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Ultrasmall optical logic gates based on silicon periodic dielectric waveguides
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Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37 www.elsevier.com/locate/photonics
Ultrasmall optical logic gates based on silicon periodic dielectric waveguides Shunquan Zeng a, Yao Zhang a, Baojun Li a,*, Edwin Yue-Bun Pun b a
State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China b Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China Received 12 November 2009; received in revised form 22 January 2010; accepted 25 January 2010 Available online 4 February 2010
Abstract An ultrasmall silicon periodic dielectric waveguides-based multimode interference all-optical logic gate has been proposed. The device consists of three 205 nm wide single-mode input waveguides, a 1.1 mm wide and 5.5 mm long multimode interference waveguide, and three 205 nm wide single-mode output waveguides. The total length and width of the device are 13.7 mm and 3.2 mm, respectively. By changing the states of the input optical signals and/or control signals launched into the device, multifunctional logic functions including OR, NAND, NOR, and NOT gates are performed, and each logic function can be realized at a specific output waveguide in accordance with the launched control signals. The ultrasmall multifunctional logic device has potential applications in high density photonic integrated circuits. # 2010 Elsevier B.V. All rights reserved. Keywords: Periodic dielectric waveguide; Optical logic gate; Multimode interference; Photonic integrated circuit
1. Introduction All-optical signal processing techniques are attractive in ultrahigh-speed and high-capacity optical networks. To avoid cumbersome optical–electrical–optical conversion, the demands for all-optical logic devices are rapidly increasing, and various approaches have been explored to realize different logic functions. Semiconductor optical amplifiers (SOAs)-based all-optical logic gates [1–3] usually have low operation speeds due to the long response time of the SOAs with unavoidable spontaneous emission noise, and the function is limited to a single gate. SOA Mach-Zehnder interferometers
* Corresponding author. Tel.: +86 20 8411 0200; fax: +86 20 8411 0200. E-mail address: [email protected] (B. Li). 1569-4410/$ – see front matter # 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2010.01.002
(MZIs) based multifunctional optical logic gates have been reported [4], but they use two parallel structures and are not single device units. Nonlinear optical component-based all-optical logic gates [5–9] have been widely investigated, but high optical power and long interaction length (several hundred microns to several millimeters) limit their applications. In addition, the number of logic gates is only one or two [5–7] in a single device chip. For multifunction devices, complex structures such as asymmetric nonlinear directional coupler and nonlinear optical loop mirror have to be used [8,9], but these devices are difficult to be integrated in high density photonic integrated circuits. Periodically poled lithium niobate (PPLN) waveguide-based optical logic gate with single function has been reported [10]. This device also has a complex configuration. The length of the PPLN waveguide is 50 mm, and the logic gate is temperature dependent. Photonic crystals (PCs)-
S. Zeng et al. / Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37
based all-optical logic gates with small sizes (estimated: 11.1 mm 5.8 mm [11], 8.3 mm 8.3 mm [12], 16.4 mm 14.6 mm [13], and 5.1 mm 5.1 mm [14] operating at 1550 nm wavelength) and simple structures are good candidates for all-optical signal processing and information communications. However, these PCsbased logic devices require a wide background region (at least several lattice constants), and the design of the PC structures depends strongly on the lattice orientation. Hence, the flexibility of the PCs-based device structure is limited by the lattice orientation in order to obtain multifunctional gates in a single device unit. Silicon multimode interference (MMI) based alloptical logic device with multifunctional performance has been reported recently [15]. The device has a simple configuration but the MMI section is 38.4 mm wide and 4980 mm long. The total length of the device is 6500 mm with an input/output waveguide width of 10 mm, thus it is not suitable for high density photonic integrated circuits. A good alternate candidate for building ultrasmall device is periodic dielectric waveguides (PDWs) [16–19]. PDWs have simple structure, occupy very little space, and can have arbitrary shape bends with high transmission. In this work, we have proposed and studied a silicon PDWs-based optical logic gate with multifunction and ultrasmall size, which is suitable for high density photonic integrated circuits and near-infrared wavelength applications. 2. Device structure Fig. 1(a) shows a schematic diagram of the proposed all-optical logic device based on silicon periodic dielectric (PD) rods in air. It consists of three singlemode input waveguides A, B, and C, a multimode waveguide, and three single-mode output waveguides D, E, and F. The waveguides A, C, D, and F are Sshaped bends and contain 19 dielectric rods. The waveguides B and E are straight waveguides and contain 18 dielectric rods. The multimode waveguide contains five parallel PD rows [19,20], and is shown in details in Fig. 1(b). The refractive index of the silicon dielectric rods n = 3.45 (at 1550 nm wavelength). The radius of all the dielectric rods r = 0.45a, where a is the center-to-center distance between two adjacent dielectric rods in each row. The center-to-center distance between two adjacent rows is 1.2a. The multimode waveguide length and width are 24a and 4.8a, respectively. To avoid the coupling effect between the two adjacent input or output waveguides, the bending angles of the S-shaped bend in the waveguides A, C, D, and F are designed to be 308. It should be noted
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Fig. 1. (a) Schematic diagram of the proposed PDWs-based alloptical logic device. (b) Multimode waveguide with five parallel periodic dielectric rows in air.
that the center-to-center distance between the two adjacent dielectric rods remains as a, thus the radius of curvature R of each bend is a/(sin 58) = 11.47a. The maximum distance between the centers of two adjacent input/output waveguides is 7a, and the total length and width of the device are 60a and 14a, respectively. The TE mode in PDWs consisting of dielectric rods tends to concentrate in air rather than in the waveguide [16], therefore, only TM mode (E-polarization) is considered in this work. Fig. 2 shows the calculated TM mode band structures in the multimode waveguide. In
Fig. 2. Calculated TM mode band structures in the multimode waveguide. The shaded region is the light cone. The solid curves located on the right hand side of the light cone are five guided modes (0th–4th modes). The inset denotes the supercell used in the calculations.
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S. Zeng et al. / Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37
the calculations, a supercell with a size of a 14.8a (Fig. 2, inset) is used. The shaded region is the light cone, denoting the extended modes which are not suitable for lightwave guiding. The solid curves located on the right hand side of the light cone are guided modes. To satisfy multimode conditions (supporting five modes) and keep imaging positions of three-fold images at the end of the multimode waveguide (with a length of 24a), an operating frequency of 0.147(a/l) with five guided modes is chosen as the incident lightwave frequency for subsequent calculations and simulations. The 0th, 2nd and 4th modes are even modes, and the 1st and 3rd modes are odd modes. For the structure shown in Fig. 1(a), when an optical signal with a frequency of 0.147(a/l) is launched into the multimode waveguide through any one of the three input waveguides, three kinds of images can be reproduced in the multimode waveguide: a single image, a two-fold image and a three-fold image [19]. To operate at wavelengths at 1550 nm, we choose a = 228 nm. Hence, the multimode waveguide length is 5.5 mm and the width is 1.1 mm. The device total length and width are 13.7 mm and 3.2 mm, respectively, and each single-mode waveguide is only 205 nm (2r = 0.9a) wide. 3. Results and discussion To illustrate the logic functions, two-dimensional (2D) finite-difference time-domain (FDTD) method with a perfectly matched layer boundary condition is used. In the simulation, the configuration of Fig. 1(a) is transformed into a FDTD computational domain, and all the input optical signals and the control signals are assumed to have the same wavelength and initial phase w0, as shown in Fig. 3. In Fig. 3, two phase shifters are used to obtain the phase differences Dw1 and Dw2, where Dw1 is the phase difference between the optical signals in waveguides A and B, and Dw2 is the phase difference between the optical signals in waveguides A and C. Thus, the phase difference between the optical signals in waveguides B and C is Dw2 Dw1. In the simulation,
Fig. 3. Schematic diagram of the proposed all-optical logic device with two phase shifters, and the phase difference between the optical signals in waveguides A and B is Dw1, and the phase difference between the optical signals in waveguides A and C is Dw2.
Fig. 4. Simulated time-averaged intensity distributions in the logic device when optical signal is launched into one of the three input waveguides only.
the phase differences Dw1 = p/2 and Dw2 = p/12. By changing the states (with or without optical signals) of the optical signals for different input waveguides, the simulated field distributions can be obtained. Fig. 4 shows the time-averaged intensity distributions in the logic device with only one optical signal launched into only one of the three input waveguides. It can be seen that there are always output optical signals at the output waveguides D, E, and F. This is attributed to the three-fold images reproduced at the end of the multimode waveguide. Based on this output condition some all-optical logic functions (OR, NAND gates, etc.) can be realized. The imaging positions of the three-fold images are calculated further. The wave vector values for the two lowest-order modes at the frequency of 0.147(a/l) are obtained from the guided mode curves in Fig. 2, and the propagation constants b0 = 0.76862(p/a) and b1 = 0.72910(p/a). By substituting the values of b0 and b1 into the formula Lp = p/(b0 b1), the beat length Lp of the two lowest-order modes is 25.3036a. The imaging position x = L of N-fold images can be approximated by the formulas L = (3PLp)/N and L = (3PLp)/(4N) expressed in Ref. [21], where P 0 and N 1 are integers having no common divisor. For asymmetric interference, as shown in Fig. 4(a) and (c), the imaging position of the three-fold images can be calculated by the formula L = (3PLp)/N with N = 3 and P = 1 (indicating the first three-fold image), and L = 25.3036a. For symmetric interference, as shown in Fig. 4(b), the imaging position of the three-fold images can be calculated by the formula L = (3PLp)/
S. Zeng et al. / Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37
(4N) with N = 3 and P = 4 (indicating the third threefold image), and L = 25.3036a. To counteract the additional coupling at the connection part of the two adjacent output waveguides, the length of the multimode waveguide was set to be 24a (a little less than 25.3036a). Assuming that the input powers in the input waveguides A, B, and C are PA, PB, and PC, respectively, the total input power will be Pin = PA + PB + PC. If the output powers in the output waveguides D, E, and F are PD, PE, and PF, respectively, the total output power will be Pout = PD + PE + PF. The calculated insertion losses, which is defined as 10 log(Pout/Pin), are 0.097 dB, 0.110 dB, and 0.097 dB in Fig. 4(a)–(c), respectively. If we define that the phase difference between the output optical signals in the waveguides D and E is Dw3 while in the waveguides D and F is Dw4, we calculated that the Dw3 are 788, 1658, and 398 while the Dw4 are 1178, 08, and 1168 for Fig. 4(a)–(c), respectively. Fig. 5 shows the time-averaged intensity distributions in the logic device when optical signals are launched into any two or all of the three input waveguides. As shown in Fig. 5(a), when two optical signals with an initial phase w0 and a phase difference Dw1 = p/2 are launched into input waveguides A and B, respectively, there will be output signals in the output
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waveguides E and F, and no output signal in the output waveguide D. As shown in Fig. 5(b), when two optical signals with an initial phase w0 and a phase difference Dw2 = p/12 are launched into input waveguides A and C, respectively, there is almost no optical signal in the waveguide F, but there are output optical signals in the waveguides D and E. As shown in Fig. 5(c), when two optical signals with a phase difference Dw2 Dw1 = 7p/ 12 are launched into input waveguides B and C, the output states are similar to those in Fig. 5(b). As shown in Fig. 5(d), when optical signals with w0, Dw1 = p/2, and Dw2 = p/12 are launched into all of the three input waveguides A, B, and C, respectively, the optical signal will be at waveguide E only. The calculated insertion losses are 0.186 dB, 0.128 dB, 0.146 dB, and 0.245 dB in the cases of Fig. 5(a)–(d), respectively. The calculated phase difference (Dw3) between the output optical signals in the waveguides D and E are 198, 238, 1018, and 438 while the calculated phase difference (Dw4) in the waveguides D and F are 1208, 48, 1778, and 248 for Fig. 5(a)–(d), respectively. By comparing the time-averaged intensity distributions in Figs. 4 and 5 four kinds of basic logic functions (OR, NAND, NOR, and NOT gates) exist, and they are summarized as follows: (1) OR gate: When an input optical signal(s) are launched into the input waveguides A, and/or B, and/or C individually or simultaneously, there is always an optical signal in output waveguide E (see Figs. 4 and 5). In this work, the power of optical signals in each input (with optical signal input) is identical P0. For each output, if the power of output signal is higher than 0.25P0, represents logic ‘1’, contrarily, represents logic ‘0’. Hence, the device operates as an OR logic gate, i.e. E = A + B + C as shown in Table 1, and the calculated average insertion loss is 0.144 dB. Table 1 OR logic gate. Input optical signal
Fig. 5. Simulated time-averaged intensity distributions in the logic device when optical signals are launched into any two or all of the three input waveguides.
Comparison
A
B
C
0 1 0 0 1 1 0 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
Output signal E
Fig. Fig. Fig. Fig. Fig. Fig. Fig.
4(a) 4(b) 4(c) 5(a) 5(b) 5(c) 5(d)
0 1 1 1 1 1 1 1
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S. Zeng et al. / Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37
Table 2 NAND logic gate. Optical signal
Control signals A
B
C
0 1 0 1
0 0 1 1
1 1 1 1
Table 4 NOT logic gate. Comparison
Fig. Fig. Fig. Fig.
4(c) 5(b) 5(c) 5(d)
Output signal
Optical signals
Control signal
D
A
B
C
1 1 1 0
1 1
1 1
0 1
Comparison
Output signal F
Fig. 5(a) Fig. 5(d)
1 0
Comparison
Output signal
Table 5 NOT logic gate.
(2) NAND gate: When an optical signal is launched into waveguide C without control signal(s) or with a control signal in waveguides A or B, there is an output optical signal in waveguide D (Figs. 4c and 5b and c). When control signals are launched simultaneously into waveguides A and B, the output signal in waveguide D is cut-off (Fig. 5d). Hence, the device operates as a NAND gate, i.e. D ¼ A B as shown in Table 2, and the calculated average insertion loss is 0.154 dB. (3) NOR gate: When an optical signal is launched into waveguide C without control signal(s), there is an optical signal in waveguide F (Fig. 4c). When control signals are launched into the waveguides A and/or B, there is no output signal in the waveguide F (Fig. 5b–d). Hence, the device operates as a NOR gate, i.e. F ¼ A þ B as shown in Table 3, and the calculated average insertion loss is also 0.154 dB. (4) NOT gates: When optical signals are launched simultaneously into input waveguides A and B without a control signal in waveguide C, there are output signals in waveguides E and F (Fig. 5a). When a control signal is launched into waveguide C, there is an output signal in waveguide E and no signal in waveguide F (Fig. 5d). Hence, the device operates as a NOT gate, i.e. F ¼ C as shown in Table 4, and the calculated average insertion loss is 0.216 dB. Similarly, when optical signals are launched simultaneously into input waveguides B and C without a Table 3 NOR logic gate. Control signals
Optical signal
A
B
C
0 1 0 1
0 0 1 1
1 1 1 1
Comparison
Output signal F
Fig. Fig. Fig. Fig.
4(c) 5(b) 5(c) 5(d)
1 0 0 0
Control signal
Optical signals B
C
A
1 1
1 1
0 1
D Fig. 5(c) Fig. 5(d)
1 0
control signal in waveguide A, there are output signals in waveguides D and E (Fig. 5c). When a control signal is launched into waveguide A, there is an output signal in waveguide E and no signal in waveguide D (Fig. 5d). Hence, the device operates as a NOT gate, i.e. D ¼ A as shown in Table 5, and the calculated average insertion loss is 0.196 dB. To calculate extinction ratios (ERs) of the logic gates, we define ER as 10 log(PON/POFF), where PON is the output power of logic ‘1’ (i.e. with optical signal), and POFF is the output power of logic ‘0’ (i.e. without optical signal). For Fig. 5(a), the calculated ER for the outputs E and D (denoted by E/D) is 14.5 dB, the ER for the outputs F/D is 14.8 dB. For Fig. 5(b), the ERs for the outputs D/F and E/F are 8.8 dB and 10.7 dB, respectively. For Fig. 5(c), the ERs for the outputs D/ F and E/F are 15.0 dB and 15.6 dB, respectively. For Fig. 5(d), the ERs for the outputs E/D and E/F are 10.6 dB and 14.5 dB, respectively. It is notable that all of the simulation results were performed with fixed values of phase differences, but it can be used for different initial phase difference with different interference phenomenon in the multimode waveguide. In this case, output optical intensity and phase in each output waveguide will be different. As a result, the logic states (‘0’ and ‘1’) may be different for a specified output waveguide. However, to perform multifunctional logic gates in a single device, the phase differences should be 858 Dw1 958 and 138 Dw2 178. In addition, considering the phase differences of the output signals are irregular, additional phase processing would be needed in a photonic integrated circuit that uses this optical logic gate. It should also be pointed out that the device structure and
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performed simulations/calculations are done in 2D, i.e. the dielectric rods in the device are considered as infinite on the vertical dimension in simulations. The difference of the results in 2D and 3D structures can be referred from the discussion in Ref. [18]. Last, in this work, a slight variation of rod radius (less than 2%) will not affect the properties (insertion losses and extinction ratios) of the device. This is because a variation of less than 2% in rod radius has little impact on the band structures of the single-row PDWs or multimode waveguide. 4. Conclusion An ultrasmall silicon PDWs-based optical logic gate has been designed based on the MMI principle. The single device can perform individually or simultaneously four different kinds of basic logic functions: OR, NAND, NOR and NOT gates. The total size of the device is 13.7 mm 3.2 mm and each input/output waveguide is only 205 nm wide. The average insertion loss of each logic gate is less than 0.22 dB and the ERs are between 8.8 dB and 15.6 dB. The proposed multifunctional logic device with extremely small feature size is an attractive candidate for high density photonic integrated circuits. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 60625404, 60577001, and 10974261).
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Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37 www.elsevier.com/locate/photonics
Ultrasmall optical logic gates based on silicon periodic dielectric waveguides Shunquan Zeng a, Yao Zhang a, Baojun Li a,*, Edwin Yue-Bun Pun b a
State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China b Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China Received 12 November 2009; received in revised form 22 January 2010; accepted 25 January 2010 Available online 4 February 2010
Abstract An ultrasmall silicon periodic dielectric waveguides-based multimode interference all-optical logic gate has been proposed. The device consists of three 205 nm wide single-mode input waveguides, a 1.1 mm wide and 5.5 mm long multimode interference waveguide, and three 205 nm wide single-mode output waveguides. The total length and width of the device are 13.7 mm and 3.2 mm, respectively. By changing the states of the input optical signals and/or control signals launched into the device, multifunctional logic functions including OR, NAND, NOR, and NOT gates are performed, and each logic function can be realized at a specific output waveguide in accordance with the launched control signals. The ultrasmall multifunctional logic device has potential applications in high density photonic integrated circuits. # 2010 Elsevier B.V. All rights reserved. Keywords: Periodic dielectric waveguide; Optical logic gate; Multimode interference; Photonic integrated circuit
1. Introduction All-optical signal processing techniques are attractive in ultrahigh-speed and high-capacity optical networks. To avoid cumbersome optical–electrical–optical conversion, the demands for all-optical logic devices are rapidly increasing, and various approaches have been explored to realize different logic functions. Semiconductor optical amplifiers (SOAs)-based all-optical logic gates [1–3] usually have low operation speeds due to the long response time of the SOAs with unavoidable spontaneous emission noise, and the function is limited to a single gate. SOA Mach-Zehnder interferometers
* Corresponding author. Tel.: +86 20 8411 0200; fax: +86 20 8411 0200. E-mail address: [email protected] (B. Li). 1569-4410/$ – see front matter # 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2010.01.002
(MZIs) based multifunctional optical logic gates have been reported [4], but they use two parallel structures and are not single device units. Nonlinear optical component-based all-optical logic gates [5–9] have been widely investigated, but high optical power and long interaction length (several hundred microns to several millimeters) limit their applications. In addition, the number of logic gates is only one or two [5–7] in a single device chip. For multifunction devices, complex structures such as asymmetric nonlinear directional coupler and nonlinear optical loop mirror have to be used [8,9], but these devices are difficult to be integrated in high density photonic integrated circuits. Periodically poled lithium niobate (PPLN) waveguide-based optical logic gate with single function has been reported [10]. This device also has a complex configuration. The length of the PPLN waveguide is 50 mm, and the logic gate is temperature dependent. Photonic crystals (PCs)-
S. Zeng et al. / Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 32–37
based all-optical logic gates with small sizes (estimated: 11.1 mm 5.8 mm [11], 8.3 mm 8.3 mm [12], 16.4 mm 14.6 mm [13], and 5.1 mm 5.1 mm [14] operating at 1550 nm wavelength) and simple structures are good candidates for all-optical signal processing and information communications. However, these PCsbased logic devices require a wide background region (at least several lattice constants), and the design of the PC structures depends strongly on the lattice orientation. Hence, the flexibility of the PCs-based device structure is limited by the lattice orientation in order to obtain multifunctional gates in a single device unit. Silicon multimode interference (MMI) based alloptical logic device with multifunctional performance has been reported recently [15]. The device has a simple configuration but the MMI section is 38.4 mm wide and 4980 mm long. The total length of the device is 6500 mm with an input/output waveguide width of 10 mm, thus it is not suitable for high density photonic integrated circuits. A good alternate candidate for building ultrasmall device is periodic dielectric waveguides (PDWs) [16–19]. PDWs have simple structure, occupy very little space, and can have arbitrary shape bends with high transmission. In this work, we have proposed and studied a silicon PDWs-based optical logic gate with multifunction and ultrasmall size, which is suitable for high density photonic integrated circuits and near-infrared wavelength applications. 2. Device structure Fig. 1(a) shows a schematic diagram of the proposed all-optical logic device based on silicon periodic dielectric (PD) rods in air. It consists of three singlemode input waveguides A, B, and C, a multimode waveguide, and three single-mode output waveguides D, E, and F. The waveguides A, C, D, and F are Sshaped bends and contain 19 dielectric rods. The waveguides B and E are straight waveguides and contain 18 dielectric rods. The multimode waveguide contains five parallel PD rows [19,20], and is shown in details in Fig. 1(b). The refractive index of the silicon dielectric rods n = 3.45 (at 1550 nm wavelength). The radius of all the dielectric rods r = 0.45a, where a is the center-to-center distance between two adjacent dielectric rods in each row. The center-to-center distance between two adjacent rows is 1.2a. The multimode waveguide length and width are 24a and 4.8a, respectively. To avoid the coupling effect between the two adjacent input or output waveguides, the bending angles of the S-shaped bend in the waveguides A, C, D, and F are designed to be 308. It should be noted
33
Fig. 1. (a) Schematic diagram of the proposed PDWs-based alloptical logic device. (b) Multimode waveguide with five parallel periodic dielectric rows in air.
that the center-to-center distance between the two adjacent dielectric rods remains as a, thus the radius of curvature R of each bend is a/(sin 58) = 11.47a. The maximum distance between the centers of two adjacent input/output waveguides is 7a, and the total length and width of the device are 60a and 14a, respectively. The TE mode in PDWs consisting of dielectric rods tends to concentrate in air rather than in the waveguide [16], therefore, only TM mode (E-polarization) is considered in this work. Fig. 2 shows the calculated TM mode band structures in the multimode waveguide. In
Fig. 2. Calculated TM mode band structures in the multimode waveguide. The shaded region is the light cone. The solid curves located on the right hand side of the light cone are five guided modes (0th–4th modes). The inset denotes the supercell used in the calculations.
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the calculations, a supercell with a size of a 14.8a (Fig. 2, inset) is used. The shaded region is the light cone, denoting the extended modes which are not suitable for lightwave guiding. The solid curves located on the right hand side of the light cone are guided modes. To satisfy multimode conditions (supporting five modes) and keep imaging positions of three-fold images at the end of the multimode waveguide (with a length of 24a), an operating frequency of 0.147(a/l) with five guided modes is chosen as the incident lightwave frequency for subsequent calculations and simulations. The 0th, 2nd and 4th modes are even modes, and the 1st and 3rd modes are odd modes. For the structure shown in Fig. 1(a), when an optical signal with a frequency of 0.147(a/l) is launched into the multimode waveguide through any one of the three input waveguides, three kinds of images can be reproduced in the multimode waveguide: a single image, a two-fold image and a three-fold image [19]. To operate at wavelengths at 1550 nm, we choose a = 228 nm. Hence, the multimode waveguide length is 5.5 mm and the width is 1.1 mm. The device total length and width are 13.7 mm and 3.2 mm, respectively, and each single-mode waveguide is only 205 nm (2r = 0.9a) wide. 3. Results and discussion To illustrate the logic functions, two-dimensional (2D) finite-difference time-domain (FDTD) method with a perfectly matched layer boundary condition is used. In the simulation, the configuration of Fig. 1(a) is transformed into a FDTD computational domain, and all the input optical signals and the control signals are assumed to have the same wavelength and initial phase w0, as shown in Fig. 3. In Fig. 3, two phase shifters are used to obtain the phase differences Dw1 and Dw2, where Dw1 is the phase difference between the optical signals in waveguides A and B, and Dw2 is the phase difference between the optical signals in waveguides A and C. Thus, the phase difference between the optical signals in waveguides B and C is Dw2 Dw1. In the simulation,
Fig. 3. Schematic diagram of the proposed all-optical logic device with two phase shifters, and the phase difference between the optical signals in waveguides A and B is Dw1, and the phase difference between the optical signals in waveguides A and C is Dw2.
Fig. 4. Simulated time-averaged intensity distributions in the logic device when optical signal is launched into one of the three input waveguides only.
the phase differences Dw1 = p/2 and Dw2 = p/12. By changing the states (with or without optical signals) of the optical signals for different input waveguides, the simulated field distributions can be obtained. Fig. 4 shows the time-averaged intensity distributions in the logic device with only one optical signal launched into only one of the three input waveguides. It can be seen that there are always output optical signals at the output waveguides D, E, and F. This is attributed to the three-fold images reproduced at the end of the multimode waveguide. Based on this output condition some all-optical logic functions (OR, NAND gates, etc.) can be realized. The imaging positions of the three-fold images are calculated further. The wave vector values for the two lowest-order modes at the frequency of 0.147(a/l) are obtained from the guided mode curves in Fig. 2, and the propagation constants b0 = 0.76862(p/a) and b1 = 0.72910(p/a). By substituting the values of b0 and b1 into the formula Lp = p/(b0 b1), the beat length Lp of the two lowest-order modes is 25.3036a. The imaging position x = L of N-fold images can be approximated by the formulas L = (3PLp)/N and L = (3PLp)/(4N) expressed in Ref. [21], where P 0 and N 1 are integers having no common divisor. For asymmetric interference, as shown in Fig. 4(a) and (c), the imaging position of the three-fold images can be calculated by the formula L = (3PLp)/N with N = 3 and P = 1 (indicating the first three-fold image), and L = 25.3036a. For symmetric interference, as shown in Fig. 4(b), the imaging position of the three-fold images can be calculated by the formula L = (3PLp)/
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(4N) with N = 3 and P = 4 (indicating the third threefold image), and L = 25.3036a. To counteract the additional coupling at the connection part of the two adjacent output waveguides, the length of the multimode waveguide was set to be 24a (a little less than 25.3036a). Assuming that the input powers in the input waveguides A, B, and C are PA, PB, and PC, respectively, the total input power will be Pin = PA + PB + PC. If the output powers in the output waveguides D, E, and F are PD, PE, and PF, respectively, the total output power will be Pout = PD + PE + PF. The calculated insertion losses, which is defined as 10 log(Pout/Pin), are 0.097 dB, 0.110 dB, and 0.097 dB in Fig. 4(a)–(c), respectively. If we define that the phase difference between the output optical signals in the waveguides D and E is Dw3 while in the waveguides D and F is Dw4, we calculated that the Dw3 are 788, 1658, and 398 while the Dw4 are 1178, 08, and 1168 for Fig. 4(a)–(c), respectively. Fig. 5 shows the time-averaged intensity distributions in the logic device when optical signals are launched into any two or all of the three input waveguides. As shown in Fig. 5(a), when two optical signals with an initial phase w0 and a phase difference Dw1 = p/2 are launched into input waveguides A and B, respectively, there will be output signals in the output
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waveguides E and F, and no output signal in the output waveguide D. As shown in Fig. 5(b), when two optical signals with an initial phase w0 and a phase difference Dw2 = p/12 are launched into input waveguides A and C, respectively, there is almost no optical signal in the waveguide F, but there are output optical signals in the waveguides D and E. As shown in Fig. 5(c), when two optical signals with a phase difference Dw2 Dw1 = 7p/ 12 are launched into input waveguides B and C, the output states are similar to those in Fig. 5(b). As shown in Fig. 5(d), when optical signals with w0, Dw1 = p/2, and Dw2 = p/12 are launched into all of the three input waveguides A, B, and C, respectively, the optical signal will be at waveguide E only. The calculated insertion losses are 0.186 dB, 0.128 dB, 0.146 dB, and 0.245 dB in the cases of Fig. 5(a)–(d), respectively. The calculated phase difference (Dw3) between the output optical signals in the waveguides D and E are 198, 238, 1018, and 438 while the calculated phase difference (Dw4) in the waveguides D and F are 1208, 48, 1778, and 248 for Fig. 5(a)–(d), respectively. By comparing the time-averaged intensity distributions in Figs. 4 and 5 four kinds of basic logic functions (OR, NAND, NOR, and NOT gates) exist, and they are summarized as follows: (1) OR gate: When an input optical signal(s) are launched into the input waveguides A, and/or B, and/or C individually or simultaneously, there is always an optical signal in output waveguide E (see Figs. 4 and 5). In this work, the power of optical signals in each input (with optical signal input) is identical P0. For each output, if the power of output signal is higher than 0.25P0, represents logic ‘1’, contrarily, represents logic ‘0’. Hence, the device operates as an OR logic gate, i.e. E = A + B + C as shown in Table 1, and the calculated average insertion loss is 0.144 dB. Table 1 OR logic gate. Input optical signal
Fig. 5. Simulated time-averaged intensity distributions in the logic device when optical signals are launched into any two or all of the three input waveguides.
Comparison
A
B
C
0 1 0 0 1 1 0 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
Output signal E
Fig. Fig. Fig. Fig. Fig. Fig. Fig.
4(a) 4(b) 4(c) 5(a) 5(b) 5(c) 5(d)
0 1 1 1 1 1 1 1
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Table 2 NAND logic gate. Optical signal
Control signals A
B
C
0 1 0 1
0 0 1 1
1 1 1 1
Table 4 NOT logic gate. Comparison
Fig. Fig. Fig. Fig.
4(c) 5(b) 5(c) 5(d)
Output signal
Optical signals
Control signal
D
A
B
C
1 1 1 0
1 1
1 1
0 1
Comparison
Output signal F
Fig. 5(a) Fig. 5(d)
1 0
Comparison
Output signal
Table 5 NOT logic gate.
(2) NAND gate: When an optical signal is launched into waveguide C without control signal(s) or with a control signal in waveguides A or B, there is an output optical signal in waveguide D (Figs. 4c and 5b and c). When control signals are launched simultaneously into waveguides A and B, the output signal in waveguide D is cut-off (Fig. 5d). Hence, the device operates as a NAND gate, i.e. D ¼ A B as shown in Table 2, and the calculated average insertion loss is 0.154 dB. (3) NOR gate: When an optical signal is launched into waveguide C without control signal(s), there is an optical signal in waveguide F (Fig. 4c). When control signals are launched into the waveguides A and/or B, there is no output signal in the waveguide F (Fig. 5b–d). Hence, the device operates as a NOR gate, i.e. F ¼ A þ B as shown in Table 3, and the calculated average insertion loss is also 0.154 dB. (4) NOT gates: When optical signals are launched simultaneously into input waveguides A and B without a control signal in waveguide C, there are output signals in waveguides E and F (Fig. 5a). When a control signal is launched into waveguide C, there is an output signal in waveguide E and no signal in waveguide F (Fig. 5d). Hence, the device operates as a NOT gate, i.e. F ¼ C as shown in Table 4, and the calculated average insertion loss is 0.216 dB. Similarly, when optical signals are launched simultaneously into input waveguides B and C without a Table 3 NOR logic gate. Control signals
Optical signal
A
B
C
0 1 0 1
0 0 1 1
1 1 1 1
Comparison
Output signal F
Fig. Fig. Fig. Fig.
4(c) 5(b) 5(c) 5(d)
1 0 0 0
Control signal
Optical signals B
C
A
1 1
1 1
0 1
D Fig. 5(c) Fig. 5(d)
1 0
control signal in waveguide A, there are output signals in waveguides D and E (Fig. 5c). When a control signal is launched into waveguide A, there is an output signal in waveguide E and no signal in waveguide D (Fig. 5d). Hence, the device operates as a NOT gate, i.e. D ¼ A as shown in Table 5, and the calculated average insertion loss is 0.196 dB. To calculate extinction ratios (ERs) of the logic gates, we define ER as 10 log(PON/POFF), where PON is the output power of logic ‘1’ (i.e. with optical signal), and POFF is the output power of logic ‘0’ (i.e. without optical signal). For Fig. 5(a), the calculated ER for the outputs E and D (denoted by E/D) is 14.5 dB, the ER for the outputs F/D is 14.8 dB. For Fig. 5(b), the ERs for the outputs D/F and E/F are 8.8 dB and 10.7 dB, respectively. For Fig. 5(c), the ERs for the outputs D/ F and E/F are 15.0 dB and 15.6 dB, respectively. For Fig. 5(d), the ERs for the outputs E/D and E/F are 10.6 dB and 14.5 dB, respectively. It is notable that all of the simulation results were performed with fixed values of phase differences, but it can be used for different initial phase difference with different interference phenomenon in the multimode waveguide. In this case, output optical intensity and phase in each output waveguide will be different. As a result, the logic states (‘0’ and ‘1’) may be different for a specified output waveguide. However, to perform multifunctional logic gates in a single device, the phase differences should be 858 Dw1 958 and 138 Dw2 178. In addition, considering the phase differences of the output signals are irregular, additional phase processing would be needed in a photonic integrated circuit that uses this optical logic gate. It should also be pointed out that the device structure and
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performed simulations/calculations are done in 2D, i.e. the dielectric rods in the device are considered as infinite on the vertical dimension in simulations. The difference of the results in 2D and 3D structures can be referred from the discussion in Ref. [18]. Last, in this work, a slight variation of rod radius (less than 2%) will not affect the properties (insertion losses and extinction ratios) of the device. This is because a variation of less than 2% in rod radius has little impact on the band structures of the single-row PDWs or multimode waveguide. 4. Conclusion An ultrasmall silicon PDWs-based optical logic gate has been designed based on the MMI principle. The single device can perform individually or simultaneously four different kinds of basic logic functions: OR, NAND, NOR and NOT gates. The total size of the device is 13.7 mm 3.2 mm and each input/output waveguide is only 205 nm wide. The average insertion loss of each logic gate is less than 0.22 dB and the ERs are between 8.8 dB and 15.6 dB. The proposed multifunctional logic device with extremely small feature size is an attractive candidate for high density photonic integrated circuits. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 60625404, 60577001, and 10974261).
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