Quantitative measurement of phase variation amplitude of ultrasonic diffraction grating based on diffraction spectral analysis Meiyan Pan, Yingzhi Zeng, and Zuohua Huang Citation: Review of Scientific Instruments 85, 093112 (2014); doi: 10.1063/1.4895649 View online: http://dx.doi.org/10.1063/1.4895649 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulation and analysis of grating-integrated quantum dot infrared detectors for spectral response control and performance enhancement J. Appl. Phys. 115, 163101 (2014); 10.1063/1.4871855 Phononic crystal diffraction gratings J. Appl. Phys. 111, 034907 (2012); 10.1063/1.3682113 Simulation of Ultrasonic Technique Using Spectral Element Method AIP Conf. Proc. 820, 111 (2006); 10.1063/1.2184518 Observing Effects of Particle Size for a Slurry Using Ultrasonic Diffraction Grating Spectroscopy AIP Conf. Proc. 760, 1729 (2005); 10.1063/1.1916879 Quantitative accuracy of the mass-spring lattice model in simulating ultrasonic waves AIP Conf. Proc. 615, 152 (2002); 10.1063/1.1472793
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 093112 (2014)
Quantitative measurement of phase variation amplitude of ultrasonic diffraction grating based on diffraction spectral analysis Meiyan Pan,a) Yingzhi Zeng, and Zuohua Huangb) Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, Guangdong 510006, China
(Received 3 February 2014; accepted 1 September 2014; published online 25 September 2014) A new method based on diffraction spectral analysis is proposed for the quantitative measurement of the phase variation amplitude of an ultrasonic diffraction grating. For a traveling wave, the phase variation amplitude of the grating depends on the intensity of the zeroth- and first-order diffraction waves. By contrast, for a standing wave, this amplitude depends on the intensity of the zeroth-, first-, and second-order diffraction waves. The proposed method is verified experimentally. The measured phase variation amplitude ranges from 0 to 2π , with a relative error of approximately 5%. A nearly linear relation exists between the phase variation amplitude and driving voltage. Our proposed method can also be applied to ordinary sinusoidal phase grating. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895649] I. INTRODUCTION
A type of ultrasonic diffraction grating, produced by ultrasonic waves traveling in a transparent medium, alters the physical properties of the medium and the refractive index in a grid-like pattern. This type of grating, similar to an ordinary phase grating, alters the light phase and diffracts light when a light beam travels perpendicularly through this grating.1 A phase grating is regarded as a type of phase object because it can only alter or influence the spatial distribution of the light phase. For an ultrasonic diffraction grating, the phase spatial distribution is usually a sine or cosine function. Determining the phase variation amplitude of the ultrasonic diffraction grating or modulation of the sine function is of great importance for studying the physical properties of the medium. Ultrasonic diffraction gratings are widely used to determine some non-acoustic properties2, 3 of liquids or characteristics of some optical elements such as lenses.4 They can also be applied in various studies such as in adulteration determination,5 particle size measurement,6 laser characteristics investigation,2 ultrasonic transmission mechanism,7 or penetration characteristics study,8 and ultrasonic highsensitivity detector development.9 However, although an ultrasonic grating is a phase object, its quantitative phase measurement that can determine the refractive index of liquids10 has not been paid much attention. Phase distribution or variation in transparent objects is usually measured by interference. Two main techniques, namely, single-point and wholefield quantitative phase microscopies, are used for quantitative phase measurement. These techniques can measure or produce an image of the phase object. Single-point quantitative phase microscopy usually adopts phase dispersion,11 phase shift,12 derivative method,13 etc., which require too much time to scan the whole image of the phase object. Wholefield quantitative phase microscopy consumes less time but requires complex algorithms such as Hilbert transform,14 poa) [email protected] b) [email protected]
0034-6748/2014/85(9)/093112/5/$30.00
larization matrix,15 Laplace operator.16 Besides, complex experimental setups and multiple images are usually necessary. These techniques are reliable for nonperiodic phase objects, such as biological cells, but they are not suitable for objects with high spatial frequency such as ultrasonic diffraction gratings. Our group have ever proposed a method for the phase measurement and the imaging of phase objects using the interference scanning technique.17 However, applying this method to an ultrasonic diffraction grating is difficult because of the large volume of the ultrasonic diffraction grating being tested. A new method based on spectral analysis for the quantitative measurement of the phase variation amplitude of an ultrasonic diffraction grating is presented in this paper. The phase variation amplitude of the grating for a traveling wave depends on the zeroth- and first-order diffraction wave intensities, whereas that for a standing wave depends on the zeroth-, first-, and second-order diffraction wave intensities. Our method can be verified experimentally. The measured phase variation amplitude ranged from 0 to 2π with a 5% average relative error. II. DIFFRACTION SPECTRAL ANALYSIS OF AN ULTRASONIC DIFFRACTION GRATING A. Formation mechanism of an ultrasonic diffraction grating
Ultrasound is an elastic stress wave. During its propagation along the x-axis in a transparent liquid, ultrasound produces different local periodic pressures that result in periodic density and refractive index distribution along the same direction. Thus, an ultrasonic diffraction grating is generated. When a parallel light perpendicular to the x-axis travels through the ultrasonic diffraction grating, the phase distribution of the light can be written as1 φ1 (x, t) = φ0 + δφm sin(ωs t − ks x),
(1)
where φ 0 is a constant that represents the phase of light without ultrasound, δφ m is the phase variation amplitude induced 85, 093112-1
© 2014 AIP Publishing LLC
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Pan, Zeng, and Huang
Rev. Sci. Instrum. 85, 093112 (2014)
by the ultrasound and ranges from 0 to 2π , ωs is the circular frequency of the ultrasonic wave, and ks is the ultrasound wave number. Equation (1) indicates a phase grating formed by a traveling wave. With a reflector set opposite the ultrasonic transducer along the x-axis to obtain the reflected wave, the phase distribution of light having traveled through the liquid can be written as φ2 (x, t) = φ0 + δφm [sin(ωs t − ks x) + sin(ωs t + ks x)] (2) = φ0 + 2δφm sin(ωs t) cos(ks x),
1
0.8
0.6
0.4
0.2
which indicates a phase grating formed by a standing wave. The phase of the ultrasonic diffraction grating evidently depends on φ 0 and δφ m . The former is irrelevant in diffraction, whereas the latter can be obtained by our method. B. Analysis of light diffraction intensity
0
0
1/3
2/3
1 δφm/π
(3)
4/3
5/3
2
(a) 2 Theoretical data Fitted curve
5/3 4/3 δ φm/π
Transparent liquid does not change the light amplitude but alters its phase. The amplitude of the light beam is assumed to be equal to one for convenience. The complex amplitude of the light after passing through the grating can be expressed as u(x) = eiφ(x) .
0th 1st 2nd 3rd 4th 5th 6th
In(Normalized)
093112-2
1
2/3 1/3
1. Grating formed by a traveling wave 0
Based on the Fourier transform theory and diffraction principle, the complex amplitude of the diffraction of the grating can be expressed as U xf = F {u(x)} ∞ xf n i(ω+nωs )t , (4) − = Jn (δφm )e δ λf λs n=−∞ where xf is the diffraction plane coordinate, λ is the light wavelength, f is the lens focal length, and Jn is a Bessel function. Thus, the nth order diffraction wave intensity can be obtained as In = Jn2 (δφm ).
(5)
The diffraction wave light intensity depends on the phase variation amplitude δφ m . The intensities of the first seven ordered diffraction waves are selected to investigate their dependences on δφ m . As Fig. 1(a) shows, all intensities decrease oscillatorily as δφ m increases. When δφ m < π /3, high intensities of the zeroth- and ±first-order diffraction waves are observed, especially that of the zeroth-order wave. The proportion of diffraction intensity for the zeroth- and first-order diffraction wave intensities can be defined as follows: ⎧ ∞ ⎪ ⎪ In , or ⎪ ⎨ R1 = (I0 + I1 ) n=−∞
⎪ ⎪ ⎪ ⎩ R1 = (I0 + I−1 )
∞
n=−∞
(6) In ,
0.2
0.4
0.6
0.8
1
R1
(b) FIG. 1. Relation between δφ m and the diffraction intensity of ultrasonic grating formed by a traveling wave. Dependence of (a) diffraction intensity on δφ m and (b) δφ m on R1 . The blue points represent the theoretical data, and the fitted curve is in purple.
which is a monotonic function of δφ m . Therefore, a monotone dependence of δφ m on R1 that may not be analytical should exist. Polynomial fitting with the least square method is adopted to obtain the fitted function [Fig. 1(b)], which is expressed as δφm (R1 ) = 2π × −0.165R110 + 0.954R19 − 2.411R18 + 3.483R17 − 3.169R16 + 1.885R15 − 0.736R14 + 0.184R13 − 0.028R12 + 0.002R1 × 105 . (7) Its correlation coefficient is 0.987. The fitting is prominent, especially when R1 ∈ (0.55, 1) and δφ m < π /2. 2. Grating formed by a standing wave
The complex amplitude of the light passing through the grating induced by a standing wave can be written as ∞ ∞ xf n U (xf ) = Jn+b (δφm )Jb (δφm )eiωn,b t δ − , λf λs n=−∞b=−∞ (8)
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093112-3
Pan, Zeng, and Huang
Rev. Sci. Instrum. 85, 093112 (2014)
1 0th 1st 2nd 3rd 4th 5th 6th
0.6
0.4
FIG. 3. Experimental setup for the diffraction spectrum observation.
n
I (Normalized)
0.8
0.2
0
which is also a monotonic function of δφ m . With polynomial fitting, the dependence of δφ m on R2 [Fig. 2(b)] can be obtained as 0
1/3
2/3
1 δφ /π
4/3
5/3
2
δφm (R2 ) = 2π × (−0.411R29 + 2.208R28 − 5.119R27
m
(a)
+ 6.688R26 − 5.412R25 + 2.806R24 − 0.933R23
2
+ 0.193R22 − 0.023R2 + 0.001) × 104 .
Theoretical data Fitted curve
5/3
Its correlation coefficient is 0.998, which indicates a prominent fitting, especially when R2 ∈ (0.55, 1) and δφ m < 2π /3.
m
δφ /π
4/3
III. EXPERIMENTS AND RESULTS
1
2/3 1/3 0 0.2
(11)
0.4
0.6 R
0.8
1
2
(b) FIG. 2. Relation between δφ m and the diffraction intensity of ultrasonic grating formed by a standing wave. Dependence of (a) diffraction intensity on δφ m and (b) δφ m on R2 . The blue points represent the theoretical data, and the fitted curve is in purple.
where ωn, b = ω + (n + 2b)ωs . Thus the intensity of the nth order diffraction wave is In =
∞
2 Jn+b (δφm )Jb2 (δφm ).
(9)
b=−∞
The diffraction wave intensities also clearly depend on δφ m . Similarly, the intensities of the first seven ordered diffraction waves were investigated [Fig. 2(a)]. The zeroth-order diffraction wave always displays the strongest intensity and the oscillations are all much weaker. When δφ m < π /3, high intensities for the zeroth-, ±first-, and ±second-order diffraction waves are observed. According to the normalized intensities of the zeroth-, first-, and second-order diffraction waves, R2 can be defined as ⎧ ∞ ⎪ ⎪ In , or ⎪ ⎨ R2 = (I0 + I1 + I2 ) n=−∞
⎪ ⎪ ⎪ + I−2 ) ⎩ R2 = (I0 + I−1
∞
n=−∞
We designed the experimental setup shown in Fig. 3 to demonstrate the feasibility of our proposed method. A He-Ne laser with a 632.8 nm wavelength is used as the light source. A beam expander and lens L1 are used for collimation. An aperture is set to obtain the paraxial light. The acousto-optic medium is water, and the ultrasonic transducer is a piezoelectric ceramic transducer (PZT) ultrasonic generator with a 2.5 MHz resonance frequency and a 20 V maximal driving voltage. A polarizer is used to adjust the light intensity. On the back focal plane of lens L2 , the diffraction spectrum appears and is recorded by a CCD camera, on which a filter is fixed to eliminate the disturbance of background illumination. A. Quantitative measurement of the phase variation amplitude of an ultrasonic diffraction grating
An image of the diffraction spectrum is shown in Fig. 4. δφ m < π /3 because diffraction wave intensities of orders higher than the second order are too weak to be observed. In the experiments, the polarizer was adjusted to keep the intensity below the unsaturated gray value18 provided by the CCD camera. The experimental measurement results are listed in Table I, where three images with different intensities
(10) In ,
FIG. 4. Image of the diffraction spectrum, with saturated gray value for convenient observation.
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093112-4
Pan, Zeng, and Huang
Rev. Sci. Instrum. 85, 093112 (2014)
TABLE I. Experimental results of the phase variation amplitude of an ultrasonic grating induced by a standing wave. Diffraction
Order −2 −1
Measured value
Im.1 Im.2 Im.3
2 3 3
22 35 46
0 111 172 223
+1 +2 22 35 46
2 3 3
R2
δφ m (π )
0.849 0.195 ± 0.009 0.846 0.20 ± 0.01 0.847 0.20 ± 0.01
are considered. δφ m = 0.20π ± 0.01π and its average relative error is 5.00%. Thus, the proposed method is feasible.
B. Influence of driving voltage on the phase variation amplitude
The power of an ultrasonic generator depends on the driving voltage (V ). Thus, the phase variation amplitude should increase with the driving voltage. The dependence of δφ m on V is illustrated in Fig. 5, which shows their nearly linear relation and indirectly indicates the feasibility of our proposed method.
0.15 0.14
Measured data Linear fitting
TABLE II. Experimental results of two different methods.
δφ m (π )
Number of grating
I
II
III
by spectrum analysis by scanning image
0.72 0.68
0.50 0.48
0.44 0.43
C. Measurement of the phase variation amplitude of ordinary sinusoidal phase gratings
To illustrate the accuracy of the proposed method, the ultrasonic diffraction gratings were replaced with ordinary sinusoidal phase gratings fabricated by the holographic technique. Three sinusoidal phase gratings have been measured by our method and the interference scanning imaging technique,17 respectively, and the experimental results are listed in Table II. The diffraction spectrum of grating I and its spacial phase distribution obtained by the scanning image technique is depicted in Fig. 6. The corresponding δφ m is so large that even ±fourth-orders can be observed [Fig. 6(a)], and it is easily obtained by Eqs. (10) and (11). We also acquired the average δφ m of grating I from its two-dimensional phase distribution shown in Fig. 6(b). Considering the experimental errors, results of the two different methods are very similar. The differences may be due to system errors or different measured areas of the gratings.
m
δφ /π
0.13 0.12 0.11 0.1 0.09 12
14
16 Driving voltage/ V
18
20
(a) 0.2 Measured data Linear fitting
δφm/π
0.15
0.1
0.05
0
5
10 15 Driving voltage/ V
20
(b) FIG. 5. Relation between δφ m and the driving voltage: gratings induced by (a) a traveling wave and (b) a standing wave.
FIG. 6. Experimental results of grating I: (a) Image of its diffraction spectrum; (b) Spacial distribution of its phase obtained using the scanning image technique.
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093112-5
Pan, Zeng, and Huang
IV. CONCLUSION
The phase variation amplitude of an ultrasonic diffraction grating can be quantitatively measured by diffraction spectral analysis. The amplitude of a traveling wave depends on the zeroth- and first-order diffraction wave intensities, whereas it can be determined by the zeroth-, first-, and second-order diffraction wave intensities for a standing wave. The proposed method requires simple procedures and algorithms. It can also be effectively verified experimentally and applied to ordinary sinusoidal gratings. The proposed method can also be applied to measuring other media properties. ACKNOWLEDGMENTS
The first author would like to thank Professor Chen Yihang for his help. The authors acknowledge the financial support provided by the National Undergraduates Innovating Experimentation Project and Guangzhou Science Research Project. The authors also thank the reviewers for their useful comments and suggestions.
Rev. Sci. Instrum. 85, 093112 (2014) 1 D.
T. Pierce and R. L. Byer, Am. J. Phys. 41, 314 (1973). A. Luna, M. A. Real, and D. V. Duran, Am. J. Phys. 70, 874 (2002). 3 K. Ferria, N. Laouar, and N. Bouaouadja, Opt. Appl. 41, 109 (2011). 4 K. Soni and R. S. Kasana, Measur. Sci. Technol. 18, 1667 (2007). 5 K. Soni and R. S. Kasana, Opt. Lasers Eng. 46, 6 (2008). 6 M. S. Greenwood and S. Ahmed, Ultrasonics 44, E1385 (2006). 7 M. S. Greenwood, J. Acoust. Soc. Am. 129, 2382 (2011). 8 M. S. Greenwood, J. Acoust. Soc. Am. 126, 2295 (2009). 9 A. Rosenthal, D. Razansky, and V. Ntziachristos, Opt. Lett. 36, 1833 (2011). 10 E. W. Max Born, Principles of Optics (Cambridge University Press, Cambridge, 1999). 11 D. Fu, C. Wonshik, S. Yongjin, O. Seungeun, Z. Yaqoob, P. Yongkeun, R. R. Dasari, and M. S. Feld, Opt. Exp. 17, 18878 (2009). 12 P. Gao, B. L. Yao, I. Harder, N. Lindlein, and F. J. Torcal-Milla, Opt. Lett. 36, 4305 (2011). 13 B. Bhaduri and G. Popescu, Opt. Lett. 37, 1868 (2012). 14 T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, Opt. Lett. 30, 1165 (2005). 15 Z. Wang, L. J. Millet, M. U. Gillette, and G. Popescu, Opt. Lett. 33, 1270 (2008). 16 T. Kim and G. Popescu, Opt. Lett. 36, 4704 (2011). 17 Z.-H. Huang and F.-C. Chen, Proc. SPIE 6150, 615030 (2006). 18 W. Lei, H. Wang, and C. Niu, Electron. Opt. Control 17, 65 (2010) (in Chinese). 2 D.
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 093112 (2014)
Quantitative measurement of phase variation amplitude of ultrasonic diffraction grating based on diffraction spectral analysis Meiyan Pan,a) Yingzhi Zeng, and Zuohua Huangb) Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, Guangdong 510006, China
(Received 3 February 2014; accepted 1 September 2014; published online 25 September 2014) A new method based on diffraction spectral analysis is proposed for the quantitative measurement of the phase variation amplitude of an ultrasonic diffraction grating. For a traveling wave, the phase variation amplitude of the grating depends on the intensity of the zeroth- and first-order diffraction waves. By contrast, for a standing wave, this amplitude depends on the intensity of the zeroth-, first-, and second-order diffraction waves. The proposed method is verified experimentally. The measured phase variation amplitude ranges from 0 to 2π , with a relative error of approximately 5%. A nearly linear relation exists between the phase variation amplitude and driving voltage. Our proposed method can also be applied to ordinary sinusoidal phase grating. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895649] I. INTRODUCTION
A type of ultrasonic diffraction grating, produced by ultrasonic waves traveling in a transparent medium, alters the physical properties of the medium and the refractive index in a grid-like pattern. This type of grating, similar to an ordinary phase grating, alters the light phase and diffracts light when a light beam travels perpendicularly through this grating.1 A phase grating is regarded as a type of phase object because it can only alter or influence the spatial distribution of the light phase. For an ultrasonic diffraction grating, the phase spatial distribution is usually a sine or cosine function. Determining the phase variation amplitude of the ultrasonic diffraction grating or modulation of the sine function is of great importance for studying the physical properties of the medium. Ultrasonic diffraction gratings are widely used to determine some non-acoustic properties2, 3 of liquids or characteristics of some optical elements such as lenses.4 They can also be applied in various studies such as in adulteration determination,5 particle size measurement,6 laser characteristics investigation,2 ultrasonic transmission mechanism,7 or penetration characteristics study,8 and ultrasonic highsensitivity detector development.9 However, although an ultrasonic grating is a phase object, its quantitative phase measurement that can determine the refractive index of liquids10 has not been paid much attention. Phase distribution or variation in transparent objects is usually measured by interference. Two main techniques, namely, single-point and wholefield quantitative phase microscopies, are used for quantitative phase measurement. These techniques can measure or produce an image of the phase object. Single-point quantitative phase microscopy usually adopts phase dispersion,11 phase shift,12 derivative method,13 etc., which require too much time to scan the whole image of the phase object. Wholefield quantitative phase microscopy consumes less time but requires complex algorithms such as Hilbert transform,14 poa) [email protected] b) [email protected]
0034-6748/2014/85(9)/093112/5/$30.00
larization matrix,15 Laplace operator.16 Besides, complex experimental setups and multiple images are usually necessary. These techniques are reliable for nonperiodic phase objects, such as biological cells, but they are not suitable for objects with high spatial frequency such as ultrasonic diffraction gratings. Our group have ever proposed a method for the phase measurement and the imaging of phase objects using the interference scanning technique.17 However, applying this method to an ultrasonic diffraction grating is difficult because of the large volume of the ultrasonic diffraction grating being tested. A new method based on spectral analysis for the quantitative measurement of the phase variation amplitude of an ultrasonic diffraction grating is presented in this paper. The phase variation amplitude of the grating for a traveling wave depends on the zeroth- and first-order diffraction wave intensities, whereas that for a standing wave depends on the zeroth-, first-, and second-order diffraction wave intensities. Our method can be verified experimentally. The measured phase variation amplitude ranged from 0 to 2π with a 5% average relative error. II. DIFFRACTION SPECTRAL ANALYSIS OF AN ULTRASONIC DIFFRACTION GRATING A. Formation mechanism of an ultrasonic diffraction grating
Ultrasound is an elastic stress wave. During its propagation along the x-axis in a transparent liquid, ultrasound produces different local periodic pressures that result in periodic density and refractive index distribution along the same direction. Thus, an ultrasonic diffraction grating is generated. When a parallel light perpendicular to the x-axis travels through the ultrasonic diffraction grating, the phase distribution of the light can be written as1 φ1 (x, t) = φ0 + δφm sin(ωs t − ks x),
(1)
where φ 0 is a constant that represents the phase of light without ultrasound, δφ m is the phase variation amplitude induced 85, 093112-1
© 2014 AIP Publishing LLC
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Pan, Zeng, and Huang
Rev. Sci. Instrum. 85, 093112 (2014)
by the ultrasound and ranges from 0 to 2π , ωs is the circular frequency of the ultrasonic wave, and ks is the ultrasound wave number. Equation (1) indicates a phase grating formed by a traveling wave. With a reflector set opposite the ultrasonic transducer along the x-axis to obtain the reflected wave, the phase distribution of light having traveled through the liquid can be written as φ2 (x, t) = φ0 + δφm [sin(ωs t − ks x) + sin(ωs t + ks x)] (2) = φ0 + 2δφm sin(ωs t) cos(ks x),
1
0.8
0.6
0.4
0.2
which indicates a phase grating formed by a standing wave. The phase of the ultrasonic diffraction grating evidently depends on φ 0 and δφ m . The former is irrelevant in diffraction, whereas the latter can be obtained by our method. B. Analysis of light diffraction intensity
0
0
1/3
2/3
1 δφm/π
(3)
4/3
5/3
2
(a) 2 Theoretical data Fitted curve
5/3 4/3 δ φm/π
Transparent liquid does not change the light amplitude but alters its phase. The amplitude of the light beam is assumed to be equal to one for convenience. The complex amplitude of the light after passing through the grating can be expressed as u(x) = eiφ(x) .
0th 1st 2nd 3rd 4th 5th 6th
In(Normalized)
093112-2
1
2/3 1/3
1. Grating formed by a traveling wave 0
Based on the Fourier transform theory and diffraction principle, the complex amplitude of the diffraction of the grating can be expressed as U xf = F {u(x)} ∞ xf n i(ω+nωs )t , (4) − = Jn (δφm )e δ λf λs n=−∞ where xf is the diffraction plane coordinate, λ is the light wavelength, f is the lens focal length, and Jn is a Bessel function. Thus, the nth order diffraction wave intensity can be obtained as In = Jn2 (δφm ).
(5)
The diffraction wave light intensity depends on the phase variation amplitude δφ m . The intensities of the first seven ordered diffraction waves are selected to investigate their dependences on δφ m . As Fig. 1(a) shows, all intensities decrease oscillatorily as δφ m increases. When δφ m < π /3, high intensities of the zeroth- and ±first-order diffraction waves are observed, especially that of the zeroth-order wave. The proportion of diffraction intensity for the zeroth- and first-order diffraction wave intensities can be defined as follows: ⎧ ∞ ⎪ ⎪ In , or ⎪ ⎨ R1 = (I0 + I1 ) n=−∞
⎪ ⎪ ⎪ ⎩ R1 = (I0 + I−1 )
∞
n=−∞
(6) In ,
0.2
0.4
0.6
0.8
1
R1
(b) FIG. 1. Relation between δφ m and the diffraction intensity of ultrasonic grating formed by a traveling wave. Dependence of (a) diffraction intensity on δφ m and (b) δφ m on R1 . The blue points represent the theoretical data, and the fitted curve is in purple.
which is a monotonic function of δφ m . Therefore, a monotone dependence of δφ m on R1 that may not be analytical should exist. Polynomial fitting with the least square method is adopted to obtain the fitted function [Fig. 1(b)], which is expressed as δφm (R1 ) = 2π × −0.165R110 + 0.954R19 − 2.411R18 + 3.483R17 − 3.169R16 + 1.885R15 − 0.736R14 + 0.184R13 − 0.028R12 + 0.002R1 × 105 . (7) Its correlation coefficient is 0.987. The fitting is prominent, especially when R1 ∈ (0.55, 1) and δφ m < π /2. 2. Grating formed by a standing wave
The complex amplitude of the light passing through the grating induced by a standing wave can be written as ∞ ∞ xf n U (xf ) = Jn+b (δφm )Jb (δφm )eiωn,b t δ − , λf λs n=−∞b=−∞ (8)
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093112-3
Pan, Zeng, and Huang
Rev. Sci. Instrum. 85, 093112 (2014)
1 0th 1st 2nd 3rd 4th 5th 6th
0.6
0.4
FIG. 3. Experimental setup for the diffraction spectrum observation.
n
I (Normalized)
0.8
0.2
0
which is also a monotonic function of δφ m . With polynomial fitting, the dependence of δφ m on R2 [Fig. 2(b)] can be obtained as 0
1/3
2/3
1 δφ /π
4/3
5/3
2
δφm (R2 ) = 2π × (−0.411R29 + 2.208R28 − 5.119R27
m
(a)
+ 6.688R26 − 5.412R25 + 2.806R24 − 0.933R23
2
+ 0.193R22 − 0.023R2 + 0.001) × 104 .
Theoretical data Fitted curve
5/3
Its correlation coefficient is 0.998, which indicates a prominent fitting, especially when R2 ∈ (0.55, 1) and δφ m < 2π /3.
m
δφ /π
4/3
III. EXPERIMENTS AND RESULTS
1
2/3 1/3 0 0.2
(11)
0.4
0.6 R
0.8
1
2
(b) FIG. 2. Relation between δφ m and the diffraction intensity of ultrasonic grating formed by a standing wave. Dependence of (a) diffraction intensity on δφ m and (b) δφ m on R2 . The blue points represent the theoretical data, and the fitted curve is in purple.
where ωn, b = ω + (n + 2b)ωs . Thus the intensity of the nth order diffraction wave is In =
∞
2 Jn+b (δφm )Jb2 (δφm ).
(9)
b=−∞
The diffraction wave intensities also clearly depend on δφ m . Similarly, the intensities of the first seven ordered diffraction waves were investigated [Fig. 2(a)]. The zeroth-order diffraction wave always displays the strongest intensity and the oscillations are all much weaker. When δφ m < π /3, high intensities for the zeroth-, ±first-, and ±second-order diffraction waves are observed. According to the normalized intensities of the zeroth-, first-, and second-order diffraction waves, R2 can be defined as ⎧ ∞ ⎪ ⎪ In , or ⎪ ⎨ R2 = (I0 + I1 + I2 ) n=−∞
⎪ ⎪ ⎪ + I−2 ) ⎩ R2 = (I0 + I−1
∞
n=−∞
We designed the experimental setup shown in Fig. 3 to demonstrate the feasibility of our proposed method. A He-Ne laser with a 632.8 nm wavelength is used as the light source. A beam expander and lens L1 are used for collimation. An aperture is set to obtain the paraxial light. The acousto-optic medium is water, and the ultrasonic transducer is a piezoelectric ceramic transducer (PZT) ultrasonic generator with a 2.5 MHz resonance frequency and a 20 V maximal driving voltage. A polarizer is used to adjust the light intensity. On the back focal plane of lens L2 , the diffraction spectrum appears and is recorded by a CCD camera, on which a filter is fixed to eliminate the disturbance of background illumination. A. Quantitative measurement of the phase variation amplitude of an ultrasonic diffraction grating
An image of the diffraction spectrum is shown in Fig. 4. δφ m < π /3 because diffraction wave intensities of orders higher than the second order are too weak to be observed. In the experiments, the polarizer was adjusted to keep the intensity below the unsaturated gray value18 provided by the CCD camera. The experimental measurement results are listed in Table I, where three images with different intensities
(10) In ,
FIG. 4. Image of the diffraction spectrum, with saturated gray value for convenient observation.
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093112-4
Pan, Zeng, and Huang
Rev. Sci. Instrum. 85, 093112 (2014)
TABLE I. Experimental results of the phase variation amplitude of an ultrasonic grating induced by a standing wave. Diffraction
Order −2 −1
Measured value
Im.1 Im.2 Im.3
2 3 3
22 35 46
0 111 172 223
+1 +2 22 35 46
2 3 3
R2
δφ m (π )
0.849 0.195 ± 0.009 0.846 0.20 ± 0.01 0.847 0.20 ± 0.01
are considered. δφ m = 0.20π ± 0.01π and its average relative error is 5.00%. Thus, the proposed method is feasible.
B. Influence of driving voltage on the phase variation amplitude
The power of an ultrasonic generator depends on the driving voltage (V ). Thus, the phase variation amplitude should increase with the driving voltage. The dependence of δφ m on V is illustrated in Fig. 5, which shows their nearly linear relation and indirectly indicates the feasibility of our proposed method.
0.15 0.14
Measured data Linear fitting
TABLE II. Experimental results of two different methods.
δφ m (π )
Number of grating
I
II
III
by spectrum analysis by scanning image
0.72 0.68
0.50 0.48
0.44 0.43
C. Measurement of the phase variation amplitude of ordinary sinusoidal phase gratings
To illustrate the accuracy of the proposed method, the ultrasonic diffraction gratings were replaced with ordinary sinusoidal phase gratings fabricated by the holographic technique. Three sinusoidal phase gratings have been measured by our method and the interference scanning imaging technique,17 respectively, and the experimental results are listed in Table II. The diffraction spectrum of grating I and its spacial phase distribution obtained by the scanning image technique is depicted in Fig. 6. The corresponding δφ m is so large that even ±fourth-orders can be observed [Fig. 6(a)], and it is easily obtained by Eqs. (10) and (11). We also acquired the average δφ m of grating I from its two-dimensional phase distribution shown in Fig. 6(b). Considering the experimental errors, results of the two different methods are very similar. The differences may be due to system errors or different measured areas of the gratings.
m
δφ /π
0.13 0.12 0.11 0.1 0.09 12
14
16 Driving voltage/ V
18
20
(a) 0.2 Measured data Linear fitting
δφm/π
0.15
0.1
0.05
0
5
10 15 Driving voltage/ V
20
(b) FIG. 5. Relation between δφ m and the driving voltage: gratings induced by (a) a traveling wave and (b) a standing wave.
FIG. 6. Experimental results of grating I: (a) Image of its diffraction spectrum; (b) Spacial distribution of its phase obtained using the scanning image technique.
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093112-5
Pan, Zeng, and Huang
IV. CONCLUSION
The phase variation amplitude of an ultrasonic diffraction grating can be quantitatively measured by diffraction spectral analysis. The amplitude of a traveling wave depends on the zeroth- and first-order diffraction wave intensities, whereas it can be determined by the zeroth-, first-, and second-order diffraction wave intensities for a standing wave. The proposed method requires simple procedures and algorithms. It can also be effectively verified experimentally and applied to ordinary sinusoidal gratings. The proposed method can also be applied to measuring other media properties. ACKNOWLEDGMENTS
The first author would like to thank Professor Chen Yihang for his help. The authors acknowledge the financial support provided by the National Undergraduates Innovating Experimentation Project and Guangzhou Science Research Project. The authors also thank the reviewers for their useful comments and suggestions.
Rev. Sci. Instrum. 85, 093112 (2014) 1 D.
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