This work was supported by National Natural Science Foundation of China (grant no. 11772045) and Education and Teaching Reform Project of University of Science and Technology Beijing (grant no. JG2017M58).

1. Experimental setup and procedures
NOTE: Set up the experimental system as shown in Figure 2.
<ol>
	<li>Preparation of the vibration system
	<ol>
		<li>Prepare three 1.0-mm-thickness mirrored circular acrylic plates with diameter of 150 mm, 200 mm and 250 mm respectively. Drill a hole of 3 mm in diameter at the center of each plate. Mark several black points every 5 mm along an arbitrary radius.</li>
		<li>Attach each plate to the actuate bar of the vibrator with a bolt in the middle point. Drive the vibrator with a sine wave using a waveform generator, and default settings will be enough for the resonance experiment. 
		NOTE: The excitation direction of the vibrator is horizontal for the convenience of moving the screen afterwards.</li>
		<li>Acquisition of the resonance frequency
		<ol>
			<li>Place the laser pen to project the laser beam to the vibrating plate perpendicularly such that the beam is reflected to the light screen in the distance. The distances between the laser pen and the plate and the light screen are 120 mm and 500 mm, respectively. 
			NOTE: The farther the distance between the light screen and the vibrating plate, the more obvious the phenomenon appears. It is also noted that the present method can be used to measure either axisymmetric or non-axisymmetric mode shapes. Due to the consideration of simplicity and convenience, the present manuscript only demonstrates the application in determining axisymmetric mode shapes of three circular plates. Then we just need to measure the vibration amplitude along any radial direction to reconstruct the two-dimensional mode shape of the plate.</li>
			<li>Move the laser pen along the direction perpendicular to its length direction to make the incident point scan over a diameter while the signal generator changing its frequency continuously. Do it quickly until the spot length is significantly stretched along the diameter when scanning in a certain frequency range, and some spots with almost no expansion appear. For the plate with a diameter of 150 mm, 200 mm and 250 mm, the frequency ranges swept are 200-400 Hz, 100-300 Hz and 50-250 Hz, respectively.</li>
			<li>Scan this certain frequency range slowly and pick out the frequency at which the spot expands most obviously. It is found that for the plate with a diameter of 150 mm, 200 mm and 250 mm, the resonance frequencies are 346 Hz, 214 Hz and 150 Hz, respectively.</li>
		</ol>
		</li>
	</ol>
	</li>
	<li>Preparation of the light path and measurement system
	<ol>
		<li>Place the light screen parallel to the vibrating plate. Mark the distance with a meter ruler, and use 500 mm as the starting distance.</li>
		<li>Place the laser pen to project the beam perpendicularly on the plate such that the beam is reflected to the light screen in the distance. Make sure that the mark made before can be scanned while the laser pen is moving. 
		NOTE: The laser beam light must be projected perpendicularly on the plate.</li>
	</ol>
	</li>
	<li>Experimental measurement
	<ol>
		<li>Turn on the signal generator and set the excitation frequency to be the same as the resonance frequency obtained in step 1.1.3.3. The signal intensity should as small as possible once the light spot on the light screen is large enough to be recorded.</li>
		<li>Adjust the laser pen to make the incident point coincide with the first marker, which is the nearest marker to the fixed point of the plate.</li>
		<li>Move the screen from a distance D of 500 mm to 1000 mm and measure the spot length L on the screen every 50 mm. Record data in tabular form.</li>
		<li>Adjust the laser pen to make the incident point adjacent to the next marker in turn and repeat step 1.3.3 until all the markers have been measured. 
		NOTE: Since acrylic plates are easily deformed plastically under excitation, the experimental measurement process of one plate cannot be paused for a long time.</li>
		<li>Replace the former plate with the next one and repeat steps 1.3.1 to 1.3.4.</li>
	</ol>
	</li>
</ol>
2. Data processing
<ol>
	<li>Determine the angle &#952; between the incident and reflected light with relationship: 
	<img alt="Equation 1" src="/files/ftp_upload/61020/61020equ01.jpg" /> 
	where D is the distance between the rest position of the vibrating plate and the light screen, w is vibrating amplitude of the plate, and L is the length of the light spot on the light screen. Several pairs of D and L are obtained in step 1.3.3.</li>
	<li>Determine the slope <img alt="Equation 2" src="/files/ftp_upload/61020/61020equ02v3.jpg" /> of the mode shape by: 
	<img alt="Equation 3" src="/files/ftp_upload/61020/61020equ03.jpg" /> 
	NOTE: The obtained slope is always positive with Eqs.(1) and (2).</li>
	<li>Use a minus sign between two zero points to obtain the true slope distribution. 
	NOTE: It does not matter whether the revision begins from the first or the second zero point.</li>
	<li>Integrate the slope distribution of each plate and determine the integral constant by the nodes to obtain the mode shape with: 
	<img alt="Equation 4" src="/files/ftp_upload/61020/61020equ04.jpg" /> 
	NOTE: Nodes correspond to the largest slope of the mode shape. is a constant determined by the location of the nodal lines of the Chladni pattern shown in Figure 2.</li>
	<li>Compute the uncertainty of the slope12 with: 
	<img alt="Equation 5" src="/files/ftp_upload/61020/61020equ05.jpg" /> 
	NOTE: t0.95(n &#8211; 2) is the t distribution factor with 95% confidence and degrees of freedom n-2, and it is about 2 here. Sr is the standard error of the linear regression with D and L, Um denotes the uncertainty of the measured distance Di, and is 0.5 mm here. The average measured distance is defined by <img alt="Equation 6" src="/files/ftp_upload/61020/61020equ06v3.jpg" />, and n denotes the total number of measured Di.</li>
</ol>

<ol>
	<li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+free+circular+plates.+Part+1:+Normal+modes.">Vibrations of free circular plates. Part 1: Normal modes.</a> Proceedings of the Physical Society. 50 (1), 70-76 (1938).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+free+square+plates:+part+I.+Normal+vibrating+modes.">Vibrations of free square plates: part I. Normal vibrating modes.</a> Proceedings of the Physical Society. 51 (5), 831-844 (1939).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+free+plates:+isosceles+right-angled+triangles.">Vibrations of free plates: isosceles right-angled triangles.</a> Proceedings of the Physical Society. 53 (1), 35-39 (1941).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+Free+Rectangular+Plates.">Vibrations of Free Rectangular Plates.</a> Proceedings of the Physical Society Section B. 62 (5), 277-285 (1949).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+Free+Elliptical+Plates.">Vibrations of Free Elliptical Plates.</a> Proceedings of the Physical Society Section B. 63 (6), 451-455 (1950).</li><li>Tuan, P. H., Wen, C. P., Chiang, P. Y., Yu, Y. T., Liang, H. C., Huang, K. F., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Exploring+the+resonant+vibration+of+thin+plates:+Reconstruction+of+Chladni+patterns+and+determination+of+resonant+wave+numbers.">Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers.</a> The Journal of the Acoustical Society of America. 137 (4), 2113-2123 (2015).</li><li>Tuan, P. H., Lai, Y. H., Wen, C. P., Huang, K. F., Chen, Y. F. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Point-driven+modern+Chladni+figures+with+symmetry+breaking.">Point-driven modern Chladni figures with symmetry breaking.</a> Scientific Reports. 8 (1), 10844 (2018).</li><li>Castellini, P., Martarelli, M., Tomasini, E. P. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Laser+Doppler+Vibrometry:+Development+of+advanced+solutions+answering+to+technology's+needs.">Laser Doppler Vibrometry: Development of advanced solutions answering to technology's needs.</a> Mechanical Systems and Signal Processing. 20 (6), 1265-1285 (2006).</li><li>Sels, S., Vanlanduit, S., Bogaerts, B., Penne, R. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Three-dimensional+full-field+vibration+measurements+using+a+handheld+single-point+laser+Doppler+vibrometer.">Three-dimensional full-field vibration measurements using a handheld single-point laser Doppler vibrometer.</a> Mechanical Systems and Signal Processing. 126, 427-438 (2019).</li><li>Georgas, P. J., Schajer, G. S. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Simultaneous+Measurement+of+Plate+Natural+Frequencies+and+Vibration+Mode+Shapes+Using+ESPI.">Simultaneous Measurement of Plate Natural Frequencies and Vibration Mode Shapes Using ESPI.</a> Experimental Mechanics. 53 (8), 1461-1466 (2013).</li><li>Luo, Y., Feng, R., Li, X. D., Liu, D. H. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=A+simple+approach+to+determine+the+mode+shapes+of+Chladni+plates+based+on+the+optical+lever+method.">A simple approach to determine the mode shapes of Chladni plates based on the optical lever method.</a> European Journal of Physics. 40, 065001 (2019).</li><li>Coleman, H. W., Steele, W. G. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=.">.</a> Experimentation and uncertainty analysis for engineer. , (1999).</li></ol>

The authors have nothing to disclose.

The optical lever method is adopted in this paper to determine the mode shape of a plate, since the Chladni pattern can only show the nodal lines of a vibrating plate. To determine the mode shape of the plate, the relationship between the slope and distance of the light screen and spot length should be obtained in advance. Then through definite integration calculation, the mode shape of the Chladni pattern could be quantitatively determined.
Generally, the whole process of the present approach includes the following steps: (1) Perform the forced vibration test to obtain the resonance frequency of the plate. (2) Conduct forced vibration test near the resonance frequency, and record the coordinates of the nodes of Chladni pattern. These data are used for calibrating the absolute mode shape obtained by experimental tests. (3) The laser spot is perpendicularly projected to different radial locations of the plate, and the length of the light spot on the light screen is measured. This test needs to be repeated several times with different distances between the vibrating plate and light screen to obtain the linear regression value of the mode shape slope with Eq. (2). (4) Obtain the experimental mode shape of the Chladni pattern with Eq. (4) through post processing the raw experimental data.
It should be pointed out that, although the present experimental demonstration only shows the measurement of axisymmetric Chladni patterns, it also could be used for the determination of nonaxisymmetric Chladni patterns in a forward manner. Not only circular plates, but also other shapes, such as triangle, rectangular, and even irregular shapes could be employed to show the beauty of Chladni patterns. Furthermore, if the measuring point density, laser source, measuring tool, as well as the integral calculation method are carefully chosen, the accuracy of the proposed method could be adapted to required level.

Chladni mode shapes are of great interest in both science and engineering applications. Chladni patterns are reactions of physical waves, and one can illustrate the wave pattern with various methods. It is a well-known method to show the various modes of vibration on an elastic plate by outlining the nodal lines. Small particles are always employed to show the Chladni patterns, since they can stop at the nodes where the relative vibrating amplitude of the plate is zero, and the positions of the nodes vary with resonant mode to form various Chladni patterns.
Many researchers have paid attention to various Chladni patterns, but they only show the nodal lines of the mode shapes, the mode shapes (i.e., vibration amplitude) between the nodal lines are not illustrated. Waller investigated the free vibrations of a circle1, a square2, an isosceles right angled triangles3, a rectangular4, elliptical5 plates, and different Chladni patterns are illustrated therein. Tuan et al. reconstructed different Chladni patterns through both experimental and theoretical approaches, and the inhomogeneous Helmholtz equation is adopted during the theoretical modeling6,7. It is a popular method of using Laser Doppler Vibrometer (LDV) or Electronic Speckle Pattern Interferometry (ESPI) to quantitatively measure the mode shapes of the Chladni patterns8,9,10. Although LDV enables femtometer amplitude resolution and very high frequency ranges, unfortunately, the price of LDV is also a little expensive for classroom demonstration and/or college physics education. With this consideration, the present paper proposed a simple approach to quantitatively determine the mode shapes of a Chladni pattern with low cost, since only an extra laser pen and a light screen are needed here.
The present measurement method is illustrated in Figure 111. The vibrating plate has three different positions: the rest position, position 1 and position 2. Position 1 and 2 represent the two maximum vibrating places of the plate. A laser pen projects a straight beam on the surface of the plate, and if the plate locates at the rest position, the laser beam will be directly reflected to the light screen. While the plate locates at position 1 and 2, then the laser beam will be reflected to point A and B on the light screen, respectively. Due to the effect of persistence of vision, there will be a bright straight line on the light screen. The length of the bright light L is related to the distance D between the light screen and the location of the laser point. Different points on the plate have different slopes, which could be determined by the relationship between L and D. After obtaining the slope of the mode shape at different points on the plate, the problem turns into a definite integral. With the help of the boundary vibration amplitude of the plate and the discrete slope data, the mode shape of the vibrating plate can be obtained easily. The whole experimental setup is given in Figure 211.
This paper describes the experimental setup and procedure for the optical lever method to measure the Chladni mode shapes. Some typical experimental results are also illustrated.

<table><tbody><tr><td>Acrylic plates</td><td>Dongguan Jinzhu Lens Products Factory</td><td></td><td>Three 1.0-mm-thickness mirrored circular acrylic plates with diameter of 150 mm, 200 mm and 250 mm respectively. They are easily deformed.</td></tr><tr><td>Laser pen</td><td>Deli Group</td><td>2802</td><td>Red laser is more friendly to the viewer. The finer the laser beam, the better.</td></tr><tr><td>Light screen</td><td>Northern Tempered Glass Custom Taobao Store</td><td></td><td>Several layers of frosted stickers can be placed on the glass to achieve the effect of frosted glass.</td></tr><tr><td>Ruler</td><td>Deli Group</td><td>DL8015</td><td>The length is 1m and the division value is 1mm.</td></tr><tr><td>Signal generator</td><td>Dayang Science Education Taobao Store</td><td>TFG6920A</td><td>Common ones in university laboratories are available.</td></tr><tr><td>Vibrator</td><td>Dayang Science Education Taobao Store</td><td></td><td>The maximum amplitude is 1.5cm.The power is large enough to cause a noticeable phenomenon when the board vibrates. Otherwise, add a power amplifier.</td></tr></tbody></table>

measurement of chladni mode shapes with an optical lever method

Quantitatively determining the Chladni pattern of an elastic plate is of great interest in both physical science and engineering applications. In this paper, a method of measuring mode shapes of a vibrating plate based on an optical lever method is proposed. Three circular acrylic plates were employed in the measurement under different center harmonic excitations. Different from a traditional method, only an ordinary laser pen and a light screen made of ground glass are employed in this novel approach. The approach is as follows: the laser pen projects a beam to the vibrating plate perpendicularly, and then the beam is reflected to the light screen in the distance, on which a line segment made of the reflected spot is formed. Due to the principle of vision persistence, the light spot could be read as a bright straight line. The relationship between the slope of the mode shape, length of the light spot and the distance of the vibrating plate and the light screen can be obtained with algebraic operations. Then the mode shape can be determined by integrating the slope distribution with suitable boundary conditions. The full-field mode shapes of Chladni plate could also be determined further in such a simple way.

The excitation frequency that can excite axisymmetric Chladni pattern is determined through the frequency sweeping test. Three circular acrylic plates with diameters of 150 mm, 200 mm and 250 mm are tested, and results show that the first order axisymmetric resonance frequencies are 346 Hz, 214 Hz and 150 Hz for the three plates respectively. It is concluded that with larger diameter, the plate is more flexible, and the corresponding resonance frequency will be smaller. The Chladni patterns of the acrylic plate with different diameters are given in Figure 311.
Under the corresponding resonant frequency, the length of the light spot on the light screen of different plates can be measured and recorded. The regression value of mode shape slope can be obtained with Eq.(1), whose distributions along the radial direction of plate A, B and C are given in Table 111, and they are determined by measuring several different light spot lengths L of the specific laser point with different distance D.
Numerical simulation with ANSYS is performed to verify the present experimental results. The script code of APDL (ANSYS Parametric Design Language) is provided as a Supplemental File 1. Figure 411 shows the comparisons of the present experimental results and numerical results on the mode shape of different plates. It is very clear that all results with different conditions compare very well, which prove the feasibility of the present method in measuring the mode shape of plates.
<img alt="Figure 1" class="xfigimg" src="/files/ftp_upload/61020/61020fig01.jpg" /> 
Figure 1: Illustration of the present measurement method. 
The basic measurement principal is illustrated in this figure, with an emphasis on the incident and reflect light beam and the relationship of different geometric parameters. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig01large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 2" class="xfigimg" src="/files/ftp_upload/61020/61020fig02.jpg" /> 
Figure 2: The experimental setup. 
The picture of experimental setup is provided for clearly understand and replicate the measurement approach easily. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig02large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 3" class="xfigimg" src="/files/ftp_upload/61020/61020fig03a.jpg" /> 
<a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig03alarge.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 3" src="/files/ftp_upload/61020/61020fig03b.jpg" /> 
<a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig03blarge.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 3" src="/files/ftp_upload/61020/61020fig03c.jpg" /> 
Figure 3: Chladni pattern of different acrylic plates: (a) 150 mm, (b) 200 mm, (c) 250 mm. 
The Chladni patterns of three different acrylic circular plates are given respectively. The brown particles are sands and clearly show the nodal line of the Chladni patterns. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig03clarge.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 4" class="xfigimg" src="/files/ftp_upload/61020/61020fig04a.jpg" /> 
<a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig04alarge.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 4" src="/files/ftp_upload/61020/61020fig04b.jpg" /> 
<a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig04blarge.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 4" src="/files/ftp_upload/61020/61020fig04c.jpg" /> 
Figure 4: Comparisons of experimental results and numerical simulation for the mode shapes of different plates: (a) 150 mm, (b) 200 mm, (c) 250 mm. 
The numerical results obtained with ANSYS and the present experimental results are compared to verify the reliability of the present experimental method. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/61020fig04clarge.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<table border="1" fo:keep-together.within-page="1" fo:keep-with-next.within-page="always">
	<tbody>
		<tr>
			<td colspan="3">Plate A 
			(Diameter=150 mm)</td>
			<td colspan="3">Plate B 
			(Diameter=200 mm)</td>
			<td colspan="3">Plate C 
			(Diameter=250 mm)</td>
		</tr>
		<tr>
			<td>r/mm</td>
			<td>Directly calculated slope</td>
			<td>Revised slope</td>
			<td>r/mm</td>
			<td>Directly calculated slope</td>
			<td>Revised slope</td>
			<td>r/mm</td>
			<td>Directly calculated slope</td>
			<td>Revised slope</td>
		</tr>
		<tr>
			<td>5</td>
			<td>0.001913</td>
			<td>0.001913</td>
			<td>7</td>
			<td>0.002668</td>
			<td>0.002668</td>
			<td>7</td>
			<td>0.0013</td>
			<td>0.0013</td>
		</tr>
		<tr>
			<td>10</td>
			<td>0.001478</td>
			<td>0.001478</td>
			<td>12</td>
			<td>0.00269</td>
			<td>0.00269</td>
			<td>12</td>
			<td>0.001613</td>
			<td>0.001613</td>
		</tr>
		<tr>
			<td>15</td>
			<td>0.00144</td>
			<td>0.00144</td>
			<td>17</td>
			<td>0.002785</td>
			<td>0.02785</td>
			<td>17</td>
			<td>0.002055</td>
			<td>0.002055</td>
		</tr>
		<tr>
			<td>20</td>
			<td>0.001088</td>
			<td>0.001088</td>
			<td>22</td>
			<td>0.00269</td>
			<td>0.00269</td>
			<td>22</td>
			<td>0.002283</td>
			<td>0.002283</td>
		</tr>
		<tr>
			<td>25</td>
			<td>0.00061</td>
			<td>0.00061</td>
			<td>28</td>
			<td>0.002543</td>
			<td>0.002543</td>
			<td>27</td>
			<td>0.002618</td>
			<td>0.002618</td>
		</tr>
		<tr>
			<td>30</td>
			<td>0.000388</td>
			<td>0.000388</td>
			<td>38</td>
			<td>0.001858</td>
			<td>0.001858</td>
			<td>32</td>
			<td>0.00256</td>
			<td>0.00256</td>
		</tr>
		<tr>
			<td>35</td>
			<td>0.000883</td>
			<td>-0.000883</td>
			<td>48</td>
			<td>0.000748</td>
			<td>0.000748</td>
			<td>37</td>
			<td>0.00209</td>
			<td>0.00209</td>
		</tr>
		<tr>
			<td>40</td>
			<td>0.001733</td>
			<td>-0.001733</td>
			<td>58</td>
			<td>0.000668</td>
			<td>0.000668</td>
			<td>42</td>
			<td>0.002128</td>
			<td>0.002128</td>
		</tr>
		<tr>
			<td>45</td>
			<td>0.002478</td>
			<td>-0.002478</td>
			<td>68</td>
			<td>0.00082</td>
			<td>-0.00082</td>
			<td>47</td>
			<td>0.001723</td>
			<td>0.001723</td>
		</tr>
		<tr>
			<td>50</td>
			<td>0.003433</td>
			<td>-0.003433</td>
			<td>72</td>
			<td>0.001583</td>
			<td>-0.001583</td>
			<td>52</td>
			<td>0.001568</td>
			<td>0.001568</td>
		</tr>
		<tr>
			<td>55</td>
			<td>0.00389</td>
			<td>-0.00389</td>
			<td>77</td>
			<td>0.00241</td>
			<td>-0.00241</td>
			<td>57</td>
			<td>0.001</td>
			<td>0.001</td>
		</tr>
		<tr>
			<td>60</td>
			<td>0.002705</td>
			<td>-0.002705</td>
			<td>82</td>
			<td>0.002813</td>
			<td>-0.002813</td>
			<td>62</td>
			<td>0.004175</td>
			<td>0.004175</td>
		</tr>
		<tr>
			<td>65</td>
			<td>0.002283</td>
			<td>-0.002283</td>
			<td>87</td>
			<td>0.0026</td>
			<td>-0.0026</td>
			<td>67</td>
			<td>0.001175</td>
			<td>0.001175</td>
		</tr>
		<tr>
			<td>70</td>
			<td>0.002223</td>
			<td>-0.002223</td>
			<td>97</td>
			<td>0.002264</td>
			<td>-0.002264</td>
			<td>72</td>
			<td>0.002825</td>
			<td>-0.002825</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>77</td>
			<td>0.000873</td>
			<td>-0.000873</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>82</td>
			<td>0.001205</td>
			<td>-0.001205</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>87</td>
			<td>0.001538</td>
			<td>-0.001538</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>92</td>
			<td>0.00176</td>
			<td>-0.00176</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>97</td>
			<td>0.001983</td>
			<td>-0.001983</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>102</td>
			<td>0.002278</td>
			<td>-0.002278</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>107</td>
			<td>0.002745</td>
			<td>-0.002745</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>112</td>
			<td>0.00269</td>
			<td>-0.00269</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>117</td>
			<td>0.002783</td>
			<td>-0.002783</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td></td>
			<td>122</td>
			<td>0.002218</td>
			<td>-0.002218</td>
		</tr>
	</tbody>
</table>
Table 1: Slope distribution of the mode shape along radial direction. The calculated slope distribution of the mode shape along the radial direction is provided, and both original and revised slope are given to illustrate the process of revision.
Supplement File 1: ANSYS script for simulating the dynamic response and mode shape of a plate. <a href="https://cloudflare-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/61020/Supplentary_File_1_ANSYS_Script.txt" target="_blank">Please click here to download this file.</a>

Watch this Scientific Journal Video about Measurement of Chladni Mode Shapes with an Optical Lever Method at JoVE.com

Measurement of Chladni Mode Shapes with an Optical Lever Method

A simple method of measuring the Chladni mode shape on an elastic plate by the principle of an optical lever is proposed.

Quantitatively determining the Chladni pattern of an elastic plate is of great interest in both physical science and engineering applications. In this ...

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Nanyang Technological University

Research

JoVE Journal

Engineering

5.0K Views.  University of Science and Technology Beijing. A simple method of measuring the Chladni mode shape on an elastic plate by the principle of an optical lever is proposed.

Video: Measurement of Chladni Mode Shapes with an Optical Lever Method

Microwave Photonics Systems Based on Whispering-gallery-mode Resonators

Microwave photonics systems rely fundamentally on the interaction between microwave and optical signals. These systems are extremely promising for various areas of technology and applied science, such as aerospace and communication engineering, sensing, metrology, nonlinear photonics, and quantum optics. In this article, we present the principal techniques used in our lab to build microwave photonics systems based on ultra-high Q whispering gallery mode resonators. First detailed in this article is the protocol for resonator polishing, which is based on a grind-and-polish technique close to the ones used to polish optical components such as lenses or telescope mirrors. Then, a white light interferometric profilometer measures surface roughness, which is a key parameter to characterize the quality of the polishing. In order to launch light in the resonator, a tapered silica fiber with diameter in the micrometer range is used. To reach such small diameters, we adopt the "flame-brushing" technique, using simultaneously computer-controlled motors to pull the fiber apart, and a blowtorch to heat the fiber area to be tapered. The resonator and the tapered fiber are later approached to one another to visualize the resonance signal of the whispering gallery modes using a wavelength-scanning laser. By increasing the optical power in the resonator, nonlinear phenomena are triggered until the formation of a Kerr optical frequency comb is observed with a spectrum made of equidistant spectral lines. These Kerr comb spectra have exceptional characteristics that are suitable for several applications in science and technology. We consider the application related to ultra-stable microwave frequency synthesis and demonstrate the generation of a Kerr comb with GHz intermodal frequency.

The customized techniques developed in our lab to build microwave photonics systems based on ultra-high Q whispering gallery mode resonators are presented. The protocols to obtain and characterize these resonators are detailed, and an explanation of some of their applications in microwave photonics is given.

Microwave photonics systems rely fundamentally on the interaction between microwave and optical signals. These systems are extremely promising for various ...

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Thermal noise in high-reflectivity mirrors is a major impediment for several types of high-precision interferometric experiments that aim to reach the standard quantum limit or to cool mechanical systems to their quantum ground state. This is for example the case of future gravitational wave observatories, whose sensitivity to gravitational wave signals is expected to be limited in the most sensitive frequency band, by atomic vibration of their mirror masses. One promising approach being pursued to overcome this limitation is to employ higher-order Laguerre-Gauss (LG) optical beams in place of the conventionally used fundamental mode. Owing to their more homogeneous light intensity distribution these beams average more effectively over the thermally driven fluctuations of the mirror surface, which in turn reduces the uncertainty in the mirror position sensed by the laser light.
We demonstrate a promising method to generate higher-order LG beams by shaping a fundamental Gaussian beam with the help of diffractive optical elements. We show that with conventional sensing and control techniques that are known for stabilizing fundamental laser beams, higher-order LG modes can be purified and stabilized just as well at a comparably high level. A set of diagnostic tools allows us to control and tailor the properties of generated LG beams. This enabled us to produce an LG beam with the highest purity reported to date. The demonstrated compatibility of higher-order LG modes with standard interferometry techniques and with the use of standard spherical optics makes them an ideal candidate for application in a future generation of high-precision interferometry.

Large laser-interferometers are being constructed to create a new type of astronomy based on gravitational waves. Their sensitivities, as for many other high-precision experiments, are approaching fundamental noise limits such as the atomic vibration of their components. We are pioneering technologies to overcome these limits using novel laser beam shapes.

Thermal noise in high-reflectivity mirrors is a major impediment for several  types of high-precision interferometric experiments that aim to reach the  ...

Fabrication and Testing of Microfluidic Optomechanical Oscillators

Cavity optomechanics experiments that parametrically couple the phonon modes and photon modes have been investigated in various optical systems including microresonators. However, because of the increased acoustic radiative losses during direct liquid immersion of optomechanical devices, almost all published optomechanical experiments have been performed in solid phase. This paper discusses a recently introduced hollow microfluidic optomechanical resonator. Detailed methodology is provided to fabricate these ultra-high-Q microfluidic resonators, perform optomechanical testing, and measure radiation pressure-driven breathing mode and SBS-driven whispering gallery mode parametric vibrations. By confining liquids inside the capillary resonator, high mechanical- and optical- quality factors are simultaneously maintained.

Parametric optomechanical excitations have recently been experimentally demonstrated in microfluidic optomechanical resonators by means of optical radiation pressure and stimulated Brillouin scattering. This paper describes the fabrication of these microfluidic resonators along with methodologies for generating and verifying optomechanical oscillations.

Cavity optomechanics experiments that parametrically couple the phonon modes and photon modes have been investigated in various optical systems including ...

Automation of Mode Locking in a Nonlinear Polarization Rotation Fiber Laser through Output Polarization Measurements

When a laser is mode-locked, it emits a train of ultra-short pulses at a repetition rate determined by the laser cavity length. This article outlines a new and inexpensive procedure to force mode locking in a pre-adjusted nonlinear polarization rotation fiber laser. This procedure is based on the detection of a sudden change in the output polarization state when mode locking occurs. This change is used to command the alignment of the intra-cavity polarization controller in order to find mode-locking conditions. More specifically, the value of the first Stokes parameter varies when the angle of the polarization controller is swept and, moreover, it undergoes an abrupt variation when the laser enters the mode-locked state. Monitoring this abrupt variation provides a practical easy-to-detect signal that can be used to command the alignment of the polarization controller and drive the laser towards mode locking. This monitoring is achieved by feeding a small portion of the signal to a polarization analyzer measuring the first Stokes parameter. A sudden change in the read out of this parameter from the analyzer will occur when the laser enters the mode-locked state. At this moment, the required angle of the polarization controller is kept fixed. The alignment is completed. This procedure provides an alternate way to existing automating procedures that use equipment such as an optical spectrum analyzer, an RF spectrum analyzer, a photodiode connected to an electronic pulse-counter or a nonlinear detecting scheme based on two-photon absorption or second harmonic generation. It is suitable for lasers mode locked by nonlinear polarization rotation. It is relatively easy to implement, it requires inexpensive means, especially at a wavelength of 1550 nm, and it lowers the production and operation costs incurred in comparison to the above-mentioned techniques.

A protocol to detect and automate mode locking in a pre-adjusted nonlinear polarization rotation fiber laser is presented. The detection of a sudden change in the output polarization state when mode locking occurs is used to command the alignment of an intra-cavity polarization controller in order to find mode-locking conditions.

When a laser is mode-locked, it emits a train of ultra-short pulses at a repetition rate determined by the laser cavity length. This article outlines a ...

Characterization of Anisotropic Leaky Mode Modulators for Holovideo

Holovideo displays are based on light-bending spatial light modulators. One such spatial light modulator is the anisotropic leaky mode modulator. This modulator is particularly well suited for holographic video experimentation as it is relatively simple and inexpensive to fabricate1-3. Some additional advantages of leaky mode devices include: large aggregate bandwidth, polarization separation of signal light from noise, large angular deflection and frequency control of color1. In order to realize these advantages, it is necessary to be able to adequately characterize these devices as their operation is strongly dependent on waveguide and transducer parameters4. To characterize the modulators, the authors use a commercial prism coupler as well as a custom characterization apparatus to identify guided modes, calculate waveguide thickness and finally to map the device's frequency input and angular output of leaky mode modulators. This work gives a detailed description of the measurement and characterization of leaky mode modulators suitable for full-color holographic video.

This work describes fabrication and characterization of anisotropic leaky mode modulators for holographic video.

Holovideo displays are based on light-bending spatial light modulators.  One such spatial light modulator is the anisotropic leaky mode modulator.  This ...

Fabrication Procedures and Birefringence Measurements for Designing Magnetically Responsive Lanthanide Ion Chelating Phospholipid Assemblies

Bicelles are tunable disk-like polymolecular assemblies formed from a large variety of lipid mixtures. Applications range from membrane protein structural studies by nuclear magnetic resonance (NMR) to nanotechnological developments including the formation of optically active and magnetically switchable gels. Such technologies require high control of the assembly size, magnetic response and thermal resistance. Mixtures of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) and its lanthanide ion (Ln3+) chelating phospholipid conjugate, 1,2-dimyristoyl-sn-glycero-3-phospho-ethanolamine-diethylene triaminepentaacetate (DMPE-DTPA), assemble into highly magnetically responsive assemblies such as DMPC/DMPE-DTPA/Ln3+ (molar ratio 4:1:1) bicelles. Introduction of cholesterol (Chol-OH) and steroid derivatives in the bilayer results in another set of assemblies offering unique physico-chemical properties. For a given lipid composition, the magnetic alignability is proportional to the bicelle size. The complexation of Ln3+ results in unprecedented magnetic responses in terms of both magnitude and alignment direction. The thermo-reversible collapse of the disk-like structures into vesicles upon heating allows tailoring of the assemblies&#39; dimensions by extrusion through membrane filters with defined pore sizes. The magnetically alignable bicelles are regenerated by cooling to 5 &#176;C, resulting in assembly dimensions defined by the vesicle precursors. Herein, this fabrication procedure is explained and the magnetic alignability of the assemblies is quantified by birefringence measurements under a 5.5 T magnetic field. The birefringence signal, originating from the phospholipid bilayer, further enables monitoring of polymolecular changes occurring in the bilayer. This simple technique is complementary to NMR experiments that are commonly employed to characterize bicelles.

Fabrication procedures for highly magnetically responsive lanthanide ion chelating polymolecular assemblies are presented. The magnetic response is dictated by the assembly size, which is tailored by extrusion through nanopore membranes. The assemblies&#39; magnetic alignability and temperature-induced structural changes are monitored by birefringence measurements, a complimentary technique to nuclear magnetic resonance and small angle neutron scattering.

Bicelles are tunable disk-like polymolecular assemblies formed from a large variety of lipid mixtures. Applications range from membrane protein structural ...

Measuring the Interaction Force Between a Droplet and a Super-hydrophobic Substrate by the Optical Lever Method

The goal of this paper is to investigate the interaction force between droplets and super-hydrophobic substrates in the air. A measurement system based on an optical lever method is designed. A millimetric cantilever is used as a force sensitive component in the measurement system. Firstly, the force sensitivity of the optical lever is calibrated using electrostatic force, which is the critical step in measuring interaction force. Secondly, three super-hydrophobic substrates with different grid fractions are prepared with nanoparticles and copper grids. Finally, the interaction forces between droplets and super-hydrophobic substrates with different grid fractions are measured by the system. This method can be used to measure the force on the scale of sub-micronewton with a resolution on the scale of nanonewton. The in-depth study of the contact process of droplets and super-hydrophobic structures can help to improve the production efficiency in coating, film and printing. The force measurement system designed in this paper can also be used in other fields of microforce measurement.

The protocol aims to investigate the interaction between droplets and super-hydrophobic substrates in the air. This includes calibrating the measurement system and measuring the interaction force at super-hydrophobic substrates with different grid fractions.

The goal of this paper is to investigate the interaction force between droplets and super-hydrophobic substrates in the air. A measurement system based on ...

Spectral and Angle-Resolved Magneto-Optical Characterization of Photonic Nanostructures

Photonic crystals are periodic nanostructures that can support a variety of confined electromagnetic modes. Such confined modes are usually accompanied by local enhancement of electric field intensity that strengthens light-matter interactions, enabling applications such as surface-enhanced Raman scattering (SERS) and surface plasmon enhanced sensing. In the presence of magneto-optically active materials, the local field enhancement gives rise to anomalous magneto-optical activity. Typically, the confined modes of a given photonic crystal depend strongly on the wavelength and incidence angle of the incident electromagnetic radiation. Thus, spectral and angular-resolved measurements are needed to fully identify them as well as to establish their relationship with the magneto-optical activity of the crystal. In this article, we describe how to use a Fourier-plane (back focal plane) microscope to characterize magneto-optically active samples. As a model system, here we use a plasmonic grating built out of magneto-optically active Au/Co/Au multilayer. In the experiments, we apply a magnetic field on the grating in situ and measure its reciprocal space response, obtaining the magneto-optical response of the grating over a range of wavelengths and incident angles. This information enables us to build a complete map of the plasmonic band structure of the grating and the angle and wavelength dependent magneto-optical activity. These two images allow us to pinpoint the effect that the plasmon resonances have on the magneto-optical response of the grating. The relatively small magnitude of magneto-optical effects requires a careful treatment of the acquired optical signals. To this end, an image processing protocol for obtaining magneto-optical response from the acquired raw data is laid out.

Photonic band structure enables understanding how confined electromagnetic modes propagate within a photonic crystal. In photonic crystals that incorporate magnetic elements, such confined and resonant optical modes are accompanied by enhanced and modified magneto-optical activity. We describe a measurement procedure to extract the magneto-optical band structure by Fourier space microscopy.

Photonic crystals are periodic nanostructures that can support a variety of confined electromagnetic modes. Such confined modes are usually accompanied by ...

Preparation of Extracellular Matrix Protein Fibers for Brillouin Spectroscopy

Brillouin spectroscopy is an emerging technique in the biomedical field. It probes the mechanical properties of a sample through the interaction of visible light with thermally induced acoustic waves or phonons propagating at a speed of a few km/sec. Information on the elasticity and structure of the material is obtained in a nondestructive contactless manner, hence opening the way to in vivo applications and potential diagnosis of pathology. This work describes the application of Brillouin spectroscopy to the study of biomechanics in elastin and trypsin-digested type I collagen fibers of the extracellular matrix. Fibrous proteins of the extracellular matrix are the building blocks of biological tissues and investigating their mechanical and physical behavior is key to establishing structure-function relationships in normal tissues and the changes which occur in disease. The procedures of sample preparation followed by measurement of Brillouin spectra using a reflective substrate are presented together with details of the optical system and methods of spectral data analysis.

We present a protocol for the application of Brillouin light scattering spectroscopy to elastin and trypsin-purified type I collagen fibers of the extracellular matrix to extract their full elastic properties.

Brillouin spectroscopy is an emerging technique in the biomedical field. It probes the mechanical properties of a sample through the interaction of ...

Bioengineering

Fabrication of a Stable, Biologically Relevant Phantom Material

A Stable Phantom Material for Optical and Acoustic Imaging

Establishing tissue-mimicking biophotonic phantom materials that provide long-term stability are imperative to enable the comparison of biomedical imaging devices across vendors and institutions, support the development of internationally recognized standards, and assist the clinical translation of novel technologies. Here, a manufacturing process is presented that results in a stable, low-cost, tissue-mimicking copolymer-in-oil material for use in photoacoustic, optical, and ultrasound standardization efforts.
The base material consists of mineral oil and a copolymer with defined Chemical Abstract Service (CAS) numbers. The protocol presented here yields a representative material with a speed of sound c(f) = 1,481 &#177; 0.4 m&#183;s-1 at 5 MHz (corresponds to the speed of sound of water at 20 &#176;C), acoustic attenuation &#945;(f) = 6.1 &#177; 0.06 dB&#183;cm-1 at 5 MHz, optical absorption &#181;a(&#955;) = 0.05 &#177; 0.005 mm-1 at 800 nm, and optical scattering &#181;s&#39;(&#955;) = 1 &#177; 0.1 mm-1 at 800 nm. The material allows independent tuning of the acoustic and optical properties by respectively varying the polymer concentration or light scattering (titanium dioxide) and absorbing agents (oil-soluble dye). The fabrication of different phantom designs is displayed and the homogeneity of the resulting test objects is confirmed using photoacoustic imaging.
Due to its facile, repeatable fabrication process and durability, as well as its biologically relevant properties, the material recipe has high promise in multimodal acoustic-optical standardization initiatives.

Author Spotlight: A Stable Phantom Material for Optical and Acoustic Imaging  

This protocol describes the fabrication of a stable, biologically relevant phantom material for optical and acoustic biomedical imaging applications, featuring independently tunable acoustic and optical properties.

Establishing tissue-mimicking biophotonic phantom materials that provide long-term stability are imperative to enable the comparison of biomedical imaging ...