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Analysis and design of hybrid ARROW-B plasmonic waveguides Shruti,* R. K. Sinha, and R. Bhattacharyya TIFAC-Centre of Relevance and Excellence in Fiber Optics and Optical Communication, Department of Applied Physics, Delhi Technological University (formerly Delhi College of Engineering, Faculty of Technology, University of Delhi), Bawana Road, Delhi 110042, India *Corresponding author: [email protected] Received February 26, 2013; revised June 11, 2013; accepted June 15, 2013; posted June 20, 2013 (Doc. ID 185952); published July 15, 2013 A hybrid antiresonant reflecting waveguide, type B (ARROW-B) plasmonic waveguide based on the resonant coupling between a guided dielectric mode and surface plasmon polariton wave is proposed. Employing the finite element method, hybrid modes including two bound supermodes are obtained at visible frequencies by varying the environmental refractive index. We investigate the propagation characteristics of hybrid modes, where the significant change of modal power by the symmetric bound mode is observed in plasmonic waveguide coupling suitable for highly sensitive detection of bulk refractive index change. Further, anomalous dispersion is shown by the antisymmetric bound mode which leads to large group velocity dispersion of −3.165 × 104 ps∕km nm and, thus, makes this hybrid plasmonic waveguide ideal for observation of soliton generation. © 2013 Optical Society of America OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (230.7370) Waveguides. http://dx.doi.org/10.1364/JOSAA.30.001502
1. INTRODUCTION The possibility of guiding electromagnetic waves beyond the diffraction limit has resulted in an unprecedented interest in the field of plasmonics. Plasmonic waveguides based on surface plasmon polaritons (SPPs) has gained unremitting attention due their ability of guiding optical signals in a deep subwavelength scale for the realization of miniaturization of high density photonic integrated circuits [1–3]. The SPPs are transverse magnetic (TM) electromagnetic surface waves resulting from an interaction of an electromagnetic wave with conduction electrons at the interface of metal and a dielectric [4,5] provided the media involved here have opposite signs of the real part of dielectric constants. Electromagnetic energy can be guided in these waveguides with subwavelength dimension by converting the optical mode into a nonradiative surface plasmon mode. In these plasmonic waveguides, there is an enhancement of mode field concentration on the metal surface that makes the SPP sensitive to the dielectric refractive index change in contact with the metal surface and, hence, essential for realizing nonlinear functionalities and sensing [6–8]. In recent years, remarkable progress has been made on the research and development of plasmonic waveguides such as metallic nanosphere chain waveguides [9,10], metallic wire, stripe and slab waveguides [11,12], channel or wedge plasmonic waveguide [13–15], plasmonic slot waveguide [16], dielectric-loaded SPP waveguide (DLSPPW) [17,18], silver (Ag) nanorod-based waveguides for plasmonic circuitry [19], etc. However, despite providing excellent mode confinement, these plasmonic waveguides suffer large propagation loss at visible and near infrared wavelengths. In order to circumvent this aforementioned issue, hybrid waveguiding schemes with the 1084-7529/13/081502-06$15.00/0
suitable choice of plasmon and index guiding are found to be promising since it may provide the best compromise between the compactness and low loss [20–23]. Recently, studies on hybrid waveguides have been proliferating and many active hybrid plasmonic-waveguide-based devices have already been demonstrated, both theoretically [24,25] and experimentally [26]. However, it is required to give further focus to the inbuilt issues associated with device compactness, efficient coupling of hybrid plasmonic waveguides with single mode fiber, and propagation loss. In view of this, an antiresonant reflection guiding mechanism is utilized in type B, antiresonant reflecting waveguide (ARROW-B) [27,28], in contrast to an index guiding mechanism in the vertical direction, to perform low-loss single mode propagation in an integrated plasmonic waveguide and to improve the coupling efficiency with a single mode fiber [29,30]. In this paper, we report the analysis and design of a hybrid ARROW-B plasmonic waveguide based on resonant interaction between a guided mode of an ARROW-B and an SPP wave. Using the finite element method (FEM), the propagation characteristics of hybrid modes including two purely bound SPP modes namely, symmetric bound (sb ) mode and antisymmetric bound (ab ) modes, are presented. It is also observed that, on varying the superstrate index, the antisymmetric bound (ab ) mode is weakly dependent on superstrate index (na ), however, at a particular (na ) and subsequently at a higher (na ), modal transition from hybrid mode to purely bound symmetric bound (sb ) mode takes place. The mode evolution is also demonstrated via plasmonic waveguide mode coupling by observing the significant transfer of modal power of a symmetric mode at a particular superstrate index. Further, the antisymmetric bound mode exhibits large © 2013 Optical Society of America
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anamolous group velocity dispersion (GVD) of −3.165 × 104 ps∕km nm ideal for observation of soliton generation.
2. WAVEGUIDE CONFIGURATION A. ARROW-B Plasmonic Waveguide A basic ARROW-B slab waveguide consists of three layers: a core layer with a high refractive index nc and thickness dc , a first cladding layer with a low refractive index n1 and thickness d1 , and a second cladding layer with high refractive index n2 and thickness d2 , respectively. The two cladding thickness should be chosen appropriately to satisfy low-loss and quasisingle mode propagation. The first cladding thickness is so thin that evanescent field reaches the second cladding and frustrated total internal reflection (FTIR) at the interface of core and cladding occurs. The second cladding thickness is chosen by the antiresonant condition [27,28] d2 ≅
−1 2 λ n2 λ2 1 − c2 2 2 ; 4n2 n2 4n2 dce
(1)
where dce is the equivalent thickness of the core. Typically, the refractive index of the second cladding is chosen as that of the core, i.e., (nc n2 ), hence, Eq. (1) can be reduced to d2
dce 2N 1; 2
N 0; 1; 2…:
When the thickness of the second cladding is half of the core, the reflection for higher-order modes become remarkably lower than that of the fundamental mode and the single mode propagation is virtually realized. [27,28]. An ARROW-B waveguide is polarization insensitive and, thus, suitable for applications such as optical interconnects [27]. However, as the SPP supports only TM modes, we focus only on the fundamental TM modes of ARROW-B structure. A schematic diagram of the proposed design of a hybrid ARROW-B plasmonic waveguide is presented in Fig. 1(a), where a thin metal film of Ag is placed on the top of the rib core of an ARROW-B waveguide. In the design of the rib core hybrid ARROW-B structure, the rib width w of ARROW-B waveguide is optimized to be w 4 μm allowing a single mode operation at λ 0.6328 μm and the rib height
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is chosen to be dr 0.85 μm. We have chosen parameters for the core and second cladding layer as nc n2 1.54 and d2 1.0 μm. The refractive index of the first cladding layer is taken as n1 1.378. The propagation loss characteristics versus the first cladding thickness (d1 ) is shown in Fig. 1(b) and it can be clearly seen that loss of each mode decreases monotonically. We have chosen d1 0.5 μm at λ 0.6328 μm as losses for the fundamental TM mode (TM0 ) were below 1 dB∕cm as shown in Fig. 1(b). The wavelength is set to be λ 0.6328 μm, corresponding to the refractive index of Ag as (0.0645 i4.02) [31] and the thickness of the Ag layer dm is set to be 40 nm for the efficient coupling of dielectric mode and SPP mode. The bound surface plasmon modes have been extensively studied and thoroughly discussed in the literature [32–34]. It is well known that for a thin metal film embedded in a symmetric dielectric slab waveguide, the SPPs on the upper and lower metal-dielectric interfaces couple and support two supermodes, namely symmetric bound (sb ) and antisymmetric bound (ab ) modes. The symmetric mode, with its field extending into surrounding dielectrics, has large beam width, thereby exhibiting long propagation lengths and, hence, is known as the long-range SPP (LRSPP) mode, whereas the antisymmetric mode, with its field tightly bound to the metal surface provides subwavelength confinement with short propagation lengths and, hence, is known as the short-range SPP (SRSPP) mode. However, in the asymmetric slab waveguide, where the dielectric below and above the metal film are different with respect to the horizontal dimension, the symmetric modes have a symmetric-like distribution with field localization along the metal-dielectric interface with the lowest dielectric constant, while the antisymmetric modes that have antisymmetric-like distribution have a field localized with the highest dielectric constant [34–36]. It is to be mentioned that in this paper, we have considered the “bound” modes as nonradiative SPP modes, and hence, the subscript “b” is added to the nomenclature of the various modes.
3. RESULTS AND DISCUSSION A. Modal Analysis of Hybrid ARROW-B Plasmonic Waveguide The hybrid ARROW-B plasmonic waveguide is analyzed using two-dimensional (2D) FEM simulation at the operating
Fig. 1. (a) Schematic diagram of the hybrid ARROW-B plasmonic waveguide. (b) Propagation loss of ARROW-B plasmonic waveguide versus first cladding thickness for first three TM modes.
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Fig. 2. (a) Transverse electric field distribution (E y ) of antisymmetric bound mode (ab ) at superstrate index na 1.5 at λ 0.6328 μm. (b) Schematic representation of (Ey ) field distribution for ab mode.
wavelength λ 0.6328 μm. For all the simulations, the nonuniform grid size is approximately 30 nm as an average. In order to visualize the mode evolution and understand the field distribution of the modes at different superstrate refractive index na , we present the exact 2D transverse electric field distribution (Ey ) of the lowest order plasmon mode (TM0 ) and higher-order hybrid modes (TM1 and (TM2 ) of the proposed waveguide, which exhibits both features of the guided mode of a dielectric waveguide and surface plasmon mode at λ 0.6328 μm for the variation of superstrate refractive index na . The transverse electric field profile of the lowest order (TM0 ) mode, known as antisymmetric (ab ) mode, at na 1.5 is shown in Fig. 2(a). It depicts that the penetration of the electric field is small in the dielectric and exhibits a zero transverse field within the metal film jE y j 0 as shown in the field profile of (ab ) mode in Fig. 2(b), thereby providing strong and subwavelength confinement at the rib core of the waveguide. The transverse electric field distribution (E y ) for the modal transformation of the higher-order hybrid modes (TM1 and TM2 ) of the plasmonic waveguide at λ 0.6328 μm are shown in Figs. 3(a), 3(b), 3(e), and 3(f). When, na 1.42, these modes (TM2 ) behave very similar to a conventional ARROW-B mode, which are also supported by the waveguide without the Ag overlayer. With increasing na , the character of TM2 mode undergoes modal transformation and it changes from that of a similar guided mode to that of a symmetric SPP mode (TM1 ). The corresponding field profiles of the evolved modes of the structure are shown in Figs. 3(c), 3(d), 3(g), and 3(h), which clearly show that the introduction of the metal on the rib core of the waveguide and the boundary condition at the metal surface effectively attract the TM radiation mode at the interface of metal and core. At a higher refractive index of na , (above 1.46), the TM1 mode exhibits features of a symmetric bound (sb ) surface plasmon mode. It can be seen that the (sb ) mode at na 1.5 in Fig. 3(h) shows a symmetric behavior of the electric field with respect to the central axis, i.e., jEy j ≠ 0 with its field penetration large in the dielectric with respect to the metal layer. Also, transverse electric field distribution (E y ) along with its schematic of the hybrid mode at na 1.46, where the coupling takes place is shown in Figs. 3(e) and 3(g). To study the propagation characteristics of lowest order (TM0 ) plasmonic and higher-order hybrid modes (TM1 and
TM2 ), we obtained the mode effective index and the propagation lengths of these modes. The measure of the propagation loss in a waveguide is the propagation length L, which is due to the power absorption by the metal and is governed by the imaginary part of the neff , which is given as L
λ : 4πImneff
The variation of real (neff ) and propagation length L of (ab ) modes and hybrid plasmonic modes with respect to the variation of superstrate refractive index na from 1.3 to 1.55 is shown in the Figs. 4(a) and 4(b). In the presence of thin metal film on the rib core of the waveguide, for na 1.3–1.45, the hybrid mode (TM2 ) is supported by the structure that is similar to the conventional TM mode of ARROW-B waveguide structure with the effective index, below the core refractive index. However, as na is increased, at the turning point, na 1.46, mode evolution takes place and at the higher na , the hybrid mode (TM2 ) becomes a nonradiative symmetric bound (sb ) mode (TM1 ). The effective index of (sb ) mode is larger than the core index as depicted in Fig. 4(a), representing the maximum coupling efficiency of waveguide to plasmon coupling. As clearly seen in Fig. 3(b), the electromagnetic field is strongly confined in the vicinity of the interface and from Fig. 4(b), it shows a sharp decrease in propagation length as superstrate index varies from na 1.3 to 1.55. However, the lowest order (TM0 ), which is an antisymmetric bound (ab ) mode, is always present and is weakly affected by na with its effective index, much large than effective index of (sb ) mode. Its fields are tightly bound to the metal, which reasserts and validates the electric field distribution (E y ) in Fig. 2(a) and, thus, exhibits subwavelength confinement. Another consequence of the field localized in metal leads to short propagation lengths (∼2 μm) for (ab ) mode as shown in Fig. 4(b). The modal transformation of the waveguide mode to plasmon mode can also be studied by observing the power transmission characteristics. The optical mode to plasmonic mode coupling is analyzed for the (sb ) mode in the proposed hybrid ARROW-B waveguide. The normalized output power (P out ) as a function of hybrid ARROW-B waveguide length for different superstrate index is shown in Fig. 5.
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Fig. 3. Transverse electric field distribution (Ey ) for the modal transformation of the hybrid modes at superstrate index (a) na 1.42, (b) na 1.44, (e) na 1.46, and (f) na 1.5 at λ 0.6328 μm. (c), (d), (g), and (h) Schematic representation of (Ey ) field distribution of the modal transformation of hybrid modes.
As depicted in Fig. 5, when na 1.4, there is no contribution to the symmetric mode (sb ) to the power coupling as the energy nearly passes through the waveguide with minimum
loss. This can also be verified for the waveguide mode having real neff < 1.54 as demonstrated in Fig. 4(a) at na 1.4. However, when na 1.46, we can observe the evolution
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Fig. 6. Variation of GVD of antisymmetric (ab ) mode as a function of superstrate index na at λ 0.6328 μm.
Fig. 4. (a) Variation of real (neff ) and (b) propagation lengths of symmetric bound (sb ) mode (TM1 ) and antisymmetric bound (ab ) mode (TM0 ) of hybrid ARROW-B plasmonic waveguide for varying superstrate index.
and excitation of (sb ) mode and the maximum amplitude of the transmitted optical power exhibits damped behavior along the propagation length of the waveguide. The damping behavior of the transmitted power is determined by the average propagation loss, which is given by the decrease in power level per micrometer. It is decided by the proportion of energy
Fig. 5. Normalized output power of symmetric bound (sb ) mode versus the interaction length of the waveguide for na 1.4 and 1.46.
coupled to the high loss metal arm and the propagation loss of the (sb ) mode for na 1.46 is calculated to be 0.016 dB∕μm. The performance parameters such as sensitivity to the external influence of superstrate index na effectively depend on the phase and group velocity of the supermodes. In Fig. 6, the variation of GVD as a function of superstrate index is shown for the antisymmetric bound (ab ) mode at wavelength, λ 0.6328 μm. For na 1.46, the strong field confinement dominates over the material dispersion and leads to an inversion of GVD for the (ab ) mode as clearly depicted in Fig. 6. This is because of the strong increase of the group index as the mode evolves from a hybrid mode to nonradiative SPP mode while being propagated in the plasmonic waveguide. As is evident from the graph in Fig. 6, the maximum anomalous GVD of −3.165 × 104 ps∕km nm is reached for the superstrate index 1.46 due to the spectral variation of the mode with respect to refractive index. Such a large value of anomalous GVD with weak higher-order dispersion makes this waveguide ideal for observation of soliton formation. It may also find potential applications as integrated plasmonic devices.
4. CONCLUSION A hybrid ARROW-B plasmonic waveguide is investigated and hybrid modes, including two superbound modes, are obtained and analyzed at the wavelength λ 0.6328 μm. The 2D modal electric field distributions of the hybrid modes as well as symmetric and antisymmetric bound supermodes are demonstrated using FEM. We investigate the propagation characteristics of the hybrid modes, including two supermodes, where the subwavelength confinement is exhibited by antisymmetric bound (ab ) mode that leads to highly confined waveguides for on-chip optical elements. The significant change of modal power by symmetric bound mode (sb ) is observed in plasmonic waveguide-mode coupling for high sensitivity detection of bulk refractive index change. Further, anomalous dispersion is shown by the antisymmetric bound mode that leads to large GVD of −3.165 × 104 ps∕km nm and, thus, makes this hybrid plasmonic waveguide ideal for soliton generation, indicating inherent potential to design highly compact integrated plasmonicwaveguide-based devices.
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ACKNOWLEDGMENTS The authors gratefully acknowledge (i) the initiatives and support toward establishment of TIFAC Centre of Relevance and Excellence in Fiber Optics and Optical Communication at Delhi College of Engineering, Delhi through Mission Reach program of Technology Vision 2020, Government of India and (ii) UGC-sponsored major research project in the area of photonic crystal fibers for sensing and telecom applications.
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17. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long range propagation,” Nat. Photonics 2, 496–500 (2008). 18. H. Chu, Y. A. Akimov, P. Bai, and E.-P. Li, “Hybrid dielectric loaded plasmonic waveguide and wavelength selective components for efficiently controlling light at subwavelength scale,” J. Opt. Soc. Am. B 28, 2895–2901 (2011). 19. V. Dillu, Shruti, T. Srivastava, and R. K. Sinha, “Propagation characteristics of silver nanorods based compact waveguides for plasmonic circuitry,” Phys. E 48, 75–79 (2013). 20. R. Slavador, A. Martinez, C. Garcia-Meca, R. Ortuno, and J. Marti, “Analysis of hybrid dielectric plasmonic waveguides,” IEEE J. Quantum Electron. 14, 1496–1501 (2008). 21. D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009). 22. M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Propagation characteristics of hybrid modes supported by metal-low-high index waveguides and bends,” Opt. Express 18, 12971–12979 (2010). 23. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10, 105018 (2008). 24. J. Grandidier, G. Colas des Francs, L. Markey, A. Bouhelier, S. Massenot, J.-C. Weeber, and A. Dereux, “Dielectric-loaded surface plasmonic polariton waveguide on a finite-width metal strip,” Appl. Phys. Lett. 96, 063105 (2010). 25. L. Chen, X. Li, and G. Wang, “A hybrid long-range plasmonic waveguide with subwavelength confinement,” Opt. Commun. 291, 400–404 (2013). 26. V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011). 27. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguidesnumerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). 28. Shruti, R. K. Sinha, and R. Bhattacharyya, “Anti-resonant reflecting photonic crystal waveguides (ARRPCW): modeling and design,” Opt. Quantum Electron. 41, 181–187 (2009). 29. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2 -Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). 30. T. Baba and Y. Kokubun, “New polarization-insensitive antiresonant reflecting optical waveguide,” IEEE Photon. Technol. Lett. 1, 232–234 (1989). 31. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985). 32. E. N. Economou, “Surface plasmon in thin films,” Phys. Rev. 182, 539–554 (1969). 33. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981). 34. J. J. Burke and G. I. Stegeman, “Surface-polariton-like waves guided by thin lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986). 35. P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express 7 329–335 (2000). 36. P. Berini, “Long-range surface plasmon polariton,” Adv. Opt. Photon. 1, 484–588 (2009).
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Analysis and design of hybrid ARROW-B plasmonic waveguides Shruti,* R. K. Sinha, and R. Bhattacharyya TIFAC-Centre of Relevance and Excellence in Fiber Optics and Optical Communication, Department of Applied Physics, Delhi Technological University (formerly Delhi College of Engineering, Faculty of Technology, University of Delhi), Bawana Road, Delhi 110042, India *Corresponding author: [email protected] Received February 26, 2013; revised June 11, 2013; accepted June 15, 2013; posted June 20, 2013 (Doc. ID 185952); published July 15, 2013 A hybrid antiresonant reflecting waveguide, type B (ARROW-B) plasmonic waveguide based on the resonant coupling between a guided dielectric mode and surface plasmon polariton wave is proposed. Employing the finite element method, hybrid modes including two bound supermodes are obtained at visible frequencies by varying the environmental refractive index. We investigate the propagation characteristics of hybrid modes, where the significant change of modal power by the symmetric bound mode is observed in plasmonic waveguide coupling suitable for highly sensitive detection of bulk refractive index change. Further, anomalous dispersion is shown by the antisymmetric bound mode which leads to large group velocity dispersion of −3.165 × 104 ps∕km nm and, thus, makes this hybrid plasmonic waveguide ideal for observation of soliton generation. © 2013 Optical Society of America OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (230.7370) Waveguides. http://dx.doi.org/10.1364/JOSAA.30.001502
1. INTRODUCTION The possibility of guiding electromagnetic waves beyond the diffraction limit has resulted in an unprecedented interest in the field of plasmonics. Plasmonic waveguides based on surface plasmon polaritons (SPPs) has gained unremitting attention due their ability of guiding optical signals in a deep subwavelength scale for the realization of miniaturization of high density photonic integrated circuits [1–3]. The SPPs are transverse magnetic (TM) electromagnetic surface waves resulting from an interaction of an electromagnetic wave with conduction electrons at the interface of metal and a dielectric [4,5] provided the media involved here have opposite signs of the real part of dielectric constants. Electromagnetic energy can be guided in these waveguides with subwavelength dimension by converting the optical mode into a nonradiative surface plasmon mode. In these plasmonic waveguides, there is an enhancement of mode field concentration on the metal surface that makes the SPP sensitive to the dielectric refractive index change in contact with the metal surface and, hence, essential for realizing nonlinear functionalities and sensing [6–8]. In recent years, remarkable progress has been made on the research and development of plasmonic waveguides such as metallic nanosphere chain waveguides [9,10], metallic wire, stripe and slab waveguides [11,12], channel or wedge plasmonic waveguide [13–15], plasmonic slot waveguide [16], dielectric-loaded SPP waveguide (DLSPPW) [17,18], silver (Ag) nanorod-based waveguides for plasmonic circuitry [19], etc. However, despite providing excellent mode confinement, these plasmonic waveguides suffer large propagation loss at visible and near infrared wavelengths. In order to circumvent this aforementioned issue, hybrid waveguiding schemes with the 1084-7529/13/081502-06$15.00/0
suitable choice of plasmon and index guiding are found to be promising since it may provide the best compromise between the compactness and low loss [20–23]. Recently, studies on hybrid waveguides have been proliferating and many active hybrid plasmonic-waveguide-based devices have already been demonstrated, both theoretically [24,25] and experimentally [26]. However, it is required to give further focus to the inbuilt issues associated with device compactness, efficient coupling of hybrid plasmonic waveguides with single mode fiber, and propagation loss. In view of this, an antiresonant reflection guiding mechanism is utilized in type B, antiresonant reflecting waveguide (ARROW-B) [27,28], in contrast to an index guiding mechanism in the vertical direction, to perform low-loss single mode propagation in an integrated plasmonic waveguide and to improve the coupling efficiency with a single mode fiber [29,30]. In this paper, we report the analysis and design of a hybrid ARROW-B plasmonic waveguide based on resonant interaction between a guided mode of an ARROW-B and an SPP wave. Using the finite element method (FEM), the propagation characteristics of hybrid modes including two purely bound SPP modes namely, symmetric bound (sb ) mode and antisymmetric bound (ab ) modes, are presented. It is also observed that, on varying the superstrate index, the antisymmetric bound (ab ) mode is weakly dependent on superstrate index (na ), however, at a particular (na ) and subsequently at a higher (na ), modal transition from hybrid mode to purely bound symmetric bound (sb ) mode takes place. The mode evolution is also demonstrated via plasmonic waveguide mode coupling by observing the significant transfer of modal power of a symmetric mode at a particular superstrate index. Further, the antisymmetric bound mode exhibits large © 2013 Optical Society of America
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anamolous group velocity dispersion (GVD) of −3.165 × 104 ps∕km nm ideal for observation of soliton generation.
2. WAVEGUIDE CONFIGURATION A. ARROW-B Plasmonic Waveguide A basic ARROW-B slab waveguide consists of three layers: a core layer with a high refractive index nc and thickness dc , a first cladding layer with a low refractive index n1 and thickness d1 , and a second cladding layer with high refractive index n2 and thickness d2 , respectively. The two cladding thickness should be chosen appropriately to satisfy low-loss and quasisingle mode propagation. The first cladding thickness is so thin that evanescent field reaches the second cladding and frustrated total internal reflection (FTIR) at the interface of core and cladding occurs. The second cladding thickness is chosen by the antiresonant condition [27,28] d2 ≅
−1 2 λ n2 λ2 1 − c2 2 2 ; 4n2 n2 4n2 dce
(1)
where dce is the equivalent thickness of the core. Typically, the refractive index of the second cladding is chosen as that of the core, i.e., (nc n2 ), hence, Eq. (1) can be reduced to d2
dce 2N 1; 2
N 0; 1; 2…:
When the thickness of the second cladding is half of the core, the reflection for higher-order modes become remarkably lower than that of the fundamental mode and the single mode propagation is virtually realized. [27,28]. An ARROW-B waveguide is polarization insensitive and, thus, suitable for applications such as optical interconnects [27]. However, as the SPP supports only TM modes, we focus only on the fundamental TM modes of ARROW-B structure. A schematic diagram of the proposed design of a hybrid ARROW-B plasmonic waveguide is presented in Fig. 1(a), where a thin metal film of Ag is placed on the top of the rib core of an ARROW-B waveguide. In the design of the rib core hybrid ARROW-B structure, the rib width w of ARROW-B waveguide is optimized to be w 4 μm allowing a single mode operation at λ 0.6328 μm and the rib height
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is chosen to be dr 0.85 μm. We have chosen parameters for the core and second cladding layer as nc n2 1.54 and d2 1.0 μm. The refractive index of the first cladding layer is taken as n1 1.378. The propagation loss characteristics versus the first cladding thickness (d1 ) is shown in Fig. 1(b) and it can be clearly seen that loss of each mode decreases monotonically. We have chosen d1 0.5 μm at λ 0.6328 μm as losses for the fundamental TM mode (TM0 ) were below 1 dB∕cm as shown in Fig. 1(b). The wavelength is set to be λ 0.6328 μm, corresponding to the refractive index of Ag as (0.0645 i4.02) [31] and the thickness of the Ag layer dm is set to be 40 nm for the efficient coupling of dielectric mode and SPP mode. The bound surface plasmon modes have been extensively studied and thoroughly discussed in the literature [32–34]. It is well known that for a thin metal film embedded in a symmetric dielectric slab waveguide, the SPPs on the upper and lower metal-dielectric interfaces couple and support two supermodes, namely symmetric bound (sb ) and antisymmetric bound (ab ) modes. The symmetric mode, with its field extending into surrounding dielectrics, has large beam width, thereby exhibiting long propagation lengths and, hence, is known as the long-range SPP (LRSPP) mode, whereas the antisymmetric mode, with its field tightly bound to the metal surface provides subwavelength confinement with short propagation lengths and, hence, is known as the short-range SPP (SRSPP) mode. However, in the asymmetric slab waveguide, where the dielectric below and above the metal film are different with respect to the horizontal dimension, the symmetric modes have a symmetric-like distribution with field localization along the metal-dielectric interface with the lowest dielectric constant, while the antisymmetric modes that have antisymmetric-like distribution have a field localized with the highest dielectric constant [34–36]. It is to be mentioned that in this paper, we have considered the “bound” modes as nonradiative SPP modes, and hence, the subscript “b” is added to the nomenclature of the various modes.
3. RESULTS AND DISCUSSION A. Modal Analysis of Hybrid ARROW-B Plasmonic Waveguide The hybrid ARROW-B plasmonic waveguide is analyzed using two-dimensional (2D) FEM simulation at the operating
Fig. 1. (a) Schematic diagram of the hybrid ARROW-B plasmonic waveguide. (b) Propagation loss of ARROW-B plasmonic waveguide versus first cladding thickness for first three TM modes.
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Fig. 2. (a) Transverse electric field distribution (E y ) of antisymmetric bound mode (ab ) at superstrate index na 1.5 at λ 0.6328 μm. (b) Schematic representation of (Ey ) field distribution for ab mode.
wavelength λ 0.6328 μm. For all the simulations, the nonuniform grid size is approximately 30 nm as an average. In order to visualize the mode evolution and understand the field distribution of the modes at different superstrate refractive index na , we present the exact 2D transverse electric field distribution (Ey ) of the lowest order plasmon mode (TM0 ) and higher-order hybrid modes (TM1 and (TM2 ) of the proposed waveguide, which exhibits both features of the guided mode of a dielectric waveguide and surface plasmon mode at λ 0.6328 μm for the variation of superstrate refractive index na . The transverse electric field profile of the lowest order (TM0 ) mode, known as antisymmetric (ab ) mode, at na 1.5 is shown in Fig. 2(a). It depicts that the penetration of the electric field is small in the dielectric and exhibits a zero transverse field within the metal film jE y j 0 as shown in the field profile of (ab ) mode in Fig. 2(b), thereby providing strong and subwavelength confinement at the rib core of the waveguide. The transverse electric field distribution (E y ) for the modal transformation of the higher-order hybrid modes (TM1 and TM2 ) of the plasmonic waveguide at λ 0.6328 μm are shown in Figs. 3(a), 3(b), 3(e), and 3(f). When, na 1.42, these modes (TM2 ) behave very similar to a conventional ARROW-B mode, which are also supported by the waveguide without the Ag overlayer. With increasing na , the character of TM2 mode undergoes modal transformation and it changes from that of a similar guided mode to that of a symmetric SPP mode (TM1 ). The corresponding field profiles of the evolved modes of the structure are shown in Figs. 3(c), 3(d), 3(g), and 3(h), which clearly show that the introduction of the metal on the rib core of the waveguide and the boundary condition at the metal surface effectively attract the TM radiation mode at the interface of metal and core. At a higher refractive index of na , (above 1.46), the TM1 mode exhibits features of a symmetric bound (sb ) surface plasmon mode. It can be seen that the (sb ) mode at na 1.5 in Fig. 3(h) shows a symmetric behavior of the electric field with respect to the central axis, i.e., jEy j ≠ 0 with its field penetration large in the dielectric with respect to the metal layer. Also, transverse electric field distribution (E y ) along with its schematic of the hybrid mode at na 1.46, where the coupling takes place is shown in Figs. 3(e) and 3(g). To study the propagation characteristics of lowest order (TM0 ) plasmonic and higher-order hybrid modes (TM1 and
TM2 ), we obtained the mode effective index and the propagation lengths of these modes. The measure of the propagation loss in a waveguide is the propagation length L, which is due to the power absorption by the metal and is governed by the imaginary part of the neff , which is given as L
λ : 4πImneff
The variation of real (neff ) and propagation length L of (ab ) modes and hybrid plasmonic modes with respect to the variation of superstrate refractive index na from 1.3 to 1.55 is shown in the Figs. 4(a) and 4(b). In the presence of thin metal film on the rib core of the waveguide, for na 1.3–1.45, the hybrid mode (TM2 ) is supported by the structure that is similar to the conventional TM mode of ARROW-B waveguide structure with the effective index, below the core refractive index. However, as na is increased, at the turning point, na 1.46, mode evolution takes place and at the higher na , the hybrid mode (TM2 ) becomes a nonradiative symmetric bound (sb ) mode (TM1 ). The effective index of (sb ) mode is larger than the core index as depicted in Fig. 4(a), representing the maximum coupling efficiency of waveguide to plasmon coupling. As clearly seen in Fig. 3(b), the electromagnetic field is strongly confined in the vicinity of the interface and from Fig. 4(b), it shows a sharp decrease in propagation length as superstrate index varies from na 1.3 to 1.55. However, the lowest order (TM0 ), which is an antisymmetric bound (ab ) mode, is always present and is weakly affected by na with its effective index, much large than effective index of (sb ) mode. Its fields are tightly bound to the metal, which reasserts and validates the electric field distribution (E y ) in Fig. 2(a) and, thus, exhibits subwavelength confinement. Another consequence of the field localized in metal leads to short propagation lengths (∼2 μm) for (ab ) mode as shown in Fig. 4(b). The modal transformation of the waveguide mode to plasmon mode can also be studied by observing the power transmission characteristics. The optical mode to plasmonic mode coupling is analyzed for the (sb ) mode in the proposed hybrid ARROW-B waveguide. The normalized output power (P out ) as a function of hybrid ARROW-B waveguide length for different superstrate index is shown in Fig. 5.
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Fig. 3. Transverse electric field distribution (Ey ) for the modal transformation of the hybrid modes at superstrate index (a) na 1.42, (b) na 1.44, (e) na 1.46, and (f) na 1.5 at λ 0.6328 μm. (c), (d), (g), and (h) Schematic representation of (Ey ) field distribution of the modal transformation of hybrid modes.
As depicted in Fig. 5, when na 1.4, there is no contribution to the symmetric mode (sb ) to the power coupling as the energy nearly passes through the waveguide with minimum
loss. This can also be verified for the waveguide mode having real neff < 1.54 as demonstrated in Fig. 4(a) at na 1.4. However, when na 1.46, we can observe the evolution
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Fig. 6. Variation of GVD of antisymmetric (ab ) mode as a function of superstrate index na at λ 0.6328 μm.
Fig. 4. (a) Variation of real (neff ) and (b) propagation lengths of symmetric bound (sb ) mode (TM1 ) and antisymmetric bound (ab ) mode (TM0 ) of hybrid ARROW-B plasmonic waveguide for varying superstrate index.
and excitation of (sb ) mode and the maximum amplitude of the transmitted optical power exhibits damped behavior along the propagation length of the waveguide. The damping behavior of the transmitted power is determined by the average propagation loss, which is given by the decrease in power level per micrometer. It is decided by the proportion of energy
Fig. 5. Normalized output power of symmetric bound (sb ) mode versus the interaction length of the waveguide for na 1.4 and 1.46.
coupled to the high loss metal arm and the propagation loss of the (sb ) mode for na 1.46 is calculated to be 0.016 dB∕μm. The performance parameters such as sensitivity to the external influence of superstrate index na effectively depend on the phase and group velocity of the supermodes. In Fig. 6, the variation of GVD as a function of superstrate index is shown for the antisymmetric bound (ab ) mode at wavelength, λ 0.6328 μm. For na 1.46, the strong field confinement dominates over the material dispersion and leads to an inversion of GVD for the (ab ) mode as clearly depicted in Fig. 6. This is because of the strong increase of the group index as the mode evolves from a hybrid mode to nonradiative SPP mode while being propagated in the plasmonic waveguide. As is evident from the graph in Fig. 6, the maximum anomalous GVD of −3.165 × 104 ps∕km nm is reached for the superstrate index 1.46 due to the spectral variation of the mode with respect to refractive index. Such a large value of anomalous GVD with weak higher-order dispersion makes this waveguide ideal for observation of soliton formation. It may also find potential applications as integrated plasmonic devices.
4. CONCLUSION A hybrid ARROW-B plasmonic waveguide is investigated and hybrid modes, including two superbound modes, are obtained and analyzed at the wavelength λ 0.6328 μm. The 2D modal electric field distributions of the hybrid modes as well as symmetric and antisymmetric bound supermodes are demonstrated using FEM. We investigate the propagation characteristics of the hybrid modes, including two supermodes, where the subwavelength confinement is exhibited by antisymmetric bound (ab ) mode that leads to highly confined waveguides for on-chip optical elements. The significant change of modal power by symmetric bound mode (sb ) is observed in plasmonic waveguide-mode coupling for high sensitivity detection of bulk refractive index change. Further, anomalous dispersion is shown by the antisymmetric bound mode that leads to large GVD of −3.165 × 104 ps∕km nm and, thus, makes this hybrid plasmonic waveguide ideal for soliton generation, indicating inherent potential to design highly compact integrated plasmonicwaveguide-based devices.
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ACKNOWLEDGMENTS The authors gratefully acknowledge (i) the initiatives and support toward establishment of TIFAC Centre of Relevance and Excellence in Fiber Optics and Optical Communication at Delhi College of Engineering, Delhi through Mission Reach program of Technology Vision 2020, Government of India and (ii) UGC-sponsored major research project in the area of photonic crystal fibers for sensing and telecom applications.
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