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We have analytically investigated the polarization dependence of surface plasmon resonance in fiber structures having strong asymmetry. From our simulation experiments it is found that the resonance wavelength coincides with the zero-birefringence point of two degenerate modes, consequently demonstrating a new approach through which one can accurately locate the resonance peak of the system without having to analyze the loss spectrum. Results obtained using the new technique also reveal better performance in terms of accuracy and computation efficiency. Application of this approach in the analysis of refractive index and pressure sensors based on the single core D-shaped and symmetric multiple air-hole fibers respectively is presented as a demonstration. The proposed technique, which primarily involves the search of zero-birefringence point, may be generalized for the study of other plasmonic waveguide structures.
©2010 Optical Society of America
1. Introduction
Optical excitations of surface plasmon waves (SPWs) on metallic surfaces have been applied extensively in sensing technology [1]. One conventional approach, which is typically bulky and expensive, uses attenuated total reflection (ATR) to excite SPWs on a gold film at the bottom of a prism [2,3]. Another popular technique for the excitation of SPWs is based on directional coupling between waveguide modes and SPWs. Among waveguide-coupling devices, fiber-based SPW sensors offer unique advantages such as miniaturization, ease of implementation, fast response, cost-effectiveness, remote sensing and the potential for multiplexing [4,5]. Indeed, optical fiber sensors have been increasingly explored for the high sensitivity measurement in different application fields: temperature sensors, electric field sensors, biosensors and gas sensors [6–9].
For the study of the resonance of SPW excitation in optical fiber waveguides, common analytical models usually simplify the metallic coated waveguide into a multi-layer structure [10]. In addition, coupled mode theory has also been deployed to investigate the coupling between surface plasmon and confined core modes [11]. The intersection point of the dispersion curves for the plasmonic mode and core mode, which corresponds to the peak transmission loss of the waveguide, is used for locating the resonance wavelength [12]. However, coupled mode theory is limited especially when the weakly coupled assumption is not satisfied. Moreover, when the fiber structure is strongly asymmetric, two degenerate fundamental modes will further result in more complicated mode coupling conditions in two orthogonal directions. Pone et al. have demonstrated an elliptical holey fiber based pressure sensor by measuring the splitting of resonance wavelengths of the two orthogonal polarizations in the loss spectrum. The splitting value increases with the ellipticity of the air holes [13]. Lee et al. also investigated the optical properties of asymmetric photonic crystal fibers with a single gold nanowire introduced into the core. Strongly polarization dependent resonance transmission was reported at surface plasmon resonance wavelength [14]. However, when the transmission spectrum is flattened, as in many real cases, it becomes increasingly difficult to obtain sizeable signal change from simply measuring the loss spectrum. On the other hand, until now there exist limited reported studies on polarization dependent SPW excitation in asymmetric optical fibers.
In this paper, we numerically investigate the correlation between resonance coupling and degenerate mode properties. We report a new approach for determining the resonance wavelength when the fiber structure is strongly asymmetric. Two typical asymmetric fiber structures including a D-shaped fiber with gold coating and a photonic crystal fiber with elliptical air holes coated with silver have been demonstrated for the verification of this proposed approach. Polarization- dependent resonance condition is obtained by calculating the birefringence (B) of the fiber with metal inclusions, defined as the difference between the real part of the effective refractive indices of two degenerate modes in the x- and y-polarization direction, respectively. We identify the resonance by looking for the corresponding resonance wavelength when birefringence is zero. The new technique offers several advantages over the conventional approach in terms of the accuracy in determining the resonance wavelength and the efficiency in computational time and memory utilization.
2. Analytical approach
Finite element method (FEM) has been used widely for the analytical study of waveguide with complex geometries and/or high index contrast materials i.e. with metallic inclusions [11–14]. In the case of a D-shaped fiber with a thin layer of gold deposited on the flat surface as shown in Fig. 1(a) , d1 is the gold layer thickness, d2 is the polishing depth from the fiber center to the polished surface. The Drude model is used in the simulation to account for dispersion in metal [15]. The background material is pure silica (n1 = 1.45). The high-index region in the core with a diameter of 10µm is silica doped with germanium (n2 = 1.47). The guided core mode propagates in the fiber and excites SPWs at the outer interface of the gold layer if they are phase-matched [16]. The aforementioned FEM based simulation tool was used to find the complex effective mode index of the coupled mode over a wide wavelength range, i.e. neff = Re(neff) + jIm(neff), where Re(neff) = βλ/2π and Im(neff) = αλ/2π [17,18], α is the attenuation constant and β is the propagation constant. The real part of the effective refractive index reflects the propagation constant while the imaginary part is proportional to the confinement loss. Two orthogonally polarized coupled modes were obtained. From the intensity distribution of the Poynting vector shown in Fig. 1(b) and (c), only the y-polarized mode has a localized field distribution on the surface of the gold layer. Moreover, when we plot the loss spectrum for the two degenerated modes, only the y-polarized coupled mode experiences a peak loss (approximately 42.8dB/cm) at 637nm while the x-polarized mode is a monotonically rising curve as shown in Fig. 2(a) . In the y-direction, the gold coated layer contributes to the excitation of SPWs, while there is no such effect from the metal in the x-direction.
Fig. 1 (a) Cross-section of the D-shaped fiber with 30 nm gold coating on the flat surface. Intensity distribution of Sz for (b) x-polarized mode (c) y-polarized mode at the resonance wavelength.
Fig. 2 FEM simulation results (a) black solid curve is the imaginary part of the y-polarized coupled mode effective index; black dotted curve is the imaginary part of the x-polarized coupled mode effective index; blue solid curve plots the modal birefringence. (b) Real parts of the effective indices for two orthogonal modes.
However, the real parts of the effective indices for two orthogonal modes exhibit a crossing point at exactly 637 nm as indicated in Fig. 2(b). Before this specific wavelength, x-polarized mode propagates slower than the y-polarized mode. This abrupt change of phase velocity at the resonance wavelength for the y-polarized mode implies the occurrence of a phase shift that also reflects the nature of the resonance behavior. This zero-birefringence situation also indicates that the x-polarized and y-polarized modes are degenerate and their phase velocities are equivalent. The D-shaped fiber introduced a structural birefringence. However, this birefringence should not have an abrupt change with wavelength, especially in a very narrow range. Therefore, the great phase shift could only be resulted from the metal layer on the polished side. It is corresponding to the phase shift on reflection at the Silica-Gold interface. At the SPR wavelength, where the absorption is the largest, the complex refractive index will contribute to the large variation of phase at this specific wavelength [19]. It should be noted that the above wavelength range was obtained with a gold layer thickness of 30 nm. The new technique has also been verified with several other thickness values as well. We find excellent agreement on the determination of resonance wavelength using both techniques, thus confirming that the new zero-birefringence approach is indeed valid. In fact, Bienstman et al. have compared both in-house developed and commercial mode solvers, and concluded that generally the calculated imaginary part of the effective index varies much more significantly than the real part for different mode solvers [20]. Hence we can infer that the new technique would provide more accurate and consistent results than the conventional approach. In addition, a mode solver for only the real part of the propagation constant is sufficient. While the complex value is not a required parameter, it becomes quite clear that the new approach also have the merits of efficient computation as well as memory utilization.
3. Simulation results
Attempts were made to facilitate zero-birefringence point at visible wavelength range by increasing the ambient index on the gold surface from 1.41 to 1.42 as shown in Fig. 3 . This zero-birefringence point exhibits a wavelength red-shift from 637 nm to 670 nm. By assuming a 0.01 nm spectral resolution, we can readily obtain a 10−6 refractive index unit measurement sensitivity from this structure. The sensitivity obtained is comparable with other fiber based refractive index sensors reported in the literature [21] including single-mode polarization maintaining fiber based SPR sensor [22] and tilted Bragg grating based SPR sensor [23]. Moreover, the steep change at the zero-crossing point should lead to a more accurate measurement of the wavelength shift. Furthermore, devices with such polarization dependent property can be applied as a new kind of in-fiber wavelength dependent notch filter and polarizer [24].
Apart from the ambient refractive index, structure parameters such as the gold layer thickness will also affect the resonance wavelength significantly. With a 5 nm change of gold coating thickness, as shown in Fig. 4(a) , may result in a resonance shift as large as 40 nm. This indicates that the structure might be potentially useful for metallic thin-film characterization with high precision. Similarly, when the polishing depth, d2 is increased from 5.4 µm to 5.8 µm, the resonance wavelength is blue-shifted as presented in Fig. 4(b). Since the gold layer is now further away from the center core, which means that less core energy is transferred to the SPW energy leading to weaker coupling efficiency. The increase of the overall effective index of waveguide will hence shift the resonance to a shorter wavelength [25]. However, the resonance shift is less significant than the previous two cases. Further polishing the device by 0.4 µm will red-shift the zero birefringence point by around 10 nm. This makes the device more robust to the practical polishing processes. Another interesting observation is that the birefringence value is inversely related to the polishing depth. This is expected because a larger value of polishing depth d2 means that the fiber structure is more towards symmetric, which results in a smaller structural birefringence. In Ref. 25, a similar relationship between the birefringence and polishing depth has been reported, which is useful for the understanding of coupling characteristics in polarization-preserving and polarization-selective directional couplers [26]. Their observation agrees well with our simulation results.
4. Extended modeling of photonic crystal fiber
Another typical structure with asymmetric properties is a photonic crystal fiber with elliptical air holes. If the inner surface of these air holes is coated with silver for example, the surface plasmons on the silver layer will be excited by the evanescent field penetrated from the silica/silver interface. The peak loss of the x-polarized coupled mode matches with the zero-birefringence point at ~1030 nm as shown in Fig. 5 . Moreover, the steep birefringence curve should provide more accurate measurement of the resonance wavelength compared to the one based on finding the loss peak from a relatively flat transmission spectrum.
Fig. 5 Matching of zero birefringence with peak coupling loss in a photonic crystal fiber (inset) with elliptical air holes: elliptical ratio a/b = 0.82, uniform silver coating (red) thickness is 100 nm, pitch size is 1.5 µm.
Because of structural asymmetry, two resonances are found for the x- and y-polarized modes respectively. Without loss of generality, we shall only discuss the case involving a shorter resonance wavelength in this paper, i.e. λrs≈1030 nm for the x-polarized mode, while the same arguments can be completely applicable to the long wavelength case. The resonance condition can be calculated using two techniques: (i) our new birefringence approach and (ii) loss spectrum analysis. Firstly, the core-guided confinement mode field distribution before resonance is obtained as shown in Fig. 6(a) . At λrs, the sudden jump from the low to high birefringence value corresponds to the SPR point of the x-polarized mode pattern evolving from Fig. 6(a) to (b). After passing the resonance point, the core-guided confinement mode distribution will remain the same as shown in Fig. 6(b). In addition, it is observed that most of the plasmon intensity is concentrated away from the fiber core leading to considerably more field penetration into the cladding holes for Fig. 6(b) than (a) [18]. Secondly, we can also interpret the resonance process in terms of loss coupling of the core confinement mode. Figure 6(c) & (d) show the intensity distribution of the Poynting vector (along the fiber axis) of the same core confinement mode as it transforms from core-guided mode into the plasmonic regime passing through the resonance wavelength [27]. As can be seen clearly from the mode intensity profiles, most of the core energy is coupled to the silver/air interface when SPR occurs.
Fig. 6 (a) & (b) Two different x-polarized core confinement mode distribution patterns before and after resonance; (c) & (d) the same core confinement mode (insets) before and after resonance wavelength with their corresponding mode profiles along the horizontal cut at Y = 0.
Figure 7 shows the change of birefringence value for three cases with different hole ellipticity ratios. The resonance wavelength is blue-shifted as the hole ellipticity ratio decreases. This is expected because a larger hole diameter in the x-direction will lead to a decrease in the resonance wavelength and an enhancement of field confinement [21]. In principle, hole ellipticity can be detected by measuring the resonance wavelength change using the zero-birefringence point technique. The steep change of the birefringence around the zero value in PCF structure is due to the limited calculation points. But it is worth to note that the birefringence value in PCF is two orders of magnitude higher compared with that in D-shaped fiber. This is due to the effective elliptical core can induce a large structural birefringence if the surrounding air-holes are elliptical. While in the case of a D-shaped fiber, the polishing side in the cladding will not vary the core shape at all, therefore, not affecting the structural birefringence significantly.
Fig. 7 Calculated birefringence for different hole elliptical ratio with Λ constant, a = 0.8 µm-δ and b = 0.8 µm + δ, where δ is the hole diameter change [13].
5. Conclusion
We have analytically investigated the polarization dependence of surface plasmon resonance in two different asymmetric fiber structures. The resonance wavelength is obtained from measuring the zero-birefringence point of the two degenerate core confinement modes. It is applicable for all structures, especially when the asymmetry is large. We have also found excellent agreement between the zero-birefringence approach and the conventional technique based on locating the maximum point of transmission loss spectrum. Two different structures including a metal-coated D-shaped fiber and a photonic crystal fiber with elliptical air holes have also been intensively investigated for verification of this new method. It has been demonstrated that the new approach is more computation efficient as compared to the one based on searching for the peak in spectral transmission loss plot.
Acknowledgement
This work is supported by Singapore A*STAR SERC grant 0921450031, and SIMTech-NTU-CUHK collaborative research project grant U09-P-007SU.
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