Triple-core collinear and noncollinear plasmonic photonic crystal fiber couplers Susobhan Das, Raktim Haldar, and Shailendra K. Varshney* Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India *Corresponding author: [email protected] Received 26 July 2013; revised 22 October 2013; accepted 25 October 2013; posted 28 October 2013 (Doc. ID 194718); published 22 November 2013
Numerical analysis of single and multiple gold nanowires embedded in triple cores arranged in collinear and noncollinear configurations in photonic crystal fibers (PCFs) is reported. A full-vectorial finite element method is used to achieve coupling characteristics of plasmonic PCF couplers for both x and y polarizations. It is demonstrated numerically that the PCF plasmonic couplers exhibit polarizationindependent tunable broadband filter characteristics that can be tuned according to the diameter of the embedded gold rod(s). © 2013 Optical Society of America OCIS codes: (060.5295) Photonic crystal fibers; (240.6680) Surface plasmons; (060.1810) Buffers, couplers, routers, switches, and multiplexers. http://dx.doi.org/10.1364/AO.52.008199
1. Introduction
The field of plasmonics has triggered new hopes and new device concepts because of their ability to confine strongly the light at the subwavelength scale [1], which is not possible using dielectric waveguides due to the diffraction limit [2]. Plasmonic devices like sensors, power dividers, and frequency splitters are the foremost applications [3–6]. The plasmons are the collective oscillations of electron gas density, which can be excited either along a metal–dielectric interface or in bulk materials (metals and semiconductors). The plasmon at the metal–dielectric interface is referred to as the surface plasmon (SP), which turns into a surface plasmon polariton (SPP) when it is coupled to externally applied electromagnetic waves. Several approaches, such as Otto configuration and the Kretschmann–Reather attenuated total reflection (ATR) method, have been employed to excite the SPs on a metal–dielectric interface. The dielectric medium can be air, planar waveguide structures, or optical fibers. Among these, optical 1559-128X/13/348199-06$15.00/0 © 2013 Optical Society of America
fiber has been a great host to incorporate metallic nanowires, trading optical fiber as plasmonic devices like polarizers, etc. [7]. With continuing development in optical fiber research, photonic crystal fibers (PCFs) show great potential to act as hosts for metallic wires, as the cladding in a PCF is comprised of air channels running down its length. The presence of air holes in the cladding makes them versatile platforms for infiltrating with liquids, gases, semiconductors, and metals [8–12]. Metals such as gold and silver have been used to fill either a single air hole or multiple air holes at different locations. This gives rise to interesting plasmonic behavior of so-called plasmonic PCF. By removing one or many air holes from selective locations in the cladding of PCFs, we can turn the PCF to operate as a coupler [13–15]. In the recent past, the coupling characteristics of plasmonic PCFs have been investigated numerically [16–20] where the polarization-dependent behavior and transmission characteristics were addressed. Conventional color optical filter works under the principle of reflection and transmission. Depending upon the behavior of the filter, the required wavelength band is reflected or transmitted. The bandwidth of such color optical 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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filters is very narrow and strongly dependent on material characteristics. One can also achieve narrow bandwidth optical filters based on optical fiber Bragg gratings [21]. In this paper, we numerically demonstrate a new approach to achieve both band rejection and bandpass filter characteristics in a single fiber. A basic physical phenomenon of the proposed idea is based on the plasmon interaction with the fundamental guided core mode of a PCF selectively filled with gold. The rigorous calculation of SPP modes can be approximated by a spiraling plasmon model, where the phase matching condition must satisfy to generate SPs along the gold rod. If εd and εm are the electric permittivity of dielectric and metal, then the phase-matching condition to excite the SPP can be written as [22] 2πakt 2πm − 1;
(1)
where k2t k2SPP − β2 ; k0 2π∕λ;
β k0 nSPP ;
kSPP
r εd εm : k0 εd εm
(2.1)
(2.2)
(2.3)
Using Eqs. (2.1)–(2.3) into Eq. (1), the axial refractive index for the SPP mode can be given as
nSPP
s εd εm λm − 1 2 − ; εd εm 2πa
2. Single-Core Plasmonic PCF with Dual Nanowires
The schematic of a single-core PCF geometry with two selectively filled gold wires of different diameters is shown in Fig. 1(a), where Λ is the separation between two air holes, d is the air-hole diameter, and d1 , d2 are the diameters of the two gold wires, respectively. The two gold wires are kept far apart from each other in such a way that an interaction takes place between the guided core mode and the SPP modes surrounding the gold wire without any interaction between the gold wires. A full-vectorial finite element method (FEM)based solver (COMSOL Multiphysics) is used to compute the plasmon-like and dielectric-like modes in selectively gold-filled PCF structures. To validate our approach, we have verified the results of PCF with gold nanowire as given in [20]. After validation, we proceeded with our proposed structure. For modal analysis, we have considered a PCF with Λ 2.3 μm and d 1 μm, while the diameter of the gold rods were varied as d1 0.4 μm and d2 1 μm. A simulation was carried out for a wavelength range of 600–1600 nm. Figure 1(b) shows the dispersion diagrams and corresponding loss spectra. The solid red and blue curves denote the effective index of the fundamental guided dielectric mode of PCF for both x and y polarizations. The green dashed curve with filled blue circles shows the dispersion of the SPP mode of second order (m 2) for the isolated excitation of different radii of gold wires embedded in the PCF silica matrix. Both the dielectric guided mode and SPP mode interact at a particular wavelength, which is called the “resonance” where both the fundamental guided mode and the SPP mode exist simultaneously. The SPP resonance mode occurs when the phase matching condition is satisfied. This
(3)
where a is the radius of the gold rod, and m ≥ 1 is the order of the SP mode, which holds true for higherorder modes. We consider silica as a dielectric medium whose refractive index is considered according to the Sellmeier equation [23]. The best-fitted analytical model [24] for the wavelength-dependent permittivity of gold is given by εAu λ ε∞ −
λ2p
1 1 λ2
X Aj λ j1;2 j
λγi p
! eiϕj e−iϕj 1 1 i ; 1 1 i λj − λ − γ j λj λ γ j
(4)
where ε∞ 1.54, λp 143 nm, γ p 14500 nm, A1 1.27, A2 1.1, ϕ1 ϕ2 −π∕4, λ1 470 nm, λ2 325 nm, γ 1 1900 nm, and γ 2 1060 nm are the parameters. 8200
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Fig. 1. (a) Cross-sectional view of single-core PCF geometry with two gold wires of different diameters, d1 0.4 μm and d2 1 μm, whereas the air-hole diameter d 1 μm and pitch constant Λ 2.3 μm. (b) Dispersion diagrams and corresponding loss spectra for both x and y polarizations of the PCF filled with two different gold rods. The loss peaks correspond to 693 and 1070 nm. (c) Snapshots of the normalized electric field distribution at the resonance condition as mentioned.
gives rise to a strong coupling between the SPP and guided modes. A sudden discontinuity on the dispersion curve is noticed. Since the SPP modes are highly lossy, the corresponding loss curves indicate high peaks at resonance wavelengths, as depicted in the lower portion of Fig. 1(b). The snapshots of the normalized electric field distribution at the resonance wavelengths for both x and y polarizations are displayed in Fig. 1(c). It can be interpreted that at the resonance condition, the maximum electric field of the fundamental guided core mode strongly couples to the second-order SPP mode for both polarizations. The resonance wavelengths for d1 0.4 μm and d2 1 μm are 693 and 1070 nm, respectively. We notice that the resonance wavelength is dependent on the size of the gold rod. If we further include gold rods in the PCF geometry such that no gold rods interact with each other, one would expect n number of resonances for n number of gold rods for a particular SPP mode. 3. Triple-Core PCF Coupler with Single/Multiple Gold Rod(s)
In this section, we examine the effect of plasmonic interaction in a three-core PCF coupler. Previously, dual-core plasmonic PCF couplers have been reported [17]. The power coupling characteristic between two cores with coupling length and transmission for the x and y polarization was investigated. Here we are interested in dividing the power in different cores. A three-core PCF is the platform to fulfill the requirement and the proper arrangements of the cores provide the particular power-coupling ratio. We have chosen two types of configurations, namely, collinear and noncollinear, which depend on the position of the dielectric cores from the plasmonic rod(s). We have achieved filtering characteristics of the proposed device along with power splitting functionality. A.
Collinear Configuration
The cross-sectional view of a three-core collinear PCF coupler with two gold rods of diameter d1 and d2 is exhibited in Fig. 2(a). The blue shaded regions adjacent to the gold rods (shown by filled red circles) constitute three dielectric cores. For three collinear cores, six types of supermodes, as shown in Fig. 2(b), will exist according to different polarization orientations. We have used the following parameters to
Fig. 2. (a) Cross-sectional view of the three-core collinear PCF configuration with two gold rods and (b) six possible supermodes with different orientation of the electric field polarization.
obtain the dispersion diagrams and loss characteristics of the coupled system: d 1 μm, d1 0.4 μm, d2 0.6 μm, and Λ 2.3 μm. We have assumed the wavelength band from 650 to 850 nm for the sake of the computations only. Figure 3 shows the dispersion for the x and y polarizations of the electric field with the corresponding losses. The phase-matching conditions for both gold nanowires fulfill at different orientations of the electric fields for the x and y polarizations. Two resonances occur at 693 and 825 nm, corresponding to the d1 and d2 diameters of the gold rod, respectively. Within the range of the wavelength, only the SPP mode (m 2) couples with the fundamental guided core mode. From the dispersion diagrams and loss characteristics, it is observed that only two orientations of the electric field for both x and y polarizations excite the SP. At a 693 nm wavelength, x-polarized β3 and y-polarized β2 supermodes interact with the SPP mode of second order, which is due to the small gold rod (d1 ). Similarly, at an 825 nm wavelength, the x-polarized β2 and y-polarized β3 supermodes interact with the second-order SPP mode due to the larger gold rod size. At resonance wavelengths, the dielectric-like fundamental mode suffers high losses, as evident from the solid black and blue curves in Fig. 3. At a resonance condition, for a particular polarization, there is almost no electric field in the core region, as displayed in Fig. 4(a). By combining the losses for different orientations of electric fields, the total transmission loss for the fiber of length 0.1 mm is calculated and depicted in Fig. 4(b). Transmission losses for the x and y polarizations are different and can be used as the polarization-independent bandpass or band-rejection filter, depending on the wavelength of operation. For example, if the wavelength of the operation is near the resonance wavelength where a deep notch is observed in the transmission loss, then the device can be used as a band-rejection filter, whereas for other wavelengths, the same can be used as a bandpass filter. It can be revealed from the plots
Fig. 3. Dispersion and loss spectrum for a three-core collinear PCF with two gold rods for the (a) x and (b) y polarizations. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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wavelength region. The normalized power coupling characteristics as a function of z∕Lc of the device at a 760 nm wavelength is exhibited in Fig. 4(c), where Lc is the coupling length. B. Noncollinear Configuration
Fig. 4. (a) Snapshots of normalized electric field distribution, (b) transmission loss spectrum of a 0.1 mm long fiber, and (c) the normalized power variation at 760 nm wavelength, a function of z∕Lc in a three-collinear core PCF coupler with two gold rods, where Lc is the coupling length.
that two notches, as shown in Fig. 4(b), can be utilized to reject certain bands of wavelengths. The notch depth can be enhanced by taking a longer length of the fiber. On the other hand, we can see that the same fiber can also be employed to pass a certain wavelength band with a 3 dB bandwidth of ∼100 nm. It is clearly observed that the filter characteristics remain polarization independent in both cases. Figure 4(c) shows the power coupling characteristics of the device that were calculated using the coupled mode theory [25] for the arrangements of three collinear identical cores. The coupled mode equations for the collinear arrangement can be written as ∂a1 −jβa1 − jκa2 ; ∂z ∂a2 −jβa2 − jκa1 a3 ; ∂z ∂a3 −jβa3 − jκa2 ; ∂z
The schematic cross section of a three-noncollinearcore PCF coupler with a single gold rod is displayed in Fig. 5(a). Three cores are designed in such a way that they are 120° apart from the center of the PCF structure denoted by 1, 2, and 3. The gold rod is placed at the center of the fiber, as shown by the red-filled circle. The architecture resembles a satellite-core PCF. For simulation purposes, we took an air hole diameter d 1 μm, pitch constant Λ 2.3 μm, and the gold nanowire diameter was the same as the air-hole diameter. A cartoon picture of the possible supermodes with field orientations is shown in Fig. 5(b). Figure 6 shows the dispersion diagrams and loss characteristics of a three-noncollinear-core PCF coupler with a single gold rod for different electric field orientations. We find that the SPP modes of third and second orders are excited in this configuration
Fig. 5. Cross-sectional view of a three-noncollinear-core PCF configuration with a single gold rod, d 1 μm, Λ 2.3 μm.
(5)
where a1 , a2 , and a3 are the real electric field amplitudes of core 1, core 2, and core 3, respectively, β is the propagation constant of the dielectric-like mode for each core, and κ is the coupling coefficient between any two cores. The coupling coefficient is given by κpq
R∞ R∞ 2 2 ωε0 −∞ −∞ N − N q Ep × Eq dxdy R R ∞ ∞ ; −∞ −∞ uz × Ep × H p Ep × H p dxdy
(6)
where Ep , H p are the electric and magnetic field at core p, and Eq is the electric field at core q. The coupling coefficient κpq is calculated according to the given expression and [26]. Within the bandpass region from the central wavelength around 760 nm, the coupling coefficient varies like 0.56 0.3 mm−1 , which remains nearly constant for the bandpass 8202
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Fig. 6. Dispersion diagram and loss spectra for a threenoncollinear-core PCF coupler with a single gold rod for the (a) x and (b) y polarizations. The PCF parameters are d 1 μm and Λ 2.3 μm. The gold rod diameter d1 is the same as the air-hole diameter.
Fig. 7. (a) Snapshots of normalized electric field distribution at resonance wavelengths, showing the SPP modes of the third and second orders. (b) Transmission loss spectrum for x and y polarizations; the device acts as a band rejection filter of 100 nm 3 dB bandwidth. (c) Normalized power variation at 860 nm wavelength in three cores as a function of z∕Lc , where Lc is the coupling length.
at different wavelengths. At a 750 nm wavelength, x-polarized β3 and y-polarized β2 supermodes interact with the SPP mode of third order. Similarly, at a 1050 nm wavelength, the x-polarized β2 and y-polarized β3 supermodes interact with the secondorder SPP mode. The corresponding peaks are observed in the loss spectra as dictated by the solid blue and black curves in Fig. 6. In Fig. 7(a), the snapshots of the normalized electric field distribution are displayed, where we can see the excitation and interaction of the SPP modes of third and second orders with the particular guided supermode as indicated by the red arrows. The resonance wavelengths are 750 and 1050 nm, respectively, for third-order and second-order SPP modes. Next, we investigate the transmission loss and normalized power variation in the proposed device. The transmission loss in the 0.1-mm-long fiber that accounts for the loss contribution from all supermodes is shown in Fig. 7(b). We observe two notches corresponding to two different SPP modes. We can see that for both x and y polarizations, the transmission notches are the same at the resonant frequencies. If we further include more gold rods in PCF geometry, then polarization dependence may occur [7]. However, in our case, we find that the device is polarization independent. The 3 dB bandwidth for this band-rejection filter is ∼100 nm. The normalized power variation as a function of z∕Lc is calculated using coupled mode equations, where Lc is the coupling length as defined before. The coupled mode equations for three noncollinear identical cores can be written as ∂a1 −jβa1 − jκa2 a3 ; ∂z ∂a2 −jβa2 − jκa1 a3 ; ∂z ∂a3 −jβa3 − jκa1 a2 ; ∂z
(7)
where a1 , a2 , and a3 are the real electric field amplitude of core 1, core 2, and core 3, respectively, β is the propagation constant of the dielectric-like mode for each core, and κ is the coupling coefficient between any two cores. The coupling coefficient at the 860 nm wavelength is 0.82 mm−1 . The power coupling characteristics at 860 nm (i.e., the wavelength in the bandpass region) for the threenoncollinear-core PCF is shown in Fig. 7(c) as a function of z∕Lc. The power coupling characteristics are obtained by solving the coupled mode equations numerically using the Runge–Kutta algorithm with the unit input excitation at core 1. Note that we have considered the losses due to the presence of a gold rod. The power in cores 2 and 3 are equal and decreases as a function of the length, contrary to what we see in a sinusoidal variation in the absence of a gold rod. 4. Conclusion and Discussion
To conclude our work, we have numerically investigated in detail the plasmonic interaction in PCFs with triple cores arranged in collinear and noncollinear configurations with single and multiple gold rods. It has been shown that the air-hole cladding acts as a platform to confine the light beyond the diffraction limit due to the presence of gold wires embedded in the air channels. Through numerical simulations based on the FEM, we have achieved distinctive features such as bandpass, band-rejection filter characteristics along with the power division capabilities of the proposed plasmonic PCF coupler devices. We have examined the gold rod size dependency on the SP resonance frequency. Through proper selection of the gold rod diameter, it is possible to tune the filter response at a particular wavelength and hence the bandwidth. Collinear structure can be used as both a bandpass and band-rejection filter. For a broadband source, it behaves as a bandpass filter with bandwidth of ∼100 nm. The presence of a single notch in the loss spectrum suggests its utility as a band rejection too with a low rejection bandwidth. The noncollinear PCF structure provides broadband rejection filter characteristics with ∼100 nm bandwidth. It was noticed that the devices were polarization independent, which would alleviate their usage. References 1. L. Novotny and B. Hecht, Principle of Nano-Optics (Cambridge University, 2006). 2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2006). 3. A. V. Krasavin and A. V. Zayats, “Electro-optic switching element for dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 97, 041107 (2010). 4. F. Bilotti, S. Tricarico, and L. Vegni, “Plasmonic metamaterial cloaking at optical frequencies,” IEEE Trans. Nanotechnol. 9, 55–61 (2010). 5. A. Hassani and M. Skorobogatiy, “Design criteria for microstructured-optical-fiber-based surface-plasmon-resonance sensors,” J. Opt. Soc. Am. B 24, 1423–1429 (2007). 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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6. C. H. Chen and K. S. Liao, “1×N plasmonic power splitters based on metal-insulator-metal waveguides,” Opt. Express 21, 4036–4043 (2013). 7. A. Nagasaki, K. Saitoh, and M. Koshiba, “Polarization characteristics of photonic crystal fibers selectively filled with metal wires into cladding air holes,” Opt. Express 19, 3799–3808 (2011). 8. P. J. A. Sazio, A. A. Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D.-J. Won, F. Zhang, E. R. Margine, V. Gopalan, V. H. Crespi, and J. V. Badding, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311, 1583–1586 (2006). 9. H. Tyagi, M. Schmidt, L. P. Sempere, and P. Russell, “Optical properties of photonic crystal fiber with integral micron-sized Ge wire,” Opt. Express 16, 17227–17236 (2008). 10. H. Tyagi, H. Lee, P. Uebel, M. Schmidt, N. Joly, M. Scharrer, and P. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35, 2573–2575 (2010). 11. G. Ren, P. Shum, X. Yu, J. J. Hu, G. Wang, and Y. Gong, “Polarization dependent guiding in liquid crystal filled photonic crystal fibers,” Opt. Commun. 281, 1598–1606 (2008). 12. H. W. Lee, M. A. Schmidt, R. F. Russell, N. Y. Joly, H. K. Tyagi, P. Uebel, and P. St. J. Russell, “Pressure-assisted melt-filling and optical characterization of Au nanowires in microstructured fibers,” Opt. Express 19, 12180–12189 (2011). 13. K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express 11, 3188–3195 (2003). 14. K. Saitoh, Y. Sato, and M. Koshiba, “Polarization splitter in three-core photonic crystal fibers,” Opt. Express 12, 3940–3946 (2004). 15. S. K. Varshney, K. Saitoh, R. K. Sinha, and M. Koshiba, “Coupling characteristics of multicore photonic crystal
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fiber-based 1×4 power splitters,” J. Lightwave Technol. 27, 2062–2068 (2009). S. Zhang, X. Yu, Y. Zhang, P. Shum, Y. Zhang, L. Xia, and D. Liu, “Theoretical study of dual-core photonic crystal fibers with metal wire,” IEEE Photon. J. 4, 1178–1187 (2012). P. Li and J. Zhao, “Polarization-dependent coupling in goldfilled dual-core photonic crystal fibers,” Opt. Express 21, 5232–5238 (2013). B. Sun, M. Y. Chen, J. Zhou, and Y. K. Zhang, “Surface plasmon induced polarization splitting based on dual-core photonic crystal fiber with metal wire,” Plasmonics 8, 1253–1258 (2013). P. Uebel, M. A. Schmidt, H. W. Lee, and P. St. J. Russell, “Polarization-resolved near-field mapping of a coupled plasmonic waveguide array,” Opt. Express 20, 28409–28417 (2012). H. W. Lee, M. A. Schmidt, and P. St. J. Russell, “Excitation of a nanowire ‘molecule’ in gold-filled photonic crystal fiber,” Opt. Lett. 37, 2946–2948 (2012). Y. H. Ja, “Optical vernier filter with fiber grating Fabry–Perot resonators,” Appl. Opt. 34, 6164–6167 (1995). M. A. Schmidt and P. S. J. Russell, “Long-range spiralling surface plasmon modes on metallic nanowires,” Opt. Express 16, 13617–13623 (2008). A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics (Cambridge University, 1988). P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006). K. Okamoto, Fundamental of Optical Waveguides, 2nd ed. (Academic, 2006). N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theor. Tech. 36, 1861–1868 (1988).
Numerical analysis of single and multiple gold nanowires embedded in triple cores arranged in collinear and noncollinear configurations in photonic crystal fibers (PCFs) is reported. A full-vectorial finite element method is used to achieve coupling characteristics of plasmonic PCF couplers for both x and y polarizations. It is demonstrated numerically that the PCF plasmonic couplers exhibit polarizationindependent tunable broadband filter characteristics that can be tuned according to the diameter of the embedded gold rod(s). © 2013 Optical Society of America OCIS codes: (060.5295) Photonic crystal fibers; (240.6680) Surface plasmons; (060.1810) Buffers, couplers, routers, switches, and multiplexers. http://dx.doi.org/10.1364/AO.52.008199
1. Introduction
The field of plasmonics has triggered new hopes and new device concepts because of their ability to confine strongly the light at the subwavelength scale [1], which is not possible using dielectric waveguides due to the diffraction limit [2]. Plasmonic devices like sensors, power dividers, and frequency splitters are the foremost applications [3–6]. The plasmons are the collective oscillations of electron gas density, which can be excited either along a metal–dielectric interface or in bulk materials (metals and semiconductors). The plasmon at the metal–dielectric interface is referred to as the surface plasmon (SP), which turns into a surface plasmon polariton (SPP) when it is coupled to externally applied electromagnetic waves. Several approaches, such as Otto configuration and the Kretschmann–Reather attenuated total reflection (ATR) method, have been employed to excite the SPs on a metal–dielectric interface. The dielectric medium can be air, planar waveguide structures, or optical fibers. Among these, optical 1559-128X/13/348199-06$15.00/0 © 2013 Optical Society of America
fiber has been a great host to incorporate metallic nanowires, trading optical fiber as plasmonic devices like polarizers, etc. [7]. With continuing development in optical fiber research, photonic crystal fibers (PCFs) show great potential to act as hosts for metallic wires, as the cladding in a PCF is comprised of air channels running down its length. The presence of air holes in the cladding makes them versatile platforms for infiltrating with liquids, gases, semiconductors, and metals [8–12]. Metals such as gold and silver have been used to fill either a single air hole or multiple air holes at different locations. This gives rise to interesting plasmonic behavior of so-called plasmonic PCF. By removing one or many air holes from selective locations in the cladding of PCFs, we can turn the PCF to operate as a coupler [13–15]. In the recent past, the coupling characteristics of plasmonic PCFs have been investigated numerically [16–20] where the polarization-dependent behavior and transmission characteristics were addressed. Conventional color optical filter works under the principle of reflection and transmission. Depending upon the behavior of the filter, the required wavelength band is reflected or transmitted. The bandwidth of such color optical 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
8199
filters is very narrow and strongly dependent on material characteristics. One can also achieve narrow bandwidth optical filters based on optical fiber Bragg gratings [21]. In this paper, we numerically demonstrate a new approach to achieve both band rejection and bandpass filter characteristics in a single fiber. A basic physical phenomenon of the proposed idea is based on the plasmon interaction with the fundamental guided core mode of a PCF selectively filled with gold. The rigorous calculation of SPP modes can be approximated by a spiraling plasmon model, where the phase matching condition must satisfy to generate SPs along the gold rod. If εd and εm are the electric permittivity of dielectric and metal, then the phase-matching condition to excite the SPP can be written as [22] 2πakt 2πm − 1;
(1)
where k2t k2SPP − β2 ; k0 2π∕λ;
β k0 nSPP ;
kSPP
r εd εm : k0 εd εm
(2.1)
(2.2)
(2.3)
Using Eqs. (2.1)–(2.3) into Eq. (1), the axial refractive index for the SPP mode can be given as
nSPP
s εd εm λm − 1 2 − ; εd εm 2πa
2. Single-Core Plasmonic PCF with Dual Nanowires
The schematic of a single-core PCF geometry with two selectively filled gold wires of different diameters is shown in Fig. 1(a), where Λ is the separation between two air holes, d is the air-hole diameter, and d1 , d2 are the diameters of the two gold wires, respectively. The two gold wires are kept far apart from each other in such a way that an interaction takes place between the guided core mode and the SPP modes surrounding the gold wire without any interaction between the gold wires. A full-vectorial finite element method (FEM)based solver (COMSOL Multiphysics) is used to compute the plasmon-like and dielectric-like modes in selectively gold-filled PCF structures. To validate our approach, we have verified the results of PCF with gold nanowire as given in [20]. After validation, we proceeded with our proposed structure. For modal analysis, we have considered a PCF with Λ 2.3 μm and d 1 μm, while the diameter of the gold rods were varied as d1 0.4 μm and d2 1 μm. A simulation was carried out for a wavelength range of 600–1600 nm. Figure 1(b) shows the dispersion diagrams and corresponding loss spectra. The solid red and blue curves denote the effective index of the fundamental guided dielectric mode of PCF for both x and y polarizations. The green dashed curve with filled blue circles shows the dispersion of the SPP mode of second order (m 2) for the isolated excitation of different radii of gold wires embedded in the PCF silica matrix. Both the dielectric guided mode and SPP mode interact at a particular wavelength, which is called the “resonance” where both the fundamental guided mode and the SPP mode exist simultaneously. The SPP resonance mode occurs when the phase matching condition is satisfied. This
(3)
where a is the radius of the gold rod, and m ≥ 1 is the order of the SP mode, which holds true for higherorder modes. We consider silica as a dielectric medium whose refractive index is considered according to the Sellmeier equation [23]. The best-fitted analytical model [24] for the wavelength-dependent permittivity of gold is given by εAu λ ε∞ −
λ2p
1 1 λ2
X Aj λ j1;2 j
λγi p
! eiϕj e−iϕj 1 1 i ; 1 1 i λj − λ − γ j λj λ γ j
(4)
where ε∞ 1.54, λp 143 nm, γ p 14500 nm, A1 1.27, A2 1.1, ϕ1 ϕ2 −π∕4, λ1 470 nm, λ2 325 nm, γ 1 1900 nm, and γ 2 1060 nm are the parameters. 8200
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Fig. 1. (a) Cross-sectional view of single-core PCF geometry with two gold wires of different diameters, d1 0.4 μm and d2 1 μm, whereas the air-hole diameter d 1 μm and pitch constant Λ 2.3 μm. (b) Dispersion diagrams and corresponding loss spectra for both x and y polarizations of the PCF filled with two different gold rods. The loss peaks correspond to 693 and 1070 nm. (c) Snapshots of the normalized electric field distribution at the resonance condition as mentioned.
gives rise to a strong coupling between the SPP and guided modes. A sudden discontinuity on the dispersion curve is noticed. Since the SPP modes are highly lossy, the corresponding loss curves indicate high peaks at resonance wavelengths, as depicted in the lower portion of Fig. 1(b). The snapshots of the normalized electric field distribution at the resonance wavelengths for both x and y polarizations are displayed in Fig. 1(c). It can be interpreted that at the resonance condition, the maximum electric field of the fundamental guided core mode strongly couples to the second-order SPP mode for both polarizations. The resonance wavelengths for d1 0.4 μm and d2 1 μm are 693 and 1070 nm, respectively. We notice that the resonance wavelength is dependent on the size of the gold rod. If we further include gold rods in the PCF geometry such that no gold rods interact with each other, one would expect n number of resonances for n number of gold rods for a particular SPP mode. 3. Triple-Core PCF Coupler with Single/Multiple Gold Rod(s)
In this section, we examine the effect of plasmonic interaction in a three-core PCF coupler. Previously, dual-core plasmonic PCF couplers have been reported [17]. The power coupling characteristic between two cores with coupling length and transmission for the x and y polarization was investigated. Here we are interested in dividing the power in different cores. A three-core PCF is the platform to fulfill the requirement and the proper arrangements of the cores provide the particular power-coupling ratio. We have chosen two types of configurations, namely, collinear and noncollinear, which depend on the position of the dielectric cores from the plasmonic rod(s). We have achieved filtering characteristics of the proposed device along with power splitting functionality. A.
Collinear Configuration
The cross-sectional view of a three-core collinear PCF coupler with two gold rods of diameter d1 and d2 is exhibited in Fig. 2(a). The blue shaded regions adjacent to the gold rods (shown by filled red circles) constitute three dielectric cores. For three collinear cores, six types of supermodes, as shown in Fig. 2(b), will exist according to different polarization orientations. We have used the following parameters to
Fig. 2. (a) Cross-sectional view of the three-core collinear PCF configuration with two gold rods and (b) six possible supermodes with different orientation of the electric field polarization.
obtain the dispersion diagrams and loss characteristics of the coupled system: d 1 μm, d1 0.4 μm, d2 0.6 μm, and Λ 2.3 μm. We have assumed the wavelength band from 650 to 850 nm for the sake of the computations only. Figure 3 shows the dispersion for the x and y polarizations of the electric field with the corresponding losses. The phase-matching conditions for both gold nanowires fulfill at different orientations of the electric fields for the x and y polarizations. Two resonances occur at 693 and 825 nm, corresponding to the d1 and d2 diameters of the gold rod, respectively. Within the range of the wavelength, only the SPP mode (m 2) couples with the fundamental guided core mode. From the dispersion diagrams and loss characteristics, it is observed that only two orientations of the electric field for both x and y polarizations excite the SP. At a 693 nm wavelength, x-polarized β3 and y-polarized β2 supermodes interact with the SPP mode of second order, which is due to the small gold rod (d1 ). Similarly, at an 825 nm wavelength, the x-polarized β2 and y-polarized β3 supermodes interact with the second-order SPP mode due to the larger gold rod size. At resonance wavelengths, the dielectric-like fundamental mode suffers high losses, as evident from the solid black and blue curves in Fig. 3. At a resonance condition, for a particular polarization, there is almost no electric field in the core region, as displayed in Fig. 4(a). By combining the losses for different orientations of electric fields, the total transmission loss for the fiber of length 0.1 mm is calculated and depicted in Fig. 4(b). Transmission losses for the x and y polarizations are different and can be used as the polarization-independent bandpass or band-rejection filter, depending on the wavelength of operation. For example, if the wavelength of the operation is near the resonance wavelength where a deep notch is observed in the transmission loss, then the device can be used as a band-rejection filter, whereas for other wavelengths, the same can be used as a bandpass filter. It can be revealed from the plots
Fig. 3. Dispersion and loss spectrum for a three-core collinear PCF with two gold rods for the (a) x and (b) y polarizations. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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wavelength region. The normalized power coupling characteristics as a function of z∕Lc of the device at a 760 nm wavelength is exhibited in Fig. 4(c), where Lc is the coupling length. B. Noncollinear Configuration
Fig. 4. (a) Snapshots of normalized electric field distribution, (b) transmission loss spectrum of a 0.1 mm long fiber, and (c) the normalized power variation at 760 nm wavelength, a function of z∕Lc in a three-collinear core PCF coupler with two gold rods, where Lc is the coupling length.
that two notches, as shown in Fig. 4(b), can be utilized to reject certain bands of wavelengths. The notch depth can be enhanced by taking a longer length of the fiber. On the other hand, we can see that the same fiber can also be employed to pass a certain wavelength band with a 3 dB bandwidth of ∼100 nm. It is clearly observed that the filter characteristics remain polarization independent in both cases. Figure 4(c) shows the power coupling characteristics of the device that were calculated using the coupled mode theory [25] for the arrangements of three collinear identical cores. The coupled mode equations for the collinear arrangement can be written as ∂a1 −jβa1 − jκa2 ; ∂z ∂a2 −jβa2 − jκa1 a3 ; ∂z ∂a3 −jβa3 − jκa2 ; ∂z
The schematic cross section of a three-noncollinearcore PCF coupler with a single gold rod is displayed in Fig. 5(a). Three cores are designed in such a way that they are 120° apart from the center of the PCF structure denoted by 1, 2, and 3. The gold rod is placed at the center of the fiber, as shown by the red-filled circle. The architecture resembles a satellite-core PCF. For simulation purposes, we took an air hole diameter d 1 μm, pitch constant Λ 2.3 μm, and the gold nanowire diameter was the same as the air-hole diameter. A cartoon picture of the possible supermodes with field orientations is shown in Fig. 5(b). Figure 6 shows the dispersion diagrams and loss characteristics of a three-noncollinear-core PCF coupler with a single gold rod for different electric field orientations. We find that the SPP modes of third and second orders are excited in this configuration
Fig. 5. Cross-sectional view of a three-noncollinear-core PCF configuration with a single gold rod, d 1 μm, Λ 2.3 μm.
(5)
where a1 , a2 , and a3 are the real electric field amplitudes of core 1, core 2, and core 3, respectively, β is the propagation constant of the dielectric-like mode for each core, and κ is the coupling coefficient between any two cores. The coupling coefficient is given by κpq
R∞ R∞ 2 2 ωε0 −∞ −∞ N − N q Ep × Eq dxdy R R ∞ ∞ ; −∞ −∞ uz × Ep × H p Ep × H p dxdy
(6)
where Ep , H p are the electric and magnetic field at core p, and Eq is the electric field at core q. The coupling coefficient κpq is calculated according to the given expression and [26]. Within the bandpass region from the central wavelength around 760 nm, the coupling coefficient varies like 0.56 0.3 mm−1 , which remains nearly constant for the bandpass 8202
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Fig. 6. Dispersion diagram and loss spectra for a threenoncollinear-core PCF coupler with a single gold rod for the (a) x and (b) y polarizations. The PCF parameters are d 1 μm and Λ 2.3 μm. The gold rod diameter d1 is the same as the air-hole diameter.
Fig. 7. (a) Snapshots of normalized electric field distribution at resonance wavelengths, showing the SPP modes of the third and second orders. (b) Transmission loss spectrum for x and y polarizations; the device acts as a band rejection filter of 100 nm 3 dB bandwidth. (c) Normalized power variation at 860 nm wavelength in three cores as a function of z∕Lc , where Lc is the coupling length.
at different wavelengths. At a 750 nm wavelength, x-polarized β3 and y-polarized β2 supermodes interact with the SPP mode of third order. Similarly, at a 1050 nm wavelength, the x-polarized β2 and y-polarized β3 supermodes interact with the secondorder SPP mode. The corresponding peaks are observed in the loss spectra as dictated by the solid blue and black curves in Fig. 6. In Fig. 7(a), the snapshots of the normalized electric field distribution are displayed, where we can see the excitation and interaction of the SPP modes of third and second orders with the particular guided supermode as indicated by the red arrows. The resonance wavelengths are 750 and 1050 nm, respectively, for third-order and second-order SPP modes. Next, we investigate the transmission loss and normalized power variation in the proposed device. The transmission loss in the 0.1-mm-long fiber that accounts for the loss contribution from all supermodes is shown in Fig. 7(b). We observe two notches corresponding to two different SPP modes. We can see that for both x and y polarizations, the transmission notches are the same at the resonant frequencies. If we further include more gold rods in PCF geometry, then polarization dependence may occur [7]. However, in our case, we find that the device is polarization independent. The 3 dB bandwidth for this band-rejection filter is ∼100 nm. The normalized power variation as a function of z∕Lc is calculated using coupled mode equations, where Lc is the coupling length as defined before. The coupled mode equations for three noncollinear identical cores can be written as ∂a1 −jβa1 − jκa2 a3 ; ∂z ∂a2 −jβa2 − jκa1 a3 ; ∂z ∂a3 −jβa3 − jκa1 a2 ; ∂z
(7)
where a1 , a2 , and a3 are the real electric field amplitude of core 1, core 2, and core 3, respectively, β is the propagation constant of the dielectric-like mode for each core, and κ is the coupling coefficient between any two cores. The coupling coefficient at the 860 nm wavelength is 0.82 mm−1 . The power coupling characteristics at 860 nm (i.e., the wavelength in the bandpass region) for the threenoncollinear-core PCF is shown in Fig. 7(c) as a function of z∕Lc. The power coupling characteristics are obtained by solving the coupled mode equations numerically using the Runge–Kutta algorithm with the unit input excitation at core 1. Note that we have considered the losses due to the presence of a gold rod. The power in cores 2 and 3 are equal and decreases as a function of the length, contrary to what we see in a sinusoidal variation in the absence of a gold rod. 4. Conclusion and Discussion
To conclude our work, we have numerically investigated in detail the plasmonic interaction in PCFs with triple cores arranged in collinear and noncollinear configurations with single and multiple gold rods. It has been shown that the air-hole cladding acts as a platform to confine the light beyond the diffraction limit due to the presence of gold wires embedded in the air channels. Through numerical simulations based on the FEM, we have achieved distinctive features such as bandpass, band-rejection filter characteristics along with the power division capabilities of the proposed plasmonic PCF coupler devices. We have examined the gold rod size dependency on the SP resonance frequency. Through proper selection of the gold rod diameter, it is possible to tune the filter response at a particular wavelength and hence the bandwidth. Collinear structure can be used as both a bandpass and band-rejection filter. For a broadband source, it behaves as a bandpass filter with bandwidth of ∼100 nm. The presence of a single notch in the loss spectrum suggests its utility as a band rejection too with a low rejection bandwidth. The noncollinear PCF structure provides broadband rejection filter characteristics with ∼100 nm bandwidth. It was noticed that the devices were polarization independent, which would alleviate their usage. References 1. L. Novotny and B. Hecht, Principle of Nano-Optics (Cambridge University, 2006). 2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2006). 3. A. V. Krasavin and A. V. Zayats, “Electro-optic switching element for dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 97, 041107 (2010). 4. F. Bilotti, S. Tricarico, and L. Vegni, “Plasmonic metamaterial cloaking at optical frequencies,” IEEE Trans. Nanotechnol. 9, 55–61 (2010). 5. A. Hassani and M. Skorobogatiy, “Design criteria for microstructured-optical-fiber-based surface-plasmon-resonance sensors,” J. Opt. Soc. Am. B 24, 1423–1429 (2007). 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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