Effect of bending on surface plasmon resonance spectrum in microstructured optical fibers

Maciej Napiorkowski; Waclaw Urbanczyk

doi:10.1364/OE.21.022762

1. Introduction

Surface plasmon resonance (SPR) based sensors are frequently used to detect various chemical and biological analytes changing a refractive index as a result of molecular interactions [1–5]. Subtle variations of the refractive index affect the resonance between the high-loss surface plasmon wave excited at the metal-dielectric interface and the incident wave. Excitation of the surface plasmon by a TM-polarized incident beam with a matched wave vector results in lower reflectivity. By measuring light reflectance, the resonance position related to the analyte refractive index is determined either as a function of wavelength at a fixed angle of incidence (spectral interrogation) or as a function of incidence angle for fixed wavelength (angular interrogation).

Conventional SPR sensors employing angular or spectral interrogation are based on Kretschmann’s configuration [6], in which a prism is used to excite the surface plasmon. Such devices are relatively complex and require fine tuning, therefore they can operate only in laboratory environment. Fiber-optic SPR sensors [7–20] overcome these limitations thanks to a simple design, high degree of integration, miniaturization and possibility of remote sensing even in harsh environment. A low cost and potentially broad applicability make them nowadays the promising class of the SPR sensors. The fiber-optic SPR sensors are limited only to wavelength interrogation but thanks to carefully designed sensing probes their performance can be competitive with conventional SPR sensors.

The simplest fiber-optic SPR sensors are based on a step index fiber with an exposed core coated with a metal layer [7,8] to allow for coupling between the guided modes and the surface plasmons. A performance of the sensing probes with removed cladding may be further increased by fiber tapering [9,10] or bending [11,12]. Theoretical studies on the U-shaped SPR probe [11] based on ray optics formalism, applicable only to multi-mode fibers of simple geometry, show that the sensitivity of the fiber-optic SPR probe increases with bending curvature. The increase in the sensitivity was accompanied by significant broadening of the resonance curve and the change in the resonance wavelength. The bend-induced change in the characteristics of the hetero-core SPR sensor was studied experimentally in [12]. It was shown that an increase in the curvature of the sensing probe amplifies the SPR loss and shifts the resonance wavelength.

In this work we numerically analyzed the effect of bending on SPR resonance in single-mode microstructured fibers. It has been already shown that microstructured fiber-optic SPR probes [14–20] containing an integrated metal-clad analyte channel allow for flexible shaping of the resonance characteristics. We show that much higher degree of bent-induced tunability of the SPR can be obtained in microstructured fibers than in conventional fibers reported in [11,12]. This effect is achieved by facilitating the power flow of the fundamental mode towards the metal layer by enlarging the pitch distance and reducing the size of selected air holes in the microstructured cladding. As the simplified ray-optics method proposed in [11] can’t be used to analyze the effect of bending in microstructured fibers with metal inclusions, we applied a rigorous numerical approach based on the finite element method (FEM). It employs the concept of the equivalent straight fiber with electric permittivity and magnetic permeability tensors derived in [21] using the transformation optics formalism [22].

2. Numerical method

Numerical modeling of straight microstructured fibers [23,24], including the SPR probes [14,16], is usually conducted using FEM, which allows to analyze waveguides with arbitrary geometry and boundary conditions. In principle, a 3-D FEM can be used to model directly the properties of bent waveguides but its high memory consumption strongly encourages the usage of two dimensional models based on the concept of an equivalent straight fiber [21,25], in which the change in the fiber geometry due to bending is transformed into appropriate distribution of material properties.

The most frequently used method of this type was proposed by Heiblum et al. in [25]. The authors showed that conformal coordinate transformation applied to the scalar Helmholtz equation is equivalent to the isotropic change in the refractive index in Cartesian coordinates. In this method, the fiber curvature is represented as an exponential change of the equivalent refractive index in the bending plane. Despite its usefulness, this method does not describe bend-induced transformation of permittivity and permeability and therefore cannot be used to model bent metal layers, which are key elements of the SPR sensors. Furthermore, the method is accurate only for the scalar Helmholtz equation, which is an approximation of the vectorial wave equation valid only for waveguides with low contrast of the refractive index. To model the effect of surface plasmon resonance in bent fibers, we adopted a recently proposed rigorous 2D representation of the curved waveguides [21], which overcomes the limitations of the Heiblum approach by use of the transformation optics formalism. In this approach, the transformed ε and µ tensors in the equivalent fiber are derived from Maxwell’s equations written in arbitrary coordinates.

In [22] Schurig et al. showed that the transformation between the coordinate systems x = {x¹,x²,x³} and x’ = {x^1’,x^2’,x^3′} is equivalent to the following change of the permittivity ε and permeability µ tensors:

ε^{i' j'} = {| \det (Λ^{i'}_{i}) |}^{- 1} Λ^{i'}_{i} Λ^{j'}_{j} ε^{i j},

μ^{i' j'} = {| \det (Λ^{i'}_{i}) |}^{- 1} Λ^{i'}_{i} Λ^{j'}_{j} μ^{i j},

where

Λ^{i'}_{i}

is the transformation (Jacobian) matrix:

Λ^{i'}_{i} = \frac{\partial x^{i'}}{\partial x^{i}} .

Equations (1) and (2) can be used to determine the material properties in the equivalent straight fiber. In our model we used the transformation from Cartesian to cylindrical coordinate system proposed in [21], which does not deform the cross-section of the waveguide and conserves its length measured along its symmetry axis:

\begin{array}{l} u = \sqrt{x^{2} + z^{2}} - R \\ v = R \tan^{- 1} (\frac{z}{x}) \\ y = y \end{array},

where x = {x,y,z} are Cartesian coordinates, x’ = {u,v,y} are transformed coordinates and R is the bending radius shown in Fig. 1.

Fig. 1 Transformation from Cartesian to cylindrical coordinates preserving the waveguide’s width and length.

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For the proposed transformation of the coordinate systems, the matrix $Λ^{i'}_{i}$ takes the following form:

Λ^{i'}_{i} = (\begin{matrix} \frac{x}{r} & \frac{z}{r} & 0 \\ \frac{- R z}{r^{2}} & \frac{R x}{r^{2}} & 0 \\ 0 & 0 & 1 \end{matrix}),

where r is the distance from the origin of the Cartesian coordinate system to the considered point:

r = \sqrt{x^{2} + z^{2}} = u + R .

According to Eqs. (1) and (2), the matrix

Λ^{i'}_{i}

defines the transformation rules for the permittivity and the permeability tensors (Fig. 2):

Fig. 2 Distribution of electric permittivity in straight fiber (a) and spatial dependence of the elements of ε and µ tensors in the equivalent fiber (b-c).

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ε^{i' j'} = (\begin{matrix} \frac{R + u}{R} ε^{X X} & \frac{R + u}{R} ε^{X Y} & ε^{X Z} \\ \frac{R + u}{R} ε^{Y X} & \frac{R + u}{R} ε^{Y Y} & ε^{Y Z} \\ ε^{Z X} & ε^{Z Y} & \frac{R}{R + u} ε^{Z Z} \end{matrix}),

μ^{i' j'} = (\begin{matrix} \frac{R + u}{R} μ^{X X} & \frac{R + u}{R} μ^{X Y} & μ^{X Z} \\ \frac{R + u}{R} μ^{Y X} & \frac{R + u}{R} μ^{Y Y} & μ^{Y Z} \\ μ^{Z X} & μ^{Z Y} & \frac{R}{R + u} μ^{Z Z} \end{matrix}),

Transformation rules established by Eq. (7) and (8) can be used to model the bent waveguides made of any materials, including conducting and anisotropic media. We therefore adopted this approach to represent bent metal layers in the SPR sensors and bent perfectly matched layers (PML) [21] that are used to model bending loss.

3. Microstructured fiber design

Propagation of guided modes in the bent fiber of low aperturedepends mainly on equivalent values of ε_eq and µ_eq in XY plane, which increase linearly towards the bent top according to the following relation:

\frac{ε_{e q}}{ε} = \frac{μ_{e q}}{μ} = 1 + \frac{u}{R} .

Assuming that the waveguide before bending is isotropic, the spatial variation of the refractive index across the equivalent fiber can be expressed as:

n_{e q} (u) = \sqrt{{(1 + \frac{u}{R})}^{2} ε μ} = n (1 + \frac{u}{R}),

where n is the function representing the refractive index distribution in the fiber before bending. The above relation allows to intuitively understand two mechanisms responsible for the change in the surface plasmon resonance characteristics due to bending. Bent-induced amplification of the resonance strength is caused by an increase in the equivalent refractive index towards the bent top. Field distribution in the fundamental mode shifts to the region of the higher equivalent refractive index (depending upon spatial coordinate as u/R), located closer the metal layer and therefore couples stronger to the surface plasmon. Enhancement of the coupling strength caused by the displacement of the modal field can be optimized by appropriate modification of the cladding microstructure. Simultaneously, the resonance wavelength increases because the equivalent refractive index of the analyte located at a certain distance from the fiber core experiences a high bending-induced increase also governed by u/R.

Basing on this intuitive understanding, we proposed three variants of the microstructured fiber showed in Fig. 3, in which the SPR effect is highly tunable by bending.

Fig. 3 Microstructured fiber designs with high tunability of the surface plasmon resonance by bending.

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The fibers are based on the hexagonal lattice with the pitch distance of Λ = 8µm and the outside diameter equal to 125 μm, Fig. 4. A solid core is surrounded by air holes of the diameter d₁ = 0.6Λ. The analyte channel of the diameter d_c = 4Λ located at 4Λ from the center of the fiber core is coated with a 40 nm gold layer. The diameter of three holes located between the core and the analyte channel is reduced to d₂ = 0.2Λ to increase a bent-induced leakage of the fundamental mode towards the gold layer.

Fig. 4 Geometry of the microstructured fiber. Placing the analyte channel far from the core and reducing the diameter of selected holes increases bent-induced tunability of the surface plasmon resonance. Dotted line marks the bending plane.

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The fiber no. 1 has the simplest structure with the core surrounded only by two rings of air holes. The fiber no. 2 has an open analyte channel, which allows for measurements of the substances located outside of the fiber as well as for local deposition of the gold layer in the sensing region by external vacuum evaporation. In the fiber no. 3, a number of air hole rings was increased to reduce propagation and bending loss. The proposed fibers are tunable by bending radii of the order of few centimeters, facilitate the flow of the analyte due to large size of the analyte channel and are relatively easy tofabricate.

4. Simulation results

To calculate the complex propagation constants of the guided modes, we used the finite element method with perfectly matched layers boundary conditions. Wavelength dependence of the refractive index of silica glass was determined using the following Sellmeier’s equation [26]:

ε (λ) = 1 + \sum_{i = 1}^{3} \frac{A_{i} λ^{2}}{(λ^{2} - Z_{i}^{2})},

where wavelength is given in microns, while A₁ = 0.6961663, A₂ = 0.4079426, A₃ = 0.8974794, Z₁ = 0.0684043, Z₂ = 0.1162414 and Z₃ = 9.896161. Complex permittivity of the gold layer was approximated by the Drude model [27]:

ε (ω) = ε^{\infty} - \frac{ω_{p}^{2}}{ω (ω + i ω_{τ})},

where e^∞ = 9.75, ω_p = 1.36x10¹⁶ [rad/s] and ω_τ = 1.45x10¹⁴ [rad/s].

The calculations were performed for the half of the fiber structure taking into account its plane symmetry. In the next paragraphs, we present the SPR spectra calculated for the mode with the electric field parallel to the bending plane, which is coupled more efficiently to the surface plasmons. In Figs. 5, 6, and 7 we show the results of simulations only for the fiber no.1, because the resonance characteristics for the other fibers are nearly identical as the differences in fibers’ microstructure are located far away from the region in which the surface plasmon resonance occurs. The comparison of the SPR spectra calculated for all the fibers is shown in Fig. 8. Discrepancies between the SPR spectra in the three fibers rise with the bending curvature but are practically negligible even for the lowest considered bending radius R = 5 mm.

Fig. 5 SPR spectra calculated for n_a = 1.33 (red) and n_a = 1.34 (blue) in the wide wavelength range (a) and in the resonance neighborhood (b).

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Fig. 6 Amplification of the SPR spectrum induced by bending (a) and comparison of the normalized SPR spectra showing that the shape of the resonance curves is conserved for different bending radii (b).

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Fig. 7 Calculated field distribution of the fundamental mode for different bending radii. Smaller diameter of air holes enhances the fundamental mode leaking towards the analyte channel, which results in strong amplification of the SPR spectrum.

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Fig. 8 SPR spectra calculated for R = 5 mm have almost identical height, width at half-maximum and the resonance wavelength for three variants of the proposed fiber design. Spectra for the fiber no. 1 and no. 2 overlap.

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As it is shown in Fig. 5, the straight fiber has a single, well defined plasmon resonance in the visible range. The resonance occurs at λ_peak = 588 nm for the analyte of the refractive index n_a = 1.33 and shifts by 14 nm to λ_peak = 602 nm for n_a = 1.34.

The resonance wavelength shift observed in response to the change in the refractive index of the analyte determines the sensitivity $S_{λ}$ of the sensing probe:

S_{λ} = | \frac{δ λ_{p e a k}}{δ n_{a}} | = 1400 [\frac{n m}{R I U}],

which is equivalent to the sensor resolution of 7.2x10⁻⁵ RIU assuming that the position of the resonance wavelength is determined with the resolution of 0.1 nm. This figure is of the same order as the resolution of other microstructured SPR sensors reported earlier in literature [13–16].

Our simulations show that the bent-induced amplification of the resonance loss in the analyzed microstructured fibers is much higher than in the conventional fiber of a comparable diameter reported in [12]. In particular, for the bending radius R = 10 mm, the SPR loss is enhanced only by the factor of 2.5 in [12] compared to the factor of 200 in case of the analyzed fiber. For the considered microstructured fiber, the SPR spectrum is amplified 20 times for much larger bending radius R = 50 mm and up to 700 times for R = 5 mm, which is assumed to be the smallest bent radius the fiber can sustain without breaking. The fiber sensitivity does not change considerably upon bending. The resonance wavelength shifts by about 6 nm for the bending radius R = 5 mm but the shape of the normalized resonance curve does not broaden as it is in [11]. This is because the considered fiber is single mode and only the fundamental mode is coupled to the plasmon, Fig. 6.

The resonance wavelength shift Δλ calculated using FEM agrees relatively well with intuitive prediction based on Eq. (7), in which the wavelength shift Δλ is attributed solely to the bending-induced variation of the equivalent refractive index of the analyte:

Δ λ = S_{λ} δ n_{a_{e f f}} = S_{λ} n_{a} \frac{u}{R} = 5.9 n m .

An increase in the attenuation at the resonance is caused by the leakage of the fundamental mode towards the metal layer enhanced by reduction of the diameter of selected air holes in the region between the core and the analyte channel. As a side effect, the reduced confinement of the fundamental mode and proximity of the metal layer lead to the higher loss in the whole analyzed spectral range.

As it is shown in Fig. 7, the field distribution of the fundamental mode changes significantly due to bending. Transition loss Γ between the fiber segments of different bending radii (R₁,R₂) can be calculated using the following relation:

Γ (R_{_{1}}, R_{2}) = - 10 \log_{10} [\frac{{| \iint E_{R_{1}} (x, y) E_{R_{2}} (x, y) d x d y |}^{2}}{\iint {| E_{R_{1}} (x, y) |}^{2} d x d y \iint {| E_{R_{2}} (x, y) |}^{2} d x d y}] .

In Table 1 we have shown the calculated transition loss caused by the modal filed mismatch between the segments of different curvature in the fiber no.1.

Table 1. Transition loss in dB between segments of different curvature in the fiber no. 1

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Abrupt changes in the fiber curvature leads to high transition loss, which, however, can be significantly reduced by a gradual increase in the bending radius. As it is shown in Table 1, the transition loss between the straight and the bent fiber with R = 5 mm is 13.2 dB, but if the additional segment of R = 50 mm is formed between them, it drops to 5.5 dB. The use of three additional segments of R = 50 mm, R = 25 mm and R = 10 mm decreases it further to 3.2 dB.

The confinement loss in the fiber without the metal layer can be greatly reduced by use of more air hole rings. The confinement loss in the straight fiber no.1 (with only two air hole rings) is about 2x10⁻² dB/m in the resonance wavelength range, whereas for the fiber no. 3 (four air hole rings) it is reduced to about 4x10⁻⁹ dB/m. For the bent fiber a greater loss occurs because the fundamental mode leaks out of the microstructured cladding through the region with smaller air holes, Fig. 7. In this case, for R = 25 mm the loss rises to approximately 0.75 dB/m in the fiber no.1 and to 4x10⁻³ dB/m in the fiber no.3. Bending the fiber in the same plane but in the opposite direction (in this case R is denoted as negative) results in the decrease in bending loss because the fundamental mode is moved towards the holes with a larger filling factor. As it is shown in Table 2, for large bending radii, the loss can be reduced in both fibers below the level of the straight fiber, i.e., nearly two orders of magnitude for R = −50 mm.

Table 2. Confinement loss calculated for straight and bent fiber near the resonance wavelength λ = 590 nm

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In Fig. 8, we show the SPR spectra calculated for three variants of the proposed fiber design for the bending radius R = 5 mm. The differences are small enough to treat the results obtained for the fiber no.1 as universal, even in case of the small bending radii.

5. Summary

To analyze the effect of bending on the SPR spectra, we adopted a recently proposed rigorous approach, which employs the concept of equivalent fiber with appropriate spatial distribution of electric permittivity and magnetic permeability tensors derived on the ground of transformation optics formalism. We proposed and analyzed a design of the microstructured fiber, in which bending leads to strong amplification of the SPR spectrum caused by leaking of the fundamental mode towards the metal layer.The mode leakage is enhanced by increasing the pitch distance and by reducing the size of selected air holes in the region between the core and the analyte channel. As a result, the SPR resonance arising in the visible range is amplified by the factor of up to 700 for bending radius R = 5 mm. The shape of the resonance curve does not change upon bending, while the resonance wavelength raises by about 6 nm for R = 5 mm as a result of equivalent change in the refractive index of the analyte. The obtained sensitivity of 1400 [nm/RIU] is of the same order as reported earlier in [13–16] for other microstructured probes.

The proposed fiber designs can be used as integrated SPR sensors with an internal analyte channel (fiber no.1) or open structure (fiber no.2) capable of sensing in situ and operating on various length scales. Additionally, the fiber no. 2 can benefit from the possibility of local deposition of the metal layer by external vacuum evaporation, which will reduce the propagation losses in the fiber section without metal layer and allow for remote sensing using a single fiber. The simulations conducted for the fiber no.3 show that the propagation loss and bending loss in the leading sections of the microstructured fiber without a metal layer can be significantly reduced by adding additional rings of holes, which does not affect the SPR characteristics of the sensing probe.

Acknowledgment

This work was supported by Wroclaw Research Center EIT + Ltd. in the frame of the NanoMat project “Application of Nanotechnology in Advanced Materials”, within the European Funds for Regional Development, POIG, Sub-action 1.1.2.

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	Straight	50 mm	25 mm	10 mm	5 mm
Straight	0	1.56	5.13	10.32	13.19
50 mm	1.56	0	0.93	2.55	3.94
25 mm	5.13	0.93	0	0.41	1.25
10 mm	10.32	2.55	0.41	0	0.29
5 mm	13.19	3.94	1.25	0.29	0

	Straight	−50 mm	−25 mm	25 mm	50 mm
	dB/m
Fiber no. 1 (2 rings)	2.2x10⁻²	5.1x10⁻⁴	4.4x10⁻³	7.6x10⁻¹	3.4x10⁻¹
Fiber no. 3 (4 rings)	4.3x10⁻⁹	1x10⁻¹⁰	1.4x10⁻⁴	3.7x10⁻⁵	8.8x10⁻⁵

	Straight	50 mm	25 mm	10 mm	5 mm
Straight	0	1.56	5.13	10.32	13.19
50 mm	1.56	0	0.93	2.55	3.94
25 mm	5.13	0.93	0	0.41	1.25
10 mm	10.32	2.55	0.41	0	0.29
5 mm	13.19	3.94	1.25	0.29	0

	Straight	−50 mm	−25 mm	25 mm	50 mm
	dB/m
Fiber no. 1 (2 rings)	2.2x10⁻²	5.1x10⁻⁴	4.4x10⁻³	7.6x10⁻¹	3.4x10⁻¹
Fiber no. 3 (4 rings)	4.3x10⁻⁹	1x10⁻¹⁰	1.4x10⁻⁴	3.7x10⁻⁵	8.8x10⁻⁵

Effect of bending on surface plasmon resonance spectrum in microstructured optical fibers

Abstract

1. Introduction

2. Numerical method

3. Microstructured fiber design

4. Simulation results

5. Summary

Acknowledgment

References and links

Cited By

Figures (8)

Tables (2)

Equations (15)

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