⇦Chinese Physics Letters, 2016, Vol. 33, No. 7, Article code 074206 Angle Compensation and Asymmetry Effect of Light Diffracted by Millimeter Liquid Surface Slosh Wave * Yang Miao(苗扬)1,2**, Can Wu(吴灿)3, Ning Wang(王宁)4, Jia-Qi You(游佳琪)1 Affiliations 1College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124 2State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences, Xi'an 710119 3Beijing Aerospace Automatic Control Institute, China Academy of Launch Vehicle Technology, Beijing 100039 4Beijing Institute of Astronautical Systems Engineering, China Academy of Launch Vehicle Technology, Beijing 100076 Received 5 May 2016 *Supported by the Open Research Fund of State Key Laboratory of Transient Optics and Photonics of Chinese Academy of Sciences under Grant No SKLST201508, the China Postdoctoral Science Foundation Funded Project under Grant No 2015M580945, and the Government of Chaoyang District Postdoctoral Research Foundation.
**Corresponding author. Email: miaoyang@bjut.edu.cn Citation Text: Miao Y, Wu C, Wang N and You J Q 2016 Chin. Phys. Lett. 33 074206 Abstract The angle compensation method is adopted to detect sloshing waves by laser diffraction, in the case that the wavelength of the sloshing waves is much greater than that of the incident light. The clear diffraction pattern is observed to be of asymmetry, involving orders, position and interval of the diffraction spots that are discovered during the light grazing incidence. It is found that the larger the angle of incidence is, the more obvious the asymmetry is. The higher the negative diffraction orders are, the smaller the intervals between spots are. On the contrary, in the positive region, the higher the diffraction orders are, the larger the spot intervals are. The positive interval is larger than that of the same negative diffraction order. If the incident angle reaches 1.558 rad in the experiment, all positive diffraction orders completely vanish. Based on the mechanism of phase modulation and with the Fourier transform method, the relations between the incident angle and position, interval spaces, and orders of diffraction spots are derived theoretically. The theoretical calculations are compared with the experimental data, and the comparison shows that the theoretical calculations are in good agreement with the experimental measurement. DOI:10.1088/0256-307X/33/7/074206 PACS:42.87.-d, 43.35.-c, 42.25.Fx © 2016 Chinese Physics Society Article Text Based on an optical effect of surface acoustic wave (SAW), a measurement technique is effectively established to detect physical parameters of millimeter liquid surface slosh waves.[1] In classic surface acoustic-optical effect experiments, to clearly observe significant optical patterns, wavelengths of surface acoustic waves need be as small as possible. Therefore, early research of SAW mainly focused on the solid surface of the ultrasonic wave.[2-6] The SAW effect of liquid surface and its applications have been developed greatly.[7-14] In traditional experiments, wavelengths of SAWs are in micron orders and wavelengths of light is in submicrons. However, in our liquid slosh wave experiment, the wavelength of SAW is measured in millimeters. This is three orders of magnitude higher than those of traditional ultrasonic waves in earlier experiments. This means that, in our experiment, the SAW wavelength is far larger than that of light. According to diffraction conditions, it would be difficult to observe diffraction patterns in the situation that wavelengths of SAWs are four orders of magnitude higher than that of laser light. To achieve a clear diffraction pattern, it is found that the incidence angle of light can be used to compensate to obtain diffraction patterns. The larger the angle of incidence is, the more obvious the effect of the compensation is. When the angle of incidence reaches a certain level, the optical pattern appears to be more obviously asymmetric. The positions, the intervals and the order of the spots all are asymmetric (classical theory of acousto-optic diffraction states that they are all symmetric). Based on the slosh wave modulation function in the laser phase and the Fourier transform calculation, the analytical diffraction light distribution is derived under an oblique incident laser light condition. The mechanism of angle compensation and diffraction distribution asymmetry is clarified. The laser detection setup for liquid slosh waves is established and the experiments show that when the wavelength of surface sloshing wave is of millimeters, clear diffraction patterns can be observed to be clearly asymmetric by incident angle compensation. The comparison of theoretical and experimental results shows that they agree well with each other. It is supposed that the laser beam is obliquely incident to the surface slosh wave at an angle $\theta$ in Fig. 1. The phase of the incident light is modulated by the liquid surface wave which is similar to a sinusoidal reflection type phase grating. According to Ref. [1], the wave function modulated by liquid surface wave can be written as $$\begin{alignat}{1} \!\!\!\!\!\!u(x)\!=\!\!\!\!\sum\limits_{n=-\infty}^\infty {J_n} \Big(\frac{4\pi A\cos \theta}{\lambda}\Big)\exp \!\Big[2\pi j\Big(\frac{\sin \theta}{\lambda}\!+\!\frac{n}{{\it \Lambda}}\Big)x\Big],~~ \tag {1} \end{alignat} $$ where $J_n$ is the Bessel function of the first kind, $n$ is the integer, $A$ is the amplitude of liquid surface wave, $\lambda$ is the wavelength of the incident laser beam, ${\it \Lambda}$ is the wavelength of the liquid surface wave, and $x$ is the coordinate of the propagation direction of the liquid surface wave.
cpl-33-7-074206-fig1.png
Fig. 1. Principle diagram of light by SAW.
The diffraction intensity $I^{\rm (o)}(\phi)$ can be achieved by the Fourier transform[15] as $$\begin{alignat}{1} I^{\rm (o)}(\phi)=\,&\sum\limits_{n=-\infty}^\infty J_n^2 \Big(\frac{4\pi A\cos \theta}{\lambda}\Big)\sin c^2\\ &\cdot\Big\{w\Big[\frac{\sin \Big(\theta -\phi)}{\lambda}-\frac{\sin \theta}{\lambda}-\frac{n}{{\it \Lambda}}\Big]\Big\},~~ \tag {2} \end{alignat} $$ where $w$ is half the width of the laser beam which spreads along a liquid surface wave propagation direction. The first factor is about the intensity which determines the maximum intensity of spots in any order. When the incident angle and the amplitude are fixed, it can be confirmed that the maximum intensity of the diffraction spots at $n$th series are $J_n (\frac{4\pi A\cos \theta}{\lambda} )$. The second factor $\sin c^2 \{w [\frac{\sin (\theta -\phi)}{\lambda}-\frac{\sin \theta}{\lambda}-\frac{n}{{\it \Lambda}} ] \}$ is the related position. This factor includes two aspects. First, when this factor takes a maximum value, the center position of $n$th spots can be derived from $\phi$ according to the maximum value. Secondly, the intensity of the spots can be obtained from Eq. (1) with the position factor and $\phi$. In regard to these above-mentioned factors, the characteristics of diffraction spots are discussed as follows: (1) compensation of incidence angle. According to Eq. (2), the diffraction angle $\phi _n$ corresponding to the maximum value of $n$th diffraction spots can be written as $$\begin{align} \frac{\sin (\theta -\phi _n)}{\lambda}-\frac{\sin \theta}{\lambda}-\frac{n}{{\it \Lambda}}=0.~~ \tag {3} \end{align} $$ The interval spaces between adjacent diffraction spots are expressed as $$\begin{align} \sin \phi _n -\sin \phi _{n+1} =\,&\frac{1}{\cos \theta}\Big[\frac{\lambda}{{\it \Lambda}}+\sin \theta (\cos \phi _n\\ &-\cos \phi _{n+1})\Big],~~ \tag {4} \end{align} $$ where angle $\phi _n$ and $\phi _{n{\rm +}1}$ are very small compared with $\theta$. Equation (4) can be approximated to $$\begin{align} \Delta \phi _{n+1,n} \approx \frac{\lambda}{{\it \Lambda} \cos \theta},~~ \tag {5} \end{align} $$ where $\Delta \phi _{n+1,n}=\phi _n -\phi _{n+1}$ represents the angular spacing between the $n$th spots and the $n+1$th spots. It can be seen from Eq. (5) that angular spacing is not only related to $\frac{\lambda}{{\it \Lambda}}$, but also to $\theta$. The angular spacing can be adjusted by incident angle of laser light. The method of increasing the incident angle of laser light to increase the angular spacing is called the angle compensation. (2) Asymmetry of diffraction orders: According to Eq. (3), the positive and negative diffraction orders can be written respectively as follows: $$\begin{align} \sin (\theta -\phi _{j_-})-\sin \theta =\,&-j_- \frac{\lambda}{{\it \Lambda}}, \\ \sin (\theta +\phi _{j_+})-\sin \theta =\,&j_+ \frac{\lambda}{{\it \Lambda}},~~ \tag {6} \end{align} $$ where $j_- =j_+ =1,2,3,\ldots$ when the incident angle of laser light is given. Since $[\sin (\theta -\phi _{j_-})]_{\min} =0$, $[\sin (\theta +\phi _{j_+})]_{\max} =1$, from Eq. (6) we can obtain $$\begin{align} j_- < \frac{{\it \Lambda}}{\lambda}\sin \theta, ~~~~ j_+ < \frac{{\it \Lambda}}{\lambda}(1-\sin \theta).~~ \tag {7} \end{align} $$ It can be deduced from Eq. (7) that the numbers of positive and negative diffraction orders are restrained when the incident angle of laser light is large enough. The constraint conditions of positive and negative diffractions are different. This phenomenon indicates that the number of positive diffraction orders are not equal to that of negative diffraction orders at a given incident angle of laser light. In other words, the number of diffraction orders shows asymmetry. (3) Asymmetry of position of diffraction spots: according to Eq. (6) we have $$\begin{alignat}{1} \sin \phi _{j_-} =\,&\frac{1}{\cos \theta}\Big[j_- \frac{\lambda}{{\it \Lambda}}-\sin \theta (1-\cos \phi _{j_-})\Big], \\ \sin \phi _{j_+} =\,&\frac{1}{\cos \theta}\Big[j_+ \frac{\lambda}{{\it \Lambda}}+\sin \theta (1-\cos \phi _{j_+})\Big].~~ \tag {8} \end{alignat} $$ It is clear that $\sin \theta (1-\cos \phi _{j_-})>0$, $\sin \theta (1-\cos \phi _{j_+})>0$ and $j_- =j_+$ for the same order of positive and negative diffraction spots. From Eq. (8) we can derive $$\begin{align} \sin \phi _{j_-} < \sin \phi _{j_+}.~~ \tag {9} \end{align} $$ Since diffraction angles $\phi _{j_-}$ and $\phi _{j{\rm +}}$ are rather small, $\sin \theta (1-\cos \phi _{j_-})$ and $\sin \theta (1-\cos \phi _{j_+})$ are also small. In classical ultrasonic wave diffraction experiments, the wavelengths of laser and ultrasonic waves have the same order of magnitude. Here $\sin \theta (1-\cos \phi _{j_-})$ and $\sin \theta (1-\cos \phi _{j_+})$ are much smaller than $j\frac{\lambda}{{\it \Lambda}}$, Eq. (8) is then recommended to $\sin \phi _{j_-} =\sin \phi _{j_+}$. Therefore, the traditional theory states that the positions of the diffraction spots are symmetric. However, in our experiment, the wavelength of laser is much smaller than that of the slosh wave. The items $\sin \theta (1-\cos \phi _{j_-})$ and $\sin \theta (1-\cos \phi _{j_+})$ will not be ignored. Thus $\sin \phi _{j_-} \ne \sin \phi _{j_+}$. This shows that the same order of both positive and negative diffraction orders are asymmetric. (4) Asymmetry of interval space between diffraction spots: In our experiment, the wavelength of the laser is much smaller than that of the slosh wave. From Eq. (8) we have $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\Delta \phi _{j_-} =\,&\frac{1}{\cos \theta}\Big[\frac{\lambda}{{\it \Lambda}}+\sin \theta (\cos \phi _{(j+1)-} -\cos \phi _{j_-})\Big], \\ \!\!\!\!\!\!\!\!\Delta \phi _{j_+} =\,&\frac{1}{\cos \theta}\Big[\frac{\lambda}{{\it \Lambda}}-\sin \theta (\cos \phi _{(j+1)+} -\cos \phi _{j_+})\Big],~~ \tag {10} \end{alignat} $$ where $\Delta \phi _{j_-} =\sin \phi _{(j+1)_-} -\sin \phi _{j_-}$ and $\Delta \phi _{j+} =\sin \phi _{(j+1)_+} -\sin \phi _{j_+}$ are the interval space between adjacent diffraction spots corresponding to both negative and positive diffraction orders, respectively. In classical acoustic-optical diffraction experiments, the wavelength of laser is in submicrons while the wavelength of the ultrasonic wave is in microns. In this situation, $\frac{\lambda}{{\it \Lambda}}\gg \sin \theta (\cos \phi _{j_-} -\cos \phi _{(j+1)_-})$ and $\frac{\lambda}{{\it \Lambda}}\gg\sin \theta (\cos \phi _{j_+} -\cos \phi _{(j+1)_+})$. Equation (8) indicates that $\Delta \phi _{j_-} =\Delta \phi _{j_+} =\frac{\lambda}{{\it \Lambda} \cos \theta}$. This shows that the interval spaces of diffraction spots are only symmetric when grating constant and wavelength of laser are of the same order of magnitude. However, in the liquid slosh experiment, $\frac{\lambda}{{\it \Lambda}}\approx 10^{-4}$. This shows that items $\sin \theta (\cos \phi _{j_-} -\cos \phi _{(j+1)_-})$ and $\sin \theta (\cos \phi _{j_+} -\cos \phi _{(j+1)_+})$ in Eq. (10) will not be ignored when compared with $\frac{\lambda}{{\it \Lambda}}$. Then, it can be derived from Eq. (10) that $\Delta \phi _{j_-} \ne \Delta \phi _{j_+}$ and $\Delta \phi _{j_-} < \Delta \phi _{j_+}$. This shows that the interval spaces of diffraction order are asymmetric in our experiment. The depiction of the experimental setup is shown schematically in Fig. 2, which includes excitation, sample cell, light source and light path, optical electronic detection and a data processing system. A weight ball (WB in Fig. 1) of 0.5 kg, that is, 500 mm from the base plate, is hung on the lower end of the controller. The liquid sample is a rectangular shape, whose length and width are 400 mm and 300 mm, respectively. The cell is on the base plate and the liquid is distilled water. The instantaneous momentum induces the liquid surface sloshing. A He–Ne laser beam is divided by using a beam splitter. One of the beams is used to monitor the laser output stability. The other is directly upon the water surface where the sloshing wave is travelling. The laser beam is used in the method of grazing incidence and several different incident angles were selected in the experiment. For the oblique incidence, the shape of the illuminating area on the liquid surface is an ellipse whose major axis and minor axis are approximately 24 mm and 2.3 mm, respectively. The major axis is parallel to the direction in which the sloshing wave is travelling. The distance between the incident point to the observation screen is approximately 8.5 m. A CCD camera is used to detect the scattering pattern on the observation screen. The model is pike F-421B manufactured by AVT Germany. The CCD sensor is KAI-4022 and a black response. The pixel size is 7.4 μm $\times$ 7.4 μm and the resolution is $2048\times 2048$. The sampling rate is 16 fps and the minimum exposure time is 70 μs. The focal length of the camera is 50 mm.
cpl-33-7-074206-fig2.png
Fig. 2. Schematic diagram of the experimental setup. WB: weight ball, BS: beam splitter, OS: oblique surface
cpl-33-7-074206-fig3.png
Fig. 3. Light pattern diffracted from liquid slosh wave: (a) 1.435 rad, (b) 1.548 rad, and (c) 1.558 rad.
The experimental setup and optical path are adjusted according to Fig. 2. Before the weight ball falls, the liquid in the container is stationary. A stable circular spot can be observed in the CCD camera. When the weight ball collides with the base plate, the liquid surface wave is generated by this instantaneous impulse. Therefore, the amplitude of the surface slosh wave changes from small to large, and then descends during the collision process. The number of diffraction orders also changes from small to large then it descends, corresponding to the process. The optical patterns corresponding to the maximum slosh wave are shot by the CCD camera. The results of the different incident angle of the laser are shown in Fig. 3. Figure 3 shows the optical patterns under three different grazing angles, with Figs. 3(a), 3(b) and 3(c) corresponding to the incident angles in 1.435 rad, 1.548 rad and 1.558 rad, respectively. The zeroth orders of optical patterns are corresponding to zeroth position of the vertical axis. The diffraction spots above the zeroth in the vertical axis represent negative diffraction orders. The greater the distances of diffraction spots from the zeroth are, the higher the diffraction orders are. On the contrary, the diffraction spots below the zeroth in the vertical axis represent positive diffraction orders.
cpl-33-7-074206-fig4.png
Fig. 4. Variation between diffraction spot position and the incidence angle.
Figure 4 shows the variation between the positions of the diffraction spots and the incidence angle, where abscissa represents the incident angle and the ordinate represents the diffraction angle. The solid blue line and the red dotted line represent the theoretical curves of the positive and negative diffraction spots, respectively. Dots and crosses represent the experimental value of the positive and negative diffraction spots. It can be seen from Fig. 4 that the incident angle varies slowly from the diffraction angle at small angles, especially when less than 1.500 rad, although the diffraction angle increases when the incident angle is enhanced. This means that the angle compensation effect is not obvious in the case of small incidents. However, when the incident angle increases to a certain value, especially under grazing incidence conditions, the diffraction position with the incident angle greatly varies. When the incident angle increases to a certain value, the two curves do not overlap. These indicate that the positions of positive and negative diffraction orders relative to the zeroth spot are asymmetrical. It can be seen that theoretical and experimental values are in good agreement with each other in Fig. 4. the values show not only the angle compensation but also the position asymmetry of negative and positive diffraction spots of the first order. These are also deduced from Fig. 3. The clear diffraction pattern appears in the case of the incident angle 1.435 rad and the interval space of adjective spots is easily resolved. However, in comparison of three pictures in Fig. 3, it can be seen that the interval space increases with the incident angle. The space in Fig. 3(a) is smallest and then space in Fig. 3(b). The greatest space appears in Fig. 3(c). These do show the incident angle compensation effect. According to Eq. (7), the relationship between the incident angle and the maximum diffraction order can be derived theoretically as shown in Fig. 5. Figure 5 includes two parts, where the upper part represents the negative diffraction order and the lower part represents positive diffraction order. The abscissa represents the incident angle and the ordinate is the diffraction order number. The integers of the ordinates in the shadows represent possible diffraction orders. From the comparison of these two parts, it can be seen that the numbers of maximum diffraction orders corresponding to the positive and negative are not equal for one incident angle. The relationships between the incident angle and diffraction order corresponding to positive and negative are also different. Under grazing incidence condition, the theoretical number of positive diffraction order is rather small. If the number of diffraction spots detectable is greater than that of theoretical positive diffraction orders, then the diffraction orders observed are asymmetric for both positive and negative.
cpl-33-7-074206-fig5.png
Fig. 5. Diffraction order changing with the incidence angle.
It can be seen from Fig. 3(a) that diffraction effect is very obvious when multi-order diffraction spots appear in incident angle 1.435 rad. Five spots of negative diffraction orders and two spots of positive diffraction orders can be observed in the incident angle 1.548 rad in Fig. 3(b). When incident angle increases to 1.558 rad, it can be seen from Fig. 3(c) that more than four spots of negative diffraction orders appear. However, all positive diffraction spots vanish. Under grazing incidence conditions, asymmetric effect of diffraction orders appears. The larger the incident angle is, the more obvious the asymmetry effect of diffraction orders is. The relationship between the first and second spots of diffraction orders is shown in Fig. 6, where blue lines and a red dotted line represent $\Delta \phi _{2,1}$ and $\Delta \phi _{-1,-2}$ of theoretical results, respectively. Meanwhile, the experimental results corresponding to $\Delta \phi _{2,1}$ and $\Delta \phi _{-1,-2}$ are dot and cross in Fig. 6. It is obvious in Fig. 6 that the interval space varies with the incident angle. The interval spacing is with slight change in small incident angles. This shows that the interval spaces are equal under the conventional paraxial condition. However, interval space increases rapidly with an increasing incident angle under the graze incident condition. The interval space of the positive diffraction orders is much larger than that of the negative ones. This means that interval spaces are asymmetric.
cpl-33-7-074206-fig6.png
Fig. 6. Relationship between the interval space and the incident angle.
Table 1. Experimental and theoretical interval space $\theta=1.548$ rad.
Angle width Theoretical results Experimental data
$\Delta \phi _{-5,-4} $ 0.00261 0.00259
$\Delta \phi _{-4,-3} $ 0.00281 0.00285
$\Delta \phi _{-3,-2} $ 0.00307 0.00310
$\Delta \phi _{-2,-1} $ 0.00342 0.00349
$\Delta \phi _{-1,0} $ 0.00393 0.00389
$\Delta \phi _{0,1} $ 0.00477 0.00463
$\Delta \phi _{1,2} $ 0.00661 0.00709
According to Eqs. (8) and (10), the theoretical angular interval can be calculated at the incident angle of 1.548 rad. Meanwhile, the angular interval can be determined by the experimental results shown in Fig. 3(b) and other experimental data. The angular intervals in both theory and experiment are listed in Table 1. The results display that interval space corresponding to different diffraction orders are different. With negative diffraction orders, the higher the orders of diffraction are, the smaller the interval spaces are. In summary, the experimental results in grazing incident angles show asymmetry, which include the positions of diffraction spots, interval space and diffraction orders. It is shown theoretically and experimentally that the diffraction angular interval changes slowly with incident angle and the spot positions of negative and positive orders corresponding to that of zeroth order are symmetric in the case of small incident angles. It increases rapidly when the incident angle is under in the grazing condition. The positions of the spots of negative and positive orders corresponding to that of the zeroth order are asymmetric in this situation. The relationships between the orders and incident angle corresponding to positive and negative are different. The detectable orders of positive and negative are asymmetric at grazing incidence. All positive orders are limited when the incident reaches a certain degree. It is shown that the spot interval varies with the incident angle. The spot interval changes with the incident angle rapidly at grazing incidence. References Small amplitude liquid surface sloshing process detected by optical methodLight Diffraction by Ultrasonic Surface WavesMEASUREMENT OF ACOUSTIC SURFACE WAVE PROPAGATION CHARACTERISTICS BY REFLECTED LIGHTLight Diffraction by an Acoustic Surface Wave in Fused QuartzINTERNAL AND SURFACE CONTRIBUTION TO LIGHT DIFFRACTION BY SURFACE-ACOUSTIC WAVESReal-time convolution using acousto-optic diffraction from surface wavesObservation of diffraction-free propagation of surface acoustic waves around a homogeneous isotropic solid spherePrecise velocity measurement of surface acoustic waves on a bearing ballVisualization of surface acoustic waves by means of synchronous amplitude-modulated illuminationVisualization of low-frequency liquid surface acoustic waves by means of optical diffractionUnusual distribution of the diffraction patterns from liquid surface wavesA simple experiment on diffraction of light by interfering liquid surface wavesNonlinear Acoustic-Optical Effect and Extraordinary Diffraction Distribution in Liquid Surface
[1] Miao Y and Wang S 2014 Opt. Commun. 315 91
[2] Mayer W G and Lamers G B 1966 J. Acoust. Soc. Am. 40 1261
[3] Korpel A, Laub L J and Sievering H C 1967 Appl. Phys. Lett. 10 295
[4] Montgomery R M and Young E H 1971 J. Appl. Phys. 42 2585
[5] Alippi A, Palma A, Palmieri L and Socino G 1971 Appl. Phys. Lett. 18 552
[6] Kramer C J, Araghi M N and Das P 1974 Appl. Phys. Lett. 25 180
[7]Brier R, Leroy O and Devolder S 1997 Appl. Phys. Lett. 75 599
[8] Tsukahara Y, Nakaso N, Cho H and Yamanaka K 2000 Appl. Phys. Lett. 77 2926
[9] Yamanaka K and Cho H 2000 Appl. Phys. Lett. 76 2797
[10] Duncan B D 2000 Appl. Opt. 39 2888
[11] Miao R, Yang Z and Zhu J 2002 Appl. Phys. Lett. 80 3033
[12] Dong J, Miao R and Qi J 2006 J. Appl. Phys. 100 033108-033108
[13] Barik T K, Roy A and Kar S 2005 Am. J. Phys. 73 725
[14] Miao Y and Wang S 2013 Chin. Phys. Lett. 30 124304
[15]Goodman J W 1968 Introduction to Fourier Optics (San Francisco: McGraw-Hill) p 62