- Email: [email protected]
A hybrid long-range plasmonic waveguide with sub-wavelength confinement
Optics Communications 291 (2013) 400–404
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A hybrid long-range plasmonic waveguide with sub-wavelength confinement Lin Chen a,n, Xun Li b, Guoping Wang c a
Wuhan National Laboratory for Optoelectronics, School of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 c Key Laboratory of Acoustic and Photonic Materials and Devices, Ministry of Education and Department of Physics, Wuhan University, Wuhan 430072, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 October 2012 Received in revised form 6 November 2012 Accepted 10 November 2012 Available online 29 November 2012
A symmetric three dielectric layer stack with a low–high–low dielectric constant distribution is proposed to replace the conventional single dielectric layer on both sides of the thin metal layer to guide the long range surface plasmonic polaritons. Our simulation result shows that the new design is capable of providing more compact modal confinement as well as low loss transmission due to the coupling between the plasmonic and dielectric waveguide modes. & 2012 Elsevier B.V. All rights reserved.
Keywords: Plasmons Optical waveguides Nanophotonics
1. Introduction The simultaneous realization of a strongly confined light beam and low propagation loss for practical applications is of central importance in various nanophotonic circuits. With only evanescent wave distribution in the cross section, surface plasmon polaritons (SPPs) can localize light energy in a nanoscale domain, which have aroused tremendous research interest in constructing various metal nanostructures for guiding SPPs with nanometric cross section [1]. However, SPPs inevitably experience significantly high propagation attenuation due to the intrinsic oscillation damping loss in metals [2]. A thin metal film sandwiched between a homogeneous dielectric medium with a symmetric structure can support long-range SPP (LRSPP) mode, which supports a relatively longer propagation length compared with that of the SPPs traveling along the surface of the bulk metal in a dielectric medium [3]. In many applications, however, the propagation length still needs to be extended. A simple but effective strategy for increasing the propagation distance of LRSPP mode is to reduce the thickness of metal layer. Nevertheless, to deposit a uniform metal layer with extremely small thickness is difficult because metals tend to form nanoscale islands during the fabrication process [4]. Other than the low-loss requirement, strongly confined SPP field in the metal–dielectric interface is essential for the development of compact plasmonic circuits since it ensures
smaller bending loss and higher component density [5]. However, as the metal layer thickness reduces, LRSPP field extends farther into the dielectric layer, which limits its practical applications in the relevant realms. The propagation loss of LRSPP mode in a dielectric–metal– dielectric structure can be significantly reduced by lowering the refractive index of the clad material [6]. However, as the dielectric constant reduces in adjacent layers, the SPP field extends farther, which brings in a large cross-sectional modal spot in such waveguide and limits its application on miniaturized devices where the size is crucial. Recently, hybrid plasmonic waveguides, consisting of highindex-contrast dielectric waveguide placed closed to a plasmonic waveguide, have been demonstrated to be capable of simultaneously achieving sub-wavelength confinement and long-range propagation due to the coupling between plasmonic and dielectric waveguide modes [2,7–26]. Based on a similar principle, in this letter we will propose a symmetric three dielectric layer stack with a low–high–low dielectric constant distribution to replace the conventional single dielectric layer on both sides of the thin metal layer to guide LRSPP mode with strong confinement and low propagation loss. It is shown that the proposed hybrid plasmonic waveguide has the potential to localize the LRSPP mode in a sub-wavelength scale and to maintain a long propagation distance as opposed to the conventional LRSPP waveguide.
2. The proposed hybrid LRSPP waveguide n
Corresponding author. Tel.: þ86 13545048010. E-mail addresses: [email protected], [email protected] (L. Chen). 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.11.031
Fig. 1 schematically shows the cross-section of the proposed one-dimensional hybrid plasmonic waveguide, where three layers
L. Chen et al. / Optics Communications 291 (2013) 400–404
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Fig. 1. One-dimensional hybrid plasmonic waveguide consisting of a thin metal film (thickness t, em ), two inner dielectric layers of low-index of refraction (thickness L1 , eL1 ), two outer dielectric layers of high-index of refraction (thickness LH , eH ), and two outer dielectric claddings of low-index of refraction (thickness L2 , eL2 ). The distribution of the main transverse electric field component of the mode is shown as the blue curve. The propagation direction is along the z direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of dielectric (eL1–eH–eL2) are placed on both sides of a thin metal film with a symmetry. The thickness of outermost cladding is infinite in the y direction and the hybrid plasmonic modes propagate along the z direction. Besides, we assume that the width of the metal film is infinite in the x direction. For this structure, when we consider TM polarization (the electric field is along the y direction), there will be a field enhancement in the inner low-index region due to a combination of plasmon and high-index-contrast effects [2, 7–26], which is somewhat similar to a field enhancement in a dielectric slot waveguide due to the strong discontinuity of the normal component of the electric field at the high-index-contrast interface [27,28]. For the proposed hybrid structure, there exist two fundamental modes: symmetric mode and asymmetric SPP mode. Here we only consider the symmetric SPP mode, which can be termed the hybrid LRSPP mode because of the symmetric dielectric distributions on each side of a thin metal film. Compared with the conventional LRSPP waveguide, the mode size in the y direction for the proposed hybrid plasmonic waveguide could be much smaller. Other than smaller mode sizes, low refractive index of inner dielectric layer will ensure low propagation loss because the field inside the metal would be weaker according to the boundary condition of Maxwell’s equations. In our work, the operating wavelength l is assumed to be 1550 nm. A thin gold film with a thickness of 20 nm is chosen as the metal layer, its dielectric constant is em ¼ 114.9 þ11.09j [29]. Silicon (eH ¼ n2H ¼ 12:1) is selected as the high-index dielectric layer and silicon dioxide (eL1 ¼ eL2 ¼2.1) as the inner and outer low-index dielectric layers. The mode field distribution, mode effective index, and propagation distance are investigated by finite element method (FEM) based mode solver using COMSOL. The propagation length is defined as the distance at which the field amplitude drops to 1/e of its initial value. Here the effective mode size along the y direction Lm is defined as the ratio of the total modal energy and the peak energy density along the y direction [2,12]: Z þ1 wm 1 Lm ¼ ¼ wðyÞdy, ð1Þ maxwðyÞ 1 maxwðyÞ where wm and wðyÞ are the total electromagnetic energy and energy density along the y direction, respectively (per unit length along the propagation direction). The normalized mode size is defined as Lm =L0 (L0 ¼ l=2). In the following study, we will vary the thickness of the inner SiO2 layer, L1 , and Si layer, LH , to control the mode size, mode effective index, and propagation length for the hybrid LRSPP mode. Fig. 2(a and b) show the normalized mode size and propagation
Fig. 2. (a) Normalized mode size (Lm =L0 ) and propagation length (b) versus LH , for different L1 . Points (1–4) indicated in (a) correspond to [L1 ,LH ] ¼[20 nm,100 nm] (1), [L1 ,LH ] ¼ [100 nm,100 nm] (2), [L1 ,LH ] ¼ [50 nm,150 nm] (3), and [L1 ,LH ] ¼ [50 nm,300 nm] (4), respectively. The normalized mode size of LRSPP mode in Si–Ag–Si is denoted as the blue solid line in (a). The propagation lengths of LRSPP modes in Si–Ag–Si and SiO2–Ag–SiO2 are denoted as black dashed line and black solid line in (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
length versus the Si thickness, LH , for different thickness of the inner SiO2 layer, L1 . It can be seen that, for a small value of L1 , the hybrid structure tends to confine light energy in the gap between the metal and Si layer, which can be reflected by the energy density distribution for [L1 ,LH ]¼[20 nm,100 nm] in Fig. 3(a). At this time, increasing LH has little influence on the mode size [Fig. 2(a)]. It should be noted that, while the mode size for a small value of L1 can be made much smaller than that of the conventional LRSPP mode in homogenous Si medium [Fig. 2(a)], the propagation loss can be comparable [Fig. 2(b)]. However, increasing L1 will lead to a much weaker field confinement; hence, a much larger mode size can be obtained [Fig. 3(e)]. In this situation, mode coupling results in a hybrid mode that features both guidedlike and SPP characteristics, namely, the light energy is distributed in both the Si layer and metal–SiO2 interface [Fig. 3(b, c)]. For large L1 and LH , the hybrid structure is shown to support low-loss guided-like dielectric mode with much light is localized in the Si layer [Fig. 3(d, e)]. It is easily understood that only a small portion of light distributed inside the metal film, inducing a low absorption loss. We have noted, in a recent study on hybrid plasmonic mode in a similar structure [30], the authors systematically analyzed the propagation loss of the symmetrical mode and asymmetrical mode, but the present mode confinement is much weaker than the result in our work. This is because that a much higher thickness of inner-cladding and lower refractive index difference between the inner-cladding and core layer are assumed in that paper. Figure of merit (FoM) has been widely employed to compare and assess various SPP waveguides, which is useful for comparing waveguides in applications where achieving prescribed or small
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Fig. 3. (a)–(d) Electromagnetic energy density distributions for [L1 ,LH ]¼[20 nm,100 nm] (a), [L1 ,LH ]¼ [100 nm,100 nm] (b), [L1 ,LH ]¼[50 nm,150 nm] (c), and [L1 ,LH ]¼ [50 nm,300 nm] (d), corresponding to the points (1–4) indicated in Fig. 2(a). (e) Normalized energy density wðyÞ =wðmÞ along y direction for the hybrid LRSPP waveguides and conventional LRSPP waveguides. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
mode width is important. Here the FoM is defined as [31] FoM ¼
1
az Lm
,
ð2Þ
where az is the imaginary part of the propagation constant. We make a direct comparison between the hybrid LRSPP mode and traditional LRSPP mode in a homogeneous medium. We can see from Fig. 4 that the LRSPP mode in Si–Ag–Si structure shows the worst propagation property because the FoM is the smallest, which can be attributed to the fact that a LRSPP mode in Si–Ag–Si structure suffers a much larger propagation loss than that of a hybrid LRSPP mode. The hybrid LRSPP mode’s FoM is smaller than that of a LRSPP mode in SiO2–Ag–SiO2 structure as LH is below a specific point, but one can increase LH to enhance the FoM. Consequently, the FoM can be made much bigger than that of a LRSPP mode in SiO2–Ag–SiO2 structure.
Fig. 4. The dependence of the figure of merit (FoM) of the hybrid LRSPP modes on LH for different L1 . The corresponding FoM of the LRSPP modes in Si–Ag–Si and SiO2–Ag–SiO2 are denoted as black dashed line and black solid line, respectively.
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Fig. 5. (a) The dependence of the mode effective index of the hybrid LRSPP mode, nhyb , on LH for different L1 . As a comparison, the mode effective index of the pure dielectric waveguides, nd , and LRSPP mode in SiO2–gold–SiO2 waveguides, nI , are depicted in the solid black line and dashed black line, respectively. (b) The mode character derived from Eq. (2).
Due to the coupling between the dielectric waveguide mode and LRSPP mode, the refractive index of the hybrid LRSPP mode is quite different from that of dielectric waveguide mode or LRSPP mode. Fig. 5(a) shows the dependence of the mode effective index of the hybrid LRSPP mode on L1 and LH . Obviously, in the limit of dielectric-like and SPP-like modes, the effective refractive index approaches that of pure dielectric waveguide mode, nd , or conventional LRSPP mode, nI . Besides, the hybrid LRSPP mode’s effective index is always larger than that of the conventional LRSPP mode or dielectric waveguide mode, which can be attributed to the fact that the LRSPP modes are coupled with the dielectric waveguide modes [2]. Furthermore, for a fixed value of LH , the mode effective index of the hybrid LRSPP mode increases with the reduced L1 because that the LRSPP mode couples with dielectric waveguide mode more effectively as L1 reduces. In order to describe the mode characteristics of the hybrid 2 LRSPP mode, the mode character, a þ ðLH ,L1 Þ , is introduced to evaluate the degree to which the hybrid mode is a dielectric waveguide mode or an SPP mode [2]. The hybrid mode is more like an SPP mode for a smaller mode character and a dielectric waveguide mode for a much larger mode character. a þ ðLH ,L1 Þ2 ¼
nhyb ðLH ,L1 ÞnI : ð3Þ ½nhyb ðLH ,L1 Þnd ðLH Þ þ ½nhyb ðLH ,L1 ÞnI 2 Fig. 5(b) shows the dependence of a þ ðLH ,L1 Þ on LH for different L1 . We can see that for a fixed value of L1 , a larger LH leads to a greater mode character, hence the hybrid LRSPP mode is more like a dielectric waveguide mode, and most of light is localized in the Si layer. For a fixed value of LH , the mode character increases with L1 . This means that, by modulating the value of L1 , the hybrid LRSPP mode can be made more like a dielectric waveguide mode with a large mode size or a conventional LRSPP mode with a small mode size. For example, the mode area for L1 ¼ 100 nm is 0.526 times smaller than the diffraction limit [Fig. 3(b)]. By decreasing L1 to 20 nm, mode size of about 0.115 times smaller than the diffraction limit is achievable [Fig. 3(a)]. As for the fabrication of the present structure, an SOI wafer can be purchased on the marked with a SiO2–Si layer stack on the Si substrate, such wafer is obtained by SIMOX (Separation by Implantation of Oxygen) technology. Through rather standard clean-room technology, we can deposit the SiO2 layer, the metal
layer, and the SiO2–Si–SiO2 top layer stack through PECVD, Ebeam evaporation and/or sputtering, respectively. Since the metal layer is very thin and the stripe is very narrow, the thermal stress may not bring in any major issue after an appropriate annealing process. However, the fabrication process is by no means easy since special cares must be taken to relax the thermal stress after deposition.
3. Conclusion In conclusion, we have proposed a hybrid plasmonic waveguide to guide the LRSPP mode with compact modal size and low propagation loss. The ultra-small modal size is achieved due to the coupling between the plasmonic and dielectric waveguide modes. By comparing the hybrid waveguide with the conventional LRSPP waveguide in terms of the modal size and propagation distance, we find that the hybrid waveguide has advantages of guiding LRSPP mode with compact modal size as well as long propagation distance. While it does not seem to have substantial difficulty in fabricating the hybrid waveguide, our proposed structure may serve as promising design for guiding SPPs in nanophotonic circuits.
Acknowledgement Lin Chen is supported by NSFC (Grant No. 11104093), and ‘the Fundamental Research Funds for the Central Universities’, HUST: 2011QN041. Guoping Wang is supported by NSFC (Grant No. 60925020). References [1] W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2003) 824. [2] R.F. Oulton, V.J. Sorger, D.A. Genov, D.F.P. Pile, X. Zhang, Nature Photonics 2 (2008) 496. [3] D. Sarid, Physical Review Letters 47 (1981) 1927. [4] L. Holland, Vacuum Deposition of Thin Films, Chapman and Hall, London, 1966. [5] T. Holmgaard, S.I. Bozhevolnyi, Physical Review B: Condensed Matter 75 (2007) 245405. [6] P. Berini, Advances in Optics and Photonics 1 (2009) 484. [7] R. Salvador, A. Martinez, C. Garcia-Meca, R. Ortuno, J. Marti, IEEE Journal on Selected Topics in Quantum Electronics 14 (2008) 1496.
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[8] R.F. Oulton, G. Bartal, D.F.P. Pile, X. Zhang, New Journal of Physics 10 (2008) 105018. [9] M. Fujii, J. Leuthold, W. Freude, IEEE Photonics Technology Letters 21 (2009) 362. [10] D. Dai, S. He, Optics Express 17 (2009) 16646. [11] Y. Zhao, L. Zhu, Journal of the Optical Society of America B: Optical Physics 27 (2010) 1260. [12] H. Benistya, M. Besbes, Journal of Applied Physics 108 (2010) 063108. [13] X. Zhang, A. Hu, J.Z. Wen, T. Zhang, X. Xue, Y. Zhou, W.W. Duley, Optics Express 18 (2010) 18945. [14] R.F. Oulton, V.J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, Nature 461 (2009) 629. [15] D. Chen, Applied Optics 49 (2010) 6868. [16] X. Yang, Y. Liu, R.F. Oulton, X. Yin, X. Zhang, Nano Letters 11 (2011) 321. [17] Y. Bian, Z. Zheng, Y. Liu, J. Liu, J. Zhu, T. Zhou, Optics Express 19 (2011) 22417. [18] V.D. Ta, R. Chen, H.D. Sun, Optics Express 19 (2011) 13598. [19] Y. Bian, Z. Zheng, Y. Liu, J. Zhu, T. Zhou, IEEE Photonics Technology Letters 23 (2011) 884.
[20] M.Z. Alam, J. Stewart Aitchison, M. Mojahedi, Optics Letters 37 (2012) 55. [21] V.J. Sorger, Z. Ye, R.F. Oulton, Y. Wang, G. Bartal, X. Yin, X. Zhang, Nature Communications 2 (2011) 1. [22] J.T. Kim, S. Choi, IEEE Photonics Technology Letters 24 (2012) 170. [23] Y. Bian, Z. Zheng, X. Zhao, J. Zhu, T. Zhou, Optics Express 17 (2009) 21320. [24] B. Yun, G. Hu, Y. Ji, Y. Cui, Journal of the Optical Society of America B: Optical Physics 26 (2009) 1924. [25] L. Chen, X. Li, G. Wang, W. Li, S. Chen, L. Xiao, D. Gao, IEEE Journal of Lightwave Technology 30 (2012) 163. [26] L. Chen, T. Zhang, X. Li, W.P. Huang, Optics Express 20 (2012) 20535. [27] V.R. Almeida, Q. Xu, C.A. Barrios, M. Lipson, Optics Letters 29 (2004) 1209. [28] C. Koos, L. Jacome, C. Poulton, J. Leuthold, W. Freude, Optics Express 15 (2007) 5976. [29] P.B. Johnson, R.W. Christy, Physical Review B: Condensed Matter 6 (1972) 4370. [30] J.T. Kim, Optics Communication 284 (2011) 4171. [31] P. Berini, Optics Express 14 (2006) 13030.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A hybrid long-range plasmonic waveguide with sub-wavelength confinement Lin Chen a,n, Xun Li b, Guoping Wang c a
Wuhan National Laboratory for Optoelectronics, School of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 c Key Laboratory of Acoustic and Photonic Materials and Devices, Ministry of Education and Department of Physics, Wuhan University, Wuhan 430072, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 October 2012 Received in revised form 6 November 2012 Accepted 10 November 2012 Available online 29 November 2012
A symmetric three dielectric layer stack with a low–high–low dielectric constant distribution is proposed to replace the conventional single dielectric layer on both sides of the thin metal layer to guide the long range surface plasmonic polaritons. Our simulation result shows that the new design is capable of providing more compact modal confinement as well as low loss transmission due to the coupling between the plasmonic and dielectric waveguide modes. & 2012 Elsevier B.V. All rights reserved.
Keywords: Plasmons Optical waveguides Nanophotonics
1. Introduction The simultaneous realization of a strongly confined light beam and low propagation loss for practical applications is of central importance in various nanophotonic circuits. With only evanescent wave distribution in the cross section, surface plasmon polaritons (SPPs) can localize light energy in a nanoscale domain, which have aroused tremendous research interest in constructing various metal nanostructures for guiding SPPs with nanometric cross section [1]. However, SPPs inevitably experience significantly high propagation attenuation due to the intrinsic oscillation damping loss in metals [2]. A thin metal film sandwiched between a homogeneous dielectric medium with a symmetric structure can support long-range SPP (LRSPP) mode, which supports a relatively longer propagation length compared with that of the SPPs traveling along the surface of the bulk metal in a dielectric medium [3]. In many applications, however, the propagation length still needs to be extended. A simple but effective strategy for increasing the propagation distance of LRSPP mode is to reduce the thickness of metal layer. Nevertheless, to deposit a uniform metal layer with extremely small thickness is difficult because metals tend to form nanoscale islands during the fabrication process [4]. Other than the low-loss requirement, strongly confined SPP field in the metal–dielectric interface is essential for the development of compact plasmonic circuits since it ensures
smaller bending loss and higher component density [5]. However, as the metal layer thickness reduces, LRSPP field extends farther into the dielectric layer, which limits its practical applications in the relevant realms. The propagation loss of LRSPP mode in a dielectric–metal– dielectric structure can be significantly reduced by lowering the refractive index of the clad material [6]. However, as the dielectric constant reduces in adjacent layers, the SPP field extends farther, which brings in a large cross-sectional modal spot in such waveguide and limits its application on miniaturized devices where the size is crucial. Recently, hybrid plasmonic waveguides, consisting of highindex-contrast dielectric waveguide placed closed to a plasmonic waveguide, have been demonstrated to be capable of simultaneously achieving sub-wavelength confinement and long-range propagation due to the coupling between plasmonic and dielectric waveguide modes [2,7–26]. Based on a similar principle, in this letter we will propose a symmetric three dielectric layer stack with a low–high–low dielectric constant distribution to replace the conventional single dielectric layer on both sides of the thin metal layer to guide LRSPP mode with strong confinement and low propagation loss. It is shown that the proposed hybrid plasmonic waveguide has the potential to localize the LRSPP mode in a sub-wavelength scale and to maintain a long propagation distance as opposed to the conventional LRSPP waveguide.
2. The proposed hybrid LRSPP waveguide n
Corresponding author. Tel.: þ86 13545048010. E-mail addresses: [email protected], [email protected] (L. Chen). 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.11.031
Fig. 1 schematically shows the cross-section of the proposed one-dimensional hybrid plasmonic waveguide, where three layers
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Fig. 1. One-dimensional hybrid plasmonic waveguide consisting of a thin metal film (thickness t, em ), two inner dielectric layers of low-index of refraction (thickness L1 , eL1 ), two outer dielectric layers of high-index of refraction (thickness LH , eH ), and two outer dielectric claddings of low-index of refraction (thickness L2 , eL2 ). The distribution of the main transverse electric field component of the mode is shown as the blue curve. The propagation direction is along the z direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
of dielectric (eL1–eH–eL2) are placed on both sides of a thin metal film with a symmetry. The thickness of outermost cladding is infinite in the y direction and the hybrid plasmonic modes propagate along the z direction. Besides, we assume that the width of the metal film is infinite in the x direction. For this structure, when we consider TM polarization (the electric field is along the y direction), there will be a field enhancement in the inner low-index region due to a combination of plasmon and high-index-contrast effects [2, 7–26], which is somewhat similar to a field enhancement in a dielectric slot waveguide due to the strong discontinuity of the normal component of the electric field at the high-index-contrast interface [27,28]. For the proposed hybrid structure, there exist two fundamental modes: symmetric mode and asymmetric SPP mode. Here we only consider the symmetric SPP mode, which can be termed the hybrid LRSPP mode because of the symmetric dielectric distributions on each side of a thin metal film. Compared with the conventional LRSPP waveguide, the mode size in the y direction for the proposed hybrid plasmonic waveguide could be much smaller. Other than smaller mode sizes, low refractive index of inner dielectric layer will ensure low propagation loss because the field inside the metal would be weaker according to the boundary condition of Maxwell’s equations. In our work, the operating wavelength l is assumed to be 1550 nm. A thin gold film with a thickness of 20 nm is chosen as the metal layer, its dielectric constant is em ¼ 114.9 þ11.09j [29]. Silicon (eH ¼ n2H ¼ 12:1) is selected as the high-index dielectric layer and silicon dioxide (eL1 ¼ eL2 ¼2.1) as the inner and outer low-index dielectric layers. The mode field distribution, mode effective index, and propagation distance are investigated by finite element method (FEM) based mode solver using COMSOL. The propagation length is defined as the distance at which the field amplitude drops to 1/e of its initial value. Here the effective mode size along the y direction Lm is defined as the ratio of the total modal energy and the peak energy density along the y direction [2,12]: Z þ1 wm 1 Lm ¼ ¼ wðyÞdy, ð1Þ maxwðyÞ 1 maxwðyÞ where wm and wðyÞ are the total electromagnetic energy and energy density along the y direction, respectively (per unit length along the propagation direction). The normalized mode size is defined as Lm =L0 (L0 ¼ l=2). In the following study, we will vary the thickness of the inner SiO2 layer, L1 , and Si layer, LH , to control the mode size, mode effective index, and propagation length for the hybrid LRSPP mode. Fig. 2(a and b) show the normalized mode size and propagation
Fig. 2. (a) Normalized mode size (Lm =L0 ) and propagation length (b) versus LH , for different L1 . Points (1–4) indicated in (a) correspond to [L1 ,LH ] ¼[20 nm,100 nm] (1), [L1 ,LH ] ¼ [100 nm,100 nm] (2), [L1 ,LH ] ¼ [50 nm,150 nm] (3), and [L1 ,LH ] ¼ [50 nm,300 nm] (4), respectively. The normalized mode size of LRSPP mode in Si–Ag–Si is denoted as the blue solid line in (a). The propagation lengths of LRSPP modes in Si–Ag–Si and SiO2–Ag–SiO2 are denoted as black dashed line and black solid line in (b), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
length versus the Si thickness, LH , for different thickness of the inner SiO2 layer, L1 . It can be seen that, for a small value of L1 , the hybrid structure tends to confine light energy in the gap between the metal and Si layer, which can be reflected by the energy density distribution for [L1 ,LH ]¼[20 nm,100 nm] in Fig. 3(a). At this time, increasing LH has little influence on the mode size [Fig. 2(a)]. It should be noted that, while the mode size for a small value of L1 can be made much smaller than that of the conventional LRSPP mode in homogenous Si medium [Fig. 2(a)], the propagation loss can be comparable [Fig. 2(b)]. However, increasing L1 will lead to a much weaker field confinement; hence, a much larger mode size can be obtained [Fig. 3(e)]. In this situation, mode coupling results in a hybrid mode that features both guidedlike and SPP characteristics, namely, the light energy is distributed in both the Si layer and metal–SiO2 interface [Fig. 3(b, c)]. For large L1 and LH , the hybrid structure is shown to support low-loss guided-like dielectric mode with much light is localized in the Si layer [Fig. 3(d, e)]. It is easily understood that only a small portion of light distributed inside the metal film, inducing a low absorption loss. We have noted, in a recent study on hybrid plasmonic mode in a similar structure [30], the authors systematically analyzed the propagation loss of the symmetrical mode and asymmetrical mode, but the present mode confinement is much weaker than the result in our work. This is because that a much higher thickness of inner-cladding and lower refractive index difference between the inner-cladding and core layer are assumed in that paper. Figure of merit (FoM) has been widely employed to compare and assess various SPP waveguides, which is useful for comparing waveguides in applications where achieving prescribed or small
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Fig. 3. (a)–(d) Electromagnetic energy density distributions for [L1 ,LH ]¼[20 nm,100 nm] (a), [L1 ,LH ]¼ [100 nm,100 nm] (b), [L1 ,LH ]¼[50 nm,150 nm] (c), and [L1 ,LH ]¼ [50 nm,300 nm] (d), corresponding to the points (1–4) indicated in Fig. 2(a). (e) Normalized energy density wðyÞ =wðmÞ along y direction for the hybrid LRSPP waveguides and conventional LRSPP waveguides. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
mode width is important. Here the FoM is defined as [31] FoM ¼
1
az Lm
,
ð2Þ
where az is the imaginary part of the propagation constant. We make a direct comparison between the hybrid LRSPP mode and traditional LRSPP mode in a homogeneous medium. We can see from Fig. 4 that the LRSPP mode in Si–Ag–Si structure shows the worst propagation property because the FoM is the smallest, which can be attributed to the fact that a LRSPP mode in Si–Ag–Si structure suffers a much larger propagation loss than that of a hybrid LRSPP mode. The hybrid LRSPP mode’s FoM is smaller than that of a LRSPP mode in SiO2–Ag–SiO2 structure as LH is below a specific point, but one can increase LH to enhance the FoM. Consequently, the FoM can be made much bigger than that of a LRSPP mode in SiO2–Ag–SiO2 structure.
Fig. 4. The dependence of the figure of merit (FoM) of the hybrid LRSPP modes on LH for different L1 . The corresponding FoM of the LRSPP modes in Si–Ag–Si and SiO2–Ag–SiO2 are denoted as black dashed line and black solid line, respectively.
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Fig. 5. (a) The dependence of the mode effective index of the hybrid LRSPP mode, nhyb , on LH for different L1 . As a comparison, the mode effective index of the pure dielectric waveguides, nd , and LRSPP mode in SiO2–gold–SiO2 waveguides, nI , are depicted in the solid black line and dashed black line, respectively. (b) The mode character derived from Eq. (2).
Due to the coupling between the dielectric waveguide mode and LRSPP mode, the refractive index of the hybrid LRSPP mode is quite different from that of dielectric waveguide mode or LRSPP mode. Fig. 5(a) shows the dependence of the mode effective index of the hybrid LRSPP mode on L1 and LH . Obviously, in the limit of dielectric-like and SPP-like modes, the effective refractive index approaches that of pure dielectric waveguide mode, nd , or conventional LRSPP mode, nI . Besides, the hybrid LRSPP mode’s effective index is always larger than that of the conventional LRSPP mode or dielectric waveguide mode, which can be attributed to the fact that the LRSPP modes are coupled with the dielectric waveguide modes [2]. Furthermore, for a fixed value of LH , the mode effective index of the hybrid LRSPP mode increases with the reduced L1 because that the LRSPP mode couples with dielectric waveguide mode more effectively as L1 reduces. In order to describe the mode characteristics of the hybrid 2 LRSPP mode, the mode character, a þ ðLH ,L1 Þ , is introduced to evaluate the degree to which the hybrid mode is a dielectric waveguide mode or an SPP mode [2]. The hybrid mode is more like an SPP mode for a smaller mode character and a dielectric waveguide mode for a much larger mode character. a þ ðLH ,L1 Þ2 ¼
nhyb ðLH ,L1 ÞnI : ð3Þ ½nhyb ðLH ,L1 Þnd ðLH Þ þ ½nhyb ðLH ,L1 ÞnI 2 Fig. 5(b) shows the dependence of a þ ðLH ,L1 Þ on LH for different L1 . We can see that for a fixed value of L1 , a larger LH leads to a greater mode character, hence the hybrid LRSPP mode is more like a dielectric waveguide mode, and most of light is localized in the Si layer. For a fixed value of LH , the mode character increases with L1 . This means that, by modulating the value of L1 , the hybrid LRSPP mode can be made more like a dielectric waveguide mode with a large mode size or a conventional LRSPP mode with a small mode size. For example, the mode area for L1 ¼ 100 nm is 0.526 times smaller than the diffraction limit [Fig. 3(b)]. By decreasing L1 to 20 nm, mode size of about 0.115 times smaller than the diffraction limit is achievable [Fig. 3(a)]. As for the fabrication of the present structure, an SOI wafer can be purchased on the marked with a SiO2–Si layer stack on the Si substrate, such wafer is obtained by SIMOX (Separation by Implantation of Oxygen) technology. Through rather standard clean-room technology, we can deposit the SiO2 layer, the metal
layer, and the SiO2–Si–SiO2 top layer stack through PECVD, Ebeam evaporation and/or sputtering, respectively. Since the metal layer is very thin and the stripe is very narrow, the thermal stress may not bring in any major issue after an appropriate annealing process. However, the fabrication process is by no means easy since special cares must be taken to relax the thermal stress after deposition.
3. Conclusion In conclusion, we have proposed a hybrid plasmonic waveguide to guide the LRSPP mode with compact modal size and low propagation loss. The ultra-small modal size is achieved due to the coupling between the plasmonic and dielectric waveguide modes. By comparing the hybrid waveguide with the conventional LRSPP waveguide in terms of the modal size and propagation distance, we find that the hybrid waveguide has advantages of guiding LRSPP mode with compact modal size as well as long propagation distance. While it does not seem to have substantial difficulty in fabricating the hybrid waveguide, our proposed structure may serve as promising design for guiding SPPs in nanophotonic circuits.
Acknowledgement Lin Chen is supported by NSFC (Grant No. 11104093), and ‘the Fundamental Research Funds for the Central Universities’, HUST: 2011QN041. Guoping Wang is supported by NSFC (Grant No. 60925020). References [1] W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2003) 824. [2] R.F. Oulton, V.J. Sorger, D.A. Genov, D.F.P. Pile, X. Zhang, Nature Photonics 2 (2008) 496. [3] D. Sarid, Physical Review Letters 47 (1981) 1927. [4] L. Holland, Vacuum Deposition of Thin Films, Chapman and Hall, London, 1966. [5] T. Holmgaard, S.I. Bozhevolnyi, Physical Review B: Condensed Matter 75 (2007) 245405. [6] P. Berini, Advances in Optics and Photonics 1 (2009) 484. [7] R. Salvador, A. Martinez, C. Garcia-Meca, R. Ortuno, J. Marti, IEEE Journal on Selected Topics in Quantum Electronics 14 (2008) 1496.
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