Binary-phase acoustic passive logic gates

The recent rapid development of acoustic logic devices has opened up the possibilities of sound computing and information processing. However, simultaneous realization of acoustic logic devices with subwavelength size, broad bandwidth and passive structure still poses a great challenge. To overcome it, we propose a subwavelength acoustic logic gate which consists of binary-phase passive unit cells placed into a multi-port waveguide. Based on the phase manipulations of the unit cells, we experimentally and numerically realize three basic logic gates OR, NOT and AND, and a composite logic gate XOR with a uniform threshold of 0.4 Pa based on linear acoustic interferences. More importantly, We also design a composite logic gate XNOR by a four-port waveguide, and composite logic gates NOR and NAND and a logic operation A⊙(B+C) based on two logic gates. We demonstrate a 0.6λ-length, 0.3λ-width, and 0.2-fractional bandwidth acoustic logic gate constructed by passive structures, which may lead to important advances in various applications, such as acoustic computing, acoustic information processing and integrated acoustics.

In this work, we propose a subwavelength acoustic logic gate which consists of binary-phase passive unit cells placed into a multi-port waveguide. Based on the phase manipulations of the unit cells, we design the basic logic gates OR, NOT and AND, and two composite logic gates XOR and XNOR with a uniform threshold of 0.4 Pa based on the linear acoustic interferences, and the fractional bandwidth (the ratio of the bandwidth to the center frequency) can reach about 0.2. The measured results agree well with the simulated ones. Finally, we also present the realizations of two composite logic gates NOR and NAND and a logic operation A⊙(B+C) based on two logic gates.

Results and Discussions
Design of acoustic logic gate. As shown in Fig. 1a, the unit cell (with length l = λ/2 and width w = λ/ 10) consists of two symmetric Helmholtz resonator arrays and a straight channel 50 with a tunable width w 1 , where λ is the wavelength. The cavity width w 2 is determined by w 1 , and the other parameters l 1 = 0.03 l, l 2 = 0.225 l, w 3 = 0.05w and w 2 = w − w 1 − 4w 3 . The unit cell filled with air is fabricated with epoxy resin to satisfy the sound hard boundary condition. The wavelength λ of input acoustic signals is selected as 10 cm (viz., frequency f 0 = 3430 Hz in air). Figure 1b shows the phase delays (blue solid line) and transmissions (red dashed line) created by the unit cells with different width ratios (w 1 /w). Note that, in the range 0.12-0.70, the phase delay of the unit cell can cover the whole 2π range, and the transmission is larger than 0.85. Here, the two selected unit cells I and II (black hollow dots) with the parameter w 1 /w of 0.22 and 0.69, which correspond to the transmissions T I = 0.94 and T II = 0.99 and the difference of the phase delays ϕ I − ϕ II = π, are used to design acoustic logic gates.
As shown in Fig. 1c, we design an acoustic logic gate which consists of a symmetric three-port waveguide with a rectangular solid made of epoxy resins and two types of passive unit cells placed into two channels, in which the parameters d = 0.1λ and d 1 = 0.055λ. Note that the width and length of the logic gate structure are only about 0.3λ and 0.6λ, respectively. When the unit cells in the channels A and B are different (shown in Fig. 1c), two output signals with out-phase characteristic produce the interference cancellation at the port O. However, the interference enhancement is obtained at the port O by placing the same unit cells into the channels A and B www.nature.com/scientificreports www.nature.com/scientificreports/ owing to the in-phase characteristic of output signals. Based on the linear acoustic interferences, we can realize all acoustic logic gates and complex logic operations. Figure 2 shows the simulated results for the logic gates OR and XOR, in which the unit cells II is placed into both channels A and B for the logic gate OR, but the unit cells I and II are used to design the logic gate XOR. Here, the same incident signals (with the initial amplitude of 1.0 Pa) are placed at the input ports A and B. The input state is expressed as {Q 1 , Q 2 }, in which Q i represent if there is (encode "1") or not (encode "0") acoustic signal from the respective input port. In addition, to determine the output states, we introduce a uniform threshold A t = 0.4 Pa. When the output amplitude is larger than A t , the output state is 1/TRUE, while the output is 0/FALSE with the output amplitude smaller than A t .

Logic oR and XoR gates.
As shown in Fig. 2a,b, the output amplitude for the input state {1, 1} is about two times larger than those for the input states {1, 0} and {0, 1} in the logic gate OR (Fig. 2a), while that is almost zero for the input state {1, 1} in the logic gate XOR (Fig. 2b). Such a phenomenon is attributed to the interference enhancement and cancellation at the output port O owing to the phase manipulation of the unit cells, which is different from previous works based on external active manipulations of signal phases 31,35,49 . Figure 2c shows the bar charts of the output amplitudes and the true Logic Not gate. We also realize the logic gate NOT with the same structure as the logic gate XOR, in which A is the input port, and B is the control port with a constant signal. As shown in Fig. 3a, the output amplitude approaches zero for the input state {1} owing to the interference cancellation, while that is larger than 0.7 Pa for the input state {0}. With the uniform threshold of 0.4 Pa, the corresponding output states are {0} and {1} for the input states {1} and {0} (Fig. 3b), respectively, realizing the logic gate NOT. The corresponding input and output states are also clearly shown in the truth table.
Logic AND gate. In addition to the logical gates constructed by the three-port waveguide structure, we design the logic gate AND based on a four-port waveguide structure, in which the rectangular solid at the center of the three-port waveguide ( Fig. 1) is replaced by the channel C with the same size as the channel A. To realize the logic function AND, we introduce another passive unit cell placed into the channel C as the control port. Figure 4 shows the phase delays (blue solid line) and transmissions (red dashed line) of the unit cell with different cavity widths (l 2 ), in which the parameter w 1 /w is a constant of 0.12. Here, we select the unit cell III (the parameters w 1 /w = 0.12 and l 2 = 6.5 mm) with the transmission T III = 0.71 and the phase delay ϕ III = ϕ I + π/6, which is utilized to realize the output amplitude less than 0.4 Pa for the input state {0, 0} and the interference cancellation between the input and control ports.
As shown in Fig. 5a, the output amplitude is about 0.92 Pa for the input state {1, 1}, which is smaller than that of the logic gate OR (Fig. 2a). Such energy reduction arises from the weak interference cancellation induced by the control signal from the port C. Moreover, the output amplitudes are smaller than 0.4 Pa for the input states  Working bands of basic logic gates. The proposed logic gates have broadband characteristic. As shown in Fig. 6a, for the logic gate OR, the output amplitudes for the three input states are larger than 0.4 Pa, and its working band far exceeds the range 3140-3840 Hz (shadowed region) owing to the interference enhancement based on the phase manipulation of two same unit cells. However, the working bands are fixed in the ranges 3140-3840 Hz for the logic gates XOR (Fig. 6b) and NOT (Fig. 6c) and 3270-3880 Hz for the logic gate AND (Fig. 6d). This is because the design mechanisms are closely related to the interference cancellation based on two different unit cells with the out-phase characteristic, and the phase difference of the two unit cells is determined by the frequency. Therefore, we have demonstrated the broad bandwidth of the logic gates, in which the fractional bandwidth can reach about 0.2 for the logic gates OR, XOR and NOT, and is 0.17 for the logic gate AND.   www.nature.com/scientificreports www.nature.com/scientificreports/ experimental measurements. To further demonstrate the aforementioned logic gates, we experimentally measure output time-domain signals for these logic gates. The measurement setup and the photograph of the sample are shown in the Fig. 7. Figure 8a-e shows the measured time-domain signals at the input ports A and B and the output port O for the logic gates OR, XOR and AND, respectively, in which all measured signals are normalized by the maximum amplitude in Fig. 8a. As shown in Fig. 8a   Moreover, as shown in Fig. 9a,b, the input signal at the port A is modulated to realize two input states {1} and {0} in the time domains t = 0-3 ms and 3-6 ms, respectively, and the constant signal placed at the port B acts as a control port (Fig. 9b). Figure 9c shows the measured output signals for the logic gate NOT, in which the output states are {0} and {1} based on the uniform threshold of 0.4 Pa, and the experiment results agree well with the logic function NOT. However, there exist weak output signals for the input states {1, 1} (Fig. 8d) and {1} (Fig. 9c), which arises from non-complete interference cancellation of two signals induced by considerable fabrication errors of the passive unit cells. Therefore, we have experimentally demonstrated the logic gates OR, XOR, AND and NOT.
Logic XNoR gate. We can design the logic gate XNOR based on the four-port waveguide, in which the unit cell II is placed into the channels A and B as the input ports, but the unit cell I is placed into the channel C as the control port. As shown in Fig. 10a, the output amplitude at the port O for the input state {1, 1} is almost the same as that for {0, 0}, which arises from the fact that the interference cancellation exists between the signals from the ports A and C for {1, 1}, and therefore the output signals are from the ports B and C for the input states {1, 1} and {0, 0}, respectively. In addition, the output amplitudes are smaller than 0.4 Pa for the input states {1, 0} and {0, 1} owing to the interference cancellation of the out-phase signals created by the phase manipulation of the unit cells. As shown in Fig. 10b Logic NoR gate. Owing to the easy connection characteristic of the multi-port waveguide, we can realize the composite logic gates NOR and NAND based on two basic logic gates. Figure 11 shows the simulated results for the composite logic gate NOR composed of the logic gates OR and NOT, in which A and B are the input ports, and C is the control port. As shown in Fig. 11a, the output amplitude for the input state {0, 0} at the port O is about 0.7 owing to the signal energy from the control port C. However, the output amplitudes are smaller than 0.4 Pa for the input states {1, 1}, {0, 1}, and {1, 0}, which is attributed to the interference cancellation induced by two signals from output port of the logic gate OR and the control port C. As shown in Fig. 11b   www.nature.com/scientificreports www.nature.com/scientificreports/ Logic NAND gate. Furthermore, as shown in Fig. 12, the composite logic gate NAND consists of the basic logic gates NOT and AND, in which A and B are the input ports, and C and D are the control ports. It is noted that the output amplitude for the input state {1, 1} is small than 0.4 Pa, which stems from the interference cancellation between the output signal of the logic gate AND and the control signal at the port D. Besides, the output amplitudes are large than 0.4 Pa for the input states {0, 1}, {1, 0} and {0, 0}. This is because the signal energy at the port D is stronger than that at the output port of the logic gate AND, and thus the weak interference cancellation is obtained at the port O. Thus, as shown in Fig. 12b Acoustic logic operation. We can also carry out complex acoustic logic operations based on the designed logic gates. As an example, we realize the logic operation A⊙(B+C) based on the logic gates OR and XNOR (shown in Fig. 13), in which A, B, and C are the input ports, and D is the control port. As shown in Fig. 13a Working bands of composite logic gates and logic operation. Finally, we discuss the bandwidth of the composite logic gates XNOR, NOR and NAND and the logic operation A⊙(B+C), which is shown in Fig. 14. Compared with the working bands of the basic logic gates (shown in Fig. 6), the band of the logic gate XNOR (3100-3740 Hz) remains almost unchanged, but those of the logic gates NOR (3350-3900 Hz) and NAND (3300-3750 Hz) and the logic operation A⊙(B+C) (3160-3500 Hz) change greatly. This is because the logic functions www.nature.com/scientificreports www.nature.com/scientificreports/ NOR and NAND and the logic operation are realized based on two different logic gates, and there exists a mismatch between the output and input amplitudes of both the logic gates.

Conclusion
In conclusion, we have demonstrated the binary-phase acoustic logic gate based on the multi-port waveguide structure, in which the width and length of the logic gate are only about 0.3λ and 0.6λ. The results show that the basic logic gates OR, NOT and AND, and the composite logic gate XOR with a uniform threshold of 0.4 Pa are realized based on the linear acoustic interferences, which arises from the phase manipulations of the binary-phase unit cells. The measured results agree well with the simulated ones, and the fractional bandwidth can reach about 0.2. Moreover, we design the composite logic gate XNOR by using the four-port waveguide, and realize the composite logic gates NOR and NAND and the logic operation A⊙(B+C) based on two different logic gates, corresponding to the logic gates NOT and OR, the logic gates NOT and AND, and the logic gates OR and XNOR, respectively. The proposed acoustic logic gates have advantages of the subwavelength size, broad bandwidth, and passive structure, which have important potential applications in acoustic computing, acoustic information processing and integrated acoustics.

Methods
sample and experimental setup. The designed acoustic logic gates are fabricated with epoxy resin by three-dimensional printing technology. To obtain the same input signals, two same cylindrical acoustic sources are placed on the right side of the input ports, and the distance between the unit cell and the cylindrical source is 5 mm. The input signals are generated from the cylindrical sources driven by a power amplifier. On the left side of the output port, a 1/4 inch microphone (Brüel & Kjaer type-4961) is adopted to measure output signals. The measured data is recorded by the Brüel&Kjaer 3160-A-022 module, and is analyzed by the software PULSE Labshop.
Numerical simulations. The finite element method (COMSOL Multiphysics software 5.2a) is utilized to numerically simulate the characteristics of the acoustic logic gates. The pressure field is calculated in the Acoustic-Solid interaction module, and the Helmholtz resonator arrays filled with air are fabricated with epoxy