Ultrashort hybrid metal–insulator plasmonic directional coupler Mahmoud Talafi Noghani and Mohammad Hashem Vadjed Samiei* School of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran *Corresponding author: [email protected] Received 12 June 2013; revised 29 September 2013; accepted 4 October 2013; posted 7 October 2013 (Doc. ID 191985); published 24 October 2013
An ultrashort plasmonic directional coupler based on the hybrid metal–insulator slab waveguide is proposed and analyzed at the telecommunication wavelength of 1550 nm. It is first analyzed using the supermode theory based on mode analysis via the transfer matrix method in the interaction region. Then the 2D model of the coupler, including transition arms, is analyzed using a commercial finite-element method simulator. The hybrid slab waveguide is composed of a metallic layer of silver and two dielectric layers of silica (SiO2 ) and silicon (Si). The coupler is optimized to have a minimum coupling length and to transfer maximum power considering the layer thicknesses as optimization variables. The resulting coupling length in the submicrometer region along with a noticeable power transfer efficiency are advantages of the proposed coupler compared to previously reported plasmonic couplers. © 2013 Optical Society of America OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (230.7400) Waveguides, slab; (060.1810) Buffers, couplers, routers, switches, and multiplexers. http://dx.doi.org/10.1364/AO.52.007498
1. Introduction
Plasmonics, as a frontier for the ever-expanding field of photonics, deals with the generation, manipulation, and detection of surface plasmon–polaritons (SPPs) in nanoscaled subwavelength structures. Although subdiffraction transverse dimensions are achieved in plasmonic waveguides, metallic loss is the major challenge. Consequently, many designs are proposed to create a better balance between confinement and loss (C-L) in plasmonic waveguiding structures inclusive of different metal and dielectric arrangements. The proposed waveguide arrangements are insulator–metal–insulator (IMI) [1,2], metal–insulator–metal (MIM) [3,4], or metal– insulator [5]. In this regard, it is shown that hybrid plasmonic structure has a better ability to provide the C-L balance [6–10]. In a hybrid configuration, the dielectric layer next to the metal is decomposed
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into three distinct layers of low index dielectric (L-layer), high index dielectric (H-layer), and cladding. The main feature of such an arrangement is the efficient coupling of the SPP wave at the metal and L-layer interface with the dielectric wave in the H-layer. The hybrid coupled mode provides a mechanism to control the amount of energy penetration into the metal as well as the spatial extent of the wave. Directional couplers (DCs) are among the most applicable devices in optical integrated circuits utilized as power splitters, switches, modulators, etc. Reducing the overall length yet having efficient power transfer is the major goal in the design of DCs. The subwavelength sized nature of plasmonic waveguides makes them hopeful candidates for providing very compact couplers that are several orders of magnitude smaller than dielectric DCs. Different coupling structures based on nonhybrid plasmonic waveguides have been analyzed previously utilizing: IMI-metal stripe [11,12], MIM-slot [13–15], combined IMI–MIM [16], and MI-dielectric-loaded
waveguide [17]. Hybrid plasmonic DCs based on hybrid MI (HMI) waveguides have also been investigated previously [8,18]. To the best of our knowledge, the minimum reported coupling length in previous works is 1 μm (at λ 1550 nm) [16]. In this paper, we propose and analyze a new DC based on the hybrid plasmonic metal–insulator slab waveguide (HMISW), which provides submicrometer coupling lengths with suitable power transfer efficiencies. The supermode theory (SMT) is used to describe the coupling process in the interaction region mathematically, and the transfer matrix method (TMM) is used to calculate the propagation parameters (propagation constants and electromagnetic fields). A 2D model of the coupler that includes the connecting input/output arms is simulated using a commercial finite-element method (FEM) software (COMSOL Multiphysics). The overall performance of the 2Dhybrid DC is promising and it is potentially suitable for application in ultracompact plasmonic integrated circuits. 2. HMI Slab Waveguide Directional Coupler Structure
The hybrid plasmonic slab waveguide (Fig. 1, left) is a four-layer structure composed of a metal (εM ), a low index dielectric (or insulator) (εL ), a high index dielectric (εH ), and a cover layer (εC ) (usually air). It is invariant in the y and z directions. An SPP wave could potentially propagate at the metal/L-layer interface with transverse magnetic (TM) polarization. According to the boundary condition of the normal component of the electric field, high index contrast between the L- and H-layers causes the field to be highly confined in the L-layer, resulting in a transversal compact structure, providing a high degree of confinement and a long propagation length (i.e., low loss). In plasmonic waveguides, the propagation length is the length at which the wave power reduces to 1∕e of its initial value and is calculated by PL 1∕2 Imγ 0 in which γ 0 is the complex propagation constant of the wave. On the other hand, the H-layer can support a guided mode (nonplasmonic) independently. It is shown that by suitably adjusting the layers’ thicknesses, an efficient coupling could be provided between the dielectric mode and the SPP
mode, thus controlling the propagation properties of the waveguide. The hybrid waveguide may be used as a suitable platform for designing various plasmonic devices such as DCs. Figure 1 shows a schematic representation of the proposed hybrid DC. Coupling occurs when two HMISWs are joined together from the metal side resulting in a single (sufficiently thin) metal layer. 3. Method of Analysis
As it is shown in Fig. 1, the DC is composed of two totally identical waveguides. This results in a phase matched coupling system and a total power transfer. Waveguides are assumed to operate in the singlemode region. Two supermodes, a symmetric one and an antisymmetric one, will propagate on the coupled system [Es x; Ea x; H s x, and H a x]. According to the SMT, the total electric field of the coupler could be written as a linear combination of orthogonal supermode fields Ex; z AzEs x BzEa x;
(1)
in which A and B are amplitudes of the symmetrical and antisymmetrical modes, respectively, as follows: Az A0 exp−jγ s z; Bz B0 exp−jγ a z:
(2)
γ s and γ a are the complex propagation constants and are written as γ s βs − jαs and γ a βa − jαa . The electric field distribution of the waveguides may be expressed as 1 E1 x p Es x Ea x; 2 1 E2 x p Es x − Ea x: 2
(3)
The following relation for the total electric field of the coupled system will result: Ex; z CzE1 x DzE2 x:
(4)
If at z 0, the optical power is incident only in waveguide one, C0 1, D0 0, then the power coupled to the second waveguide at an arbitrary distance of z will be 2 1 P2 z jDzj exp−jγ s z − exp−jγ a z 2 αs − αa jβs − βa 2 − z : ≅ exp−αsαa z sinh 2 2 2
(5) Fig. 1. Schematic of the DC based on hybrid metal–insulator slab waveguide.
Assuming βs − βa ∕2 ≫ αs − αa ∕2, Eq. (5) leads to 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
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2 π P2 z ≅ exp − z · sin2 z Lp 2Lc Lc π∕jβs − βa j;
Lp 2∕jαs αa j;
(6)
where Lc is the “coupling length” and Lp is the “mean attenuation length.” Equation (6) describes the transferred power with an error of less than 1% since β is at least 2 orders of magnitude larger than α in HMISWs. It is obvious that the transferred power decreases exponentially with a factor of (αs αa ) and the transfer is not complete. The maximum transferred power (Pmax ) corresponds to a point zmax . They are found by calculating the extrema of Eq. (6) πLp 2 zmax Lc · arctan 2Lc π Pmax
exp−2x arctanx−1 ; 1 x2
(7)
in which x 2Lc ∕πLp . x is normally very small (βs − βa ≫ αs αa ) in plasmonic (as well as hybrid) structures; thus zmax and Pmax become Lc and exp−2Lc ∕Lp , respectively. The complex propagation constants of supermodes, γ s and γ a , completely describe the electric field (and power) distribution in the coupler. Their values may be obtained using the TMM. Based on this method, the dispersion equation of a TM-polarized mode is given as [19] γ sub γc γ γ m11 m22 − m21 sub c m12 0 Fγ −j εsub εc εsub εc (8) with γ sub
q γ 2 − k20 εsub ;
γc
q γ 2 − k20 εc ;
(9)
where γ is the complex propagation constant (γ β − jα), and indices “sub” and “c” refer to substrate and cover, respectively, and mij indicates an element in M T ; the total transfer matrix is given by M T M 1 M 2 …M N " Mn
m11 m21
m12 ; m22
coskn dn
− jεknn sinkn dn
− jkεnn sinkn dn
coskn dn
# ;
(10)
where n 1; …; N and kn
q k20 εn − γ 2 :
(11)
εn and dn refer to dielectric constant and thickness of the nth layer. The roots of the dispersion equation, γ q βq − jαq (m 1; …; Q), are found using a suitable zero searching algorithm. The tangential electric and 7500
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magnetic field distributions in the nth layer corresponding to a given γ q are found as a function of fields at the bottom of that layer by
H yn x
ωε0 εn Ezn x − coskn x − xn − kjn sinkn x − xn −1 −jkn sinkn x − xn coskn x − xn H yn−1 xn ; (12) · ωε0 εn Ezn−1 xn where xn refers to the x coordinate of the bottom of the nth layer. 4. Analysis of HMI Directional Coupler in the Interaction Region
Utilizing the method described in Section 3, the interaction region of the hybrid DC is analyzed at the wavelength of λ0 1550 nm. Silver is used for the metallic layer with a dielectric constant of εAg −127.1 − j3.46 at λ0 . Materials of L, H, and C dielectric layers are silica (SiO2 ), silicon (Si), and air with constants 2.1, 12.1, and 1 correspondingly. A. HMI Slab Waveguide
At the first step and in order to recognize the propagation properties of the HMISW as the building block of the hybrid coupler, it is analyzed by the TMM. Results are shown in Fig. 2. Effective refractive index curves of HMISW for the first two TM-polarized modes [Fig. 2(a)] indicate that for H-layer thicknesses (dH ) up to 200 nm, the waveguide operates in the single-mode region, when L-layer thickness (dL ) is less than 100 nm. On the other hand, the minimum propagation lengths of the fundamental mode for dL 2; 25, and 100 nm are 10, 25, and 62 μm, respectively [Fig. 2(b)]. A suitable DC should have a coupling length much smaller than the propagation length. B. HMI Directional Coupler
Single-mode HMISW is used according to results of Section 4.A. Effective index variation of the symmetrical and antisymmetrical modes versus dM , metal thickness, in the range 20–90 nm is depicted in Fig. 3. Transverse profiles of the E-field major component (Ex ) for the supermodes are also shown in the inset. High confinement of E-field is clearly seen in the L-layer. Figure 4 depicts the coupling length versus H-layer thickness in the range of 20–200 nm when L-layer thicknesses are 2, 25, and 100 nm. Metal thickness is 20 nm. The maximum transferred power is shown in Fig. 5 for the same parameters. A very important factor in design of DCs is to minimize the coupling length. It is seen that Lc versus dH curves take minimum values of 0.46, 1.17, and 2.82 μm when dH 90 nm, dL 2 nm; dH 80 nm, dL 25 nm; and dH 60 nm, dL 100 nm.
2.1 3.5
Antisymmetric Symmetric
2
2
2.5
2
TM0 , dL=2 nm
Ex [V/m]
1.9
Effective Index
Effective Index
3
1.8
0 -2
1.7
-200
0
200 x [nm]
400
70
80
1.6
TM0 , dL=25 nm
1.5
TM0 , dL=100 nm
1.5
TM1 , dL=2 nm
1.4
TM1 , dL=25 nm TM1 , dL=100 nm
1 0
100
200
300
400
500
600
1.3 20
700
30
40
50
90
M
H
(a)
Fig. 3. Effective index of the symmetrical and antisymmetrical modes of the HMI DC at dM 20–90 nm, dL 25 nm, and dH 80 nm. The transverse profile of the E-field major component (Ex ) at dM 70 nm is also depicted in the inset.
300 dL=2 nm dL=25 nm
250
dL=100 nm
200
150
since it leads to shorter coupling length, as shown in Fig. 6. The curves of Figs. 2(b) and 6 emphasize the obvious requirement for coupler function: that coupling length should be very small with respect to propagation length. 5. 2D Analysis of HMI Directional Coupler
50
40
60
80
100
120
140
160
180
200
d [nm] H
(b) Fig. 2. Results of propagation analysis for HMISW. (a) Effective index of the first two TM-polarized modes. (b) Propagation length of the fundamental mode (TM0 ).
The corresponding values for maximum normalized transferred powers are 0.904, 0.905, and 0.903. Regarding the simultaneity of minimum Lc and maximum Pmax (taking the corresponding values of 0.910, 0.911, and 0.905; see Fig. 5), we may consider the coupler as a nearly optimum design. It should be noted that Lc versus dH curves at their minima are relatively insensitive to dH variation. In addition, the same behavior is true for Lc versus dM curves (not indicated in figures). Regarding the priority of coupling length minimization, Lc and Pmax curves versus dM , in the range 20–70 nm, are plotted in Figs. 6 and 7, respectively, for dH 90, 80, and 60 nm and dL 2, 25, and 100 nm. Increasing dM decreases coupling, which leads to an increase in coupling length that means a rise in propagation loss in the coupler or a decrease in Pmax . The maximum transferred power Pmax is relatively insensitive to dL when dH takes the values indicated in Fig. 7. Hence smaller dL is preferred
Schematic representation of the 3D HMI DC is depicted in Fig. 8(a). It is assumed that the coupler has sufficiently large height (in the y direction) so that a 2D analysis (based on the modal analysis of Section 4) is adequate to model the electromagnetic wave propagation in the structure. Figure 8(b) represents the details of the 2D model of the HMI coupler, carrying the wave in the z direction, confining it in the x direction (TM-polarized lateral confinement), and ignoring the y direction due to (quasi-) translational symmetry. The structure includes the 14 dL=2 nm dL=25 nm
12
dL=100 nm
Coupling Length [ µm]
100
0 20
60
d [nm]
d [nm]
Propagation Length [ µm]
8
x 10
10 8 6 4 2 0 20
40
60
80
100
120
140
160
180
200
d [nm] H
Fig. 4. Coupling length versus H-layer thickness for dL 2; 25, and 100 nm at dM 20 nm. 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
7501
0.92
Maximum Trans. Power Ratio
0.9 0.88 0.86 0.84
y
0.82
x
dL=2 nm
z
dL=25 nm
0.8
(a)
dL=100 nm 0.78 20
40
60
80
100
120
140
160
180
200nm
500nm
Interaction Length (L)
200
d [nm] H
Air A
Fig. 5. Maximum transferred power ratio versus H-layer thickness for dL 2; 25, and 100 nm at dM 20 nm.
Input
Through B
Silver
Silica Silicon
100nm
35 dL=2 nm , dH=90 nm
D
dL=25 nm , dH=80 nm
30
Coupling Length [ µm]
dL dL
dM
dH Coupled
Isolated
C
x z
25
(b)
20
Fig. 8. Schematic representation of: (a) 3D and (b) 2D model of HMI DC.
15 10 5 0 20
30
40
50
60
70
d [nm] M
Fig. 6. Coupling length versus metal thickness for optimum values of Fig. 4. 0.95 dL=2 nm , dH=90 nm 0.9
Maximum Trans. Power Ratio
dH
Air
dL=100 nm , dH=60 nm
dL=25 nm , dH=80 nm dL=100 nm , dH=60 nm
0.85 0.8 0.75 0.7 0.65 0.6 0.55 20
30
40
50
60
70
d [nm] M
Fig. 7. Maximum transferred power ratio versus metal thickness for optimum values of Fig. 4. 7502
Silicon Silica
APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013
interaction region (with length L) and transition regions (with length 500 nm), at a distance of 200 nm from input, A and D, and output, B and C, ports. The separation distance of A and D (as well as B and C) ports is selected to be 200 nm to provide sufficient isolation between them, based on the result indicated in Fig. 3 (waveguides are almost decoupled for dM > 100 nm). The interaction region parameters (layer thicknesses dL , dH , and dM ) are chosen according to the results of Section 4. For a submicrometer coupling length of about 600 nm, we must have dL 10 nm, dH 85 nm, and dM 20 nm. The commercial FEM simulation software, COMSOL Multiphysics, is adopted to simulate and analyze the performance of the device. Figure 9 depicts the percentage of transmitted power from the input port (A) to the through (B), coupled (C), and isolated (D) ports as a function of interaction length (L). Simulation results (discrete points) are interpolated to show the sinusoidal behavior of the coupling power. The peak values of each output port decay almost exponentially as predicted by Eq. (6). A maximum power percentage of about 75% is coupled from the first waveguide to the second one when L ≈ 600 nm, which is 5 and 1.7 times shorter than the co-directional and contra-DCs reported in [13] and [16], respectively. A 15% decrease in the coupled power efficiency, with respect to 90% as indicated in Fig. 7, is due to an increase in the total length of the structure, the inclusion of bending
80
60 Transmission (%)
waveguides. In this paper, an ultrashort DC is proposed based on the HMISW and is analyzed theoretically (using SMT and TMM) and numerically (using FEM). It is shown that very short submicrometer coupling lengths (as low as 600 nm) are obtained with suitable power transfer efficiencies (75% and above) and relatively high isolations (>25 dB). Consequently, the hybrid coupler based on HMISW is very advantageous over nonhybrid plasmonic DCs.
Through Coupled Isolated
70
50 40 30 20 10
References
0 0
500 1000 1500 Interaction Length (L) [nm]
2000
Fig. 9. Percentage of the transmitted power to the through, coupled, and isolated ports of the HMI DC as a function of interaction length (L).
2
x 10
5
A'
A'D'
B'
1.8 Input 1.6
D'
B'C'
C'
|E| [V/m]
1.4 1.2
Coupled
1 0.8 0.6 0.4 0.2
Through
Isolated 0 -500
0
500
1000
x [nm] Fig. 10. Electric field distribution (jEj) at the input (A0 D0 ) and output (B0 C0 ) cross sections of the HMI DC (as indicated in the inset) when L ∼ 600 nm.
sections, and the decreasing metallic layer thickness in the transition region near the interaction region, which leads to a decrease in A-D ports isolation. The electric field distribution (jEj) at the input (A0 D0 ) and output (B0 C0 ) cross sections of the coupler is also depicted in Fig. 10, when L ≈ 600 nm. The input electric field distribution is mainly retained at the coupled port, except for an amplitude reduction. Suitable isolation between input and output waveguide pairs is clearly seen. The excess loss (EL) and isolation (Iso), as two common figures of merit for DCs defined by EL −10 LogPB PC ∕PA and Iso −10 LogPD ∕PA (in dB), are 1.04 and 25.12 dB, respectively, at the interaction length of 600 nm, and are reasonable for the proposed ultrashort DC. 6. Conclusion
The hybrid structure is a promising choice for providing better confinement-loss balance in plasmonic
1. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). 2. R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and bound modes of surface plasmon waveguides,” Phys. Rev. B 71, 165431 (2005). 3. R. Zia, M. D. Selker, P. B. Catrysse, and M. I. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2004). 4. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30, 3359–3361 (2005). 5. A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90, 211101 (2007). 6. 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). 7. R. Salvador, A. Martinez, C. G. Meca, and J. Marti, “Analysis of hybrid dielectric plasmonic waveguides,” IEEE J. Sel. Top. Quantum Electron. 14, 1496–1501 (2008). 8. H.-S. Chu, E.-P. Li, P. Bai, and R. Hegde, “Optical performance of single-mode hybrid dielectric-loaded plasmonic waveguidebased components,” Appl. Phys. Lett. 96, 1–3 (2010). 9. M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Theoretical analysis of hybrid plasmonic waveguide,” IEEE J. Sel. Top. Quantum Electron. 19, 4602008 (2013). 10. M. T. Noghani and M. H. V. Samiei, “Analysis and optimum design of hybrid plasmonic slab waveguides,” Plasmonics 8, 1155–1168 (2013). 11. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23, 413–422 (2005). 12. A. Boltasseva and S. I. Bozhevolnyi, “Directional couplers using long-range surface plasmon polariton waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1233–1241 (2006). 13. H. Zhao, X. G. Guang, and J. Huang, “Novel optical directional coupler based on surface plasmon polaritons,” Physica E 40, 3025–3029 (2008). 14. P. Dastmalchi, N. Granpayeh, and M. Rasouli Disfani, “Investigation of coupling length in a semi-cylindrical surface plasmonic coupler,” Appl. Phys. A 103, 741–744 (2011). 15. P. Chen, R. Liang, Q. Huang, Z. Yu, and X. Xu, “Plasmonic filters and optical directional couplers based on wide metalinsulator-metal structure,” Opt. Express 19, 7633–7639 (2011). 16. Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express 14, 413–422 (2006). 17. T. Holmgaard, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Design and characterization of dielectric-loaded plasmonic directional couplers,” J. Lightwave Technol. 27, 5521–5528 (2009). 18. M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “Polarizationindependent hybrid plasmonic coupler for a silicon on insulator platform,” Opt. Lett. 37, 3417–3419 (2012). 19. E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
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An ultrashort plasmonic directional coupler based on the hybrid metal–insulator slab waveguide is proposed and analyzed at the telecommunication wavelength of 1550 nm. It is first analyzed using the supermode theory based on mode analysis via the transfer matrix method in the interaction region. Then the 2D model of the coupler, including transition arms, is analyzed using a commercial finite-element method simulator. The hybrid slab waveguide is composed of a metallic layer of silver and two dielectric layers of silica (SiO2 ) and silicon (Si). The coupler is optimized to have a minimum coupling length and to transfer maximum power considering the layer thicknesses as optimization variables. The resulting coupling length in the submicrometer region along with a noticeable power transfer efficiency are advantages of the proposed coupler compared to previously reported plasmonic couplers. © 2013 Optical Society of America OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (230.7400) Waveguides, slab; (060.1810) Buffers, couplers, routers, switches, and multiplexers. http://dx.doi.org/10.1364/AO.52.007498
1. Introduction
Plasmonics, as a frontier for the ever-expanding field of photonics, deals with the generation, manipulation, and detection of surface plasmon–polaritons (SPPs) in nanoscaled subwavelength structures. Although subdiffraction transverse dimensions are achieved in plasmonic waveguides, metallic loss is the major challenge. Consequently, many designs are proposed to create a better balance between confinement and loss (C-L) in plasmonic waveguiding structures inclusive of different metal and dielectric arrangements. The proposed waveguide arrangements are insulator–metal–insulator (IMI) [1,2], metal–insulator–metal (MIM) [3,4], or metal– insulator [5]. In this regard, it is shown that hybrid plasmonic structure has a better ability to provide the C-L balance [6–10]. In a hybrid configuration, the dielectric layer next to the metal is decomposed
1559-128X/13/317498-06$15.00/0 © 2013 Optical Society of America 7498
APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013
into three distinct layers of low index dielectric (L-layer), high index dielectric (H-layer), and cladding. The main feature of such an arrangement is the efficient coupling of the SPP wave at the metal and L-layer interface with the dielectric wave in the H-layer. The hybrid coupled mode provides a mechanism to control the amount of energy penetration into the metal as well as the spatial extent of the wave. Directional couplers (DCs) are among the most applicable devices in optical integrated circuits utilized as power splitters, switches, modulators, etc. Reducing the overall length yet having efficient power transfer is the major goal in the design of DCs. The subwavelength sized nature of plasmonic waveguides makes them hopeful candidates for providing very compact couplers that are several orders of magnitude smaller than dielectric DCs. Different coupling structures based on nonhybrid plasmonic waveguides have been analyzed previously utilizing: IMI-metal stripe [11,12], MIM-slot [13–15], combined IMI–MIM [16], and MI-dielectric-loaded
waveguide [17]. Hybrid plasmonic DCs based on hybrid MI (HMI) waveguides have also been investigated previously [8,18]. To the best of our knowledge, the minimum reported coupling length in previous works is 1 μm (at λ 1550 nm) [16]. In this paper, we propose and analyze a new DC based on the hybrid plasmonic metal–insulator slab waveguide (HMISW), which provides submicrometer coupling lengths with suitable power transfer efficiencies. The supermode theory (SMT) is used to describe the coupling process in the interaction region mathematically, and the transfer matrix method (TMM) is used to calculate the propagation parameters (propagation constants and electromagnetic fields). A 2D model of the coupler that includes the connecting input/output arms is simulated using a commercial finite-element method (FEM) software (COMSOL Multiphysics). The overall performance of the 2Dhybrid DC is promising and it is potentially suitable for application in ultracompact plasmonic integrated circuits. 2. HMI Slab Waveguide Directional Coupler Structure
The hybrid plasmonic slab waveguide (Fig. 1, left) is a four-layer structure composed of a metal (εM ), a low index dielectric (or insulator) (εL ), a high index dielectric (εH ), and a cover layer (εC ) (usually air). It is invariant in the y and z directions. An SPP wave could potentially propagate at the metal/L-layer interface with transverse magnetic (TM) polarization. According to the boundary condition of the normal component of the electric field, high index contrast between the L- and H-layers causes the field to be highly confined in the L-layer, resulting in a transversal compact structure, providing a high degree of confinement and a long propagation length (i.e., low loss). In plasmonic waveguides, the propagation length is the length at which the wave power reduces to 1∕e of its initial value and is calculated by PL 1∕2 Imγ 0 in which γ 0 is the complex propagation constant of the wave. On the other hand, the H-layer can support a guided mode (nonplasmonic) independently. It is shown that by suitably adjusting the layers’ thicknesses, an efficient coupling could be provided between the dielectric mode and the SPP
mode, thus controlling the propagation properties of the waveguide. The hybrid waveguide may be used as a suitable platform for designing various plasmonic devices such as DCs. Figure 1 shows a schematic representation of the proposed hybrid DC. Coupling occurs when two HMISWs are joined together from the metal side resulting in a single (sufficiently thin) metal layer. 3. Method of Analysis
As it is shown in Fig. 1, the DC is composed of two totally identical waveguides. This results in a phase matched coupling system and a total power transfer. Waveguides are assumed to operate in the singlemode region. Two supermodes, a symmetric one and an antisymmetric one, will propagate on the coupled system [Es x; Ea x; H s x, and H a x]. According to the SMT, the total electric field of the coupler could be written as a linear combination of orthogonal supermode fields Ex; z AzEs x BzEa x;
(1)
in which A and B are amplitudes of the symmetrical and antisymmetrical modes, respectively, as follows: Az A0 exp−jγ s z; Bz B0 exp−jγ a z:
(2)
γ s and γ a are the complex propagation constants and are written as γ s βs − jαs and γ a βa − jαa . The electric field distribution of the waveguides may be expressed as 1 E1 x p Es x Ea x; 2 1 E2 x p Es x − Ea x: 2
(3)
The following relation for the total electric field of the coupled system will result: Ex; z CzE1 x DzE2 x:
(4)
If at z 0, the optical power is incident only in waveguide one, C0 1, D0 0, then the power coupled to the second waveguide at an arbitrary distance of z will be 2 1 P2 z jDzj exp−jγ s z − exp−jγ a z 2 αs − αa jβs − βa 2 − z : ≅ exp−αsαa z sinh 2 2 2
(5) Fig. 1. Schematic of the DC based on hybrid metal–insulator slab waveguide.
Assuming βs − βa ∕2 ≫ αs − αa ∕2, Eq. (5) leads to 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
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2 π P2 z ≅ exp − z · sin2 z Lp 2Lc Lc π∕jβs − βa j;
Lp 2∕jαs αa j;
(6)
where Lc is the “coupling length” and Lp is the “mean attenuation length.” Equation (6) describes the transferred power with an error of less than 1% since β is at least 2 orders of magnitude larger than α in HMISWs. It is obvious that the transferred power decreases exponentially with a factor of (αs αa ) and the transfer is not complete. The maximum transferred power (Pmax ) corresponds to a point zmax . They are found by calculating the extrema of Eq. (6) πLp 2 zmax Lc · arctan 2Lc π Pmax
exp−2x arctanx−1 ; 1 x2
(7)
in which x 2Lc ∕πLp . x is normally very small (βs − βa ≫ αs αa ) in plasmonic (as well as hybrid) structures; thus zmax and Pmax become Lc and exp−2Lc ∕Lp , respectively. The complex propagation constants of supermodes, γ s and γ a , completely describe the electric field (and power) distribution in the coupler. Their values may be obtained using the TMM. Based on this method, the dispersion equation of a TM-polarized mode is given as [19] γ sub γc γ γ m11 m22 − m21 sub c m12 0 Fγ −j εsub εc εsub εc (8) with γ sub
q γ 2 − k20 εsub ;
γc
q γ 2 − k20 εc ;
(9)
where γ is the complex propagation constant (γ β − jα), and indices “sub” and “c” refer to substrate and cover, respectively, and mij indicates an element in M T ; the total transfer matrix is given by M T M 1 M 2 …M N " Mn
m11 m21
m12 ; m22
coskn dn
− jεknn sinkn dn
− jkεnn sinkn dn
coskn dn
# ;
(10)
where n 1; …; N and kn
q k20 εn − γ 2 :
(11)
εn and dn refer to dielectric constant and thickness of the nth layer. The roots of the dispersion equation, γ q βq − jαq (m 1; …; Q), are found using a suitable zero searching algorithm. The tangential electric and 7500
APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013
magnetic field distributions in the nth layer corresponding to a given γ q are found as a function of fields at the bottom of that layer by
H yn x
ωε0 εn Ezn x − coskn x − xn − kjn sinkn x − xn −1 −jkn sinkn x − xn coskn x − xn H yn−1 xn ; (12) · ωε0 εn Ezn−1 xn where xn refers to the x coordinate of the bottom of the nth layer. 4. Analysis of HMI Directional Coupler in the Interaction Region
Utilizing the method described in Section 3, the interaction region of the hybrid DC is analyzed at the wavelength of λ0 1550 nm. Silver is used for the metallic layer with a dielectric constant of εAg −127.1 − j3.46 at λ0 . Materials of L, H, and C dielectric layers are silica (SiO2 ), silicon (Si), and air with constants 2.1, 12.1, and 1 correspondingly. A. HMI Slab Waveguide
At the first step and in order to recognize the propagation properties of the HMISW as the building block of the hybrid coupler, it is analyzed by the TMM. Results are shown in Fig. 2. Effective refractive index curves of HMISW for the first two TM-polarized modes [Fig. 2(a)] indicate that for H-layer thicknesses (dH ) up to 200 nm, the waveguide operates in the single-mode region, when L-layer thickness (dL ) is less than 100 nm. On the other hand, the minimum propagation lengths of the fundamental mode for dL 2; 25, and 100 nm are 10, 25, and 62 μm, respectively [Fig. 2(b)]. A suitable DC should have a coupling length much smaller than the propagation length. B. HMI Directional Coupler
Single-mode HMISW is used according to results of Section 4.A. Effective index variation of the symmetrical and antisymmetrical modes versus dM , metal thickness, in the range 20–90 nm is depicted in Fig. 3. Transverse profiles of the E-field major component (Ex ) for the supermodes are also shown in the inset. High confinement of E-field is clearly seen in the L-layer. Figure 4 depicts the coupling length versus H-layer thickness in the range of 20–200 nm when L-layer thicknesses are 2, 25, and 100 nm. Metal thickness is 20 nm. The maximum transferred power is shown in Fig. 5 for the same parameters. A very important factor in design of DCs is to minimize the coupling length. It is seen that Lc versus dH curves take minimum values of 0.46, 1.17, and 2.82 μm when dH 90 nm, dL 2 nm; dH 80 nm, dL 25 nm; and dH 60 nm, dL 100 nm.
2.1 3.5
Antisymmetric Symmetric
2
2
2.5
2
TM0 , dL=2 nm
Ex [V/m]
1.9
Effective Index
Effective Index
3
1.8
0 -2
1.7
-200
0
200 x [nm]
400
70
80
1.6
TM0 , dL=25 nm
1.5
TM0 , dL=100 nm
1.5
TM1 , dL=2 nm
1.4
TM1 , dL=25 nm TM1 , dL=100 nm
1 0
100
200
300
400
500
600
1.3 20
700
30
40
50
90
M
H
(a)
Fig. 3. Effective index of the symmetrical and antisymmetrical modes of the HMI DC at dM 20–90 nm, dL 25 nm, and dH 80 nm. The transverse profile of the E-field major component (Ex ) at dM 70 nm is also depicted in the inset.
300 dL=2 nm dL=25 nm
250
dL=100 nm
200
150
since it leads to shorter coupling length, as shown in Fig. 6. The curves of Figs. 2(b) and 6 emphasize the obvious requirement for coupler function: that coupling length should be very small with respect to propagation length. 5. 2D Analysis of HMI Directional Coupler
50
40
60
80
100
120
140
160
180
200
d [nm] H
(b) Fig. 2. Results of propagation analysis for HMISW. (a) Effective index of the first two TM-polarized modes. (b) Propagation length of the fundamental mode (TM0 ).
The corresponding values for maximum normalized transferred powers are 0.904, 0.905, and 0.903. Regarding the simultaneity of minimum Lc and maximum Pmax (taking the corresponding values of 0.910, 0.911, and 0.905; see Fig. 5), we may consider the coupler as a nearly optimum design. It should be noted that Lc versus dH curves at their minima are relatively insensitive to dH variation. In addition, the same behavior is true for Lc versus dM curves (not indicated in figures). Regarding the priority of coupling length minimization, Lc and Pmax curves versus dM , in the range 20–70 nm, are plotted in Figs. 6 and 7, respectively, for dH 90, 80, and 60 nm and dL 2, 25, and 100 nm. Increasing dM decreases coupling, which leads to an increase in coupling length that means a rise in propagation loss in the coupler or a decrease in Pmax . The maximum transferred power Pmax is relatively insensitive to dL when dH takes the values indicated in Fig. 7. Hence smaller dL is preferred
Schematic representation of the 3D HMI DC is depicted in Fig. 8(a). It is assumed that the coupler has sufficiently large height (in the y direction) so that a 2D analysis (based on the modal analysis of Section 4) is adequate to model the electromagnetic wave propagation in the structure. Figure 8(b) represents the details of the 2D model of the HMI coupler, carrying the wave in the z direction, confining it in the x direction (TM-polarized lateral confinement), and ignoring the y direction due to (quasi-) translational symmetry. The structure includes the 14 dL=2 nm dL=25 nm
12
dL=100 nm
Coupling Length [ µm]
100
0 20
60
d [nm]
d [nm]
Propagation Length [ µm]
8
x 10
10 8 6 4 2 0 20
40
60
80
100
120
140
160
180
200
d [nm] H
Fig. 4. Coupling length versus H-layer thickness for dL 2; 25, and 100 nm at dM 20 nm. 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
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0.92
Maximum Trans. Power Ratio
0.9 0.88 0.86 0.84
y
0.82
x
dL=2 nm
z
dL=25 nm
0.8
(a)
dL=100 nm 0.78 20
40
60
80
100
120
140
160
180
200nm
500nm
Interaction Length (L)
200
d [nm] H
Air A
Fig. 5. Maximum transferred power ratio versus H-layer thickness for dL 2; 25, and 100 nm at dM 20 nm.
Input
Through B
Silver
Silica Silicon
100nm
35 dL=2 nm , dH=90 nm
D
dL=25 nm , dH=80 nm
30
Coupling Length [ µm]
dL dL
dM
dH Coupled
Isolated
C
x z
25
(b)
20
Fig. 8. Schematic representation of: (a) 3D and (b) 2D model of HMI DC.
15 10 5 0 20
30
40
50
60
70
d [nm] M
Fig. 6. Coupling length versus metal thickness for optimum values of Fig. 4. 0.95 dL=2 nm , dH=90 nm 0.9
Maximum Trans. Power Ratio
dH
Air
dL=100 nm , dH=60 nm
dL=25 nm , dH=80 nm dL=100 nm , dH=60 nm
0.85 0.8 0.75 0.7 0.65 0.6 0.55 20
30
40
50
60
70
d [nm] M
Fig. 7. Maximum transferred power ratio versus metal thickness for optimum values of Fig. 4. 7502
Silicon Silica
APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013
interaction region (with length L) and transition regions (with length 500 nm), at a distance of 200 nm from input, A and D, and output, B and C, ports. The separation distance of A and D (as well as B and C) ports is selected to be 200 nm to provide sufficient isolation between them, based on the result indicated in Fig. 3 (waveguides are almost decoupled for dM > 100 nm). The interaction region parameters (layer thicknesses dL , dH , and dM ) are chosen according to the results of Section 4. For a submicrometer coupling length of about 600 nm, we must have dL 10 nm, dH 85 nm, and dM 20 nm. The commercial FEM simulation software, COMSOL Multiphysics, is adopted to simulate and analyze the performance of the device. Figure 9 depicts the percentage of transmitted power from the input port (A) to the through (B), coupled (C), and isolated (D) ports as a function of interaction length (L). Simulation results (discrete points) are interpolated to show the sinusoidal behavior of the coupling power. The peak values of each output port decay almost exponentially as predicted by Eq. (6). A maximum power percentage of about 75% is coupled from the first waveguide to the second one when L ≈ 600 nm, which is 5 and 1.7 times shorter than the co-directional and contra-DCs reported in [13] and [16], respectively. A 15% decrease in the coupled power efficiency, with respect to 90% as indicated in Fig. 7, is due to an increase in the total length of the structure, the inclusion of bending
80
60 Transmission (%)
waveguides. In this paper, an ultrashort DC is proposed based on the HMISW and is analyzed theoretically (using SMT and TMM) and numerically (using FEM). It is shown that very short submicrometer coupling lengths (as low as 600 nm) are obtained with suitable power transfer efficiencies (75% and above) and relatively high isolations (>25 dB). Consequently, the hybrid coupler based on HMISW is very advantageous over nonhybrid plasmonic DCs.
Through Coupled Isolated
70
50 40 30 20 10
References
0 0
500 1000 1500 Interaction Length (L) [nm]
2000
Fig. 9. Percentage of the transmitted power to the through, coupled, and isolated ports of the HMI DC as a function of interaction length (L).
2
x 10
5
A'
A'D'
B'
1.8 Input 1.6
D'
B'C'
C'
|E| [V/m]
1.4 1.2
Coupled
1 0.8 0.6 0.4 0.2
Through
Isolated 0 -500
0
500
1000
x [nm] Fig. 10. Electric field distribution (jEj) at the input (A0 D0 ) and output (B0 C0 ) cross sections of the HMI DC (as indicated in the inset) when L ∼ 600 nm.
sections, and the decreasing metallic layer thickness in the transition region near the interaction region, which leads to a decrease in A-D ports isolation. The electric field distribution (jEj) at the input (A0 D0 ) and output (B0 C0 ) cross sections of the coupler is also depicted in Fig. 10, when L ≈ 600 nm. The input electric field distribution is mainly retained at the coupled port, except for an amplitude reduction. Suitable isolation between input and output waveguide pairs is clearly seen. The excess loss (EL) and isolation (Iso), as two common figures of merit for DCs defined by EL −10 LogPB PC ∕PA and Iso −10 LogPD ∕PA (in dB), are 1.04 and 25.12 dB, respectively, at the interaction length of 600 nm, and are reasonable for the proposed ultrashort DC. 6. Conclusion
The hybrid structure is a promising choice for providing better confinement-loss balance in plasmonic
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