Open Access
Add to BibSonomyAbstract
We introduce a kind of surface plasmonic waveguide with double elliptical air cores. The dependence of distribution of longitudinal energy flux density, effective index and propagation length of the fundamental mode supported by this waveguide on geometrical parameters and working wavelengths are analyzed using the finite-difference frequency-domain (FDFD) method. Results show that the longitudinal energy flux density distributes mainly in the two wedged corners which are formed by two elliptical air cores, and the closer to the corners the stronger the longitudinal energy flux density. The effective index and propagation length of the fundamental mode can be adjusted by the centric distance of two ellipses as well as the size of the two semiaxis. At the certain working wavelength, relative to the case of a=b, in the case of a>b, the energy in the metal is small, then the interaction of field and silver is weak, and the effective index becomes small, and the propagation length becomes large. With certain geometric parameters, relative to the case of λ=632.8nm, in the case of larger λ, the area of field distribution is large, and the energy in the metal is small, then the interaction of field and silver is weak, and the effective index becomes small, and the propagation length becomes large. This kind of hollow surface plasmonic waveguide can be applied to the field of photonic device integration and sensors.
©2008 Optical Society of America
1. Introduction
During past few years, surface plasmon polariton (SPP)-based waveguides have been a subject of intensive research [1-4]. Such waveguides provide a new mechanism of guiding the propagation of lights in the form of SPP along surface of metals. Because SPP has lateral scale of subwavelength, it can be used to overcome the diffraction limit that exists in conventional or photonic crystal waveguides, and to fulfill the further miniaturization of photonic devices and high integration density of photonic chips.
Using thin metal films as surface plasmon waveguidguides (SPWs) [5] was first proposed, but they have limited applications due to the lack of lateral confinement. Therefore, metal films with finite size [6], metal rod with square cross section [7] and metal strip with finite size and width [8] were proposed. At present, SPWs with various geometries that could confine lights in 2D have been reported, such as gap SPWs [9-11], slot SPWs [12, 13], wedge SPWs [14-16], trench SPWs [17-20] and mixed SPWs [21].
Now, structures of SPWs, which have already been proposed and investigated, are few and simple, and people’s attention are mostly focused on very few simple kinds of SPWs. In order to find SPWs with much better propagation properties, it is necessary to design and investigate much more novel structures. In this paper, a kind of surface plasmonic waveguide with double elliptical air cores is designed and the propagation properties are investigated using the finite-difference frequency-domain (FDFD) method. We will discuss the longitudinal energy flux density, the effective index and propagation length influenced by geometrical parameters and working wavelength of the waveguide.
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 two elliptical air holes with centric distance 2c and semiaxis a, b respectively, which are hollowed in the silver cladding. Apparently, it can be separated into three cases of a<b, a=b and a>b, as is shown in Fig. 1(a), Fig. 1(b) and Fig. 1(c) respectively.
Fig. 1. Crosssection of plasmonic waveguides with double elliptical air cores. The centric distance of two ellipses is 2c and semiaxis of two ellipses is a, b respectively.
Fabrication steps of the proposed surface plasmonic waveguide are suggested and shown schematically in Fig. 2. At first, a pair of W-suface plasmonic waveguides can be fabricated using the standard lithographic and etching techniques [16], then a suface plasmonic waveguide with double elliptical air cores can be obtained by combination of them.
Fig. 2. Schematic of the fabrication steps: (1) fabricate a pair of W-suface plasmonic waveguides, (2) combine them into a surface plasmonic waveguide with double elliptical air cores
Because the primary materials that be made of the SPWs are noble metals, there is complex relation between the dielectric constant and frequency of optical wave. The main methods of investigating propagation properties of SPWs are classified as two kinds, namely time domain method, such as the finite-difference time-domain (FDTD) method, and frequency domain method, such as the finite-difference frequency-domain (FDFD) method and the finite-element method (FEM). In this paper, the 2D full-vectorial FDFD method [22-24] is used to study propagation properties of our proposed PWs structure. This is a simple and effective numerical simulation approach.
Supposing the propagation constant in the z direction is β, each field component is denoted by ϕ(x, y, z)=ϕ(x, y) exp(jβz), and here ϕ denotes arbitrary field component. According to the theory of 2D-FDFD, we can get two eigenvalue equations:
Where Pij and Qij are coefficient matrix elements, Ex and Ey are electric field components, and Hx, Hy are magnetic field components respectively in the x direction and y direction. Setting geometrical parameters, electromagnetic parameters and the working wavelength, propagation constant β, effective index neff and the distribution of magnetic field of each mode at the working wavelength can be obtained by solving the eigenvalue Eq (1). This eigenvalue problem is solved by Arnoldi arithmetic [25], which can deal with matrix eigenvalue problem with large complex coefficient matrix. Then taking propagation constant β into Eq. (2), the electric field distribution in the cross section of corresponding mode can be obtained by solving this linear algebraic equations.
In this paper, 601×601 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 these lattices. Spatial discretization distance is Δx=Δy=1.0nm.
Yan and Qiu claimed that the corner angle and the corner tip sharpness would affect the mode properties of a Λ-wedge waveguide considerably [26]. Although the FDFD method use orthogonal mesh and corner tips are not rounded in this paper, we find that, generally, for example, in the range of b=a-40nm to b=a+40nm, corner angles are not too small, corner tips are not too sharp, so the situation is in the safe case. The accuracy of the FDFD code adopted in this paper has been validated in our previous works [27-29].
In the calculation, the dielectric constant of silver εm is -14.8817+0.3858j (λ=0.5487µm), -18.0550+0.4776 j (λ=0.6328µm), -23.4046+0.3870 j (λ=0.7050µm) and -31.0784+0.4118 j (λ=0.8000µm) [27] respectively, and the background is vacuum.
3. Results and discussions
Mode characteristies of the surface plasmonic waveguide with double elliptical air cores shown in Fig. 1 are investigated firstly. Results show that the distribution of Hx, Hy, and the longitudinal energy flux density Sz of the fundamental mode in the cases of a=130nm, c=120nm, λ=632.8nm and b=90nm, 130nm and 170nm respectively are shown in Fig. 3, Fig. 4, Fig. 5. Here the longitudinal energy flux density is defined as Sz=Re(ExHy-EyHx), and Ex, Ey, Hx, Hy are components of electrical and magnetic field of Eqs. (1) and (2) respectively.
Fig. 3. The distribution of the field of (a)Hx, (b)Hy and (c)Sz at the cross section when a=130nm, b=170nm, c=120nm and λ=632.8nm. Dashed line in (a) and (b) indicate the outline of the structure.
Fig. 4. The distribution of the field of (a)Hx, (b)Hy and (c)Sz at the cross section when a=130nm, b=130nm, c=120nm and λ=632.8nm. Dashed line in (a) and (b) indicate the outline of the structure.
Fig. 5. The distribution of the field of (a)Hx, (b)Hy and (c)Sz at the cross section when a=130nm, b=90nm, c=120nm and λ=632.8nm. Dashed line in (a) and (b) indicate the outline of the structure.
One can find that Hx components of fields shown in Fig. 3(a), Fig. 4(a) and Fig. 5(a) are antisymmetric distribution with x axes and y axes, however, Hy components of fileds shown in Fig. 3(b), Fig. 4(b) and Fig. 5(b) are symmetric distribution with x axes and y axes. From Fig. 3(c), Fig. 4(c) and Fig. 5(c), one can find that the longitudinal energy flux density mainly distributes in the tips of the two wedged region that were formed by two hollow ellipses. Relative to the case of a=b, in the case of a>b, the energy in the metal is small. In the case of a<b, the energy in the metal is large.
Since effective index Re(neff) and propagation length Lprop are two most important physical quantities that describe the propagation properties of surface plasmonic waveguides, we then investigate the dependence of Re(neff) and Lprop of the fundamental mode shown in Fig. 3, Fig. 4 and Fig. 5 on geometrical parameters of the waveguide. Here Re(neff)=Re(β)λ/2π, Lprop=1/Im(β).
Graphs of Re(neff) and Lprop varied with c in three cases of b=a-40nm, a, a+40nm at λ=632.8nm are shown in Fig. 6(a) and Fig. 6(b). One can see from these two figures, curves can be obviously separated into three groups according to different a. In each group, Re(neff) increases as c increases, however Lprop decreases as c increases. Parameter b has little influence on the position of curves. One can also see from Fig. 6(a), when c≪a, the corresponding mode is cut off. One can also see from Fig. 6(b), curves corresponding to b=a-40nm are commonly on the upward side of each group of curves. This phenomenon can be explained by the field distribution of Fig. 3, Fig. 4 and Fig. 5: because the field mainly centralize on the two sides that are close to the tip of the wedged region. Relative to the case of a=b, in the case of a>b, the energy in the metal is small, then the interaction of field and silver is weak, and the effective index becomes small, and the propagation length becomes large. However, in the case of a<b, the energy in the metal is large, then the interaction of field and silver is strong, and the effective index becomes large, and the propagation length becomes short.
In order to find out the dependence of field distribution of the fundamental mode shown in Fig. 3, Fig. 4 and Fig. 5 on the working wavelength, the distribution of Hx, Hy, and the longitudinal energy flux density Sz in the case of a=b=130nm, c=120nm and λ=548.7nm, 705.0nm and 800.0nm are calculated. Here we extend the calculation to longer wavelength 800nm where vertical cavity surface emitting lasers (VCSELs) are available. Results are shown in Fig. 7, Fig. 8 and Fig. 9.
Fig. 7. The distribution of the field of (a)Hx, (b)Hy and (c)Sz at the cross section when a=b=130nm, c=120nm and λ=548.7nm. Dashed line in (a) and (b) indicate the outline of the structure.
Fig. 8. The distribution of the field of (a)Hx, (b)Hy and (c)Sz at the cross section when a=b=130nm, c=120nm and λ=705.0nm. Dashed line in (a) and (b) indicate the outline of the structure.
Fig. 9. The distribution of the field of (a)Hx, (b)Hy and (c)Sz at the cross section when a=b=130nm, c=120nm and λ=800.0nm. Dashed line in (a) and (b) indicate the outline of the structure.
One can see from Fig. 7, Fig. 8 and Fig. 9, field distributions are similar to that shown in the Fig. 4. Relative to the case of λ=632.8nm (Fig. 4), in the case of λ=548.7nm (Fig. 7), the area of field distribution is small, and the field are more confined near to the wedged region. However, in the case of λ=705.0nm (Fig. 8) or λ=800.0nm (Fig. 9), the area of field distribution is large, and the field is less confined near to the wedged region.
Fig. 10. Dependence of (a) Re(neff) and (b) Lprop on c when a=b=120nm,130nm,140nm and λ=632.8nm, 705.0nm and 800.0nm
Figure 10(a) and Figure 10(b) show the dependence of Re(neff) and Lprop on c when semiaxis a=b=120nm, 130nm, 140nm and λ=548.7nm, 632.8nm, 705.0nm and 800.0nm respectively. One can also see from these two figures, curves of Re(neff) and curves of Lprop could also be separated into three groups according to the parameter a. In each group, Re(neff) increases as c increase, however, Lprop decreases as c increase. Parameter λ has considerable effect on position of the curve. From Fig. 10(a), one can also see that, in the case of c≪a, the corresponding mode is cut off. One can also see from Fig. 10 (b), curves corresponding to λ=800.0nm are always on the upward side of each group of curves. The phenomenon can be explained by the different field distribution at different wavelength. Because the field mainly centralize on the tip of the wedged region, relative to the case of λ=632.8nm, when λ is small, the area of field distribution is small, the energy in the metal is large, then the interaction of field and silver is strong, and the effective index becomes large, and the propagation length becomes short. However, when λ is large, the area of field distribution is large, and the energy in the metal is small, then the interaction of field and silver is weak, and the effective index becomes small, and the propagation length becomes large.
Structures studied here can be seen as a kind of couple of two metallic wedges that have been studied in [15,16,26]. Although we adopted parameters a, b and c, rather then the wedge angle and the wedge height, to describe the geometric structure, our obtained results are still agree with results obtained in [15,16,26].
Experimentalist maybe concern how to couple SPP efficiently from an external source to our waveguide structure. We notice that, in the case of small cross-section, Silicon waveguides used in the context of Silicon photonics, one can use tapers to increase coupling efficiency. We think that this problem maybe solved by the method proposed in [15], viz., one can use a geometry-driven convertor to couple SPP efficiently from an external source to our waveguide structure.
4. Conclusion
In this paper, we have designed a kind of surface plasmonic waveguide with double elliptical air cores. The structure of this kind of waveguide can be separated into three cases of a<b, a=b and a>b. Numerical calculation indicates that the longitudinal energy flux density distributes mainly in two wedged corners which are formed by two elliptical air cores, and the closer to the corners the stronger longitudinal modal energy flux density. At the certain working wavelength, relative to the case of a=b, in the case of a>b, the energy in the metal is small, then the interaction of field and silver is weak, and the effective index becomes small, and the propagation length becomes large. With certain geometric parameters, relative to the case of λ=632.8nm, in the case of larger λ, the field distribution is large, and the energy in the metal is small, then the interaction of field and silver is weak, and the effective index becomes small, and the propagation length becomes large. Since the effective index and propagation length of the mode can be adjusted by the centric distance of two ellipses as well as the size of the two semiaxis, this kind of hollow surface plasmonic waveguide can be applied to the field of photonic device integration and sensors.
Acknowledgment
This work is supported by the National Fundamental Fund of Personnel Training (Grant No. J0730317).
References and links
1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830(2003). [CrossRef] [PubMed]
2. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193(2006). [CrossRef] [PubMed]
3. 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, 508–511(2006). [CrossRef] [PubMed]
4. S. A. Maier, “Plasmonics: The promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12, 1671–1677(2007).
5. E. N. Economou, “Surface plasmon in thin films,” Phys. Rev. 182, 539–554(1969). [CrossRef]
6. P. Berini, “Plasmon polariton modes guided by a metal film of finite width,” Opt. Lett. 24, 1011–1013(1999). [CrossRef]
7. 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, 035434-10(2007). [CrossRef]
8. J. Guo and R. Adato, “Control of 2D plasmon-polariton mode with dielectric nanolayers,” Opt. Express 16, 1232–1237(2008). [CrossRef] [PubMed]
9. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82, 1158–1160 (2003). [CrossRef]
10. F. Kusunokia, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86, 211101-3(2005). [CrossRef]
11. R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express 13, 1933–1938(2005). [CrossRef] [PubMed]
12. 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, 261114-3(2005). [CrossRef]
13. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650(2005). [CrossRef] [PubMed]
14. 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 triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106-3(2005). [CrossRef]
15. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. M. Moreno, and F. J. G. Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons”, Phys. Rev. Lett. 100, 023901-4(2008). [CrossRef] [PubMed]
16. 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, 5252–5260(2008). [CrossRef] [PubMed]
17. J. Q. Lu and A. A. Maradudin, “Channel plasmons,” Phys. Rev. B 42, 11159–11165(1990). [CrossRef]
18. S. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31, 2133–2135(2006). [CrossRef] [PubMed]
19. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon—polariton in a triangular groove on a metal surface,” Opt. Lett. 29, 1069–1071(2004). [CrossRef] [PubMed]
20. 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, 16596–16603(2007). [CrossRef] [PubMed]
21. D. Arbel and M. Orenstein, “Plasmonic modes in w-shaped metal-coated silicon grooves,” Opt. Express 16, 3114–3119(2008). [CrossRef] [PubMed]
22. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864(2002). [PubMed]
23. S. Guo, F. Wu, and S. Albin, “Loss and dispersion analysis of microstructured optical fibers by finite-difference method,” Opt. Express 12, 3341–3352(2004). [CrossRef] [PubMed]
24. C. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundaray conditions for optical waveguides and photonic craystal fibers,” Opt. Express 12, 6165–6177(2004). [CrossRef] [PubMed]
25. W. E. Arnoldi, “The principle of minimized iteration in the solution of matrix eigenvalue problems,” Q. Appl. Math. 9, 17–29(1951).
26. M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal coners,” J. Opt. Soc. Am. B 24, 2333–2342(2007). [CrossRef]
27. H. Wang, W. Xue, and W. Zhang, “Negative dispersion properties of photonic crystal fiber with dual core and composite lattice,” Acta Optica Sinica 28, 27–30 (2008). [CrossRef]
28. L. Guo, Y. Wu, W. Xue, and G. Zhou, “Dispersion properties of photonic crystal fiber with composite hexagonal air hole lattice,” Acta Optica Sinica 27, 935–939 (2007).
29. Y. Wu, L. Guo, W. Xue, and G. Zhou, “Photonic crystal fiber with single polarization,” Acta Optica Sinica 27, 593–597(2007).
30. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4317–4379(1972). [CrossRef]
