- Email: [email protected]
Visualized measurement of the acoustic levitation field based on digital holography with phase multiplication
Optics Communications 282 (2009) 4339–4344
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Visualized measurement of the acoustic levitation field based on digital holography with phase multiplication Puchao Zheng, Enpu Li, Jianlin Zhao *, Jianglei Di, Wangmin Zhou, Hao Wang, Ruifeng Zhang Institute of Optical Information Science and Technology, Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 30 July 2008 Received in revised form 18 June 2009 Accepted 5 August 2009
PACS: 42.40.Ht 42.40.Kw 87.50.Kk Keywords: Digital holography Holographic interferometory Acoustic levitation Sound pressure distribution Phase multiplication
a b s t r a c t By using digital holographic interferometory with phase multiplication, the visualized measurement of the acoustic levitation field (ALF) with single axis is carried out. The digital holograms of the ALF under different conditions are recorded by use of CCD. The corresponding digital holographic interferograms reflecting the sound pressure distribution and the interference phase distribution are obtained by numerical reconstruction and phase subtraction, which are consistent with the theoretical results. It indicates that the proposed digital holographic interferometory with phase multiplication can successfully double the fringe number of the interference phase patterns of the ALF and improve the measurement precision. Compared with the conventional optical holographic interferometory, digital holographic interferometory has the merits of quasi real-time, more exactitude and convenient operation, and it provides an effective way for studying the sound pressure distribution of the ALF. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Since the charge coupled device (CCD) has been used to record the digital hologram in 1994 at first time [1], with the development of the computer and the CCD technology, the digital holography attracts more and more attentions. The best merits of the digital holography are that it can use optoelectronic imaging devices such as CCD in stead of film as recording medium and numerically reconstruct the object field. It leaves out the complex wet physical and chemical operations required in processing the film, so that it not only improves the measurement efficiency but also permits the measurement process in real-time. Acoustic levitation [2] is an important technique for containerless solidification of materials, which uses the pressure of the acoustic standing wave with high intensity, namely acoustic radiation pressure. Under certain conditions this acoustic radiation pressure can levitate an object with density many times more than atmosphere in the air. For convenience, here we call the acoustic standing wave field as acoustic levitation field (ALF). The acoustic levitator consists of a magnetostrictive ultrasonic transducer, a sound emitter and a reflector. The emitter and reflector are coaxial * Corresponding author. Tel./fax: +86 29 88495724 801. E-mail address: [email protected] (J. Zhao). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.08.008
and the axis is parallel to the gravitational direction. The sound emitter is driven by the magnetostrictive ultrasonic transducer. So the acoustic radiation pressure will be enhanced with increasing the inspiriting current of the ultrasonic transducer. When the acoustic levitator is working under certain resonant state, an acoustic standing wave is formed between the emitter and the reflector. The object will be levitated at the node of the standing wave. The number of the nodes and the resonant modes are identical and each resonant mode is corresponding to a certain distance between the emitter and the reflector [3]. Because the ALF is invisible and the traditional direct detection would disturb the actual distribution of the acoustic radiation pressure, it is important to develop a visible, nondestructive and full-field measurement method for ALF. In recent years, digital holography has been widely developed. Many new approaches are proposed, such as digital shearographic holography [4], digital phase-shifting holography [5], digital microscopic holography [6], and digital holographic interferometry [7–10]. The primary applications of these techniques are focused on strain and vibration analysis, nondestructive testing, the display and identification of three-dimensional objects. Besides these above, the digital holographic interferometry can also be used to measure temperature field, refractive index distribution, flow field and so on [11–13]. Zhang et al. [14] primarily measured the ALF by
4340
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
using traditional optical holography. The variation of the sound pressure distribution [15] will change the refractive index distribution of the air between the reflector and the emitter, and then change the optical path of the object wave in the holographic setup. So the sound pressure distribution in the ALF can be nondestructively detected by measuring the index distribution of the air. But the precision of the measurement results by Zhang et al. was not ideal because of the traditional optical reconstruction process of the holographic interferograms. In this paper, we present a new way for simplifying the measurement procedure and improving the measurement sensitivity of the ALF. The digital holographic interferograms can directly reflect the change of the refractive index (or the phase distribution of the wavefront) of the air in ALF. The measurement processes are as follows. Firstly, by use of double-exposure and phase multiplication operations, the digital holograms taking the information of the ALF are recorded at different inspiriting currents, different resonant modes, and with or without levitated object. Then the complex amplitude distributions of the wavefront describing the corresponding ALF are numerically reconstructed. By phase subtraction the phase differences of the wavefront would be obtained, which depicts the distribution difference of the ALF at various states. 2. Experimental principle and setup 2.1. Double-exposure holography For a phase object, the complex amplitude distribution of the object wave in initial state can be expressed as
U 1 ðx; yÞ ¼ u0 ðx; yÞ exp½j/1 ðx; yÞ:
ð1Þ
Changing the phase object means that the phase distribution will be changed but the amplitude of the corresponding object wave remains stationary. So the complex amplitude distribution of the changed object wave will be expressed as
U 2 ðx; yÞ ¼ u0 ðx; yÞ exp½j/2 ðx; yÞ:
ð2Þ
The superposition intensity of the object waves in both types of the states before and after being changed is written as
Tðx; yÞ ¼ jU 1 ðx; yÞ þ U 2 ðx; yÞj2 / ðx; yÞ /1 ðx; yÞ ¼ 4u20 ðx; yÞ cos2 2 2 D /ðx; yÞ ; ¼ 4u20 ðx; yÞ cos2 2
ð3Þ
where D/ðx; yÞ ¼ /2 ðx; yÞ /1 ðx; yÞ depicts the phase difference of the two object waves [16,17]. Eq. (3) shows that the superposition intensity distribution reveals an interferogram reflecting some variations of the object.
Digital hologram i1(k,l)
Superposition i1(k,l)+ i2(k,l)
Digital hologram i2(k,l)
Numerical Reconstruction
Numerical Reconstruction
Numerical Reconstruction
Object wave o′1(m,n)
Object waves o′1(m,n)+ o2′(m,n)
Object wave o2′(m,n)
Phase distributionφ1
Interference pattern T=o*o
Phase distribution φ2
Phase unwrapped Phase difference
φ Fig. 1. Flow chart of the numerical reconstruction of the digital hologram with double-exposure.
gram from the superposition of the complex amplitude distributions of the two object fields by Eq. (3). The first approach requires one cycle of the numerical reconstruction, while the second one requires two cycles. If the purpose of the experiment is only to get the numerical holographic interferogram, the approach one can save the processing time, but it can not obtain the absolute phase difference. If we want to get the absolute phase difference, the approach two is preferred. In other words, the change of the phase distribution of the object wavefront directly reflects the change of the object field. For this reason, the change of the object field can be obtained by demodulating interference phase difference. For a phase object, the phase difference between the waves passing through (in z direction) in its two density states can be described by Du(x,y) = (2p/k)DL(x,y). The optical path difference DL(x,y) is generally described as a line integral of the refractive inR dex variation DLðx; yÞ ¼ Dnðx; y; zÞdz, where Dn(x,y,z) = [n(x,y,z) n0] and n0 and n(x,y,z) denote the initial and changed refractive indexes, respectively. Once obtained, Dn(x,y,z) can be further used for calculating the sound pressure distribution. As shown in Fig. 2, supposing that the distribution of the ALF to be measured in the experiment is axisymmetrical, and the incident beam is perpendicular to the symmetrical axis x, in a cross section, the relationship of the phase difference vs. the change of the refractive index induced by the sound pressure is expressed as [20]
Du0 ðx; yÞ ¼
2p k
Z
½nðx; y; zÞ n0 dz:
ð4Þ
Under the condition of the same entropy, by Gladstone–Dale equation [21] we can get
r P n1 ¼ ; P0 n0 1
ð5Þ
2.2. Implement method According to double-exposure holographic interferometory, there are two approaches [18,19] to get the digital holographic interferograms from the holograms i1 and i2 of the object recorded by CCD under various states, as shown in Fig. 1, where the symbol ‘‘” depicts the complex conjugation operation. For the approach one, firstly we compose the two digital holograms i1 and i2 by simply taking the sum of their gray distributions, and then the digital holographic interferogram can be formed by synchronously reconstructing two object waves from the composed digital hologram. For the approach two, firstly we reconstruct the object field in two states from the digital holograms i1 and i2 separately, and then form the digital holographic interfero-
(a) Beam
x
(b)
y
Emitter
Beam r
z Reflector
o
z
Fig. 2. Track of the beam passing through the ALF. ALF. (a) Front view and (b) planform.
4341
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
where, P0 and P are the pressure before and after changing the acoustic field, respectively; n0 and n are the corresponding refractive index of the atmosphere. Combining Eq. (4) with Eq. (5), we obtain
2p Du0 ðx; yÞ ¼ k
Z
" 1 # P r ðn0 1Þ 1 dz; P0
ð6Þ
where r = 1.4. From above analysis, it is known that the refractive index distribution of the air in ALF can be converted into the phase distribution of the object wavefront. From Eq. (6), the distribution of the sound pressure can be obtained.
composite lenses. The angle between the object and the reference beams is about 1.10o. By adjusting the distances between CCD, lens L4 and the ALF, we can receive the interferogram between the reference and the object beams. It is obvious that, to make the reflected beam and the incident beam superposable, the mirror M3 must be perpendicular to the incident beam. Compared with the setup in Fig. 3a, the phase variation of the object beam in the setup in Fig. 3b will be doubled. By such a phase multiplication we can obtain more holographic interference fringes. And when different modes (resonance spaces are different and the areas of measured field are changing) are measured, clear hologram can be attained by adjusting the position of the L4 lens instead of changing the light path.
2.3. Experimental setup 3. Experimental results and discussions Fig. 3 depicts the experimental setups for measuring the ALF, where Fig. 3a shows the generally used one and Fig. 3b is the new proposed one with phase multiplication. In the setups, a Mach–Zehnder interferometer is employed for recording image plane hologram. As shown in Fig. 3b, a He–Ne laser with wavelength of 632.8 nm is used as the coherent light source. A black and white CCD with 1392 1040 pixels is used for recording the hologram. The output beam is passed through an attenuator and reflected by mirror M1, then divided by tunable beam splitter BS1 into reference beam and object beam. The reference beam is expanded and collimated by an inversed telescope T1 and then received by CCD after being successively reflected by mirror M2 and beam splitter BS3 and passed though the composite lenses L1, L3 and L4. The object beam is expanded and collimated by an inversed telescope T2 and then passed through the ALF twice by beam splitter BS2 and mirror M3. The object beam taking the information of the ALF will be finally received by CCD after being reflected by mirror M3 and passed through the
(a)
3.1. Comparison of the experimental results obtained by two setups In order to verify the availability of the proposed method, we recorded the holograms under the same inspiriting current of ultrasonic transducer employing the setups in Fig. 3a and b, respectively. Fig. 4 shows the numerical reconstruction results of the holographic interferogram reflecting the refractive index distribution of the air in ALF. It is shown that the interferent fringes in Fig. 4a obtained by the general setup are sparse and the fringes in Fig. 4b obtained by use of the setup with phase multiplication are compact. It indicates that the reconstructed interferogram employing the new proposed setup can make the phase of ALF doubled successfully, enhancing measure precision. So in following experiments we use the new setup with phase multiplication. Fig. 5a shows the measurement result of the refractive index distribution of the air under the third resonant mode in ALF without object, which reflects the sound pressure distribution in ALF.
(b)
P M1
He-Ne Laser
P M1
He-Ne Laser
T1
M2
T1
M2
BS1
BS1 T2
T2
Computer
Computer L1
CCD
L1
BS3 BS2
BS3
CCD L4
L3
L2 Object
M3
L4
L3
L2
Object M3
Fig. 3. Experimental setups. (a) For general use and (b) with phase multiplication
Fig. 4. Comparison of the experimental results obtained using two setups. (a) Using the setup in Fig. 3a and (b) using the setup in Fig. 3b.
4342
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
(b)
(a)
+
1.5
1.0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
x/λa
+
0.5
+
0.0 -1.0
-0.5
0.0
0.5
-----------
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
1.0
r/λa
normalization intensity
(c)
theoretical distribution measured distribution
1.0
0.8
0.6
0.4
I
0.2
0.0 0.0
0.5
1.0
1.5
x/λα Fig. 5. Comparison of the measurement result with theory of the sound pressure distribution of the third resonant mode in ALF without object. (a) Measurement result; (b) theoretical result; and (c) comparison of the sound pressure distribution curves.
Fig. 5b shows the theoretical result of the sound pressure distributions under the same condition calculated by standing wave formula of the doubled-bicylinder sound wave mode with the help of Origin software, where the gray value represents the magnitude of the sound pressure (or the refractive index), the symbol ‘‘ + ” represents the node of the standing wave. Fig. 5c depicts the comparison of the measurement results (dashed line) in Fig. 5a with theory (solid line) in Fig. 5b, where the sound pressure distribution is along the symmetric axis and the vertical coordinate represents normalization intensity. It is obvious that the two curves are iden-
tical. It also shows that the measurement result obtained by phase multiplication is more precise than that using general setup. 3.2. Sound pressure distributions of the second resonant mode at different inspiriting currents Fig. 6 are the digital holographic interferograms of the second resonant mode under the condition that the ultrasonic transducer’s inspiriting currents are 75 mA, 100 mA, 125 mA, 150 mA, respectively. It is shown that in the area of the maximum sound pressure,
Fig. 6. Digital holographic interferograms of the second resonant modes at different inspiriting currents. (a) I = 75 mA; (b) I = 100 mA; (c) I = 125 mA; (d) I = 150 mA.
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
with increasing the ultrasonic transducer’s current, the fringes tend to be closer, and the area held by the outer fringe tends to be larger (sound pressure becomes strong). 3.3. Sound pressure distributions of different resonant modes at same inspiriting currents Fig. 7 shows the digital holographic interferograms obtained under three different resonant modes with the constant current (I = 75 mA). It can be seen that in the area of the maximum sound pressure, there are four fringes for the first resonant mode (Fig. 7a), two fringes for the second resonant mode (Fig. 7b), and only one fringe for the third resonant mode (Fig. 7c). The reason would be that, with increasing the number of the resonant modes, the refrac-
4343
tive index change induced by the sound pressure decreases, so does the phase difference. When the phase difference is less than 2p, the number of the fringes appearing in the area of the maximum sound pressure will be less than one. The decrease of the phase difference by increasing the mode order shows the decrease of the peak pressure. It is also found that there exists always an extremum of the sound pressure near the reflector, under whatever resonant mode. 3.4. Sound pressure distributions of the second resonant mode with and without object Fig. 8 depicts the digital holographic interferograms of the second resonant mode with and without levitated object at the inspir-
Fig. 7. Digital holographic interferograms of different resonant modes at same output currents. (a) First mode; (b) second mode and, (c) third mode.
Fig. 8. Digital holographic interferograms of the second resonant mode at the same inspiriting currents. (a) Without levitated object and (b) With levitated object.
Fig. 9. Unwrapped phase patterns. (a) For the object wave without sound pressure; (b) for the object wave under the second resonant mode without levitated object; (c) for the object wave under the second resonant mode with levitated object; (d) subtraction between (b) and (a); and (e) subtraction between (c) and (a).
4344
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
iting current of 75 mA. Comparing Fig. 8a with b, we can see that the fringe number increases, which indicates that the sound pressure has a slight increasing with levitating the spherule. 3.5. Phase distribution of the ALF with and without object Fig. 9a is the unwrapped phase pattern of the object wave without ALF. Fig. 9b and c are the unwrapped phase patterns of the second resonant mode without and with object at the inspiriting current of 75 mA, respectively. Fig. 9d is the phase difference between Fig. 9b and a, for the similar case, Fig. 9e is the phase difference between Fig. 9c and a, demonstrating the changes of the phase difference induced by levitating object, showing the change of the phase difference induced by levitating a plastic spherule. It is easy to get the phase difference between any two ALFs by subtraction of the corresponding unwrapped phase patterns and then the variation information of the refractive index as well as the sound pressure from Eqs. (4) and (5). And thus, the ALF could be quantificationally understood. 4. Conclusions Visualized measurement of ALF, carried out by using digital holographic interferometer, directly demonstrates the sound pressure distributions. Besides, the measurement sensitivity can be increased through multiplying the light path of the object wave in ALF. The experimental results show that the measured sound pressure distribution along with the mid-axis in the ALF is similar to the ideal sound pressure distribution obtained from the wave function and more accurate. For a certain resonant mode, the higher the inspiriting currents are, the larger the maximum of the sound pressure is. For certain inspiriting current, the higher the mode orders are, the less the maximum about the ALF is. When the resonant mode and inspiriting current are both invariable, the maximum will change a little with spherule in levitation. To get the distribution of the interference phase in different cases when the resonant mode and current are both constant, the way is subtracting the unwrapped phase patterns.
As for the absolute accuracy, digital holography is not as good as traditional optical holography, but digital holography is less affected by artificial and environmental error. Compared with the traditional optical holographic interferometry, the digital holographic interferometry can not only simplify the operations of the ALF measurement, but also make the measurement operation more convenient and rapid, and the measurement results would be more straightforward. So it should be a practical way to quantificationally measure a three-dimensional ALF. Acknowledgements This work is supported by the Science Foundation of Aeronautics of China under grants No. 2006ZD53042. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
U. Schnars, W.P.O. Jüptner, Appl. Opt. 33 (2) (1994) 179. E.H. Brandt, Science 243 (4889) (1989) 349. W.J. Xie, B. Wei, Phys. Rev. E 66 (2) (2002) 1. U. Schnars, W.P.O. Jüptner, Appl. Opt. 33 (20) (1994) 4373. I. Yamaguchi, T. Zhang, Opt. Lett. 22 (16) (1997) 1268. T. Zhang, I. Yamaguchi, Opt. Lett. 23 (15) (1998) 1221. G. Pedrini, P. Froning, H. Fessler, H.J. Tiziani, Appl. Opt. 37 (26) (1998) 6262. M.K. Kim, Opt. Lett. 24 (23) (1999) 1693. D. Dirksen, H. Droste, B. Kemper, B. Kemper, H. Deleré, M. Deiwick, H.H. Scheld, G. von Bally, Opt. Laser Eng. 36 (3) (2001) 241. G. Pedrini, S. Schedin, H.J. Tiziani, Opt. Commun. 171 (1) (1999) 29. J.L. Zhao, H.Y. Tan, Acta Opt. Sin. 22 (12) (2001) 1447 (in Chinese). J.L. Zhao, P. Zhang, J.B. Zhou, D.X. Yang, D.S. Yang, E.P. Li, Chin. Phys. Lett. 20 (10) (2003) 1748. W. Feng, E.P. Li, Q. Fan, L. Zhang, J.L. Zhao, Acta Photon. Sin. 34 (12) (2005) 1833 (in Chinese). L. Zhang, E.P. Li, W. Feng, Z.Y. Hong, W.J. Xie, Y.H. Ma, Acta Phys. Sin. 54 (5) (2005) 2038 (in Chinese). W.J. Xie, B. Wei, Physics 31 (9) (2002) 551 (in Chinese). S.F. Wang, Z.Q. Zhu, Press of University of Electronic Science Technology of China (2001) 195 (in Chinese). B.H. Xiong, Z.R. Wang, Lesser J. 20 (3) (1999) 3 (in Chinese). H. Sekanina, S. Hledik, Acta Phys. Slov. 51 (4) (2001) 261. U. Schnars, J. Opt. Soc. Am. A 11 (7) (1994) 2011. C. Shakher, A.K. Nirala, Opt. Laser Eng. 31 (6) (1999) 455. X.D. Li, S. Schedin, P. Gren, Acta Opt. Sin. 18 (12) (1998) 1669 (in Chinese).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Visualized measurement of the acoustic levitation field based on digital holography with phase multiplication Puchao Zheng, Enpu Li, Jianlin Zhao *, Jianglei Di, Wangmin Zhou, Hao Wang, Ruifeng Zhang Institute of Optical Information Science and Technology, Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 30 July 2008 Received in revised form 18 June 2009 Accepted 5 August 2009
PACS: 42.40.Ht 42.40.Kw 87.50.Kk Keywords: Digital holography Holographic interferometory Acoustic levitation Sound pressure distribution Phase multiplication
a b s t r a c t By using digital holographic interferometory with phase multiplication, the visualized measurement of the acoustic levitation field (ALF) with single axis is carried out. The digital holograms of the ALF under different conditions are recorded by use of CCD. The corresponding digital holographic interferograms reflecting the sound pressure distribution and the interference phase distribution are obtained by numerical reconstruction and phase subtraction, which are consistent with the theoretical results. It indicates that the proposed digital holographic interferometory with phase multiplication can successfully double the fringe number of the interference phase patterns of the ALF and improve the measurement precision. Compared with the conventional optical holographic interferometory, digital holographic interferometory has the merits of quasi real-time, more exactitude and convenient operation, and it provides an effective way for studying the sound pressure distribution of the ALF. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Since the charge coupled device (CCD) has been used to record the digital hologram in 1994 at first time [1], with the development of the computer and the CCD technology, the digital holography attracts more and more attentions. The best merits of the digital holography are that it can use optoelectronic imaging devices such as CCD in stead of film as recording medium and numerically reconstruct the object field. It leaves out the complex wet physical and chemical operations required in processing the film, so that it not only improves the measurement efficiency but also permits the measurement process in real-time. Acoustic levitation [2] is an important technique for containerless solidification of materials, which uses the pressure of the acoustic standing wave with high intensity, namely acoustic radiation pressure. Under certain conditions this acoustic radiation pressure can levitate an object with density many times more than atmosphere in the air. For convenience, here we call the acoustic standing wave field as acoustic levitation field (ALF). The acoustic levitator consists of a magnetostrictive ultrasonic transducer, a sound emitter and a reflector. The emitter and reflector are coaxial * Corresponding author. Tel./fax: +86 29 88495724 801. E-mail address: [email protected] (J. Zhao). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.08.008
and the axis is parallel to the gravitational direction. The sound emitter is driven by the magnetostrictive ultrasonic transducer. So the acoustic radiation pressure will be enhanced with increasing the inspiriting current of the ultrasonic transducer. When the acoustic levitator is working under certain resonant state, an acoustic standing wave is formed between the emitter and the reflector. The object will be levitated at the node of the standing wave. The number of the nodes and the resonant modes are identical and each resonant mode is corresponding to a certain distance between the emitter and the reflector [3]. Because the ALF is invisible and the traditional direct detection would disturb the actual distribution of the acoustic radiation pressure, it is important to develop a visible, nondestructive and full-field measurement method for ALF. In recent years, digital holography has been widely developed. Many new approaches are proposed, such as digital shearographic holography [4], digital phase-shifting holography [5], digital microscopic holography [6], and digital holographic interferometry [7–10]. The primary applications of these techniques are focused on strain and vibration analysis, nondestructive testing, the display and identification of three-dimensional objects. Besides these above, the digital holographic interferometry can also be used to measure temperature field, refractive index distribution, flow field and so on [11–13]. Zhang et al. [14] primarily measured the ALF by
4340
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
using traditional optical holography. The variation of the sound pressure distribution [15] will change the refractive index distribution of the air between the reflector and the emitter, and then change the optical path of the object wave in the holographic setup. So the sound pressure distribution in the ALF can be nondestructively detected by measuring the index distribution of the air. But the precision of the measurement results by Zhang et al. was not ideal because of the traditional optical reconstruction process of the holographic interferograms. In this paper, we present a new way for simplifying the measurement procedure and improving the measurement sensitivity of the ALF. The digital holographic interferograms can directly reflect the change of the refractive index (or the phase distribution of the wavefront) of the air in ALF. The measurement processes are as follows. Firstly, by use of double-exposure and phase multiplication operations, the digital holograms taking the information of the ALF are recorded at different inspiriting currents, different resonant modes, and with or without levitated object. Then the complex amplitude distributions of the wavefront describing the corresponding ALF are numerically reconstructed. By phase subtraction the phase differences of the wavefront would be obtained, which depicts the distribution difference of the ALF at various states. 2. Experimental principle and setup 2.1. Double-exposure holography For a phase object, the complex amplitude distribution of the object wave in initial state can be expressed as
U 1 ðx; yÞ ¼ u0 ðx; yÞ exp½j/1 ðx; yÞ:
ð1Þ
Changing the phase object means that the phase distribution will be changed but the amplitude of the corresponding object wave remains stationary. So the complex amplitude distribution of the changed object wave will be expressed as
U 2 ðx; yÞ ¼ u0 ðx; yÞ exp½j/2 ðx; yÞ:
ð2Þ
The superposition intensity of the object waves in both types of the states before and after being changed is written as
Tðx; yÞ ¼ jU 1 ðx; yÞ þ U 2 ðx; yÞj2 / ðx; yÞ /1 ðx; yÞ ¼ 4u20 ðx; yÞ cos2 2 2 D /ðx; yÞ ; ¼ 4u20 ðx; yÞ cos2 2
ð3Þ
where D/ðx; yÞ ¼ /2 ðx; yÞ /1 ðx; yÞ depicts the phase difference of the two object waves [16,17]. Eq. (3) shows that the superposition intensity distribution reveals an interferogram reflecting some variations of the object.
Digital hologram i1(k,l)
Superposition i1(k,l)+ i2(k,l)
Digital hologram i2(k,l)
Numerical Reconstruction
Numerical Reconstruction
Numerical Reconstruction
Object wave o′1(m,n)
Object waves o′1(m,n)+ o2′(m,n)
Object wave o2′(m,n)
Phase distributionφ1
Interference pattern T=o*o
Phase distribution φ2
Phase unwrapped Phase difference
φ Fig. 1. Flow chart of the numerical reconstruction of the digital hologram with double-exposure.
gram from the superposition of the complex amplitude distributions of the two object fields by Eq. (3). The first approach requires one cycle of the numerical reconstruction, while the second one requires two cycles. If the purpose of the experiment is only to get the numerical holographic interferogram, the approach one can save the processing time, but it can not obtain the absolute phase difference. If we want to get the absolute phase difference, the approach two is preferred. In other words, the change of the phase distribution of the object wavefront directly reflects the change of the object field. For this reason, the change of the object field can be obtained by demodulating interference phase difference. For a phase object, the phase difference between the waves passing through (in z direction) in its two density states can be described by Du(x,y) = (2p/k)DL(x,y). The optical path difference DL(x,y) is generally described as a line integral of the refractive inR dex variation DLðx; yÞ ¼ Dnðx; y; zÞdz, where Dn(x,y,z) = [n(x,y,z) n0] and n0 and n(x,y,z) denote the initial and changed refractive indexes, respectively. Once obtained, Dn(x,y,z) can be further used for calculating the sound pressure distribution. As shown in Fig. 2, supposing that the distribution of the ALF to be measured in the experiment is axisymmetrical, and the incident beam is perpendicular to the symmetrical axis x, in a cross section, the relationship of the phase difference vs. the change of the refractive index induced by the sound pressure is expressed as [20]
Du0 ðx; yÞ ¼
2p k
Z
½nðx; y; zÞ n0 dz:
ð4Þ
Under the condition of the same entropy, by Gladstone–Dale equation [21] we can get
r P n1 ¼ ; P0 n0 1
ð5Þ
2.2. Implement method According to double-exposure holographic interferometory, there are two approaches [18,19] to get the digital holographic interferograms from the holograms i1 and i2 of the object recorded by CCD under various states, as shown in Fig. 1, where the symbol ‘‘” depicts the complex conjugation operation. For the approach one, firstly we compose the two digital holograms i1 and i2 by simply taking the sum of their gray distributions, and then the digital holographic interferogram can be formed by synchronously reconstructing two object waves from the composed digital hologram. For the approach two, firstly we reconstruct the object field in two states from the digital holograms i1 and i2 separately, and then form the digital holographic interfero-
(a) Beam
x
(b)
y
Emitter
Beam r
z Reflector
o
z
Fig. 2. Track of the beam passing through the ALF. ALF. (a) Front view and (b) planform.
4341
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
where, P0 and P are the pressure before and after changing the acoustic field, respectively; n0 and n are the corresponding refractive index of the atmosphere. Combining Eq. (4) with Eq. (5), we obtain
2p Du0 ðx; yÞ ¼ k
Z
" 1 # P r ðn0 1Þ 1 dz; P0
ð6Þ
where r = 1.4. From above analysis, it is known that the refractive index distribution of the air in ALF can be converted into the phase distribution of the object wavefront. From Eq. (6), the distribution of the sound pressure can be obtained.
composite lenses. The angle between the object and the reference beams is about 1.10o. By adjusting the distances between CCD, lens L4 and the ALF, we can receive the interferogram between the reference and the object beams. It is obvious that, to make the reflected beam and the incident beam superposable, the mirror M3 must be perpendicular to the incident beam. Compared with the setup in Fig. 3a, the phase variation of the object beam in the setup in Fig. 3b will be doubled. By such a phase multiplication we can obtain more holographic interference fringes. And when different modes (resonance spaces are different and the areas of measured field are changing) are measured, clear hologram can be attained by adjusting the position of the L4 lens instead of changing the light path.
2.3. Experimental setup 3. Experimental results and discussions Fig. 3 depicts the experimental setups for measuring the ALF, where Fig. 3a shows the generally used one and Fig. 3b is the new proposed one with phase multiplication. In the setups, a Mach–Zehnder interferometer is employed for recording image plane hologram. As shown in Fig. 3b, a He–Ne laser with wavelength of 632.8 nm is used as the coherent light source. A black and white CCD with 1392 1040 pixels is used for recording the hologram. The output beam is passed through an attenuator and reflected by mirror M1, then divided by tunable beam splitter BS1 into reference beam and object beam. The reference beam is expanded and collimated by an inversed telescope T1 and then received by CCD after being successively reflected by mirror M2 and beam splitter BS3 and passed though the composite lenses L1, L3 and L4. The object beam is expanded and collimated by an inversed telescope T2 and then passed through the ALF twice by beam splitter BS2 and mirror M3. The object beam taking the information of the ALF will be finally received by CCD after being reflected by mirror M3 and passed through the
(a)
3.1. Comparison of the experimental results obtained by two setups In order to verify the availability of the proposed method, we recorded the holograms under the same inspiriting current of ultrasonic transducer employing the setups in Fig. 3a and b, respectively. Fig. 4 shows the numerical reconstruction results of the holographic interferogram reflecting the refractive index distribution of the air in ALF. It is shown that the interferent fringes in Fig. 4a obtained by the general setup are sparse and the fringes in Fig. 4b obtained by use of the setup with phase multiplication are compact. It indicates that the reconstructed interferogram employing the new proposed setup can make the phase of ALF doubled successfully, enhancing measure precision. So in following experiments we use the new setup with phase multiplication. Fig. 5a shows the measurement result of the refractive index distribution of the air under the third resonant mode in ALF without object, which reflects the sound pressure distribution in ALF.
(b)
P M1
He-Ne Laser
P M1
He-Ne Laser
T1
M2
T1
M2
BS1
BS1 T2
T2
Computer
Computer L1
CCD
L1
BS3 BS2
BS3
CCD L4
L3
L2 Object
M3
L4
L3
L2
Object M3
Fig. 3. Experimental setups. (a) For general use and (b) with phase multiplication
Fig. 4. Comparison of the experimental results obtained using two setups. (a) Using the setup in Fig. 3a and (b) using the setup in Fig. 3b.
4342
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
(b)
(a)
+
1.5
1.0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
x/λa
+
0.5
+
0.0 -1.0
-0.5
0.0
0.5
-----------
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
1.0
r/λa
normalization intensity
(c)
theoretical distribution measured distribution
1.0
0.8
0.6
0.4
I
0.2
0.0 0.0
0.5
1.0
1.5
x/λα Fig. 5. Comparison of the measurement result with theory of the sound pressure distribution of the third resonant mode in ALF without object. (a) Measurement result; (b) theoretical result; and (c) comparison of the sound pressure distribution curves.
Fig. 5b shows the theoretical result of the sound pressure distributions under the same condition calculated by standing wave formula of the doubled-bicylinder sound wave mode with the help of Origin software, where the gray value represents the magnitude of the sound pressure (or the refractive index), the symbol ‘‘ + ” represents the node of the standing wave. Fig. 5c depicts the comparison of the measurement results (dashed line) in Fig. 5a with theory (solid line) in Fig. 5b, where the sound pressure distribution is along the symmetric axis and the vertical coordinate represents normalization intensity. It is obvious that the two curves are iden-
tical. It also shows that the measurement result obtained by phase multiplication is more precise than that using general setup. 3.2. Sound pressure distributions of the second resonant mode at different inspiriting currents Fig. 6 are the digital holographic interferograms of the second resonant mode under the condition that the ultrasonic transducer’s inspiriting currents are 75 mA, 100 mA, 125 mA, 150 mA, respectively. It is shown that in the area of the maximum sound pressure,
Fig. 6. Digital holographic interferograms of the second resonant modes at different inspiriting currents. (a) I = 75 mA; (b) I = 100 mA; (c) I = 125 mA; (d) I = 150 mA.
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
with increasing the ultrasonic transducer’s current, the fringes tend to be closer, and the area held by the outer fringe tends to be larger (sound pressure becomes strong). 3.3. Sound pressure distributions of different resonant modes at same inspiriting currents Fig. 7 shows the digital holographic interferograms obtained under three different resonant modes with the constant current (I = 75 mA). It can be seen that in the area of the maximum sound pressure, there are four fringes for the first resonant mode (Fig. 7a), two fringes for the second resonant mode (Fig. 7b), and only one fringe for the third resonant mode (Fig. 7c). The reason would be that, with increasing the number of the resonant modes, the refrac-
4343
tive index change induced by the sound pressure decreases, so does the phase difference. When the phase difference is less than 2p, the number of the fringes appearing in the area of the maximum sound pressure will be less than one. The decrease of the phase difference by increasing the mode order shows the decrease of the peak pressure. It is also found that there exists always an extremum of the sound pressure near the reflector, under whatever resonant mode. 3.4. Sound pressure distributions of the second resonant mode with and without object Fig. 8 depicts the digital holographic interferograms of the second resonant mode with and without levitated object at the inspir-
Fig. 7. Digital holographic interferograms of different resonant modes at same output currents. (a) First mode; (b) second mode and, (c) third mode.
Fig. 8. Digital holographic interferograms of the second resonant mode at the same inspiriting currents. (a) Without levitated object and (b) With levitated object.
Fig. 9. Unwrapped phase patterns. (a) For the object wave without sound pressure; (b) for the object wave under the second resonant mode without levitated object; (c) for the object wave under the second resonant mode with levitated object; (d) subtraction between (b) and (a); and (e) subtraction between (c) and (a).
4344
P. Zheng et al. / Optics Communications 282 (2009) 4339–4344
iting current of 75 mA. Comparing Fig. 8a with b, we can see that the fringe number increases, which indicates that the sound pressure has a slight increasing with levitating the spherule. 3.5. Phase distribution of the ALF with and without object Fig. 9a is the unwrapped phase pattern of the object wave without ALF. Fig. 9b and c are the unwrapped phase patterns of the second resonant mode without and with object at the inspiriting current of 75 mA, respectively. Fig. 9d is the phase difference between Fig. 9b and a, for the similar case, Fig. 9e is the phase difference between Fig. 9c and a, demonstrating the changes of the phase difference induced by levitating object, showing the change of the phase difference induced by levitating a plastic spherule. It is easy to get the phase difference between any two ALFs by subtraction of the corresponding unwrapped phase patterns and then the variation information of the refractive index as well as the sound pressure from Eqs. (4) and (5). And thus, the ALF could be quantificationally understood. 4. Conclusions Visualized measurement of ALF, carried out by using digital holographic interferometer, directly demonstrates the sound pressure distributions. Besides, the measurement sensitivity can be increased through multiplying the light path of the object wave in ALF. The experimental results show that the measured sound pressure distribution along with the mid-axis in the ALF is similar to the ideal sound pressure distribution obtained from the wave function and more accurate. For a certain resonant mode, the higher the inspiriting currents are, the larger the maximum of the sound pressure is. For certain inspiriting current, the higher the mode orders are, the less the maximum about the ALF is. When the resonant mode and inspiriting current are both invariable, the maximum will change a little with spherule in levitation. To get the distribution of the interference phase in different cases when the resonant mode and current are both constant, the way is subtracting the unwrapped phase patterns.
As for the absolute accuracy, digital holography is not as good as traditional optical holography, but digital holography is less affected by artificial and environmental error. Compared with the traditional optical holographic interferometry, the digital holographic interferometry can not only simplify the operations of the ALF measurement, but also make the measurement operation more convenient and rapid, and the measurement results would be more straightforward. So it should be a practical way to quantificationally measure a three-dimensional ALF. Acknowledgements This work is supported by the Science Foundation of Aeronautics of China under grants No. 2006ZD53042. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
U. Schnars, W.P.O. Jüptner, Appl. Opt. 33 (2) (1994) 179. E.H. Brandt, Science 243 (4889) (1989) 349. W.J. Xie, B. Wei, Phys. Rev. E 66 (2) (2002) 1. U. Schnars, W.P.O. Jüptner, Appl. Opt. 33 (20) (1994) 4373. I. Yamaguchi, T. Zhang, Opt. Lett. 22 (16) (1997) 1268. T. Zhang, I. Yamaguchi, Opt. Lett. 23 (15) (1998) 1221. G. Pedrini, P. Froning, H. Fessler, H.J. Tiziani, Appl. Opt. 37 (26) (1998) 6262. M.K. Kim, Opt. Lett. 24 (23) (1999) 1693. D. Dirksen, H. Droste, B. Kemper, B. Kemper, H. Deleré, M. Deiwick, H.H. Scheld, G. von Bally, Opt. Laser Eng. 36 (3) (2001) 241. G. Pedrini, S. Schedin, H.J. Tiziani, Opt. Commun. 171 (1) (1999) 29. J.L. Zhao, H.Y. Tan, Acta Opt. Sin. 22 (12) (2001) 1447 (in Chinese). J.L. Zhao, P. Zhang, J.B. Zhou, D.X. Yang, D.S. Yang, E.P. Li, Chin. Phys. Lett. 20 (10) (2003) 1748. W. Feng, E.P. Li, Q. Fan, L. Zhang, J.L. Zhao, Acta Photon. Sin. 34 (12) (2005) 1833 (in Chinese). L. Zhang, E.P. Li, W. Feng, Z.Y. Hong, W.J. Xie, Y.H. Ma, Acta Phys. Sin. 54 (5) (2005) 2038 (in Chinese). W.J. Xie, B. Wei, Physics 31 (9) (2002) 551 (in Chinese). S.F. Wang, Z.Q. Zhu, Press of University of Electronic Science Technology of China (2001) 195 (in Chinese). B.H. Xiong, Z.R. Wang, Lesser J. 20 (3) (1999) 3 (in Chinese). H. Sekanina, S. Hledik, Acta Phys. Slov. 51 (4) (2001) 261. U. Schnars, J. Opt. Soc. Am. A 11 (7) (1994) 2011. C. Shakher, A.K. Nirala, Opt. Laser Eng. 31 (6) (1999) 455. X.D. Li, S. Schedin, P. Gren, Acta Opt. Sin. 18 (12) (1998) 1669 (in Chinese).