Acoustic logic gates and Boolean operation based on ...

Acoustic logic gates and Boolean operation based on self-collimating acoustic beams Ting Zhang, Ying Cheng, Jian-zhong Guo, Jian-yi Xu, and Xiao-jun Liu Citation: Applied Physics Letters 106, 113503 (2015); doi: 10.1063/1.4915338 View online: http://dx.doi.org/10.1063/1.4915338 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/106/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonvolatile “AND,” “OR,” and “NOT” Boolean logic gates based on phase-change memory J. Appl. Phys. 114, 234503 (2013); 10.1063/1.4852995 Fluorescence resonance energy transfer-based molecular logic circuit using a DNA scaffold Appl. Phys. Lett. 101, 233703 (2012); 10.1063/1.4769812 Phase-controlling phononic crystals: Realization of acoustic Boolean logic gates J. Acoust. Soc. Am. 130, 1919 (2011); 10.1121/1.3631627 Logic gates with a single graphene transistor Appl. Phys. Lett. 94, 073305 (2009); 10.1063/1.3079663 Nanoelectronic logic device based on the manipulation of magnetic and electric barriers J. Appl. Phys. 103, 054310 (2008); 10.1063/1.2838211

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APPLIED PHYSICS LETTERS 106, 113503 (2015)

Acoustic logic gates and Boolean operation based on self-collimating acoustic beams Ting Zhang,1 Ying Cheng,1,2,a) Jian-zhong Guo,3 Jian-yi Xu,1 and Xiao-jun Liu1,2,b) 1

Key Laboratory of Modern Acoustics, Department of Physics and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China 2 State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing 100190, China 3 School of Physics and Information Technology, Shaanxi Normal University, Xian 710119, China

(Received 7 January 2015; accepted 6 March 2015; published online 18 March 2015) The reveal of self-collimation effect in two-dimensional (2D) photonic or acoustic crystals has opened up possibilities for signal manipulation. In this paper, we have proposed acoustic logic gates based on the linear interference of self-collimated beams in 2D sonic crystals (SCs) with line-defects. The line defects on the diagonal of the 2D square SCs are actually functioning as a 3 dB splitter. By adjusting the phase difference between two input signals, the basic Boolean logic functions such as XOR, OR, AND, and NOT are achieved both theoretically and experimentally. Due to the non-diffracting property of self-collimation beams, more complex Boolean logic and algorithms such as NAND, NOR, and XNOR can be realized by cascading the basic logic gates. The achievement of acoustic logic gates and Boolean operation provides a promising approach for C 2015 AIP Publishing LLC. acoustic signal computing and manipulations. V [http://dx.doi.org/10.1063/1.4915338]

The Boolean logic gates, which are idealized or physical devices implementing Boolean functions, are the most important components in modern electronic integrated circuits. Recently, motivated by the significant potential in future photonic integrated circuits, the Boolean operation has been introduced into the optical fields.1–4 The optical logic gates, such as logical conjunction by AND gate, disjunction by OR gate, and exclusive disjunction by XOR gate, have been achieved both theoretically and experimentally.2,3 Based on the linear interferences of light, which comprise the constructive and destructive interferences, the logic function can be realized by careful control of the phase difference between the input light beam and the control beam.4 For linear logic gates, the inevitable diffraction broaden of light beams propagating in waveguide has been solved by the introduction of self-collimation effect, which shows merits of good stability and extensibility.5–7 The approaches of acoustic wave manipulation have also attracted much attention due to numerous applications such as acoustic communication, medical ultrasonic imaging/therapy, and sensor technologies, which require flexible control of acoustic waves by switching their propagation.8–13 Specially, in underwater and medical applications, optic signals exhibit high attenuation and short transmission distance. Furthermore, the integration of optic logic gates in acoustic structures is challenging due to the distinct physical properties such as wavelength and propagation medium between optic and acoustic waves. Thus, the acoustic logic gates show the potential applications. In the last few years, novel acoustic devices such as acoustic switches and acoustic diodes have been proposed and demonstrated.14–17 Series of rectifiers and switches have been proposed based on the a)

[email protected] [email protected]

b)

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nonlinear effects of bubbly liquids.14,15 Besides, the acoustic switch has also been realized by the use of the control sound beam or rotating the rods of sonic crystals (SCs) owing to the special propagation properties of sound waves in twodimensional (2D) SCs.16,17 However, most of these pioneering works are based on the nonlinear effects in bubbly liquid or complex metamaterials, which inevitably suffers certain limitations such as power consumption, frequency shift, waveform distortion, structure complexity, and instantaneity. In the other hand, the NAND, XOR, and NOT acoustic logic operation has been achieved via the phase-control of incidents waves.18,19 The logic operations were realized by the interference of four waves (two inputs and two additional complementary waves) with certain phase shift as they propagate through different paths in the SC.18 The significant wave diffraction, signal aliasing, and attenuation are the main limitations in realizing practical acoustic logic gates. Besides, the whole basic logic operations cannot be implemented based on the previous work. Li et al. further realized the AND and OR logic operations based on a driven chain of spherical particles with a nonlinear contact force.20 Moreover, due to the structure characterizes or the introduction of additional complementary waves, the logic gates are not feasible to be cascaded to implement more complex logic operations. The self-collimation effect of acoustic waves in 2D SCs overcomes the above-mentioned limitations.21 Here, we demonstrate acoustic logic computing based on suitably designed SC blocks that can perform basic Boolean mathematical operations such as XOR, OR, AND, and NOT. By appropriately selecting the structure parameters of scatterers on the diagonal of SC blocks, the line defects in SCs can function as a 3 dB splitter. Through the linear interference between the transmission and reflection waves, which, respectively, split from two input self-collimated ultrasonic

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waves, the basic acoustic switch and logic gates are achieved by adjusting the phase difference between two input signals. We present both theoretical model and experimental demonstration of this mechanism in a logic gate operating on the waveforms of input ultrasonic signals as they propagate through the device. Furthermore, the proposed acoustic logic gate fully preserves the information representation from input to output with minor waveform distortion and signal attenuation. Thus, the gates can be cascaded, allowing implementation of combinatorial and sequential Boolean operation with complex logic function. These acoustic logic gates developed here may enable engineering ultrasonic computations for underwater communication and clinical diagnosis. Here, we consider a square lattice 2D SCs composed of epoxy resin rods immersed in water with the filling ratios of 0.196. The density and acoustic longitudinal speed are qe ¼ 1180 kg=m3 and ce ¼ 2700 m=s for epoxy, and qw ¼ 998 kg=m3 and cw ¼ 1483 m=s for water, respectively. The band structures and relevant equifrequency contours (EFCs) in the first Brillouin zone of the employed SCs are shown in Figs. 1(a) and 1(b), respectively. It is found that normalized frequency f ¼ 0:578cw =a is located at the first band, as shown in Fig. 1(a), while the EFCs at f ¼ 0:578cw =a is flat along the C-M direction, as shown by a red arrow in Fig. 1(b). Here, a is the lattice constant. Thus, the acoustic waves at f ¼ 0:578cw =a should propagate along the C-M direction without diffraction.21 We also show the simulation result for sound propagation at f ¼ 0:578cw =a in finite-sized SCs in Fig. 1(c). It is clear that after a few periods, the incident monochromatic acoustic beam propagates along the C-M direction with minor diffraction, as so called self-collimation phenomenon. pffiffiffiffi pffiffiffi Figure 2(a) shows the schematics of a 12 2 a  12 2 a 2D SC block used in building acoustic switch and basic logic gates. For the SC, the radius of epoxy resin rods is r0 ¼ 0:25 a, and the radius of the rods on the diagonal is 1:3 r0 . There are four faces on this double-input and doubleoutput structure. Among them, two adjacent faces separated

FIG. 1. Acoustic self-collimation effect. (a) The band structure of the 2D SCs). Here, the frequency is normalized by 2pcw =a. (b) The EFCs in the first Brillouin zone of the SCs. (c) Simulation result for acoustic beam at f ¼ 0:578cw =a propagation in the SCs.

Appl. Phys. Lett. 106, 113503 (2015)

by the line defect indicated by the dashed box are input faces (I1, I2), and the others are output faces (O1, O2). The selfcollimated beams can be bent and split by introducing the line defects in 2D SCs structure, and then the power ratio between two split self-collimated beams can be controlled systematically by varying the radii of rods in the line defect.7 Expectedly, there should be a phase lag between the transmission and reflection beams. Thus, if another selfcollimated beam with appropriate initial phase is introduced, the interference between two transmission and reflection beams should realize the switching and logic gate functions. Here, two incident acoustic beams pI1 ¼ A1 e j ðxtu1 Þ and pI2 ¼ A2 e j ðxtu2 Þ are transferred into the faces I1 and I2, respectively. A1 (¼A2 ¼ A) is the pressure amplitude of acoustic beam, while u1 and u2 are the initial phase of two incident beams. In this case, we set the transmission coefficient is equal to reflection coefficient, and then the transmission and reflection beams of two incidences can be written as pffiffiffi pT1;2 ¼ Aejðxtu1;2 þuÞ = 2 ; (1) pffiffiffi pR1;2 ¼ Aejðxtu1;2 þuþp =2 Þ = 2: From Eq. (1), there is a phase difference of p = 2 between the transmission and reflection beams. The output beams from faces O1 and O2 can be written as the superposition of the transmission and reflection beams   pffiffiffi u  u2 p jðxtþu þ Þ þ e pO1 ¼ pT1 þ pR2 ¼ 2 A cos 1 2 4  jðxtþu 2 þp4Þ ¼ A3 e ;   pffiffiffi u  u2 p jðxtþu  þp4Þ þ e pO2 ¼ pT2 þ pR1 ¼ 2 A cos  1 2 4  jðxtþu 2 þp4Þ ¼A e ; (2) u1 u2 2

u1

p 4

u2

u1

u2

2

u1

u2

4

where A3 and A4 are the pressure amplitudes of the output acoustic beams from O1 and O2, respectively. Here, we can assume A as nonzero constant, and then the output amplitudes are governed by the phase of two incident beams. pffiffidifference ffi Thus, p Affiffi3ffi ¼ 0 and A4 ¼ 2A when u1  u2 ¼ p = 2, while A3 ¼ 2A and A4 ¼ 0 when u1  u2 ¼ p = 2. We further investigate the switch function by adjusting the phase difference between the two incident beams. Figure 2(b) [or Fig. 2(c)] shows the sound pressure distribution under only one incident beam pI1 (or pI2 ). It is noted that the incident beam is split into transmission and reflection beams with nearly equal power, and the phase difference between two beams is exactly p = 2. Figures 2(d) and 2(e) give the sound pressure distribution under two incident beams ( pI1 and pI2 ) with phase differences p = 2 and p = 2, respectively. As shown in Fig. 2(d), the transmission beam of pI1 and reflection beam of pI2 are destructively interfered by each other and no acoustic wave outputs from O1 consequently. By contrast, the constructive interference of reflection beam of pI1 and the transmission beam of pI2 amplifies the output wave from O2. When the phase difference between two incident beams becomes p = 2 as shown in Fig. 2(e), the output wave from O1 is amplified, while there is almost no output acoustic wave

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FIG. 2. Design and behavior of the acoustic switch and basic logic gate. (a) Schematic diagram of the acoustic switch and logic gate in cross-section view. The diagonal line defect is detailed by the inset. (b) and (c) show the sound pressure distribution under only one incident sound beam from the input faces I1 and I2, respectively. (d) and (e) show the sound pressure distribution under two incident sound beams with the phase difference u1  u2 ¼ p = 2 and u1  u2 ¼ p = 2, respectively.

from O2. The results indicate that acoustic switch function can be realized by adjusting the phase between two incident acoustic beams. We further realize the acoustic logic functions such as AND, OR, and XOR. For the output waves, the Boolean value is determined by the output acoustic pressure amplitudes A3 and A4 . If the output amplitude is larger than the threshold At , the output value is “1/TURE.” If the output amplitude is smaller than At , the output is “0/FALSE.” Here, we define pffiffiffitwo threshold values pffiffiffi for the output signals as At1 ¼ A =2 2 and At2 ¼ A = 2. In the case of the phase difference u1  u2 ¼ p = 2 as shown in Fig. 2(d), if the threshold At1 is adopted, the output state of pO1 is “1” for inputs {A, 0} and {0, A} and “0” for inputs {0, 0} and {A, A}, realizing XOR logic function. Meanwhile, the output state of pO2 is “1” for inputs {A, 0}, {0, A}, and {A, A}, realizing OR logic function. If the threshold At2 is adopted, the output of pO2 is “1” only for {A, A}, corresponding to AND logic operation. By contrast, in the case of the phase difference u1  u2 ¼ p = 2 as shown in Fig. 2(e), for At1 , the output states of pO1 and pO2 operate as OR and XOR logic gates, respectively, while the pO1 state realizes AND

logic function with the threshold At2 .22 Therefore, the logic AND, OR, XOR functions are achieved. To verify above conceptual design and simulation results, we demonstrate the acoustic switch and basic logic gates in experiments. The measurement set-up is shown in Fig. 3. In order to adjust the phase arbitrarily, one of the emitting transducers was fixed on a stepping motor controlled by a computer, by which we can move the source along the incident direction. In the experiment, the incident source is moved in 1-mm interval. The acoustic signal launched by the transducer through the SC sample was detected by the other two transducers. The acquired signals were sent to a low noise preamplifier, and then processed by the LabVIEW application. Time domain data were finally analyzed after averaging 20 measurements. Taking the bias between the simulation model and experimental prototype into account, the optimal effect is achieved when the incident beam frequency is set as 0:559 cw =a with 2 Vpp amplitude. Figure 4(a) shows the output acoustic waveforms measured from O1 and O2. In Fig. 4(a), it is found that the output acoustic wave from O2 is a general sine signal, while the output wave from O1 is much weaker (with the amplitude ratio

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FIG. 3. Experimental setup and prototype. (a) Schematic of the experimental setup. The control source can move along the incident direction, as shown by the arrow. (b) The photograph of the prototype fabricated by using epoxy resin rods.

around 4.14) than that from O2. When the control source is moved by 7 mm along the incident direction (corresponding to phase change of p), O1 outputs normal sine wave while O2 outputs a weak signal, as shown in Fig. 4(b). In this case, the amplitude ratio between two output beams is 3.22. Figure 4(c) shows the peak pressures variation of two output beams when the control source is moved up to 20 mm with 1-mm intervals. From Fig. 4(c), we note that the amplitude of each output beam changes periodically with the movement of control source, and the variations of two output beams show contrary trend. The period of the variation is about 15 mm, approximate one wavelength of 14.54 mm, meaning a phase change of 2p. Therefore, the output should switch between two faces O1 and O2 when the control source moves a half wavelength. Thus, the acoustic switching is obtained. Figures 5(a)–5(d) show the measured input signals with several different operation states and the output signals from the output faces. The corresponding Vin-Vout curves

FIG. 4. Experimental demonstration of tunable acoustic switch effect. (a) Normalized output signals pO1 and pO2 . (b) Normalized output signals pO1 and pO2 when the control source moves a half wavelength (7 mm) along the incident direction. (c) Variation of amplitudes A3 for pO1 and A4 for pO2 with the movement of the control source.

illustrating the relationship between output amplitude and two variable input amplitudes are also measured, as shown in Figs. 5(e)–5(h). Figures 5(a) and 5(b) present the input signals pI1 and pI2 , which are amplitude-modulated signals with different trigger periods (200 ls and 100 ls). Orderly, the input states are {1, 1}, {1, 0}, {0, 1}, and {0, 0} for t ¼ 0–100, 100–200, 200–300, and 300–400 ls, as shown in Figs. 5(e) and 5(f). Figures 5(c) and 5(d) show the output signals from O1 and O2, respectively. With the threshold At1 , the output states of O1 are “0,” “1,” “1,” and “0,” as shown in Fig. 5(g), operating as a XOR logic gate. Meanwhile, the output states of O2 correspond to “1,” “1,” “1,” and “0,” realizing an OR logic function, as shown in Fig. 5(h). With the threshold At2 , the output states of O2 are “1,” “0,” “0,” and “0,” as shown in Fig. 5(h), indicating an AND logic operation. From Fig. 5(g) we also note that if input pI1 works as a constant signal, the output pO1 actually corresponds to the NOT of pI2 . Therefore, the basic acoustic logic gate functions XOR, OR, AND, and NOT are achieved. Moreover, as a consequence of the unique low diffraction of self-collimated acoustic beam, more complex Boolean logic and algorithms can be realized by cascading the above basic logic gates.22 We can further construct a NAND gate by combining the AND and NOR gates, a NOR gate by the OR and NOR gates, and a XNOR gate by the XOR and NOR gates. Note that for the given operation frequency, precise phase locking in combining logic gates can be achieved by optimizing the length of self-collimation interval between two cascaded SCs. For instance, at the normalized operation frequency of 0.559 cw/a, a phase shift of p/2 can be achieved by setting the interval as a/0.559/4. As the requisite components are equipped, integrated acoustic circuits are theoretically realizable. Based on self-collimation effect of ultrasonic waves in SCs, acoustic switch and logic gates are proposed and demonstrated. The basic building block is composed of 2D SCs with a line defect on the diagonal. In this paper, the switching effect has been achieved by adjusting the phase difference between two incident beams (adjusting the control source). We further find that at certain phase difference (such as p = 2 or p = 2), the acoustic output from the faces

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FIG. 5. Basic acoustic logic gates implementing AND, OR, XOR, and NOT functions. (a) and (b) show time dependence of input signals pI1 and pI2 , respectively. (c) and (d) show time dependence of output signals pO1 and pO2 , respectively. (e)–(h) show Vin-Vout curves of two input and output amplitudes. The dashed line and the dotted line in (g) and (h) indicate the thresholds At1 and At2 , respectively.

O1, O2 can function as a series of logic gates such as AND, OR, XOR, and NOT gates. Moreover, due to the high transmission efficiency, the proposed logic gates can be cascaded in the same way that Boolean functions can be composed without amplification, which allows the construction of a physical model of complex Boolean logic and algorithms. The effective switching and logic gate functions and the simple structure make this acoustic device a strong candidate for ultrasonic computations in underwater communication and medical ultrasound diagnosis. This work was supported by the National Basic Research Program of China under Grant No. 2012CB921504, NSFC (11474162, 11274171, 11274099, and 11204145), and SRFDP (20110091120040 and 20120091110001). 1

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