1. Introduction

Biological or chemical sensors become a fast expanding research field as a result of the rapidly growths of the healthcare industry and the environmental monitoring. However, the technology to create a feasible biosensor with advantages including convenience of use, low cost, high sensitivity, specificity, and stability etc. is still a challenge for scientists and engineers. To obtain a biosensing system with high sensitivity, local-field enhancement of metallic nanoparticles caused by localized surface plasmon excitation have been widely used recently [1, 2, 3, 4]. For examples, an immunoassay system based on gold nanoparticles coated with protein antigens is used for monitoring the immune-aggregation process by Rosenzweig et al. [1]. Further, a rapid blood immunoassay technique using gold nanoshells has been proposed by Halas and co-workers [2]. The fluorescence enhancement from localized surface plasmon (LSP) resonance of metallic nanoparticles have been applied for detection of DNA hybridization by Lakowicz and co-workers [3].

On the other hand, fiber-optic biosensing systems have also been well developed [5]. The evanescent waves from the decladded optical fiber interact with the analytes in vicinity of the fiber core and the fluorescence signals of fluorophores on the analytes are excited and detected [6]. The fiber-optic biosensor can be fabricated and miniaturized at low cost and is versatile to overcome various difficult measurement conditions. It allows a large adjustable distance between the detected assay and the optoelectronic components of the biosensor system in order to avoid the electrical interference and to provide more freedom during the detection. Compared with the conventional surface plasmon resonance (SPR) biosensing system, the fiber-based biosensor is suitable for in-vivo detection in commercial application, however, the low detection sensitivity becomes one of serious disadvantages. To increase the sensitivity of the fiber-optic biosensor, a fiber-optic evanescent-wave biosensor with self-assembled gold colloids was studied by Chau et al. [7]. Gold colloids are attached to the surface of a multimode decladded silica optical fiber and the attenuated total reflection (ATR) spectrum of the gold colloids is changed when the index of refraction of the surrounding medium changed. The limit of detection of the sensor is about 9.8 × 10^-11 mole for biotin-streptavidin interaction. Furthermore, a high sensitivity fiber-optic biosensor based on the localized surface plasmon of gold nanoparticles coupled fluorescence (LSPCF) system was demonstrated by Chou and co-workers [8, 9]. The mouse immunoglobulin G (IgG) with very low concentration of 1pg/ml was detected experimentally by the LSPCF biosensor [8]. For clinical applications, the LSPCF biosensor has detected the alpha fetoprotein (AFP) in human serum and the fluorescence signals of AFP with concentrations from 2.33 ng/ml to 143.74 ng/ml have be observed [9]. The evanescent waves from the surface of a decladded fiber form an interactive region and only the fluorescent signals of fluorophores within the interactive region are enhanced significantly. In other words, the LSPCF biosensor is insensitive to the influence of non-interacting background molecules and possesses high specificity and sensitivity comparing with conventional SPR biosensor [10].

According to previous experimental results, the fluorescence signal enhancement of the LSPCF biosensor associated closely with the highly enhanced local field in the vicinity of metallic nanoparticles [3, 11, 12, 13]. Enhanced local fields are generated around metallic nanoparticles for localized surface plasmon excitation by illuminating evanescent waves from the decladded fiber. In order to obtain high-performance biosensors, it is necessary to have a detailed understanding of the mechanism of local-field-enhanced fluorescence of the LSPCF biosensor with metallic nanoparticles and their dependence on the size, geometrical shape, the refractive index of metallic nanoparticles, and the radiative efficiency of fluorophores in the presence of the nanoparticles. It is not easy to explore the relationship between them directly during measurement, therefore, a theoretical study based on the scattering theory of evanescent waves by an individual spherical particle is necessary to establish a model for local-field enhanced fluorescence signals of the LSPCF biosensor. The averaged near-field enhancement around the surface of a gold nanoparticle has been used to estimate the fluorescence enhancement of the LSPCF biosensor [8]. Nevertheless, the previous theoretical model only told half of the story, since the fluorescence signals of fluorophores are influenced by the scattering properties of the fluorophores in the presence of metallic nanoparticles. In order to compare with detected signals in the experiment, the far-field properties of a fluorophore-nanoparticle coupled system must be considered in the theoretical model. Furthermore, the performance of the biosensor could be optimized easily by selecting metallic nanoparticles of special characteristics so that the sensitivity of the biosensor is increased significantly via maximizing the fluorescence enhancement of fluorophores near the nanoparticles.

In this paper, the influences of the near fields and scattering properties of the metallic nanoparticle on the fluorescence enhancement of the LSPCF biosensor are discussed. The strong local field around metallic nanoparticles enhances the field intensity of the fluorophore, meanwhile the far-field detected signals of fluorophores is also enhanced by large scattering cross section of the nanoparticles. In our theoretical study, we assume that each fluorophore is treated as an emitting dipole and the intensity of the reflected dipole field by a metallic nanoparticle is negligible since it is much weaker than the enhanced local field of the metallic nanoparticle. Therefore, the procedure of calculations for metallic nanoparticles and fluorophores can be done separately. In other words, the averaged field intensity around a metallic nanoparticle is calculated firstly, and then the enhanced local field as a local excitation field ${\stackrel{\rightharpoonup }{E}}_{loc}$ to the fluorophores and leads to fluorescence emission of the fluorophores. Finally, the emission signals can be obtained by calculating scattering cross section (or radiative rate) of the fluorophore-nanoparticle coupled system. Since a dilute solution of the fluorescence probe is used in the experiment [8], the near-field interaction between metallic nanoparticles is weak and is neglected here. The fluorescence enhancements of the biosensor can be described by the collective effect of the local-field enhancement of individual metallic nanoparticle. Consequently, the scattering theory for evanescent waves can be treated as a first-order approximation to calculate the local-field enhancement around metallic nanoparticles and the enhancement of fluorescence signals is estimated from the theoretical calculations to compare with experimental results.

2. Models

2.1. Scattering of evanescent waves by a metallic nanoparticle

The analytical solutions for the scattering of evanescent waves by a sphere were studied theoretically by Chew et al. in 1979 [14]. Evanescent waves are generated from the dielectric/air plane interface due to total internal reflection and are scattered by a dielectric sphere. The total cross section for extinction and scattering of evanescent waves by a small metal particle was derived by Quinten et al. [15]. In previous studies, a particle which is much smaller than the illuminating wavelength was placed away from the plane interface, and the effect of the multiple scattering between the particle and the plane interface was neglected. Besides, the results also showed that the extinction cross section for extinction of p-polarized evanescent waves by a metallic nanoparticle is larger than those from s-polarized evanescent waves and plane waves, since evanescent fields increase the contributions of multipoles to the cross section and the local-field enhancement of nanoparticles. In other words, a strongly enhanced local field is excited around the metallic nanoparticle by p-polarized evanescent waves and the local-field enhancement is higher than using s-polarized evanescent-waves or plane-wave illumination.

In the experiment, the gold nanoparticles are far from the dielectric interface of fiber core since the size of IgG and anti-IgG is 6.5 ± 0.9 nm according to atomic force microscope (AFM) tapping mode image [8]. The sandwich immunocomplex height is estimated to be about 20 nm or more. When the fiber core is considered as a dielectric substrate, and the distance between gold nanoparticles and interface of the fiber is larger than the diameter of the nanoparticle, the influence of the fiber’s interface from induced image charges on the optical properties of metallic nanoparticles is negligible [16]. However, when the particle diameter is larger than d, the localized surface plasmon spectra of metallic nanoparticles are red-shifted caused by the interaction between the fiber core and nanoparticles [17]. It implies that when the wavelength of the excited light is longer than the resonant wavelength, the substrate effect of the fiber core leads to increase the intensity of enhanced local fields of metallic nanoparticles.

 

Fig. 1. Cross section of a decladded multimode fiber. metallic nanoparticles are distributed arbitrarily in the detected solution and only the metallic nanoparticles in the vicinity of the fiber core can be excited by evanescent waves from the core.

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The cross section of a decladded multimode fiber is shown in Fig.1. Metallic nanoparticles are distributed arbitrarily in the detected solution and the localized surface plasmons of metallic nanoparticles in the vicinity of the fiber core (interactive region) are excited by evanescent waves from the core. Evanescent waves are scattered by an individual metallic nanoparticle. As shown in Fig. 2, the core of the fiber is simplified as a semi-infinite dielectric plane medium since the size of nanoparticles is much smaller than the core (about 1:100000) and the dielectric plane and the surrounding medium are separated by a plane interface (y-z plane). The refractive indices of the fiber core (n_core) and of the surrounding medium (water) of a metallic nanoparticle (n_w) are set to be 1.492 and 1.33, respectively. A spherical metallic nanoparticle of radius a is placed at a distance d above the interface. Evanescent waves ${\stackrel{\rightharpoonup }{E}}_{inc}^{s,p}$ with s- or p- polarization from the interface of the fiber core couple with the localized surface plasmon of the nanoparticle when total internal reflection of the waves propagating with various propagation constants β in the fiber occurs. θ_k is the angle between transmitted direction and the interface of the core. The total electric field ${\stackrel{\rightharpoonup }{E}}_{tot}^{s,p}$ at the outside of the particle is the sum of the incident field and the scattered field,

 

Fig. 2. Scattering of evanescent waves by a metallic nanosphere with radius a. The refractive indices of the core of the fiber (n_core) and of the medium around the nanosphere (n_w) are 1.492 and 1.33, respectively. The nanosphere is placed at a distance d above the surface of the fiber.

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The illuminated evanescent field with s- or p- polarization is expanded in spherical coordinates by spherical Bessel functions of the first kind j_n (ρ) and normalized vector spherical harmonics ${\stackrel{\rightharpoonup }{X}}_{nm}$ (θ, ϕ) [14]:

Similar to the expression in (2), the scattered electric field of a nanoparticle with s- or p-polarization in spherical coordinates is obtained by replacing j_n with spherical Hankel functions of the first kind h _n ⁽¹⁾, that is

where a_n and b_n are scattered coefficients of a metallic sphere as derived by Bohren [18]. The functions α^s,p _TM(n,m) and α^s,p _TE(n,m) are expansion coefficients and they are found by solving the Maxwell’s equations with boundary conditions at the surface of the sphere [15]. ρ = k_wa and k_w is the wave number of incident light propagates in water. The normalized vector spherical harmonics ${\stackrel{\rightharpoonup }{X}}_{\mathrm{nm}}(\theta ,\varphi )=-i\left(\stackrel{\rightharpoonup }{r}\times \nabla \right)Y_{\mathrm{nm}}(\theta ,\varphi )/\sqrt{n\left(n+1\right)}$ are described by Jackson [19] and Y_nm (θ, ϕ) are scalar spherical harmonics [20]. The electric field around the nanoparticle can be calculated analytically from Eqs. (2) and (3) and the intensity of the electric field directly affects the intensity of fluorescence signals of the LSPCF biosensor.

2.2. Averaged local-field enhancement of metallic nanoparticles

Consider an incident light with wavelength of 650 nm propagates in the multimode fiber with various propagation constants β (sinθ_inc) and β ranging from 0.0 to NA (see Fig. 1). Where NA is the numerical aperture of the incident laser beam and is set to be 0.45. The amplitudes of modes in the fiber are assumed uniformly in calculations. When the incident light propagates in the fiber with different β, the decay length of the evanescent waves from the interface will increase with a larger β and the range of the decay lengths will be 153 nm to 205 nm. It indicates that the decay lengths are much longer than the distance between the interface of the core and bound metallic nanoparticles, and therefore the nanoparticles within the interactive region are illuminated under strong enough evanescent fields. In this study, gold nanoparticles are used, as in Ref. [8]. It is noted that the incident wavelength is not the LSP resonance wavelength of gold nanoparticles (λ_LSP = 550 nm) since the gold nanoparticles exhibit strongly absorption at resonance and the fluorescence signals will be absorbed by gold nanoparticles when the emission wavelength of fluorophores near to the resonant wavelength of gold nanoparticles. However, even it is slightly off-resonant in the experiment and in this theoretical study, the local field around gold nanoparticles is still strongly enhanced for broad excitation of surface plasmon. The frequency-dependent optical constant of gold nanoparticles is from the experimental data of Palik [21].

The averaged scattered electric field in the vicinity of the metallic nanoparticle by including various β is calculated as

where ${\stackrel{\rightharpoonup }{E}}_{sca}^p$ is the scattered electric fields of the metallic nanoparticle by illuminating p-polarized incident light. Hence, the total averaged electric field in the outside of the nanoparticle is

To obtain a better fluorescence signal detection performance, the detector of the LSPCF biosensor is located beside the decladded fiber [8]. Because the fluorescence signal from the bottom half surface of a metallic nanoparticle is screened by the metallic nanoparticle itself, only the field intensity on top half surface of the metallic nanoparticle is considered. Since the orientation of fluorophores is the same with the excitation field and the location of the detector leads to that only the fluorescence signals of the fluorophores of orientation parallel to the interface of the fiber are detectable, therefore only the electric fields of direction on y-z plane are considered. Since the fluorophores are attached arbitrarily on the surface of metallic nanoparticles, averaged local-field enhancements around an individual metallic nanoparticle are calculated to relate to the fluorescence enhancement of experiments. The averaged local-field intensity ratio Λ of the averaged-field intensity of a metallic nanoparticle to the averaged-field intensity of evanescent waves (without nanoparticles) is given by

where E ^p,∥ _tot is total electric field of direction parallel to the interface of the fiber and the reference ∣〈E ^p,∥ _inc〉∣² is calculated at a distance d above the interface of the fiber.

2.3. Radiative properties of fluorophores near metallic nanoparticles

 

Fig. 3. An excited fluorophore is placed in the vicinity of a metallic nanosphere is treated as an emitting dipole $\stackrel{\rightharpoonup }{p}$. Where $\stackrel{\rightharpoonup }{r}$ and $\stackrel{\rightharpoonup }{x}$ are the positions of the dipole and the detector, respectively.

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If a fluorophore is placed close to the surface of metallic nanoparticles, a dramatic fluorescence signal quenching is observed since the interaction between the fluorophore and the free electrons of the metal surface leads to increasing total decay rate of the fluorophore. [11, 22, 23, 24] The total decay rate includes the radiative rate (W_r) and nonradiative rate (W_nr). As a result of increasing total decay rate, the lifetime of a fluorophore in the presence of a metallic nanoparticle are shortened significantly. By considering the photostability of fluorophores, principally more excitation-emission cycles of fluorophores of shorter lifetimes occurs before photobleaching [25], so the fluorescence signal of fluorophores in the presence of metallic nanoparticles is increased comparing to the fluorophores in free space. Besides, fluorescence radiation of fluorophores is scattered by metallic nanoparticles with large scattering cross section and the scattering comes around to increase the far-field signals of the biosensor.

To explore the radiative properties of fluorophores near metallic nanoparticles, a simple approach based on classical electromagnetic theory for scattering of an emitting dipole near a metallic nanoparticle was used [26]. Consider a fluorophore of emission wavelength λ_em = 680 nm is excited by the local excitation field ${\stackrel{\rightharpoonup }{E}}_{loc}$ at excitation wavelength λ_ex = 650 nm and behaves as an emitting dipole $\stackrel{\rightharpoonup }{p}$ of orientation same with the polarization of the excitation field. It is located in the vicinity of a metallic nanosphere of radius a at location $\stackrel{\rightharpoonup }{r}$ (see Fig. 3) from the origin of the nanosphere and the total electric field ${\stackrel{\rightharpoonup }{E}}_{tot}$ at the outside of the particle is equal to the dipole field ${\stackrel{\rightharpoonup }{E}}_{dip}$ plus the scattered field of the nanosphere ${\stackrel{\rightharpoonup }{E}}_{sca}$,

The dipole and scattered fields are expanded in spherical coordinates as similar as Eqs. 2 and 3. The radiative energy rate W_r is calculated from the total electric field by integrating energy flux through the surface of an infinite large sphere. There are two orientations of the dipole moment are discussed: the emitting dipole is oriented (a) radially, and (b) tangentially to the nanosphere. The W_r of the emitting dipole in the presence of the metallic nanosphere with different orientations are given by [26]

for dipole oriented radially and by

for dipole oriented tangentially. Here a_n and b_n are the scattered coefficients. W^rad ₀ and W^tan ₀ are the radiative rate of the emitting dipole oriented radially and tangentially, respectively, in free space and their magnitude is directly proportional to the ∣E^rad_loc∣² and ∣E^tan_loc∣², respectively. k = 2π/λ_em. For an emitting dipole with arbitrarily orientation, the total radiative rate of the dipole is W_r = W^rad_r + W_r^tan and the field-intensity ratio Λ_r of the averaged radiative rate 〈W_r〉 of an emitting dipole in the presence of the metallic nanosphere to the averaged radiative rate 〈W ₀〉 of the dipole at a distance d above the interface of the fiber (without nanoparticles) can be obtained by

The reference 〈W ₀〉 corresponds to the result from a conventional fluorescence probe without gold nanoparticle (the first kind of fluorescence probe in the experiment. [8])

3. Results and discussion

According to the experiments, the fluorescence probe of the LSPCF biosensor is constructed using Cy5-labeled antibody bounded to protein A molecules conjugated with gold nanoparticles. Therefore the fluorophore should be maintained few nanometers distance from the surface of the nanoparticles. First, the position-dependent field enhancements around a gold nanosphere are considered. The calculated results correspond to the results from a fluorophore coupled with a gold nanoparticle in the experiment. Averaged local-field enhancements Λ and averaged radiative-rate enhancements Λ_r on the top half surface of a gold nanosphere of radius a as a function of the fluorophore-particle separation r′ are shown in Fig. 4 and the cases of a dielectric nanosphere of a = 10 nm and refractive index n = 2.0 are given as a reference.

When the nanoparticle is a dielectric nanosphere, the local-field enhancement of the nanosphere almost equals to the reference 〈W ₀〉 since localized surface plasmon cannot be excited on the surface of the dielectric nanosphere. In the case of a gold nanosphere of a= 10 nm, the averaged-field intensities are enhanced from 20 to 2 times of the averaged-field intensity of the reference when the fluorophore-particle separation is changed from 1 to 9 nm. When a = 20 nm, the local-field enhancements are enhanced from 28 to 5 times when the fluorophore-particle separation is increased from 1 to 9 nm. As a result of localized surface plasmon excitation, the electric field exhibits highly localized distribution around the gold nanosphere and the intensity decreased dramatically with increasing r′ only in a few nanometers. The decreasing of the averaged-field enhancement of a smaller particle is more rapid than a larger one. Figure 4(a) indicates that the averaged-field intensity around the gold nanosphere is enhanced by increasing radius of the nanosphere and the averaged-field enhancements from larger particles are still strong at large fluorophore-particle separation comparing with those from smaller particles. It is noted that when the radius of the gold nanosphere is larger than 10 nm, the dielectric fiber may influences the local-field enhancement in the vicinity of the nanosphere. It is reasonable that the local-field enhancement to be increased by considering substrate effect of the dielectric fiber core, since the surface plasmon resonant wavelength will shift to longer wavelength as described in section 2.1

 

Fig. 4. (a) Averaged local-field enhancements Λ, and (b) averaged radiative-rate enhancements Λ_r in the vicinity of the top half surface of a gold nanosphere of radius a as a function of the distance r′ from the surface of the nanosphere. The fiber-particle separation d = 20 nm. The results using dielectric nanosphere with refractive index n = 2.0 are given as a reference.

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In order to compare with experimental detected signals of the LSPCF biosensor, the averaged radiative-rate enhancements of fluorophores are calculated using Eqs. 10. The magnitude of the radiative-rate enhancement of fluorophores strongly depends on the local-field enhancement of the gold nanoparticle. Figure 4(b) demonstrates the same trend as in Fig. 4(a), except the magnitude in Fig. 4(b) is higher than that in 4(a), especially for smaller fluorophore-particle separation. Results in Fig. 4(b) show that the radiative rate of a fluorophore binding with a dielectric nanosphere is similar to 〈W ₀〉 due to weak local-field enhancement of the nanosphere, which is shown in Fig. 4(a). For gold nanosphere, the radiative rate is enhanced from 63 to 2 times for a = 10 nm and from 125 to 7 times for a = 20 nm when the fluorophore-particle separation is changed from 1 to 9 nm. Remarkably, Fig. 4(a) and 4(b) tell that the enhancement of the averaged radiative rate is higher than the enhancement of the averaged local field of the gold nanoparticle at smaller fluorophore-particle distance. The increasing of radiative-rate enhancement results from the scattering of the strongly radiation of the emitting fluorescent molecule by gold nanoparticles with large scattering cross section at short fluorophore-particle distance and the interaction between the molecule and nanoparticle will become weaker with increasing the separation. These results indicate that the local-field enhancement of metallic nanoparticles not only can enhance the radiation intensity of fluorophores but also can help to enhance the far-field fluorescence signals of the LSPCF biosensor.

In the experiment, the detectable signal of the LSPCF biosensor results from specific binding to the fiber, and only the fluorescence signal of fluorophores within the interactive region are enhanced. The distance between the fiber and the gold nanoparticles is estimated about 20 nm under binding state [8]. The influence of non-interacting background molecules outside the range is negligible. It is necessary to study the influences of the fiber-particle separation on the fluorescence enhancement. Figure 5 displays the averaged local-field and averaged radiative-rate enhancements of fiber-particle separations ranging from 20 to 200 nm as a function of the fluorophore-particle separation. The radius of the gold nanosphere is 10 nm. Figure 5(a) shows that when the fiber-particle separation is increased from 20 to 200 nm, the averaged local-field enhancement for r′ = 1 nm is decreased from 20 to 2.7 times. By analyzing the calculated results in Fig. 5(a), the local-field enhancement of the gold nanosphere decreases to 7.4 times of the enhancement for d = 20 nm when the fiber-particle separation is increased from 20 to 200 nm. The averaged radiative-rate enhancements of fluorophores in Fig. 5(b) are higher than those in Fig. 5(a) due to enhanced radiative rate by gold nanoparticles. The averaged radiative-rate enhancement for r′ = 1 nm decreases from 63 to 8.5 times when d is increased from 20 to 200 nm. Similar to the results in Fig. 5(a), the radiative rate enhancement decreases about 7.4 times of the enhancement for d = 20 nm when the fiber-particle separation is increased from 20 to 200 nm. The local-field enhancement from the illuminated evanescent waves decays exponentially with the distance from the interface of the fiber. When d = 200 nm, it is larger than the averaged decay length of evanescent waves at λ = 650 nm. The effects of conjugated forms of d larger than 200 nm is negligible in the experiment.

 

Fig. 5. (a) Averaged local-field enhancements Λ, and (b) averaged radiative-rate enhancements Λ_r of a gold nanosphere of radius 10 nm as a function of the distance r′. The nanosphere is placed at various distances d above the interface of the fiber.

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Fig. 6. Fluorophore-particle separation averaged values of the averaged radiative rate enhancements as shown in Fig. 5(b) as a function of the distance d above the interface of the fiber. The radius of the nanoparticle is 10 nm.

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To make the observations from Fig. 5(b) clearer, the radiative rate enhancements 〈Λ_r〉 are calculated from results in Fig. 5(b) after fluorophore-particle separation averaging. Although the exact distance between fluorophores and gold nanoparticles cannot be confirmed, the best conjecture is the separations ranging from 1 to 9 nm. When a fluorescence probe is captured by the capture bio-recognition antibody which was immobilized on the decladded fiber, the distance between the fiber and the conjugated form is 20 nm and the r′-averaged radiative rate enhancement is about 11 times the 〈W ₀〉 (see Fig. 6) and the enhancement decreased to 1.5 times when the distance increased to 200 nm. The calculated result corresponds to the second kind of fluorescence probe in the experiment [8] and is consistent with the experimental result. It is noted that, when the fiber-particle separation is 100 nm, r′-averaged radiative rate of the fluorescence probe still is enhanced to about 5 times the 〈W ₀〉. However, the enhanced fluorescence signals of those non-captured fluorescence probes will not affect too much for the readout of the LSPCF biosensor since the probes are distributed uniformly in the detected solution with dilute concentration, the number of non-captured probe suspend within d = 100 nm is fewer than the number of captured probe.

If there are multiple fluorophores on a gold nanoparticle, the fluorescence signal is proportional to the number of fluorophores. A fluorescence probe with 40 Cy5 molecules attached to a gold nanoparticle are considered in calculations to compare with the results of the third kind of fluorescence probe in the experiment [8]. It is estimated that there are 20 or more Cy5 molecules on the upper half of the nanoparticle. The r′-averaged radiative rate of the fluorescence probe with 20 Cy5 molecules is estimated straightforwardly and the enhancement is about 11×20 = 220 times the 〈W ₀〉. By summarizing our calculated results, the ratio of intensities of fluorescence signals versus the three different fluorescence probes is 1:11:220, and the ratio from the experimental measurement is 1:12:245. The results indicate that the estimated ratio agrees well with the experimental measurements. Although the r′-averaged radiative rate enhancement should be modified by considering various r′-interval averaging, the order of magnitude of the theoretical prediction is still in agreement with the experimental result. It should be noted that the geometric shape of a realistic gold nanoparticle is usually not a perfect sphere; rather its shape is more likely to be irregular. Generally, the local-field enhancement of non-spherical gold nanoparticle is stronger than that of the gold nanosphere since the nanoparticle of complicated geometrical shape has complex curved surface and dramatically local-field enhancement can be generated by the bounded charge oscillation on the surface [27]. However, the first-order approximation based on scattering of evanescent waves of spherical particles still provide a good qualitative understanding for mechanism of the fluorescence enhancement of the LSPCF biosensor.

4. Conclusions

In this paper, the mechanism of fluorescence enhancements of the LSPCF biosensor is investigated by scattering theory of evanescent waves by a spherical metallic nanoparticle in dilute approximation and fluorescence quenching. The fluorescence enhancement of the LSPCF biosensor is closely related to radius of gold nanoparticles, fluorophore-particle separation, and fiber-particle separation. The averaged local-field enhancement in the vicinity of larger gold nanoparticles is higher than the smaller nanoparticles and the fluorescence signal of fluorophores near the larger particle become stronger. Because highly enhanced local field concentrated on the surface of gold nanoparticles, fluorescence enhancements of the fluorescence probes with smaller fluorophore-particle separation are higher than those with larger separation. When the fiber-particle separation is larger than the averaged decay length of evanescent waves, the fluorescence enhancement of the fluorescence probe decreased significantly. The boundary of the interactive region of the LSPCF biosensor is estimated theoretically and it can be tuned by controlling the propagating constant of the incident light coupling into the decladded fiber. Decreasing the interactive region helps to reduce the signals from non-captured fluorescence probes and to increase the accuracy of the LSPCF biosensor. This theoretical study provides a practical way to estimate the performance of the LSPCF biosensor system and the understanding of the mechanism of the fluorescence enhancement can help to develop high performance biosensing system.