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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 14 — Jul. 6, 2009
  • pp: 11822–11833
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Numerical Simulations of a Surface Plasmonic Waveguide with three circular air cores

Ya-nan Guo, Wenrui Xue, Rongcao Yang, and Wenmei Zhang  »View Author Affiliations


Optics Express, Vol. 17, Issue 14, pp. 11822-11833 (2009)
http://dx.doi.org/10.1364/OE.17.011822


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Abstract

In this paper, a kind of surface plasmonic waveguide (SPW) with three circular air cores is presented. Based on the finite-difference frequency-domain (FDFD) method, dependence of the distribution of energy flux density, effective index, propagation length and mode area of the fundamental mode on the geometrical parameters and the working wavelengths is analyzed firstly. Then, comparison with the SPW which was proposed in our previous work has been carried out. Results show that this kind of three cores structure has better propagation properties than the double cores structure. To investigate the relative advantages of this kind of SPW over other previous reported SPWs, comparison with the SPW with a single wedge has been carried out. Results show that this kind of SPW has shorter propagation length and larger mode area. Finally, the possibility to overcome the large propagation loss by using a gain medium as core material is investigated. Since the propagation properties can be adjusted by the geometrical and electromagnetic parameters, this kind of surface plasmonic waveguide can be applied to the field of photonic components in the integrated optical circuits and sensors.

© 2009 OSA

1. Introduction

Recently, there has been an increase of interest in surface plasmon polaritons (SPPs) [1

1. H. A. Atwater, “The promise of plasmonics,” Science 296, 56–63 (2007).

]. SPPs are electromagnetic wave that are bound to a metal-dielectric interface and are coupled to the oscillations of the free electrons in the metal [2

2. H. Rather, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1988).

]. Because SPPs have lateral dimensions on the order of subwavelength, it overcomes the diffraction limit that exists in conventional or photonic crystal waveguides, and fulfills the further miniaturization of photonic devices and high integration density of photonic chips. So during the past few years, SPPs-based waveguides have also been a subject of intensive research [3

3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef]

6

6. S. A. Maier, “Plasmonics: The promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1671–1677 (2006).

].

Up to now, several kinds of SPWs structures have been proposed, such as nanoparticle SPWs [7

7. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef]

,8

8. H. X. Zhang, Y. Gu, and Q. H. Gong, “A visible-near infrared tunable waveguide based on plasmonic gold nanoshell,” Chinese Physics B 17(7), 2567–2573 (2008).

], film SPWs [9

9. E. N. Economou, “Surface plasmon in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

,10

10. P. Berini, “Plasmon polariton modes guided by a metal film of finite width,” Opt. Lett. 24(15), 1011–1013 (1999). [CrossRef]

], rod SPWs [11

11. J. Jung, T. Sondergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76(3), 035434 (2007). [CrossRef]

,12

12. J. Guo and R. Adato, “Control of 2D plasmon-polariton mode with dielectric nanolayers,” Opt. Express 16(2), 1232–1237 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-2-1232. [CrossRef]

], gap SPWs [13

13. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]

15

15. R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express 13(6), 1933–1938 (2005), http://www.opticsexpress.org/abstract.cfm?uri=oe-13-6-1933. [CrossRef]

], slot SPWs [16

16. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two- dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87(26), 261114 (2005). [CrossRef]

,17

17. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005), http://www.opticsexpress.org/abstract.cfm?uri=oe-13-17-6645. [CrossRef]

], wedge SPWs [18

18. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on riangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87(6), 061106 (2005). [CrossRef]

21

21. W. R. Xue, Y. N. Guo, P. Li, and W. M. Zhang, “Propagation Properties of a Surface Plasmonic Waveguide with double elliptical air cores,” Opt. Express 16(14), 10710–10720 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-14-10710. [CrossRef]

], channel SPWs [22

22. J. Q. Lu and A. A. Maradudin, “Channel plasmons,” Phys. Rev. B 42(17), 11159–11165 (1990). [CrossRef]

25

25. I. Lee, J. Jung, J. Park, H. Kim, and B. Lee, “Dispersion characteristics of channel plasmon polariton waveguides with step-trench-type grooves,” Opt. Express 15(25), 16596–16603 (2007), http://www.opticsexpress.org/abstract.cfm?uri=oe-15-25-16596. [CrossRef]

], heterostructured SPWs [26

26. G. P. Wang and B. Wang, “Metal heterostructure-based nanophotonic devices: finite-difference time-domain numerical simulations,” J. Opt. Soc. Am. B 23(8), 1660–1665 (2006). [CrossRef]

,27

27. B. Wang and G. P. Wang, “Planar metal heterostructures for nanoplasmonic waveguides,” Appl. Phys. Lett. 90(1), 013114 (2007). [CrossRef]

], and mixed SPWs [28

28. D. Arbel and M. Orenstein, “Plasmonic modes in W-shaped metal-coated silicon grooves,” Opt. Express 16(5), 3114–3119 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-5-3114. [CrossRef]

].

2. Structure and simulation method

The cross section of our proposed surface plasmonic waveguide in this paper is shown in Fig. 1. It is composed of three circular air cores with the same radius r in the silver cladding, and 2a is the centric distance of the upper two air cores, 2h is the vertical distance from the centre of the nether circular air core to the horizontal line which is passed through the centres of the upper two air cores. Obviously, it can be separated into three cases of 2a>r, 2a=r and 2a<r, when 2h is a fixed value.

Noble metals are usually used to make SPWs. At optical frequencies, the dielectric constants of these metals are complex. In calculation, the dielectric constant of silver εclad is chosen as - 18.0550+0.4776j (λ=632.8 nm), - 23.4046+0.3870j (λ=705.0 nm), and - 31.0784+0.4118j (λ=800.0 nm) [36

36. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] respectively. Here, λ is the working wavelength in the vacuum.

In this paper, the 2D full-vectorial FDFD method [37

37. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=oe-10-17-853.

39

39. C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express 12(25), 6165–6177 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-25-6165. [CrossRef]

] is used to study the propagationproperties of our proposed SPW structure. This is a simple and effective numerical simulation approach. Setting the geometrical parameters, electromagnetic parameters and working wavelength, an eigenvalue equation can be obtained. Solving the eigenvalue equation by Arnoldi arithmetic [40

40. W. E. Arnoldi, “The principle of minimized iteration in the solution of matrix eigenvalue problems,” Q. Appl. Math. 9, 17–29 (1951).

], which can deal with large matrix eigenvalue problem with complex coefficient matrix, propagation constant and the distribution of field of each mode at the working wavelength can be obtained. In our calculation, 601×601 Yee’s lattices are adopted to discretize the whole computational domain, and 20 layers of them are perfectly matched layer absorbing boundary layers (APML) that used to truncate the lattices. Spatial discretization distance is Δxy=1.0 nm.

Fig. 1. Cross section of the proposed surface plasmonic waveguide with three circular air cores. (a) 2a>r, (b) 2a=r and (c) 2a<r

To validate our FDFD code, an optical fiber with a finite metallic core [41

41. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef]

] is analyzed. The working wavelength is chosen as λ 0=633nm. The relative dielectric constant of the metal and the cladding is taken as εcore=19.00-0.53 j and εclad=4.00 respectively. In the calculation, the radius of the core is assumed from 0.05λ 0 to 0.25λ 0. Here we have taken the notation of the working wavelength in reference [41

41. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef]

] as λ 0. In the later text, we will still use λ as the working wavelength in the vacuum. For clarity, we show results in the figure form. As shown in Fig. 2, the analytic [41

41. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef]

] and simulated propagation constants for the 0th-order TM mode exhibit good agreement. This fact shows that our FDFD code can be used to simulate plasmonic structures.

Fig. 2. Analytic and simulated normalized (a) real part and (b) imaginary part of propagation constant of an optical fiber [41] with a finite metallic core for the 0th-order TM mode. Here, a is the radius of the core, λ 0 is the working wavelength, and k 0=2π/λ 0.

3. Results and discussion

3.1. Modal characteristics

Characteristics of the modes supported by the surface plasmonic waveguide with three circular air cores shown in Fig. 1 are investigated firstly. We find that there is a kind of mode which have longer propagation length among many modes supported by this kind of SPW, and call this mode the fundamental mode. The distribution of field Hx, Hy, and energy flux density Sz in the cases of a=60 nm, h=85 nm, and r=120 nm at λ=705.0 nm is shown in Fig. 3. Here the energy flux density is defined as Sz=Re(ExHy-EyHx), Ex and Ey, Hx and Hy are components of electrical and magnetic field respectively.

Fig. 3. The distribution of the field (a) Hx, (b) Hy and (c) Sy on the cross section when a=60 nm, h=85 nm, r=120 nm at λ=705.0 nm. Dashed lines in (a), (b) and (c) indicate the outline of the structure.

It can be seen from Fig. 3(a) that the component of field x H is symmetric distribution with y axes, however, component of field Hy shown in Fig. 3(b) is antisymmetric distribution with y axes. From Fig. 3(c), we can find that the energy flux density mainly distributes near the left and right wedged corner which are formed by three circular air cores. Among the three cases of 2a>r, 2a=r and 2a<r, the two corner angles are smallest and the two corner tips are sharpest in the case of 2a>r, so the degree of localization of field is the largest.

3.2. Effect of geometrical parameters of the waveguide

Since the effective index Re(neff), propagation length Lprop and mode area Am are the three most important physical quantities that describe the propagation properties of surface plasmonic waveguides, then we investigate the dependence of Re(neff), Lprop and Am of the fundamental mode shown in Fig. 3 on the geometrical parameters. Here Re(neff) is defined as Re(β)λ/2π, Lprop is defined as 1/Im(β), and Am is defined as the area in which the energy flux density Sz descend from 100% to 10% of its maximum value.

The relational graphs of Re(neff), Lprop and Am varied with h in the three cases of r=2a-5 nm, 2a and 2a+5 nm, and 2a=100 nm, 120 nm, 140 nm at λ=705.0 nm are shown in Fig. 4(a–c), Fig. 4(d–f) and Fig. 4(g–i) respectively. It can be seen from these figures that curves can be obviously separated into three groups according to different a. In each group, Re(neff) increases as h increases, however Lprop on the whole lessly varied with h, and Am increases as h decreases. The position of curves is influenced by parameter r. We can also find that curves corresponding to r=2a+5 nm are commonly on the upward side of each group of curves. The above phenomena can be explained by the different field distribution with different geometrical parameters. Because the field is mainly centralized near the left and right wedged corner which are formed by three circular air cores. Relative to the case of r=2a, when r is small, the area of field distribution is small. Namely the degree of localization of field is large. In this instance, the interaction of field and silver is strong. The effective index increases. Then the propagation length and mode area decrease. However, when r is large, the area of field distribution is large. Namely the degree of localization of field is small. In this instance, the interaction of field and silver becomes weak. The effective index decreases. Then the propagation length and mode area increase.

Fig. 4. Dependence of (a–c) Re(neff), (d–f) Lprop and (g–i) Am on h when r=2a - 5 nm, 2a and 2a+5 nm at λ=705.0 nm.

3.3. Effect of working wavelengths

In order to find out the dependence of field distribution of the mode shown in Fig. 3 on the working wavelength, the distribution of field Hx, Hy, and energy flux density Sz in the cases of r=2a=120 nm, h=85 nm at λ=632.8 nm and λ=800.0 nm is calculated respectively. Here we extend the calculation to longer wavelength 800 nm where the vertical-cavity surface-emitting lasers (VCSELs) are available. Results shown that field distribution is similar to the Fig. 3. Relative to the case of λ=705.0 nm, in the case of λ=632.8 nm, the area of field distribution is small. And the field is mainly confined near the left and right wedged corner which are formed by three circular air cores. However, in the case of λ=800.0 nm, the area of field distribution is large, and the field is less confined.

Fig. 5. Dependence of (a–c) Re(neff), (d–f) Lprop and (g–i) Am on h when r=2a=100 nm, 120 nm and 140 nm at λ=632.8 nm, 705.0 nm, 800.0 nm.

3.4. Comparation with the SPW with two air cores

Fig. 6. Dependence of (a–b) Re(neff), (c–d) Lprop and (e–f) Am on h when a=0 nm, 60 nm, r=120 nm at λ=632.8 nm, 705.0 nm, 800.0 nm.

3.5. Comparation with the SPW with a single wedge

Dependence of Re(neff), Lprop and Am on c when r=100 nm and c=85 nm at λ=705.0 nm is shown in Fig. 8 respectively. From Fig. 8, we can see that Re(neff) increases as c increases, however Lprop and Am decrease as c increase. Results also show that the single wedge SPW has longer propagation length and smaller mode area, but the double cores structure has shorter propagation length and larger mode area. This phenomenon can also be explained by the interactional intensity between the field and metal. For the single wedge SPW, the field is confined only on one single wedge. The interaction of the field and siver is small. So the propagation length is large, but the mode area is small. For the double cores structure, the field is coupled and confined on two wedges. The interaction of the field and siver is large. So the propagation length is short, but the mode area is large. This fact implies that the double cores structure as well as the three cores structure can be used as a key component of a sensor although it is difficult to fabricate than the single wedge SPW.

Fig. 7. The cross section of the SPW with a single wedge (a), the distribution of the field (b) Hx, (c) Hy and (d) Sz on the cross section when r=100 nm, c=85 nm, at λ=705.0 nm. Dashed lines in (b), (c) and (d) indicate the outline of the structure.
Fig. 8. Dependence of (a) Re(neff), (b) Lprop and (c) Am on c (or h) when r=100 nm at λ=705.0 nm.

Table 1. The dielectric constants of the SPW.

table-icon
View This Table

3.6. Effect of Gain in the Core Dielectric

To investigate the effect of gain in the core dielectric, we fill the whole air cores of the structure shown in Fig. 1(b) with a kind of available gain medium [34

34. S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006). [CrossRef]

,35

35. D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31(1), 98–100 (2006). [CrossRef]

,43

43. E. D. Palik, Handbook of Optical Constants of Solids(Academic, New York, 1985).

], and the dielectric constants of the SPW are shown in Table 1. Here, the corresponding gain coefficirnt γ=298cm -1 (εcore=11.56-0.025j) and 596cm -1(εcore=11.56-0.050j) [34

34. S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006). [CrossRef]

]. As an example, in the case of λ=1550 nm, a=60 nm, r=120 nm, dependence of Re(neff), Lprop and Am on h and the gain of the core dielectric are shown in Fig. 9. The corresponding distribution of the field z S are shown in Fig. 10.

As shown in Fig. 9(a), Fig. 9(b), and Fig. 9(c), the propagation length can be extended obviously with the help of the gain medium. It demonstrated that the presence of the gain medium result in an increase of the propagation length Lprop. However, the effective index is almost kept invariably with different εcore, here Am not obviously varied with εcore.

Fig. 9. Dependence of (a) Re(neff), (b) Lprop and (c) Am on h in the different cases.

Since the core material has been changed from air to a gain medium with higher dielectric constant, confinement is expected to increase. As shown in Fig. 10, when the core is filled with gain medium (Fig. 10(a), Fig. 10(b) and Fig. 10(c)), the field Sz is more concentrated on the surface of the metal than that with air cores (Fig. 10(d)).

Fig. 10. The distribution of the field Sz when λ=1550 nm, a=60 nm, r=120 nm with (a) εcore=11.56-0.000j, (b) εcore=11.56-0.025j, (c) εcore=11.56-0.050j, and (d) εcore=1.00-0.000j.

The presence of the gain medium to compensate for the absorption loss in propagating [30

30. A. N. Sudarkin and P. A. Demkovich, “Excitation of surface electromagnetic waves on the boundary of a metal with an amplifying medium,” Sov. Phys. Tech. Phys. 34, 764–766 (1989).

32

32. I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B 70(15), 155416 (2004). [CrossRef]

] SPPs, although very small, it has been a possible solution, and at the same time localized [33

33. N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004). [CrossRef]

] the surface plasmons. The influence of the gain medium on SPPs propagation has received some attention previously. There are much work to do in later time.

4. Conclusions

In this paper, we have designed a kind of surface plasmonic waveguide with three circular air cores. The structure of the waveguide can be separated into three cases of 2a>r, 2a=r and 2a<r. Numerical calculation show that:

(2) At the certain working wavelength, in the case of 2a<r, the degree of localization of field is the smallest, and the interaction of field and silver becomes small. The effective index decreases. Then the propagation length and mode area become large.

(3) With the certain geometric parameters, in the case of λ=800.0 nm, the area of field distribution is large. Namely, the degree of localization of field is small, and the interaction of field and silver becomes small. The effective index decreases. Then the propagation length and mode area become large.

(4) Comparison with the double air cores structure, this kind of SPW has better propagation properties. Comparison with the SPW with a single wedge, it has shorter propagation length and larger mode area.

(5) The propagation length can be extended obviously with the help of the gain dielectric medium.

Since the effective index, propagation length and the mode area can be adjusted by the geometrical and electromagnetic parameters, this kind of surface plasmonic waveguide can be applied to the field of photonic components in the integrated optical circuits and sensors.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 60878008, 60771052) and the Natural Science Foundation of Shanxi Province (Grant No. 2008012002-1, 2006011029).

References and links

1.

H. A. Atwater, “The promise of plasmonics,” Science 296, 56–63 (2007).

2.

H. Rather, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1988).

3.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef]

4.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef]

5.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef]

6.

S. A. Maier, “Plasmonics: The promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1671–1677 (2006).

7.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef]

8.

H. X. Zhang, Y. Gu, and Q. H. Gong, “A visible-near infrared tunable waveguide based on plasmonic gold nanoshell,” Chinese Physics B 17(7), 2567–2573 (2008).

9.

E. N. Economou, “Surface plasmon in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

10.

P. Berini, “Plasmon polariton modes guided by a metal film of finite width,” Opt. Lett. 24(15), 1011–1013 (1999). [CrossRef]

11.

J. Jung, T. Sondergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76(3), 035434 (2007). [CrossRef]

12.

J. Guo and R. Adato, “Control of 2D plasmon-polariton mode with dielectric nanolayers,” Opt. Express 16(2), 1232–1237 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-2-1232. [CrossRef]

13.

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]

14.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86(21), 211101 (2005). [CrossRef]

15.

R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express 13(6), 1933–1938 (2005), http://www.opticsexpress.org/abstract.cfm?uri=oe-13-6-1933. [CrossRef]

16.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two- dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87(26), 261114 (2005). [CrossRef]

17.

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005), http://www.opticsexpress.org/abstract.cfm?uri=oe-13-17-6645. [CrossRef]

18.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on riangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87(6), 061106 (2005). [CrossRef]

19.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100(2), 023901 (2008). [CrossRef]

20.

A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, “Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths,” Opt. Express 16(8), 5252–5260 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-8-5252. [CrossRef]

21.

W. R. Xue, Y. N. Guo, P. Li, and W. M. Zhang, “Propagation Properties of a Surface Plasmonic Waveguide with double elliptical air cores,” Opt. Express 16(14), 10710–10720 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-14-10710. [CrossRef]

22.

J. Q. Lu and A. A. Maradudin, “Channel plasmons,” Phys. Rev. B 42(17), 11159–11165 (1990). [CrossRef]

23.

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef]

24.

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004). [CrossRef]

25.

I. Lee, J. Jung, J. Park, H. Kim, and B. Lee, “Dispersion characteristics of channel plasmon polariton waveguides with step-trench-type grooves,” Opt. Express 15(25), 16596–16603 (2007), http://www.opticsexpress.org/abstract.cfm?uri=oe-15-25-16596. [CrossRef]

26.

G. P. Wang and B. Wang, “Metal heterostructure-based nanophotonic devices: finite-difference time-domain numerical simulations,” J. Opt. Soc. Am. B 23(8), 1660–1665 (2006). [CrossRef]

27.

B. Wang and G. P. Wang, “Planar metal heterostructures for nanoplasmonic waveguides,” Appl. Phys. Lett. 90(1), 013114 (2007). [CrossRef]

28.

D. Arbel and M. Orenstein, “Plasmonic modes in W-shaped metal-coated silicon grooves,” Opt. Express 16(5), 3114–3119 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-5-3114. [CrossRef]

29.

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef]

30.

A. N. Sudarkin and P. A. Demkovich, “Excitation of surface electromagnetic waves on the boundary of a metal with an amplifying medium,” Sov. Phys. Tech. Phys. 34, 764–766 (1989).

31.

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-17-4072. [CrossRef]

32.

I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B 70(15), 155416 (2004). [CrossRef]

33.

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004). [CrossRef]

34.

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006). [CrossRef]

35.

D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31(1), 98–100 (2006). [CrossRef]

36.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

37.

Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=oe-10-17-853.

38.

S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, “Loss and dispersion analysis of microstructured fibers by finite-difference method,” Opt. Express 12(15), 3341–3352 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-15-3341. [CrossRef]

39.

C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express 12(25), 6165–6177 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-25-6165. [CrossRef]

40.

W. E. Arnoldi, “The principle of minimized iteration in the solution of matrix eigenvalue problems,” Q. Appl. Math. 9, 17–29 (1951).

41.

J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef]

42.

M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal coners,” J. Opt. Soc. Am. B 24(9), 2333–2342 (2007). [CrossRef]

43.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, New York, 1985).

OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.6680) Optics at surfaces : Surface plasmons
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Optics at Surfaces

History
Original Manuscript: April 30, 2009
Revised Manuscript: June 5, 2009
Manuscript Accepted: June 14, 2009
Published: June 29, 2009

Citation
Ya-nan Guo, Wenrui Xue, Rongcao Yang, and Wenmei Zhang, "Numerical Simulations of a Surface 
Plasmonic Waveguide with three circular 
air cores," Opt. Express 17, 11822-11833 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11822


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  31. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-17-4072 . [CrossRef]
  32. I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B 70(15), 155416 (2004). [CrossRef]
  33. N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004). [CrossRef]
  34. S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006). [CrossRef]
  35. D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31(1), 98–100 (2006). [CrossRef]
  36. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
  37. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?uri=oe-10-17-853 .
  38. S. Guo, F. Wu, S. Albin, H. Tai, and R. S. Rogowski, “Loss and dispersion analysis of microstructured fibers by finite-difference method,” Opt. Express 12(15), 3341–3352 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-15-3341 . [CrossRef]
  39. C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express 12(25), 6165–6177 (2004), http://www.opticsexpress.org/abstract.cfm?uri=oe-12-25-6165 . [CrossRef]
  40. W. E. Arnoldi, “The principle of minimized iteration in the solution of matrix eigenvalue problems,” Q. Appl. Math. 9, 17–29 (1951).
  41. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef]
  42. M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal coners,” J. Opt. Soc. Am. B 24(9), 2333–2342 (2007). [CrossRef]
  43. E. D. Palik, Handbook of Optical Constants of Solids(Academic, New York, 1985).

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