## Numerical Simulations of a Surface Plasmonic Waveguide with three circular air cores

Optics Express, Vol. 17, Issue 14, pp. 11822-11833 (2009)

http://dx.doi.org/10.1364/OE.17.011822

Acrobat PDF (1558 KB)

### 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

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

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]

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]

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]

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]

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]

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]

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]

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]

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

## 2. Structure and simulation method

*r*in the silver cladding, and 2

*a*is the centric distance of the upper two air cores, 2

*h*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 2

*a*>

*r*, 2

*a*=

*r*and 2

*a*<

*r*, when 2

*h*is a fixed value.

*ε*is chosen as - 18.0550+0.4776j (

_{clad}*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]

*λ*is the working wavelength in the vacuum.

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. 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]

*x*=Δ

*y*=1.0 nm.

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]

*λ*

_{0}=633

*nm*. The relative dielectric constant of the metal and the cladding is taken as

*ε*=19.00-0.53

_{core}*j*and

*ε*=4.00 respectively. In the calculation, the radius of the core is assumed from 0.05

_{clad}*λ*

_{0}to 0.25

*λ*

_{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]

*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]

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

## 3. Results and discussion

### 3.1. Modal characteristics

*H*, and energy flux density

_{x}, H_{y}*S*in the cases of

_{z}*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

*S*=

_{z}*R*(

_{e}*E*),

_{x}H_{y}-E_{y}H_{x}*E*and

_{x}*E*,

_{y}*H*and

_{x}*H*are components of electrical and magnetic field respectively.

_{y}*x H*is symmetric distribution with y axes, however, component of field

*H*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 2

_{y}*a*>

*r*, 2

*a*=

*r*and 2

*a*<

*r*, the two corner angles are smallest and the two corner tips are sharpest in the case of 2

*a*>

*r*, so the degree of localization of field is the largest.

### 3.2. Effect of geometrical parameters of the waveguide

*Re*(

*n*), propagation length

_{eff}*L*and mode area

_{prop}*A*are the three most important physical quantities that describe the propagation properties of surface plasmonic waveguides, then we investigate the dependence of

_{m}*Re*(

*n*),

_{eff}*L*and

_{prop}*A*of the fundamental mode shown in Fig. 3 on the geometrical parameters. Here

_{m}*Re*(

*n*) is defined as

_{eff}*Re*(

*β*)

*λ*/2

*π*,

*L*is defined as 1/

_{prop}*Im*(

*β*), and

*A*is defined as the area in which the energy flux density

_{m}*S*descend from 100% to 10% of its maximum value.

_{z}*Re*(

*n*),

_{eff}*L*and

_{prop}*A*varied with

_{m}*h*in the three cases of

*r*=2

*a*-5 nm, 2

*a*and 2

*a*+5 nm, and 2

*a*=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*(

*n*) increases as

_{eff}*h*increases, however

*L*on the whole lessly varied with

_{prop}*h*, and

*A*increases as

_{m}*h*decreases. The position of curves is influenced by parameter

*r*. We can also find that curves corresponding to

*r*=2

*a*+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*=2

*a*, 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.

### 3.3. Effect of working wavelengths

*H*, and energy flux density

_{x}, H_{y}*S*in the cases of

_{z}*r*=2

*a*=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.

*Re*(

*n*),

_{eff}*L*and

_{prop}*A*on

_{m}*h*when radius

*r*=2

*a*=100 nm, 120 nm, 140 nm and

*λ*=632.8 nm, 705.0 nm, 800.0 nm respectively. It can be seen that curves of

*Re*(

*n*),

_{eff}*L*and

_{prop}*A*could also be separated into three groups according to different

_{m}*λ*. And in each group,

*Re*(

*n*) increases along with the increase of

_{eff}*h*, however,

*L*has many trends that varied with

_{prop}*h*, and

*A*decreases as

_{m}*h*increases. Parameter

*r*has effect on the position of curves. It can be seen that curves according to

*r*=2

*a*=140 nm are always on the upward side of each group of curves. The above phenomena can be explained by the different field distribution at different wavelength. 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

*λ*=705.0 nm, when

*λ*is small, the area of field distribution is small. Namely the degree of localization of the field is large. In this instance, the interaction of the field and silver is strong. The effective index increases. Then the propagation length and mode area decrease. However, when

*λ*is large, the area of field distribution is large. Namely the degree of localization of field is small. In this instance, the interaction of the field and silver becomes weak. The effective index decreases. Then the propagation length and mode area increase.

### 3.4. Comparation with the SPW with two air cores

**16(14)**, 10710–10720 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-14-10710. [CrossRef]

*a=b*. Here the parameter

*r*of the three cores structure is equal to

*a*and

*b*, and

*h*is equal to

*c*of the double cores structure. Dependence of

*Re*(

*n*),

_{eff}*L*and

_{prop}*A*on

_{m}*h*when

*a*=0 nm, 60 nm,

*r*=120 nm, and

*λ*=632.8 nm, 705.0 nm, 800.0 nm is shown in Fig. 6 respectively. From Fig. 6, we can see that

*Re*(

*n*) (Fig. 6a and Fig. 6b) increases as h increases, however Lprop (Fig. 6c and Fig. 6d) increases as h increases when

_{eff}*λ*=800.0 nm, and in the other two cases the propagation length not obviously varied with

*h*. And

*A*(Fig. 6e and Fig. 6f) increases as

_{m}*h*decreases. Parameter

*λ*has effect on the position of the curves in Fig. 6. Results also show that SPW with three circular air cores (Fig. 6b, Fig. 6d and Fig. 6f) has longer propagation length, smaller effective index and mode area compared with the two circular air cores (Fig. 6a, Fig. 6c and Fig. 6e) with the same parameters. In other words, the proposed SPW in this paper has better propagation properties than that of our previous SPW.

### 3.5. Comparation with the SPW with a single wedge

*c*on the silver slab. This structure not only can be seen as a representative SPW with a single wedge [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]

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]

**16(14)**, 10710–10720 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-14-10710. [CrossRef]

*r*and

*c*is equal to

*r*and

*c*of the double cores structure [21

**16(14)**, 10710–10720 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-14-10710. [CrossRef]

*H*and

_{x}, H_{y}*z*

*S*of the fundamental mode supported by this single wedge structure is shown in Fig. 7(b), Fig. 7(c) and Fig. 7(d). It can be seen that the energy flux density mainly distributes in the wedged corner. The component of field

*H*is symmetric distribution with y axes, however, component of field

_{x}*H*is antisymmetric distribution with y axes.

_{y}*Re*(

*n*),

_{eff}*L*and

_{prop}*A*on

_{m}*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*(

*n*) increases as

_{eff}*c*increases, however

*L*and

_{prop}*A*decrease as

_{m}*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.

### 3.6. Effect of Gain in the Core Dielectric

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]

*γ*=298

*cm*

^{-1}(εcore=11.56-0.025j) and 596

*cm*

^{-1}(

*ε*=11.56-0.050j) [34

_{core}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]

*λ*=1550 nm,

*a*=60 nm,

*r*=120 nm, dependence of

*Re*(

*n*),

_{eff}*L*and

_{prop}*A*on

_{m}*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.

*L*. However, the effective index is almost kept invariably with different

_{prop}*ε*, here

_{core}*A*not obviously varied with

_{m}*ε*.

_{core}*S*is more concentrated on the surface of the metal than that with air cores (Fig. 10(d)).

_{z}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]

**16(14)**, 10710–10720 (2008), http://www.opticsexpress.org/abstract.cfm?uri=oe-16-14-10710. [CrossRef]

## 4. Conclusions

*a*>

*r*, 2

*a*=

*r*and 2

*a*<

*r*. Numerical calculation show that:

*a*<

*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.

*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.

## Acknowledgments

## References and links

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34. | S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. |

35. | D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. |

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40. | W. E. Arnoldi, “The principle of minimized iteration in the solution of matrix eigenvalue problems,” Q. Appl. Math. |

41. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

42. | M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal coners,” J. Opt. Soc. Am. B |

43. | E. D. Palik, |

**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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