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Optical coupling between two nanobelts
Physics Letters A 373 (2009) 2061–2064
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Optical coupling between two nanobelts Changyong Lan, Qingping Ding, Yuwen Jiang, Hongbo Huang, Shaoguang Yang ∗ National Laboratory of Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
a r t i c l e
i n f o
Article history: Received 21 January 2009 Received in revised form 1 April 2009 Accepted 10 April 2009 Available online 18 April 2009 Communicated by R. Wu
a b s t r a c t In this study, we have investigated effective refractive index (ERI) and power confinement factor (PCF) in single nanobelt and coupling between two parallel nanobelts using finite-difference time-domain (FDTD) method. As the width of nanobelt increases, both the effective refractive index (ERI) and power confinement factor (PCF) are increased till the saturation values. The ERI changes linearly with refractive index in our simulation range, while the PCF shows nonlinear characteristic. The coupling length (L T ) between two nanobelts is much shorter than that in weakly coupled waveguide, which may find applications in solving the cross talk problem. Nanobelt behaves like slab waveguide when the width becomes big enough. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Accompanied with the development of integrated optics, the size of the unit cells and connections between the cells become smaller and smaller which have come into nano-scale. For both application and fundamental purpose, it is worth to studying the light transmission in nanoscale. Up to now, nanobelts have been investigated in many areas, such as nanolasers [1,2], nanosensors [3], field effect transistors [4], etc. They are also good candidates as optical dielectric waveguides and components in integrated optics for their excellent waveguide effect [5–7]. Some works have been reported on the coupling between two single mode nanowires [8]. To our best knowledge, few works have been reported on the study of optical transmission in nanobelts. The transmission mode of nanobelts and nanowires is different, it is necessary to study the light transmission and optical coupling between two nanobelts. Here we report our studies on the light transmission and coupling in nanobelts by simulation method. In this study, we found that both effective refractive index (ERI) and power confinement factor (PCF) increase with the width of nanobelts till a saturation value, and the minimum transfer length (L T ) for energy exchange between two nanobelts is much shorter than that in weakly coupled waveguide. 2. Model for simulation Nanobelt is a kind of one-dimensional nanostructure with rectangular cross section [9]. Most of them have a width of several hundred nanometers, and the thickness of tens of nanometers
*
Corresponding author. Tel.: +86 25 83597483; fax: +86 25 83595535. E-mail address: [email protected] (S. Yang).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.012
Fig. 1. Cross section of a single nanobelt.
Fig. 2. Model for coupling of two parallel nanobelts.
[10–13]. In this study, we fix the thickness as 50 nm for simplicity in our simulation. Finite-difference time-domain (FDTD) simulation is a useful tool for studying electromagnetic problems [8,14,15]. FDTD method is applied to the study of the light transmission in nanobelts. The schematic maps for numerical analysis are shown in Fig. 1 and Fig. 2. Fig. 1 is the cross section of a single nanobelt
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C. Lan et al. / Physics Letters A 373 (2009) 2061–2064
Fig. 3. Effective refractive index (ERI) as a function of the width of a single nanobelt with refractive index of 2.5, 3.0, 3.5 and 4.0, respectively.
Fig. 4. Power confinement factor (PCF) as a function of the width of a single nanobelt with refractive index of 2.5, 3.0, 3.5 and 4.0, respectively.
for calculation with nanobelt as the core and air as the cladding. Fig. 2 is the model of coupling between two parallel nanobelts. The distance between the two nanobelts is represented by D. For simplicity, we apply double frequency of the YAG laser (with wavex length of 532 nm) as the source. Only E 11 mode was selected as the transmission mode in case of cutting off. The computational domain is discretized into a uniform orthogonal three-dimensional mesh with cell size of 5 nm, terminated by perfectly matched layer boundaries. 3. Transmission of light in a single nanobelt In order to study the optical coupling between two nanobelts, the transmission of light in a single nanobelt is investigated first. ERI and PCF are the main factors that represent the properties of a guided mode in the waveguide [16]. In order to study the characteristic of the light transmission in a single nanobelt, the property of the ERI and PCF were investigated. This is especially important for coupling applications, because the coupling relations concerned much with evanescent wave which is related to PCF. The thickness of the nanobelt is fixed as 50 nm, we studied the PCF and ERI at different width and refractive index. Fig. 3 and Fig. 4 show us the simulation results. It can be found that the ERI and PCF both increase with the width of the nanobelts for all refractive indexes. The ERI and PCF increase with the width of the nanobelts, which will be saturated when the width is large enough. The saturated values of ERI and PCF are shown in Table 1. From this table, we can find out that the Saturated ERI and PCF increase with the refractive indexes of the nanobelts. The saturated ERI seems to be a linear relation of refractive index. The PCF here is very large due to the high refractive index. This result shows that dielectric waveguides with high refractive index contains large confinement effect on light. This gives us some revelation on the application of dielectric waveguide in integrated optics. In Table 2, the ERI and PCF of the basic TE mode of slab waveguides with different refractive indexes are given, where the width of the slab waveguide is 50 nm. The slab waveguides are surrounded with air. Compared Table 1 with Table 2, it is clear that the ERI and PCF of slab waveguides and nanobelts are almost the same. This explained why there exist saturation phenomena in nanobelts. For understanding the relationship of ERI and PCF with the refractive index of nanobelts, we fixed the width and thickness of the nanobelts to be 500 nm and 50 nm respectively in this study. The
Fig. 5. Effective refractive index as a function of the refractive index of a single nanobelt. The cross section of the nanobelt is 500 nm × 50 nm. Table 1 Saturation values of a single nanobelt. Refractive index
Saturation value of ERI
Saturation value of PCF (%)
2.5 3.0 3.5 4.0
1.58 2.07 2.46 3.05
72.9 83.9 89.8 93.2
Table 2 ERI and PCF of the basic TE mode of slab waveguide with its width of 50 nm. Refractive index
ERI
PCF (%)
2.5 3.0 3.5 4.0
1.59 2.00 2.46 2.95
70.6 81.6 88.1 92.0
simulation results are shown in Fig. 5 and Fig. 6. From Fig. 5, we can find out that the ERI almost linearly increase with refractive index. From Fig. 6, it can be fitted by: PCF = 96.26 − 178.47/{1 + exp[(RI − 1.17)/0.71]}, in which RI presents the refractive index of the nanobelts.
C. Lan et al. / Physics Letters A 373 (2009) 2061–2064
Fig. 6. Power confinement factor as a function of the refractive index of a single nanobelt. The cross section of the nanobelt is 500 nm × 50 nm.
Fig. 7. Typical coupling between two nanobelts. Coupling efficiency as a function of overlapping distance. The cross section of the nanobelts is fixed to 500 nm × 50 nm, and the distance between two nanobelts is 0 nm.
4. Coupling between two parallel nanobelts In order to show the coupling between two parallel nanobelts, the width, thickness and refractive index of each nanobelt are fixed as 500 nm, 50 nm and 3.0 respectively. For comparison, we have tried to do some simulations with setting the narrow surfaces face to face. It showed no clear coupling behavior even the overlapping length comes to tens of micrometer. So we set the broad surfaces of the two nanobelts face to face as shown in Fig. 2 in the simulation. Fig. 7 is the typical coupling relations of two nanobelts in our simulation. The coupling efficiency is defined as the power of output divided by input. The coupling efficiency (η ) oscillate with overlapping length (L), and the minimum coupling efficiency (ηmin ) is considerably higher than zero, especially when the distance (D) between two nanobelts is much shorter (as shown in Fig. 8). This is similar to the previous work discussing coupling relations between two parallel nanowires [8]. It is regarded that the radiation of light at the end of the upper nanobelt may be the reason for ηmin higher than zero. ηmin decreases when D increases,
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Fig. 8. The minimum and maximum coupling efficiency as a function of the distance of two nanobelts.
Fig. 9. Coupling length as a function of the distance between two nanobelts.
which is due to the power of radiation from the upper nanobelt coming to nether nanobelt becoming weak. As part of the radiation from the upper nanobelt not reach the nether one, the maximum coupling efficiency (ηmax ) is lower than 100% as shown in Fig. 8. It is worthy to note that ηmax almost keeps the same as D changes, which may be due to most of light has been coupled to the nether nanobelt and the radiation from the end of the upper one is very weak when the η comes to its maximum. Fig. 9 shows the coupling length (L T ) as a function of D. The L T becomes larger when D increases. This is because the evanescent wave reaching the nether nanobelt becomes weak as D increases, and it needs a longer distance to accumulate power from the evanescent wave. This result resembles that of previous works which discussing slab waveguide [17]. This result indicates that it is necessary to make the D small enough when fabricating a small coupler using nanobelts. L T comes to tens of micrometers when D is more than 150 nm, this tells us that the cross talk problem can be solved if the distance big enough but very small compared with weakly coupled waveguides. This shows a very important potential application for minimizing the size of integrated optics circuit.
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C. Lan et al. / Physics Letters A 373 (2009) 2061–2064
5. Summary
References
In summary, we have investigated ERI and PCF in a single nanobelt and coupling relations between two nanobelts using FDTD method. The relations between width and ERI/PCF are obtained. Compared with weakly coupled waveguides, high refractive index dielectric waveguides have strong power confinement effect and a much shorter coupling length without sacrificing the high coupling efficiency. The coupling length increases with the distance between two nanobelts. Nanobelts behave like slab waveguides when the width become big enough. The characteristics of nanobelts discussed here suggest that nanobelts can be used as waveguides and components in integrated optics to reduce its size.
[1] M.H. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, P. Yang, Science 292 (2001) 1897. [2] K. Bando, T. Sawabe, K. Asaka, Y. Masumoto, J. Lumin. 108 (2004) 385. [3] S. Kaciulis, L. Pandolfi, E. Comini, G. Faglia, M. Ferroni, G. Sberveglieri, S. Kandasamy, M. Shafiei, W. Wlodarski, Surf. Interface Anal. 40 (2008) 575. [4] M.S. Arnold, P. Avouris, Z.W. Pan, Z.L. Wang, J. Phys. Chem. B 107 (2003) 659. [5] C.J. Barrelet, A.B. Greytak, C.M. Lieber, Nano Lett. 4 (2004) 1981. [6] L. Tong, R.R. Gattass, J.B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, Nature 426 (2003) 816. [7] M. Law, D.J. Sirbuly, J.C. Johnson, J. Goldberger, R.J. Saykally, P. Yang, Science 305 (2004) 1269. [8] K. Huang, S. Yang, L. Tong, Appl. Opt. 46 (2007) 1429. [9] Y. Ding, Z.L. Wang, J. Phys. Chem. B 108 (2004) 12280. [10] F. Lu, W. Cai, Y. Zhang, Y. Li, F. Sun, S.H. Heo, S.O. Cho, Appl. Phys. Lett. 89 (2006) 231928. [11] T. Gao, T. Wang, J. Phys. Chem. B 108 (2004) 20045. [12] Y.B. Li, Y. Bando, D. Golberg, K. Kurashima, Appl. Phys. Lett. 81 (2002) 5048. [13] J. Duan, S. Yang, H. Liu, J. Gong, X. Zhao, R. Zhang, Y. Du, J. Am. Chem. Soc. 127 (2005) 6180. [14] C. Manolatou, S.G. Johnson, S. Fan, P.R. Villeneuve, H.A. Haus, J.D. Joannopoulos, J. Lightwave Technol. 17 (1999) 1682. [15] X. Wang, Q. Ding, H. Huang, S. Yang, Phys. Lett. A 365 (2007) 175. [16] T. Tamir, Integrated Optics, Springer, Berlin–Heidelberg, 1975. [17] W.P. Huang, J. Opt. Soc. Am. A 11 (1994) 963.
Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 60577002), New Century Excellent Talents in University (07-0430) and the National Key Project for Basic Research (No. 2007CB936302).
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Optical coupling between two nanobelts Changyong Lan, Qingping Ding, Yuwen Jiang, Hongbo Huang, Shaoguang Yang ∗ National Laboratory of Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
a r t i c l e
i n f o
Article history: Received 21 January 2009 Received in revised form 1 April 2009 Accepted 10 April 2009 Available online 18 April 2009 Communicated by R. Wu
a b s t r a c t In this study, we have investigated effective refractive index (ERI) and power confinement factor (PCF) in single nanobelt and coupling between two parallel nanobelts using finite-difference time-domain (FDTD) method. As the width of nanobelt increases, both the effective refractive index (ERI) and power confinement factor (PCF) are increased till the saturation values. The ERI changes linearly with refractive index in our simulation range, while the PCF shows nonlinear characteristic. The coupling length (L T ) between two nanobelts is much shorter than that in weakly coupled waveguide, which may find applications in solving the cross talk problem. Nanobelt behaves like slab waveguide when the width becomes big enough. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Accompanied with the development of integrated optics, the size of the unit cells and connections between the cells become smaller and smaller which have come into nano-scale. For both application and fundamental purpose, it is worth to studying the light transmission in nanoscale. Up to now, nanobelts have been investigated in many areas, such as nanolasers [1,2], nanosensors [3], field effect transistors [4], etc. They are also good candidates as optical dielectric waveguides and components in integrated optics for their excellent waveguide effect [5–7]. Some works have been reported on the coupling between two single mode nanowires [8]. To our best knowledge, few works have been reported on the study of optical transmission in nanobelts. The transmission mode of nanobelts and nanowires is different, it is necessary to study the light transmission and optical coupling between two nanobelts. Here we report our studies on the light transmission and coupling in nanobelts by simulation method. In this study, we found that both effective refractive index (ERI) and power confinement factor (PCF) increase with the width of nanobelts till a saturation value, and the minimum transfer length (L T ) for energy exchange between two nanobelts is much shorter than that in weakly coupled waveguide. 2. Model for simulation Nanobelt is a kind of one-dimensional nanostructure with rectangular cross section [9]. Most of them have a width of several hundred nanometers, and the thickness of tens of nanometers
*
Corresponding author. Tel.: +86 25 83597483; fax: +86 25 83595535. E-mail address: [email protected] (S. Yang).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.012
Fig. 1. Cross section of a single nanobelt.
Fig. 2. Model for coupling of two parallel nanobelts.
[10–13]. In this study, we fix the thickness as 50 nm for simplicity in our simulation. Finite-difference time-domain (FDTD) simulation is a useful tool for studying electromagnetic problems [8,14,15]. FDTD method is applied to the study of the light transmission in nanobelts. The schematic maps for numerical analysis are shown in Fig. 1 and Fig. 2. Fig. 1 is the cross section of a single nanobelt
2062
C. Lan et al. / Physics Letters A 373 (2009) 2061–2064
Fig. 3. Effective refractive index (ERI) as a function of the width of a single nanobelt with refractive index of 2.5, 3.0, 3.5 and 4.0, respectively.
Fig. 4. Power confinement factor (PCF) as a function of the width of a single nanobelt with refractive index of 2.5, 3.0, 3.5 and 4.0, respectively.
for calculation with nanobelt as the core and air as the cladding. Fig. 2 is the model of coupling between two parallel nanobelts. The distance between the two nanobelts is represented by D. For simplicity, we apply double frequency of the YAG laser (with wavex length of 532 nm) as the source. Only E 11 mode was selected as the transmission mode in case of cutting off. The computational domain is discretized into a uniform orthogonal three-dimensional mesh with cell size of 5 nm, terminated by perfectly matched layer boundaries. 3. Transmission of light in a single nanobelt In order to study the optical coupling between two nanobelts, the transmission of light in a single nanobelt is investigated first. ERI and PCF are the main factors that represent the properties of a guided mode in the waveguide [16]. In order to study the characteristic of the light transmission in a single nanobelt, the property of the ERI and PCF were investigated. This is especially important for coupling applications, because the coupling relations concerned much with evanescent wave which is related to PCF. The thickness of the nanobelt is fixed as 50 nm, we studied the PCF and ERI at different width and refractive index. Fig. 3 and Fig. 4 show us the simulation results. It can be found that the ERI and PCF both increase with the width of the nanobelts for all refractive indexes. The ERI and PCF increase with the width of the nanobelts, which will be saturated when the width is large enough. The saturated values of ERI and PCF are shown in Table 1. From this table, we can find out that the Saturated ERI and PCF increase with the refractive indexes of the nanobelts. The saturated ERI seems to be a linear relation of refractive index. The PCF here is very large due to the high refractive index. This result shows that dielectric waveguides with high refractive index contains large confinement effect on light. This gives us some revelation on the application of dielectric waveguide in integrated optics. In Table 2, the ERI and PCF of the basic TE mode of slab waveguides with different refractive indexes are given, where the width of the slab waveguide is 50 nm. The slab waveguides are surrounded with air. Compared Table 1 with Table 2, it is clear that the ERI and PCF of slab waveguides and nanobelts are almost the same. This explained why there exist saturation phenomena in nanobelts. For understanding the relationship of ERI and PCF with the refractive index of nanobelts, we fixed the width and thickness of the nanobelts to be 500 nm and 50 nm respectively in this study. The
Fig. 5. Effective refractive index as a function of the refractive index of a single nanobelt. The cross section of the nanobelt is 500 nm × 50 nm. Table 1 Saturation values of a single nanobelt. Refractive index
Saturation value of ERI
Saturation value of PCF (%)
2.5 3.0 3.5 4.0
1.58 2.07 2.46 3.05
72.9 83.9 89.8 93.2
Table 2 ERI and PCF of the basic TE mode of slab waveguide with its width of 50 nm. Refractive index
ERI
PCF (%)
2.5 3.0 3.5 4.0
1.59 2.00 2.46 2.95
70.6 81.6 88.1 92.0
simulation results are shown in Fig. 5 and Fig. 6. From Fig. 5, we can find out that the ERI almost linearly increase with refractive index. From Fig. 6, it can be fitted by: PCF = 96.26 − 178.47/{1 + exp[(RI − 1.17)/0.71]}, in which RI presents the refractive index of the nanobelts.
C. Lan et al. / Physics Letters A 373 (2009) 2061–2064
Fig. 6. Power confinement factor as a function of the refractive index of a single nanobelt. The cross section of the nanobelt is 500 nm × 50 nm.
Fig. 7. Typical coupling between two nanobelts. Coupling efficiency as a function of overlapping distance. The cross section of the nanobelts is fixed to 500 nm × 50 nm, and the distance between two nanobelts is 0 nm.
4. Coupling between two parallel nanobelts In order to show the coupling between two parallel nanobelts, the width, thickness and refractive index of each nanobelt are fixed as 500 nm, 50 nm and 3.0 respectively. For comparison, we have tried to do some simulations with setting the narrow surfaces face to face. It showed no clear coupling behavior even the overlapping length comes to tens of micrometer. So we set the broad surfaces of the two nanobelts face to face as shown in Fig. 2 in the simulation. Fig. 7 is the typical coupling relations of two nanobelts in our simulation. The coupling efficiency is defined as the power of output divided by input. The coupling efficiency (η ) oscillate with overlapping length (L), and the minimum coupling efficiency (ηmin ) is considerably higher than zero, especially when the distance (D) between two nanobelts is much shorter (as shown in Fig. 8). This is similar to the previous work discussing coupling relations between two parallel nanowires [8]. It is regarded that the radiation of light at the end of the upper nanobelt may be the reason for ηmin higher than zero. ηmin decreases when D increases,
2063
Fig. 8. The minimum and maximum coupling efficiency as a function of the distance of two nanobelts.
Fig. 9. Coupling length as a function of the distance between two nanobelts.
which is due to the power of radiation from the upper nanobelt coming to nether nanobelt becoming weak. As part of the radiation from the upper nanobelt not reach the nether one, the maximum coupling efficiency (ηmax ) is lower than 100% as shown in Fig. 8. It is worthy to note that ηmax almost keeps the same as D changes, which may be due to most of light has been coupled to the nether nanobelt and the radiation from the end of the upper one is very weak when the η comes to its maximum. Fig. 9 shows the coupling length (L T ) as a function of D. The L T becomes larger when D increases. This is because the evanescent wave reaching the nether nanobelt becomes weak as D increases, and it needs a longer distance to accumulate power from the evanescent wave. This result resembles that of previous works which discussing slab waveguide [17]. This result indicates that it is necessary to make the D small enough when fabricating a small coupler using nanobelts. L T comes to tens of micrometers when D is more than 150 nm, this tells us that the cross talk problem can be solved if the distance big enough but very small compared with weakly coupled waveguides. This shows a very important potential application for minimizing the size of integrated optics circuit.
2064
C. Lan et al. / Physics Letters A 373 (2009) 2061–2064
5. Summary
References
In summary, we have investigated ERI and PCF in a single nanobelt and coupling relations between two nanobelts using FDTD method. The relations between width and ERI/PCF are obtained. Compared with weakly coupled waveguides, high refractive index dielectric waveguides have strong power confinement effect and a much shorter coupling length without sacrificing the high coupling efficiency. The coupling length increases with the distance between two nanobelts. Nanobelts behave like slab waveguides when the width become big enough. The characteristics of nanobelts discussed here suggest that nanobelts can be used as waveguides and components in integrated optics to reduce its size.
[1] M.H. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, P. Yang, Science 292 (2001) 1897. [2] K. Bando, T. Sawabe, K. Asaka, Y. Masumoto, J. Lumin. 108 (2004) 385. [3] S. Kaciulis, L. Pandolfi, E. Comini, G. Faglia, M. Ferroni, G. Sberveglieri, S. Kandasamy, M. Shafiei, W. Wlodarski, Surf. Interface Anal. 40 (2008) 575. [4] M.S. Arnold, P. Avouris, Z.W. Pan, Z.L. Wang, J. Phys. Chem. B 107 (2003) 659. [5] C.J. Barrelet, A.B. Greytak, C.M. Lieber, Nano Lett. 4 (2004) 1981. [6] L. Tong, R.R. Gattass, J.B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, Nature 426 (2003) 816. [7] M. Law, D.J. Sirbuly, J.C. Johnson, J. Goldberger, R.J. Saykally, P. Yang, Science 305 (2004) 1269. [8] K. Huang, S. Yang, L. Tong, Appl. Opt. 46 (2007) 1429. [9] Y. Ding, Z.L. Wang, J. Phys. Chem. B 108 (2004) 12280. [10] F. Lu, W. Cai, Y. Zhang, Y. Li, F. Sun, S.H. Heo, S.O. Cho, Appl. Phys. Lett. 89 (2006) 231928. [11] T. Gao, T. Wang, J. Phys. Chem. B 108 (2004) 20045. [12] Y.B. Li, Y. Bando, D. Golberg, K. Kurashima, Appl. Phys. Lett. 81 (2002) 5048. [13] J. Duan, S. Yang, H. Liu, J. Gong, X. Zhao, R. Zhang, Y. Du, J. Am. Chem. Soc. 127 (2005) 6180. [14] C. Manolatou, S.G. Johnson, S. Fan, P.R. Villeneuve, H.A. Haus, J.D. Joannopoulos, J. Lightwave Technol. 17 (1999) 1682. [15] X. Wang, Q. Ding, H. Huang, S. Yang, Phys. Lett. A 365 (2007) 175. [16] T. Tamir, Integrated Optics, Springer, Berlin–Heidelberg, 1975. [17] W.P. Huang, J. Opt. Soc. Am. A 11 (1994) 963.
Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 60577002), New Century Excellent Talents in University (07-0430) and the National Key Project for Basic Research (No. 2007CB936302).