and

*S*

_{2}refers to the displacement along the $\overrightarrow{\text{OQ}}$ and $\overrightarrow{\text{OP}}$ direction, and the two directions of the lattice do not need to be orthogonal in nature. The resulting displacement component either in the

*x*or

*y*directions are regardless of the principal directions of the lattice whereas, for the FCM, only one displacement in the single principal direction of the fringe can be obtained.

Besides eliminating the rotational angle errors, the developed method can obtain the displacement fields in both the *x* and *y* directions at the same time. Thus the Newton iteration algorithm can be performed along multiple directions, such as the positive and negative *x* and *y* directions, and the overall morphology can be calculated with a weighted average to improve precision.

## 4. Technique verification

A dynamic process is consisting of a series static views at different moments, i.e. the static measurement can be used to verify the validity of the method. In this section, a validation experiment for measuring the 3D shape of a transparent plano-concave spherical lens was carried out to test the feasibility and the effectiveness of the TLGPA method. The test layout is shown in Fig. 5.

A printed periodic lattice was placed underneath the spherical lens to be tested. When observed from direct above of the lens, a distorted lattice image could be seen. By capturing this distorted image using a CCD camera, a digital lattice image for calculation the 3D shape of the lens was obtained, as shown in Fig. 6(a). Here a center area covered with 15 × 15 mm^{2} (framed by the red square) was selected to be calculated. The calculation steps for the 3D shape of lens were the same as Subsection 3.1. After performing FFT and IFFT to the calculation region with selecting small aperture around a strong peak value in the diffraction spectrum (see bottom right corner of Fig. 6(a)), the U and V fields can be calculated, as shown in Figs. 6(b) and 6(c). In this study, the directional of iteration is from the middle to the border, and weighted average with U and V fields was applied to reduce errors. Then the height distribution of the lens’ spherical surface can be calculated point by point using the Newton iteration algorithm with the U and V displacement field, as shown in Fig. 6(d). The comparison between the measurement value and real height value was presented in downside of Fig. 6(d). Figure 6(e) shows the quantitative comparison between the real height of the spherical lens and the measurement value along three different cross sections.

Figure 6(f) shows the relative measurement errors distribution within the calculation region. It can be seen that relatively large errors usually occur at the boundaries due to the Fourier algorithm is often sensitive to the image boundaries [25]. The maximum relative errors was located at the upper right part of the calculation region, and was no more than 3%. The root-mean-square of the measurement errors was about 0.62%, which demonstrates the superior effectiveness and feasibility of the developed method. It should be noted that the measurement uncertainties due to the manufacturing errors of the test object are not taken into account at this point.

## 5. Practical application with TLGPA for measuring dynamic water ripples

The morphology and dynamic evolution of a ripple was investigated experimentally using the developed TLGPA method. The ripple was generated by a water droplet dropped into still water, and the experimental layout used is shown in Fig. 2. The high-speed camera had a resolution of 1,024 × 1,024 pixels with a 1000 fps frame rate. The procedure designed for the experiment is listed as follows:

- 1) Establish the experiment system (transparent groove, light illuminators and high speed camera) according to the experiment layout in Fig. 2;
- 2) Place a physical periodic lattice below the transparent groove within the water, with its pitch pre-set as required;
- 3) Select the appropriate optical lens and adjust the optical layout such that the whole transmission image of the lattice can be observed clearly, and then captured on an original transmission-lattice image when the water surface is flat, as a reference for subsequent calculations and eliminating the environmental influence on the final results.
- 4) Drop a water droplet on the still water surface to generate a ripple pattern.
- 5) Record a series of deformed transmission-lattice images during the ripple evolution for the following computation.

The displacement field or phase in *x* and *y* directions were calculated using a series of transmission-lattice images captured using the GPA method. The height distribution of the liquid surface was then calculated according the Eq. (9) using the Newton iteration pixel by pixel. For the calculations, all the displacement data from four directions (i.e. both positive and negative *x* and *y* directions) were used to increase the accuracy using the weighted average method. The ripple photographs and the liquid surface morphology and their evolution at five moments are shown in Fig. 7.

Figure 7 shows the deformation process of a liquid surface. The deformation field is continuous and smooth which indicate the TLGPA method is feasible to measure the dynamic deformation of a liquid surface. In this experiment, the sensitivity of measuring the liquid surface height is 2.5 μm under the condition of a lattice frequency of 0.4 line/mm with a calculation step of 0.147 mm and a water depth of 6 mm.

## 6. Conclusion

A simple and feasible TLGPA method has been proposed for measuring the dynamic deformation of a transparent liquid surface. The complete quantitative relationship between the height changes of the liquid surface and the phase variation (or in-plane displacement) of transmission-lattice image were systematically and theoretically investigated. The developed method has the advantage of self-adaption to the initial lattice rotational errors. A validation experiment for measuring the 3D shape of a transparent plano-concave spherical lens was carried out and validated the feasibility and the accuracy of the TLGPA method. Combined with multi-directional Newton iteration algorithms, the dynamic deformation field of the liquid surface is obtained from the phase variation (or in-plane displacement vector field) of the transmission-lattice images captured at different moments. The dynamic deformation and propagation process of ripples on the water surface caused by a droplet were investigated. The developed method has the following advantages:

- 1) The principle of the method is easy to understand and the devices required are simple and convenient to operate, thus experimental efficiency is improved.
- 2) The deformation measurement sensitivity of a liquid surface is in micron scale and dependant on the phase measurement sensitivity, pitch size of lattice used, liquid depth, and calculating sub-step length.
- 3) The developed TLGPA method has a large height measurement range for transparent liquid surfaces and high calculation accuracy due to its self-adaption in the initial lattice rotational errors, as well as the multi-directional iterative algorithm through reduced error accumulation.

With this, it is possible to implement the automatic measurement of a liquid surface deformation by phase analysis for the first time, instead of a semi-automatic processing by extracting center lines which, to a large extent, saves on calculating time making it extremely suitable for the dynamic measurement of a transparent liquid surface.

## Acknowledgments

The authors are grateful to the financial support from the National Natural Science Foundation of China (Nos. 91216301, 11232008, 11072033 and 11372037), Program for New Century Excellent Talents in University (NCET-12-0036) and the Natural Science Foundation of Beijing, China (grant No. 3122027).

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