Theoretical analysis of a fiber optic surface plasmon resonance sensor utilizing a Bragg grating

Barbora Špačková; Jiří Homola

doi:10.1364/OE.17.023254

1. Introduction

In the last two decades, optical sensors based on surface plasmon resonance (SPR) have been widely researched and numerous SPR sensor configurations have been developed. SPR biosensors have become an essential tool for the study of biomolecular interactions and have been applied in the detection of chemical and biological analytes in areas including proteomics, medical diagnostics, environmental monitoring, food safety and security [1]. In order to obtain localized measurements and remote sensing, miniature fiber optic SPR sensors were developed in early 1990. They were based on multimode optical fibers with an exposed core region symmetrically coated with a thin layer of metal [2] or side-polished single-mode optical fibers [3]. In the following years, the designs of these sensors have been refined and their performance has been considerably improved [4–6]. The best fiber optic SPR sensors achieved a refractive index resolution as low as 10⁻⁶ [7]. In the last decade, SPR sensors employing planar waveguides or optical fibers with imprinted long-period gratings [8,9] or fiber Bragg gratings (FBG) [10,11] were proposed. SPR sensors based on FBGs have also been demonstrated experimentally using a standard single-mode optical fiber with a tilted FBG [12]. One major limitation of all of the fiber optic SPR sensors remains the lack of referencing channels which are necessary for reference-compensated biosensing. To overcome this limitation, multichannel sensing systems based on a series of FBG SPR sensing elements employing FBG with different periods along a single optical fiber while utilizing wavelength division multiplexing has been proposed [8,13]. Typically, fiber optic SPR sensors with FBGs are simulated as equivalent planar 2-D structures waveguide [11,13] or the diameter of the fiber is reduced to allow 3-D analysis without computational problems [10].

In this paper we present a rigorous theoretical analysis of a fiber optic SPR employing an FBG to excite back-propagating mixed surface plasmon – fiber cladding (SP-FC) modes. The implemented method is based on the Coupled Mode Theory (CMT) [14], which is a technique for obtaining quantitative information about the diffraction efficiency and spectral dependence of the fiber grating. The field distribution and modal dispersion, used in the calculation, are calculated using a method based on a field expansion approach for matching the field continuity at the boundary of the layers. Since this method on the basic level is numerically unstable for the designed parameters of the structure proposed here (fiber cladding of 125 μm), a modified approach has been proposed. Using this approach transmission spectrum was simulated. The sensitivity and resolution of the sensor are discussed in relation to the design parameters of the sensor.

2. Description of the structure and the principle of operation

Figure 1 shows the design of the investigated fiber optic SPR sensor. It consists of a single-mode step-index optical fiber with a FBG inscribed in the fiber core. The fiber cladding is coated with a thin layer of gold (gold is chosen for its excellent chemical resistance and stability). The sensing structure is surrounded by the analyte which is considered to be in an aqueous environment (the most common medium in SPR biosensing).

Fig. 1 Schematic of the sensing fiber optic structure

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When the axial component of the propagation constant of the cladding mode matches that of the SP, coupling between the SP mode and the cladding mode occurs. Due to the coupling with the SP, the propagating constant of this hybrid mode is highly sensitive to refractive index changes of the analyte. In addition, FBG can couple the light from the forward-propagating fundamental (core) mode of the optical fiber into the SP-FC modes. The transfer of optical power between the modes can occur when the propagation constant of the fundamental mode diffraction-altered by the FBG is equal to the propagation constant of the SP-FC mode. This coupling condition is fulfilled at the resonance wavelength which can be expressed as

λ_{r e s}^{μ} = ℜ (n_{e f f}^{1} + n_{e f f}^{μ}) Λ,

where n_eff¹ and n_eff^μ are the effective refractive indices of the fundamental mode and the μ-th SP-FC mode, respectively, and Λ is the period of the FBG. The coupling gives rise to dips in the spectrum of transmitted light.

If two Bragg gratings of different periods are inscribed into two different regions of the fiber core, in each region, the back-propagating SP-FC modes are excited at different wavelengths. If the structure is illuminated with a polychromatic light source and the transmitted light is detected by a spectrum analyzer, it is possible to discriminate between changes in the spectrum corresponding to different fiber regions. Therefore, one of the regions can act as a sensing channel while the other can serve as a reference channel.

3. Theory

The most widely used technique to simulate the propagation of the light is the CMT. In this approach, the modes of the structure have to be determined and then the coupling between the fundamental mode and the counter-propagating cladding modes is obtained. Several numerical approaches have been used to determine the modes in the structure depicted in Fig. 1. These include the Finite Elements Method (FEM), and the Transmission Matrix Method (TMM) [15]. Since the functional section of the structure has a rather large volume, discretization of the structure for the subsequent application of FEM would require extensive computing power to obtain a reasonably accurate result. Although TMM is often the method of choice for determining the modes of a waveguiding structure, it is rather unstable for multilayer structures containing extremely thick layers (cladding) and metals (gold layer). To avoid this issue, a modified approach was used based on the field expansion approach for matching the field continuity at the boundary of the layers.

3.1 Guided modes of the structure

In the calculation of the modes of the sensing structure, a straight four-layer cylindrical structure consisting of a fiber core, cladding, gold layer and an aqueous environment was assumed. Since the field distribution of the fundamental mode is concentrated mainly in the core, the approximation of a two-layer structure (core-cladding) was used to describe the behavior of the fundamental mode [15]. Since the Bragg grating is assumed to be a cylindrically symmetric perturbation of the refractive index of the core, the transfer of power occurs only between the fundamental mode and the cladding modes of the azimuthal order l = 1; therefore in the following analysis we shall focus only on these cladding modes. The electromagnetic field of these cladding modes (specifically the tangential components of magnetic and electric field vectors) can be expressed as follows:

\begin{array}{l} E_{z} = [A I_{1} (γ r) + B K_{1} (γ r)] \cos (φ) \exp (i β z) \\ H_{z} = [C I_{1} (γ r) + D K_{1} (γ r)] \sin (φ) \exp (i β z) \\ E_{φ} = [\frac{a}{r} A I_{1} (γ r) + \frac{a}{r} B K_{1} (γ r) + b C I_{1}' (γ r) + b D K_{1}' (γ r)] \sin (φ) \exp (i β z) \\ H_{φ} = [c A I_{1}' (γ r) + c B K_{1}' (γ r) + \frac{a}{r} C I_{1} (γ r) + \frac{a}{r} D K_{1} (γ r)] \cos (φ) \exp (i β z) \\ I_{1}' (γ r) = γ I_{0} - \frac{I_{1}}{r}, K_{1}' (γ r) = - γ K_{0} - \frac{K_{1}}{r} \\ γ = k_{0} \sqrt{n_{e f f}^{2} - ε}, a = \frac{β}{γ^{2}}, b = i \frac{ω μ}{γ^{2}}, c = - i \frac{ω ε}{γ^{2}}, \end{array}

where I₁, K₁ are a modified Bessel function of the first and second kind, respectively, β is the propagation constant, ε is permittivity, μ is permeability, and ω is the angular frequency of light.

Then the requirement of the continuity of the tangential components of the magnetic and electric intensity vectors at the boundaries yields:

\det (\begin{matrix} A^{11} & - B^{12} & 0 & 0 \\ 0 & B^{22} & - B^{23} & 0 \\ 0 & 0 & B^{33} & - C^{34} \end{matrix}) = 0

A^{11} = (\begin{matrix} I_{1} (γ_{1} r_{1}) & 0 \\ 0 & I_{1} (γ_{1} r_{1}) \\ \frac{a_{1}}{r_{1}} I_{1} (γ_{1} r_{1}) & b_{1} I_{1}^{'} (γ_{1} r_{1}) \\ c_{1} I_{1}^{'} (γ_{1} r_{1}) & \frac{a_{1}}{r_{1}} I_{1} (γ_{1} r_{1}) \end{matrix})

B^{k l} = (\begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \frac{a_{l}}{r_{k}} & \frac{a_{l}}{r_{k}} & b_{l} \frac{I_{1}^{'} (γ_{l} r_{k})}{I_{1} (γ_{l} r_{k})} & b_{l} \frac{I_{1}^{'} (γ_{l} r_{k})}{I_{1} (γ_{l} r_{k})} \\ c_{l} \frac{I_{1}^{'} (γ_{l} r_{k})}{I_{1} (γ_{l} r_{k})} & c_{l} \frac{K_{1}^{'} (γ_{l} r_{k})}{K_{1} (γ_{l} r_{k})} & \frac{a_{l}}{r_{k}} & \frac{a_{l}}{r_{k}} \end{matrix}), k \neq l

B^{k k} = (\begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \frac{a_{k}}{r_{k}} \frac{I_{1}^{} (γ_{k} r_{k})}{I_{1} (γ_{l} r_{k})} & \frac{a_{k}}{r_{k}} \frac{K_{1}^{} (γ_{k} r_{k})}{K_{1} (γ_{l} r_{k})} & b_{k} \frac{I_{1}^{'} (γ_{k} r_{k})}{I_{1} (γ_{l} r_{k})} & b_{k} \frac{I_{1}^{'} (γ_{k} r_{k})}{I_{1} (γ_{l} r_{k})} \\ c_{k} \frac{I_{1}^{'} (γ_{k} r_{k})}{I_{1} (γ_{l} r_{k})} & c_{k} \frac{K_{1}^{'} (γ_{k} r_{k})}{K_{1} (γ_{l} r_{k})} & \frac{a_{k}}{r_{k}} \frac{I_{1}^{} (γ_{k} r_{k})}{I_{1} (γ_{l} r_{k})} & \frac{a_{k}}{r_{k}} \frac{K_{1}^{} (γ_{k} r_{k})}{K_{1} (γ_{l} r_{k})} \end{matrix}), k \neq l

C^{34} = (\begin{matrix} K_{1} (γ_{4} r_{3}) & 0 \\ 0 & K_{1} (γ_{4} r_{3}) \\ \frac{a_{4}}{r_{3}} K_{1} (γ_{4} r_{3}) & b_{4} K_{1}^{'} (γ_{4} r_{3}) \\ c_{4} K_{1}^{'} (γ_{4} r_{3}) & \frac{a_{4}}{r_{3}} K_{1} (γ_{4} r_{3}) \end{matrix})

where r₁ = r_co, r₂ = r_cl, r₃ = r_cl + r_g, ε₁ = ε_co, ε₂ = ε_cl, ε₃ = ε_g, ε₄ = ε_sur,The ratio of two Bessel functions of almost the same large complex argument can be expressed as [16]

\begin{matrix} \frac{K_{ν} (α)}{K_{η} (β)} = \sqrt{\frac{β}{α}} \frac{S_{ν}^{-} (α)}{S_{η}^{-} (β)} \exp (β - α) \\ \frac{I_{ν} (α)}{I_{η} (β)} = \sqrt{\frac{β}{α}} \frac{S_{ν}^{+} (α) \exp (α) + i \frac{| α |}{α} S_{ν}^{-} (α) \exp (- α)}{S_{η}^{+} (β) \exp (β) + i \frac{| β |}{β} S_{η}^{-} (β) \exp (- β)} \\ \overset{| α | \to \infty, | β | \to \infty}{\to} \sqrt{\frac{β}{α}} \frac{S_{ν}^{+} (α)}{S_{η}^{+} (β)} \exp (α - β) \\ S_{ν}^{\pm} (α) = \sum_{k = 0}^{n} \frac{{(ν + \frac{1}{2})}_{k} {(\frac{1}{2} - ν)}_{k}}{k!} {(\pm \frac{1}{2 α})}^{k} + O (\frac{1}{α^{n + 1}}) . \end{matrix}

After these manipulations, effective refractive indices of modes n_eff can be found by solving Eq. (3). Initially, material losses are “switched off” and effective refractive indices of modes of the lossless structure are calculated by Newton-Raphson method [17]. Subsequently, these values are used as a starting point of the process in which the losses are gradually increased up to their actual value and the effective refractive indices of modes are tracked in the complex plane. Determinant in Eq. (3) is calculated using the standard numerical LU decomposition method [17].

3.2 Transmission spectrum

To calculate the coupling between the forward-propagating fundamental mode and the backward-propagating mode, an unconjugated form of the coupled-mode equations was used [18]. After solving this set of differential equations, the transmission spectrum can be expressed as:

T_{μ} = {| \frac{δ \exp (- \frac{1}{2} i L Δ β_{1 μ})}{\frac{1}{2} i Δ β_{1 μ} \sin (δ L) - δ \cos (δ L)} |}^{2}

δ = \sqrt{{(\frac{Δ β_{1 μ}}{2})}^{2} - {| κ_{1 μ} |}^{2}},

where Δβ_1μ is the phase detuning between the core mode and the μth cladding mode

Δ β_{1 μ} = β_{1} + β_{μ} - \frac{2 π}{Λ} .

κ_1μ is the coupling constant between the fundamental mode and the μth cladding mode

κ_{1 μ} = \frac{i ω ε_{0}}{4} \frac{| β_{μ} |}{β_{μ}} \iint_{A} Δ ε e_{1 ⊥}^{} \cdot e_{μ ⊥},

β is the propagation constant,

e_{μ ⊥}

is the transverse field distribution of the μth mode, L denotes the length of the grating,

Δ ε

is the perturbation of permittivity due to the presence of the grating.

At the resonance Δβ_1μ = 0 and

T_{μ}^{r e s} = | \frac{1}{\cos (i | κ_{1 μ} | L)} |,

which defines the transmittance corresponding to the μth mode.

4. Numerical results

As most of the existing SPR sensors operate at wavelengths between 600 and 1000 nm [1–5], we investigated the fiber optic SPR sensor designed for the operating wavelength of 800 nm to allow for direct comparison. For other operating wavelengths, sensor analysis and optimization can be performed using the same approach. In order to ensure excitation of surface plasmons at 800 nm, the period of the FBG was set to Λ = 287 nm. To avoid technological issues associate with the development of special optical fibers, we investigated a standard silica single-mode optical fiber with d_cl = 125μm and n_cl = 1.4440 and the refractive index and diameter of the core fulfilling the single-mode condition λ_cut-off = 750 nm. Refractive index of gold was assumed to be n_g = 0.18 + 5.13i. The surrounding medium was considered to be a water-like medium with a refractive index n_sur = 1.32 – 1.33, as this is a typical scenario for most biosensing applications.

4.1 Mixed surface plasmon – fiber cladding modes

SP-FC modes were found to be a solution of the dispersion equation Eq. (3). In principle, there are two kinds of solutions of the dispersion Eq. (-) even and odd modes. Even modes with the axial component of the propagation constant matching the propagation constant of SP can couple with SP forming SP-FC modes. The distribution of the electromagnetic field of such modes exhibit (local) maxima at the interface of the gold layer and decay exponentially into the surrounding medium (Fig. 2 ). Figure 3 shows attenuation of the modes (at wavelength λ = 800 nm) for all the 178 cladding modes supported by the structure as a function of the gold layer thickness. Clearly the modes coupled with SP exhibit the highest attenuation. In addition, the modal attenuation increases with the increasing thickness of the metal film. On the other hand, odd modes do not couple with SP and their field is concentrated mainly in the cladding. Attenuation of the odd modes is rather low (inset of Fig. 3).

Fig. 2 Azimuthal distribution of the magnetic field of the 138th – 140th mode (top). Distribution of the real part of the refractive index (bottom).

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Fig. 3 Attenuation of the even and odd (inset) modes for different r_g.

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Due to the strong coupling of certain cladding modes with SPs (134th – 146th mode, in particular), the properties of these modes are highly sensitive to changes in the refractive index of the surrounding medium. This effect can be exploited for sensing.

4.2 Transmission spectrum analysis

Considering the transmittance of the fiber with the Bragg grating inscribed into the fiber core (Eq. (14), the product |κ|L determines the magnitude of the power transfer between the coupled modes. A higher |κ|L results in a higher reflectivity of the grating and a more pronounced dip in the spectrum of transmitted light. Figure 4 depicts κ of SP-FC modes for different values of r_g. |κ| decreases with increasing r_g, |κ| of the odd modes is generally higher than |κ| of the even modes – moreover, the mode is lossy, i.e. more light coupled in the mode is lost while propagating the structure. The coupling strength κ was optimized in terms of the refractive index and diameter of the fiber core assuming only those combinations of these parameters which fulfill the single-mode condition λ_cut-off = 750 nm. The coupling strength was found to reach a maximum for d_co = 4.775 μm and n_co = 1.449.

Fig. 4 Coupling constant of the 134th – 146th mode for r_g = 20 - 40 nm. Coupling constant for all the cladding modes for r_g = 40 nm (inset).

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The same behavior can be seen in the transmission spectrum (Fig. 5 ). Two sets of dips appear. The dips of the transmission spectrum which correspond to the even modes are generally less deep and wider than the ones which correspond to the odd modes because of their lossy behavior. The depth of the transmission dip can significantly influence the final resolution of the sensing device. To obtain a reasonable reflectivity for the SP-FC modes, relatively low κ can be compensated by high L and Δε. The length L is limited by practical usage, the sensing area should be as short as possible (optimally 2 mm) to maintain homogeneity. High Δε (of the order of 10⁻²) can be realized through hydrogenation of the fibers prior to UV exposure [19]. On the other hand, if Δε becomes high, the dips suffer from increased widening.

Fig. 5 Part of the transmission spectrum which corresponds to 134th - 146th mode for r_g = 20 - 40 nm. Transmission spectrum which corresponds to the core and all the cladding modes for r_g = 40 nm (inset). L = 2 mm, Δε = 10⁻², n_sur = 1.325.

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Propagation of light through a similar structure was modeled earlier using the 2D equivalent planar waveguide approach [13]. Although transmission spectra predicted by the 2D approach and the rigorous 3D models described herein exhibit similar general properties, there are considerable differences in the coupling strength κ.

5. SPR sensor based on spectroscopy of mixed surface plasmon – fiber cladding modes

If the above described fiber optic structure is to be used as a sensor to detect changes in the refractive index of the surrounding medium, a normalized, rather than absolute, transmission is important. One approach is to normalize the spectrum of the transmitted light using transmission of the structure surrounded by a medium (e.g. air) with a substantially different refractive index than the (water-like) medium under study. As follows from Fig. 6 , such normalization produces a periodic series of peaks and dips in the spectrum of transmitted light. The peaks are associated with the odd modes and the dips are due to the SP-coupling sensitive to the refractive index of the surrounding medium. Figure 7 illustrates the principle of operation of the proposed sensor – the wavelengths at which the dips occur increase with the increasing refractive index of the surrounding medium while the spectral position of the peaks in the transmitted light remains constant.

Fig. 6 Transmission spectrum which corresponds to 134th - 146th cladding modes. r_g = 30nm, n_sur = 1.325, L = 2mm, Δε = 10⁻².

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Fig. 7 Normalized transmission spectrum as a function of refractive index of surrounding medium. r_g = 30nm, L = 2mm, Δε = 10⁻².

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5.1 Performance characteristics of the sensor

In this section we shall investigate key performance characteristics of the sensor – sensitivity and resolution. The sensitivity of the μth mode (its resonant wavelength) to a change in the refractive index of the surrounding medium is defined as

S_{μ} = \frac{d λ_{r e s}^{μ}}{d n_{s u r}}

Using Eq. (1), the sensitivity can be expressed as:

S_{μ} = Λ \frac{d ℜ {n_{e f f}^{μ} (λ_{r e s}^{μ})}}{d n_{s u r}} .

Figure 8 shows the sensitivity of the spectral position of the transmission dips corresponding to 134th – 140th cladding modes as a function of the thickness of the gold layer, r_g. The sensitivity for different modes exhibit similar behavior – when the thickness of the gold film increases, the sensitivity of the mode increases to a certain value of the thickness of the gold layer, beyond which it decreases (negative sensitivity may be also observed; it corresponds to situation where an increase in the refractive index of surrounding medium produces a decrease in the resonant wavelength as illustrated in the inset of Fig. 8). The maximum sensitivity and the gold thickness at which it occurs depend on the mode order. Clearly, for the cladding modes coupled most strongly with SP, the sensitivity is highest (Fig. 8). This behavior is in a good agreement with the results obtained using the 2D equivalent planar waveguide approach [13]. The predicted sensitivity values also agree well – the sensitivity for the most attenuated mode of the equivalent planar waveguide was calculated to be S₄₇ = 74 nm/RIU [13].

Fig. 8 Sensitivity of 134th - 140th cladding mode as a function of r_g. n_sur = 1.325. Resonant wavelength of 136th and 137th cladding modes as a function of r_g (inset).

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The transmittance at a resonance corresponding to the μth mode $T_{μ}^{r e s}$ , and the full-width-half-maximum (FWHM) w_μ of the transmission dip follow the same trends (Fig. 9 ). As the thickness of the gold layer decreases, the imaginary part of the propagation constant decreases (the mode becomes less attenuated), which results in a narrowing and deepening of the transmission dips.

Fig. 9 FWHM w_μ of the transmission dip corresponding to 134th - 140th cladding mode as a function of r_g. (solid line, left axes). The transmittance at a resonance $T_{μ}^{r e s}$ corresponding to the mode 134th - 140th cladding mode as a function of r_{g. .} (dashed line, right axes). n_sur = 1.325.

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According to the theoretical analysis presented in [20], the resolution of SPR sensors σ_SI can be expressed as a product of two terms - $σ_{S I} = A B$ , where A represents the effect of noise in the SPR sensor and B describes the effect of the sensor configuration. The first term can be written as

A = K \frac{r_{s} σ_{I}^{(t h)}}{I_{0} \sqrt{N}},

where K depends on the statistical properties of the noise; r_S describes the effect of the noise correlation;

σ_{I}^{(t h)}

represents the shot noise of light at the threshold of the SPR dip; and N is the total number of intensities involved in the determination of the SPR dip position. The second term can be expressed as:

B = \frac{w_{μ}}{1 - T_{μ}^{r e s}} \frac{1}{S_{μ}} .

Based on our theoretical analysis, it was determined that B reaches its minimum value for the 138th mode and r_g = 35 nm which yields w_μ = 88 pm, $T_{μ}^{r e s} = 0.74$ , and S_μ = 19 nm/RIU. In order to estimate the resolution of the proposed fiber optic SPR sensor, the following parameters were used: K = 0.5 (homogenous noise); $σ_{I}^{(t h)} = 0.6 % \cdot I_{0}$ (a scientific grade CCD detector); N = N_t⋅N_p = 1000 (N_t = 100 frames averaged for each time record, N_p = 10 pixels) [20]. Using these parameters, the resolution of the sensor was estimated to be approximately 1.6 × 10⁻⁶ RIU which makes it comparable with the best fiber optic SPR sensors.

6. Conclusion

A rigorous theoretical analysis of the fiber optic SPR sensor is reported. The structure consists of a standard single-mode optical fiber, covered with a thin gold film, with an FBG inscribed into the core. A new method for the determination of fiber modes, which avoids numerical instability, is proposed. The transmission spectrum is simulated using CMT. The effect of the thickness of the gold layer is discussed in detail, as this is the most critical parameter in controlling the properties of the SP-FC modes. Finally, the refractive index resolution of the sensor is analyzed and estimated to be better than 2 × 10⁻⁶ RIU.

Acknowledgments

This research was supported by the Academy of Sciences of the Czech Republic under contract KAN200670701, by the Ministry of Health of the Czech Republic (IGA MHCR) under contract NR/9322-3, and by the Ministry of Education, Youth and Sports under contract OC09058.

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Theoretical analysis of a fiber optic surface plasmon resonance sensor utilizing a Bragg grating

Abstract

1. Introduction

2. Description of the structure and the principle of operation

3. Theory

3.1 Guided modes of the structure

3.2 Transmission spectrum

4. Numerical results

4.1 Mixed surface plasmon – fiber cladding modes

4.2 Transmission spectrum analysis

5. SPR sensor based on spectroscopy of mixed surface plasmon – fiber cladding modes

5.1 Performance characteristics of the sensor

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (17)

Optics Express